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Calculation of the CPPI value at any time t in the period [0,T]

Submitted by cahn on 16 June, 2007 - 12:57am

Bertrand and Prigent, while analysing and comparing two portfolio insurance methods, derive the value of a CPPI value at any time t in the period [0,T] as

\[<br /> V_{t}^{CPPI}\left(m,S_{t}\right)=F_{0}e^{rt}+\alpha_{t}S_{t}^{m}\]

Full proof below:

The value at $ t $ of the CPPI portfolio is given by $ dV_{t}=\left(V_{t}-e_{t}\right)\frac{dB_{t}}{B_{t}}+e_{t}\frac{dS_{t}}{S_{t}} $.

Recall that $ V_{t}=C_{t}+F_{t} $, $ e_{t}=mC_{t} $ and $ dF_{t}=rdt $, where $ m $ is a fixed constant, called a multiplier. Thus, the cushion value $ C $ must satisfy:

\[<br /> \begin{array}{ccc}<br /> dC_{t} & = & d\left\left(V_{t}-F_{t}\right)\left\\<br /> & = & \left(V_{t}-e_{t}\right)\frac{dB_{t}}{B_{t}}+e_{t}\frac{dS_{t}}{S_{t}}-dF_{t}\\<br /> & = & \left(C_{t}+F_{t}-mC_{t}\right)\frac{dB_{t}}{B_{t}}+mC_{t}\frac{dS_{t}}{S_{t}}-dF_{t}\\<br /> & = & \left(C_{t}-mC_{t}\right)\frac{dB_{t}}{B_{t}}+mC_{t}\frac{dS_{t}}{S_{t}}\\<br /> & = & C_{t}\left[\left(m\left(\mu-r\right)+r\right)dt+m\sigma dW_{t}\right]\end{array}\]

Thus: $ C_{t}=C_{0}\exp\left[\left(m\left(\mu-r\right)+r-\frac{m^{2}\sigma^{2}}{2}\right)t+m\sigma W_{t}\right] $. By using the relation: $ S_{t}=S_{0}\exp\left[\left(\mu-\frac{1}{2}\sigma^{2}\right)t+\sigma W_{t}\right] $, it can be deduced that: $ W_{t}=\frac{1}{\sigma}\left[\ln\left(\frac{S_{t}}{S_{0}}\right)-\left(\mu-\frac{1}{2}\sigma^{2}\right)t\right] $.

Substituting this expression for $ W_{t} $ into the expression for $ C_{t} $ leads to:

\[<br /> \begin{array}{ccc}<br /> C_{t}\left(m,S_{t}\right) & = & C_{0}\left(\frac{S_{t}}{S_{0}}\right)^{m}\exp\left[\left(r-m\left(r-\frac{1}{2}\sigma^{2}\right)-\frac{m^{2}}{2}\sigma^{2}\right)t\right]\\<br /> & = & \alpha_{t}S_{t}^{m}\end{array}\]

where $ \alpha_{t}=\left(\frac{C_{0}}{S_{0}^{m}}\right)\exp\left[\beta t\right] $ and $ \beta=\left(r-m\left(r-\frac{1}{2}\sigma^{2}\right)-\frac{m^{2}}{2}\sigma^{2}\right) $.

The portfolio value is then obtained:

\[<br /> V_{t}^{CPPI}\left(m,S_{t}\right)=F_{0}e^{rt}+\alpha_{t}S_{t}^{m}\]
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Portfolio Insurance Strategies - OBPI versus CPPI.pdf337.4 KB
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What did I say then?

It was US subprime mortgages, then it was money markets in US, then it was...

Submitted by cahn on 5 October, 2008 - 6:44pm

It was the US subprime mortgages, then it was money markets in the US, then it was money markets in the Europe, then it was the collapse of Lehman Brothers, the collapse of Washington Mutual, the run on Bank of East Asia, the collapse of Fortis, the $700bn bailout of the US, ... , and is now the collapse of the Hypo Real Estate rescue and the gloom in the Chinese art auction.

If the rest of the world were to go through what the US had to go through, I would not know if we, the rest of the world, would ever do better than what the US has done.

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