The
Black-Litterman (B-L) model follows a complex portfolio composition approach
developed by Fischer Black and Robert Litterman in 1990 (Idzorek4, 2005). This
model uses the Bayesian approach to combine the views of an investor’s expected
returns from various assets and the market equilibrium’s expected returns,
generating a new and consolidated estimation of expected returns. By applying
the B-L asset allocation model, it can alleviate the three major problems in MV
models used above, “insensitive and highly-concentrated portfolio,
input-sensitivity, and error in estimated mean” (Idzorek4, 2005).
The B-L model
can mitigate the problem of input sensitivity via reverse optimization. In
other words, by creating consistent mean-variance efficient portfolios based on
a diversified market portfolio, reverse engineering the expected returns.
Furthermore, the model primarily improves the estimation error in maximization
by spreading the errors throughout the calculated expected returns. This is
extremely beneficial, since the expected return is the most important input for
mean-variance optimization (Lee5, 2000). This model also looks into a number of
alternative methods to forecast and generate extreme portfolios, which include
historical returns, risk adjusted equal mean returns, and non-risk adjusted
equal mean returns. Extreme portfolios are portfolios that have large long and
short positions when unconstrained, or they are portfolios concentrated with relatively
small amount of assets when subject to long constraint.
The B-L model
begins with an unbiased starting point, the equilibrium returns. The
equilibrium returns are derived from the following formula using reverse
optimization.
The risk
aversion coefficient (λ) scales the reverse optimization estimate of excess
returns based on risk, the greater the weighted reverse optimized excess
returns per unit of risk the higher the estimated excess returns.
By re-arranging
the first formula and substituting μ (any vector of excess return) for Π gives
the second formula, which alleviate the problem of an unconstrained
maximization: .
Using the second
formula, the optimum weights for the portfolios can be found based on the
expected excess return vectors calculated using the formulas abovementioned.
The results of the new recommended weights is then calculated to compare the
highest and lowest between weights based on the
different vectors.
The
Black-Litterman formula is then introduced; it is the new Combined Return
Vector (E(R)) as follows.
(Note: K represents the number of views and N represents the number of assets in the formula)
The next step is
to translate an investor’s views into input for the B-L formula; this can be
done through matrix. The variance of each individual view portfolio can be
calculated. The scalar (τ) is comparatively inversely proportional to relative
weight on Π, but process to determine the scalar (τ) is unclear. The calculated
scalar value (τ) is normally set from between 0.01 and 0.05. This would allow
the model to be calibrated and according to the target level of tracking error
(Idzorek4, 2005. Once the scalar (τ) and covariance matrix of error term (Ω) is
defined, this is inputted into the B-L formula to calculate the New Combined
Return Vector (E(R)). The newly calculated results should now show that the
weights of stocks are different from their original market capitalisation
weights. Hence, it is recommended to add investment constraints such as risk,
beta and short selling constraints to help alleviate the intuitiveness of the
B-L model and generating a well-informed estimation.
