I've been trying to work through a simple example using A Step-by-Step Guide to the Black Litterman Model, but I'm having trouble understanding implied risk aversion.

Say I have two uncorrelated assets, cash as my stand in for the risk free rate and my market capitalization weight happens to conveniently be equal weighted.

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According to the paper my implied risk aversion $\lambda$ should be the portfolio excess return (5%) over the portfolio variance (0.0066)


which gives $\lambda = 0.05 / 0.0066 \approx 7.6$.

However, when I try check my work by calculating the equilibrium excess returns

equilibrium excess returns

I get an equity excess return (again assuming uncorrelated assets) of $$ r_e = \lambda * \sigma_e^2 * w_e = 9.7 \% $$

rather than the expected $7\%$.

Is there a reason the implied risk aversion does give implied equity excess returns? Something to do with requiring fully funded portfolios maybe? Though that doesn't seem to tie out either.


It's because the model assumes that the market will maximize its Sharpe ratio and your weights don't do that. Essentially, your example assumes investors are irrational in their allocation. If you solve for the weights that maximize the Sharpe ratio, the implied returns will equal the given returns.

In your example, the Sharpe Ratio reaches a maximum value of 1.091516 when weights of 7.58% and 92.42% are given to equity and bonds, respectively. This implies a λ of 36.06771. Taking 36.06771*[16%^2,3%^2]*[7.58%,92.42%] gives [7%,3%] implied returns, which match the original implied returns.

The risk aversion you solved for is indeed the aversion implied by a 50/50 weighting, but the market participants would be better off using the weights that maximize the Sharpe Ratio and then allocating more or less of their assets to the portfolio vs cash. In the ideal world of the BL framework, market participants can borrow at the risk free rate, so someone with a lower risk aversion than that of the (Sharpe optimized) market portfolio would simply take a leveraged position in the market portfolio.

  • $\begingroup$ I'm confused. Maximizing the Sharpe ratio would be equivalent to having $\lambda=1$ in this problem and then solving for the weights. I thought the whole point of finding $\lambda$ would be you find a risk aversion that gives you the 50/50 breakdown. Can you show how your solution solves the problem round trip? $\endgroup$ – rhaskett Mar 6 at 0:18
  • $\begingroup$ I've edited my solution to address your questions. $\endgroup$ – Kalev Maricq Mar 6 at 1:00
  • $\begingroup$ Right. So I was stuck on the implicit assumption of Sharpe optimization. Turns out with a bit of trouble this can be solved in the fully-funded/no-leverage case as well. Adding that constraint, the risk aversion can be solved numerically for the 50/50 portfolio to give $\lambda = 3.2$ and then the constrained equilibrium return (second link in the OP) can be used to round trip back to [7%, 3%]. $\endgroup$ – rhaskett Mar 7 at 19:03

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