# relation between risk averson coefficient and maximum Sharp ratio in Black-Litterman context

BL model compute the implied returns based on the reverse optimization where the objective is:

$${\underbrace U_{{\rm{investor's \ risk \ utility}}} \buildrel \Delta \over = {\bf{w}}_M^T{\bf{\Pi }} - \frac{\delta }{2}{\bf{w}}_M^T{\bf{\Sigma w}}_M^{}}$$

A mentioned here, we can compute the risk aversion parameter by multiplying both sides of $${\bf{\Pi }} = \delta \times {\bf{\Sigma }} \times {\bf{w}}_M^{}$$ with $${{\bf{w}}_M^T}$$, to output the following relation:

$$\delta = \frac{{Sharp \ Ratio}}{{\sqrt {{\bf{w}}_M^T{\bf{\Sigma w}}_M^{}} }}$$

As we know

$$Sharp \ Ratio= \frac{{\bf{\Pi }}}{{\sqrt {{\bf{w}}_M^T{\bf{\Sigma w}}_M^{}} }} = \frac{{\mu _M^{} - {r_f}}}{{\sigma _M^{}}}$$

However, I do not know how we can reach from $${\bf{w}}_M^T{\bf{\Pi }} = \delta {\bf{w}}_M^T{\bf{\Sigma w}}_M^{}$$ to the relation above.

## Solving it algebraically:

As seen in the above provided reference (just above " 1) "), the general formulation for the unconstrained Markowitz portfolio optimization scheme, is given by:

\begin{align} &\text{arg}\max_{w} \; w^T\mu-\frac{\delta}{2} w^T\Sigma w.\\ \end{align} In absence of any constraints, the above optimization scheme have the closed-form solution:

$$w = \frac{1}{\delta} \Sigma^{-1}\mu$$.

Now, solving for the expected excess returns $$\mu$$ and we see something recognisable, $$\mu = \delta \Sigma w$$. In essence, if the weights equal the market weights, $$w=w_m$$, then the implied excess equilibrium returns equals the expected excess returns, $$\Pi = \mu$$ (this is also written here on p. 5).

### The derivations:

Now, let $$\Pi = \delta \Sigma w_m$$ be the implied excess equilibrium returns, then multiplying both sides with $$w^T_m$$ we get:

$$$$w^T_m \Pi = \delta w^T_m\Sigma w_m \qquad \iff \qquad \delta = \frac{w^T_m \Pi}{ w^T_m\Sigma w_m}$$$$

Since we are working with the market weights, it implies that $$\Pi = \mu$$ and thus we can do the following algebraic calculations:

\begin{align*} \delta &= \frac{w^T_m \Pi}{ w^T_m\Sigma w_m} \\ &= \frac{w^T_m \mu}{ w^T_m\Sigma w_m}\\ &= \frac{\frac{w^T_m \mu}{\sqrt{w^T_m\Sigma w_m}}}{ \sqrt{w^T_m\Sigma w_m}}\\ &=\frac{\text{Sharpe}}{\sigma_m}, \end{align*}

which is the same expression as stated in your reference. I hope this provide some help.

• Thank you for your answer. And I should add that in Mathworks's help, the value of risk-free rate is zero. Now everything seems clear, but I was wondering what is the relation between the expected market return $\mu_M$ in the CAPM model and implied return $\Pi$? Another question, what is the relation between the $\mu$ in your answer and $\mu_M$ here ${\rm{E}}[{R_i}] - {r_f} = \beta \left( {{{\bf{\mu }}_M} - {r_f}} \right)$?
– sci9
Jul 16, 2021 at 19:43
• Hi @sci9. I will check back on these questions later. However, in the mean time I encourage you to post these sub-questions in a separate post on Quant SE. This will give your questions more visibility and a higher chance for an answer. Wrt. your second question, then we could have denoted $w_m^T \mu = \mu_m$ since this is the expected excess return for the market portfolio (and $w^T\mu$ for a general portfolio). Related to the CAPM then $\mu_m$ is the same as $\mu_M - r_f$ (in the above equation).
– Pleb
Jul 16, 2021 at 20:25
• I think your comment answered my questions quite clearly. Merci beaucoup +1.
– sci9
Jul 16, 2021 at 20:41