CFA Level 1 – Portfolio Management · CFA
The Risk Aversion Coefficient: A Parameter, Not a Personality
The risk aversion coefficient is usually described as a measure of how uncomfortable an investor is with uncertainty. That is what it represents. It is not what it is.
It is a parameter inside a specific equation, and every property it has follows from where it sits in that equation. Read it that way and the exam questions become arithmetic. Read it as a personality description and the questions stay guesswork, because the descriptions do not tell you which of two portfolios a given investor picks, and the equation does.
Key Takeaways
- A is the coefficient in U = E(R) minus one half A times variance, and every property it has follows from that position.
- Keep expected return and standard deviation in decimals: percentages inflate the variance term by 10,000 against 100 for the return term.
- A greater than zero is risk averse, A equal to zero is risk neutral and ranks on expected return alone, A below zero is risk seeking.
- The utility score is a certainty equivalent return, so it is directly comparable with a guaranteed rate.
- A higher coefficient means steeper indifference curves, because more return is demanded per unit of risk.
- The optimal risky weight is the excess return divided by A times variance, and a value above one is a leveraged position rather than an error.
What the coefficient actually is
The standard mean variance utility function assigns a single score to a portfolio from its expected return and its variance.
U = E(R) − ½ A σ²
U is the utility score, E(R) is the expected return, σ² is the variance of returns, and A is the risk aversion coefficient
Three things follow directly from the structure and none of them needs to be memorised separately.
Expected return enters positively and variance enters negatively, so a portfolio is preferred for its return and penalised for its risk. A is the exchange rate between the two: it says how much expected return the investor requires as compensation for a unit of variance. And because A multiplies the variance rather than the standard deviation, the penalty grows with the square of the risk. Doubling the standard deviation quadruples the penalty.
The use of variance rather than standard deviation is deliberate and it is not merely a convention. Standard deviation treats a portfolio twice as volatile as twice as bad. Variance treats it as four times as bad, which is a stronger statement about how quickly discomfort grows, and it is the assumption that gives indifference curves their characteristic upward bend rather than a straight line. A framework built on standard deviation would produce straight indifference lines and a very different set of optimal portfolios.
An investor is not choosing a portfolio with the highest return or the lowest risk. They are choosing the portfolio with the highest U, and A is what decides which portfolio that turns out to be.
The units trap that changes the answer by a factor of 100
Before any calculation, fix the units. This is the single most common way candidates lose a mark on this reading, and it is entirely avoidable.
The formula as written above assumes returns and standard deviations expressed as decimals: 12% enters as 0.12 and 20% enters as 0.20. Enter them instead as whole numbers, 12 and 20, and the return term grows by a factor of 100 while the variance term grows by a factor of 10,000. The variance term then swamps the return term and every portfolio looks catastrophic.
Mixing conventions inside one question, typically by taking A from a question written in decimals and applying it to figures entered as percentages. If a question insists on whole number percentages, the coefficient has to be divided by 100 to give the same ranking, because the two forms are related by U in decimals equalling one hundredth of the expression in percentages with A replaced by A over 100. Rather than remember that, keep everything in decimals and the problem does not arise.
A itself is a pure number with no units. Curriculum problems typically use values between 1 and 10, with 2 to 4 describing a moderately risk averse investor. Treat those as teaching conventions rather than measured population parameters.
Same two portfolios, three different investors
The clearest way to see what A does is to hold the portfolios fixed and vary only the investor.
Portfolio A has an expected return of 12% and a standard deviation of 20%. Portfolio B has an expected return of 8% and a standard deviation of 10%. A risk-free asset returns 3%. Which does each investor choose?
- Investor with A = 2. UA = 0.12 − 0.5 × 2 × 0.04 = 0.08. UB = 0.08 − 0.5 × 2 × 0.01 = 0.07. The risk-free asset scores 0.03. Chooses Portfolio A.
- Investor with A = 4. UA = 0.12 − 0.08 = 0.04. UB = 0.08 − 0.02 = 0.06. Chooses Portfolio B.
- Investor with A = 8. UA = 0.12 − 0.16 = −0.04. UB = 0.08 − 0.04 = 0.04. Still chooses Portfolio B, and now scores Portfolio A below zero.
- Find the switching point by setting the two equal: 0.12 − 0.5A(0.04) = 0.08 − 0.5A(0.01), which gives 0.04 = 0.015A, so A = 2.67.
- At A = 10, UB = 0.08 − 0.05 = 0.03, exactly the risk-free score. This investor is indifferent between Portfolio B and holding cash.
Answer: below A = 2.67 the investor prefers Portfolio A, above it Portfolio B, and at A = 10 even Portfolio B has stopped being worth the risk. Nothing about either portfolio changed. A negative utility score is not an error and does not mean the portfolio loses money; it means this investor would rather hold nothing than hold it.
| Investor | Portfolio A 12% return, 20% risk | Portfolio B 8% return, 10% risk | Risk-free 3% | Choice |
|---|---|---|---|---|
| A = 2 | 0.080 | 0.070 | 0.030 | Portfolio A |
| A = 4 | 0.040 | 0.060 | 0.030 | Portfolio B |
| A = 8 | −0.040 | 0.040 | 0.030 | Portfolio B |
| A = 10 | −0.080 | 0.030 | 0.030 | Indifferent |
Read the table by column rather than by row and the mechanism is visible. Portfolio B’s score falls by 0.01 for every increase of 2 in A, while Portfolio A’s falls by 0.04, four times as fast, because its variance is four times as large. That ratio is fixed by the portfolios and has nothing to do with the investor. All A does is decide how far down the two columns to travel before the ranking flips.
Risk neutral and risk seeking
The sign and size of A partition investors into three cases, and the equation makes each one obvious.
| Value of A | Investor type | What happens to the variance term | Choice between two portfolios with equal expected return |
|---|---|---|---|
| A > 0 | Risk averse | Subtracts from utility | Prefers the lower variance one |
| A = 0 | Risk neutral | Disappears entirely | Indifferent. Ranks purely on expected return |
| A < 0 | Risk seeking | Adds to utility | Prefers the higher variance one |
The risk neutral case is worth dwelling on, because it explains why the whole framework exists. If every investor had A equal to zero, risky assets and the risk-free asset would be ranked on expected return alone, and there would be no reason for a risky asset to offer more than the risk-free rate. The equity risk premium exists because A is positive in aggregate. The coefficient is not a description of individual temperament that happens to appear in a formula; it is the term that produces risk premiums at all.
The certainty equivalent, which is what the number means
A utility score on its own is hard to interpret. Is 0.04 good? The question has an answer, and it comes from noticing what the formula does to a risk-free asset. With a variance of zero the second term vanishes and utility equals the return itself.
So the utility score of a risky portfolio is directly comparable with a guaranteed return. It is the certain return that would leave this investor exactly as well off as holding the risky portfolio, which is the certainty equivalent.
In the example above, the investor with A = 4 scored Portfolio A at 0.04. That means they are indifferent between a portfolio with a 12% expected return and 20% volatility, and a guaranteed 4%. The 8 percentage point gap is the risk premium this specific investor demands for that specific amount of risk. Since the risk-free rate is 3%, they still prefer the portfolio, but by only one percentage point. The utility score is not an abstract index. It is a return.
This also explains the negative score in the A = 8 case. A certainty equivalent of −4% says the investor would accept a guaranteed loss of 4% rather than hold that portfolio. That is an extreme position and it follows mechanically from a large A applied to a large variance.
Indifference curves, where the slope carries the meaning
Hold U constant and the equation becomes a curve in risk and return space: E(R) = U + ½Aσ². Every portfolio on that curve gives this investor the same satisfaction, which is what makes it an indifference curve.
Because A multiplies the squared term, it controls how steeply the curve rises. A more risk averse investor demands more extra return for each additional unit of risk, so their curves are steeper. Curves further up and to the left represent higher utility, since they deliver more return for the same risk.
The curve through Portfolio A belongs to the investor with A = 2, and Portfolio B sits below it, meaning lower utility for that investor. The curve through Portfolio B belongs to the investor with A = 8, and Portfolio A sits a long way below it. Neither investor is making a mistake and the two portfolios have not changed. Only the slope has.
The same coefficient also fixes how much of a risky portfolio to hold alongside the risk-free asset, through the optimal weight in the risky asset.
y* = [E(Rp) − Rf] / (A σ²p)
y* is the proportion placed in the risky portfolio, with the remainder in the risk-free asset
Using Portfolio A, with an expected return of 12%, a standard deviation of 20% and a risk-free rate of 3%, how much does each investor place in it?
- A = 2: y* = (0.12 − 0.03) / (2 × 0.04) = 0.09 / 0.08 = 1.125.
- A = 4: y* = 0.09 / 0.16 = 0.5625.
- A = 8: y* = 0.09 / 0.32 = 0.28125.
Answer: 112.5%, 56.25% and 28.1% respectively. The first investor holds more than 100% of their wealth in the risky portfolio, funded by borrowing 12.5% at the risk-free rate. A weight above one is not an error in the arithmetic, it is a leveraged position, and it appears whenever the excess return is large relative to A times the variance.
What the coefficient cannot tell you
The framework is clean and it rests on assumptions that should be stated rather than absorbed silently.
Mean variance utility uses only the first two moments, so it is a complete description of preferences only if returns are elliptically distributed, of which the normal distribution is the familiar case, or if the investor’s underlying utility function is quadratic. Neither holds exactly. Real return distributions are skewed and fat tailed, and an investor who dislikes a large loss far more than they dislike an equally large gain is expressing something this equation cannot represent. Two portfolios with identical mean and variance but different skewness score identically here and would not be regarded as identical by any actual investor.
A is also not observable. It is inferred, usually from a questionnaire or from choices the investor has already made, and inferred values move around. They shift with wealth, with age, with the recency of a market fall, and with how the question was worded. A number estimated in a calm market is not the number that will govern behaviour in a disorderly one, which is exactly when it matters.
Almost every question on this topic is one of four things: compute U for a portfolio, rank portfolios for a given A, find the A at which an investor is indifferent between two portfolios, or compute the optimal weight y*. Each is a direct substitution into one of the two equations here. Fix the units first, keep everything in decimals, and treat a negative utility score as a valid answer rather than a signal that something went wrong.
One further limitation is worth naming because it is easy to miss inside a well behaved equation. The framework is single period. It scores a portfolio on the distribution of returns over one horizon and says nothing about the path taken to get there, which is why it cannot represent an investor who could tolerate the end result but not the drawdown along the way. For a pension fund with a fixed horizon that omission is minor. For an individual who may be forced to sell at the worst moment, it is not.
None of this makes the coefficient useless. It makes it a model parameter rather than a measurement, which is the correct way to hold it. It answers precisely one question well, which is how a stated tolerance for risk translates into a portfolio choice, and it is the mechanism through which risk premiums exist in the first place.
The short version
- A is the coefficient in U = E(R) − ½Aσ². Every property it has follows from that position.
- Keep expected return and standard deviation in decimals. Percentages inflate the variance term by 10,000 against 100 for the return term.
- A > 0 is risk averse, A = 0 is risk neutral and ranks on expected return alone, A < 0 is risk seeking.
- The utility score is a certainty equivalent return, so it is directly comparable with a guaranteed rate.
- Higher A means steeper indifference curves, because more return is demanded per unit of risk.
- The optimal risky weight is y* = [E(Rp) − Rf] / (Aσ²p), and a value above one is a leveraged position rather than an error.
Frequently Asked Questions
What is the risk aversion coefficient?
It is the parameter A in the mean variance utility function U = E(R) minus one half A times variance. It sets the exchange rate between expected return and variance: how much extra expected return the investor requires as compensation for a unit of variance. Because it multiplies variance rather than standard deviation, the penalty rises with the square of the risk.
Should returns be entered as decimals or percentages?
Decimals, unless a question states otherwise. Entering 12 and 20 instead of 0.12 and 0.20 scales the return term by 100 and the variance term by 10,000, so the variance term dominates and every portfolio scores badly. If percentages must be used, the coefficient has to be divided by 100 to preserve the ranking.
What does a negative utility score mean?
That the investor would rather hold nothing than hold that portfolio. It is a valid answer, not an arithmetic error, and it does not mean the portfolio is expected to lose money. Read the score as a certainty equivalent: a utility of minus 0.04 says the investor would accept a guaranteed loss of 4% instead of taking that portfolio.
What is a certainty equivalent return?
The guaranteed return that leaves the investor exactly as well off as holding the risky portfolio. Under this utility function it is simply the utility score, because a risk-free asset has zero variance and its utility therefore equals its return. This makes the score directly comparable with the risk-free rate.
Why are indifference curves steeper for a more risk averse investor?
Holding utility constant gives E(R) = U plus one half A times variance, so A controls how fast required return rises with risk. A larger A means more extra return demanded for each additional unit of risk, which is a steeper curve. Curves higher and to the left represent greater utility.
What are the limitations of this measure?
It uses only mean and variance, so it fully describes preferences only under elliptical return distributions or quadratic utility. It cannot represent a preference over skewness, so two portfolios with the same mean and variance but different tail shapes score identically. A is also not observable. It is inferred, and inferred values move with wealth, age and recent market experience.
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