Risk aversion, utility and indifference curves

What does it mean to be risk averse?

Offer an investor two choices with the same expected value: a sure USD 2,000, or a draw that pays USD 5,000 with a 40% chance and nothing otherwise (0.4 × 5,000 = USD 2,000 on average). The choice shows the investor’s attitude to risk.

• A risk averse investor takes the sure USD 2,000. The draw must offer a higher expected value before it is worth taking.
• A risk neutral investor is indifferent. Only the expected return matters, never the risk.
• A risk seeking investor takes the draw, and might even prefer a draw worth USD 1,800 on average to the sure USD 2,000, because the uncertainty itself is enjoyable.

A sure USD 2,000, or a 40% chance of USD 5,000?same expected value, different risk

• Risk averse Memorize takes the sure USD 2,000; A > 0
  • Indifference curve slopes up and is convex; steeper as A rises
• Risk neutral Memorize indifferent; cares only about return; A = 0
  • Indifference curve horizontal
• Risk seeking Memorize takes the draw; enjoys risk; A < 0
  • Indifference curve slopes down

One choice does not classify an investor, but the curriculum, like the investment industry, assumes the representative investor is risk averse. That is consistent with the historical record: prices have rewarded higher risk with higher return.

Risk aversion does not mean refusing all risk. A risk-averse investor holds risky assets if the expected return is high enough. Risk tolerance is the flip side: the amount of risk an investor can accept to reach a goal. The more risk tolerant, the less risk averse.

How is risk aversion turned into a number?

The standard tool is the utility function. Utility is a measure of the satisfaction an investor gets from a portfolio. Here it is a simple score built from expected return and variance.

$$U = E(r) - \tfrac{1}{2}\,A\,\sigma^{2}$$

The risk aversion coefficient A is the extra return the investor demands for taking additional risk. Use returns and standard deviations as decimals (8% is 0.08, and a σ of 15% gives σ² = 0.0225).

U = E(r) − ½Aσ²a ranking device, not a measure of happiness

• Higher E(r) Use it raises U
• Higher σ² Use it lowers U; the loss is multiplied by A
• Risk-free asset (σ = 0) Memorize U = Rf for every investor
• Only ranks Use it U = 4 is not twice U = 2; not comparable across investors

Move the risk aversion slider and watch the best fund change. As A falls toward zero the curves flatten and the highest-return fund wins; below zero they slope down.

Four facts about the score are worth keeping.

1. Utility is unbounded: it can be large and positive or negative. Fund R at A = 8 scores 0.12 − 4 × 0.0576 = −0.1104.
2. It only ranks. A utility of 4 does not mean twice the satisfaction of a utility of 2, and one investor’s utility cannot be compared with another’s.
3. The risk-free asset has σ = 0, so its utility is its return for every investor, whatever A is.
4. More return always raises utility. For a risk-averse investor, less risk raises it too, so the investor wants to move north-west.

Go deeper What a risk-free equivalent tells you

Because a risk-free return has no variance penalty, the utility of a risky portfolio equals the sure return that would make the investor equally happy. This is the risk-free equivalent return . For E(r) = 9%, σ = 16% and A = 2.5, it is 0.09 − 1.25 × 0.0256 = 5.80%: a sure 5.80% is worth the same to this investor as the risky portfolio with a 9% expected return. The gap, 3.20 points, is the price the investor puts on the risk.

$$R_{eq} = U = E(r) - \tfrac{1}{2}\,A\,\sigma^{2}$$

On your calculator

TI BA II Plus HP 12C

1. 0.09−(0.5×2.5×0.16x²)= 0.058

x² square

1. 0.16ENTER×0.5×2.5×CHS0.09+ 0.058

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

What are indifference curves?

An indifference curve plots all the combinations of risk and return that give an investor the same utility. The investor does not care where on one curve they sit. Because there are infinitely many such combinations, the curve is continuous.

Investor	Indifference curve (σ on the x-axis, E(r) on the y-axis)
Risk averse (A > 0)	upward sloping and convex; steeper the higher A
Risk neutral (A = 0)	horizontal, because only return matters
Risk seeking (A < 0)	downward sloping: would give up return to get more risk

Three rules govern how to read them.

• North-west is better. More return for less risk means higher utility, so the investor wants the highest curve reachable.
• Curves never cross. Investors rank portfolios consistently (if X beats Y and Y beats Z, then X beats Z), which rules out intersections.
• Convex, and steeper for higher A. Each extra unit of risk needs a bigger return reward than the last, and a more risk-averse investor needs a bigger reward at every level of risk.

Exam tip Match the picture to A: slope up means averse, flat means neutral, slope down means seeking. The risk-neutral curve is horizontal, never vertical.

Go deeper Why the curve bends: the slope is Aσ

Hold U fixed and solve for the return: E(r) = U + ½Aσ². The slope of that curve in the risk–return plane is Aσ. It grows as σ grows, which is why the curve is convex, and it is larger for a larger A, which is why a more risk-averse investor’s curves are steeper. The ½ is a scaling convention, and the minus sign on the variance is what makes risk a cost.

Go deeper Measuring A in practice

Advisers use questionnaires and conversations, and they separate willingness to take risk from the capacity to bear it. A young investor with a stable income may have high capacity and low willingness, or the reverse. The investment policy statement lesson picks this up. Behavioral research also finds that people react more to losses than to variance as such, a point the behavioral biases lesson covers.