Portfolio optimization inputs

• Set of expected returns for assets

• Set of std deviations (variances) for assets

• Set of correlations (covariances) across assets

Three-asset example

Sensitivity to expected returns

Sensitivity to standard deviations

Sensitivity to correlations

The Error-Maximization Problem

Mean-variance portfolio optimization:

• Will tilt too heavily toward assets with estimated expected returns above true expected returns \((\hat{\mu}>\mu)\)  

• Will tilt too heavily toward assets with diversification benefits greater than true benefits \((\widehat{\text{cov}}_{ij}<\text{cov}_{ij})\)  

• May try to short assets with diversification benefits lower than true benefits \((\widehat{\text{cov}}_{ij}>\text{cov}_{ij})\)

Position limits

Short-selling constraints

  Prevent hedging positions due to overestimated covariances and/or underestimated \(E[r]\)

Maximum positions

  Prevent overweighting due to overestimated \(E[r]\) and/or underestimated covariances

Shrinkage

• Shrink extreme inputs toward some more moderate input
• Example: CAPM betas \[ \beta_{\text{adj}}= 0.67\cdot \hat{\beta} + 0.33\cdot 1 \]
• There are some fairly sophisticated shrinkage techniques for the covariance matrix.

Use models to infer expected returns

Black-Litterman

  Use market value weights to back out \(E[r_i]\)’s via CAPM
  Then add alphas to expected returns

Treynor-Black

  Consider benchmark index as an asset
  Use expected alphas to create an active portfolio
  Combine index and active portfolio optimally

Factor models

• Can be used to estimate both \(E[r]\)’s and correlations

• Market Model/CAPM: \[E[r_i]=r_f + \beta E[r_{\text{mkt}}-r_f]\] \[\text{cov}_{ij}=\beta_i\beta_j \sigma^2_{\text{mkt}}\]

• Can dramatically reduce the number of estimated parameters

• We will discuss (multi-)factor models beyond CAPM

Don’t even try to estimate some inputs!

Global minimum variance

  assume all \(E[r_i]\)’s equal

Risk parity

  assume all \(E[r_i]\)’s equal and all \(\rho_{ij}=0\)

Equal-weighted portfolio

  assume all \(E[r_i]\)’s, \(\text{sd}[r_i]\)’s equal; all \(\rho_{ij}=0\)

Historical Plug-in Estimators

Expected return

  use historical arithmetic average return

Standard deviation

  use historical standard deviation

Correlations

  use historical pair-wise correlation

Stocks, Bonds, and Gold

Let’s run a backtest of annual optimization of portfolios of the following asset classes:

• Stocks
• Treasury bonds
• Corporate bonds
• Gold

Strategy 1: Est-All

• use historical data to estimate expected returns, standard deviations, and correlations.
  
• optimal risky portfolio is the tangency portfolio
• scale tangency up or down depending on risk aversion or target expected return

Strategy 2: Est-SD-Corr

• use historical data to estimate standard deviations and correlations
• assume expected returns are the same across all assets.
  
• optimal risky portfolio is the global minimum variance portfolio.
• for the purposes of determining optimal capital allocation, use the cross-sectional average of the historical time-series average return as the expected return input.

Strategy 3: Est-SD

• use historical data to estimate standard deviations only
• assume correlations across assets are zero
• assume expected returns are the same across all assets
• for the purposes of determining optimal capital allocation, use the cross-sectional average of the historical time-series average return as the expected return input.

Strategy 4: Est-None

• do not use historical data to estimate expected returns, standard deviations, or correlations.
• the optimal portfolio is an equal-weighted portfolio of the assets (\(1/N\) portfolio).
• for the purposes of determining optimal capital allocation, use the cross-sectional average of the historical time-series average return as the expected return input.

To notebook #1

Industry Portfolios

Let’s return to our 48 industry portfolio example.

• Using full sample means, standard deviations, and correlations suggested that allowing short selling could improve mean-variance efficiency.

• Let’s consider how this would have fared in an out-of-sample context.

• We will use expanding windows to estimate inputs.

To notebook #2

Length of Estimation Window

• Use last \(T\) years to estimate inputs (rebalance each year)

• Consider windows of 10, 20, 30, 40, and 50 years

• Scenarios with more or less dispersion in true expected returns

Higher \(E[r]\) dispersion

Lower \(E[r]\) dispersion

Number of Assets

• 3, 5, or 10 assets

• Estimation window of 30 years

• Investment period of 50 years

• Theoretical Sharpe ratio of tangency portfolio is the same

Number of Assets

For next time: Market Model Regression