# Black Litterman - Equilibrium Returns

(Searched for this specific question, no dice) I understand that the BL model (not UBL) is “backwards” solving to find the equilibrium returns for all assets/indices involved. It uses the covariance matrix and market weightings as equilibium weightings. These equilibrium returns cause the CML line to touch the market portfolio on the efficient frontier. This is how I picture it: MVO optimizes: Portfolio return § = weighting i * E(r of i) + weighting j * E(r of j) +…etc Relative to the risk in the portfolio standard deviation formula. So the textbook (CFAI and Schweser) tell us that it only uses covariance and weightings, but doesn’t it also need to use the target index/portfolio’s “overall” return? If you use just weights, variances and covariances, you omit “Portfolio return §”. Hence, doesn’t the method also require us to use the “market portfolio” return? Any help would be much appreciated!

The method calculates the equilibrium returns vector based on observed weights of asset classes and the covariance matrix ( also needs a risk aversion level , delta . The exact steps are not shown , and most probably not needed for the test. This delta is a portfolio risk premium δ = µP/ (σ P*σ P).Based on prices and returns in the immediate past , we can get an approximation of the delta. Then the expected returns vector = delta * sigma covariance matrix * weights vector. The sigma covariance matrix is an estimate of future covariances . Covariance is typically an unstable , noisy matrix and never easy to estimate So you see the ex-post result of the application of weights yielding a market return and a market risk ( which is based on current prices). first step of b-l assumes that the left side (Portfolio return § ) is the market ( i.e. benchmark) return at current prices , given weights of asset classes in the market and their covariances of prices and the current risk premium( i.e. delta) Under the circumstances , the market is the benchmark for us, and it is telling us that the expected asset class returns= current market returns, under the assumption of an efficient allocation . Now we can back-calculate what the market opinion of expected asset class returns should be under the assumption that future=present, i.e. ex-ante returns would be similar to realized returns.

I agree with janakisri. Mean-variance optimization gives w = (1/gamma)*B^-1*mu, where B is the covariance matrix, gamma - risk aversion coefficient, mu - vector of excess returns. Given B (which is estimated using historical numbers), estimating gamma and using actual market weights gives mu = gamma*B*w. At this point MV optimization would give market weights as optimal. that’s the first step of BL. Second step, introduce opinions and run MV optimization that incorporates those opinions to get final weights.

Thanks to both of you, that helped a lot.