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: \(r \sim N \Big(\bar{\mu} = \Pi, \bar{\Sigma} = (1+ \tau) \Sigma \Big)\)
    • 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: \(w\)

Library

Step 1

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

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