Mauricio Hilbck Rios Thesis MSC CASO VAR

CRANFIELD UNIVERSITY
CRANFIELD SCHOOL OF MANAGEMENT
FINANCE & ACCOUNTING GROUP
MSc IN FINANCE & MANAGEMENT
Academic Year 2009 - 2010
MAURICIO HILBCK RIOS
Managing and Hedging the Foreign Exchange Risk of a
Multinational UK NGO
Supervisor:
Tarik Driouchi
September 2010
This thesis is submitted in partial fulfilment of the requirements for the degree of
Master of Science
© Cranfield University 2010. All rights reserved. No part of this publication may be
reproduced without the written permission of the copyright owner.
ABSTRACT
The following paper develops a comprehensive risk management guideline in order to
identify, measure, monitor and report the foreign exchange (FX) risk exposure of a
Multinational UK Corporation which has to manage more than 80 different currencies
during its regular operations. The paper implements a number of models going from MeanVariance Analysis (Modern Portfolio Theory) to Value-at-Risk (VaR) models, finishing with
a framework for risk management including hedging by using linear (forwards) and nonlinear (options) financial derivatives. Results suggest that the main FX risk factors for the
company are Euro and US Dollars and that its current FX portfolio is not efficient. VaR results
could serve as a reference to set aside some FX risk reserve (e.g. its current 3 times 10-day
99% VaR is currently £2.8m). Also, Historical and Monte Carlo VaR allow us to identify the
presence of “fat tails” which are measured by the calculation of a monthly Conditional VaR
(CVaR) as an additional £0.3m. Moreover, a Stress Test suggests that if the last crisis
scenario repeats, there is 1% chance that the company loses an additional 1.8% of its
monthly net positions. Finally, a Full Valuation model shows that the GBP hedge reached by
using a portfolio of EUR and USD options is well higher than the hedge given by forwards
only.
KEYWORDS: Risk Management, Portfolio Optimization, Value-at-Risk, Currency Forwards,
Currency Options.
A mis adorados padres,
Cecilia y Kike,
A mis queridos hermanos,
Joche y Kiko,
Y a mis bellos abuelos
en la eternidad del cielo,
All we need is Love
ACKNOWLEDGMENTS
My first thanks go to all the Cranfield SoM’s people, professors, lecturers, staff,
students…it’s been an amazing year and they’ve been always nice and helpful. I’m leaving
campus, 100% satisfied with both my master degree and the extraordinary friends I made
here. I’m sure most of these new friendships will be last forever as well as my future life as
proud Cranfield alumni.
In particular, I want to thank Tarik not only as his student but also, if he let me say so, as his
friend. He is an admirable teacher and tutor and an excellent and reliable friend. I’ll be
grateful always for his academic support during the entire programme and during the
development of this project and I will be especially grateful as his friend, for his honest and
helpful advices that always will be taken into account for my future decisions.
I also would like to thank our Director’s programme Sunil. He’s been always very helpful and
interested in our progress as students and also as professionals. In fact, He was the person
who allows me to be part of this interesting project which has let me learn more than I
expected. Furthermore, it’s been a great experience to work with a great company and
great people like Paul. I am very happy that they would use my work as the beginning of
their FX risk management’s improvements.
Last but not least: Thank you very much to my family! They are not only my lovely parents
and brothers but also my loyal best friends. It’s been a great year but I was able to enjoy
this experience only because all the love I always receive from my family. Thanks to my
father Kike for his strength and his eternal support and lucid advices, to my mother Cecilia
for her eternal love and concern about me and my life and thanks to my brothers, Joche y
Kiko, for their eternal and truly love and friendship. Together we are the strongest team I’ve
ever seen. I love you so much!
Mauricio
TABLE OF CONTENTS
INTRODUCTION ................................................................................................................................... 1
Chapter 1 RESEARCH CONTEXT AND LITERATURE REVIEW ................................................................ 4
1.
Foreign Exchange Risk ............................................................................................................. 4
2.
The Origins of Risk Measures: Modern Portfolio Theory and Mean-Variance Analysis.......... 6
3.
Measuring Downside Risk: from MVA to VaR ....................................................................... 10
4.
To Hedge or not to Hedge, that is the question .................................................................... 15
Chapter 2 RESEARCH QUESTION AND OBJECTIVES........................................................................... 19
1.
Research Questions ............................................................................................................... 19
2.
Objectives .............................................................................................................................. 20
3.
A brief description of the Company ...................................................................................... 21
Chapter 3 DATA AND METHODOLOGY.............................................................................................. 22
1.
Data Gathering ...................................................................................................................... 22
2.
Economic Exposure ............................................................................................................... 24
3.
Portfolio Optimization (MVA) ............................................................................................... 26
4.
Value-At-Risk (VaR) Methodology ......................................................................................... 32
a.
Parametric (Variance-Covariance) VaR ......................................................................... 33
b.
Historical VaR ................................................................................................................ 35
c.
Monte Carlo VaR ........................................................................................................... 36
5.
Improving VaR: Conditional Value-at-Risk (CVaR) and Stress Testing ................................... 40
6.
Hedging the Portfolio: Options and Forwards ...................................................................... 42
a.
Definitions ..................................................................................................................... 42
b.
Derivatives Valuation and Optimal Hedge .................................................................... 44
c.
Reassessing the risk of the hedged portfolio ................................................................ 46
Chapter 4 ANALYSIS OF RESULTS AND DISCUSSION ......................................................................... 49
1.
Regression Analysis Results................................................................................................... 49
2.
Portfolio Optimisation Results .............................................................................................. 51
a.
3.
Allowing Short Selling .................................................................................................... 55
Value-at-Risk Results ............................................................................................................. 58
a.
Parametric VaR results .................................................................................................. 58
b.
Historical VaR results..................................................................................................... 60
c.
Monte Carlo VaR results................................................................................................ 62
d.
A comparison between different approaches and their results ................................... 63
4.
CVaR and Stress Testing results............................................................................................. 66
5.
Portfolio Hedging Results ...................................................................................................... 68
a.
Valuation ....................................................................................................................... 68
b.
Risk measures after hedging ......................................................................................... 69
Chapter 5 CONCLUSIONS, RECOMMENDATIONS AND CRITIQUE ..................................................... 73
REFERENCES ...................................................................................................................................... 77
APPENDICES ...................................................................................................................................... 81
a.
Appendix A: List of Currencies ...................................................................................... 81
b.
Appendix B: Variance-Covariance Matrix ..................................................................... 84
c.
Appendix C: Historical VaR, Portfolio’s values distribution .......................................... 85
d.
Appendix D: Monte Carlo VaR, Portfolio’s values distribution..................................... 86
e.
Appendix E: Correlation Matrix .................................................................................... 87
f.
Appendix F: Regressions Results ................................................................................... 88
g.
Appendix G: Optimal Weights (Without short selling) .................................................. 92
h.
Appendix H: Optimal Weights (With short selling) ....................................................... 93
i.
Appendix I: Individual VaRs ........................................................................................... 94
LIST OF FIGURES
Figure 1-Portfolio Optimization Example ............................................................................................ 8
Figure 2-Asymmetric Distributions.................................................................................................... 11
Figure 3-Normal Distribution ............................................................................................................ 34
Figure 4-Exchange Rate’s Volatilities................................................................................................. 41
Figure 5-Derivative’s Pay-offs............................................................................................................ 48
Figure 6-Efficient Frontier, Income Weights ..................................................................................... 52
Figure 7-Portfolio’s Comparison, Main Currencies ........................................................................... 55
Figure 8-Efficient Frontier, Short Selling ........................................................................................... 56
Figure 9- Parametric VaR Results, Distribution ................................................................................. 59
Figure 10-Historical VaR Results, Distribution................................................................................... 61
Figure 11-Monte Carlo VaR Results, Distribution.............................................................................. 63
Figure 12-VaR % Results Comparison................................................................................................ 65
Figure 13-Stress Testing Results ........................................................................................................ 67
Figure 14-Hedged and Non-hedged Portfolio Comparison ............................................................... 70
LIST OF TABLES
Table 1-Income/Expenditure Data Summary - £m............................................................................ 22
Table 2-Portfolio Return.................................................................................................................... 29
Table 3-Regression Results................................................................................................................ 50
Table 4-MVA, Income Weights .......................................................................................................... 51
Table 5-MVA, Efficient Income Weights............................................................................................ 53
Table 6- MVA, Main Currencies......................................................................................................... 54
Table 7-MVA, short selling ................................................................................................................ 56
Table 8-Efficient Income Weights, Short Selling ............................................................................... 57
Table 9-Parametric VaR Results ........................................................................................................ 58
Table 10-Parametric VaR, Benefits of Diversification........................................................................ 60
Table 11-Historical VaR Results ......................................................................................................... 60
Table 12-Monte Carlo VaR Results .................................................................................................... 62
Table 13-Comparison of VaR Results ................................................................................................ 64
Table 14-CVaR Results ....................................................................................................................... 66
Table 15-Stress Testing Results ......................................................................................................... 67
Table 16-Derivative’s Valuation ........................................................................................................ 68
Table 17-Hedged VaR Comparison.................................................................................................... 69
Table 18-Derivative’s Hedging .......................................................................................................... 71
LIST OF FORMULAS and OPTIMIZATION PROBLEM
Formula 1-Economic Exposure Regression ....................................................................................... 24
Formula 2-Adjusted CF ...................................................................................................................... 24
Formula 3-Exchange Rate’s Return ................................................................................................... 28
Formula 4-Portfolio Return ............................................................................................................... 28
Formula 5-Portfolio Variance ............................................................................................................ 29
Formula 6-General Quadratic Utility Function .................................................................................. 30
Formula 7-Particular Quadratic Utility Function ............................................................................... 30
Formula 8-VaR, General Form ........................................................................................................... 32
Formula 9-VaR, Parametric Form ...................................................................................................... 34
Formula 10-Geometric Brownian motion Model .............................................................................. 37
Formula 11-Simulated Return’s Generator ....................................................................................... 37
Formula 12-Univariate Standardized Normal Distribution ............................................................... 38
Formula 13-Conditional VaR ............................................................................................................. 40
Formula 14-Forward Exchange Rate ................................................................................................. 44
Formula 15-Optimal Hedge ............................................................................................................... 44
Formula 16-Call and Put Valuation Model (BSM) ............................................................................. 45
Formula 17-Portfolio Value, Linear Model ........................................................................................ 46
Optimization Problem 1- MVA .......................................................................................................... 26
INTRODUCTION
Within a more globalised world, Multinational Corporations (MNC) are continuously
expanding their operations internationally. Therefore, private, public and nongovernmental Institutions across the world are increasing their concerns about the efficient
managing of their foreign exchange (FX) risk. In particular, a company is exposed to FX risk
when its operations involve more than one currency and hence its performance could be
affected by exchange rate movements. For instance, a firm could have a mismatch between
the currency at which its assets are denominated and the currency at which its liabilities
are. Therefore, depreciation in the asset’s currency could lead to a loss in value of the
company’s assets in relation to their liabilities.
The aim of this paper is to develop a comprehensive risk management system in order to
identify, measure, monitor and report the FX risk exposure of a UK MNC by using a number
of frameworks and models to finally recommend the best alternative in order to reduce its
exposure by using linear and non-linear financial derivative’s hedging. In particular, the
paper focuses on the Economic and Transaction FX risk exposure of the company. This
particular company operates in 99 countries and as a result, it manages 80 different
currencies during its regular operations. Consequently, assessing and controlling its FX
exposure in the most efficient and accurate way becomes a vital task for them.
As described by Buehler, K. (2008), there has been a “risk revolution” not only within
Financial Services Institutions but also within Corporation’s Finance Departments during
last decades. Following his description of the “Evolution of Risk Management”, this paper
implements a number of models going from Mean-Variance Analysis/Portfolio Optimization
(or Modern Portfolio Theory) to Value-at-Risk (VaR) and Option Pricing models, finishing
with a framework for risk management including hedging by using linear
1
(forwards) and non-linear (options) financial derivatives. All of these models were
implemented by using Computational and Programming tools. Hence, it will be noted that
the Methodology Chapter is extensive given that another particular objective of this project
is to serve as a wide-ranging guideline for the company to be able to successfully apply
theoretical/mathematical models into a computational system in order to efficiently
manage its FX risk.
In general, results suggest that the main FX risk factors for the company are Euro and US
Dollars and that its current FX portfolio is not efficient according to the portfolio
optimization solution. Also, VaR results are meaningful in terms of applicability for the
company since they could serve as a reference to set aside some FX risk reserve (e.g. its
current 3 times 10-day 99% VaR is currently £2.8m). Furthermore, Historical and Monte
Carlo VaR allow us to identify the presence of “fat tails” which are measured by the
calculation of a monthly Conditional VaR (CVaR) as an additional £0.3m. In addition, a stress
test suggests that if the last crisis scenario repeats in the near future, there is 1% chance
that the company loses an additional 1.8% of its monthly net positions. Finally, a Full
Valuation model shows that the GBP hedge reached by using a portfolio of EUR and USD
options is well higher that the hedge given by forwards only. The latter point is in line with
some Literature’s findings about the fact that hedging does add value to the firm and that
Options are more effective than Forwards while hedging against FX risk.
The paper is organized as follows. The first Chapter put the readers in the context of the
research issue by reviewing the relevant literature about it. Chapter 2 states the research
question, objectives and the scope of the following paper. Then, Chapter 3 describes data
collection process and main data sources along with a detailed description of the
methodologies, models applied and justification of choice. The fourth Chapter provides the
reader with a very illustrative analysis of empirical results, interpretations and discussions.
Lastly, Chapter 5 summarizes the main findings with a particular reference to
2
questions and objectives and some recommendations for the company along with a
critique of the work, limitations and some personal reflections.
3
Chapter 1 RESEARCH CONTEXT AND LITERATURE REVIEW
1.
Foreign Exchange Risk
In general, companies use to face two different kinds of risk. On one hand, Business Risk
derived from business decisions (e.g. strategic risks and product, marketing and
organizational risks) and business environment (e.g. macroeconomic risks and technological
changes risks) and on the other hand, Non-business Risk derived from financial risks (e.g.
market risk, liquidity risk, credit risk and operational risk) and from other risks like
reputational, regulatory and political risk (Jorion, P., 2007).
The focus of the following paper will be on the Financial Risk faced by companies and in
particular the management of foreign exchange risks. A company’s assets could be assumed
as a large portfolio of investments whereas their liabilities could be assumed as short
positions within the same portfolio (Freixas, X. et al 1997), and hence a company itself is
exposed to market risk. Precisely, Hull, J. (2007) defines Market Risk as the risk related to
movements in market variables including foreign exchange rates (i.e. FX Risk), interest rates
(Interest Rate Risk), commodity or stock prices (Price Risk), etc. Specifically, a company is
exposed to FX risk when its operations involve more than one currency. For instance, a firm
could have a mismatch between the currency at which its assets are denominated and the
currency at which its liabilities are. Therefore, depreciation in the asset’s currency could
lead to a loss in value of the company’s assets in relation to their liabilities. Broadly speaking,
“exchange rate risk can be defined as the risk that a company’s performance will be affected
by exchange rate movements” (Madura, J. et al, 2007).
4
For Madura, J. et al (2007), Multinational Corporations1 (MNCs) are exposed to exchange
rate fluctuations in three forms:
▪
Transaction Exposure: The degree of exposure of the cash inflows and outflows to
their respective exchange rates when converting to the domestic currency. This
exposure is closely related to the market risk exposure.
▪
Economic Exposure: The degree of exposure of company’s present value of cash
flows to exchange rate fluctuations.
▪
Translation Exposure: The degree of exposure of a company owners’ equity to
exchange rates fluctuations after consolidating financial statements.
1
MNCs are defined as companies that are operating globally.
5
2.
The Origins of Risk Measures: Modern Portfolio Theory and Mean-Variance
Analysis.
The modern analysis of risk in every investment decision was considered for the first time
in the former formulation of the Modern Portfolio Theory developed by Markowitz, H.
(1952). Markowitz simplified the optimal selection of a portfolio assuming that the
preferences of investors depend only of the first and the second moment (i.e. mean and
variance) of the random value of a portfolio’s returns distribution (Freixas, X. et al, 1997).
The Mean-Variance Analysis (MVA) or the Markowitz’s “expected return-variance of return
rule” set the idea of the trade-off between risk and return i.e. the more risk an investor
takes, the most return this investor would expect. It was also the first analytical review of
variance (or standard deviation2) as a measure of risk in modern finance.
Markowitz also showed that the idea of taking advantage of diversification with a portfolio
which gives both the maximum expected return and a minimum variance is not feasible
because the portfolio with maximum return is not the one with minimum variance. The
reason is that the correlations between the securities included in the portfolio are not equal
to zero and therefore to reduce the risk and take advantage of diversification, it is better to
invest in securities with low covariance among themselves. As a result, the MVA gives to
the investors the decision of maximize the expected return given a level of risk or minimize
the risk given a level of return (i.e. a dual optimization problem).
Later, Sharpe (1964), by working around Markowitz’s framework, developed what is known
as the Capital Asset Pricing Model (CAPM). The model basically relates the expected return
of a particular asset and the systematic risk (i.e. the “residual risk” after
2
In fact, both measures of risk can be considered as only one since the standard deviation is just the square root of the variance. However,
Lintner, (1965) showed, that under the portfolio optimization framework, the indifference functions of investors are linear only between
their expected returns and their variance and not their standard deviation, especially when all covariances are invariant (or zero).
6
diversification which results from movements in the economic activity and remains even
after efficient combination of securities within a portfolio) or “β”- Market Risk.
As we have seen, the portfolio optimization framework could be used as a tool for
minimizing the risk of the portfolio. In particular, since the securities within a portfolio are
exposed to the fluctuations of market variables, the major risk is the one called Market Risk.
Therefore, the first model to be used for assessing the transaction exposure of the
company, will be the MVA framework by minimizing the risk exposure or standard deviation
of their FX portfolio (σp) subject to a given level of expected return or E(rp). However, that’s
only one side of the problem. Apart from the set of portfolios we can construct based on
the minimum variance (i.e. the efficient frontier) for any given expected results, we also
have to take into account the preferences of investors. Since no portfolio on the efficient
frontier is superior or inferior to another one, the choice of the optimal portfolio depends
on the investor’s risk preference. As Elton, E. et al (1995) have explained, utility functions3
have four main economic properties:
▪
The nonsatiation property states that the utility increases as wealth increases (and
therefore the first derivative of utility with respect to wealth is positive).
▪
The investor’s taste for risk could take the form of an investor who is either risk
averse (i.e. the investor will reject a fair gamble and hence the second derivative of
utility with respect to wealth is negative), risk neutral or risk seeking.
▪
Investors can exhibit either an increasing absolute risk aversion (as wealth increases,
the investor holds fewer dollars in risky assets), constant absolute risk aversion (as
wealth increases, the investor hold the same amount of dollars in risky assets) or a
decreasing absolute risk aversion (as wealth increases, the investor holds more
dollars in risky assets). According to the author, most evidence suggests that
investors should show a decreasing absolute risk aversion.
3
An alternative framework is the one relating to behavioural finance and prospect theory (Khaneman and Tversky, 1979). We don’t
consider these aspects here because we examine our research problem from the perspective of a rational risk manager.
7
▪
Investors can exhibit an increasing relative risk aversion (as wealth increases,
percentage invested in risky assets declines), constant relative risk aversion (as
wealth increases, the percentage invested in risky assets is unchanged) or
decreasing absolute risk aversion (as wealth increases, the percentage invested in
risky assets increases). According to the author despite less general agreement
about this assumption, often a constant relative risk aversion is assumed.
As a result, after making the relevant assumptions behind the utility of investors, the
optimal portfolio will be the point where the investor’s utility function is tangential to the
efficient set. As showed graphically in the following figure, the red points show optimal
portfolios given a particular level of risk aversion (A) i.e. the more risk aversion, the lower
the risk taken by the investor.
Figure 1-Portfolio Optimization Example
14.0%
A=1
12.0%
A=2
Expected return
10.0%
8.0%
A=5
6.0%
A=10
4.0%
2.0%
0.0%
0.0%
5.0%
10.0%
15.0%
Standard deviation
8
20.0%
25.0%
It has to be noted that since the original objective of using the portfolio optimization
framework is to build a guideline in order to choose between different asset classes within
a portfolio, most of the literature is mainly focused on the selection between bonds and
stocks or between domestic and foreign assets in order to increase portfolio’s
diversification. Therefore, most of the literature about portfolio optimization including
currencies has been done by implementing a portfolio optimization of “real” assets plus the
use of currencies as hedging tools. For instance, Cantaluppi, L. (1994), showed that currency
hedge could lower the risk of an internationally diversified portfolio. He developed a
methodology where currency hedges are treated as a separate “pseudo- assets” with their
own weights, returns and risk. Later, Adcock, C. (2003), reported a performance of currencyhedged portfolios constructed by using MVA. The results showed the superiority of
optimally determined hedges over a fixed (1 to 1 ratio) hedge.
However, as remarked by Ratner, M. et al (2007) among others4, professional investors are
now treating currencies as a new asset class in order to take advantage of diversification. In
particular, He applied a portfolio optimization technique to a portfolio of
U.S and non-U.S. equity and foreign currencies taking past information between 1975 and
2006. He found that foreign currencies actually improve the Sharpe Ratio5 of the portfolio
since they show low correlation with U.S. equities.
4
5
See, for instance, Laise, E. (2006) and Vames, S. (2006).
It is a risk-adjusted return measure, defined as (E(Rp)-Rf)/σp, where Rf is the risk-free rate.
9
3.
Measuring Downside Risk: from MVA to VaR
One of the main assumptions of the MVA framework is that returns are normally distributed
and therefore the first two moments of the distribution (i.e. mean and variance) contain all
the relevant information about it (Elton, E. et al, 1995). As pointed out by Harvey, C. et al
(2010), Markowitz’s first paper didn’t analyze the effect of higher moments in the
distribution like skewness and kurtosis over the portfolio’s optimal selection and, in fact,
Markowitz himself highlighted that when an investor’s utility is a function of mean, variance
and skewness, mean-variance efficient portfolios would not be optimal6.
Moreover, variance (and standard deviation) is just a measure of dispersion so it only
examines the difference between the actual outcomes and the average return of the assets
and hence risk is measured by using both upside and downside movements. However,
“ceteris paribus, investors prefer a high probability of an extreme event in the positive
direction over a high probability of an extreme event in the negative direction” (Harvey, C.
et al, 2010) or, in other words, investors show a preference for a positively skewed return’s
distribution (See: Roy, A., 1952 and Sortino, F. et al, 1994).
For instance, the following Figure shows two different distributions. Both of them are
asymmetric but one of them is skewed towards gains whereas the second one is skewed
towards losses. Therefore, while using symmetric measures like variance, we obtain the
same level of risk for both of the distributions but while using asymmetric measures, the
outcomes show the second distribution as the riskiest one.
6
However, in subsequent years, Markowitz used optimization methods that also included the use of negative semi-variance in place of
variance. See: Markowitz, H. (1959) and Markowitz, H. et al, (1993).
10
Figure 2-Asymmetric Distributions
Asymmetric distribution
towards gains
Expected Results
Asymmetric distribution
towards losses
Furthermore, as pointed out by Blair, N. (2008) the focus on the minimization of variance
as a main risk management’s objective is not consistent with the financial management of
corporations (or MNCs) since their goals “are designed to reduce the expected costs of
financial trouble while preserving the company’s ability to exploit any comparative
advantage in risk bearing it may have” (Blair, N., 2008).
Therefore, “downside risk” measures can result to be better proxies for investor’s and
company’s exposures. The use of downside risk measures has been widely studied and
implemented and it seems to be even a better measure to implement, for the purpose of
portfolio optimizations strategies. For instance, Harlow, V. (1991) showed that downside
11
risk frameworks resulted in a better asset allocation and ultimately higher returns for
investors.
Since banks manage a large portfolio of tradable securities with daily price’s changes, the
need for a tool which can easily measure their daily market downside risk was the main
motivation for the creation of Value-at-Risk (VaR) model. JP Morgan introduced the model
in 1994 (J. P. Morgan Co., 1994) and now it’s very popular among practitioners and financial
regulators like the Basel Committee on Banking Supervision. As explained by Jorion, P.
(2007), “VaR summarizes the worst loss over a target horizon that will not be exceeded with
a given level of confidence”. For instance, if the daily VaR of an institution’s trading portfolio
is £ 100 million at the 95% confidence, there is a chance of 5 in a 100 that the loss could
exceed £ 100 million.
The advantage is that VaR is an aggregated measure which takes into account correlations
and current positions within a portfolio. VaR can be computed by taking into account either
the actual empirical distribution of returns of the portfolio to be assessed (“Historical” VaR)
or by assuming a Normal distribution (“Parametric” VaR) (Jorion, P., 2007). In addition,
distributions can be simulated by using Monte Carlo experiments (“Monte Carlo” VaR).
Madura, J. et al (2007) has developed some examples on how to use VaR models to asses
transaction exposure of MNCs derived from FX risk. However, authors like Shimko, D.
(1998), Jorion, P. (2007) and Blair, N. (2008) have studied the suitability of Cash Flow-atRisk (CFAR) instead of the traditional VaR models in the case of Non-financial Corporations
or MNCs. The main difference is that VaR consider the company’s market value at a
particular point in time and assumed that the portfolio of net assets is fixed over the shortterm horizon during which the risk exposure will be assessed whereas CFAR considers the
volatility and risk exposure of a flow of funds.
12
On the other hand, VaR models have shown a number of drawbacks that ultimately could
lead to inaccurate risk measurements. As highlighted by Cheng, S. (2004), Hull, J. (2006) and
Jorion, P. (2007), VaR measure takes into account only the frequency of losses but not their
size hence it doesn’t capture extreme losses with small probability of occurrence. In
particular, within a parametric VaR framework the assumption of a normal (and symmetric)
distribution doesn’t take into account higher moments like its skewness and kurtosis (i.e.
“fat tails”). Even more, as remarked by Stulz, R. (2009), by examining the historical volatility
of prices, investor’s could underestimate the probability of a severe drop in prices by
underestimating kurtosis and giving a positive skew to the distribution. Furthermore, from
an empirical perspective, Basak, S. et al (2001) found that VaR could lead investors to suboptimal investment allocations and in fact they could incur larger losses than non riskadverse investors.
From a more analytical perspective, Artzner, P. et al (1999) showed that VaR is not a
“coherent” measure of risk7 since it shows to be a not convex and also a not subadditive
measure of risk. Non-convexity means that VaR doesn’t take into account potential nonlinear increments of risk relative to the notional position whereas non-subadditivity implies
that the risk measure (i.e. VaR) of a portfolio could be larger than the sum of individual risk
measures (i.e. individual VaRs) of its components8.
As a result, a measure called Conditional-Value-at-Risk (CVaR), Expected Shortfall or Tail
Loss was developed by Artzner, P. et al (1999). CVaR, at a given confidence level, is the
expected loss over the losses which are higher than the VaR or the expected loss which is
equal or higher than the VaR (Rockafellar, R. T. et al, 2000). In other words, is an average of
the losses that exceed VaR and hence it takes into consideration the extreme losses
7
These authors defined a number of particular mathematical conditions that a good measure of risk must have to be considered as
such.
8
In particular, in the case of a portfolio formed by the securities X and Y, could occurs that: VaR(X+Y)>VaR(X) +VaR(Y). This is particular
important for Financial Institutions since it could happen that a well diversified portfolio require more regulatory capital than a less
diversified portfolio (See: Cheng, S. et al, 2004).
13
within the distribution. For instance, with a confidence level of 99% and 10 days of time
horizon, “CVaR is the average amount we lose over 10 days assuming the 1% worst-case
event occurs” (Hull, J., 2006).
14
4.
To Hedge or not to Hedge, that is the question
After assessing its risk exposure, a company could decide to hedge its positions by operating
or financial hedges. However, if we take into account the Modigliani and Miller (M-M)
theorem (Modigliani, F. et al, 1958), which states that under some particular conditions
companies’ financial policies shouldn’t affect their values, it is not clear if risk management
could or not add value to the company. In particular, some theoretical literature has
remarked the irrelevance of risk management, as summarized by Hull, J. (2007) and
Madura, J. et al (2007); the main arguments are as follows:
▪
Purchase Power Parity (PPP) argument: When markets are in equilibrium with
respect to parity conditions, the value of hedging is zero since the effect of exchange
risk would be offset by the change in prices.
▪
The investor hedge argument: Shareholders are able to diversify their risk by
themselves. If they don’t want to take currency risk in some specific investment,
they can diversify their portfolios in order to reduce it; therefore companies don’t
need to concern about FX risk.
▪
Currency diversification argument: As long as MNC are already diversified across a
number of countries, there is no need to hedge their positions.
However, Madura, J. et al (2007) has developed strong contra arguments against those
described above. Regarding the PPP for instance, the author highlighted the fact that the
parity doesn’t necessarily hold in the short term and even if it holds over a long period of
time, managers are always specially concern about the “next period” (e.g. next quarter
figures). Moreover, there has been a large research around the empirical testing of the PPP.
For instance, Miskhin et al (1984) and Abuaf, N. et al (1990) found significant deviation of
exchange that persisted for specific periods. In particular, the latter paper
15
found that deviations from PPP are substantial in the short run and could take about 3 years
to be reduced in half.
Regarding the investor hedge argument, asymmetric information has to be taken into
account. Company managers have comparative advantage in knowing the actual exposure
of the firm and, in fact, their expertise allows them to hedge at a lower cost than investors
could. Finally, perfect diversification and hence perfect offset between currencies changes
is not usually possible since correlations can be volatile and high during particular periods
and therefore the need for hedging is still there.
Other arguments that proponents of hedging give are more related to the financial
management of the company. For instance, by reducing the volatility of cash flows, a firm
can improve its planning capability and also reduces the likelihood that firm value fall below
a predefined minimum target. Also, Jorion, P. (2007), remarked that the real usefulness of
the M-M theorem is that it allows us to remark the actual market imperfections. Therefore,
He summarized a number of reasons that different authors have remarked since the
appearance of the M-M theorem. For instance, “Hedging can lower the cost of financial
distress” by reducing the probability of left-tail results; “Hedging can lower taxes” (only
when firm’s tax function is convex) by reducing the volatility of cash flows; “Hedging can
lower agency costs” between managers and stakeholders by reducing volatility of earnings
and therefore making earnings more informative and easy to evaluate; and “Hedging can
facilitate optimal investments” by providing steady cash flows needed for investing in R&D
programs, for instance.
As we have seen, despite the fact that hedging not only protects a firm against potential
losses (i.e. downside risk) but also could eliminates any gain (upside movement), its
implementation could reduce the uncertainty about the cash flows of the firm and
ultimately increase its value. Precisely, there are a number of empirical studies around the
16
question of “Does risk management add value?” In particular, Smithson, C. et al (2005)
made a remarkable effort to summarize the results of the empirical evidence. For instance,
under the question “Is the use of Risk management tools associated with lower levels of
risk?” They found that in the case of non-financial corporations, almost 90% of the studies
showed that the use of financial derivatives reduced the effect of currency’s changes on the
volatility of their stock returns. For instance, Guay, W. (1999) by studying the effect of
interest rate and FX derivatives of new users (i.e. firms that previously had not reported
using derivatives) between 1990 and 1994, found a decline in sensitivities along with a
reduction of beta (i.e. market risk). Furthermore, they reported that for the case of
nonfinancial companies studied, all of them indicated that hedging by using FX derivatives,
added value to the firm (as measured by Tobin Q9). In particular, Allayannis, et al (2004) by
studying the impact of the use of FX risk on firm value for 379 firms between 1990 and 1999,
they found a significant positive premium for users of derivatives with FX risk exposures.
After reviewing the goodness of hedging (i.e. actions that help the company to lower the
volatility of cash flows or firm value), firms can opt between, broadly speaking, two ways
for hedging: on one hand, a natural or operating hedge which refers to an offsetting
operating cash flow with another one that arises from the conduct of business itself and on
the other hand, a financial hedge which refers to a money market hedge or a financial
derivative like futures, forwards, or options (Madura, J. et al, 2007). In particular, the
following paper focuses in the use of currency forwards and options as financial hedge tools
against FX risk.
Finally, there is some literature related to the comparison between different hedging
strategies. For instance, Allayanis, G. et al (2001) by analyzing a sample of U.S. multinational
nonfinancial firms between 1996 and 1998, found that operational hedging
9
Ratio of a company’s market value to the replacement value of its assets.
17
(measured by using geographic dispersion metrics as a proxy) didn’t reduce their exchange
rate exposure nor added value and neither maximized shareholder value whereas financial
hedging (i.e. use of financial derivatives) was related to lower exposures and higher firm
values. Furthermore, Maurer, R. et al (2007), found that by using in-the- money options and
forwards an investor could build a much more diversified bond and stocks portfolio that
when no hedging strategy is taken. Also, He showed that by dedicating a smaller part of the
investment in European Options, they have the potential to substitute an optimally
forward-hedge portfolio.
18
Chapter 2 RESEARCH QUESTION AND OBJECTIVES
1.
Research Questions
The main question of the following research document would be “how different risk
measures and models could be successfully applied in order to assess and reduce foreign
exchange risk of a Multinational Corporation?”
In line with the latter general interrogate, the company provided us with the following
questions to be analysed during the developing of this project:
a. Can we mathematically describe the foreign exchange risks that we face?
▪
How do the specifics of the company funding affect this modelling of risk?
▪
What data do we need to do this?
▪
How can we use the models in the future?
b. How can we use this to help us manage our foreign exchange risk?
▪
What risk management tools should we use?
(How much can we absorb
internally and what external tools should we use?)
▪
How can we reflect this management of risk within our systems?
▪
How do we use market intelligence to help with risk management?
19
2.
Objectives
The aim of this paper is to develop a comprehensive risk management system in order to
identify, measure, monitor and report the foreign exchange risk exposure of a UK MNC by
using a number of frameworks and models to finally recommend the best alternative in
order to reduce its exposure by using linear and non-linear financial derivative’s hedging. In
particular, the paper focuses on the Economic and Transaction FX risk exposure of the
company.
It will be noted that the Methodology Chapter is extensive given that another particular
objective of this project is to serve as a wide-ranging guideline for the company to be able
to successfully apply theoretical/mathematical models into a computational system by
using Excel and its tools in order to efficiently manage its FX risk.
20
3.
A brief description of the Company10
The company analysed is a multinational UK Non-Governmental Organisation (NGO), we
call it MNGO hereafter, which works as an international confederation of 14 organizations
working together in 99 countries and with partners and allies around the world. As a result
of its truly global work, the company manages more than 80 different currencies during its
regular operations. Consequently, assessing and controlling its FX exposure in the most
efficient and accurate way becomes a vital task for them.
It has to be pointed out that the cycle of income and expenditure is different to that of a
commercial company. The company’s Income (which increased £8.6m to £308.3m between
2008 and 2009) comes predominantly from the activities of trading (e.g. commercial shops)
and fundraising (e.g. donors). Fundraising could be divided in unrestricted and restricted
with a regular ratio of 40:60 between them. Unrestricted funding come from regular givers
and legacies and could be spent for different purposes whereas restricted funding comes
from governments and institutional donors and have to be used for specific projects
(programmes) in particular currencies and countries around the world. Even unrestricted
income, which is predominantly in GBP, have to be spent mostly in (a very wide range of)
non-GBP currencies. Consequently, the company operates in a permanent mismatch
between currencies in its “asset” and “liability” side, increasing its exposure to FX risk and
the need for its efficient management.
10
Most of the information in this section has been extracted from the Company’s Official Web Page and Annual Report and also from
some internal reports provided.
21
Chapter 3 DATA AND METHODOLOGY
1.
Data Gathering
Data about MNGO’s positions in different currencies was obtained from the reports of
monthly Income and Expenditures between 2006 and 2010 financial years, provided by the
point of contact. Financial periods run to April up to 2009 and then, they run to March only
in 2010. A list of the currencies (plus codes and countries of origin) with which the company
operates, could be seen in Appendix A. It should be noted thought that the best
approximation for assessing the market risk of a company could be performed by taking
into account its balance sheet information (i.e. assets and liabilities). However, the balance
sheet of this company in particular doesn’t reflect the difference in domestic and foreign
currencies and hence the only meaningful data is on the income/expenditure side. As it will
be discussed later, one of the main assumptions is that the flows of income/expenditure
funds remain fixed over the period to be analysed (i.e. monthly). The following table
summarizes income/expenditure data from 2006 to 2010:
Table 1-Income/Expenditure Data Summary - £m
Source
Item
Income
From Annual Report Expenditure
Income
From Currency
Analysis
Expenditure
2006
310.5
298.0
216.2
216.7
2007
290.7
297.2
203.1
209.9
2008
299.7
298.4
206.8
210.9
2009
308.3
318.6
215.4
227.9
2010
318.0
294.8
232.3
212.8
It should be noted that the currency analysis is only for MNGO’s International Division;
therefore, there is a gap between this information and the Organisational Income
Statements shown in the financial statements. Since the analysis of this gap is not easy to
get, it can be assumed that it be all denominated in Sterling Pounds-GBP (as it will all be
22
UK expenditure) and spread evenly over the 12 months and consequently it doesn’t
represent a risk factor for the company in terms of FX risk.
Data of daily exchange rates for 80 different currencies relative to GBP was obtained from
Thomson Reuters DataStream for a period of 10 years. However, data for Eritrean Nakfa
(ERN) was obtained from Oanda.com11 due to the unavailability of a time series in
DataStream. All the exchange rates were downloaded as indirect quotes i.e. as a foreign
currency price of a unit of home currency or GBP. For instance, on July 30 th the exchange
rate U.S dollar to Sterling Pound (USD/GBP) was 1.4949. Also, for the aim of simplicity and
availability of data, only mid rates were taken (average between bid and ask rates). Finally,
UK Inflation and interest rates were collected from the Office for National Statistics 12 and
FT.com, respectively.
On the other hand, all of the data analysis was conducted by using Microsoft Excel, its tools
complements (e.g. Data Analysis and Solver) and VBA macros. Also, regression analysis was
carried out by using E-Views 4.0.
In the following sections we will discuss in detail the different methodologies used for assess
and measure the economic and transaction exposure of the company along with the model
used for hedging the FX portfolio.
11
“OANDA is a market maker and a trusted source for currency data. It has access to one of the world's largest historical, high
frequency, filtered currency databases”. (www.oanda.com).
12
http://www.statistics.gov.uk/statbase/TSDdownload2.asp
23
2.
Economic Exposure
In order to assess the economic FX exposure of the firm, a sensitivity analysis of cash flows
to exchange rates is carried out first. As proposed by Madura, J. (2007) this could be done
by implementing a regression analysis of cash flows and exchange rates according to the
following model:
Formula 1-Economic Exposure Regression
∆CFt = α + þERt + et
Where, ∆CFt is the percentage change in the inflation-adjusted cash flows measured in GBP
for the period t; α and β are the intercept and the slope coefficient, respectively; ER t is the
percentage change in the exchange rate of the currency over the period t and, finally, et is
the error term.
As commented in the last section, monthly data of Net Cash Flows (Income minus
Expenditures) from June 2005 to March 2010 was collected giving a total of 58 observations
for each variable. The adjusted Cash Flows figures were obtained by the following formula:
Formula 2-Adjusted CF
∆CFt =
(1 + dCFt)
(1 + INFt)
Where, dCFt is the monthly percentage change of the firm’s net cash flows and INF t is the
monthly UK inflation.
Madura, J. (2007) proposed the inclusion of more than one currency as independent
variables if that is the case, however, to avoid problems of multicolinearity and
24
autocorrelation (because of the high correlation among all the currencies as show in the
next section); a univariate regression is carried out independently for each of the main
currencies within the portfolio. This allows us to check the size of the sensitivity of CF to
exchange rates by looking at the slope coefficient of the regression.
In the following section, we will start the Transaction FX Exposure Analysis by using the
Modern Portfolio Theory Framework. In particular, a Mean-Variance Analysis allows us to
minimize the variance of the FX portfolio while getting its efficient frontier.
25
3.
Portfolio Optimization (MVA)
As commented in the Literature Review, Portfolio Optimization framework is developed by
minimizing the risk exposure or standard deviation of the portfolio (σp) subject to a given
particular level of expected return or E(Rp). Hence, by using Solver tool of MS Excel, we
identified the corresponding minimum variance portfolios by increasing the required return
of the portfolio in discrete quantities. As a result, the efficient frontier will be a concave
function within the expected return-standard deviation plane which extends from the
Minimum Variance Portfolio (MVP) to the Maximum Return Portfolio (MRP) (see: Elton, E.
et al, 1995) as showed before in Figure 1.
In particular, in the case of a portfolio with “N” assets, the Optimisation Problem is given as
following:
Optimization Problem 1- MVA
N
N
N
Minimize: JΣ w 2 o 2 + Σ Σ wi wj oij
ii
i=1 j=1
i*j
i=1
Subject to:
N
(1) Σ wi R̄i = R̄p
i=1
N
(2)
Σ wi = 1
i=1
(3)
wi ≥ 0,
i = 1, … , N
Where, wi is the weight of asset i in the portfolio, σij is the covariance between asset i and
j (i ≠ j) and Ri is the returns of asset i.
26
Restriction (1) simply reflects that the weighted average of given returns is equal to the
given return of the portfolio. Also by varying it between the return of the MVP and the
return of the MRP, the Efficient Frontier could be obtained (Elton, E. et al, 1995).
Restrictions (2) and (3) reflect the assumptions that there is no short selling and no riskless
lending and borrowing. However, results obtained when the assumption of no short selling
is relaxed, are showed later in this paper.
As commented in the last chapter, it has to be noted that the portfolio optimization
framework is originally used as a guideline for the optimal selection between different asset
classes to be included within a portfolio of investment (i.e. asset allocation) and also for the
selection between domestic or foreign investments in order to take advantage of
diversification. In this particular case though, our parameter to make the asset allocation
will be the currency at which company’s positions are denominated, in other words,
different currencies are treated as different asset classes with their own weights, returns,
correlations and risk as developed by Cantaluppi, L. (1994) and VanderLinden, D. (2002) and
therefore MVA is developed over a Portfolio of Currencies.
At first, taking into account the restrictions in the company’s expenditures funds, no short
positions will be assumed. Therefore the portfolio optimization is performed by taking only
the “asset side” (i.e. income) of the portfolio of currencies of the company at the end of
March 2010. Consequently, the weights of different currencies within the portfolio are
calculated by using the Income Data which includes operations with 20 different currencies.
Daily exchange rate’s returns were calculated by using the continuous compounded return
Rt, given by the following formula:
27
Formula 3-Exchange Rate’s Return
St
R = ln (
)
t
St–1
Where, St is the exchange rate at time t and St-1 is the exchange rate at the previous date.
Then, the expected return E(Ri) of each currency was obtained from historical data of one
year period before the end of June 2010. As a result, the expected return of the portfolio or
E(Rp) was calculated by multiplying the vector of weights by the respective expected return
of every currency as shown in restriction (2) of the Optimization Problem 1. In terms of
matrix algebra the Portfolio Return is calculated as following:
Formula 4-Portfolio Return
E(Rp) = [W’] [E(Ri)]
Where W’ is the transpose of the weight’s vector with a dimension of (1x20) in the case of
income position’s weights.
It has to be highlighted that in this particular case, as the analysis is done over currencies’
long positions, all returns were calculated after transforming indirect exchange rates quotes
into direct exchange rates quotes (i.e. their inverse). So that, a positive change means an
appreciation of the foreign currency and therefore a positive return for the company and
vice versa.
The following table summarizes the calculation of the expected Portfolio Returns:
28
Table 2-Portfolio Return
Curr. Code
Income£
Weight
Exp Return
USD
EUR
PKR
BRL
INR
CAD
AUD
BDT
ZAR
TZS
ILS
SLL
YER
HTG
PHP
KES
XOF
COP
AFO
NPR
Portfolio
7,406,410
6,705,768
331,837
311,944
149,508
146,477
133,848
105,994
73,387
40,654
33,550
29,266
18,623
16,711
8,584
5,420
3,830
972
516
502
15,523,799
47.710%
43.197%
2.138%
2.009%
0.963%
0.944%
0.862%
0.683%
0.473%
0.262%
0.216%
0.189%
0.120%
0.108%
0.055%
0.035%
0.025%
0.006%
0.003%
0.003%
100.000%
0.039%
-0.014%
0.019%
0.070%
0.052%
0.071%
0.054%
0.038%
0.046%
-0.005%
0.043%
-0.029%
-0.005%
0.038%
0.054%
0.018%
-0.014%
0.084%
0.054%
0.052%
0.016%
On the other hand, Portfolio Variance is calculated by using the target of the optimization
problem 1 showed before. In matrix algebra the calculation is as follows:
Formula 5-Portfolio Variance
σ2p = [W’] ∑ [W]
Where, ∑ is the variance-covariance matrix of the 20 currencies derived from the correlation
matrix which are also attached in Appendix B. Then the standard deviation is obtained from
the square root of the variance.
Finally, as commented in the previous chapter investor’s utility has to be taken into account
as well. In particular, under our portfolio optimization framework, investors are assumed to
be both risk averse (i.e. the decision is restricted to the efficient frontier since
29
investors want to minimize variance) and utility maximisers (i.e. non-satiation property) and
therefore while keeping risk constant they prefer highest possible return and also the more
risk they take, higher the compensation they expect. Finally, investors are indifferent
between alternatives that offer the same utility.
In particular, as also explained by Elton, E. et al (1995) and Freixas, X. et al (1997), a
quadratic utility function is usually used in MVA because the use of it, leads to optimal MVA
outcomes. In its general form, a quadratic utility function could be defined as:
Formula 6-General Quadratic Utility Function
U(W) = W − AW2
And its first and second derivative, respectively, as:
U′(W) = 1 − 2AW
U"(W) = −2A
And therefore, the investor shows risk aversion since the second derivative is negative for
all positive “A”. Also, to be consistent with non-satiation property, the first derivative should
be positive. That is always true if and only if: 1-2A>0 or W<(A/2). Therefore, as usually used
in MVA, the following particular form of a quadratic function is used in this project:
Formula 7-Particular Quadratic Utility Function
A 2
U = E(Rp) − ( ) op
2
Where, A is the company’s level of risk aversion. This number is subjective and could be
assigned so that a comparison between different levels of risk aversion affects the final
30
asset allocation within the portfolio. Therefore, is a relative metric instead of an absolute
one.
It has to be noted also that the quadratic function exhibits increasing absolute risk aversion
(i.e. as wealth increases, the investor holds fewer dollars in risky assets) and also increasing
relative risk aversion (i.e. as wealth increases, percentage invested in risky assets declines).
Finally, by using Solver the maximization of formula 7 is carried out subject to the
parameters of portfolios within the efficient frontier in order to get the portfolio which is
optimal (i.e. risk-return combination where the investor’s utility function is tangential to the
efficient frontier).
As commented in Chapter 1, variance analysis could be a biased or a not accurate measure
of risk and therefore in the following sections, we will describe the methodology of a
downside risk measure like the Value-at-Risk (VaR) model.
31
4.
Value-At-Risk (VaR) Methodology
In a formal expression, VaR can be defined as the lower quartile q-th “q” of the inverse
return’s distribution function “F” of a portfolio, as showed as follows:
Formula 8-VaR, General Form
VaR(q) = − F–1(q)
As commented in Chapter 1, there are basically three methods to be implemented in order
to calculate the VaR of a portfolio. The main difference between them is principally the
assumption behind the distribution of the portfolio’s return. In this paper, the Parametric,
Historical and Monte Carlo Simulations Approaches are implemented.
As a first step in order to calculate VaR the risk factors have to be identified. In this particular
case, we need to assess the FX risk of the company’s Cash Flows of Income and
Expenditures. Therefore, every currency in which the company has position will be a risk
factor and hence the movements of these exchange rates will affect the value of the
portfolio. It has to be pointed out again that the calculation of VaR was made originally to
take into account of the market risk exposure of an asset’s portfolio (e.g. a portfolio of
securities which is affect to market’s prices movements) however, in this particular case we
will carried out a sort of Cash-Flow-at-Risk (CFAR) assuming that the only risk which could
affect Cash Flows is the FX risk.
Moreover, the assumption again is that the flow of income and expenses funds will freeze
during the period (i.e. a month). Also, since the company has income and expenditures (i.e.
long and short positions), the VaR is calculated over both the gross position (i.e. sum of long
and short positions) and absolute net position (i.e. absolute value of the difference between
long and short positions). The former calculation takes into the effect of exchange rate
volatility without differencing the position, whereas the second one takes
32
into account the offset effect between income and expenditures denominated in the same
currency during the period to be analysed.
Finally, as commented earlier the latest data available of the company’s currencies positions
is as March 2010, however, volatilities and correlations are calculated by taking into account
information between June 2009 and June 2010, in order to reflect the effect of recent
market (i.e. exchange rates) movements over the Foreign Exchange position of the
company. In other words, it has been assumed that March position will be approximately
the same as would it be in June. It has to be noted however that the correct way to calculate
VaR is by taking both market and position data for the same month but in this particular
case that wouldn’t be a good approximation of the current Company’s FX Risk exposure.
a.
Parametric (Variance-Covariance) VaR
The implementation of the Parametric VaR is carried out by following the methodologies
explained by Hull, J. (2006), Madura, J. et al (2007) and Jorion, P. (2007). The main
assumption is that exchange rates returns follow a normal distribution (i.e. a symmetric
distribution) with known mean and variance. Then, it is possible to calculate the lowest
percentile of the distribution at a particular level of significance (e.g. 1%, 5% or 10%).
The two main components are the mean and the variance of the distribution. These two
components are calculated over a period of one year before June 2010 by using formulas 4
and 5 and including weights for both gross and net positions. Also, VaR parameters have to
be chosen, the time horizon (T) and the Confidence Level (CL). Regarding the time horizon,
as commented before, the FX exposure of the company changes monthly and therefore a
T=20 (i.e. 20 working days) is chosen. Finally, regarding the confidence level, Basel
Committee (BIS, 1996), proposed a 99% (i.e. 1% probability of loss-level of
33
significance or -2.33 standard deviations) within a period of 10 days whereas JP Morgan
(1996) proposed a 95% (i.e. 5% probability of loss or -1.65 standard deviations) within one
day. In the following paper we will use predominantly a CL of 99%, 95% and 90% within 1
day and 20 days.
Therefore the corresponding VaR of the portfolio will be calculated by using the following
formula:
VaRp =
Formula 9-VaR, Parametric Form
£ Position x CL NorNalized Factor x op x √T
Where, the £ Position is the gross or net position expressed in the domestic currency (i.e.
GBP), the CL normalized factor is the critical value (measured in standard deviations) which
corresponds to a certain one-tail probability within a normal distribution. The following
figure shows the standard deviations and their corresponding probability within a normal
distribution:
Figure 3-Normal Distribution
Probability
Values of the
Random Variable
68.26%
95.44%
99.74%
34
Finally, σp is the standard deviation of the portfolio and the holding period of the portfolio
of currencies are assumed to be 20 days. As explained by Hull, J. (2006), it can we assumed
that random variables (as exchange rates) follow a Markov Stochastic Process13 and that its
change in the value during 1 day is φ (0, 1), where φ (μ, σ) is a normally distributed
probability function with mean μ and standard deviation σ. Therefore, the change in 20 days
will be the sum of 20 normal distributions with mean zero and standard deviation of 114 and
since the variable follows the Markovian property, these distributions are independent and
their addition will result in a normal distribution with the corresponding sum of means and
variances. As a result, the change in the variable over 20 days is φ (0,√20) since its mean is
zero and its standard deviation is just the square of its variance which is equal to 20.
b.
Historical VaR
In contrast with the previous approach, there is no assumption behind a particular
distribution of portfolio’s returns, however, the assumption that historical information is a
good proxy of the future, holds. The empirical histogram of historical returns is used to
calculate the lower percentiles of losses and the variance-covariance patterns of different
assets are incorporated in the procedure itself.
Historical VaR is implemented by following the methodology described by Hull, J. (2006).
The first step is to again identify the risk factors or market variables affecting the portfolio
which is in our case are the exchange rates movements within the company’s portfolio.
Then, 500 observations before June 30th 2010 (starting on July, 30th 2008 as Day 0) were
selected. As commented by Jorion, P. (2007), there is always a trade-off between take a
series longer enough to be precise but not too long to become irrelevant, therefore, a series
between 250 and 750 observations is usually recommended. The daily percentage
13
14
Particular type of stochastic process where only the present value of the variable is relevant to predict the future (Hull, J., 2006).
And a variance of 1 since variance= σ2
35
changes in exchange rates values during that period are assumed to occur again during the
following 500 days starting on July, 1st 2010. For example, if EUR changed from 1.2723 in
Day 0 to 1.2717 in Day 1 hence the change on our first scenario will be the change
percentage of the exchange rate multiplied by the exchange rate on our last day of
observations (1.2218 on June, 30th 2010) as following:
1.2218 x
1.2717
= 1.2212
1.2723
The same method is assumed for each exchange rate during the following 499 scenarios.
Thirdly, the whole portfolio is revaluated by using the corresponding weights (net and gross)
and exchange rates for each currency. Appendix C shows the frequency distribution of net
position weighted portfolio’s values over the scenarios generated for 500 days ahead.
Finally, taking into account the empirical distribution of portfolio’s changes the lower 1, 5
or 10-percentile within the left tail of the distribution (i.e. losses) could be taken. For
instance, in the case of 1% (5 out of 500 scenarios), the fifth worst number from value’s
changes will be taken. It should be pointed out that this is a daily or 1-day VaR and therefore
the 20-day VaR is calculated by multiplying the result by √20 as in the previous approach.
c.
Monte Carlo VaR
In contrast with the previous framework, the assumption that historical information is a
good proxy of the future doesn’t hold. We can actually randomly generate a probability
distribution for changes in the portfolio’s value and then calculate the lower percentiles of
losses.
36
Monte Carlo simulations are implemented by following the methodology explained by Hull,
J. (2006) and Jorion, P. (2007). As explained by the latter author, the first step is to assume
a particular stochastic behaviour for the financial variable’s prices to be analyzed (i.e.
exchange rates in this particular case). In particular, a Geometric Brownian Motion Model
will be assumed in this paper. This model is commonly used in order to characterize a stock
price process and includes most of the assumptions behind option pricing theory (i.e. Black
Scholes model). The model assumes that small changes in prices could be described by the
following formula:
Formula 10-Geometric Brownian motion Model
dSt = µtStdt + otStdz
Where, St is the price at time t, µt and ot stand for the expected return and volatility,
respectively and dz is a normally distributed random variable with mean zero and variance
dt. As explained by the author the model is Brownian in the sense that the variance
decrease continuously with time (i.e. there are no jumps) and it is geometric since all other
parameters are scaled by the current price.
Then, in order to generate scenarios for exchange rates over the assumed time horizon for
the company (i.e. 20 days), we can assume that exchange rates follow a lognormal
distribution15 and hence, after integrated the last formula (dS/S) over a finite period of time,
future exchange rates can be generated by using the following formula 16:
Formula 11-Simulated Return’s Generator
ST = Stexp ((µ − 0.5o2)T + so√T)
15
However, as described by Hull, J. (2006), it has to be noticed that the lognormal assumption could be not suitable for exchange rates
since they usually exhibit jumps (the price doesn’t change smoothly) and it doesn’t exhibit constant volatility.
16
dS is obtained by (ST-St)/St and therefore, the right hand side of the formula will be given by exp ((µ − 0.5o2)T + so√T)-1.
37
Where, μ is the annual expected returns and σ is the annual standard deviation of
exchange rates, T is the time horizon and ε is the standard normal random variable.
As explained by Hull, J. (2006) a univariate standardized normal distribution could be
obtained by the following formula:
Formula 12-Univariate Standardized Normal Distribution
12
s = Σ Ri − 6
i=1
Where, Ri are the independent random numbers between 0 and 1.
In this paper, as an alternative way, single random numbers are created in excel by using
the formula = RAND() and hence a random sample from a standard normal distribution was
obtained by using =NORMSINV(RAND()).
Then, in order to generate correlated random numbers among all the currencies company’s
portfolio (while using net positions), Cholesky decomposition or factorization will be used.
This procedure was named after the French mathematician André-Louis Cholesky and
stated that any positive definite symmetric matrix (R) can be decomposed into R = LL’ where
L is the lower triangular matrix with zeros in the upper right corners. As explained by Hull,
J. (2006), the coefficients of the lower matrix have to be selected so that correlations and
variances are correct. For instance, if we define three correlated random samples, with the
correlation between sample i and j being ρij, as follows: s1 = α11x1, s2 = α21x1 + α22x2
and s3 = α31x1 + α32x2 + α33x23. Hence, we have to
set α11 = 1, choose α21 so that α21α11 = q21, choose α22 so that α2 + α2 = 1 and so
21
22
on till choosing α33 so that α2 + α2 + α2 = 1. Then, correlated random numbers can
31
32
33
38
be generated by using: rxL’ where, r is the row vector of random numbers generated for
each currency.
After generating 500 scenarios by using formula 11, we obtained the change in the
portfolio’s value by multiplying changes of each individual exchange rate change by its
corresponding weight within the portfolio. Finally, a simulated distribution is obtained and
the specific lower quartile is calculated in order to get Monte Carlo VaR for 20 days. The
frequency of the resulted distribution for the value of net position weighted portfolio is
included in Appendix D.
39
5.
Improving VaR: Conditional Value-at-Risk (CVaR) and Stress Testing
As mentioned in Chapter 1, VaR doesn’t capture extreme losses with small probability of
occurrence and in fact, could have be seen as the minimum loss that could happen since it
doesn’t say anything about losses when the threshold given by the selected confidence level
is trespassed. Therefore, taking into account the scenarios build for the Historical approach,
the Conditional-Value-at-Risk (CVAR) is calculated, at a given confidence level, as the
expected loss over the losses which are higher than the historical VaR. In other words, it will
be an average of the losses that exceed VaR and hence it will take into consideration the
extreme losses (i.e. “fat tails”) within the empirical distribution.
As explained by McNeil, A. (1999), the expected shortfall (ES) or CVaR is related to VaR by
the following formula:
Formula 13-Conditional VaR
CVaRq = VaRq + E[X − VaRq|X > VaRq]
Where, X corresponds to the losses that occur when the particular quartile q-th is
trespassed. Hence the second term of the formula reflects the expected excess losses
distribution above the VaRq. Following this methodology, the mean of the excess losses is
calculated and added to the original Historical VaR calculated.
Another option in order to measure the impact of extreme events over the portfolio of the
company is by using a non-statistical risk measure such as Stress Testing. Stress Testing is a
sort of Scenario or Sensitivity Analysis to measure the risk under non-normal market
conditions (see; Jorion, P., 2007). For instance, a historical scenario of financial turmoil could
be taking into account in order to re-evaluate potential losses of the portfolio.
40
In our particular case, for example, the last few years have been a scenario of global market
turmoil and therefore the period between July 2008 and July 2009 is taken. The following
figure demonstrates that in fact, this period shows the largest episode of volatility
(measured by both monthly and yearly standard deviation of exchange rate’s returns) for
the two main currencies within the company’s portfolio (i.e. EUR and USD) during almost
the last 10 years:
Volatility
Figure 4-Exchange Rate’s Volatilities
Consequently, this period is used in order to calculate the standard deviation and
correlations of the portfolio’s currencies. Finally, a Parametric VaR is obtained by using
current weights along with these stressed parameters.
41
6.
Hedging the Portfolio: Options and Forwards
a.
Definitions
EUR/GBP and USD/GBP are not only two of the most liquid exchange rates traded in the FX
markets but also they are the main FX risk factors of the company’s portfolio. In fact, as the
correlation matrix, used for portfolio optimization, shown (Attached in Appendix E), all of
the currencies including in the portfolio have a positive and relatively high correlation with
USD and EUR. For the previous reasons, for the purpose of hedging the company’s portfolio
against exchange risk, only EUR and USD financial instruments are considered. In particular,
we will analyse the case of hedging with both FX Options and FX Forwards. Both of them
are financial derivatives and therefore their values are derived from the price’s movements
of a particular underlying asset (i.e. exchange rates in this particular case). It also involves a
transaction today but with a future settlement date at a pre- established price.
In the case of forwards, a company agrees with a commercial bank to exchange a specified
amount of a currency at a specified exchange rate (i.e. forward rate) on a specified date in
the future. Forward contracts are often valued at £ 1 million and trade in maturities up to
12 months and in the over-the-counter (OTC) market (i.e. private agreement between two
parties like banks and companies). When a company anticipated need of hedging for a
future receipt (i.e. long position) or payment (i.e. short position) of foreign currency, they
can sell forward contracts or buy forwards contracts, respectively in order to “lock in” the
exchange rate in the future. It has to be noted that futures contracts have the same dynamic
as forwards however, they differ from forwards since future contracts are standardized in
terms of size, futures have fixed maturities, and they are traded in organized exchanges
with a need for initial margins.
42
On the other hand, a FX option is a contract giving the company (the buyer) the right, but
not the obligation, to buy (i.e. call option) or sell (i.e. put option) a given amount of currency
at a fixed price (i.e. exercise or strike price) until the maturity date. There is a difference
between European Options and American options since the first one could be exercised
only on its expiration date and not before as in the case of the American options. Currency
options are traded in the OTC market and in the Philadelphia Stock Exchange in the U.S.
When a company is due to receive foreign currency in the future, it can hedge its FX
exposure by buying FX put options in that currency with the same maturity date in the
future while when the company is due to pay in foreign currency, the hedge can be done by
buying call options.
In the particular case of this company, it has a long position of EUR and a short position of
USD and hence they will be hedge by short forwards/long a put and long forwards/long a
call, respectively. The length of the contracts will be assumed to be one month as the
company resets its position (income/expenditures) on a monthly basis.
Finally it has to be highlighted that by hedging with forwards, futures or options, the
company is able to guarantee the value or cost of the foreign currency at a certain point in
time in the future. However, since the FX option gives the company the right but not the
obligation to buy or sell the foreign currency, it also allows the company to benefit from
favorable exchange rate movement while forwards or futures don’t give that flexibility. This
sort of insurance given by options has a cost which is reflected in the premium, cost or price
of the option.
43
b.
Derivatives Valuation and Optimal Hedge
The value of a forward exchange rate is derived from the interest rate parity condition, by
adjusting the spot rate with the interest rate differential between the foreign and domestic
currencies, as described in the following formula:
Formula 14-Forward Exchange Rate
FO = SO e (r–rf )T
Where, F0 is the forward exchange rate, S0 is the spot exchange rate, r the domestic interest
rate, rf is the foreign interest rate and T the time of the contract.
On the other hand, as showed by Hull, J. (2006), the optimal hedge ratio-h* (or the minimum
variance hedge ratio) while using futures, is obtained by the following formula:
Formula 15-Optimal Hedge
ℎ∗ = q
oS
oF
Where, ρ is the coefficient of correlation between the change in spot rate and the change
in forward rates, σS and σF is the standard deviation of the change in the spot and forward
rate respectively.
In this particular case, an h*= 1 can be assumed i.e. It is assumed that futures price mirrors
the spot price perfectly and hence has a delta “δ” (defined as the rate of change in the value
of the derivative with respect to the change of the underlying asset, dF/dS) of 1. In fact,
Hull, J. (2006) has demonstrated that in the case of a forward contract on a non- dividendpaying stock, the value of delta is always equal to one and there is no need of change the
hedge ratio during the life of the contract (this is called hedge-and-forget
44
schemes). Moreover, in the case of FX forwards the same author shows that the forward
contract’s delta is e –rf T and as it will be shown in a following Chapter, a delta of 1 is found
for both EUR and USD forward contracts.
On the other hand, in order to value FX options, the Black-Scholes Model (BSM) is used. As
showed by Hull, J. (2006), the following model provides the pricing formulas for currency
options:
Formula 16-Call and Put Valuation Model (BSM)
c = SO e –rf T N(d1 ) − Ke –rT N(d2 )
p = Ke –rT N(−d2 ) − SO e –rf T N(d1 )
Where,
d1 =
ln(SO/K) + (r − rf − o/2)T
o√T
d2 = d1 − o√T
And, S0 is the spot exchange rate, K is the strike price, c is the call option price, p is the put
option price, r is the domestic risk-free interest rate, rf is the foreign risk-free interest rate,
T is the time to maturity (in years), σ is the Standard deviation of exchange rates
(annualized) and N (.) is the Cumulative Normal Probability Distribution.
As in the case of forwards, a delta hedge will be assumed for the case of options.
Therefore, as showed in Hull, J. (2006), the delta of a FX call option is calculated by using
e –rf T N(d1 ) and the delta of a put option on a currency will be given by[N(d1 ) − 1]e –rf T .
45
c.
Reassessing the risk of the hedged portfolio
Two approaches are implemented in order to recalculate the VaR of both a portfolio hedged
with forwards and a second one hedged with options. First, the Linear Model and Options
developed by Hull, J. (2007) is applied and hence the change of the portfolio’s value, ∆P, is
given by the following formula:
Formula 17-Portfolio Value, Linear Model
n
∆P = Σ Siði∆xi
i=1
Where, Si is the current exchange rate for currency i, δi is the delta of the portfolio with
respect to currency i and ∆Xi is the GBP change of the exchange rate.
Therefore, the original weights (as in June, 30th 2010) could be adjusted by the delta of
forwards and options to be bought. After that, the standard deviation of the portfolio could
be calculated by using formula 5 and, finally, by setting a confidence level, Parametric VaR
could be obtained for both cases (i.e. forwards and options). For instance, in the case of
forwards, a full hedge will be assumed at the beginning of the contract. Therefore, current
position’s weights in EUR and USD within the portfolio will be fully offset by taking the
inverse positions in forwards. While in the case of options, the short position in USD will be
partially offset by a positive delta of a call option whereas the long position will be partially
offset by the negative delta of a put option.
However, it has to be pointed out that by assuming this linear approach, only a static
analysis could be done and moreover, this new risk measure will include only the hedge of
the current portfolio without taking into account potential profit/losses generated by new
positions in the portfolios of forwards and options. Therefore, also a scenario/dynamic
hedge approach is implemented by taking the scenarios already built for the calculation of
46
the Historical VaR. This method is in line with the difference between partial valuation and
full valuation VaR models explained by Jorion, P. (2007). The first type of models such us
the parametric VaR (delta-normal approach) doesn’t take into account non-linear pay-offs
of financial instruments (e.g. options and mortgages) and therefore underestimate the VaR
measure whereas full valuation models like Historical or Monte Carlo VaR do take into
consideration non-linear exposures and even capture Gamma and Vega risk17.
Therefore, by assuming a full hedge made at the end of June 2010, the payoffs of both
forwards and options are calculated during the life of the monthly contracts across the daily
500 scenarios and then the portfolio’s value is adjusted by including any gain or loss derived
from the derivative’s portfolio. In particular, the payoff on a long call and a long put at time
T are calculated by the following formulas: max (ST – K, 0) and max (K – ST, 0), respectively
where, ST is the spot exchange rate at time T, and K is the strike price. Whereas the payoffs
of a long and short forward are calculated by the following differences: ST - F and F - ST,
respectively and where, F is the forward exchange rate.
So that, the flexibility embedded in the options contracts will be taken into account since
any spot exchange rate’s movement against the strike price will have a payoff of zero
whereas in the case of forwards this movement will have a negative impact over the
forwards portfolio’s value. The following figure shows the limited potential loss (downside
protection) given by call options and the larger potential lose that could occur when hedging
by buying forwards (the blue line represents the pay-offs given by the option and the
pointed line represented the pay-offs derived from the forward):
17
Gamma is defined as the second derivative of the value function with respect to the underlying asset and Vega as the derivative of
the option value with respect to the volatility of the underlying asset.
47
Figure 5-Derivative’s Pay-offs
Pay-off
Unlimited
potential gain
Premium
Option Limited
potential loss
Breakeven
price
Forward
Potential Loss
48
Chapter 4 ANALYSIS OF RESULTS AND DISCUSSION
1.
Regression Analysis Results
By looking at the first ten main currencies within the firm’s portfolio (approximately 80% of
portfolio’s gross position), we found that all the coefficients are significant at 1% with only
two exceptions (BRL and ZAR). Not surprisingly, USD and EUR are within the top 5 currencies
which variation has a positive effect over the variation of Cash Flows with coefficients of
0.98 and 0.51, respectively. This result is positively correlated with the large weights of
these currencies within the portfolio.
However, in the case of currencies like BDT, PKR, PHP, KES and XOF, their considerable
positive effect over the variability of Cash Flows might be explained by their particular
importance within receiving funding (income) or projects implementations (expenditure).
It has to be noted that all coefficients are positive since all exchange rate’s variations were
calculated from indirect quotes (as explained in the third chapter) and therefore an
exchange rate’s positive movement means a depreciation of the foreign currency in relation
to GBP and consequently a positive effect over cash flows (denominated in GBP).
As explained by Madura, J. (2007), these sensitivities of cash flows to exchange rates could
be a good proxy of future relationships as long as the operating structure
(income/expenditure structure) of the company doesn’t suffer big changes. That seems to
be the case of this particular company since both the origin of funding and project’s
implementation expenditures, seems to be stable during long periods of time. The following
table summarizes main findings of the analysis:
49
Currency
USD
BDT
PKR
PHP
EUR
KES
XOF
UGX
BRL
ZAR
Coefficient
0.976513
0.771494
0.767494
0.569952
0.513695
0.482161
0.45889
0.42862
0.141711
0.037586
Table 3-Regression Results
Std. Error
t-Statistic
0.113087
8.635023
0.115368
6.687223
0.145406
5.278267
0.14217
4.008947
0.183969
2.792295
0.130344
3.69914
0.179002
2.563598
0.129914
3.29926
0.119961
1.181314
0.114259
0.328956
Prob.
0.00000
0.00000
0.00000
0.00020
0.00750
0.00060
0.01360
0.00190
0.24330
0.74360
Weight
43.4%
1.7%
3.1%
3.5%
18.8%
3.1%
2.5%
2.1%
2.3%
1.9%
Finally, Appendix F includes individual E-views outcomes of regression analysis. All the
models show to be significant as a whole by looking at the F-test and there seems to be no
serial correlation problems by looking at the Durbin-Watson tests (i.e. all values near to 2).
Also, independent variables show good explanatory power since models show high R 2
results are higher (at least for the main currencies in terms of weight within the portfolio).
However, this analysis is not taking into account the joint effect of exchange rates over cash
flow’s variation and hence R2 is not a key reference.
50
2.
Portfolio Optimisation Results
Daily Returns and Standard Deviations of the company’s current portfolio were calculated
by using formulas 4 and 5, showed in the previous chapter. Currently, the company’s
currency portfolio has a return of 0.016% with a standard deviation of 0.54%.
Then, and as commented in the Methodology section, the first optimisation exercise was
carried out by taking Income positions as weights of the currencies’ portfolio. At first short
selling is constrained. The following table show the combination of σp and E(Rp) for different
portfolios sorted from the MVP to the MRP. The last three portfolios show the optimal
portfolios when the level of risk aversion is 10, 5 and 1, respectively:
Table 4-MVA, Income Weights
σ
0.47%
E(Rp)
0.020%
0.48%
0.025%
0.48%
0.030%
0.49%
0.040%
0.51%
0.050%
0.54%
0.060%
0.61%
0.070%
0.66%
0.79%
0.89%
0.074%
0.081%
0.084%
Graphically, the efficient frontier is defined by the following concave function. Blue points
show the optimal portfolios for the three different levels of risk aversion:
51
Figure 6-Efficient Frontier, Income Weights
Expected Return
Efficient Frontier and Optimal Portfolios
(under short selling constraints)
Current Portfolio
It has to be pointed out that the MRP is the same as the portfolio with lower level of risk
aversion (i.e. when A=1). That is explained by the fact that a higher expected return is always
accompanied for the corresponding higher level of risk taken.
Results also show that the current portfolio of the Company is inefficient since with the
same level of risk already taken by the company (i.e. σ=0.54%), a higher level of return could
be obtained (i.e. 0.06% vs. 0.016%). In other words, a lower level of risk (less than 0.47%)
could be taken in order to get the same level of return (0.15%). The following table show
the comparison between the current weights and the ones needed to get an efficient riskreturn combination (i.e. 0.054%, 0.06%)18.
18
Appendix G shows all the optimal weights for every corresponding level of Expected Return.
52
Table 5-MVA, Efficient Income Weights
Curr. Code Original Weight
Rp=0.06%
AFO
AUD
BDT
BRL
CAD
COP
EUR
HTG
ILS
INR
KES
NPR
PHP
PKR
SLL
TZS
USD
XOF
YER
ZAR
0.00%
0.86%
0.68%
2.01%
0.94%
0.01%
43.20%
0.11%
0.22%
0.96%
0.03%
0.00%
0.06%
2.14%
0.19%
0.26%
47.71%
0.02%
0.12%
0.47%
25.9%
8.9%
6.7%
16.3%
13.1%
10.3%
0.0%
4.7%
0.0%
0.0%
0.0%
10.2%
3.8%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
TOTAL
100.00%
100.00%
It is clear that due to their higher historical returns and lower correlations, the company
could start investing more in currencies like Brazilian Pesos (BRL), Canadian Dollars (CAD) or
Colombian Pesos (COP). In fact, as showed in Appendix G, even if we assumed a high level
of risk aversion for the company (i.e. A=10), the optimal portfolio with a risk-return
combination of 0.66%-0.074% indicates that approximately 85% of the funding received by
the company should be invested in these three currencies.
However, considering that i) FX trading is not a core competence of the company, ii) the
difficulties which could arise when looking for a liquid market makers in order to trade with
emerging economies’ currencies and that iii) Euro (EUR) and U.S Dollars (USD) represents
more than 90% of its assets, optimal portfolios were also found by using the four main
currencies within the current portfolio (only EUR, UDS, BRL and Pakistan Rupee- PKR counts
for approximately 95% of total assets within the portfolio).
53
The risk-return combination of the current portfolio of these four currencies is 0.61%0.015% (even lower than the whole portfolio return). Again, after solving the optimisation
problem we found that the current company’s portfolio is inefficient since a higher level of
return could be obtained (0.052%) by taking the same level of risk. Also, we found that the
optimal portfolio with high level of risk aversion (i.e. A=10) has a risk-return combination of
0.66%-0.06%. Once more, a suggested switch from a currency with negative short term
historical return (i.e. EUR) to a more profitable asset like BRL is the main finding. However,
in any case, MNGO could minimize its current risk and even increase its return by allocating
the investments according to the structure of the MVP which would give risk- return
combination of 0.52%-0.019% and which also includes less risky currencies such us EUR. The
following table shows a summary of the results including the weights obtained by solving
the optimisation problem:
Portfolios
Current W
Efficient W
MVP
MRP
Max U (A=10)
USD
50.19%
56.89%
14.16%
0.00%
3.26%
Table 6- MVA, Main Currencies
EUR
PKR
BRL
45.44%
2.25%
2.11%
0.00%
0.00%
43.11%
51.38%
6.31%
28.15%
0.00%
0.00%
100.00%
0.00%
0.00%
96.74%
σ
0.61%
0.61%
0.52%
0.675%
0.659%
Rp
0.015%
0.052%
0.019%
0.070%
0.069%
Also, the following figure shows the potential strategies that the company could follow to
optimise its FX portfolio. They can move from their current portfolio to either an MVP
(which reduces its risk considerable and even increases its return) or to a more efficient
portfolio (i.e. more return for the same current level of risk taken):
54
Figure 7-Portfolio’s Comparison, Main Currencies
Expected Return
Main Currencies' Portfolio Comparison
a.
Allowing Short Selling
As we have seen in both of the cases analysed earlier, MRP are obtained by investing all the
income funding in currencies with the highest historical return such us COP (0.084%) and
BRL (0.07%). The reason is due to short selling constraints. By relaxing this restriction, a
company theoretically could take a short position in one currency (e.g. borrow a specific
amount of money in that currency) and then invest more money in a more profitable asset.
Both the following table and figure show the different combinations of risk-return for each
corresponding efficient portfolio (sorted from the MVP to the MRP) within the efficient
frontier:
55
Table 7-MVA, short selling
σ
0.467%
0.468%
0.469%
0.472%
0.476%
0.481%
0.484%
E(Rp)
0.030%
0.040%
0.050%
0.060%
0.070%
0.080%
0.084%
Figure 8-Efficient Frontier, Short Selling
Expected Return
Efficient Frontier and Optimal Portfolios
(without short selling constraints)
It is possible to see that by allowing for short selling strategies, a company could obtain even
higher levels of return by taken the same levels of risk while comparing with the constrained
efficient frontier (Figure 6). For instance, the MRP could be obtained by assuming a standard
deviation of only 0.49% vs. the 0.89% needed when restricted for short selling.
56
In fact, the current portfolio is still inefficient (and even more far from the efficient frontier)
since a well higher return (0.13% vs. 0.015%) could be obtained by taking the same level of
risk according to the optimization solution. The weights obtained for the latter comparison
are showed in the following table while the total numbers of portfolios within the efficient
frontier is show in Appendix H.
Table 8-Efficient Income Weights, Short Selling
Curr. Code
Income
Current W
Rp= 0.13%
AFO
AUD
BDT
BRL
CAD
COP
EUR
HTG
ILS
INR
KES
NPR
PHP
PKR
SLL
TZS
USD
XOF
YER
ZAR
£
516.19
£
133,847.74
£
105,993.66
£
311,944.46
£
146,476.86
£
971.53
£ 6,705,768.16
£
16,711.34
£
33,549.55
£
149,507.61
£
5,420.38
£
501.82
£
8,583.53
£
331,836.58
£
29,266.44
£
40,654.23
£ 7,406,410.15
£
3,829.68
£
18,622.95
£
73,386.52
0.00%
0.86%
0.68%
2.01%
0.94%
0.01%
43.20%
0.11%
0.22%
0.96%
0.03%
0.00%
0.06%
2.14%
0.19%
0.26%
47.71%
0.02%
0.12%
0.47%
18.1%
9.2%
96.1%
13.1%
3.5%
-1.2%
-14.4%
95.5%
-1.8%
-9.7%
-15.9%
11.3%
7.1%
-4.7%
-31.5%
-18.3%
57.1%
41.3%
-160.8%
6.2%
TOTAL
£15,523,799.38
100.00%
100.0%
As commented before, the company could take short positions in a number of different
currencies in order to invest more in more profitable currencies.
57
3.
Value-at-Risk Results
a.
Parametric VaR results
According to the Parametric VaR results, there is 99% chance that the portfolio will not lose
more than £295k (1.25%) in the following day and not more than £1.3m (5.6%) over the
following 20 working days (i.e. month) after the end of the analyzed period. In other words,
there is 1% chance than the portfolio could lose more than these amounts over 1 and 20
days, respectively. The following table summarizes the findings of the current VaR of the
company in terms of FX risk exposure. As commented in the last chapter, the analysis of
VaR in terms of GBP would be more accurate if we take values as a percentage of the
absolute net position since it does take into account the offset effect between income and
expenditures denominated in the same currency during the analyzed period.
Table 9-Parametric VaR Results
Position Denom.
%
GBP
Gross
%
Net
GBP
90% (Factor: 1.28σ)
Daily
Monthly
0.7%
3.2%
314,415 1,406,106
0.7%
3.1%
163,006
728,987
95% (1.64σ)
Daily
Monthly
0.9%
4.1%
403,547 1,804,718
0.9%
4.0%
209,217
935,645
99% (2.33σ)
Daily
Monthly
1.3%
5.9%
570,744 2,552,447
1.3%
5.6%
295,899 1,323,300
The table shows that the VaR increases while moving to the left hand side of the assumed
normal distribution (from 90% to 99% of confidence level) of the FX returns of the company.
For instance, daily VaR increases from £163k (0.7%) to £295k (1.3%). Graphically, after
assume that 1-day exchange rate’s returns within the portfolio are normally distributed, we
can look at the values that corresponds to 1%, 5% and 10% left tails:
58
Figure 9- Parametric VaR Results, Distribution
Probability
1-day :
-1.3%
-0.9%
-0.7%
20-day:
-5.6%
-4%
-3.1%
Portfolio’s Returns
It has to be noted that the figures presented here are Relative VaR figures since the actual
expected return of the portfolio is ignored (i.e. the mean of returns is assumed as zero).
However, the expected return can be incorporated by calculating the Absolute VaR which
is just the difference between the Relative VaR and the expected return and therefore
typically makes a small difference. For instance, since the expected return of the portfolio
is 0.023%, the Absolute 20-day 99% VaR for the portfolio would be: 5.61%-0.023%*20 =
5.15%.
Finally, from the analysis of the data it is possible to calculate the benefits of diversification
given by the current portfolio of the firm. As we have seen in the first chapter of this paper,
Markowitz, H. (1952) has been one of the first authors to study the benefits of
diversification. If the correlation between all of the currencies within the firm’s portfolio
were perfect (i.e. equal to 1), the VaR of the portfolio including these currencies should be
equal to the sum of the individual VaRs for each currency (See also: Hull, J., 2006). However,
as Markowitz showed, since the correlation between all the currencies is less than 1, there
is a benefit of diversification as showed in the following table. All individual VaR’s results
are shown in Appendix I.
59
Table 10-Parametric VaR, Benefits of Diversification
VaR
Sum of individual VaRs
VaR of the Portoflio
Benefits of diversification
1-day 99% VaR£
£
390,258
£
295,899
£
94,359
20-day 99% VaR£
£
1,745,288
£
1,323,300
£
421,988
Therefore, there is a benefit of diversification of around £94k and £422k for 1-day and 20day VaR, respectively. This is obtained by calculating the difference between the sum of
individual VaRs and the VaR of the diversified portfolio19.
b.
Historical VaR results
According to the Historical VaR, firm’s FX risk exposure is even higher than the one obtained
with the previous approach20. For instance, by analyzing net positions, the 1-day 99% VaR
is around £420k (1.8%) whereas the 20-day 99% VaR falls around £1.9m (8.0%). So that
there is 1% chances that the portfolio could lose more than these amounts over 1 and 20
days, respectively. The following table summarizes the findings of the current VaR of the
company in terms of FX risk exposure.
Table 11-Historical VaR Results
Position Denom.
%
GBP
Gross
%
GBP
Net
90% (Factor: 1.28σ)
Daily
Monthly
0.6%
2.7%
262,785 1,175,209
1.0%
4.3%
227,675 1,018,195
95% (1.64σ)
Daily
Monthly
0.9%
4.0%
386,993 1,730,684
1.3%
5.9%
311,756 1,394,217
99% (2.33σ)
Daily
Monthly
1.4%
6.3%
610,208 2,728,933
1.8%
8.0%
420,347 1,879,849
As under the parametric approach, the previous table shows that the VaR increases while
moving to the left hand side of the distribution (from 90% to 99% of confidence level) of the
FX returns of the company. As an example, daily VaR increases from £228k (1.0%) to
19
However, as showed in the first Chapter, sin VaR is not a “coherent” measure of risk; it could happen that the VaR of a portfolio could
be larger than the sum of risk measures VaRs of its components.
20
A further comparison between each approach is included in a following section.
60
£312k (1.3%) while moving from a 90% to a 95% of confidence level. However, in contrast
with the parametric approach, the distribution is not assumed as normal but instead it
assumes an empirical distribution taken from historical information of exchange rate’s
movements. Therefore, we can look at the values that correspond to 1%, 5% and 10% left
tails of this particular distribution. In particular, the following figure shows the 1% tail with
the corresponding 1-day 99% VaR and the observations that lie in the space after the
threshold of 1% (i.e. evidence of “fat tail”):
Figure 10-Historical VaR Results, Distribution
Empirical Histogram
90
80
70
Frequency
60
50
40
30
1-Day 99%VaR:
-£420k
20
10
"Fat Tail"
1%
Portfolio's Value in GBP
61
c.
Monte Carlo VaR results
The implementation of Monte Carlo simulations was done by taking only net position
weights within the firm’s portfolio. The following table summarizes the main results at 90%,
95% and 99% of confidence level:
Table 12-Monte Carlo VaR Results
monthly
Conf. Level
90% (1.28σ)
95% (1.64σ)
99% (2.33σ)
%
2.5%
3.7%
5.3%
daily
GBP
589,629
876,764
1,241,011
%
0.6%
0.8%
1.2%
GBP
131,845
196,050
277,498
The results show that the Parametric and Monte Carlo approaches give virtually the same
current firm’s FX risk exposure. According to the latter, there is 1 in 100 chance that the
portfolio could lose more than £ 277k (1.2%) on a daily basis and more than £1.2m (5.3%)
over the following 20 working days (i.e. monthly basis) after the end of June 2010. As in the
historical approach there is no assumption behind the distribution of exchange rate’s
returns (i.e. it could be non-normal). As explained in Chapter 3, the first step was to choose
a model in order to simulate the stochastic behaviors of exchange rates and finally
according to the simulations, the distribution could be obtained. For instance, the following
figure shows the simulated distributions of portfolio’s returns and the 1% left tail with the
corresponding VaR value (as in the previous case, there is evidence of a “fat tail” with values
lower than -5.3%):
62
Figure 11-Monte Carlo VaR Results, Distribution
Simulated Histogram
60
50
Frequency
40
30
20
1-Day 99%VaR:
-5.3%
10
%Changes in Portfolio's Value
d.
A comparison between different approaches and their results
In terms of practicability, the parametric approach is the easiest and fastest way to
implement a VaR model while keeping it easily updated with latest market information and
also its concept is easy to understand when explaining it to different levels of professionals
within the company. On the other hand, both the Historical and Monte Carlo approaches
involve quite large computational and programming effort and even more they could lead
to operational risks derived from human errors where implementing them.
However, the main disadvantage of the Parametric VaR is the assumption of normality
behind the distributions of asset’s returns. In fact, as we have seen in the previous section,
both historical and simulated distributions are able to capture “fat tails” since they are not
normal distributions by construction as in the case of the parametric one.
63
In terms of our results, the Historical VaR seems to be the highest since the outcome rely
on recent historical information. As we have seen in figure 4, the period taken for this
approach includes most of the last crisis volatilities and therefore the higher VaR is related
to the higher standard deviation of the portfolio. On the other hand, Parametric and Monte
Carlo approaches show pretty much the same level of risk. This is explained by the fact that
the portfolio doesn’t include non-linear instruments (i.e. options) and therefore, in this
particular case, there is no difference between partial and full valuation VaR models as
explained in the previous Chapter. However, as it could be seen in the previous section,
Monte Carlo approach allows us to assess potential “fat tails” of the distribution whereas
Parametric VaR doesn’t. The following table and figure summarize the latter comparison
between models. For instance, the 99% Monthly Historical VaR (red bar) is around 8% while
the Parametric and Monte Carlo (blue and green bars) are around 5.6% and 5.3%,
respectively:
Table 13-Comparison of VaR Results
Approach
Parametric
Historical
Monte Carlo
90% (Factor: 1.28σ)
Daily
Monthly
0.7%
3.1%
1.0%
4.3%
0.6%
2.5%
95% (1.64σ)
Daily
Monthly
0.9%
4.0%
1.3%
5.9%
0.8%
3.7%
64
99% (2.33σ)
Daily
Monthly
1.3%
5.6%
1.8%
8.0%
1.2%
5.3%
Figure 12-VaR % Results Comparison
VaR %
VaR Results Comparison
Monthly
Monthly
Historical
65
Monthly
4.
CVaR and Stress Testing results
As commented in Chapter 3, CVaR is calculated in order to take into account potential
extreme losses or “fat tails” in the distributions of the portfolio returns and as it could have
been seen in figures 10 and 11, firm’s current portfolio show this situation. By using the
methodology already explained before, we define VaRq (where q=1%) as the 1-day 99%
Historical VaR (i.e. around £420k). The, we define the vector X as all the losses that are
higher than VaR as shown in the following table:
Table 14-CVaR Results
X£
Diff: X-VaR1%
587,958
167,611
484,559
64,212
482,047
61,700
470,637
50,290
444,029
23,682
E[Diff]
73,499
VaR1%
1-day CVaR
20-day CVaR
420,347
493,846
2,208,547
The expected (i.e. mean) difference between the vector of these extreme losses and the
actual VaR is the additional Expected Shortfall which is added to the traditional VaR. As a
result, a higher and therefore more conservative measure of risk is obtained. As showed in
the previous table the 1-Day CVaR is now around £494K and the corresponding monthly
CVaR is approximately £2.2m (vs. £1.9m obtained by using the “normal” measure of VaR).
CVaR is a way to answer the question: “how much can we expect to lose if things go bad?”
and is important for a more prudent company’s FX risk management.
Another important non-statistical approach to risk that could help to improve firm’s risk
management systems is the Stress Testing. As commented in the previous Chapter, the
66
period between July 2008 and July 2009 is taken as the stress scenario and the portfolio
variance is recalculated by using the Parametric VaR framework. The results showed an
increase of 0.17% in the daily portfolio variance by moving from 0.54% to 0.71% under the
stress scenario. Moreover, the daily VaR and monthly VaR increases in 95k and 423k,
respectively under the new scenario as showed in the following table and figure:
Table 15-Stress Testing Results
VaR
1-Day 99%VaR
20-Day 99%VaR
Stressed VaR
%
GBP
1.7%
390,456
7.4%
1,746,174
Differences
%
GBP
0.4%
94,558
1.8%
422,874
Figure 13-Stress Testing Results
Thousands GPB
Stress Testing
Therefore, if the last crisis scenario repeats in the near future, there is 1% chance that the
company loses an additional 1.8% of its net positions during the following month. Again,
the company could set a number of different scenarios in order to build a more prudent risk
management system.
67
5.
Portfolio Hedging Results
a.
Valuation
The following table summarizes the results obtained for the valuation of USD and EUR
options and forwards:
Table 16-Derivative’s Valuation
Derivatives
S
X
USD
0.6689
0.6689
EUR
0.8185
0.8185
rd
0.67%
0.67%
rf
T
σ
N(d1)
N(d2)
Call Value
Put Value
Delta Call
Delta Put
0.45%
0.0833
10.71%
50.85%
49.62%
0.83%
-0.83%
50.83%
-49.13%
0.78%
0.0833
9.12%
50.39%
49.34%
0.86%
-0.86%
50.35%
-49.58%
Forward Price
Delta Forward
0.6691
100.0%
0.8184
99.9%
By using formulas 14 and 15, forward exchange rate is obtained (very close to the current
spot price, since the period taken is only one month or 0.083 years) along with the optimal
hedge which, as commented earlier, is almost 1 for both cases. Also, by using BSM (formula
16), a value of a USD call options is found to be 0.83 and the value of a EUR put option is
found to be 0.86. Also, Delta of the USD call option is around 51% and delta of the EUR put
option is around -50%. Volatility figure has been annualized and rf shows the risk free rate
for USD and EUR, respectively.
68
b.
Risk measures after hedging
Results show that after hedging the portfolio with either FX Forwards or FX Options for USD
and EUR net positions, the VaR measure diminishes considerable. The following table
summarizes these results for both the Parametric and the Historical approaches:
Table 17-Hedged VaR Comparison
VaR
Parametric VaR
Options VaR
Forward VaR
Historical VaR
Options VaR
Forward VaR
1-Day 99% VaR
£
295,899
£
240,033
£
187,140
20-Day 99% VaR
£
1,323,300
£
1,073,462
£
836,917
£
£
£
£
£
£
420,347
370,538
413,255
1,879,849
1,657,095
1,848,133
It is showed that, by using the linear model and the parametric (delta-normal) VaR, forwards
reduce the monthly FX risk exposure of the currency firm’s portfolio in approximately £489k
whereas by hedging with options, the hedge is only around £250k.
On the other hand, when implementing the full valuation methodology by using Historical
VaR approach, the GBP hedge reached by using a portfolio of options is still around £223k
while the hedge given by forwards reduces to only £31k of the monthly FX firm’s exposure.
As explained in the previous chapter, the latter methodology allow us to take into account
non-linear payoff as those calculated in the case of options and therefore it takes into
consideration the fact that by using options, the company could limit a potential downside
loss and take benefit from favorable exchange rates movements. The following figure
remarks the latter difference, it could be seen that under a partial valuation model, hedging
with forwards seems to be the best option by reducing VaR in 0.46% of net positions,
however, by using a full valuation model, the best choice is hedging by options since it
reduces the value-at-risk of the portfolio in 0.21%:
69
Figure 14-Hedged and Non-hedged Portfolio Comparison
Hedged Portfolio VaR
2.00%
1.78%
1.75%
1.75%
1.57%
% Net Position
1.50%
1.25%
1.25%
1.02%
1.00%
0.79%
0.75%
0.50%
0.25%
0.00%
Unhedged VaR Options VaR
Forward VaR Unhedged VaR Options VaR
Partial-Valuation Model
Forward VaR
Full-Valuation Model
One real-life example could be useful to show not only hedging tools and mechanisms but
also to clearly state the difference between options and forwards. In particular, MNGO has
a current short position of -USD 5,979,600 (GBP 4m converted at the spot price on June,
30th). If the company decides to hedge the position it should take the inverse position (i.e.
long/buy) in derivatives. Therefore, a long position in call options and forwards is analyzing
along with the decision of not to hedge.
In the case of Options, the company has to pay a premium for that. According to the Nasdaq
stock exchange21 each contract of USD/GBP Options is of USD 10,000 (we rounded to 600
contracts as the amount to be hedged is around USD 6m) and a premium of approximately
USD 100 (100 points) per contract. In the case of forwards there is no need for premiums
except for potential commissions in the OTC market (private agreements with commercial
banks) that won’t be taking into consideration. Finally, we assume a
21
http://www.nasdaq.com/includes/british-pound-specifications.stm
70
strike price equal to the current spot price to make the comparison straightforward. The
results of the scenario analysis are summarized in the following table. The figures showed
in both scenarios are the amounts of USD received in each case whereas the “Hedge Payoff” is the difference between these amounts and the original position for each scenario.
Table 18-Derivative’s Hedging
So = 0.6689
K (GBP/USD)
No. Contracts
Premium
Scenario 1 (+5%)
Scenario 2 (-5%)
Hedge Pay-off S1
Hedge Pay-off S2
Unhedge
-
Options
0.6689
600
60,000
5,694,857
5,979,600
6,294,316
6,294,316
284,743 60,000
314,716
254,716
Forwards
0.6689
Customized
5,979,600
5,979,600
-
As showed in the table, under the scenario of a 5% appreciation of the USD (or a higher
GBP/USD exchange rate), the company fully off-set its short position in USD by buying USD
at a cheaper (strike/forward) price than the spot price (0.7024) in the case of both
derivatives (with a premium paid up front for Options) whereas if the company decides not
to hedge, it could lose USD 285k in regards of FX Risk by buying dollars at a more expensive
price than in the spot market.
On the other hand, an appreciation of USD shows that by hedging with forwards, the
position continues to be fully offset in comparison with the position and time zero (that
means actually a negative payoff in the forward transaction, obtained by calculating the
difference between ST and K); however, by using Options, the company could actually make
a profit by not exercising the option and go to the spot market. It has to be highlighted
though that counterparty risk22 should be also considered in the risk assessment of this
operation. This profit equals the money exchanged at a 5% cheaper
22
Is the possibility that one of the parties within the contract could default on its debt obligations before or at the liquidation’s date. In
the case of derivatives, this is a bilateral risk since derivatives market value could be positive or negative for both counterparties,
irrespectively.
71
price in the spot market minus the USD 60k premium paid up front (resulting in USD 255k
approximately).
Therefore, as noted earlier, while considered all the scenarios and taking into consideration
the non-linear pay-offs of FX options, it is possible to realize that Options, after paying a
premium, could give the MNGO the flexibility of protecting its downside risk while also
taking the benefits of an upside movement in exchange rates.
In the following Chapter, we will discuss main conclusions, recommendations and
limitations derived from the comprehensive analysis made throughout this paper.
72
Chapter 5 CONCLUSIONS, RECOMMENDATIONS AND CRITIQUE
According to their weights within the portfolio and also from the sensitivity analysis’ results,
it is clear that the main FX risk factors for the company are exchange rate movements in
EUR and USD hence, and taking advantage of the liquid markets for those currencies, an
efficient hedge in EUR and USD could protect the entire firm’s portfolio from FX risk.
However, since the portfolio includes a large number of currencies, the assessment of firm’s
FX risk also includes correlations and dynamics between all the currencies within the
portfolio.
First, MVA results show that the current distribution of the portfolio is not efficient (i.e. a
higher level of return could be obtained by taking the same current level of risk and by
switching the composition of the portfolio). Despite the fact that it is usually complicated
for the company to select the currency of donations and that FX trading is not its core
competence, it is pertinent to recommend that the company uses MVA as a guideline of FX
portfolio’s efficiency. In fact, when analyzing the portfolio of main currencies, the company
could reduce its current FX risk exposure from 9.7% p.a. to 8.2% p.a. by taking the
composition suggested by the Minimum Variance Portfolio (MVP) which even could give
MNGO an annual return of 4.8% (higher than the current 3.8%). This new portfolio includes
approximately 65% of its weights distributed between USD and EUR as showed in table 6.
Even more, the company could receive a 13.2% of annual return with the current level of
risk by compose the portfolio as suggested by the optimization problem solution.
Second, as commented before, parametric VaR could be the easiest way to implement an
FX risk measure, however, Historical and Monte Carlo VaR would help to identify potential
kurtosis of the distribution (i.e. the presence of “fat tails”) since they don’t assume return’s
normal distribution. In terms of results, the Historical VaR seems to be the highest since the
outcome rely on recent historical information (i.e. last financial crisis). 1-day 99%
73
historical VaR is around £420k (1.8%) whereas the 20-day 99% VaR falls around £1.9m
(8.0%). So that there is 1% chances that the portfolio could lose, because exchange rate
movements, more than these amount over 1 and 20 days, respectively.
On the other hand, Parametric and Monte Carlo approaches show pretty much the same
level of risk. This is explained by the fact that the portfolio doesn’t include non-linear
instruments. According to the Parametric VaR results, there is 99% chance that the portfolio
will not lose more than £295k (1.25%) in the following day and not more than
£1.3m (5.6%) over the following 20 working days (i.e. month) after the end of the analyzed
period. According to the Monte Carlo simulation, there is 1 in 100 chance that the portfolio
could lose more than £ 277k (1.2%) on a daily basis and more than £1.2m (5.3%) over the
following 20 working days (i.e. monthly basis).
To take into account “fat tails”, CVaR is also calculated and the results showed that the 1Day CVaR is now around £494K and the corresponding monthly CVaR is approximately
£2.2m (vs. £1.9m obtained by using the “normal” measure of Historical VaR). Another
important non-statistical approach to risk that could help to improve firm’s risk
management systems is the Stress Testing. Hence, we found that if the last crisis scenario
repeats in the near future, there is 1% chance that the company loses an additional 1.8% of
its net positions during the following month.
These results are meaningful in terms of applicability for the company since they could serve
as a reference of the potential losses derived from FX risk. In fact, nowadays, financial
institutions use VaR measures to set aside their minimum regulatory capital requirements.
The current capital standard (i.e. Basel II) typically requires banks to set aside at least 3
times a 10-day 99% VaR (which is currently £2.8m for this company). This capital could serve
as a “cushion” for the company in order to “absorb” some potential losses (in terms of GBP)
derived from foreign exchange rates movements.
74
Third, in terms of hedging, a Full-Valuation model allow us to compare the effectiveness of
the use of either options or forwards by taking into account the non-linear pay-offs of the
former. The GBP hedge reached by using a portfolio of EUR and USD options is around
£223k while the hedge given by forwards is only around £31k of the monthly FX firm’s net
exposure. Therefore, it is advisable for the company that its hedging strategy takes into
account that options would allow them to limit its potential downside loss while taking
benefit from favorable exchange rates movements. In any case, as commented earlier, the
company should hedge its portfolio by using only EUR and USD forwards or European
options. This could be done either on a monthly basis or on a longer period basis. In the
former case, MNGO should take care of roll-over risk or the risk that market conditions and
prices would change between renovations of derivatives contracts (see: Hull, J., 2006) and
in the latter case, the company has to be able to estimate its potential positions in both
currencies during the following quarter, semester or year.
Fourth, in terms of the research questions stated in Chapter 2, as showed in the paper,
different risk measures could be applied by implementing models in MS Excel that could be
easily updated with new market information. In particular, MVA could serve not only as a
tool to find a Minimum Variance Portfolio but also a reference to increase return derived
from FX positions. Also, VaR calculations and stress testing could serve as a reference to set
aside some reserve in order to face potential Foreign Exchange losses.
Regarding the questions formulated by the company, along this document it has been
showed that indeed it is possible to mathematically describe the FX risk faced by them by
using a number of different models. The specificity of the company funding affects the
modeling of risk since instead of a Value-at-Risk, the measuring of FX risk could be done by
implementing a sort of Cash Flow-at-Risk on a monthly basis and taking into consideration
Income/Expense figures and foreign exchange market information. Also, these models
could be easily updated by including new market data. Finally, these models could
75
become an effective tool for the company in order to manage its FX risk. As commented
before, MVA could help the company to get an efficient portfolio in terms of risk-return and
the VaR could serve as a guideline to set aside a reserve to face potential losses derived
from FX positions (Internal absorption of FX risk) and also to assess how effective is the use
of financial hedging by using FX options or forwards (External absorption of FX risk). Also,
these FX risk management tools could be included in the company’s systems by the means
of permanent calculation of FX reserve and/or limits in certain currency’s positions.
Moreover, the company could use market intelligence in terms of foreign exchange
forecasting and even in terms of some currency trading operations (by using Portfolio
Optimization as a framework).
Finally, it should be pointed out that there is always some space for improvements within a
project of this nature. For instance, the regression analysis could be improved in order to
look at the joint effect of exchange rates on the variability of cash flows (e.g. a composite
index of currencies could be constructed or found in order to serve as an independent
variable of the regression). In the case of MVA and VaR measures, there could be a mix of
these models by including VaR figures as a measure of risk (instead of variance) and look at
the optimal weights of portfolios in order to minimize VaR. Also, VaR results need to be Back
Tested in the future to check for the accuracy and robustness of the model. In addition, a
more comprehensive scenario analysis could be included within the stress testing and
derivative’s hedging valuation by, for instance, implementing an exchange rate’s forecasting
model. Last but not least, the MNGO has to bear always in mind that each of these models
are just an approximation of real figures. As commented all over this paper, they have many
drawbacks since they rely on historical information and/or distribution’s assumptions.
Therefore, quantitative tools have to be analyzed along with an adequate professional
judgment and a comprehensible reporting of FX risk exposures across the whole
organization.
76
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www.ft.com
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80
APPENDICES
a.
Appendix A: List of Currencies
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
Code
USD
EUR
AFO
ALL
AON
AMD
AUD
AZN
BDT
BBD
BOB
BRL
BIF
KHR
CAD
CLP
COP
XOF
XAF
CDF
DKK
DOP
XCD
EGP
SVC
ERN
ETB
GEL
GHS
GTQ
GYD
HTG
HNL
HKD
Country
USA
Euro
Afghanistan
Albania
Angola
Armenia
Australia
Azerbaijan
Bangladesh
Barbados
Bolivia
Brazil
Burundi
Cambodia
Canada
Chile
Colombia
Communauté Financière Africaine BCEAO
Communauté Financière Africaine BEAC
DR Congo (Congo - Kinshasa)
Denmark
Dominican Republic
East Caribbean Dollar
Egypt
El Salvador
Eritrea
Ethiopia
Georgia
Ghana
Guatemala
Guyana
Haitia
Honduras
Hong Kong
81
Currency
US Dollar
Euro
Afghan Afghani (Official)
Albanian Lek
Angolan New Kwanza
Armenian Dram
Australian Dollar
Azerbaijan New Manat
Bangladesh Taka
Barbados Dollar
Boliviano
Brazilian Real
Burundi Franc
Cambodian Riel
Canadian Dollar
Chilean Peso
Colombian Peso
CFA Franc BCEAO
CFA Franc BEAC
Franc Congolais
Danish Krone
Dominican Peso
East Caribbean Dollar
Egyptian Pound
El Salvador Colon
Eritrean Nakfa
Ethiopian Birr
Georgian Lari
Ghana Cedi (New)
Guatemalan Quetzal
Guyana Dollar
Haitian Gourde
Honduran Lempira
Hong Kong Dollar
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
INR
IDR
IRR
IQD
ILS
JMD
JPY
JOD
KES
KWD
LBP
LRD
MWK
MYR
MRO
MXN
India
Indonesia
Iran
Iraq
Israel
Jamaica
Japan
Jordan
Kenya
Kuwait
Lebanon
Liberia
Malawi
Malaysia
Mauritania
Mexico
51
52
53
54
55
56
57
58
59
60
61
MZN
MMK
NPR
NZD
NIO
NGN
NOK
PGK
PKR
PEN
PHP
Mozambique
Myanmar
Nepal
New Zealand
Nicaragua
Nigeria
Norway
Papua New Guinea
Pakistan
Peru
Philippines
62
63
64
65
66
67
68
69
70
71
72
RUR
RWF
SLL
SGD
SOS
ZAR
LKR
SDG
SEK
CHF
SYP
Russia
Rwanda
Sierra Leone
Singapore
Somalia
South Africa
Sri Lanka
Sudan
Sweden
Switzerland
Syria
82
Indian Rupee
Indonesian Rupiah
Iranian Rial
Iraqi Dinar
New Israeli Sheqel
Jamaican Dollar
Japanese Yen
Jordanian Dinar
Kenyan Shilling
Kuwaiti Dinar
Lebanese Pound
Liberian Dollar
Malawian Kwacha
Malaysian Ringgit
Mauritainian Ouguiya
Mexican Peso
Mozambique New
Metical
Myanmar Kyat
Nepalese Rupee
New Zealand Dollar
Nicaraguan Cordoba Oro
Nigerian Naira
Norwegian Krone
Kino
Pakistan Rupee
Peruvian Nuevo Sol
Philippine Peso
Russian Federation
Rouble
Rwanda Franc
Leone
Singapore Dollar
Somali Shilling
South African Rand
Sri Lanka Rupee
Sudanese Pounds
Swedish Krona
Swiss Franc
Syrian Pound
73
74
75
76
77
78
79
80
TJS
TZS
THB
UGX
GBP
VND
YER
ZMK
Tajikistan
Tanzania
Thailand
Uganda
United Kingdom
Vietnam
Yemen
Zambia
83
Tajik Somoni
Tanzanian Shilling
Thai Baht
Uganda Shilling
Pound Sterling
Vietnamese Dong
Yemeni Rial
Zambian Kwacha
b.
Currency
AFO
AUD
BDT
BRL
CAD
COP
EUR
HTG
ILS
INR
KES
NPR
PHP
PKR
SLL
TZS
USD
XOF
YER
ZAR
Appendix B: Variance-Covariance Matrix
AFO
0.00606%
0.00034%
0.00270%
0.00091%
0.00127%
0.00240%
0.00118%
0.00270%
0.00235%
0.00240%
0.00245%
0.00218%
0.00288%
0.00315%
0.00288%
0.00295%
0.00319%
0.00100%
0.00275%
0.00115%
AUD
0.00034%
0.00681%
0.00101%
0.00464%
0.00459%
0.00290%
0.00240%
0.00101%
0.00237%
0.00302%
0.00188%
0.00255%
0.00179%
0.00110%
0.00091%
0.00187%
0.00114%
0.00202%
0.00095%
0.00450%
BDT
0.00270%
0.00101%
0.00463%
0.00112%
0.00192%
0.00227%
0.00186%
0.00463%
0.00257%
0.00229%
0.00287%
0.00347%
0.00280%
0.00351%
0.00478%
0.00385%
0.00355%
0.00200%
0.00471%
0.00109%
BRL
0.00091%
0.00464%
0.00112%
0.00725%
0.00420%
0.00351%
0.00247%
0.00112%
0.00188%
0.00246%
0.00170%
0.00199%
0.00171%
0.00114%
0.00095%
0.00175%
0.00117%
0.00182%
0.00107%
0.00446%
CAD
0.00127%
0.00459%
0.00192%
0.00420%
0.00588%
0.00383%
0.00240%
0.00192%
0.00285%
0.00306%
0.00258%
0.00253%
0.00248%
0.00212%
0.00206%
0.00261%
0.00224%
0.00196%
0.00193%
0.00378%
COP
0.00240%
0.00290%
0.00227%
0.00351%
0.00383%
0.00799%
0.00224%
0.00227%
0.00341%
0.00379%
0.00346%
0.00275%
0.00389%
0.00330%
0.00230%
0.00345%
0.00330%
0.00198%
0.00230%
0.00309%
EUR
0.00118%
0.00240%
0.00186%
0.00247%
0.00240%
0.00224%
0.00330%
0.00186%
0.00221%
0.00210%
0.00190%
0.00211%
0.00199%
0.00188%
0.00191%
0.00220%
0.00188%
0.00267%
0.00191%
0.00215%
HTG
0.00270%
0.00101%
0.00463%
0.00112%
0.00192%
0.00227%
0.00186%
0.00463%
0.00257%
0.00229%
0.00287%
0.00347%
0.00280%
0.00351%
0.00478%
0.00385%
0.00355%
0.00200%
0.00471%
0.00109%
ILS
0.00235%
0.00237%
0.00257%
0.00188%
0.00285%
0.00341%
0.00221%
0.00257%
0.00487%
0.00328%
0.00338%
0.00281%
0.00345%
0.00336%
0.00255%
0.00403%
0.00342%
0.00189%
0.00255%
0.00237%
INR
0.00240%
0.00302%
0.00229%
0.00246%
0.00306%
0.00379%
0.00210%
0.00229%
0.00328%
0.00506%
0.00362%
0.00386%
0.00397%
0.00337%
0.00232%
0.00435%
0.00345%
0.00195%
0.00231%
0.00268%
84
KES
0.00245%
0.00188%
0.00287%
0.00170%
0.00258%
0.00346%
0.00190%
0.00287%
0.00338%
0.00362%
0.00533%
0.00283%
0.00392%
0.00398%
0.00282%
0.00453%
0.00412%
0.00167%
0.00287%
0.00181%
NPR
0.00218%
0.00255%
0.00347%
0.00199%
0.00253%
0.00275%
0.00211%
0.00347%
0.00281%
0.00386%
0.00283%
0.00483%
0.00324%
0.00294%
0.00351%
0.00410%
0.00293%
0.00238%
0.00352%
0.00239%
PHP
0.00288%
0.00179%
0.00280%
0.00171%
0.00248%
0.00389%
0.00199%
0.00280%
0.00345%
0.00397%
0.00392%
0.00324%
0.00602%
0.00398%
0.00291%
0.00488%
0.00399%
0.00174%
0.00281%
0.00179%
PKR
0.00315%
0.00110%
0.00351%
0.00114%
0.00212%
0.00330%
0.00188%
0.00351%
0.00336%
0.00337%
0.00398%
0.00294%
0.00398%
0.00507%
0.00362%
0.00477%
0.00443%
0.00152%
0.00355%
0.00098%
SLL
0.00288%
0.00091%
0.00478%
0.00095%
0.00206%
0.00230%
0.00191%
0.00478%
0.00255%
0.00232%
0.00282%
0.00351%
0.00291%
0.00362%
0.00648%
0.00387%
0.00364%
0.00210%
0.00483%
0.00098%
TZS
0.00295%
0.00187%
0.00385%
0.00175%
0.00261%
0.00345%
0.00220%
0.00385%
0.00403%
0.00435%
0.00453%
0.00410%
0.00488%
0.00477%
0.00387%
0.00931%
0.00487%
0.00221%
0.00391%
0.00159%
USD
0.00319%
0.00114%
0.00355%
0.00117%
0.00224%
0.00330%
0.00188%
0.00355%
0.00342%
0.00345%
0.00412%
0.00293%
0.00399%
0.00443%
0.00364%
0.00487%
0.00455%
0.00161%
0.00357%
0.00113%
XOF
0.00100%
0.00202%
0.00200%
0.00182%
0.00196%
0.00198%
0.00267%
0.00200%
0.00189%
0.00195%
0.00167%
0.00238%
0.00174%
0.00152%
0.00210%
0.00221%
0.00161%
0.00328%
0.00206%
0.00173%
YER
0.00275%
0.00095%
0.00471%
0.00107%
0.00193%
0.00230%
0.00191%
0.00471%
0.00255%
0.00231%
0.00287%
0.00352%
0.00281%
0.00355%
0.00483%
0.00391%
0.00357%
0.00206%
0.00495%
0.00114%
ZAR
0.00115%
0.00450%
0.00109%
0.00446%
0.00378%
0.00309%
0.00215%
0.00109%
0.00237%
0.00268%
0.00181%
0.00239%
0.00179%
0.00098%
0.00098%
0.00159%
0.00113%
0.00173%
0.00114%
0.00777%
c.
Appendix C: Historical VaR, Portfolio’s values distribution
Value
Frequency
-£587,957.84
1
-£528,976.22
0
-£469,994.61
3
-£411,013.00
2
-£352,031.38
9
-£293,049.77
15
-£234,068.16
17
-£175,086.55
54
-£116,104.93
72
-£ 57,123.32
83
£ 1,858.29
69
£ 60,839.91
72
£119,821.52
44
£178,803.13
16
£237,784.74
17
£296,766.36
10
£355,747.97
8
£414,729.58
2
£473,711.20
2
£532,692.81
1
£591,674.42
0
£650,656.03
2
and higuer…
1
Scenarios
500
85
d.
Appendix D: Monte Carlo VaR, Portfolio’s values distribution
Value's Changes Frequency
-6.9%
1
-6.2%
0
-5.5%
2
-4.8%
4
-4.1%
8
-3.4%
16
-2.8%
11
-2.1%
32
-1.4%
41
-0.7%
39
0.0%
53
0.7%
47
1.4%
54
2.0%
46
2.7%
40
3.4%
47
4.1%
20
4.8%
16
5.5%
11
6.1%
3
6.8%
5
7.5%
3
8.2%
1
Scenarios
500
86
e.
Appendix E: Correlation Matrix
code
AFO ALL AMD AON AUD AZN BBD BDT BOB BRL CDF CHF CLP COP DOP EGP ERN ETB EUR GEL GHS GTQ HNL HTG IDR ILS INR JOD KES KHR LBP LKR LRD MMK MRO MWK M XN MZN NGN NIO NPR PEN PHP PKR RUR RWF SDG SLL THB TJS TZS UGX USD VND XOF YER ZAR ZMK CAD HKD JMD JPY
AFO
1.000 0.251 0.395 0.553 0.053 0.509 0.580 0.510 0.510 0.137 0.373 0.294 0.328 0.344 0.340 0.591 0.508 0.305 0.263 0.433 0.475 0.520 0.510 0.510 0.456 0.432 0.434 0.592 0.431 0.542 0.602 0.582 0.430 0.601 0.311 0.360 0.201 0.226 0.470 0.509 0.403 0.505 0.477 0.568 0.296 0.500 0.508 0.460 0.562 0.315 0.393 0.429 0.606 0.537 0.224 0.503 0.168 0.255 0.213 0.604 0.505 0.470
ALL 1.000 0.496 0.421 0.353 0.597 0.484 0.598 0.598 0.303 0.471 0.643 0.476 0.333 0.463 0.468 0.381 0.356 0.722 0.556 0.568 0.563 0.598 0.598 0.409 0.502 0.441 0.453 0.382 0.404 0.454 0.438 0.390 0.477 0.264 0.314 0.372 0.341 0.416 0.597 0.595 0.599 0.368 0.410 0.442 0.600 0.381 0.512 0.464 0.173 0.418 0.338 0.462 0.441 0.863 0.601 0.248 0.278 0.405 0.459 0.569 0.353
AMD
1.000 0.589 0.066 0.812 0.591 0.812 0.812 0.098 0.641 0.346 0.467 0.277 0.661 0.588 0.474 0.556 0.326 0.711 0.768 0.783 0.812 0.812 0.377 0.388 0.339 0.603 0.428 0.559 0.599 0.588 0.474 0.604 0.351 0.390 0.261 0.435 0.507 0.812 0.561 0.724 0.384 0.585 0.225 0.809 0.474 0.750 0.564 0.299 0.459 0.408 0.609 0.559 0.404 0.800 0.048 0.274 0.236 0.602 0.774 0.434
AON
1.000 0.211 0.736 0.790 0.736 0.736 0.220 0.539 0.471 0.474 0.431 0.583 0.835 0.711 0.580 0.414 0.612 0.674 0.712 0.736 0.736 0.684 0.616 0.588 0.840 0.691 0.783 0.843 0.826 0.642 0.853 0.514 0.569 0.354 0.376 0.680 0.735 0.571 0.687 0.608 0.798 0.402 0.721 0.711 0.649 0.810 0.193 0.618 0.660 0.854 0.790 0.357 0.714 0.180 0.512 0.370 0.849 0.734 0.594
AUD
1.000 0.179 0.236 0.181 0.181 0.660 0.132 0.410 0.340 0.392 0.132 0.228 0.182 0.095 0.506 0.167 0.166 0.162 0.181 0.181 0.364 0.411 0.514 0.215 0.312 0.181 0.183 0.195 0.134 0.216 0.128 0.105 0.630 0.115 0.196 0.177 0.444 0.279 0.280 0.187 0.487 0.182 0.182 0.137 0.277 0.051 0.235 0.253 0.205 0.212 0.428 0.164 0.618 0.246 0.725 0.206 0.181 -0.017
AZN
1.000 0.731 1.000 1.000 0.191 0.802 0.486 0.593 0.373 0.815 0.750 0.606 0.689 0.473 0.878 0.927 0.953 1.000 1.000 0.541 0.540 0.475 0.759 0.575 0.711 0.757 0.745 0.602 0.766 0.429 0.474 0.360 0.530 0.607 0.998 0.735 0.928 0.530 0.723 0.328 0.994 0.606 0.872 0.728 0.331 0.585 0.535 0.772 0.714 0.511 0.982 0.181 0.384 0.366 0.763 0.970 0.576
BBD
1.000 0.731 0.731 0.222 0.567 0.566 0.478 0.513 0.539 0.903 0.733 0.528 0.506 0.608 0.677 0.699 0.731 0.731 0.754 0.703 0.683 0.903 0.789 0.837 0.896 0.903 0.685 0.913 0.562 0.621 0.375 0.395 0.753 0.728 0.637 0.695 0.687 0.858 0.509 0.722 0.733 0.641 0.889 0.209 0.707 0.716 0.919 0.851 0.431 0.714 0.203 0.569 0.419 0.915 0.719 0.661
BDT
1.000 1.000 0.193 0.802 0.489 0.594 0.372 0.815 0.749 0.606 0.690 0.476 0.878 0.928 0.954 1.000 1.000 0.541 0.540 0.474 0.759 0.577 0.712 0.757 0.745 0.602 0.766 0.430 0.474 0.361 0.529 0.607 0.999 0.734 0.928 0.530 0.723 0.329 0.995 0.606 0.873 0.729 0.335 0.586 0.534 0.772 0.714 0.512 0.982 0.182 0.384 0.368 0.763 0.970 0.577
BOB
1.000 0.193 0.802 0.489 0.594 0.372 0.815 0.749 0.606 0.690 0.476 0.878 0.928 0.954 1.000 1.000 0.541 0.540 0.474 0.759 0.577 0.712 0.757 0.745 0.602 0.766 0.430 0.474 0.361 0.529 0.607 0.999 0.734 0.928 0.530 0.723 0.329 0.995 0.606 0.873 0.729 0.335 0.586 0.535 0.772 0.714 0.512 0.982 0.182 0.384 0.368 0.763 0.970 0.577
BRL
1.000 0.149 0.368 0.256 0.462 0.162 0.223 0.208 0.083 0.504 0.213 0.168 0.198 0.192 0.193 0.282 0.317 0.406 0.228 0.273 0.201 0.186 0.191 0.122 0.212 0.054 0.134 0.649 0.142 0.167 0.192 0.337 0.276 0.259 0.189 0.413 0.190 0.208 0.138 0.228 0.073 0.213 0.193 0.204 0.200 0.374 0.179 0.594 0.220 0.643 0.199 0.197 -0.087
CDF
1.000 0.379 0.474 0.258 0.666 0.546 0.426 0.536 0.367 0.701 0.753 0.762 0.802 0.802 0.375 0.383 0.343 0.554 0.394 0.529 0.559 0.558 0.503 0.567 0.244 0.345 0.277 0.430 0.423 0.802 0.558 0.741 0.346 0.542 0.181 0.798 0.426 0.747 0.523 0.258 0.425 0.397 0.573 0.558 0.374 0.785 0.134 0.249 0.303 0.560 0.777 0.434
CHF
1.000 0.396 0.470 0.346 0.555 0.428 0.338 0.850 0.408 0.460 0.466 0.489 0.489 0.514 0.557 0.555 0.552 0.490 0.497 0.539 0.535 0.415 0.562 0.334 0.363 0.413 0.313 0.511 0.491 0.530 0.517 0.506 0.522 0.519 0.485 0.428 0.439 0.587 0.122 0.421 0.421 0.560 0.541 0.705 0.489 0.399 0.327 0.483 0.554 0.491 0.552
CLP
1.000 0.364 0.471 0.506 0.366 0.382 0.416 0.544 0.557 0.577 0.594 0.594 0.419 0.413 0.456 0.489 0.416 0.443 0.489 0.481 0.418 0.500 0.336 0.277 0.430 0.298 0.397 0.590 0.578 0.614 0.404 0.485 0.404 0.598 0.366 0.497 0.496 0.183 0.425 0.383 0.501 0.449 0.442 0.598 0.307 0.346 0.422 0.503 0.575 0.307
COP
1.000 0.248 0.550 0.423 0.192 0.435 0.335 0.341 0.358 0.372 0.372 0.562 0.546 0.597 0.547 0.530 0.477 0.535 0.526 0.382 0.544 0.371 0.349 0.497 0.250 0.394 0.377 0.443 0.446 0.561 0.518 0.464 0.359 0.423 0.319 0.561 0.116 0.400 0.425 0.548 0.498 0.387 0.365 0.392 0.390 0.559 0.549 0.371 0.305
DOP
1.000 0.565 0.488 0.572 0.365 0.669 0.746 0.775 0.815 0.815 0.403 0.432 0.319 0.570 0.403 0.548 0.561 0.568 0.420 0.582 0.349 0.384 0.286 0.461 0.456 0.813 0.573 0.747 0.343 0.527 0.206 0.812 0.488 0.718 0.539 0.267 0.477 0.386 0.580 0.536 0.384 0.791 0.162 0.271 0.257 0.569 0.792 0.418
EGP
1.000 0.798 0.559 0.489 0.631 0.685 0.717 0.749 0.749 0.848 0.729 0.744 0.972 0.841 0.894 0.975 0.963 0.734 0.984 0.628 0.652 0.404 0.386 0.773 0.748 0.640 0.718 0.774 0.914 0.521 0.740 0.798 0.649 0.969 0.224 0.752 0.758 0.989 0.918 0.431 0.731 0.205 0.630 0.440 0.989 0.745 0.707
ERN
1.000 0.453 0.401 0.505 0.556 0.593 0.606 0.606 0.700 0.588 0.553 0.794 0.712 0.761 0.792 0.766 0.618 0.811 0.533 0.541 0.303 0.296 0.650 0.605 0.480 0.604 0.621 0.757 0.411 0.597 1.000 0.488 0.783 0.192 0.601 0.604 0.807 0.752 0.338 0.587 0.184 0.538 0.332 0.803 0.606 0.544
ETB
1.000 0.319 0.597 0.629 0.659 0.690 0.690 0.443 0.372 0.375 0.562 0.467 0.534 0.563 0.530 0.447 0.572 0.313 0.358 0.244 0.371 0.472 0.691 0.514 0.633 0.411 0.613 0.248 0.685 0.453 0.594 0.559 0.249 0.452 0.373 0.577 0.520 0.307 0.682 0.109 0.346 0.211 0.569 0.671 0.456
EUR
1.000 0.443 0.445 0.458 0.476 0.476 0.474 0.552 0.513 0.486 0.452 0.434 0.464 0.473 0.354 0.499 0.244 0.342 0.489 0.286 0.452 0.477 0.527 0.513 0.447 0.459 0.533 0.475 0.401 0.412 0.515 0.120 0.397 0.410 0.486 0.478 0.810 0.473 0.425 0.306 0.545 0.478 0.477 0.404
GEL
1.000 0.826 0.837 0.878 0.878 0.451 0.461 0.407 0.629 0.511 0.609 0.638 0.627 0.506 0.643 0.343 0.383 0.332 0.475 0.499 0.877 0.656 0.813 0.489 0.593 0.311 0.874 0.505 0.745 0.617 0.272 0.482 0.466 0.649 0.589 0.497 0.868 0.171 0.406 0.343 0.642 0.848 0.486
GHS
1.000 0.880 0.928 0.928 0.504 0.495 0.427 0.693 0.511 0.639 0.694 0.677 0.548 0.702 0.387 0.443 0.342 0.499 0.557 0.927 0.672 0.857 0.502 0.658 0.301 0.925 0.556 0.800 0.663 0.321 0.538 0.527 0.707 0.653 0.479 0.908 0.156 0.347 0.330 0.699 0.900 0.515
GTQ
1.000 0.954 0.954 0.519 0.524 0.452 0.724 0.555 0.691 0.732 0.716 0.581 0.737 0.428 0.447 0.360 0.486 0.584 0.952 0.693 0.886 0.503 0.694 0.321 0.948 0.593 0.848 0.701 0.345 0.563 0.501 0.743 0.682 0.472 0.933 0.180 0.343 0.362 0.733 0.925 0.562
HNL
1.000 1.000 0.541 0.540 0.474 0.759 0.577 0.712 0.757 0.745 0.602 0.766 0.430 0.475 0.361 0.529 0.608 0.999 0.734 0.928 0.529 0.723 0.329 0.995 0.606 0.873 0.729 0.335 0.586 0.534 0.772 0.714 0.512 0.982 0.182 0.384 0.368 0.763 0.970 0.577
HTG
1.000 0.541 0.540 0.474 0.759 0.577 0.712 0.757 0.745 0.602 0.766 0.430 0.474 0.361 0.529 0.607 0.999 0.734 0.928 0.530 0.723 0.329 0.995 0.606 0.873 0.729 0.335 0.586 0.534 0.772 0.714 0.512 0.982 0.182 0.384 0.368 0.763 0.970 0.577
IDR
1.000 0.710 0.795 0.810 0.778 0.758 0.813 0.804 0.621 0.818 0.556 0.566 0.414 0.237 0.641 0.541 0.647 0.552 0.789 0.775 0.565 0.531 0.700 0.485 0.848 0.181 0.663 0.694 0.825 0.775 0.425 0.524 0.320 0.640 0.491 0.833 0.538 0.545
ILS
1.000 0.661 0.713 0.664 0.601 0.719 0.723 0.533 0.715 0.473 0.485 0.464 0.285 0.548 0.545 0.580 0.559 0.638 0.676 0.566 0.535 0.588 0.453 0.722 0.168 0.599 0.598 0.727 0.689 0.472 0.519 0.386 0.460 0.532 0.723 0.540 0.429
INR
1.000 0.716 0.696 0.650 0.702 0.690 0.567 0.711 0.510 0.504 0.537 0.255 0.604 0.472 0.781 0.529 0.719 0.665 0.616 0.469 0.553 0.405 0.754 0.181 0.635 0.634 0.718 0.703 0.479 0.462 0.427 0.565 0.561 0.726 0.462 0.426
JOD
1.000 0.808 0.893 0.970 0.950 0.744 0.980 0.618 0.664 0.402 0.392 0.762 0.758 0.620 0.720 0.752 0.913 0.480 0.749 0.794 0.664 0.955 0.214 0.751 0.723 0.983 0.916 0.403 0.742 0.201 0.599 0.426 0.981 0.757 0.707
KES
1.000 0.752 0.817 0.800 0.604 0.835 0.570 0.549 0.432 0.295 0.710 0.578 0.557 0.571 0.693 0.766 0.544 0.565 0.712 0.481 0.843 0.135 0.644 0.720 0.837 0.777 0.400 0.559 0.282 0.657 0.460 0.844 0.572 0.536
KHR
1.000 0.907 0.887 0.708 0.913 0.550 0.655 0.357 0.390 0.693 0.706 0.573 0.675 0.676 0.834 0.418 0.704 0.761 0.609 0.884 0.220 0.702 0.678 0.907 0.845 0.365 0.703 0.174 0.556 0.367 0.906 0.706 0.669
LBP
1.000 0.955 0.758 0.980 0.613 0.662 0.374 0.402 0.777 0.756 0.596 0.720 0.748 0.911 0.479 0.747 0.792 0.654 0.956 0.234 0.747 0.718 0.985 0.920 0.403 0.739 0.175 0.588 0.410 0.983 0.755 0.714
LKR
1.000 0.728 0.964 0.615 0.648 0.359 0.383 0.758 0.743 0.603 0.694 0.742 0.893 0.475 0.736 0.766 0.656 0.943 0.216 0.724 0.730 0.970 0.901 0.400 0.723 0.166 0.566 0.432 0.968 0.741 0.707
LRD
1.000 0.755 0.444 0.542 0.302 0.296 0.575 0.601 0.480 0.556 0.581 0.697 0.366 0.599 0.618 0.529 0.717 0.194 0.600 0.521 0.753 0.697 0.318 0.585 0.103 0.436 0.359 0.748 0.547 0.581
MMK
1.000 0.619 0.673 0.398 0.400 0.782 0.766 0.617 0.731 0.749 0.915 0.495 0.756 0.811 0.665 0.967 0.232 0.746 0.741 0.994 0.923 0.434 0.746 0.192 0.600 0.437 0.992 0.763 0.724
MRO
1.000 0.493 0.199 0.247 0.513 0.431 0.447 0.424 0.499 0.572 0.358 0.429 0.533 0.356 0.628 0.073 0.516 0.492 0.625 0.571 0.258 0.412 0.094 0.433 0.230 0.633 0.429 0.452
MWK
1.000 0.238 0.327 0.524 0.467 0.400 0.447 0.476 0.598 0.297 0.460 0.541 0.402 0.648 0.099 0.530 0.538 0.669 0.628 0.319 0.446 0.110 0.411 0.284 0.666 0.471 0.456
MXN
1.000 0.199 0.313 0.359 0.458 0.448 0.385 0.354 0.449 0.361 0.303 0.296 0.431 0.029 0.315 0.352 0.391 0.364 0.393 0.350 0.583 0.276 0.696 0.388 0.353 0.065
MZN
1.000 0.316 0.529 0.374 0.500 0.240 0.379 0.203 0.521 0.296 0.457 0.386 0.186 0.307 0.245 0.403 0.381 0.336 0.530 0.049 0.209 0.252 0.400 0.524 0.327
NGN
1.000 0.607 0.513 0.588 0.579 0.720 0.447 0.603 0.650 0.521 0.780 0.219 0.600 0.580 0.787 0.735 0.374 0.594 0.159 0.473 0.392 0.787 0.615 0.588
NIO
1.000 0.732 0.928 0.529 0.725 0.331 0.993 0.605 0.871 0.728 0.331 0.584 0.536 0.771 0.712 0.512 0.981 0.182 0.382 0.368 0.762 0.969 0.576
NPR
1.000 0.750 0.601 0.594 0.556 0.736 0.480 0.627 0.642 0.253 0.611 0.547 0.624 0.587 0.598 0.720 0.390 0.488 0.475 0.626 0.699 0.396
PEN
1.000 0.540 0.684 0.397 0.926 0.604 0.787 0.714 0.324 0.564 0.524 0.735 0.694 0.545 0.919 0.273 0.367 0.451 0.727 0.907 0.514
PHP
1.000 0.721 0.541 0.523 0.621 0.466 0.774 0.219 0.652 0.632 0.763 0.705 0.392 0.515 0.262 0.548 0.417 0.767 0.510 0.521
PKR
1.000 0.476 0.713 0.757 0.631 0.896 0.231 0.694 0.702 0.923 0.851 0.374 0.709 0.156 0.589 0.389 0.920 0.711 0.660
RUR
1.000 0.326 0.411 0.283 0.528 0.036 0.474 0.474 0.494 0.485 0.494 0.311 0.406 0.394 0.515 0.498 0.332 0.263
RWF
1.000 0.597 0.868 0.722 0.339 0.583 0.525 0.762 0.704 0.514 0.977 0.177 0.385 0.365 0.753 0.963 0.575
SDG
1.000 0.488 0.783 0.192 0.601 0.604 0.807 0.752 0.338 0.587 0.184 0.538 0.332 0.803 0.606 0.544
SLL
1.000 0.627 0.350 0.499 0.418 0.670 0.616 0.455 0.852 0.138 0.305 0.334 0.663 0.838 0.520
THB
1.000 0.210 0.733 0.739 0.973 0.907 0.442 0.711 0.231 0.607 0.483 0.972 0.724 0.730
TJS
1.000 0.131 0.140 0.231 0.221 0.135 0.331 0.098 0.044 0.126 0.230 0.333 0.166
TZS
1.000 0.600 0.748 0.684 0.399 0.576 0.187 0.508 0.352 0.742 0.570 0.524
UGX
1.000 0.746 0.719 0.353 0.511 0.161 0.592 0.351 0.749 0.521 0.497
USD
1.000 0.929 0.416 0.752 0.190 0.605 0.434 0.998 0.768 0.729
VND
1.000 0.392 0.692 0.194 0.565 0.422 0.928 0.713 0.662
XOF
1.000 0.510 0.342 0.264 0.447 0.415 0.491 0.332
YER
1.000 0.184 0.382 0.357 0.744 0.952 0.569
ZAR
1.000 0.147 0.559 0.190 0.196 0.012
ZMK
1.000 0.311 0.611 0.372 0.347
CAD
1.000 0.430 0.374 0.128
HKD
1.000 0.760 0.726
JMD
1.000 0.575
JPY
1.000
87
f.
Appendix F: Regressions Results
Dependent Variable: CF
Method: Least Squares
Date: 08/03/10 Time: 19:23
Sample(adjusted): 2005:06 2009:06
Included observations: 49 after adjusting endpoints
Variable
C
USD
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Coefficient
1.000147
0.976513
0.613371
0.605145
0.023593
0.026161
115.0868
1.983074
Std. Error
t-Statistic
0.003375
296.3344
0.113087
8.635023
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
Prob.
0.0000
0.0000
0.998617
0.037546
-4.615787
-4.538570
74.56363
0.000000
Dependent Variable: CF
Method: Least Squares
Date: 08/03/10 Time: 19:55
Sample(adjusted): 2005:06 2009:07
Included observations: 50 after adjusting endpoints
Variable
C
BDT
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Coefficient
0.998034
0.771494
0.482306
0.471521
0.027016
0.035034
110.6397
1.957506
Std. Error
t-Statistic
0.003821
261.1655
0.115368
6.687223
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
Prob.
0.0000
0.0000
0.998560
0.037163
-4.345589
-4.269109
44.71895
0.000000
Dependent Variable: CF
Method: Least Squares
Date: 08/03/10 Time: 19:53
Sample(adjusted): 2005:06 2009:07
Included observations: 50 after adjusting endpoints
Variable
C
PKR
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Coefficient
0.994432
0.767494
0.367256
0.354074
0.029868
0.042820
105.6227
1.706050
Std. Error
t-Statistic
0.004296
231.4945
0.145406
5.278267
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
88
Prob.
0.0000
0.0000
0.998560
0.037163
-4.144907
-4.068427
27.86011
0.000003
Dependent Variable: CF
Method: Least Squares
Date: 08/03/10 Time: 19:27
Sample(adjusted): 2005:06 2009:07
Included observations: 50 after adjusting endpoints
Variable
C
PHP
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Coefficient
1.000642
0.569952
0.250839
0.235231
0.032499
0.050698
101.4005
1.656630
Std. Error
t-Statistic
0.004625
216.3400
0.142170
4.008947
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
Prob.
0.0000
0.0002
0.998560
0.037163
-3.976019
-3.899538
16.07165
0.000212
Dependent Variable: CF
Method: Least Squares
Date: 08/03/10 Time: 19:24
Sample(adjusted): 2005:06 2009:07
Included observations: 50 after adjusting endpoints
Variable
C
EUR
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Coefficient
1.000724
0.513695
0.139737
0.121815
0.034826
0.058216
97.94338
1.711171
Std. Error
t-Statistic
0.004986
200.7182
0.183969
2.792295
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
Prob.
0.0000
0.0075
0.998560
0.037163
-3.837735
-3.761254
7.796911
0.007493
Dependent Variable: CF
Method: Least Squares
Date: 08/03/10 Time: 19:28
Sample(adjusted): 2005:06 2009:07
Included observations: 50 after adjusting endpoints
Variable
C
KES
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Coefficient
0.999108
0.482161
0.221836
0.205624
0.033122
0.052661
100.4509
1.612041
Std. Error
t-Statistic
0.004687
213.1860
0.130344
3.699140
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
89
Prob.
0.0000
0.0006
0.998560
0.037163
-3.938035
-3.861554
13.68363
0.000556
Dependent Variable: CF
Method: Least Squares
Date: 08/04/10 Time: 05:01
Sample(adjusted): 2005:06 2009:06
Included observations: 49 after adjusting endpoints
Variable
C
XAF
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Coefficient
1.000560
0.458890
0.122677
0.104010
0.035540
0.059364
95.01122
1.690601
Std. Error
t-Statistic
0.005133
194.9125
0.179002
2.563598
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
Prob.
0.0000
0.0136
0.998617
0.037546
-3.796376
-3.719159
6.572033
0.013618
Dependent Variable: CF
Method: Least Squares
Date: 08/03/10 Time: 19:54
Sample(adjusted): 2005:06 2009:06
Included observations: 49 after adjusting endpoints
Variable
C
UGX
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Coefficient
0.997786
0.428620
0.188047
0.170771
0.034190
0.054941
96.90833
1.575568
Std. Error
t-Statistic
0.004891
204.0146
0.129914
3.299260
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
Prob.
0.0000
0.0019
0.998617
0.037546
-3.873809
-3.796592
10.88511
0.001854
Dependent Variable: CF
Method: Least Squares
Date: 08/03/10 Time: 19:54
Sample(adjusted): 2005:06 2009:07
Included observations: 50 after adjusting endpoints
Variable
C
BRL
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Coefficient
0.999389
0.141711
0.028252
0.008007
0.037014
0.065761
94.89690
1.391240
Std. Error
t-Statistic
0.005281
189.2322
0.119961
1.181314
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
90
Prob.
0.0000
0.2433
0.998560
0.037163
-3.715876
-3.639395
1.395503
0.243298
Dependent Variable: CF
Method: Least Squares
Date: 08/03/10 Time: 19:55
Sample(adjusted): 2005:06 2009:07
Included observations: 50 after adjusting endpoints
Variable
C
ZAR
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
Coefficient
0.998485
0.037586
0.002249
-0.018537
0.037506
0.067521
94.23674
1.554463
Std. Error
t-Statistic
0.005309
188.0721
0.114259
0.328956
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
91
Prob.
0.0000
0.7436
0.998560
0.037163
-3.689469
-3.612989
0.108212
0.743620
g.
Appendix G: Optimal Weights (Without short selling)
Code Current Income Original Weight 0.020% 0.025% 0.030% 0.040% 0.050% 0.060% 0.070% 0.074% 0.081% 0.084%
AFO
AUD
BDT
BRL
CAD
COP
EUR
HTG
ILS
INR
KES
NPR
PHP
PKR
SLL
TZS
USD
XOF
YER
ZAR
£
516.19
£
133,847.74
£
105,993.66
£
311,944.46
£
146,476.86
£
971.53
£ 6,705,768.16
£
16,711.34
£
33,549.55
£
149,507.61
£
5,420.38
£
501.82
£
8,583.53
£
331,836.58
£
29,266.44
£
40,654.23
£ 7,406,410.15
£
3,829.68
£
18,622.95
£
73,386.52
TOTAL £15,523,799.38
0.00%
0.86%
0.68%
2.01%
0.94%
0.01%
43.20%
0.11%
0.22%
0.96%
0.03%
0.00%
0.06%
2.14%
0.19%
0.26%
47.71%
0.02%
0.12%
0.47%
100.00%
19.7%
8.5%
3.7%
5.4%
0.0%
0.0%
9.8%
3.7%
0.0%
0.0%
4.3%
0.0%
0.0%
5.8%
0.0%
0.0%
0.0%
33.8%
0.0%
5.3%
20.4%
9.7%
4.7%
7.3%
0.0%
0.0%
6.0%
4.7%
1.2%
0.0%
2.6%
0.0%
0.7%
2.9%
0.0%
0.0%
2.9%
32.1%
0.0%
4.9%
20.9%
10.8%
5.3%
9.0%
0.0%
0.0%
2.2%
5.3%
2.5%
0.0%
0.0%
0.0%
2.1%
0.0%
0.0%
0.0%
6.7%
30.5%
0.0%
4.7%
22.8%
11.6%
8.0%
11.4%
3.5%
1.7%
0.0%
8.0%
4.3%
0.0%
0.0%
0.0%
5.0%
0.0%
0.0%
0.0%
0.0%
20.3%
0.0%
3.4%
23.8%
12.4%
8.8%
13.7%
6.4%
4.6%
0.0%
8.8%
3.9%
0.0%
0.0%
2.4%
5.0%
0.0%
0.0%
0.0%
0.0%
7.9%
0.0%
2.3%
25.9%
8.9%
6.7%
16.3%
13.1%
10.3%
0.0%
4.7%
0.0%
0.0%
0.0%
10.2%
3.8%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
23.1%
0.0%
0.0%
18.1%
24.5%
30.1%
0.0%
0.0%
0.0%
0.0%
0.0%
4.3%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
14.7%
0.0%
0.0%
16.8%
24.8%
43.7%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
9.9%
15.4%
74.7%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
100.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
0.0%
100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00% 100.00%
92
h.
Appendix H: Optimal Weights (With short selling)
Code
Income
AFO
AUD
BDT
BRL
CAD
COP
EUR
HTG
ILS
INR
KES
NPR
PHP
PKR
SLL
TZS
USD
XOF
YER
ZAR
£
516.19
£
133,847.74
£
105,993.66
£
311,944.46
£
146,476.86
£
971.53
£ 6,705,768.16
£
16,711.34
£
33,549.55
£
149,507.61
£
5,420.38
£
501.82
£
8,583.53
£
331,836.58
£
29,266.44
£
40,654.23
£ 7,406,410.15
£
3,829.68
£
18,622.95
£
73,386.52
Current W 0.030% 0.040% 0.050% 0.060% 0.070% 0.080% 0.084% 0.13%
0.00%
0.86%
0.68%
2.01%
0.94%
0.01%
43.20%
0.11%
0.22%
0.96%
0.03%
0.00%
0.06%
2.14%
0.19%
0.26%
47.71%
0.02%
0.12%
0.47%
18.4%
10.1%
20.5%
6.1%
-2.5%
-5.2%
6.6%
19.9%
1.5%
3.9%
5.6%
-12.0%
3.7%
11.0%
-5.0%
-8.5%
-0.1%
40.1%
-20.8%
6.7%
18.6% 18.7% 18.9% 19.1% 19.3% 19.3%
9.7%
9.3%
9.0%
8.6%
8.3%
8.2%
28.8% 37.2% 45.6% 54.0% 62.5% 66.1%
6.3%
6.3%
6.4%
6.5%
6.5%
6.5%
-1.6% -0.8% 0.1%
0.9%
1.7%
2.2%
-4.8% -4.4% -4.0% -3.5% -3.1% -2.9%
5.6%
4.5%
3.4%
2.2%
1.0%
0.4%
28.2% 36.6% 45.0% 53.4% 61.9% 65.5%
1.3%
1.2%
1.2%
1.0%
1.0%
0.9%
3.8%
3.9%
4.0%
4.0%
4.2%
4.3%
4.5%
3.4%
2.2%
1.0%
-0.1% -0.7%
-11.1% -10.3% -9.5% -8.7% -8.1% -7.8%
4.2%
4.6%
5.0%
5.5%
5.9%
6.1%
9.9%
8.9%
7.8%
6.7%
5.8%
5.5%
-7.7% -10.5% -13.2% -16.0% -18.7% -19.9%
-9.1% -9.7% -10.3% -10.9% -11.5% -11.7%
1.7%
3.4%
4.9%
6.7%
8.1%
8.6%
39.6% 39.1% 38.6% 38.2% 37.8% 37.8%
-34.5% -48.2% -61.9% -75.6% -89.3% -95.2%
6.6%
6.6%
6.7%
6.7%
6.8%
6.8%
18.1%
9.2%
96.1%
13.1%
3.5%
-1.2%
-14.4%
95.5%
-1.8%
-9.7%
-15.9%
11.3%
7.1%
-4.7%
-31.5%
-18.3%
57.1%
41.3%
-160.8%
6.2%
1.88%
3.74%
18.56%
49.8%
-54.2%
1578.0%
21.0%
154.6%
72.4%
-201.9%
1577.3%
-17.5%
15.9%
-207.4%
135.1%
84.2%
-181.6%
-513.9%
-119.3%
302.4%
-43.8%
-2561.2%
10.1%
81.2%
-118.5%
3135.6%
35.8%
311.6%
150.0%
-409.7%
3134.7%
-36.3%
28.4%
-420.4%
281.7%
165.0%
-375.1%
-1022.6%
-230.1%
604.7%
-128.4%
-5101.1%
13.5%
332.5%
-632.8%
15594.1%
154.6%
1568.2%
770.9%
-2076.3%
15592.5%
-187.8%
124.1%
-2123.8%
1459.1%
808.9%
-1916.5%
-5092.7%
-1116.8%
3023.1%
-800.8%
-25421.8%
41.2%
100.0%
100.0%
TOTAL £15,523,799.38 100.00% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0%
93
i.
Appendix I: Individual VaRs
Currency
EUR
USD
PHP
KES
XOF
UGX
PKR
MWK
ZAR
IDR
ILS
BDT
TZS
LKR
BRL
COP
ERN
AFO
HTG
THB
NPR
SDG
CAD
VND
INR
ZMK
YER
AUD
NGN
SLL
RUR
PEN
AZN
GEL
BOB
MMK
AMD
ALL
KHR
TJS
DOP
GHS
CLP
RWF
HNL
AON
GTQ
ETB
MXN
NIO
BBD
CDF
MZN
HKD
EGP
JPY
JMD
JOD
LRD
CHF
LBP
MRO
1-day 99% VaR%
W£
1-day 99% VaR£ 20-day VaR£
1.34%
5,228,008
69,907
312,636
1.57%
4,117,599
64,633
289,049
1.80%
1,489,799
26,888
120,246
1.70%
1,348,670
22,912
102,464
1.33%
1,101,435
14,679
65,645
2.04%
909,554
18,531
82,873
1.66%
682,061
11,301
50,540
2.68%
681,116
18,251
81,621
2.05%
660,489
13,548
60,590
1.74%
645,507
11,222
50,187
1.62%
579,390
9,407
42,070
1.58%
547,030
8,662
38,738
2.24%
466,827
10,477
46,854
1.63%
394,478
6,421
28,715
1.98%
377,319
7,474
33,423
2.08%
351,073
7,302
32,657
1.35%
334,229
4,497
20,112
1.81%
332,637
6,024
26,938
1.58%
306,556
4,854
21,709
1.56%
256,334
3,998
17,878
1.62%
249,588
4,037
18,053
2.08%
242,315
5,034
22,513
1.78%
146,477
2,613
11,687
1.66%
141,035
2,347
10,495
1.65%
139,950
2,316
10,356
2.19%
138,361
3,025
13,527
1.64%
131,983
2,161
9,664
1.92%
131,911
2,533
11,328
2.02%
128,883
2,598
11,620
1.87%
127,296
2,383
10,658
1.71%
117,100
2,002
8,955
1.53%
112,095
1,716
7,674
1.58%
103,243
1,634
7,306
1.92%
102,133
1,959
8,763
1.58%
95,291
1,509
6,748
1.57%
82,641
1,294
5,788
1.95%
77,162
1,504
6,726
1.39%
60,129
838
3,745
1.67%
59,809
999
4,468
1.82%
58,019
1,058
4,730
1.91%
55,160
1,053
4,710
1.69%
51,446
867
3,878
1.86%
51,259
954
4,268
1.60%
45,353
725
3,241
1.58%
42,090
667
2,981
1.91%
23,715
452
2,021
1.68%
16,679
281
1,256
2.15%
9,553
205
918
1.77%
8,092
143
639
1.59%
6,936
110
492
1.56%
6,196
97
433
1.98%
2,745
54
243
3.05%
2,014
61
274
1.57%
684
11
48
1.57%
560
9
39
2.39%
366
9
39
1.62%
252
4
18
1.60%
244
4
17
1.93%
241
5
21
1.40%
18
0
1
1.56%
2
0
0
2.20%
Total
23,579,136
390,258
1,745,288
99% VaR of the Portoflio
295,899
1,323,300
Benefits of diversification
94,359
421,988
94