FINM7008 Chap.5 Capital Allocation and the Capital Allocation Line
Capital Allocation and the Capital Allocation Line
How much to put at risk
Capital allocation decides how wealth is divided between one risky portfolio and the risk-free asset. Every mix lies on the capital allocation line, whose slope is the risky portfolio's Sharpe ratio.
The optimal weight
Maximising utility along the line gives the optimal weight y*: the risk premium divided by the product of risk aversion and the risky portfolio's variance.
The chapter interprets leverage, target-return and risk-cap questions, the kinked line created by a higher borrowing rate, and the largest fee an active fund can charge.
The question types
The tutorial solutions work a single risky fund through a sequence of questions: the complete portfolio for a given split, its Sharpe ratio, the split needed for a target return, the best split under a risk cap, the optimum for a given A, the range of A that holds the fund alone on a kinked line, and the largest fee a client will pay.
What this chapter covers
- 01
The complete portfolio
- 02
The capital allocation line and its slope
- 03
Deriving the optimal weight
- 04
Leverage when y* exceeds one
- 05
Target returns and risk caps
- 06
Borrowing rates and the kinked line
- 07
Maximum fees for active funds
Worked example · free
The optimal point on a capital allocation line
- 1y* = (0.11 − 0.03) / (4 × 0.202) = 0.08 / 0.16 = 0.5.
- 1Mean: 3% + 0.5 × 8% = 7%.
- 1Risk: 0.5 × 20% = 10%.
- 1Utility: 0.07 − 2 × 0.102 = 0.05, against 0.03 for bills alone.
Key terms
- Optimal Risky Weight
- The fraction y* of wealth in the risky portfolio that maximises an investor's utility.
- Leverage
- Borrowing to hold more than 100% of wealth in the risky portfolio, so that y exceeds one.
- Kinked Capital Allocation Line
- A line that becomes flatter beyond the risky portfolio because borrowing costs more than lending.
- Risk-Free Asset
- An asset, typically a Treasury bill, whose return over the holding period is known in advance.
Capital Allocation and the Capital Allocation Line FAQ
Why is the capital allocation line straight?
Both the expected return and the standard deviation of the complete portfolio change in proportion to y, because the risk-free asset adds no variance or covariance. Plotting a quantity that is linear in y against another linear in y gives a straight line.
What does an optimal weight above one mean?
It means the investor should borrow at the risk-free rate and invest more than all of their wealth in the risky portfolio. Low risk aversion or a high expected risk premium can make that optimal, which is why over-optimistic forecasts can inflate leverage.
Does mixing a portfolio with bills change its Sharpe ratio?
No. The risk premium and the standard deviation are both scaled by the same weight y, so their ratio is unchanged. Every complete portfolio on one capital allocation line has the line's slope as its Sharpe ratio.
How do I find the split for a target expected return?
Set the capital allocation line equal to the target and solve for y: the target minus the risk-free rate, divided by the risky portfolio's risk premium. Multiply each asset's weight inside the risky portfolio by y to get its overall weight, and put the rest in bills.
What is the largest fee an active fund can charge?
The fee that makes the fund's Sharpe ratio after fees equal to the Sharpe ratio of a passive index. Any higher fee leaves the client better off on the index's line, and the answer differs for clients who borrow, because their line is flatter.
What happens to y* when the risky portfolio becomes more volatile?
It falls with the square of the standard deviation. Doubling the risky portfolio's volatility, with the same risk premium and risk aversion, cuts the optimal weight to a quarter of its previous value, moving the investor down the line towards bills.
Exam move
Memorise the two CAL equations and y*, then solve one problem of each kind: an optimal weight, a target return with overall asset weights, a risk cap and a kinked line with two rates. Check every answer by confirming that the complete portfolio lies on the same line, with the same Sharpe ratio as the risky portfolio.
Write y* with its interpretation in words every time, lending below one and borrowing above one, so that a leveraged answer never looks like an error.
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