Jensen's inequality is probably the most well-known inequality in economics. Students stumble at least over the following two straightforward applications during their studies:

- Microeconomics: E[U(w)] < U(E[w]) if U''(w) < 0: For a risk averse agent, the expected utility of wealth is less than the utility of expected wealth. The reason this is so: If wealth has diminishing marginal utility, losses cost more utility than equivalent monetary gains provide. Consequently, a risk averse agent is better of to receive a given amount of wealth
*with certainty*than the same amount of wealth*on average*but with variance around this quantity. <>.

- International Macroeconomics/Finance: E[1/S] > 1/E[S]: The expectation of the reciprocal of an exchange rate is greater than the reciprocal of the expectation of the exchange rate. Jeremy Siegel showed that as a result of Jensen's inequality, the currency forward rate cannot be an unbiased estimate of the future spot rate because an expected increase in one exchange rate implies an expected decrease of smaller magnitude in its reciprocal. Therefore, even if expected changes in the spot rate are distributed symmetrically around the forward rate from the perspective of one investor, "Siegel's paradox" guarantees that the forward rate will be biased from the perspective of the investor on the other side of the exchange rate. <>

The sample variance is an unbiased estimate of the true variance, but due to Jensen's Inequality, the sample standard deviation is biased low as an estimate of the true volatility, because the square root is a concave function:

Mahalanobis - am 2006-12-15 22:10 - Rubrik: EconoSchool

HedgeFundGuy meinte am 15. Dec, 23:48:

ok...

average stock has a volatility of 30% ... how much is the bias?
Mahalanobis antwortete am 16. Dec, 01:26:

Let's say

that depends on the distribution of the returns and on the sample size...I've heard that when r ~ N(0,σ

^{2}), then E[|r|] = sqrt(2/π)σ.

But unbiasedness isn't the only issue and I leave calculating the distribution of E[|r|]/sqrt(2/π) as an exercise...

Mahalanobis antwortete am 16. Dec, 14:43:

Couldn't resist...

Ok, for r ~ N(0,1) I get the following bias for the sample standard deviation:Return Sample Size (Bias)

2 (0.20)

3 (0.11)

4 (0.08)

5 (0.06)

.

10 (0.03)

.

20 (0.013)

.

30 (0.009)

Here is the R output:

> n <- 10

> v <- n - 1

> sampvar <- rchisq(10000000,v)/v

> mean(sampvar)

[1] 0.9997442

> mean(sqrt(sampvar))

[1] 0.9725524

> approx.bias <- 1 - mean(sqrt(sampvar))

> approx.bias

[1] 0.02744761

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a1 (guest) meinte am 2. Jul, 07:57:

<=?

I thought Jensen's inequality was <= rather than <?
tikhonov antwortete am 2. Jul, 17:00:

I guess its < for strictly convex functions...
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