Financial Mathematics for Actuarial Practice 2015 Assessment

Pontificia Universidad Católica de Chile
Facultad de Matemáticas
Departamento de Estadı́stica
Financial Mathematics for Actuarial Practice
2015
Assessment Workout
Consider the so-called Stoodley formula for the force of interest
δt = p +
s
,
1 + rest
t ∈ [0, ∞),
with p, r ∈ [0, ∞) and s + p > 0.
1. Show that a(t) = e(p+s)t (1 + r)/(1 + rest ).
2. Show that there exists λ > 0, and functions w1 (·) and w2 (·) such that
v(t) = λw1 (t) + (1 − λ)w2 (t),
t ∈ [0, ∞).
Solution.
Rt
1. Recall that a(t) = e
0
δu du
. Note that
Z t
Z t
s
δu du =
p+
du
1
+
resu
0
0
Z t
s(1 + resu ) − resu
=
p+
du
1 + resu
0
Z t
resu
p+s−
du
=
1 + resu
0
Z t
Z t
resu
=
p + s du −
du
su
0
0 1 + re
= [(p + s)u]t0 − [log(1 + resu )]t0
= (p + s)t − log(1 + rest ) + log(1 + r)
1+r
= (p + s)t + log
,
1 + rest
and thus a(t) = exp[(p+s)t+log{(1+r)/(1+rest )}] = exp{(p+s)t} exp[log{(1+r)/(1+rest )}]
from where the result follows.
2. Note that
1
v(t) =
=
a(t)
1 + rest −(p+s)t
1
r
−(p+s)t
e
e−pt .
e
=
+
1+r
1 + r | {z }
1 + r |{z}
| {z } w1 (t)
| {z } w2 (t)
λ
1
1−λ