FIN521 Chap.7 Capital Allocation to Risky Assets
Capital Allocation to Risky Assets
The decision taken before any security is chosen
This topic occupies the fifth teaching week and the midterm covers it. It answers the first question in portfolio construction, deliberately placed first: before deciding which shares to hold, decide what proportion of the money is exposed to shares at all.
That proportion does more to determine an outcome than any individual selection made afterwards, which is why the course separates it from security analysis rather than folding the two together.
Why the arithmetic collapses to one weight
Treat the risky holding as a single fund, whatever is inside it, and the safe holding as Treasury bills.
Put a weight into the fund and the remainder into bills, and the combination is the complete portfolio. Because bills are treated as having no dispersion, both properties of the combination become simple expressions in that one weight. Expected return is the bill rate plus the weight times the risk premium on the fund.
Standard deviation is the weight times the fund's standard deviation, with no cross term, because there is nothing for a riskless asset to covary with.
Students expect variances, weights and a covariance and are suspicious when they do not appear; they do not appear because one asset's variance is zero.
The line, and what choosing a point on it means
Both properties rise linearly in the weight, so the ratio between them is constant.
That constant is the slope of the capital allocation line, equal to the risk premium divided by the standard deviation, and it is the reward per unit of risk this particular fund offers. Moving along the line changes how much risk is taken; it does not change the terms on which risk is rewarded, because the terms are a property of the fund and the bill rate rather than of the investor's appetite.
This separation is the most useful idea in the chapter: a risky fund is ranked by the steepness of its line, and only the choice of point along the winning line depends on the investor.
Weights above one, and where the line bends
Nothing in the algebra stops the weight exceeding one.
A weight of one and a half means borrowing half the investor's capital again and putting the total into the fund, which is the margin arithmetic of the trading chapter written as a portfolio weight. One complication is real and examinable: investors borrow at a higher rate than they lend at, so beyond a weight of one the line starts from the borrowing rate and is flatter.
The reward per unit of risk falls the moment you become a borrower, which is the precise version of the informal claim that leverage does not improve the terms of a bet.
What this chapter covers
- 01
Separating the allocation decision from security selection
- 02
The complete portfolio as one risky fund plus bills
- 03
Expected return and dispersion as linear functions of one weight
- 04
Why no covariance term survives when one asset is riskless
- 05
The capital allocation line and the meaning of its slope
- 06
Setting a risk budget and reading the weight off it
- 07
Weights above one, borrowing, and the flatter segment
- 08
Ranking two risky funds by slope rather than by dispersion
- 09
Why the safe asset is a short bill held to maturity
Turn a return target into a weight and check it two ways
- 3Express the target as a distance above the bill rate and divide by the risk premium.
- 2Compute the standard deviation at that weight.
- 3Check the slope from the fund and from the investor's own point.
Key terms
- Complete Portfolio
- The combination of a risky holding and a risk-free asset that an investor actually owns, defined by the single weight placed in the risky part.
- Capital Allocation Line
- The set of complete portfolios available from one risky fund and the risk-free asset, drawn as a straight line in return and dispersion space.
- Slope Of The Line
- The risk premium divided by the standard deviation of the risky fund, which is the extra expected return available per unit of dispersion accepted.
- Risk Free Asset
- A short-dated government bill held to maturity, whose return over the horizon being modelled is known with certainty.
- Borrowing Rate
- The higher rate at which an investor funds a weight above one, which flattens the line beyond that point and reduces reward per unit of risk.
- Risk Budget
- A constraint expressed as a maximum acceptable standard deviation, which in this framework is a statement about the weight.
Capital Allocation to Risky Assets FAQ
Why is there no covariance term in the portfolio standard deviation here?
Because one of the two assets is risk-free. Its variance is zero and its covariance with anything is zero, so every term in the general formula vanishes except the one containing the risky fund's variance. The general expression returns as soon as both assets are risky, which is the next step in the subject. This simplified case is taught first precisely because it isolates the allocation decision from the diversification one.
Is the portfolio with the highest expected return the best one?
No, and recommending it is the most common wrong answer in this topic. The highest expected return is always the most levered position available, and every point on a given line offers identical reward per unit of risk, so choosing among them is a statement about tolerance for dispersion and about how soon the money is needed.
A question that describes an investor's circumstances is telling you which point to choose; a question that offers two funds is asking which line is steeper.
How do I compare two risky funds?
By slope, not by dispersion or by expected return alone. The fund with the steeper line dominates, because any point reachable on the shallower line can be matched at lower risk or beaten at the same risk on the steeper one. That ranking does not depend on the investor at all.
A cautious investor who prefers the lower-dispersion fund on those grounds has confused the choice of line with the choice of point, even if the answer happens to come out right.
Exam move
Set yourself three risk budgets and solve for the weight each time from the same fund, then repeat with a return target instead and confirm the two routes agree.
After that, take two funds with different premiums and dispersions and rank them by slope before looking at anything else, because the separation between ranking the line and choosing the point is what this chapter is really examining and it is invisible in the arithmetic.
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