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

  • $\begingroup$ 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


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:

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


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