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Jensen's inequality in finance.

Article Excerpt
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 < [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 < [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...