April 2011
Intermediate to advanced
328 pages
9h 5m
English
Content preview from Risk Analysis in Finance and Insurance, 2nd Edition
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Implementations of Risk Analysis in Various Areas of Financial Industry 185
Calculating
DR =
r
−∞
(r −x)
2
f
R
(x) dx =
(r−m)/σ
−∞
(r −σy − m)
2
ϕ(y) dy
= σ
2
−z
−∞
(−z −y)
2
ϕ(y) dy = σ
2
(z
2
+1)Φ(−z) −zϕ(z)
,
where Φ(x)=
1
√
2π
x
−∞
e
−y
2
/2
dy, we obtain the following expression for SoR
is terms of z:
SoR =
m − r
σ
(z
2
+1)Φ(−z) −zϕ(z)
=
z
(z
2
+1)Φ(−z) − zϕ(z)
.
Introducing the Upside Potential (UP)as
UP =
∞
r
(x −r) f
R
(x) dx,
we now define the third performance ratio
UPR =
UP
√
DR
,
which is called the Upside Potential ratio. The following calculations connect
it with the Sharpe ratio. We have
UP =
∞
r
(x −r) f
R
(x) dx =
∞
−z
(σy + m − r)ϕ(y) dy
= σ
∞
−z
(y + z)ϕ(y) dy = σ
ϕ(z)+zΦ(z)
,
and hence,
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