lecture 7 blm

8/7/2019 Lecture 7 BLM

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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!

lecture 7 blm

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