black-litterman asset allocation model

Black-Litterman Asset Allocation Model

QSS Final Project

Midas Group MembersBo Jiang, Tapas Panda, Jing Lin, Yuxin Zhang

Under the Guidance of Professor Campbell Harvey

April 27, 2005

Agenda

Part 1: Motivation and Intuition Part 2: Analytics Part 3: Numerical Example Part 4: BL in Practice Part 5: Test the Model Epilogue: 3 Recommendations

Part 5: Test the Model

The best way to test the model is…

Introspection…

Part 1: Motivation & Intuition

The Problems of Markowitz Optimization

Highly-concentrated portfolios Extreme portfolios

Input-sensitivity unstable Estimation error maximization

Unintuitive No way to incorporate investor’s view No way to incorporate confidence level No intuitive starting point for expected return. Complete set of expected return is required.

Black-Litterman Model

B-L model uses a Bayesian approach to combine the subjective views of an investor regarding the expected returns of one or more assets with the market equilibrium vector of expected returns (the prior distribution) to form a new mixed estimate of expected returns (the posterior distribution).

How does BLM work?

Start with the market returns using reverse optimization and CAPM.

Apply your own unique views of how certain markets are going to behave.

The end result includes both a set of expected returns of assets as well as the optimal portfolio weights.

Intuition of BLM If you do not have views, you hold the

market portfolio (the benchmark).

Your views will tilt the final weights away from the market portfolio, the degree to which depending on how confident you are about your views.

Road Map

Part 2: Analytics

Equilibrium Returns (1)

Equilibrium Return =current Market collective forecasts of next

period returns; i.e., the market’s collective view on future returns

=reverse optimized returns this Market View is to be combined with Our

View; and the combination (using GLS) will take the estimation error of either views into consideration.

Equilibrium Returns (2)

Assume Market has the following attributes N assets Expected Return vector μ[Nx1] Expected covariance Matrix ∑[NxN]

Equilibrium Returns (3)

Today when the trades took place, market collectively reached the equilibrium (supply = demand).

To do this it had ran the Markowitz mean-variance optimization and reached the optimized weights w[Nx1] – which are the current market capitalization weights

Equilibrium Returns (4)

Max [w’μ – (λ/2)w’∑w] Note: This is derived from the utility theory and

multivariate normal distribution – Financial Economics 101

λ = risk aversion coefficient (E(M) –rf)/σ(mkt)^2)E(M) = Expected market or benchmark total returnλ is found from historical data (approx = 3.07) Solve δw’μ /δw - δ((λ/2)w’∑w)/ δw = 0 They got μ = λ ∑w

Note: two most important matrix derivation formula δw’μ /δw = μ and δ(w’∑w)/ δw = 2∑w

Equilibrium Returns using Implied Beta

Equilibrium Returns can be calculated by using the “implied Beta” of assets. μ = β(implied)*(risk premium of market ) Implied β = ∑*w(mkt)/(w(mkt)T*∑w(mkt))

The denominator is basically the variance of market portfolio. The numerator is the covariance of the assets in the market portfolio. Asset weights are the equilibrium weights. Covariance matrix ∑ is historical covariance.

What is the estimation error of the Equilibrium Returns?

A controversial issue in BL model. Since the equilibrium returns are not actually

estimated, the estimation error cannot be directly derived.

But we do know that the estimation error of the means of returns σE[r(i,t+1)] should be less than the covariance of the returns.

A scalar τ less than 1 is used to scale down the covariance matrix (Σ) of the returns.

Some say that “τ =0.3 is plausible”.

Forming Our View (1)

Our view is: Q=Pu+η, μ~Φ(0,Ω) Note: same as Pu=Q+η, because η~Φ(0,Ω) -

η~Φ(0,Ω) u is the expected future returns (a NX1

vector of random variables). Ω is assumed to be diagonal (but is it

necessary?)

Forming Our View (2)

What does this Q=P*u+η, Or equivalently P*u=Q+ η mean?

Look at P*u:each row of P represents a set of weights on the N assets, in other words, each row is a portfolio of the N assets. (aka “view portfolio”)

u is the expected return vector of the N assets P*u means we are expressing our

views through k view portfolios.

Forming Our View (3)

Our Part 3 Numerical Example will show some examples of the process of expressing views.

The Goldman Sachs Enigma is how they express views quantitatively.

Forming Our View (4)

Why is expressing views so important?

Because the practical value of BL model lies in the View Expressing Scheme; the model itself is just a publicly available view combining engine. Our view is the source of alpha. Expressing views quantitatively means efficiently and

effectively translate fundamental analyses into Views

Forming Our View (5)

We will try to decode Goldman Sachs Enigma in Part 4 “Applications”.

Combining Views (1)

Generalized Least Square Estimator of μ

μComb

μComb

Combining Views (1)

Generalized Least Square Estimator of μ

μComb

μComb

Combining Views (2)

Var(μComb)

Now we have a combined forecast of the expected returns.

The next step is to do Markowitz Mean-Variance Optimization.

By using the combined forecasted means

and the forecasted covariance matrix ∑.

So we start with Markowitz (reverse optimization) and CAPM (implied beta).

Go though Black-Litterman View Combining engine.

And end up with Markowitz again with predictive means, (and forward looking return covariance matrix.)

Part 3: Numerical Example

An Eight Assets Example…

μHist is historical mean asset returns μp is calculated relative to the market cap. weighted portfolio

using implied betas and CAPM model. Market portfolio weights wmkt is based on market capitalization

for each of the assets

Asset Class µHist µP wmkt

US Bonds 3.15% 0.08% 19.34%Int’l Bonds 1.75% 0.67% 26.13%US Large Growth -6.39% 6.41% 12.09%US Large Value -2.86% 4.08% 12.09%US Small Growth -6.75% 7.43% 1.34%US Small Value -0.54% 3.70% 1.34%Int’l Dev. Equity -6.75% 4.80% 24.18%Int’l Emerg. Equity -5.26% 6.60% 3.49%

Weighted Average -1.97% 3.00%Standard Deviation 3.73% 2.53%

High 3.15% 7.43% 26.13%Low -6.75% 0.08% 1.34%

Market Returns П(nx1)

Market returns are derived from known information using Reverse Optimization:

П = ג ∑wmkt П (nx1) is the excess return over the risk free

rate is the risk aversion coefficient ג ∑(nxn) is the covariance matrix of excess returns Wmkt (nx1) is the market capitalization weight of

the assets

Risk Aversion Coefficient ג More return is required for more risk

σ2=Risk Premium/Variance/( E (r) – rf)=ג

Using historical risk premium and variance, we got a ג of aprrpoximately 3.07

Coviriance Matrix ∑

Coviriance Matrix ∑(nxn)

Asset Class 1 2 3 4 5 6 7 81. US Bonds 0.001005 0.001328 -0.000579 -0.000675 0.000121 0.000128 -0.000445 -0.0004372. Intl Bonds 0.001328 0.007277 -0.001307 -0.00061 -0.002237 -0.000989 0.001442 -0.0015353. US Large Growth -0.000579 -0.001307 0.059852 0.027588 0.063497 0.023036 0.032967 0.0480394. US Large Value -0.000675 -0.000610 0.027588 0.029609 0.026572 0.021465 0.020697 0.0298545. US Small Growth 0.000121 -0.002237 0.063497 0.026572 0.102488 0.042744 0.039943 0.0659946. US Small Value 0.000128 -0.000989 0.023036 0.021465 0.042744 0.032056 0.019881 0.0322357. Int'l Dev. Equity -0.000445 0.001442 0.032967 0.020697 0.039943 0.019881 0.028355 0.0350648. Int'l Emerg.Equity -0.000437 -0.001535 0.048039 0.029854 0.065994 0.032235 0.035064 0.079958

Market Returns П(nx1)

Π = λΣwmkt

Asset Class µHist µP ΠUS Bonds 3.15% 0.08% 0.08%Int’l Bonds 1.75% 0.67% 0.67%US Large Growth -6.39% 6.41% 6.41%US Large Value -2.86% 4.08% 4.08%US Small Growth -6.75% 7.43% 7.43%US Small Value -0.54% 3.70% 3.70%Int’l Dev. Equity -6.75% 4.80% 4.80%Int’l Emerg. Equity -5.26% 6.60% 6.60%

Weighted Average -1.97% 3.00% 3.00%Standard Deviation 3.73% 2.53% 2.53%

High 3.15% 7.43% 7.43%Low -6.75% 0.08% 0.08%

The Black – Litterman Model

The Black – Litterman Formula

• E[R] (nx1) is the new Combined Return Vector

• τ is a scalar

• ∑ (nxn) is the covariance matrix of excess returns

• P (kxn) is the view matrix with k views and n assets

• Ω (kxk) is a diagonal covariance matrix of error terms from the expressed views

• Π (nx1) is the implied market return vector

• Q (kx1) is the view vector

What is a view?

Opinion: International Developed Equity will be doing well

Absolute view: View 1: International Developed Equity will have an

absolute excess return of 5.25% (Confidence of view = 25%)

Relative view: View 2: International Bonds will outperform US bonds by

25 bp (Confidence of view = 50%) View 3: US Large Growth and US Small Growth will

outperform US Large Value and US Small Value by 2% (Confidence of View = 65%)

What Is The View Vector Q Like?

Unless a clairvoyant investor is 100% confident in the views, the error term ε is a positive or negative value other than 0

The error term vector does not enter the Black – Litterman formula; instead, the variance of each error term (ω) does.

5.25% ε1

Q + ε = 0.25% + ε2

2.00% ε3

What Is The View Matrix P Like?

View 1 is represented by row 1. The absolute view results in the sum of row equal to 1

View 2 & 3 are represented by row 2 & 3. Relative views results in the sum of rows equal to 0

The weights in view 3 are based on relative market cap. weights, with outperforming assets receiving positive weights and underperforming assets receiving negative weights

US Bonds Intl Bonds US Lg Growth US Lg Value US Sml Growth US Sml Value Int'l Dev. Eqt Int'l Emerg.Eqt0 0 0 0 0 0 1 0

P = -1 1 0 0 0 0 0 00 0 0.9 -0.9 0.1 -0.1 0 0

Finally, The Covariance Matrix Of The Error Term Ω

Ω is a diagonal covariance matrix with 0’s in all of the off-diagonal positions, because the model assumes that the views are independent of each other

This essentially makes ω the variance (uncertainty) of views

0.0007089 0 0Ω = 0 0.000141 0

0 0 0.000866

Go Back to B-L Formula…

First bracket “[ ]” (role of “Denominator“) : Normalisation Second bracket “[ ]” (role of “Numerator“) : Balance between

returns Π (equilibrium returns) and Q (Views). Covariance (τ Σ)-1 and confidence P’ Ω-1P serve as weighting factors, and

P’ Ω-1Q = P’ Ω-1P P-1 Q Extreme case 1: no estimates ⇔ P=0: E(R) = Π i.e. BL-returns

= equilibrium returns. Extreme case 2: no estimation errors ⇔ Ω -1→ ∞: E(R) = P -1Q

i.e. BL-returns = View returns.

Return Vector & Resulting Portfolio Weights

Π = λΣwmkt

w =(λΣ) -

1

E[R]

w =(λΣ) -1E[R]

w =(λΣ) -1Π

Asset Class E[R] Π E[R]-Π norm wmkt

US Bonds 0.07% 0.08% -0.02% 29.88% 28.83% 19.34% 10.54%Int’l Bonds 0.50% 0.67% -0.17% 15.59% 15.04% 26.13% -10.54%US Large Growth 6.50% 6.41% 0.08% 9.35% 9.02% 12.09% -2.73%US Large Value 4.32% 4.08% 0.24% 14.82% 14.30% 12.09% 2.73%US Small Growth 7.59% 7.43% 0.16% 1.04% 1.00% 1.34% -0.30%US Small Value 3.94% 3.70% 0.23% 1.65% 1.59% 1.34% 0.30%Int’l Dev. Equity 4.93% 4.80% 0.13% 27.81% 26.84% 24.18% 3.63%Int’l Emerg. Equity 6.84% 6.60% 0.24% 3.49% 3.37% 3.49% 0.00%

Sum 103.63% 100% 100% 3.63%

Combined Return E[R] vs. Equil. Return Π

0%

2%

4%

6%

8%

US Bonds Int’l Bonds US LargeGrowth

US Large Value US SmallGrowth

US Small Value Int’l Dev. Equity Int’l Emerg.Equity

Π E[R]

Resulting Asset Allocations Changed A Lot…

0%

5%

10%

15%

20%

25%

30%

35%

US Bonds Int’l Bonds US LargeGrowth

US Large Value US SmallGrowth

US Small Value Int’l Dev.Equity

Int’l Emerg.Equity

Market Cap. Weight New Weight

View 1 – Bullish view on Int’l Dev. Equity increases allocation.

View 3 – Growth tilt towards value

View 2: Int’l bonds will outperform US bonds less than market implied.

Part 4: BL in Practice

Applications Just now we presented unconstrained optimization.

Of course constraints can be added to the optimizer.

Also, the market portfolio can be replaced with any benchmark portfolio, and the Mean-Variance objective function can be replaced by any other risk models (maximize risk adjusted returns.)

Littleman, “The real power of the BL model arises when there is a benchmark, a risk or beta target, or other constraints, or when transaction costs are taken into consideration. In these more complex contexts, the optimal weights are no long obvious or intuitive”.

Wai Lee, “The model can be used to combine different models or signals, ”such as valuation model and technical analysis.

BL Limitation

What we presented is still in the mean-variance optimization framework, which cannot deal with higher moments.

For ideas of handling both estimation error and higher moments, see “Portfolio Selection With Higher Moments: A Bayesian Decision Theoretic Approach”, by our professor Campbell Harvey.

Attempt to decode GSQE (1)

Return generating model is the source of alpha.

Ideally, views and their estimation error should be generated quantitatively.

That’s what Goldman Sachs Quantitative Equity does.

How the heck do they actually do it?

Attempt to decode GSQE (2)

Credit Swisse’ sort of confirmed our decoding of GSQE. Previously we thought there was 30% chance that we have decoded GSQE; now we are 80% sure.

The two companies are doing virtually the same thing in terms of generating views quantitatively.

Attempt to decode GSQE (3)

Ri,t+1 =f(z1,z2,z3,z4,z5,z6), z is firm attributes. The factor loading is just partial derivative. Credit Swisse uses long-short to get this

partial derivative (5 long-short portfolios) Goldman Sachs has another scheme to do it:

a special kind of Characteristic Portfolio (6 view portfolios).

Whatever, the essence is still to get the partial derivative for each factor.

Epilogue: 3 Recommendations

1st: To Our Professor and Fuqua

Fuqua Course 999: “Quantitative Beauty Selection”

By the way, this is your long list.

Mahalanobis Distance

2nd: To Corporate America

Quantitative Employee Selection

Better Get Rid Of Cover Letters and Interviews !!

unless …

3rd: To Ourselves

You can quantify pretty much everything in the pragmatic world, but

Do not ‘calculate’: Compassion

Friendship and Love Aesthetic Value

Intellectual Curiosity Respect for Individuality