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Inferring Risk Aversion from Portfolio Value at Risk

Article Quant Q&A · Author: XY0
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Summary

The document considers whether a risk-aversion parameter can be inferred from a target value at risk under a continuous-time portfolio model. The risky-asset allocation follows the Merton rule, linking weight to expected return, the risk-free rate, volatility, and risk aversion. Given that weight, log portfolio returns are modeled as normally distributed, so a VaR condition can be translated into an equation for the allocation.

The answer’s method is to solve the VaR relationship first for possible portfolio weights, then equate those weights to the Merton optimal weight and solve for risk aversion. The displayed derivation simplifies the horizon to one unit and produces two candidate weights through a quadratic equation. This is an algebraic illustration, not an empirical estimation procedure. Results depend on the return and loss sign convention, the quantile definition, and model assumptions; the prompt’s stated closed-form expression and the answer’s intermediate VaR equation use signs that should be reconciled before applying the result.

Key ideas

  • The Merton rule maps a risk-aversion parameter to an optimal risky-asset weight.
  • Normally distributed log returns allow a VaR condition to be expressed using a normal quantile.
  • The proposed inversion solves for portfolio weights before equating them to the Merton weight.
  • The quadratic equation can yield two candidate weights and therefore candidate risk-aversion values.
  • Sign conventions for returns, losses, VaR, and quantiles must be consistent.

Tags

Risk managementPosition sizingStatisticsPortfolio construction
Full text
# Solving Equation for estimation risk averse parameter


# 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

## Answer by Kermittfrog (score 1, accepted)

https://quant.stackexchange.com/a/76037

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:

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

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.