Capital Allocation & the Efficient Frontier

We’ve seen that the correlation between assets determines the risk of a portfolio, and that mixing imperfectly correlated assets lowers that risk. Diversification alone does not tell us which of the many diversified mixes to hold. This post builds the machinery that picks one: the Capital Allocation Line, the efficient frontier, and the result that every investor’s optimal portfolio can be described as a blend of one risky portfolio with risk-free lending or borrowing.

The Cost of Volatility

Suppose we hold 100,000 dollars and someone offers a fair coin flip: heads adds 50,000, tails subtracts 50,000. The expected change in wealth is zero. Would we take it?

Most people would not, even though the math says the bet is fair. The jump from 100k to 150k feels good, but not as good as the drop from 100k to 50k feels bad.

Economists model this with a concave utility function: a curve that rises as wealth increases, but at a decreasing rate. That curvature is diminishing marginal utility. Each additional dollar is worth slightly less than the one before it.

That concavity has a direct consequence. A gamble with zero expected dollar gain but added uncertainty lowers expected utility, because the cost of the downside outweighs the benefit of the upside even though the probabilities are symmetric.

Investors who respond this way are risk-averse. Not irrationally fearful, but rationally aware that volatility is costly when a utility function curves.

Different investors have different degrees of curvature. Some barely care about volatility, while others lose sleep over it. But as long as the utility function is concave at all (and for nearly everyone it is), volatility has a cost.

That cost is why investors demand higher expected returns for bearing more risk. Volatility is not evil. It merely has a price, and people want to be paid for tolerating it.

The Capital Allocation Line

Start with the simplest possible portfolio problem: one risky asset and one risk-free asset. We put weight $w$ in the risky asset and $1 - w$ in the risk-free asset, and our expected portfolio return is the weighted average of the two:

E [r_{p}] = w \times E [r_{S}] + (1 - w) \times r_{f}

Because the risk-free asset has $σ = 0$ and zero correlation with everything, we drop both terms it would otherwise contribute to portfolio variance, leaving:

σ_{p} = w \times σ_{S}

Both equations are linear in $w$ . We plot $(σ_{p}, E [r_{p}])$ as $w$ varies, and the points trace a straight line, the Capital Allocation Line (CAL).

Take a risky equity fund with $E [r_{S}] = 11%$ and $σ_{S} = 18%$ , against a risk-free rate of $r_{f} = 3%$ . We vary $w$ and read off the risk and return of each blend:

Weight ( $w$ )	Risk ( $σ_{p}$ )	Return ( $E [r_{p}]$ )	Description
0%	0%	3.0%	All risk-free
50%	9%	7.0%	Half and half
100%	18%	11.0%	All risky
150%	27%	15.0%	Levered (borrowing at $r_{f}$ )

The intercept of the CAL is $r_{f}$ , and we compute its slope from the risky asset’s excess return per unit of risk:

slope = \frac{E [ r _{S} ] - r _{f}}{σ _{S}} = \frac{11 - 3}{18} = 0.44

That slope is the Sharpe ratio of the risky asset, and every point on the CAL delivers the same ratio: more risk comes with proportionally more excess return. Moving up the line means accepting more volatility in exchange for higher expected return. Past $w = 1$ , the position is levered, funded by borrowing at $r_{f}$ .

Drag the weight slider below to move along the line, and change the risky asset’s return or volatility to watch the CAL pivot.

RISKY ASSET E[r]: 11.0% σ: 18.0% RISK-FREE RATE: 3.0% WEIGHT IN RISKY: 100% 0% 2% 4% 6% 8% 10% 12% 14% 16% 0% 5% 10% 15% 20% 25% 30% 35% 40% r f Risky σ E[r] E[r] 11.0% σ 18.0% SHARPE 0.44 The slope of the CAL is the Sharpe ratio. Every point on the line has the same Sharpe.

A steeper line (higher Sharpe) is always better, delivering more expected return for each unit of risk.

Choosing Between Risky Assets

For a fixed risk-free rate, the slope of a CAL depends only on the risky asset it is built from. If two funds offer different Sharpe ratios, they trace different CALs, and the fund with the higher Sharpe dominates.

Take Fund P with $E [r] = 13%$ and $σ = 25%$ , and Fund Q with $E [r] = 9%$ and $σ = 12%$ . At $r_{f} = 3%$ we get:

Sharpe_{P} = \frac{13 - 3}{25} = 0.40 Sharpe_{Q} = \frac{9 - 3}{12} = 0.50

Fund Q’s Sharpe ratio of 0.50 beats P’s 0.40, so Q’s CAL is the steeper of the two. At any target risk level, we can reach a higher expected return by blending Q with risk-free lending or borrowing than by using P. Even though P has the higher raw return and the higher raw risk, Q is the better building block.

This generalizes into a surprisingly strong result, the Two Fund Separation Theorem: the optimal portfolio for any risk-averse investor combines (1) the risky asset with the highest Sharpe ratio and (2) risk-free lending or borrowing. Risk preference determines only the position along the resulting CAL.

Conservative investors lend ( $w < 1$ ) and aggressive investors borrow ( $w > 1$ ), but every investor holds the same risky asset. Which risky asset to hold and how much risk to take are separate decisions, made in that order: find the steepest line, then pick a point on it.

Scaling vs. Tilting

Two Fund Separation gives us a clean distinction between two kinds of portfolio change.

Scaling multiplies all risky weights by the same constant and takes an offsetting risk-free position. A risky portfolio of 60% stocks and 40% bonds, scaled by 1.5×, becomes 90% stocks and 60% bonds, funded by borrowing 50% at $r_{f}$ . The relative mix does not change, the portfolio moves along the CAL, and the Sharpe ratio stays the same.

Tilting raises the weight in one risky asset and lowers another, or adds a risk-free offset. The relative mix does change, the portfolio moves off the current CAL onto a different one, and the Sharpe ratio may rise or fall.

Scaling answers the question “how much risk?” Tilting answers “is this the right mix?” Once we hold the risky portfolio with the highest Sharpe ratio, the only move left is to scale. Tilting moves to a worse CAL. Scaling stays on the best one.

The Efficient Frontier

The “risky asset” in Two Fund Separation is itself a portfolio, and the theorem says nothing about how to find it. Evaluating the two-asset variance formula from the last post at every possible weight traces a curve in $(σ, E [r])$ space, and the shape of that curve depends on correlation: lower $ρ$ bows it further left, creating more diversification benefit.

The upper portion of that curve, from the minimum-variance portfolio upward, is the efficient frontier, which gives the highest achievable expected return at every level of risk. Any portfolio below the frontier is inefficient, since a point on the frontier offers more return for the same risk, or less risk for the same return.

Now we add the risk-free asset and construct the line from $(0, r_{f})$ tangent to the frontier. The point where the line touches the curve is the tangent portfolio, the risky portfolio with the highest Sharpe ratio. Its CAL is the steepest line from $r_{f}$ to any achievable portfolio.

By Two Fund Separation, every investor should hold the tangent portfolio combined with risk-free lending or borrowing. Cautious investors put most of their money in the risk-free asset and a small allocation in the tangent portfolio, while aggressive investors lever it up. No rational investor chooses a different risky mix.

Drag the correlation slider below and watch the frontier bow inward.

ASSET A E[r]: 10.0% σ: 15.0% ASSET B E[r]: 6.0% σ: 10.0% ρ: +0.30 RISK-FREE RATE: 2.0% 0% 2% 4% 6% 8% 10% 12% 14% 0% 5% 10% 15% 20% 25% r_f A B T σ E[r] Efficient frontier Inefficient CAL (tangent) TANGENT w(A) 53% TANGENT E[r] 8.1% TANGENT σ 10.4% SHARPE 0.59 The tangent portfolio has the highest Sharpe ratio. Every investor holds it, scaled by risk preference.

Lower $ρ$ pushes the frontier left, expanding the set of efficient portfolios. Change the risk-free rate and the tangent point shifts, since a different $r_{f}$ means a different optimal risky mix.

What’s Next

If every investor holds the tangent portfolio, and the tangent portfolio is determined by the available assets and their correlations, then in equilibrium the tangent portfolio is the market portfolio: the value-weighted combination of all risky assets.

That equivalence leads directly to the Capital Asset Pricing Model (CAPM), under which expected return depends on exposure to market risk (beta) rather than total volatility. In the next post we take up why only systematic risk is priced, what the Security Market Line looks like, and how factor models extend the CAPM beyond one market factor.