Deriving the Expectation-Variance Model « Brainrack Aborning

Deriving the Expectation-Variance Model

March 29, 2008

Last week I did some work trying to make mathematical sense of the Precautionary Principle. For better or worse, except for a couple weeks in a Stanford econ class that I had to drop, I haven’t got any formal econ. (The prof told me that I was so obstreperous that most of his colleagues would deal with me by tossing me from their offices — I got the hint.) So, every time I wrote something down I had to go check to see if it was a known result, or simply false, or maybe a little bit new. Thus, I got a quickee education in some modern economics.

Several times along the way I ran across some hand-waving arguments about the functional form of the `total marginal utility curve’ and its relationship to peoples’ tendencies towards risk-aversion. When I tried to prove what I thought various authors were suggesting about this relationship I found I couldn’t. One hint about doing proofs: when you are stuck, look for a counterexample. It often helps understand things well enough to make the proof go through. Of course, sometimes, you actually find a counterexample; as in this case!

I had been wasting time trying to prove something that wasn’t true. It happens; but I was curious as to what was going on and started looking in the literature. It turned out that I had run across one of those things that is just too beautiful to be false. In spite of the fact that everyone knows the relationship only holds in very narrow circumstances, a lot of people continue to pretend it is more true than it is.

In the hope that it saves someone else some confusion, I have written up my notes on the subject. There’s nothing particularly new here, but it was a pretty good test of my LaTex to WordPress conversion macros in emacs, which I described here. I will publish them after they have ripened a little bit.

Along these lines I will also write up my notes on the mathematics of the Precautionary Principle in a few days, and then return to describing the latest developments on the Brainrack Search Engine.

[Update: 04 April 2008 - Fixed a few typesetting errors and other minor errors in the math.]

Pitfalls in Deriving
the Expectation-Variance Model
for Investment Value

G. Neil Haven
Liberty Reach, Inc.
Clearwater, Idaho

28 March 2008

ABSTRACT

The Expectation-Variance Model for Investment Value posits that the value of an investment to an investor is a function of the investor’s sensitivity to risk, the expectation value of the return from the investment, and the volatility in that expectation value. Gossen’s First Law, or the law of diminishing marginal utility, is often used to derive this relationship between the investment value of a set of assets, on the one hand, and the expected return and volatility of the set of assets, expressed in terms of a variance, on the other hand. Misleading evidence for and derivations of this relationship are rife in the formal and informal literature.¹The primary confusions arise from two assertions: that the utility function of wealth is quadratic;² and that the Taylor expansion of the utility function may be reasonably truncated at low order.³ This note reviews the fallacies in turn, provides a relevant counterexample, and exhibits the missing constraints needed for the argument establishing the Expectation-Variance Model to go through.

Statement of the Hypothesis

The statement of the false hypothesis is the following:

Hypothesis: Given $U(x)$ , a utility function,

$U(x) \ni: \forall x\, U'(x) > 0, U''(x) < 0$

and given two distributions

$f_1, f_2: \mathfrak{R}\Rightarrow[0,1] \ni: 1=\int_{\forall x}f_1(x)\,dx = \int_{\forall x}f_2(x)\,dx$

with the same mean

$\overline{x} = \int_{\forall x}f_1(x)x\,dx = \int_{\forall x}f_2(x)x\,dx,$

the inequality

$\int_{\forall x}f_1(x)(x-\overline{x})^2\,dx - \int_{\forall x}f_2(x)(x-\overline{x})^2\,dx$

$= \sigma^2(f_1) - \sigma^2(f_2) = \Delta_{\sigma^2} > 0$

implies

$\int_{\forall x}f_1(x)U(x)\,dx < \int_{\forall x}f_2(x)U(x)\,dx.$

The economic interpretation of this hypothesis is that, given the law of diminishing marginal utility, and assuming the expected payouts from two different investments is the same, the investment with less variance will be preferred.

Plausibility of the Hypothesis — Taylor Series Expansion

Expand $U(x)$ around $\overline{x}$ as a Taylor Series:

$U(x) = U(\overline{x}) + (x-\overline{x})U^{(1)}(\overline{x}) + \frac{(x-\overline{x})^2}{2!}U^{(2)}(\overline{x}) + \sum_{j>2}\frac{(x-\overline{x})^j}{j!}U^{(j)}(\overline{x}).$

Then form the integral $\int_{\forall x}f(x)U(x)\,dx$ and evaluate

$\int_{\forall x}f(x)\left(\sum_{j}\frac{(x-\overline{x})^j}{j!}U^{(j)}(\overline{x}) \right) dx$

$= U(\overline{x}) + \frac{\sigma^2(f)}{2}U^{(2)}(\overline{x})\,dx + \int_{\forall x}f(x)\sum_{j>2}\frac{(x-\overline{x})^j}{j!}U^{(j)}(\overline{x}) \,dx.$

Calculate the increase in investment value by forming the difference:

$\int_{\forall x}(f_2-f_1)(x)U(x)\,dx$

$= \frac{\sigma^2(f_2)-\sigma^2(f_1)}{x}U^{(2)}(\overline{x}) - \int_{\forall x}(f_2(x) - f_1(x))\sum_{j>2}\frac{(x-\overline{x})^j}{j!}U^{(j)}(\overline{x}) \,dx$

$= \frac{-\Delta_{\sigma^2}}{2}U^{(2)}(\overline{x}) - \int_{\forall x}(f_2(x) - f_1(x))\sum_{j>2}\frac{(x-\overline{x})^j}{j!}U^{(j)}(\overline{x}) \,dx.$

This last expression is positive, so that the investment value increases with decreasing variance at constant expected value, when

$-U^{(2)}\Delta_{\sigma^2} > 2\int_{\forall x}(f_2(x) - f_1(x))\sum_{j>2}\frac{(x-\overline{x})^j}{j!}U^{(j)}(\overline{x}) \,dx.$

Confusion arises because, assuming the Taylor Series expansion is convergent, the right hand side of this expression is close to zero. (Since $U^{(2)} < 0$ , the left hand side is positive, by hypothesis.)

Illegitimate Derivation — Truncation of the Taylor Series Expansion

The illegitimate step from here is to assert that, being a difference of two tails from Taylor Series expansions, we may treat the right hand side as zero compared with $\Delta_{\sigma^2}$ . That is, since

$\int_{\forall x}(f_2(x) - f_1(x))\sum_{j>2}\frac{(x-\overline{x})^j}{j!}U^{(j)}(\overline{x}) \,dx \approx 0$

then (falsely)

$-U^{(2)}\Delta_{\sigma^2} - 2\int_{\forall x}(f_2(x) - f_1(x))\sum_{j>2}\frac{(x-\overline{x})^j}{j!}U^{(j)}(\overline{x}) \,dx > 0.$

Since $\Delta_{\sigma^2}$ itself is the difference of two terms, and may be arbitrarily close to zero, the simplification is illegitimate.

An extra condition is needed for the valid applicability of the Expectation-Variance Model, to wit:

$-U^{(2)}\Delta_{\sigma^2} > 2\int_{\forall x}(f_2(x) - f_1(x))\sum_{j>2}\frac{(x-\overline{x})^j}{j!}U^{(j)}(\overline{x}) \,dx.$

Illegitimate Derivation — Quadratic Utility Function

Rearranging the extra condition we can obtain

$-U^{(2)}\Delta_{\sigma^2} > 2\sum_{j>2}\frac{U^{(j)}(\overline{x})}{j!} \int_{\forall x}(f_2(x) - f_1(x))(x-\overline{x})^j \,dx.$

A simple way to satisfy this condition is to require $U^{(j)}(\overline{x})$ to be identically zero for $j>2$ . That is, $U(x)$ must be quadratic. So $U(x) = ax^2 + bx + c$ . But a quadratic $U(x)$ cannot be a utility function: diminishing marginal utility implies $U^{(2)}(x) = a \prec 0$ , but then, for $x \succ -b/(2a)$ we have $U'(x) \prec 0$ , a contradiction. $\square$

I surmise that what happens here is that commentators confuse $U(x) = ax^2$ , which has a zero second derivative but the wrong structure for a utility function, with the inverse function $U(x) = kx^{1/2}$ , which has the correct structure for a utility function but a non-zero second derivative. The needed mental hybrid cannot exist.

Legitimate Derivation — Gaussian Distribution

Look at the extra condition again

$-U^{(2)}\Delta_{\sigma^2} > 2\sum_{j>2}\frac{U^{(j)}(\overline{x})}{j!} \int_{\forall x}(f_2(x) - f_1(x))(x-\overline{x})^j \,dx.$

A little rearranging shows that the right hand side is a sum of third-order and higher moments from the distributions yielding $f_1(x)$ and $f_2(x)$ . If the distributions are gaussian, this sum is identically zero. Thus, a sufficient condition for the Expectation-Variance Model is that $f_1(x)$ and $f_2(x)$ are pdf’s from gaussian distributions.

While it is true that a gaussian distribution arises naturally as the distribution of the sum of i.i.d. random variables, and so the hypothesis that $f_1$ and $f_2$ represent normal distributions may seem initially plausible, careful consideration, a priori, indicates that the default distribution of investment returns ought to be the result of a multiplicative, not an additive, process.⁴ In that case $f_1$ and $f_2$ would be most naturally taken to be pdf’s from log-normal distributions, not gaussian ones. The higher-order moments from a log-normal distribution are not zero.

Counterexample to the Hypothesis

Take $U(x) = \lg (x+1)$ . It is trivial to verify $U(x)$ is a utility function.⁵ To contradict the hypothesis above, it is enough to exhibit two distributions with identical means and identical variances, but non-identical expectation values.

Consider the distributions $D_1 = [x=1,p=2/3; x=4,p=1/3]$ and $D_2 = [x=0,p=1/3; x=3,p=2/3]$ . For both distributions, mean = variance = 2. The expectation value of $U(x)$ in distribution $D_1$ is $1+\lg 5$ . The expectation value of $U(x)$ in distribution $D_2$ is $2$ . $\square$

Restatement

From the above discussion we may collect a true restatement of the original hypothesis.

Given $U(x)$ , a utility function,

$U(x) \ni: \forall x\, U'(x) > 0, U''(x) < 0$

and given two distributions,

$f_1(x), f_2(x) \ni: 1=\int_{\forall x}f_1(x)\,dx = \int_{\forall x}f_2(x)\,dx$

with the same mean,

$\overline{x} = \int_{\forall x}f_1(x)x\,dx = \int_{\forall x}f_2(x)x\,dx,$

define $\Delta_{\sigma^2} =$

$\int_{\forall x}f_1(x)(x-\overline{x})^2\,dx - \int_{\forall x}f_2(x)(x-\overline{x})^2\,dx$ .

The inequality

$-U^{(2)}(\overline{x})\Delta_{\sigma^2} > 2\sum_{j>2}\frac{U^{(j)}(\overline{x})}{j!} \int_{\forall x}(f_2(x) - f_1(x))(x-\overline{x})^j \,dx$

implies

$\int_{\forall x}f_1(x)U(x)\,dx < \int_{\forall x}f_2(x)U(x)\,dx.$

In particular, if $f_1$ and $f_2$ are the pdf’s from gaussian distributions, $\Delta_{\sigma^2} > 0$ is a sufficient condition for the conclusion.

Comments

It seems a more natural statement of investment value at constant expected return would be in terms of the entropy of a distribution. Entropy is a single number describing a distribution, whereas most distributions require a series of moments to describe.

¹ – This is not a new observation.

² – Although the authors do not treat the origin of the confusion, for a review containing nearly a dozen examples within 10 years from the formal agricultural economics literature alone (!) see “Quadratic Utility and Linear Mean-Variance: A Pedagogic Note” Robert A. Collins, Edward E. Gbur Review of Agricultural Economics, 13(2) (Jul., 1991), pp. 289-291

³ – For a numerical discussion of the problem see Loistl, Otto “The Erroneous Approximation of Expected Utility by Means of a Taylor’s Series Expansion” The American Economic Review, 66(5) (Dec., 1976), pp. 904-910.

⁴ – See, for example, Conover, John “The Quantitative Analysis of Non-Linear Entropic Economics” online at http://www.johncon.com/ndustrix/.

⁵ – This is a reasonable utility function. If we assume that marginal utility decreases directly with increasing wealth, so that $U(x+dx) - U(x) = k/x dx,$ then $U(x) = lg(kx+c)$ .

Posted by brainrack

Filed in Off-Topic ·Tags: expectation variance, investment value, marginal utility

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