# Jensen's inequality in finance.

Abstract The purpose of this paper is to measure the size and the statistical significance of three inequalities in the field of financial economics. The three are variants of Jensen's inequality. The first inequality is a comparison of the expected value of a ratio to the ratio of the expected value, a problem that arises in pricing foreign exchange rates. The second is a comparison of the log of the expected value to the expected value of the log, a problem that arises in testing forward market efficiency, money demand, production functions, and trade gravity models. The third is a comparison of the expected utility to the utility of the expected value, and helps in determining the importance of the expected utility paradigm, and the magnitude of the equity risk premium. The methodology used is by simulation of random normal variables, thereby introducing sampling error. Despite this sampling error the conclusion is general: all three inequalities are economically material, and stand statistically as inequalities. The major conclusions is that Jensen's inequality is not a theoretical and superfluous exercise in finance as some have advocated.

Keywords Jensen's inequality * Concave and convex functions.

Foreign exchange markets * Expected utility.

Simulation of random normal variables * Equity risk premium.

Power functions * CCAPM

JEL D81 * E44 * F31 * C15

Introduction

This paper addresses three inequalities in finance. The three are variants of Jensen's inequality, which is concerned with convex and concave functions. The purpose ofthis paper is to measure the size and statistical significance of these three inequalities.

A function f(x) is convex if:

E([florin](x))> [florin](E(x)) (1)

or if:

[lambda][florin](a) + (1 - [lambda])[florin](b)> [florin]([lambda]a + (1 - [lambda])b) (2)

where 0 < [lambda] < 1 and a and b are distinct but otherwise arbitrary points. An example of a convex function is when:

[florin](x) = 1/x (3)

An application of such a function is how a foreign exchange rate is quoted in the market. Suppose the USD per GBP rate goes up from 1.85 to 2.00. The percent appreciation of the GBP is 100(2-1.85)/1.85, or 8.11%. The percent depreciation of the USD is found by taking first the inverse of the quotes, i.e. 1/1.85, or 0.54054, and 1/2, or 0.5. The depreciation of the USD is hence 100(0.5-0.54054)/0.54054, or -750%. The difference is more than half of a percent, and the paradox is that the GBP has appreciated by 8.11% while the USD has depreciated by only 7.50%! A function f(x) is concave if:

E([florin](x))< [florin](E(x)) (4)

or if:

[lambda][florin](a) + (1 - [lambda])[florin](b)< [florin]([lambda]a + (1 - [lambda])b) (5)

where 0 < [lambda] < 1 and a and b are distinct but otherwise arbitrary points. An example of a concave function is the logarithmic function:

[florin](x) = Log(x) (6)

An application of such a function is when forward market efficiency in the foreign exchange market is tested in a log-log specification, or when production functions, money demand, and trade gravity models (Silva and Tenreyro 2006) are tested in a log-log specification, or when the Euler equations of the Consumption Capital Asset Pricing Model (CCAPM) are transformed by taking logs (see below).

Another application of a concave function is the utility of wealth, or of consumption, where the utility function takes the following two forms:

[florin](x) = U(x) = [x.sup.(1 - [gamma])/(1 - [gamma]) or [florin](x) = [x.sup.(1 - [gamma]) - 1/(1 - [gamma]) (7)

where U(x) is a concave utility function with argument x. For a practical example see the following section.

Theory

In the case of convex functions, the expected value of the ratio is greater than the ratio of the expected value:

E (1/1 + [epsilon]) - (1/E(1 + [epsilon])) > 0 (8)

Where [epsilon] is a random normal variable with mean zero and standard deviation 10%. These are approximately the first two moments of the percentage change in foreign exchange rates in developed markets. The level of the foreign exchange rate is normalized to I without loss of generality. If the inequality (Eq. 8) stands, and this is the first statistical purpose of this paper, then one cannot get the same econometric results when using the price of the foreign currency in terms of the domestic currency, and when using the price of the domestic currency in terms of the foreign currency. This is particularly true when testing for uncovered interest rate parity, or forward rate which is unbiased, and is based upon the expectation of the future spot foreign exchange rate ([S.sub.t+1]) by the current forward rate ([F.sub.t]):

E(S.sub.t + 1) = [.sub.t + 1] [F.sub.t] (9)

Inequality (Eq.8) says that it matters how the expectation of the foreign exchange rate is defined in Eq. 9, because if Eq. 9 holds then: (1)

E (1/[S.sub.t + 1]) > 1/[.sub.t + 1] [F.sub.t] = 1/E(S.sub.t + 1] (10)

In order to avoid this inequality, which "is dismissed as being empirically insignificant in the short run" (Sercu and Uppal 1995, p. 420), (2) researchers usually take logarithms of the foreign exchange rate, and state the relation as follows:

E(Log(S.sub.t + 1])) = Log([.sub.t + 1] [F.sub.t]) (11)

Equation 11 introduces another problem. The logarithmic function being concave the log of the expected value. is greater than the expected value of the log by half the variance ([sigma].sup.2]):

Log(E([S.sub.t])) = E(Log([S.sub.t])) + [[sigma].sup.2]/2 (12)

Therefore the log of the LHS of Eq. 9 is not equal to the expected value of the log, as in the LHS of Eq. 11. The difference between the two terms is equal to half the variance, which, in this paper, is 0.005, or 0.5%. The second purpose of this paper is to try to find out whether the variance term is statistically significant.

Another illustration of log-normality is when productuon functions, money demand, and trade gravity models (Silva and Tenreyro 2006) are estimated. Without loss of generality if these models have a power functional form, like a CobbDouglas function, and assuming, as an example, that M/P is real money, Y is permanent income, A is a constant, and [epsilon] is an error term:

[M/P] = A[Y.sup.[alpha]][epsilon] with E([epsilon]) = 1(13)

Then, if these models are estimated with a log-log specification, it can be shown that:

(Log (M/P|Y)) = Log (E(M/P|Y)) - 0.5[[alpha].sup.2]Variance(Log(y)) - 0.5 variance(Log([epsilon])) - [alpha] covariance(Log(y), Log([epsilon])) + [zeta] (14)

which shows clearly that the composite residual is heteroscedastic, breaching one assumption of ordinary least squares. If the functional form is different like:

[M/P] = A[Y.sup.[alpha]]exp ([epsilon]) with E([epsilon]) = 0 (15)

Then, if these models are estimated with a log-log specification, it can be shown that:

(Log(M/P|Y)) = Log(E(M/P|Y)) - 0.5 var iance([epsilon]) + [upsilon] (16)

which shows that an additional term is needed in the regression, especially if the random error [epsilon] is heteroscedastic or time-variable.

Yet another illustration is the transformation of the CCAPM Euler equations. Taking the risk-free rate Euler equation. with [beta] as the discount factor, C as per capita consumption, [R.sub.f] as the gross risk-free rate, t as the time period, and [gamma] as a constant, one has:

1 = [beta][R.sub.[florin]] E [([C.sub.t]/[C.sub.t + 1]).sup.[gamma]] (17)

If the term in parentheses follows a log-normal distribution, then the following relation stands:

Log ([R.sub.[florin]]) = - Log([beta]) + [gamma][micro] - [[[[gamma].sup.2][[sigma].sup.2]]/2] (18)

Where [micro] is the average growth rate of per capita consumption and [sigma] is the variance of this growth rate. Without the application of Jensen's inequality the above relation will not stand. Equation 18 is utilized in the derivation of the so-called risk-free rate puzzle (Weil 1989). Application of this relation serves to show that the theoretical risk-free rate, obtained from this relation, is much higher than the actual or observed risk-free rate. See however Azer (2008).

The third purpose of this paper is to test whether the utility of the expected value is greater than the expected value of the utility, when the utility function is concave, as is should be. For this the utility functions in Eq. 7 are utilized. As in Azar (2006) the coefficient or relative risk aversion (CRRA), of [gamma], is taken to be 4.5, and wealth is omitted because it cancels out. Therefore the inequality, which is the same for the two utility functions, becomes as follows:

[[(1 + E([~.r])).sup.(1 - [gamma])]/(1 - [gamma])] - E(([1 + [~.r]).sup.(1 - [gamma])]]/(1 - [gamma]))> 0 (19)

where [~ r] is a random normal variable with mean 13.3% and standard deviation 20.1%, which are the first two moments of a portfolio of stocks (Ross et al. 2002, p. 233). Inequality (Eq. 19) is at the basis of the economics of risk and time and of the expected utility paradigm (Gollier 2001; Eeckhoudt et al. 2005). Moreover this inequality measures the equity risk premium in utiles.

Empirical Results

The empirical methodology is by simulation of random normal variables. The first two examples of Jensen's inequality depend on the first two statistical moments of foreign exchange rates among developed countries. The mean is approximately zero and the standard deviation is close to 10%. A random variable is simulated from such a probability distribution. The total number of simulations is 5,000. Each round of simulation is repeated 200 times, resulting in 200 inequalities of Eqs. 8 and 12. Table 1 presents the findings.

In what concerns Eq. 8, which normalizes without loss of generality the foreign exchange rate to 1, the average difference between the expected value of the ratio and the ratio of the expected value across the 200 data points, is statistically significantly different from zero (t-statistic: 44.649). The size of the difference is around 1%, which is economically material in most applications. It is noteworthy to point out that since the variables are mean values, the standard deviation of the average difference is at the same time a standard error. Then the sampling distribution is a distribution of a difference between means, and should be a normal distribution. Any skew and kurtosis are rejected, with respective t-statistics of 1.2384 and -0.6842, under the nulls of no skew and no kurtosis. This is confirmed by the Kolmogorov-Smirnov non-parametric normality test that provides an asymptotic probability of 0.516 under the null of normality. The two run tests, relative to the mean and relative to the median, fail to reject randomness. The Ljung-Box Q-statistics (Ljung and Box 1978) on the underlying variable and its square reject higher-order serial correlation and conditional heteroscedasticity at two lags, 6 and 12, with a minimum probability of 0.626. Therefore the sampling distribution is normal and independently and identically distributed, and this is as expected. The conclusion is that this variant of Jensen's inequality stands as an inequality in statistical terms. Since the foreign exchange rate is normalized to 1, the average difference is around 1%, which is also economically significant.

In what concerns Eq. 12, which does not need normalization because the log of the mean level of the foreign exchange rate cancels, the average difference, between the log of the expected value and the expected value of the log across the 200 data points, is statistically significantly different from zero (t-statistic: 41.737). It is noteworthy to point out that since the variables are mean values the standard deviation of the average difference is at the same time a standard error. Then the sampling distribution is a distribution of a difference between means and should be a normal distribution. Skew and kurtosis are rejected, with respective t-statistics of 1.8256 and 0.0789, under the nulls of no skew and no kurtosis. This is confirmed by the Kolmogorov-Smirnov non-parametric normality test that provides an asymptotic probability of 0.515 under the null of normality. The two run tests, relative to the mean and relative to the median, fail to reject randomness. The Ljung-Box Q-statistics on the underlying variable and its square reject higher-order serial correlation and conditional heteroscedasticity at two lags, 6 and 12, with a minimum probability of 0.479. Therefore the sampling distribution is normal and independently and identically distributed, and this is as expected. The conclusion is that this variant of Jensen's inequality stands as an inequality in statistical terms. The average difference is around 0.48%, which is close to the theoretical value of 0.5%, and is also economically significant.
```Table 1 Statistical analysis of two variants of Jensen's inequality

E (1/1 +           Log(E(1 +
[epsilon]) -       [epsilon])) -
(1/E(1 +          E(Log (1 +
[epsilon])            [epsilon]))
where [epsilon]       where [epsilon]
~ N (0, 0.01)         ~ N (0, 0.01)

Number of simulations per       5,000                 5,000
run

Number of runs                    200                   200

Sample size                       200                   200

Average differences                 0.01031353            0.00480393

Standard error                      0.00023099            0.0001151

t-Statistic for the null           44.649                41.737

hypothesis that the
average difference is
zero

Skewness t-statistic                1.2384                1.8256

Kurtosis t-statistic               -0.6842                0.0789

Probability of the runs             0.406                 0.847
test with the mean

Probability of the runs             0.321                 0.887
with the median

Probability of the                  0.516                 0.515
Kolmogorov-Smirnov
normality test

Probabilities of:

Q(6)                              0.641                 0.879

Q(12)                             0.626                 0.482

Probabilities of:

[Q.sup.2](6)                      0.663                 0.870

[Q.sup.2](12)                     0.648                 0.479

All expectations E(.) are the averages of the 5,000 simulations in each
run; the random variable is simulated to be normal with mean zero and
variance 0.01
Q(k) Ljung-Box Q-statistic with k lags for the sequence of the 200
differences, [Q.sup.2](k) Ljung-Box Q-statistic with k lags for the
sequence of the square of the 200 differences
```

The empirical methodology for the last inequality is still by simulation of random normal variables. The mean is 13.3%, and the standard deviation is 20.1%, being the first two moments of a portfolio of stocks (Ross et al. 2002, p. 233). The total number of simulations is 5,000. Each round of simulation is repeated 200 times, resulting in 200 inequalities of Eq. 19. Table 2 presents the findings.

For inequality (Eq. 19) the average difference across the 200 data points between the utility of the expected value and the expected value of the utility is again statistically significantly different from zero with a t-statistic of 15.363. Although the non-parametric Kolmogorov-Smirnov test fails to reject normality both skew and kurtosis have high t-values denoting the presence of non-normality. The parametric tests are usually more powerful than the non-parametric ones although the advantage of the latter is in the fact that no distributional assumptions are made. The run tests fail to reject randomness. The Q-statistics on the variable and its square, for lags 6 and 12, reject higher-order serial correlation and conditional heteroscedasticity, with a minimum probability of 0.490. Therefore the average difference is independently and identically distributed, without however being normal. In any case one can invoke the Central Limit Theorem for asymptotic normality because the sample size is large. The average difference is 0.0667, and is both statistically and economically large. This difference, as noted above, is the equity risk premium in utiles, which corresponds to an equity premium of 9.5%. Fama and French (2002, p. 637) find an equity premium of 5.57%. The estimate in Mehra and Prescott (2003, p. 894) is 6.92%. The estimate in Goetzmann (2006, p. 17) is 6.6%. The estimate in Mehra (2008, p. 7) is 6.36%. All these estimates are close to each other. The fact that the model in this paper produced an equity premium higher than the actual one is evidence that the equity premium puzzle (Mehra and Prescott 1985) does not show up in this paper's model.
```Table 2 Statistical analysis of one variant of Jensen's inequality

[(1 + E([~
r])).sup.(1-[gamma])/(1-[gamma]) -
E([(1 + E([~
r])).sup.(1-[gamma])/(1-[gamma]))
where [gamma] = 4.5 and where [~
r] ~ N (0.133, 0.040401)

Number of simulations per run                 5000

Number of runs                                 200

Sample size                                    200

Average difference                               0.06670410

Standard error                                   0.00434188

t-statistic for the null                        15.363
hypothesis that the average
difference is zero

Skewness t-statistic                            8.7209

Kurtosis t-statistic                           12.6374

Probability of the runs test                    0.797
with the mean

Probability of the runs test                    0.395
with the median

Probability of the                              0.065
Kolmogorov-Smirnov

normality test

Probabilities of:

Q (6)                                         0.496

Q (12)                                        0.532

Probabilities of:

[Q.sub.2] (6)                                 0.490

[Q.sub.2] (12)                                0.524

All expectations E(.) are the averages of the 5000 simulations in each
run; the random variable is simulated to be normal with mean 0.133 and
variance 0.040401
Q(k) Ljung-Box Q-statistic with k lags for the sequence of the 200
differences, [Q.sub.2] (k) Ljung-Box Q-statistic with k lags for the
sequence of the square of the 200 differences
```

Conclusion

This paper has tested the statistical significance of three variants of Jensen's inequality, applied to finance, by using the methodology of simulation of random normal variables. In the three cases the average difference is statistically significantly different from zero, and is economically large. The conclusion remains strong: Jensen's inequality, applied to finance, cannot be dismissed as insignificant. Moreover the third variant of Jensen's inequality tested provides assurances that the expected utility paradigm is not just a mathematical or a theoretical exercise, but has a statistical support. Finally, this last variant of Jensen's inequality produced an equity risk premium of 9.5%. This means that a CRRA, or [gamma], of 4.5 is sufficient, as a utility curvature, to explain the actual risk premium, as has been pointed out in Azar (2006), although the literature on the equity risk premium has found that a very high and unreasonable CRRA is needed to equalize the actual with the theoretical risk premium.

References

Azar, S. (2006). Measuring relative risk aversion. Applied Financial Economics Letters, 2, 341-345.

Azar, S. (2008). The minimum required rate of return. Applied Financial Economics Letters, in press.

Eeckhoudt, L., Gollier, C., & Schlesinger, H. (2005). Economic and financial decisions under risk. Princeton: Princeton University Press.

Fama, E., & French, K. R. (2002). The equity premium. The Journal of Finance, LVII(2), 637-659.

Goetzmann, W. N. (2006). The lessons of history. In W. N. Goetzmann, & R. G. Ibbotson (Eds.), The equity risk premium, essays and explorations (pp. 15-23). Oxford: Oxford University Press.

Gollier, C. (2001). The economics of risk and time. Cambridge: MIT.

Kritzman, M. P. (2000). Puzzles of finance. New York: Wiley.

Ljung, G. M., & Box, G. E. P. (1978). On a measure of lack of fit in time series models. Biometrika, 65, 297-303.

Mehra, R. (2008). Handbook of the equity risk premium. Amsterdam: Elsevier.

Mehra, R., & Prescott, E. C. (1985). The equity premium: a puzzle. Journal of Monetary Economics, 22, 145-161.

Mehra, R., & Prescott, E. C. (2003). The equity premium in retrospect. In G. M. Constantinides, M. Harris, & R. M. Stulz (Eds.), Handbook of the economics of finance, vol. IB (pp. 889-938). Amsterdam: Elsevier.

Ross, S. A., Westerfield, R. W., & Jaffe, J. F. (2002). Corporate finance (6th ed.). Boston: McGraw-Hill.

Sercu, P., & Uppal, R. (1995). International financial markets and the firm. Cincinnati: South-Western College Publishing.

Siegel, J. (1972). Risk, interest rates and the forward exchange. Quarterly Journal of Economics, 86, 303-309.

Silva, J. M. C. S., & Tenreyro, S. (2006). The log of gravity. Review of Economics and Statistics, 88(4), 641-658.

Weil, P. (1989). The equity premium puzzle and the risk-free rate puzzle. Journal of Monetary Economics, 24, 401-421.

Published online: 7 August 2008

S.A. Azar

Faculty of Business Administration and Economics, Haigazian University, Mexique Street, Kantari, Beirut, Lebanon

e-mail: azars@haigazian.edu.lb

(1) Inequality (Eq. 10) is sometimes called "Siegel's paradox" in reference of Siegel (1972).

(2) Kritzman (2000) also minimizes the significance of "Siegel's paradox."