# Black and Litterman

# Introduction

- Black-Litterman theory and steps of implementation
- Implementation of Black and Litterman (BL) model for global equity allocation

# Black - Litterman steps

**Notation** (alphabetic order):

- \(\delta =\) risk aversion parameter, represents average risk tolerance of the world
- \(\epsilon =\) error term to reflect uncertainty
- \(K =\) number of investor’s views
- \(\mu =\) expected returns
- \(\bar{\mu} =\) BL expected returns
- \(N =\) number of assets
- \(\Omega =\) diagonal covariance matrix of error terms from investor’s view
- \(\pmb{1} =\) \(N \times 1\)matrix of 1
- \(P = K \times N\) matrix, the rows are portfolio weights to reflect investor’s view
- \(\Pi =\) equilibrium risk premiums
- \(Q = K\) vector of the expected returns on the \(P\) portfolios that reflect investor’s view
- \(r =\) vector of asset returns
- \(\Sigma =\) covariance matrix
- \(\bar{\Sigma} =\) BL covariance matrix
- \(\tau =\) a scalar indicating the uncertainty of the CAPM prior
- \(w =\) weight of optimal portfolio from mean variance optimization
- \(w_{eq} =\) weight at equilibrium, i.e. asset’s weight in market portfolio

**Steps:**

- Assume that \(r \sim N(\mu, \Sigma)\)
- Calculate CAPM equilibrium risk premium for prior belief.
- Input: \(w_{eq}, \delta, \Sigma\)
- Process: \(\Pi = \delta \Sigma w_{eq}\)
- Output: \(\Pi\)

- Using Bayesian approach, find the expected returns and returns distribution. Use the CAPM prior and additional investor’s views.
- Input: \(\Pi, \Sigma, \tau, P, Q\)
- Process:
- Bayesian prior: \(\mu = \Pi + \epsilon^{(e)}\)
- Where \(\epsilon^{(e)} \sim N(0, \tau \Sigma)\)

- Investor’s view: \(P \mu = Q + \epsilon^{(v)}\)
- Where \(\epsilon^{(v)} \sim N(0, \Omega)\)
- \(\Omega = P \Sigma P' \tau\)

- Distribution of
**Expected return**- \(\mu \sim N \Big(\bar{\mu}, \bar{M}^{-1} \Big)\)
- \(\bar{\mu} = [(\tau \Sigma )^{-1} + P' \Omega^{-1} P ]^{-1} [(\tau \Sigma )^{-1} \Pi + P' \Omega^{-1} Q ]\)
- \(\bar{M}^{-1} = [(\tau \Sigma )^{-1} + P' \Omega^{-1} P ]^{-1}\)

- Distribution of
**return**- \(r \sim N \Big(\bar{\mu}, \bar{\Sigma} \Big)\)
- where \(\bar{\Sigma} = \Sigma + \bar{M}^{-1}\)

- In case of no additional investor’s view:
- P and Q are zero, hence: \(r \sim N \Big(\bar{\mu} = \Pi, \bar{\Sigma} = (1+ \tau) \Sigma \Big)\)

- Bayesian prior: \(\mu = \Pi + \epsilon^{(e)}\)
- Output: distribution of return, posterior, \(r \sim N \Big(\bar{\mu}, \bar{\Sigma} \Big)\)

- Optimize the allocation based on the posterior estimates using the standard mean-variance optimization method
- Input: Distribution of (posterior)
**return**= \(r \sim N \Big(\bar{\mu}, \bar{\Sigma} \Big)\) - Process:
- min \(w' \bar{\Sigma} w\)
- subject to :
- \(w' \bar{\mu} = constant\)
- \(w' \pmb{1} = 1\)
- \(w_{n} \geq 0\) for \(n = 1,2,...,N\)

- Output: \(w\)

- Input: Distribution of (posterior)

# Library

# Step 1

Download some country’s (Australia, Canada, France, Germany, Japan, UK, USA):

- Market capitalization weight. Source: World Federation of Exchanges database, accessed from data.worldbank.org.
- Market capitalization for UK
- Equity index: use iShares MSCI for USD currency from Yahoo Finance.
- Risk free rate: Prof. French’s website

Correlation and covariance matrix.

# Step 2

Calculate \(\Pi = \delta \Sigma w_{eq}\). To be used as neutral reference (prior belief) of expected return.

# Step 3

Calculate \(\bar{\mu}\) and \(\bar{\Sigma}\) for:

- 1 relative view: long position in German equity and short position in UK has 5% (annual) return.
- 1 absolute view: US equity market will have (annual) return of 10%.
- 1 relative view and 1 absolute view

P is \(K \times N\) matrix. The matrix for each case is:

Calculate posterior.

# Step 4

Comparing optimization for different scenario:

- historical average,
- prior,
- posterior allocation:
- 1 relative view
- 1 absolute view
- 1 relative view and 1 absolute view

## Use Quadratic programming

No short-sale constraint

With short-sale constraint

## How the belief change from prior to posterior

Simulate data and plot distribution

# Reference and further reading

- Black and Litterman, Global Portfolio Optimization, 1992
- Cheung, The Black–Litterman model explained, 2010
- He and Litterman, The Intuition Behind Black-Litterman Model Portfolios, 2002
- Idzorek, A Step-By-Step Guide to the Black-Litterman Model Incorporating User-specified Confidence Levels, 2005
- Satchell and Scowcroft, A demystification of the Black–Litterman model: Managing quantitative and traditional portfolio construction, 2000