The capital allocation line and the optimal portfolio

What happens when a risk-free asset joins a risky portfolio?

A risk-free asset has zero standard deviation and zero correlation with everything else, so it contributes no risk of its own. Put a weight w_f in it and the rest in a risky portfolio, and the variance formula collapses: only the risky portfolio’s term survives.

$$E(R_p) = w_f R_f + (1 - w_f)\,E(R_i), \qquad \sigma_p = (1 - w_f)\,\sigma_i$$

Both equations are linear in the weight, so every mix lies on a straight line from the risk-free rate on the vertical axis through the risky portfolio. That line is the capital allocation line (CAL). Eliminating the weight gives its equation.

$$E(R_p) = R_f + \frac{E(R_i) - R_f}{\sigma_i}\,\sigma_p$$

The intercept is R_f. The slope is the extra return for each extra unit of risk, often called the market price of risk .

$$\text{Slope} = \frac{E(R_i) - R_f}{\sigma_i}$$

Where a portfolio sits relative to the CAL says everything about it.

Position against the CALthe line is the set of the best attainable portfolios

• Above the line Use it desirable but not achievable with the available assets
• On the line Memorize attainable; the optimal investor portfolio sits here
• Below the line Use it attainable but dominated: more return is available at the same risk

Which risky portfolio should everyone hold?

With many risky portfolios on the efficient frontier, draw a CAL from the risk-free rate to each of them. A steeper line gives more return for every level of risk, so the best line is the steepest one, which just touches the frontier. The touching point is the optimal risky portfolio : no other line offers more return at the same risk.

That line beats the frontier itself. Take any efficient portfolio X other than the tangency point. A mix of the risk-free asset and the optimal risky portfolio with the same risk as X (lending if X is less risky, borrowing if it is more risky) earns more than X. So the optimal CAL dominates every portfolio of risky assets alone.

Because the choice of risky portfolio does not use anyone’s preferences, every investor who shares the same expectations picks the same one. This is the two-fund separation theorem : all investors hold a combination of two funds, the risk-free asset and the optimal risky portfolio. It splits the problem in two.

Choosing a portfolio when a risk-free asset existstwo-fund separation: investment decision, then financing decision

• 1. Investment decision Memorize pick the optimal risky portfolio: where the steepest CAL touches the efficient frontier
  • Same for every investor with the same expectations, whatever the risk aversion or wealth
• 2. Financing decision Memorize pick the optimal investor portfolio: where the investor’s highest indifference curve touches the CAL
  • Lending below 100% in the risky portfolio; more risk averse
  • Borrowing above 100% in the risky portfolio, borrowed at Rf; less risk averse

Which point on the line is best for you?

The second step brings back the indifference curves from the utility lesson. The optimal investor portfolio is where an investor’s highest reachable indifference curve is tangent to the CAL. A higher curve lies out of reach, above the line; a lower curve, crossing the line, gives less utility. More risk-averse investors end up closer to the risk-free asset; less risk-averse investors sit further out and may borrow. These lending and borrowing portfolios differ only in how much of the same risky portfolio they hold.

Start on the first tab and pivot the line until it touches the frontier: the tangency is the optimal risky portfolio. On the second tab, change A to see the investor’s point slide along the line. The default already shows an investor with A = 2.5 holding 70% in the risky portfolio.

The exam method is mechanical: compute U = E(r) − 0.5Aσ² for each point on the CAL and take the highest.

Go deeper The exact best weight

Maximizing U along the CAL gives a closed form for the weight in the risky portfolio. With these inputs, 0.07 ÷ (2.5 × 0.04) = 70% (so 30% is lent): E = 7.9%, σ = 14% and U = 0.079 − 1.25 × 0.0196 = 0.0545, a little above the 60% point’s 0.0540. An investor with A = 2 would hold 87.5% and one with A = 4 would hold 43.75%, so the more risk-averse investor lends more.

$$y^{*} = \frac{E(R_i) - R_f}{A\,\sigma_i^{2}}$$

Best weight at A = 2.5

TI BA II Plus HP 12C

1. 0.07÷(2.5×0.2x²)= 0.7

x² square

1. 0.07ENTER2.5ENTER0.2ENTER××÷ 0.7

ENTER enter (on the HP, puts the number on the stack)

Rearrange itBest weight in the risky portfolio: solve for the weight or the risk aversion

The question asks for

the best weight in the risky portfoliothe risk aversion that gives a chosen weight

The risk-free rate is 2.00%. A risky portfolio has an expected return of 8.00% and a standard deviation of 15.00%. An investor with a risk aversion coefficient of 4 maximizes U = E(r) − 0.5Aσ². The weight in the risky portfolio is closest to:

Risk-free rate%Risky portfolio expected return%Risky portfolio standard deviation%Risk aversion coefficient, A

$$y^{\ast }=\frac{E(R_i)-R_f}{A{\sigma }_i^2}$$

(0.08 − 0.02) ÷ (4 × 0.15²) = 0.6667, so 66.67% in the risky portfolio

Weight in the risky portfolio (decimal)0.6667

On your calculator

TI BA II PlusHP 12C

1. 0.06÷(4×0.15x²)=a decimal, ×100 for percent

x² square

1. 0.06ENTER4ENTER0.15ENTER××÷a decimal, ×100 for percent

ENTER enter (on the HP, puts the number on the stack)

Go deeper Borrowing at a higher rate, and the market portfolio

If investors must borrow at a rate above the lending rate, the CAL bends to a flatter slope beyond the risky portfolio, and not everyone shares one risky portfolio any more. When all investors share expectations and the optimal risky portfolio is the market portfolio, the CAL becomes the capital market line, the starting point for the next module.

Exam tip Two tangencies, two answers. The CAL touching the frontier gives the one risky portfolio everyone holds. An indifference curve touching the CAL gives how much of it you hold.