Noise Traders - Detailed Derivation

University of California, Berkeley Summer 2010 ECON 138

A common response to the behavioral anomalies we have discussed in this course is that they have little impact on the long run, because investors with such anomalies are not maximizing optimally, and would thus be driven out of the market. We shall show, however, that noise traders—traders who are not maximizing—can indeed survive even in a very simple market setting. The model is taken out of De Long et al, "Noise Trader Risk in Financial Markets," The Journal of Political Economy, Aug. 1990.

I. Setting . Two-period model. Invest in the first period, consume in the second . CARA utility . One risk-free asset with return r and infinite supply

. One risky asset with price and pays a dividend of r. Supply is fixed to 1. . noise traders and rational traders, . Noise traders misperceive the expected price of the risky asset in the second

period by

II. Derivation 1. Utility Function We start off with CARA utility

Where is the rate of absolute risk aversion and is wealth in dollars.1 If the return of the risky asset is normally distributed, is also normally distributed, and thus utility is log-normally distributed. As we did in the first lecture, we can take log of the

utility function and use to get

Dividing the above by gives us .

1 This is a bit different from the CARA utility we have seen before, as we are not having a as the numerator. This is alright because is a positive constant as long as the investors are risk averse.

1 2. Maximization Problem i. Rational Traders

Let be the number of risky shares rational traders purchase and their current wealth. Rational traders maximize

Next period's wealth is the sum of wealth in risky asset and wealth in risk-free asset, dividend stock price next period wealth in risk-free asset

So the maximization problem becomes

Breaking up the expectation and variance,

Taking the first-order condition with respect to ,

We have

Which is essentially the condition we obtained in the first lecture—excess return over rate of risk aversion times variance.

ii. Noise Traders The maximization problem of noise traders is identical except they mistakenly think that

Note that is only random throughout time. For any given period it is not random. Think stock price in real life—stock price changes, but at any given moment you can just check Yahoo! Finance for the actual stock price. Noise traders' first order condition gives

2 3. Market Price for Risky Asset Price is determined by requiring the market for the risky asset to clear,

4. Steady State Assumption Price in period depends on the expected price in period . To find that we

assume that the economy is in steady state, so that and

. We can then apply the pricing formula to ,

Rearranging the terms we have

As for the variance, notice that interest rate/dividend, expectations and variances are all numbers. Numbers are not random and thus have zero variance. Therefore,

Plugging everything back to the pricing formula for we have

5. Returns We are interested in comparing the expected returns earned by the two types of traders,

is the difference in number of shares of risky asset held by the two types,

while is the expected excess return of the risky asset.

equals to

This number is positive whenever . This means that whenever the noise traders overprice the risky asset—being "bullish"—they hold more risky asset than rational traders.

For excess return,

from II.3

Excess return is increasing in variance in mispricing ( ) but decreasing in overpricing

( . Intuitively, higher variance in mispricing causes higher variance in noise traders'

4 demand for the risky asset. This in turn induces higher variance in the price of the risky asset. For market to clear, the average price for risky asset has to drop, raising its expected return along the way. As for overpricing, higher overpricing pushes up noise traders' demand for the risky asset, drives up its price and lowering expected return.

Finally, the difference in expected return is

Take expectation again. Because , we have

This formula sums up what we have discussed above. To make it easier to understand we can break it down into different parts,

A B where and .

Noise traders will have higher excess return if . They overprice the risky asset on average (A), but . Not overpricing so much that it drives price too high (B). . There is bigger variance in mispricing (C). Variance discourage rational traders from buying the risky asset, drives price of the risky asset down and thereby raising its expected return.