The Capital Allocation Line

Suppose we construct a portfolio (P) that combines a risky asset i with an expected return of r_i and standard deviation of σ_i, and a riskless asset with a return of r_f. Let w₁ represent the fraction of the total portfolio value placed in the riskless asset.

• The portfolio return E(r_p) is given by E(r_p) = w₁ r_f + (1 - w₁) E(r_i).
• The portfolio standard deviation is given by σ_p = (1 - w₁) σ_i

Example

Combine the S&P and a T-bill in a portfolio. E(r_S&P) = 13%, σ_S&P = 20.3%, and r_f = 3.8%. Some of the possible portfolios are:

• w₁ = 0, E(r_p) = E(r_S&P) = 13%, and σ_p = 20.3%.
• w₁ = 0.5, E(r_p) = 0.5 x 0.038 + 0.5 x 0.13 = 8.4%, and σ_p = 0.5 x 20.3% = 10.1%.
• w₁ = 1, E(r_p) = r_f) = 3.8%, and σ_p = 0%.
• w₁ = -0.5, E(r_p) = -0.5 x 0.038 + 1.5 x 0.13 = 17.6%, and σ_p = 1.5 x 0.203 = 30.5%. This means there is negative investment in the riskfree asset: the investor borrowed at the risk-free rate. This is called a leveraged position in the risky asset - some of the investment is financed by borrowing.

The portfolio's expected return and standard deviation obey a liner relation:

• The slope, [(E(r_i - r_f)/σ_i], is known as the portfolio's Sharpe measure or reward-to-variability ratio.
• The intercept is the risk-free rate.
• The line is called the capital allocation line (CAL). When i is a marker index portfolio, the line is called the capital market line (CML).
• A leveraged position is to the right of P.

CAL shows one simple fact: increasing the amount invested in the risky asset increases the expected return by a certain risk premium.

Now the investor must find the point of highest utility on CAL.

The optimal choice for an investor is the point of tangency of the highest indifference curve to the CAL - slope of the indifference curve is equal to the slope of the CAL.

The optimal w₁ will be higher for investors with higher A.