Saving and Borrowing

Saving

• Uninvested funds should earn money market rate.
• \(n\) risky assets, \(w_i\) = weights, \(x_s\) = fraction of portfolio earning money market rate

Expected return and risk

• Let \(r_s\) = money market rate (saving rate)
• portfolio expected return is \(x_sr_s + \bar{r}^\top w\)
• portfolio variance is still \(w^\top C w\)

Fully invested portfolio

• Because \(\sum w_i < 1\), we can view \((w_1, \ldots, w_n)\) as a scaled down version of a fully-invested portfolio.
• \(x_s\) in savings and \(1-x_s\) in the fully invested portfolio
  • \(\hat{w}_i = w_i/(1-x_s)\) are the fully invested weights
  • \(\sum \hat{w}_i = \sum w_i/(1-x_s) = 1\)
  • \(w_i = (1-x_s)\hat{w}_i\)

Risk premium and Sharpe ratio

• For any asset or portfolio, expected return minus \(r_s\) is called its risk premium
  • It is compensation for risk.
• (expected return - \(r_s\)) / std dev is called the Sharpe ratio
  • It is the reward-to-risk ratio.

Portfolio Risk premium

• weights \(w_i = (1-x_s)\hat{w}_i\) where \(\sum \hat{w}_i=1\)
• risk premium is

Standard deviation and Sharpe ratio

• variance is \((1-x_s)\hat{w}^\top C \hat{w}(1-x_s)\)

• std dev is \((1-x_s)\sqrt{\hat{w}^\top C \hat{w}}\)

• So, \(1-x_s\) is in both numerator and denominator of Sharpe ratio. Sharpe ratio doesn’t depend on \(x_s\)

Example

Borrowing (Margin loans)

• \(x_b \ge 0\) is fraction borrowed, \(r_b\) = borrowing rate
• portfolio expected return is \(w^\top \bar{r} - x_br_b\)
• portfolio variance is still \(w^\top C w\)
• weights are \(w_i = (1+x_b)\hat{w}_i\) for a fully invested portfolio \(\hat{w}\)
• portfolio mean is \((1+x_b)\bar{r}^\top \hat w - x_br_b\) and std dev is \((1+x_b)\sqrt{\hat{w}^\top C \hat{w}}\)
• Sharpe ratio (mean - \(r_b\)) / std dev is invariant to \(x_b\)

Example

Saving and borrowing opportunities