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The Black-Litterman model
was published in the Financial Analyst's journal in 1992. It
provides an estimate of future expected returns by combining
equilibrium returns with an investor's views. The resulting new
vector of returns leads to potentially well diversified, intuitive,
and stable portfolios. Implied returns are computed by using the
following equation (reverse optimization):
with risk aversion coefficient (δ), asset covariance matrix
(∑), and
equal asset weights (w). Since the true expected returns
(µ) are
unknown, it is assumed that the equilibrium returns are reasonable
estimate of the true expected returns:
where:
ϵΠ
~ N (0,τ∑).
The matrix τ∑ corresponds to
the confidence in how well the implied returns can be estimated.
K views in the model can be expressed as
where:
P is a
K x N matrix, each row of which provides a combination
of assets for which we have a view.
q is a
K-dimensional vector of the
views about expected returns.
εq is a
K-dimensional vector of errors, ϵ ~ N(0,Ω).
Ω is a
K x K matrix expressing the confidence in the views.
Equations (2) and (3) can be stacked in the form
Where:
with
I denoting the
N x N identity matrix. Calculating
the Generalised Least Squares estimator for
µ results in the
following Black-Litterman combined expected returns:
AlternativeSoft's software platform uses the Black-Litterman
combined returns as the input to compute optimal portfolios on the
efficient frontier which will have: (i) low volatility
(Mean-Variance Optimization), or (ii) small average extreme loss
(Mean-Conditional VaR optimization), or (iii) low volatility, high
skewness, and low kurtosis (mean-Modified VaR optimization). As an
example of diversification achieved by using AlternativeSoft's
Black-Litterman based optimization, two figures are provided below.
Figure 1 exhibits the portfolio weights on the efficient frontier
obtained from classical mean-variance optimization. Portfolio
weights computed by using BL model based mean-variance optimization
are shown in Figure 2. Figure 2 displays that BL model based
optimization produces a portfolio with more hedge funds in the
portfolio leading to highly diversified optimized portfolio.
Figure 1: Distribution of weights based on classical MVO
Figure 2: Distribution of weights based on BL MVO
Read the full paper
here. |
