CFA Level 1 – Portfolio Management · CFA
The Capital Allocation Line: Why the Steepest One Wins
The capital allocation line is the set of portfolios available from combining one risky portfolio with the risk-free asset. Every point on it is a different split between the two, and the line is the complete menu that split can produce.
Two properties do all the work in this topic. The line is straight, which is not obvious and is worth deriving rather than accepting. And its slope is the Sharpe ratio, which turns out to mean that the best risky portfolio is the same portfolio for every investor, whatever their appetite for risk. That second result is the one the exam is really testing, and it is not intuitive until the line is drawn.
Key Takeaways
- The CAL is every combination of one risky portfolio and the risk-free asset, and it is straight because the risk-free asset has zero variance and zero covariance.
- Its slope is the Sharpe ratio of the risky portfolio: the extra return per unit of total risk.
- All lines drawn from one risk-free rate share an intercept, so they cannot cross and the steepest is best at every risk level.
- The highest Sharpe ratio portfolio therefore suits every investor. Risk appetite is expressed through the weight, not through the choice of portfolio.
- A borrowing rate above the lending rate kinks the line at a weight of one and permanently flattens the leveraged segment.
- The CML is the CAL built on the market portfolio under homogeneous expectations. It uses total risk, while the security market line uses beta.
What the line is, and why it is straight
Let y be the proportion of wealth placed in a risky portfolio P, with the remaining 1 − y in the risk-free asset. The combined portfolio C has an expected return that is simply the weighted average of the two.
E(RC) = Rf + y[E(RP) − Rf]
the risk-free rate plus y times the risk premium of the risky portfolio
The risk of the combination is where the useful result appears. In general, combining two assets gives a standard deviation that depends on both variances and the covariance between them. Here the risk-free asset has zero variance and, because its return is certain, zero covariance with anything. The general expression collapses.
σC = y σP
risk scales linearly with the weight, with no diversification term to complicate it
Both expected return and risk are now linear in y, so eliminating y between them gives a straight line. Substituting y = σC / σP into the return equation produces the capital allocation line.
E(RC) = Rf + [(E(RP) − Rf) / σP] × σC
an intercept at the risk-free rate and a constant slope
It is worth being precise about what “risk-free” is doing here, since the whole result rests on it. The asset has to be free of default risk and free of reinvestment risk over the investor’s horizon, which in practice means a government bill whose maturity matches that horizon rather than a long government bond, whose price moves with rates. It is also risk-free only in nominal terms. Inflation still moves the real return, so a strictly risk-free real asset would have to be index linked. These are approximations the model accepts, and they are approximations rather than facts.
The straightness is entirely a consequence of one asset having no risk. Combine two risky assets instead and the covariance term survives, the relationship curves, and the familiar bowed frontier appears. The CAL is the one straight line in this part of the curriculum, and it is straight for a reason worth being able to state.
The slope is the Sharpe ratio
The bracketed term in that equation is the risk premium of the risky portfolio divided by its standard deviation, which is the Sharpe ratio. So the slope of the capital allocation line is the Sharpe ratio of the risky portfolio it is built on, and nothing else.
That gives the slope a concrete reading: it is the extra expected return the investor receives for accepting one more unit of total risk. A steeper line is a better deal at every level of risk simultaneously, not just at one point.
A consequence follows immediately and it is easy to miss. Every point on a given capital allocation line has the same Sharpe ratio, because the Sharpe ratio is the slope and the slope is constant. Take the y = 0.50 point from the next section: its risk premium is 7.5% minus 3% = 4.5% and its standard deviation is 10%, giving 0.45 again. Moving along the line by borrowing or lending changes the expected return and the risk in exactly the same proportion, so it cannot change the ratio between them. Leverage, at a single rate, buys more of the same deal rather than a better one.
The intercept of every CAL drawn from the same risk-free rate is identical. Only the slope differs. Two lines from a common intercept can therefore never cross, which means one of them lies above the other everywhere. This single geometric fact is what makes the highest Sharpe ratio portfolio best for every investor regardless of risk tolerance, and Section 4 works it through with numbers.
Reading a point on the line
The weight y locates the investor on the line, and three regions matter.
The risk-free rate is 3%. Risky portfolio P has an expected return of 12% and a standard deviation of 20%. Locate the combined portfolio for three weights.
- Sharpe ratio of P = (0.12 − 0.03) / 0.20 = 0.45, which is the slope of the line.
- y = 0.50: E(RC) = 0.03 + 0.50(0.09) = 7.5%, and σC = 0.50 × 0.20 = 10%.
- y = 1.00: E(RC) = 12%, σC = 20%. This is P itself.
- y = 1.50: E(RC) = 0.03 + 1.50(0.09) = 16.5%, and σC = 1.50 × 0.20 = 30%.
- Check the last one against the line equation: 0.03 + 0.45 × 0.30 = 0.165. It matches, as it must.
Answer: 7.5% at 10% risk, 12% at 20%, and 16.5% at 30%. The y = 1.50 investor has borrowed 50% of their wealth at the risk-free rate and invested 150% in P. Both routes to any point give the same answer, which is a useful arithmetic check under exam pressure.
| Weight y | Position | Where on the line |
|---|---|---|
| y = 0 | Entirely in the risk-free asset | The intercept |
| 0 < y < 1 | Lending at the risk-free rate | Between the intercept and P |
| y = 1 | Entirely in the risky portfolio | At P |
| y > 1 | Borrowing to invest more than wealth in P | Beyond P, a leveraged position |
Why the highest Sharpe ratio wins for every investor
A cautious investor might reasonably assume that a lower risk portfolio suits them better than a higher risk one. The capital allocation line shows why that reasoning is wrong, and the demonstration is short.
The risk-free rate is 3%. Portfolio P offers 12% at 20% volatility. Portfolio Q offers 9% at 14% volatility. A conservative investor wants total volatility of no more than 14%. Should they hold Q?
- Sharpe ratio of P = (0.12 − 0.03) / 0.20 = 0.450.
- Sharpe ratio of Q = (0.09 − 0.03) / 0.14 = 0.429.
- Holding Q outright gives 9.0% at 14% volatility.
- On P’s line, 14% volatility requires y = 0.14 / 0.20 = 0.70, so 70% in P and 30% in the risk-free asset.
- That combination returns 0.03 + 0.70(0.09) = 9.3% at exactly 14% volatility.
Answer: no. The conservative investor gets 9.3% instead of 9.0% at identical risk by holding the higher volatility portfolio in smaller size. The risk appetite is expressed through the weight, not through the choice of portfolio, and it is the weight that adjusts. The same argument works at any risk level, because the two lines share an intercept and cannot cross.
The reasoning depends on a risk-free asset being available to borrow and lend at. Remove it and the argument collapses, because there is no common intercept, the comparison between portfolios is no longer a comparison of slopes, and investors with different risk appetites genuinely do want different risky portfolios, each choosing their own point along the curved efficient frontier. The single best risky portfolio is a result about markets with a risk-free asset, not a general truth about diversification.
Extend this to every risky portfolio available and one of them produces the steepest line. That portfolio is the tangency portfolio, the point where the capital allocation line touches the efficient frontier, and no other combination of risky assets can improve on it once the risk-free asset is available.
What breaks the straight line
The derivation assumed one risk-free rate. Real investors lend at one rate and borrow at a higher one, and the moment those differ the line stops being a single straight line.
Below y = 1 the investor is lending, so the lending rate is the intercept and the slope uses that rate. Above y = 1 the investor is borrowing, so the relevant rate is the borrowing rate, which lowers the risk premium and therefore flattens the slope. The two segments meet at P, producing a kink.
Two further frictions push in the same direction and are worth naming, since exam questions occasionally introduce them. Margin requirements cap how far above y = 1 an investor can go at all, so the leveraged segment may simply end rather than continue indefinitely. And an investor whose own borrowing rate exceeds the institutional one, which is the ordinary case for a retail investor, faces a flatter upper segment still. The shape of the constraint changes; the direction never does.
Lending is available at 3% and borrowing at 6%. Portfolio P still offers 12% at 20% volatility. What does an investor targeting 30% volatility actually receive?
- 30% volatility requires y = 0.30 / 0.20 = 1.50, so the investor borrows 50% of wealth.
- The borrowing applies to the amount above wealth, so the return is 0.06 + 1.50(0.12 − 0.06) = 0.06 + 0.09 = 15.0%.
- At a single 3% rate the same position would have returned 0.03 + 1.50(0.09) = 16.5%.
- The upper slope is (0.12 − 0.06) / 0.20 = 0.30, against 0.45 below the kink.
Answer: 15.0% rather than 16.5%, so 1.5 percentage points are lost to the spread. Note that the Sharpe ratio available to a leveraged investor falls from 0.45 to 0.30, which is a third of the deal gone. Leverage does not merely scale a position up. On a kinked line it buys a worse trade-off than the one the unleveraged investor gets.
From the CAL to the CML
The capital market line is the capital allocation line in one specific case. Assume every investor has the same information and the same expectations about returns, variances and correlations. They then all identify the same tangency portfolio, and since every risky asset must be held by someone, that common portfolio has to be the market portfolio.
| Capital allocation line | Capital market line | |
|---|---|---|
| Risky portfolio used | Any risky portfolio the investor chooses | The market portfolio |
| Assumption required | None beyond a risk-free asset | Homogeneous expectations across all investors |
| Slope | Sharpe ratio of the chosen portfolio | Market Sharpe ratio, the highest available |
| How many exist | One per risky portfolio | One |
| Applies to | Combinations of the risk-free asset and that portfolio | Efficient portfolios only |
One distinction is worth guarding carefully, because it is a reliable source of lost marks. The capital market line prices efficient portfolios against total risk, measured by standard deviation. The security market line prices any asset, efficient or not, against systematic risk, measured by beta. An inefficient asset sits below the CML and still lies on the SML. The two lines answer different questions and neither is a version of the other.
Where the investor sits, and why that is a separate question
The line describes what is available. It does not say which point an investor should take, and that choice comes from their tolerance for risk. Under the standard mean variance utility function the optimal weight has a closed form.
y* = [E(RP) − Rf] / (A σ²P)
A is the risk aversion coefficient, so a more risk averse investor holds less of the risky portfolio
With P offering 12% at 20% volatility against a 3% risk-free rate, an investor with A = 2 arrives at y* = 0.09 / 0.08 = 1.125 and takes a leveraged position, while an investor with A = 8 arrives at y* = 0.09 / 0.32 = 0.281 and holds mostly cash. They sit at different points on the same line and hold the same risky portfolio.
That is the separation result, and it is the reason the capital allocation line is taught before anything about individual preferences. Finding the best risky portfolio is a technical problem with one answer that is the same for everyone. Deciding how much of it to hold is a personal problem with a different answer for each investor. The two are solved independently, in that order, and confusing them produces the mistake in Section 4, where a cautious investor picks a safer portfolio when they should have picked the better one and held less of it.
The short version
- The CAL is every combination of one risky portfolio and the risk-free asset, and it is straight because the risk-free asset has zero variance and zero covariance.
- Its slope is the Sharpe ratio of the risky portfolio: the extra return per unit of total risk.
- All CALs from one risk-free rate share an intercept, so they cannot cross and the steepest is best at every risk level.
- The highest Sharpe ratio portfolio therefore suits every investor. Risk appetite is expressed through the weight, not the portfolio.
- A borrowing rate above the lending rate kinks the line at y = 1 and permanently flattens the leveraged segment.
- The CML is the CAL built on the market portfolio under homogeneous expectations. It uses total risk, while the SML uses beta.
Frequently Asked Questions
Why is the capital allocation line straight?
Because one of the two assets is risk-free. Its variance is zero and its covariance with the risky portfolio is zero, so the standard deviation of the combination is simply the weight times the risky portfolio’s standard deviation. Both expected return and risk are then linear in the weight, so eliminating the weight gives a straight line. Combine two risky assets instead and the covariance term survives and the relationship curves.
What does the slope of the CAL represent?
The Sharpe ratio of the risky portfolio it is built on: the risk premium divided by the standard deviation. It measures the extra expected return received for accepting one more unit of total risk. A steeper line is a better deal at every level of risk at once, not only at one point.
Should a cautious investor choose a lower risk portfolio?
No. They should choose the portfolio with the highest Sharpe ratio and hold less of it, putting the rest in the risk-free asset. Because all lines from the same risk-free rate share an intercept, they cannot cross, so the steepest line offers a higher return at every level of risk including a low one. Risk appetite is expressed through the weight, not through the choice of portfolio.
What happens if borrowing costs more than lending?
The line kinks at the risky portfolio. Below it the investor lends and the slope uses the lending rate. Above it the investor borrows, which reduces the risk premium and flattens the slope permanently. A leveraged investor therefore faces a worse risk and return trade-off than an unleveraged one, not simply a scaled up version of the same one.
What is the difference between the CAL and the CML?
The CML is the special case of the CAL where the risky portfolio is the market portfolio. Getting there requires the assumption that all investors share the same expectations, which makes them all select the same tangency portfolio. There are as many CALs as there are risky portfolios, and only one CML.
How does an investor decide where to sit on the line?
Through the optimal weight, which is the risk premium divided by the risk aversion coefficient times the variance. A more risk averse investor holds less of the risky portfolio. This is a separate decision from choosing the risky portfolio itself, which has the same answer for everyone, and solving them in that order is the separation result.
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