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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.


Reference: https://www.mathworks.com/help/finance/black-litterman-portfolio-optimization.html

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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:

\begin{equation} 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} \end{equation}

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.

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  • $\begingroup$ 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 ${\rm{E}}[{R_i}] - {r_f} = \beta \left( {{{\bf{\mu }}_M} - {r_f}} \right)$? $\endgroup$
    – sci9
    Jul 16, 2021 at 19:43
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    $\begingroup$ 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). $\endgroup$
    – Pleb
    Jul 16, 2021 at 20:25
  • $\begingroup$ I think your comment answered my questions quite clearly. Merci beaucoup +1. $\endgroup$
    – sci9
    Jul 16, 2021 at 20:41

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