CM2 · explained

Utility theory and risk aversion: non-satiation, A(w) and R(w)

CM2 opens by asking not what a sum of money is worth, but what it is worth to a particular person. This page explains what a utility function is, what non-satiation and risk aversion mean as conditions on its derivatives, and how the two measures of risk aversion are built and read.

CM2 · Financial Engineering & Loss Reserving · 13 min read

After this page you can

Test a utility function for non-satiation and risk aversion, and calculate the largest premium a person would pay to remove a risk

The question utility theory answers

Two people are offered the same bet: a coin is tossed, and they win ₹10,000 or lose ₹10,000. One takes it and one refuses. Neither is being irrational. The bet has an expected value of zero to both of them, so expected value cannot be what either is deciding on.

Utility theory is the machinery that lets you say why they differ, and then calculate with it. It replaces the amount of money with a number representing how much that amount is worth to this person, and it insists on one rule: faced with a choice, a person picks the option with the highest expected value of that number. Everything in the first topic of CM2 — the shapes of utility functions, the measures of risk aversion, the price a person will pay for insurance — is a consequence of that one rule.

The number is called utility and the function that produces it is written U(w), where w is wealth. Note what w is. It is total wealth after the decision, not the gain or the loss. Setting the problem up in terms of gains is the commonest way to get a utility question wrong before any arithmetic starts.

Why the units of utility do not matter

A utility function is only ever used to rank options, never to report a quantity. So if you replace U(w) with a + bU(w) for any constants a and b with b > 0, every ranking is unchanged and every decision is the same. Utility has no units, no natural zero and no scale.

This is worth knowing for two reasons. It tells you that a question asking "what is this person's utility?" is incomplete, because the answer depends on an arbitrary choice. And it tells you that a difference of two utilities is meaningless on its own, while a comparison of two expected utilities is exactly what you are meant to compute.

The two conditions a sensible utility function must meet

Non-satiation: more money is better

U′(w) > 0

In words: the utility function is increasing, so being given more wealth never makes a person worse off.

This sounds too obvious to state until you meet a utility function that fails it. Quadratic utility, which is used all over CM2 because it makes mean-variance theory work, fails it above a certain level of wealth, and a question will ask you to find where. Failing non-satiation is not a small technical blemish: it means the function is claiming a person would pay to be made poorer.

Risk aversion: a fair bet is not worth taking

U″(w) < 0

In words: the utility function is concave — each extra rupee adds less utility than the one before it.

Concavity is risk aversion, and the connection is worth seeing rather than memorising. Take wealth w and a bet that adds or subtracts h with equal probability. The gain of h raises utility by less than the loss of h lowers it, precisely because the function is flattening out. So the expected utility of the bet is below the utility of not betting, and a risk-averse person declines. That argument used nothing but concavity, so it holds for every concave function and for every fair bet.

The three cases are worth naming, because exam wording uses them:

  • U″(w) < 0 — risk averse. Declines a fair bet.
  • U″(w) = 0 — risk neutral. Indifferent to a fair bet, and decides purely on expected value.
  • U″(w) > 0 — risk seeking. Accepts a fair bet.

Measuring how risk averse someone is

Knowing the sign of the second derivative tells you the direction. Measuring the strength needs a number, and the obvious candidate — U″(w) itself — is useless, because it changes when you rescale the function and the rankings do not. The fix is to divide by the first derivative, which rescales the same way, so the ratio does not move at all.

A(w) = −U″(w) / U′(w)    (absolute risk aversion)
R(w) = −w · U″(w) / U′(w) = w · A(w)    (relative risk aversion)

In words: absolute risk aversion measures how hard a person resists risking a fixed sum of money. Relative risk aversion measures how hard they resist risking a fixed proportion of what they have.

The minus sign is there to make both quantities positive for a risk-averse person, since U″ is negative. What matters in a question is almost never the level of A(w) but whether it rises or falls as w rises:

  • Decreasing absolute risk aversion says a richer version of this person would be readier to risk a given number of rupees. That is how people behave, and it is the realistic case.
  • Increasing absolute risk aversion says the opposite: the richer they get, the less they will stake. It is hard to defend as a description of anybody, and a question that produces it is usually asking you to say so.
  • Constant relative risk aversion says a person's willingness to risk a given percentage of their wealth does not change as they get richer. That is a common modelling assumption, because it means the portfolio does not have to be redesigned every time the fund grows.

The four utility functions to know cold

FunctionA(w)R(w)What it says about behaviour
Quadratic
U = aw − ½bw2		b / (a − bw)
increasing		bw / (a − bw)
increasing		Non-satiation only below w = a/b. Convenient, not realistic.
Logarithmic
U = ln w		1 / w
decreasing		1
constant		Decreasing absolute, constant relative. The workhorse.
Power
U = w1−γ / (1−γ)		γ / w
decreasing		γ
constant		Same shape as log, with γ dialling the strength. Log is the γ → 1 case.
Exponential
U = −e−λw		λ
constant		λw
increasing		Decisions do not depend on how rich the person already is.

Two of these are worth a second look. Quadratic utility is the one that misbehaves: differentiate and you get U′ = a − bw, which hits zero at w = a/b and goes negative above it, so the function has a peak and then falls. Above that wealth the person prefers less money to more, which is nonsense, and the honest answer in an exam is to state the bound and say the model is only usable below it.

Exponential utility is the one with the surprising property: because A(w) is a constant, the amount a person will stake does not depend on their wealth at all. A person with ₹50,000 and a person with ₹50 crore make the same bet. That is convenient algebraically and clearly wrong as a description of people, which is exactly the sort of trade-off a "comment on the realism" question is testing.

The certainty equivalent: putting a price on a risk

The certainty equivalent of a risky prospect is the certain amount of wealth that would leave the person exactly as happy:

U(certainty equivalent) = E[U(random wealth)]

In words: find the expected utility of the gamble, then ask what certain sum has that same utility. That sum is the certainty equivalent.

For a risk-averse person the certainty equivalent is always below the expected wealth, and the gap is the risk premium — the amount of expected value the person is willing to give up in exchange for certainty. That gap is where insurance comes from. A policyholder will pay more than the expected loss, and an insurer with a bigger, better-diversified balance sheet is content to accept it, and both sides are better off by their own standards. The whole industry sits in that gap.

There is a standard approximation for the risk premium when the risk is small relative to wealth:

risk premium ≈ ½ · A(w) · Var(loss)

It is worth knowing both because it is quotable and because it shows what drives the answer: the price of a risk rises with the person's risk aversion and with the variance of the risk, not with its expected size. But it is a second-order approximation, and the worked example below shows it breaking down when the risk is a quarter of the person's wealth. Use it as a sanity check on an exact answer, never as a substitute for one.

Questions students ask

Is utility a real thing that could be measured?

No, and it does not need to be. Utility is a device for representing preferences consistently; it is justified by the fact that someone whose choices obey a short list of reasonable conditions behaves as if they were maximising the expected value of some function. That is a theorem about consistency, not a claim about a quantity inside anybody's head. The practical content is the ranking it produces.

If investors are risk averse, why does anyone hold a risky asset?

Because the bets on offer in a market are not fair bets. Risk aversion says a person declines a gamble with zero expected gain; it says nothing about a gamble with a positive one. Equities are held because they are expected to return more than cash, and how much more they must offer before a risk-averse investor will hold them is the question the Capital Asset Pricing Model goes on to answer.

Which utility function should I use if a question does not say?

A question will always say, because the answer depends on it. If you are choosing for yourself in a modelling exercise, logarithmic utility is the usual starting point: it is risk averse, it has decreasing absolute and constant relative risk aversion, it needs no parameter to be estimated, and its algebra collapses neatly because the logarithm of a product is a sum.

Why does CM2 keep using quadratic utility if it is unrealistic?

Because it is the one utility function under which a decision depends on nothing but the mean and the variance of the outcome. Mean-variance portfolio theory and everything built on it needs that, and quadratic utility is the price of admission. The examinable skill is being able to use it and to state honestly what it is doing wrong.

A worked example

A family has investable wealth of ₹20,00,000. Over the coming year there is a 10% chance of a loss of ₹5,00,000 — a flat damaged in a flood, say. Their utility of wealth is U(w) = ln(w).

An insurer offers to cover the loss in full for a premium P payable now. What is the largest premium the family would accept?

Every figure here is invented for this example. It is not a real household, a real flood probability or a real premium.

  1. The expected utility of carrying the risk

    Two outcomes. With probability 0.90 wealth stays at ₹20,00,000; with probability 0.10 it falls to ₹15,00,000.

    ln(20,00,000) = 14.5086577
    ln(15,00,000) = 14.2209757
    E[U] = 0.90 × 14.5086577 + 0.10 × 14.2209757
        = 13.0577919 + 1.4220976 = 14.4798895

  2. The certainty equivalent

    Find the certain wealth with that utility. For log utility, invert by exponentiating.

    CE = e14.4798895 = ₹19,43,283.32

  3. The premium

    With insurance the family's wealth is certain, at 20,00,000 − P. They will accept any premium that leaves them at least as well off as carrying the risk, so the largest acceptable premium puts them exactly level:

    20,00,000 − P = 19,43,283.32
    Pmax = ₹56,716.68

  4. The second route, as a check

    For log utility the certainty equivalent is a weighted geometric mean of the possible wealths, with the probabilities as the weights. That follows from E[ln w] = ln(w1p1 w2p2…), and it uses no logarithm table at all:

    CE = (20,00,000)0.9 × (15,00,000)0.1
        = 20,00,000 × (15/20)0.1 = 20,00,000 × 0.750.1
        = 20,00,000 × 0.97164166 = ₹19,43,283.32

    The same figure to the paisa, by a different calculation. And a third check: the premium must by construction leave the family exactly indifferent, so ln(19,43,283.32) must equal the expected utility computed in step 1. It is 14.4798895. It does.

  5. Reading the answer

    The expected loss is 0.10 × 5,00,000 = ₹50,000. The family would pay up to ₹56,716.68, so they are willing to hand over ₹6,716.68 more than the risk costs on average. That is the risk premium, and it is the reason a household insurance market exists: there is ₹6,716.68 of value per year to be shared between this family and an insurer who can pool the risk with thousands of others.

    Notice what the answer is not sensitive to. It is not the chance of the flood alone, nor the size of the loss alone, but the way the two combine against the family's wealth. Halve the loss to ₹2,50,000 and the premium falls by far more than half, because a smaller risk is also a proportionally cheaper one.

  6. Checking against the approximation

    The small-risk approximation says the risk premium should be about ½ A(w) Var(L). For log utility A(w) = 1/w, and the variance of the loss is that of a two-point distribution:

    Var(L) = 0.1 × 0.9 × (5,00,000)2 = 0.09 × 2.5 × 1011 = 2.25 × 1010
    ½ × (1 / 20,00,000) × 2.25 × 1010 = ₹5,625

    ₹5,625 against the true ₹6,716.68 — the approximation is about a sixth too small. That is not an arithmetic error in either figure, and it is worth understanding rather than ignoring. The approximation is a second-order expansion and assumes the risk is small next to the wealth. Here the loss is a quarter of the family's wealth, which is nowhere near small, so the curvature of the logarithm over that range is not well described by a single quadratic term. Shrink the loss to ₹50,000 and the two figures converge.

    That is exactly how to use the approximation in an exam: as a check that an exact answer is in the right region, and as the thing to quote when a question asks what drives the price of a risk. It is not a shortcut to the number.

  7. And if the family were risk neutral

    With U(w) = w, the expected utility of carrying the risk is simply the expected wealth, 0.9 × 20,00,000 + 0.1 × 15,00,000 = ₹19,50,000, so the largest premium is exactly ₹50,000 — the expected loss, and not one paisa more. A risk-neutral household never buys insurance at a loaded premium. The entire ₹6,716.68 in the answer above is the concavity of the logarithm, and nothing else.

Where this fits in CM2

This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Other CM2 topics explained

Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.