Expected utility and risk aversion

Average the utility, then translate back into money

For possible final wealth levels $w_i$ with probabilities $p_i$,

Consider a fictional one-period choice with no intermediate consumption: final wealth is USD 90 or USD 110, each with probability one half. Suppose the decision maker has log utility,

with fixed reference unit $w_{\mathrm{ref}}=\text{USD }1$. This reference unit makes the logarithm dimensionless; it is not a psychological gain/loss reference point. The mean wealth is USD 100, but

Thus the model ranks USD 100 for certain above the lottery and regards USD 99.498744 for certain as equivalent to it. The premium measures the gap between mean wealth and its certainty equivalent. It is neither a realised trading loss nor an equilibrium expected excess return. These are stipulated preferences and probabilities for a teaching example, not an investment recommendation.

Why concavity creates the premium

Assume the relevant expectations exist and wealth stays in the utility function’s interval domain. Strict concavity makes the comparison strict for a genuinely random lottery. Affine increasing utility gives equality and risk neutrality. Risk aversion does not mean rejecting every risky position: a sufficiently favourable distribution can have greater expected utility than an available safe alternative.

Arrow–Pratt coefficients: fixed money versus fractions of wealth

For twice differentiable utility with $u'(w)>0$, the coefficients are

For log utility, differentiation gives $u'(w)=1/w$ and $u''(w)=-1/w^2$, hence

At USD 100, $A=0.01$ per USD. At USD 200 it is $0.005$ per USD: the same small dollar risk matters less as wealth increases. Relative risk aversion stays one, so proportional scaling has a different implication. More generally, power utility $u(w)=((w/w_{\mathrm{ref}})^{1-\gamma}-1)/(1-\gamma)$, with $\gamma>0$ and $\gamma\ne1$, has $A(w)=\gamma/w$ and constant $R(w)=\gamma$ on positive wealth (the $\gamma\to1$ limit is log utility); exponential utility $u(w)=-e^{-aw}$ with $a>0$ has constant $A(w)=a$ and $R(w)=aw$. These assumptions describe different responses to changing wealth, rather than interchangeable labels for “conservative”.

A small-risk approximation, with its boundary visible

Write wealth as $W=w+\varepsilon$, with $\mathbb{E}[\varepsilon]=0$ and variance $\sigma^2$. Taylor expansion gives

Equating these expressions produces

This is a local expansion: for example, take $\varepsilon=tX$ with bounded zero-mean $X$, smooth utility near $w$ and $t\to0$. Keeping only variance is justified as the risk shrinks; it is not an exact rule for arbitrary distributions with that variance. For a small zero-mean proportional shock $\eta$, substitute $\varepsilon=w\eta$ to obtain $\pi/w\approx\tfrac12R(w)\text{Var}(\eta)$.

In the USD 90/110 example, $w=100$ and $\sigma^2=100$ USD squared, so the local premium is USD 0.50, close to the exact USD 0.501256. Enlarging the same symmetric risk to USD 20/180 gives exact premium USD 40, while the approximation gives USD 32. Wealth remains positive, yet the local calculation has become materially inaccurate.

Keep the domain and the objective attached

Log and power utilities require positive final wealth. Applying log utility directly to a possibly negative profit is a domain error: define final wealth, including the starting resources and other exposures, first. If a log model assigns positive probability to zero wealth, expected log utility is negative infinity; even strictly positive wealth needs a finite expected log for the ordinary finite comparisons above. Changing the utility domain to accommodate debt is an economic modelling choice.

Practice: double the wealth, double the stakes

Keep log utility and equal probabilities. If final wealth becomes USD 180 or USD 220, does the risk premium halve, stay fixed, or double relative to the USD 90/110 lottery?

Sources

• Wolitzky, A. Attitudes toward Risk, MIT 14.121, Fall 2015, slides 2–11: expected utility, certainty equivalents and risk-aversion coefficients.
• Pratt, J. W. Risk Aversion in the Small and in the Large, Econometrica 32 (1964), 122–136, introduction: local absolute and proportional risk aversion, utility equivalence and wealth conventions. The numerical examples and Taylor calculations above are independently worked.
• Kahneman, D. and Tversky, A. Prospect Theory: An Analysis of Decision under Risk, Econometrica 47 (1979), pp. 263–264 and 277–279: final assets versus reference-dependent gains and losses.
• Kelly, J. L. A New Interpretation of Information Rate, Bell System Technical Journal 35 (1956), pp. 919–920 and 925–926: repeated betting, expected log wealth and growth.