Certainty Equivalent Approximation and the Missing One-Half Factor
Summary
The document examines a second-order approximation for the certainty equivalent of a risky payoff. Expanding utility around the payoff’s mean approximates expected utility using the utility curvature and payoff variance; a first-order expansion of utility at the certainty equivalent then links that value back to a cash equivalent. The questioner derives a formula with a one-half factor and suspects the textbook expression omits it.
The included answer agrees that the derivation is correct and cites a separate teaching document whose certainty-equivalent approximation includes the one-half factor. The exchange therefore supports the proposed correction, but it provides no full independent derivation or numerical example in the answer itself. The approximation relies on local Taylor expansions and a certainty equivalent near the mean, so it is a small-risk approximation rather than a general exact valuation rule.
Key ideas
- A second-order expansion of utility around the expected payoff introduces variance through the curvature of utility.
- A first-order expansion around the certainty equivalent yields a local approximation for its value.
- The accepted answer supports including a one-half factor in the variance adjustment.
- The approximation depends on local expansions and is not presented as an exact result for arbitrary risk.
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Full text
# Utility Theory - Certainty equivalent approximation formula derivation
# Utility Theory - Certainty equivalent approximation formula derivation
I have a question on an exercise from chapter 9 of D. Luenberger, Investment Science, International Edition, where I suspect there may be a typo.
> Exercise 8 (Certainty approximation) There is a useful approximation to the certainty equivalent that is easy to derive. A second-order expansion near $\bar x=E(x)$ gives $$U(x)\approx U(\bar x)+U^{'}(\bar x)(x-\bar x)+\frac12U^{''}(\bar x)(x-\bar x)^2$$ Hence, $$E[U(x)]\approx U(\bar x)+\frac 12 U^{''}(\bar x)var(x)$$ On the other hand, if we let c denote the certainty equivalent and assume it is close to $\bar x$, we can use the first-order expansion $$U(c)\approx U(\bar x)+U^{'}(\bar x)(c-\bar x)$$ Using these approximations, show that $$c \approx \bar x+{U^{''}(\bar x) \over U^{'}(\bar x)}var(x)$$
Now, I used general methods of algebra along with the fact that $E[U(x)] = U(c)$ to show directly that $$c \approx \bar x+\frac 12 ({U^{''}(\bar x) \over U^{'}(\bar x)})var(x)$$ as follows:
Take the third equation and transform it into $$ c\approx \bar x + {U(c)-U(\bar x) \over U^{'}(\bar x)}$$ Now all I have to do is show that the numerator in the fraction part is $\frac 12 U^{''}(\bar x)var(x)$ which is done by putting $E[U(x)] = U(c)$ into the second formula and you can see the result is immediately there.
On top of this work, I wrote out an example of an investment with the log utility function and showed that my approximation for c worked whereas the book's formula without the "2" didn't.
However, I would like to post all this here just to verify that this is a typo from the book and not some misunderstanding on my part.
Thanks in advance for any feedback.
## Answer by zsljulius (score 2, accepted)
https://quant.stackexchange.com/a/20629
Your calculation seems to be correct. I found this document here:http://home.uchicago.edu/rmyerson/teaching/util206.pdf. You can see that in P10, the certainty equivalence formula has that 1/2 factor there.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.