w1=-20.6%, w2=1.7%, w3=96.5%Mean-Variance Frontier
Kerry Back
Objective
For any target expected return, find the least-risk portfolio.
Frontier
- Find the least-risk portfolio for various target expected returns.
- The (expected return, least risk) pairs are called the mean-variance frontier.
- Least-risk portfolios are frontier portfolios.
Borrow and save at same rate
Assume \(r_s=r_b=r_f\) (risk-free rate).
Set \(x_f = x_s - x_b\).
portfolio expected return is \(x_f r_f + w^\top \bar{r}\).
The accounting identify \(x_s + \sum w_i = 1 + x_b\) implies \(x_f = 1-\sum w_i\), so portfolio expected return is
\[r_f + w^\top (\bar{r}-r_f1_n)\]
No Short Sales Constraints
- Assume short w/o borrowing fee
- implies return on a short is minus the return on the asset
- contribution of a short to portfolio return is \(w_i r_i\) (with \(w_i<0\))
- All previous formulas are valid with no sign constraints on \(w_i\).
No margin requirements
- Assume no Fed Reg T to limit positions.
- Assume full use of short proceeds to buy other assets or to save (get full interest).
- Only limit on positions is the accounting identity \(x_f = 1 - w^\top 1_n\), which we’ve built in to the expected return formula:
\[r_f + w^\top (\bar{r}-r_f1_n)\]
Find a frontier portfolio
- Let \(\bar{r}_{\text{targ}}\) be target expected return.
- Minimize \(w^\top C w\)
subject to
\[r_f + w^\top (\bar{r}-r_f1_n) = \bar{r}_{\text{targ}}\]
- This is like “choose output quantities to minimize production cost subject to achieving target revenue.”
Optimality principle
- Ratio of marginal benefit to marginal cost must be same for all choice variables.
- Marginal benefit of \(w_i\) is the risk premium \(\bar{r}_i-r_f\) that appears in the constraint.
- Marginal cost is how the variance changes with a small change in \(w_i\).
Marginal cost
- Variance is
\[\sum_{i=1}^n w_i^2\sigma_i^2 + 2 \sum_{i=1}^n \sum_{j=i+1}^n w_iw_j\sigma_{ij}\]
- The terms that involve any given \(w_i\) are
\[w_i^2\sigma_i^2 + 2 \sum_{j \neq i} w_iw_j\sigma_{ij}\]
- The derivative with respect to \(w_i\) is (with \(\sigma_{ii} = \sigma_i^2\))
\[2 \sum_{j=1}^n w_j \sigma_{ij}\]
- The sum equals \(C_i^\top w\) where \(C_i\) is the \(i\)th column of the covariance matrix.
Equate benefit-cost ratios
Equal benefit-cost ratios means \((\bar{r}_i-r_f)/C_i^\top w =k\).
Rearrange: \(k C_i^\top w = \bar{r}_i - r_f\)
Stack: \(k C w = \bar{r} - r_f 1_n\)
Solve:
\[w = (1/k)C^{-1}(\bar{r}-r_f1_n)\]
Find the constant
- The portfolio expected return is
\[r_f + \frac{1}{k}(\bar{r}-r_f1_n)^\top C^{-1}(\bar{r}-r_f1_n)\]
- Equating to the target gives
\[k = \frac{(\bar{r}-r_f1_n)^\top C^{-1}(\bar{r}-r_f1_n)}{\bar{r}_{\text{targ}}-r_f}\]