Black-Litterman: Method

Black-Litterman: Method

What it does

Mean-variance optimisation needs an expected return for every asset, and small changes in those estimates swing the weights wildly. Fischer Black and Robert Litterman (1992) proposed a different starting point: take the returns that would make the market portfolio the best one to hold, and move away from them only where the investor has a view and only as far as the view is trusted. The page works that calculation step by step: the implied equilibrium returns, the views written as matrices, the uncertainty of each view, the posterior returns that blend the two, and the weights that follow, shown beside the market weights.

The method

Inputs. For each asset i: its market weight w_i (its share of the total market value of the assets listed) and its volatility σ_i; for each pair, the correlation ρ_ij. The covariance is σ_ij = ρ_ij σ_i σ_j and Σ is the matrix of covariances. Returns are in percent per year; with a risk-free rate r_f of zero they are returns above the risk-free rate. The correlation matrix is entered as on the Efficient Frontier page.

Risk aversion. δ is the market portfolio's expected return above the risk-free rate divided by its variance σ_m² = w′Σw (both as decimals). Entered directly, it implies a market premium δ σ_m² and a market Sharpe ratio δ σ_m; entered as a Sharpe ratio S_m (Advanced), δ = S_m / σ_m.

Implied equilibrium returns. An investor who maximises w′μ − (δ / 2) w′Σw holds w = (δΣ)⁻¹μ. Reversing this for the market weights gives the returns under which the market portfolio is that investor's choice:

Π = δ Σ w,   that is, Π_i = δ c_i with c_i = Σ_j σ_ij w_j the covariance of asset i with the market portfolio.

Views. Each of the K views is a row p of the matrix P and an entry q of the vector Q: the view says that the combination of assets p has an expected return q. "Bonds" is a row with a 1 for Bonds (an absolute view); "Stocks − Bonds" has +1 and −1 (a relative view: Stocks will outperform Bonds by q); a number before a name is a share, so a view can be about a mix. For an absolute view the risk-free rate is subtracted from q, because the formulas work in returns above it.

Uncertainty. The implied returns are themselves uncertain, with covariance τΣ for a small scalar τ. Each view has an error variance ω_k, the k-th diagonal entry of Ω. With s_k = p_k Σ p_k′ the variance of the view's combination:

proportional form (He and Litterman, 1999): ω_k = τ s_k;   confidence form (after Idzorek, 2005): ω_k = τ s_k (1 − C_k) / C_k, with C_k the stated confidence as a fraction.

The confidence form is built so that a view held alone moves the weights the fraction C of the way from the market weights to the weights of full confidence. Idzorek finds each ω by a numerical search for that target; for one view the search has this exact answer. At C = 50 % the two forms agree, at C = 100 % ω is zero and the view holds exactly, and at C = 0 the view is left out. In both forms τ cancels out of the posterior returns.

Posterior returns (the master formula, in its two equivalent forms):

μ̄ = [(τΣ)⁻¹ + P′Ω⁻¹P]⁻¹ [(τΣ)⁻¹Π + P′Ω⁻¹Q]   =   Π + τΣP′ (PτΣP′ + Ω)⁻¹ (Q − PΠ).

With one view everything is a number. Let m = pΠ be the market-implied return of the view and v = τ s the uncertainty of that figure. Then the view gets the weight k = v / (v + ω), the posterior return of the view's combination is m + k (q − m), and each asset moves in proportion to its covariance with the view: μ̄_i = Π_i + b_i (q − m) / (v + ω), with b_i = τ (Σp′)_i.

Posterior covariance. The uncertainty of the posterior returns is M = [(τΣ)⁻¹ + P′Ω⁻¹P]⁻¹ = τΣ − τΣP′ (PτΣP′ + Ω)⁻¹ PτΣ, smaller than τΣ wherever a view speaks. The covariance of the asset returns around μ̄ is Σ + M.

Weights. w* = (δΣ)⁻¹ μ̄ = w + P′Λ, with Λ = τ (PτΣP′ + Ω)⁻¹ (Q − PΠ) / δ: the market portfolio plus a position in each view's combination. The page uses Σ here, as the two cited papers do, so that with no views w* is the market portfolio exactly; some authors use Σ + M, which scales the weights down slightly. The weights need not add to 100 %: an absolute view adds a position financed at the risk-free rate. In Advanced mode the page also finds the weights that maximise w′μ̄ − (δ / 2) w′Σw with every weight at zero or more and a sum of 100 %, by the quadratic programme of the Efficient Frontier page.

Confidence table. The calculation is repeated with every view at one common confidence of 0, 10, 25, 50, 75, 90 and 100 %.

What it assumes

The market weights entered are those of a portfolio that investors hold by choice, and one risk aversion describes them; returns are judged by mean and variance over one period; volatilities and correlations are known; the errors of the views are independent of one another and of the market; the views are linear statements about expected returns. The implied returns are not forecasts: they are the entered covariances and market weights restated as returns.

When it misleads

The implied returns can be no better than the "market portfolio" typed in. A list of a few funds is not the market, and their weights in one investor's account are not market weights; the returns then describe that list, not an equilibrium. The result also inherits every error in the volatilities and correlations, which change over time and tend to rise together in falling markets. The confidence of a view is a judgement with no objective scale: the two forms of ω on this page are conventions, and other conventions give other weights. τ has no agreed value; here it cancels out of the returns but still sets the size of the posterior covariance. Views that overlap are treated as independent pieces of information, so repeating a view counts it twice. The model does not test whether a view is right: a confident wrong view produces a confident wrong portfolio, and the weights describe the inputs rather than what will happen.

Worked example

Two assets, one view. Stocks: market weight 60 %, volatility 16 %. Bonds: 40 %, 6 %. Correlation 0.2, δ = 3, τ = 0.05, r_f = 0. View: Stocks will outperform Bonds by 3 %, proportional form.

Covariances (percent squared): σ₁₁ = 256, σ₂₂ = 36, σ₁₂ = 0.2 × 16 × 6 = 19.2. Covariance with the market: c₁ = 256 × 0.6 + 19.2 × 0.4 = 161.28; c₂ = 19.2 × 0.6 + 36 × 0.4 = 25.92. Implied returns: Π₁ = 3 × 161.28 / 100 = 4.84 %; Π₂ = 3 × 25.92 / 100 = 0.78 %.

The market already implies that Stocks outperform Bonds by m = 4.8384 − 0.7776 = 4.06 %; the view says 3 %. Variance of the view's combination: s = 256 + 36 − 2 × 19.2 = 253.6; v = 0.05 × 253.6 = 12.68; ω = 12.68; k = 12.68 / 25.36 = 0.5. Posterior outperformance: 4.0608 + 0.5 × (3 − 4.0608) = 3.53 %.

Each asset: b₁ = 0.05 × (256 − 19.2) = 11.84 and b₂ = 0.05 × (19.2 − 36) = −0.84, so μ̄₁ = 4.8384 + 11.84 × (−1.0608) / 25.36 = 4.34 % and μ̄₂ = 0.7776 + (−0.84) × (−1.0608) / 25.36 = 0.81 %. Their difference is the 3.53 % above.

Weights: Λ = 0.05 × (−1.0608) / (3 × 25.36) × 100 = −6.97 %, so Stocks go from 60 % to 53.03 % and Bonds from 40 % to 46.97 %. The view was less optimistic about Stocks than the market, so the portfolio holds fewer of them.

The default case is the example of Idzorek (2005): eight asset classes, δ = 3.0658 (a 3 % market premium divided by the market variance), τ = 0.025 and three views. The volatilities and correlations on the form are computed from the covariance matrix printed in that paper. Market volatility σ_m = √97.85 = 9.89 %, so the premium is 3.0658 × 97.85 / 100 = 3.00 % and the Sharpe ratio 0.303. For US Bonds c₁ = 2.7022 and Π₁ = 3.0658 × 2.7022 / 100 = 0.08 %.

View 1 (Intl Dev Equity will return 5.25 %) alone: m = 4.80 %, s = 283.55, v = ω = 0.025 × 283.55 = 7.089, k = 0.5, posterior 4.801 + 0.5 × (5.25 − 4.801) = 5.03 %. With the three views together it is 4.93 %, because the views overlap through the correlations. The posterior returns are 0.07, 0.50, 6.50, 4.32, 7.59, 3.94, 4.93 and 6.84 % and the weights 29.89, 15.58, 9.36, 14.82, 1.04, 1.64, 27.81 and 3.49 %, adding to 103.63 %; the paper's table prints the same returns and the weights within 0.01 point. Intl Emerg Equity, named in no view, keeps its market weight.

This page is a calculator for education and analysis, not investment advice.