0
$\begingroup$

Let the portfolio value follow the SDE:

$$V_t=(\mu w+r(1-w))\cdot V_t\cdot dt +\sigma \cdot w\cdot V_t \cdot dB_t $$

where $\mu$ = drift of the portfolio, $\sigma$=standard deviation of the portfolio, $r$ = risk free rate and the merton optimal weight for the riskly asset: $$w=\frac{\mu-r}{(1-\gamma)\cdot\sigma^2}$$ with $\gamma$ the risk aversion parameter

Knowing that

$$\log \left(\frac{Vt}{V0}\right) \sim \mathcal{N}[(\mu w + r(1-w)-0.5(w\cdot\sigma)^2)\delta,(\sigma\cdot w)^2\delta]$$

Let $VaR$ is the Value at risk for that portfolio, and $$Xt=\log \left(\frac{Vt}{V0}\right)$$ as our measure of loss, we know that the EQUATION (1) is $P(-Xk>VaR)=\alpha$

TASK: Knowing the probability distribution of the log returns show the steps of solving the EQUATION (1) for $\gamma$ (risk aversion parameter).

N.B. The right solution is the following quadratic equation:

$$\gamma = 1 + \frac{(\mu - r) \left(\mu - r + \frac{q_{\alpha} \sigma}{\sqrt{\delta}} \pm \sqrt{(\mu - r + \frac{q_{\alpha} \sigma}{\sqrt{\delta}})^2 + 2 \sigma^{2} \left(\frac{\text{VaR}}{\delta} + r\right)}\right)}{2 \sigma^{2} \left(\frac{\text{VaR}}{\delta} + r\right)}$$ where $q_{\alpha}$ is the quantile of the standard normal distribution

$\endgroup$
2
  • $\begingroup$ What have you tried so far? $\endgroup$ Jul 3, 2023 at 9:38
  • $\begingroup$ I tried starting from the equation (1) to rewrite It with the expectation of the distribution of the log returns, but i don't know how to treat It then $\endgroup$
    – XY0
    Jul 3, 2023 at 11:30

1 Answer 1

1
$\begingroup$

You can think about it like this: given $\mu,\sigma,r$, a risk aversion parameter $\gamma$ will induce an optimal weight $w(\gamma)$, which in turn will induce some value at risk $VaR_{\alpha}$.

Hence you can solve backwards. For simplicity, assume $\delta=1$. Then, since $X_1\equiv \log(V_1/V_0)\sim N(\mu(w)-\frac{1}{2}\sigma^2(w),\sigma(w))$, we have

$$ \begin{align} VaR_{\alpha}&=\mu(w)+\sigma(w)z_{\alpha}\\ \Rightarrow VaR_{\alpha}&=r+w(\mu-r+z\sigma)-\frac{1}{2}w^2\sigma^2\\ \Rightarrow 0&=\frac{1}{2}w^2\sigma^2-w(\mu-r+z\sigma)+(VaR_{\alpha}-r)\\ \Rightarrow w_{1,2}&=\frac{\mu-r+z\sigma}{\sigma^2}\pm\sqrt{\left(\frac{\mu-r+z\sigma}{\sigma^2}\right)^2-2\frac{VaR_{\alpha}-r}{\sigma^2}} \end{align} $$

We can now equate (both) weights with your optimal investment:

$$ \frac{1}{1-\gamma}\frac{\mu-r}{\sigma^2}=\frac{\mu-r+z\sigma}{\sigma^2}\pm\sqrt{\left(\frac{\mu-r+z\sigma}{\sigma^2}\right)^2-2\frac{VaR_{\alpha}-r}{\sigma^2}} $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.