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Lecture 7:
Revisiting The Black-Litterman Model
February 2009
Dr. Hao Jiang
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I. Black-Litterman ModelStep 1: Goldman uses weighted daily data to estimate volatilities and
covariance, where the weights depend on the horizon.These weights in conjunction with a global CAPM yield
equilibrium returns.
Step 2: Perceptions of relative value, momentum etc. yield views on
rofitable deviations from e uilibrium ex ected returns. Attach
weights to views and combine with CAPM to yield expected
returns.
Step 3: Impose desired target risk/beta level relative to the benchmark.
ep : mpose oun s on es re re a ve exposure o e mar e .
Step 5: Solve constrained optimization to find MVE portfolio.
Step 6: Check solution to see if it is suitably diversified.
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I. Black-Litterman ModelNote:
If the manager holds no views, he will hold the equilibrium/marketportfolio.
If his views have high variance (low certainty), he will hold close to
the equilibirum portfolio.
,
away from the market portfolio.
This method is useful in that it tells you how to optimally
incorporate your information/views to tilt your portfolio, takingadvanta e of the correlation structure to hed e lar e ositions.
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Mathematical RepresentationThere are N risky assets, with expected returns , which is an N1
v v - v , ,
We assume the investor knows V but not .
The investor believes that is normally distributed with the mean
vector and the covariance matrix .
The investor has personal views with which she updates the expected
returns. The investors view is given by
P =Q+,
where is normally distributed with means of zeros and variance
matrix of.
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The updated expected return vector is now:
E(|views)= + PT[PPT+ ] -1 [Q-P ].
A simplifying case is the investor is extremely confident in her views
so that =0, then the updated conditional expected return becomes:
E(|views)= + PT[PPT] -1 [Q-P ].
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A Seven-Country Equity Allocation Problem
Portfolio allocation using historical volatilities and assuming
- Equal returns=7% per year [purple bars]
- View: German equity will outperform European equities by 5%
per year. Translated into: Germany (+2.5%) and France and UK
(-2.5%) [red bars]
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CAPM-Based Estimates (assuming market risk premium=7.15%)
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How to back out equilibrium returns using the CAPM, given the
market weights?
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Suppose we impose the view:
E(rGer)=market cap weighted French and UK returns + 5%
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The optimal allocations are:
The changes in weights for Germany, France and UK are
expected, but why do the other weights change?
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The Black-Litterman Model
Using these expected returns (internalizing the correlation
structure), the optimal weights are then:
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The Black-Litterman Model
Generally, the optimal portfolio equals the (CAPM) market
portfolio plus a weighted sum of the portfolios about which the
nves or as v ews.
An unconstrained investor will invest first in the market
portfolio, then in the portfolios about which views are expressed.
The investor will never deviate from the market weights on
assets about which no views are held.
-- This means that an investor does not need to holdviews about each and ever asset!