How can solve black litterman under the negative expected return?

Usually, for solving reserve optimization about implied return, we use lambda. = portfolio expected return/ variance

If portfolio expected return is negative (<0), how could i make positive implied return?

If you are asking for exam prep purposes, the CFA exam writers typically will give you lambda in the question and you won’t need to calculate it yourself. They may bury it somewhere in the question text but they will give it to you. Just use their given lambda in such case and solve with that. But chances are they won’t ask you to solve it mathematically.

Black Litterman related questions on CFA Level 3 typically should be qualitative as opposed to quantitative. They will describe some process the PM is using and you will need to identify it as Black Litterman. Or they will ask you to list the shortcomings of traditional MVO that can be solved by Black Litterman etc.

Thank you. However i want to know negative lambda for real asset allocation process.

If you only had this portfolio and your expected return model calculated that expected return to decrease below the risk free rate, there would be no excess return for the numerator of Lambda. Lambda would be negative in this case. If the probability of seeing a negative Lambda is low in the eyes of the person doing the reverse optimization process, often times they can set a restriction where Lambda must be greater than or equal to zero. So if it’s a negative number you’d plug zero in for Lambda when doing the reverse optimization part of Black Litterman.

The key is that the numerator of Lamda is E(r) - rf … not just E(r)

For example, Investor will invest to S&P stocks. First, he has to gain implied return, however, S&P had minus return last year. And, Since covid 19 is spreading, expected return of S&P is minis too. In this case, no investing is best. But if Black-Litterman is launched in this case, would be processed allocation in normal?

In this case you would need to understand the time frame and all the S&P component returns and their covariances with each other. Because S&P the index is expected to be minus this year doesn’t mean there are not stocks in the S&P index that will go up that you can invest in. Each micro variant would be modeled for determining possible excess return. Also if the fund is allowed to short stocks, this can generate positive investment returns for the manager and this should be modeled.

Take hypothetical example, however, where it’s a scenario where Lambda would be zero or negative in all variants. Lambda is part of the reverse optimization process where you calculate excess returns. It is only one part of Black Litterman. In this case, if there are no excess returns possible, Lambda would either be the negative number you calculate based on your models, or a zero “plug” number because you’ve restricted Lambda in your model to be greater to or equal to zero.

Lambda is used to calculate the excess market returns over the risk free rate in the reverse optimization step of Black Litterman. If there are no excess returns, then there are no excess returns generated by reverse optimization. Combined then with the portfolio manager’s own market views, in this type of situation the result where there are no possible positive investments to make can be that either the portfolio manager decides to pull out of the market or they can form a view to short the market if their fund strategy allows them to do that.

If your fund has a shorting constraint (not allowed to short), then Black Litterman in your extreme scenario where there are no positive investments to make, should result in zero weights for assets. This is the difference between constrained Black Litterman (weights can be zero but no shorting) and unconstrained Black Litterman (weights can be negative and shorting is allowed).

portfolio expected return : -1.55%, Standard deviation = 9.53%
risk-free : 1.2%. (no short selling)

lambda : 0% ( Since excess return in zero.)

0.025720323 0.035929821 -0.000921306 0.012405324 0.003864637 0.005790221 0.000340486
0.035929821 0.070609431 -0.001050267 0.024126416 0.006254066 0.010626876 -0.000563475
-0.000921306 -0.001050267 0.002523097 0.005505919 0.002294463 0.002219539 0.000183734
0.012405324 0.024126416 0.005505919 0.027574445 0.008903208 0.011466051 0.00037154
0.003864637 0.006254066 0.002294463 0.008903208 0.003956863 0.003510804 0.000291947
0.005790221 0.010626876 0.002219539 0.011466051 0.003510804 0.006541398 3.17272E-05
0.000340486 -0.000563475 0.000183734 0.00037154 0.000291947 3.17272E-05 0.000190868
wmkt cov*wmkt Implied return
Equity 11.9% 1.0664% 1.2000%
PE 7.3% 1.8740% 1.2000%
Fixed income 6.6% 0.2147% 1.2000%
Private fixed income 24.4% 1.4358% 1.2000%
Hedge fund 4.4% 0.4554% 1.2000%
Real estate 31.5% 0.006616922 1.2000%
opportunity 14.0% 0.000151842 1.2000%

Is this implied return right?

Implied return = lambda*covariance * Wmkt + Rf

For Black Litterman you are mainly concerned with the implied excess return for your portfolio (amount in excess of the risk free rate). You use that in your reverse optimization calculations etc.

In your example above you are instead using implied total return. You would not use that in a Black Litterman reverse optimization step. But for the sake of discussion, say you wanted to side calculate the implied total return using lambdas of zero, and not implied excess return. What your results above are saying is your portfolio will have a zero allocation to risky assets and an entire portfolio allocation to risk free instruments. So your expected total return is the risk free rate from being out of the market and in treasuries etc.

But this is only if you are not allowed to short. If you can short, then unconstrained Black Litterman will calculate optimal shorting allocations based on your return allocations and market views. And if you can short, again, you are chiefly concerned with the implied excess return of your shorting allocations, over the risk free rate, for Black Litterman’s reverse optimization process.

Again, this is not a CFA exam testable question. If you would like to learn more about Black Litterman for your own purposes outside the CFA Program, here is a good paper, cheers: