# 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: $$w$$

# 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 