# Solving Equation for estimation risk averse parameter

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

• What have you tried so far? Jul 3, 2023 at 9:38
• 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
– XY0
Jul 3, 2023 at 11:30

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