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

Submitted by loner 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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