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**Asset Return Predictability**

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**Asset Return Predictability**

Chapter 1, CLM Introduce notation to a limited extent. Discuss the basic assumptions financial economists make about returns distributions. Review the various forms of the efficient markets hypothesis.

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**Chapter 2, CLM Tests of asset return predictability.**

Various forms of the random walk hypothesis. Tests of the random walk hypothesis: CJ test, runs test, technical trading rules. Variance ratio tests (LM 1988). Autocorrelations (FF 1988, Richardson 1993). Long horizon returns. Application: Momentum (JT 1993, CK 1998, DT 1985). We will have a lecture next week from Anne about access to and use of the various data bases supported here.

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Why Returns? The statistical methods you will learn in this course will be used primarily to analyze returns and the relations between different returns, not prices. This may seem paradoxical, because one might think that asset pricing models would have a lot to say about how assets are priced. Although it is true that financial economists have devised such models, dividend discount model earnings multiple valuation model they work notoriously badly, primarily because forecasting both cash flows and future interest rates is incredibly difficult.

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The Focus on Returns The price formation process is taken as given – investors have no market power. Investment technology is taken as a constant returns to scale technology so return is a scale free description of the opportunity. The focus is then on what stock returns ought to look like as a function of: Risk Information flows Risk – risky returns have higher expectations than safe returns if investors are risk averse Information – if investors are rational the contents of their information sets should be impounded into the price and so price changes should follow from new information flows.

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A Technical Issue Returns processes are thought to be stationary while price processes are not. To use the statistical techniques commonly applied by economists it is necessary that the sample moments converge to the population moments. Typically, what is assumed is covariance stationarity and ergodicity (or perhaps mixing rather than ergodicity).

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**Stationarity and Ergodicity**

The material will typically deal with the time series properties of returns. It is common to compute numbers such as: expected returns, the variances of returns, and the covariance between the returns of one asset and the returns of another. What is required for this to make sense?

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**Stationarity and Ergodicity…**

These statistics must be well defined in the sense that they do not change (except perhaps in some pre-specified way) over the course of the analysis. Covariance stationarity and ergodicity are the assumptions commonly employed to ensure this basic requirement.

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**Covariance Stationarity**

A stochastic process yt is weakly stationary or covariance stationary if 1. E(yt) is independent of t 2. Var(yt) is a finite positive constant, independent of t. 3. Cov(yt, ys) is a finite function of t-s but not of t or s. Note: stock prices do not satisfy the first requirement. Indeed, returns do not seem to but their expectation may vary in some predictable manner. The second simply says that the volatility does not change over time. The third is that the covariance between observations in a time series is independent of where they are in time and depends only on how far apart they are. If the dependence between the innovations at different times that are far apart is too big, initial conditions will matter a lot, and sample moments don’t converge to population moments. Thus we do need more.

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Ergodicity Intuitively, this means that values of the process sufficiently far apart are uncorrelated. Therefore, by averaging a series through time one is continually adding new and useful information to the average. Thus the time series average is an unbiased and consistent estimate of the population mean and estimates of the variance and autocovariances will be consistent.

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**Basics Simple return from time t-1 to t:**

where Pt is the price of the asset at time t. Simple Gross Return: 1+Rt

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Basics… Compound return over k periods:

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Annualized Returns Often the returns expressed in the popular press are annualized. Let k be the number of compounding periods and let n be the number of compounding periods in a year, so that there are N=k/n years of data. The annualized return is then defined as the geometric average of the returns:

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Annualized Returns For example, suppose the compounding interval is monthly (n = 12), the monthly return is 1%, and there are two years of data (k = 24). Then, the annualized return would be given by:

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**Annualized Returns: An Approximation**

The annualized return is often approximated using the arithmetic mean: • This approximation can be fairly poor. The problem is larger when the returns themselves are larger or when the compounding interval is smaller, but the issue is the same.

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Annualized Returns Suppose there are instead, 360 compounding periods in a year and the return in each is 1/30%. Then, the annualized return is actually while the approximated return is still 0.12.

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**The Alternative: Continuous Compounding**

The continuously compounded return, or log return ri of an asset is defined as the natural logarithm of its gross return: Why is this called the continuously compounded return?

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**Continuously Compounded Return**

For some reason, although banks calculate and pay interest more frequently, (quarterly, monthly, daily, or continuously), it is traditional to quote the rate on an annual basis. Call that rate Rnom (nominal). Then, if there are n compounding periods per year, the rate of return per period is Rnom/n. And the rate of return per year is (1+Rnom/n)n – 1.

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**Continuously Compounded Return**

If we take the limit as n goes to infinity, we get an annualized return of and the balance grows to at the end of the year, so that is the gross continuously compounded return.

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**Continuously Compounded Return**

Taking logs yields: r = Rnom. This is essentially what CLM call the continuously compounded return. Of course, this is just an approximation, because n never goes to infinity but it is usually a very good one. Ex: suppose returns were compounded monthly and the monthly return is 1%. Annualized gross return is (1.01)12 = Log monthly returns are ln(1.01), 12 of them are 12ln(1.01) = If you took this nominal return and compounded it continuously (i.e. raise e to that power) you would get the gross return of

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**Why Use Logs Conversion of products to sums**

Consider multiperiod return 1+Rt(k). It’s the product of single period returns But the log return is

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Why Use Logs The continuously compounded return is the sum of the continuously compounded single period returns. This makes some things much easier in modeling time series behavior. We will see that it is easier to model the behavior of sums than of products. We will also see that it allows the imposition of limited liability in a straightforward way.

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A Slight Problem The simple return on a portfolio is the weighted average of the simple returns on the individual securities in the portfolio: But Simply because the log of a sum is not equal to the sum of the logs.

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Example Suppose that N=2, 1+R1=1.12, 1+R2=1.08, w1 = w2 = .5, then 1+Rp = 1.10. The log portfolio return is rp = ln(1.10) = But .5ln(1.12) + .5ln(1.08) = When returns are measured over shorter intervals of time, the approximation is better. Still, it is traditional to use simple returns when a cross section of assets is studied but log return for time series studies.

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Dividends When the asset in question pays periodic dividends, the simple net return is: The simple gross return and the log returns are defined from this.

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Excess Returns An excess return is the difference between the return on an asset of interest and the return on a reference asset, R0t. Quite often the reference asset is a riskfree asset or short term T-bill, maybe a zero beta asset. The simple excess return on asset i is defined as Zit = Rit – R0t

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Excess Returns The log excess return is not the log of the excess return, but instead: zit = rit – r0t, the difference between the log return on the asset and the log return on the reference asset. The excess return can be thought of as the payoff on an arbitrage portfolio, long asset i and short the reference asset so there is no initial investment. Because the initial investment is zero the return is undefined but the payoff will be proportional to the excess return.

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“Perhaps the most important characteristic of asset returns is their randomness. The return of IBM stock over the next month is unknown today, and it is largely the explicit modeling of the sources and nature of this uncertainty that distinguishes financial economics from other social sciences…without uncertainty, much of the financial economics literature, both theoretical and empirical, would be superfluous.”

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**Common Distributional Assumptions**

Much of financial econometrics makes the assumption of normal distributions, but some care must be taken in determining what is normal. Normality is appealing because sums of normally distributed random variables are normal. If simple returns are iid normal what happens? First, this violates limited liability. Second, multiperiod simple returns then cannot be normal since they are products of simple returns.

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Lognormality Let single period simple gross returns be lognormally distributed so that the continuously compounded or log returns are normally distributed. That is, rit is i.i.d normal with mean μi and variance σi2. Then 1+Rit is lognormally distributed and thus has a minimum realization of zero.

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**Lognormality Rit has a mean and variance given by:**

The lognormal model is what is generally used.

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Data and Statistics This being an empirical course, it is important to understand the difficulties associated with estimating something as simple as the mean return. What we focus on now are the properties of estimates of Means Other moments

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Empirical Validation In the book you will see empirical properties of stock returns. They are not really consistent with either the simple normal or the lognormal models. You need to note the differences and see how things might be improved. Our continuing discussions will examine these issues.

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Empirical Validation Chapter 2 considers the predictability of asset returns. One question will be: Can past return realizations tell us anything about expected future returns. The efficient markets hypothesis (EMH) is an important aspect of this discussion.

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The Problem Estimating means requires more data than we can reasonably expect to get. That is, the time series is not likely to be stationary for long enough for us to get enough precision. Luenberger calls this “The Blur of History.”

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EMH Fama (1970) “A market in which prices always `fully reflect` available information is `efficient`.” Malkiel (1992) “A capital market is fully efficient if it correctly reflects all information in determining security prices. Formally, the market is said to be efficient with respect to some information set… if security prices would be unaffected by revealing that information to all market participants. Moreover, efficiency with respect to an information set… implies that it is impossible to make economic profits by trading on the basis of [the information in that set].”

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**In an efficient market, prices should be random**

Let the price of a security at time t be given by: Pt = E[V*|It] = Et V* The same equation holds one period ahead so that: Pt+1 = E[V*|It+1] = Et+1 V* The expectation of the price change over the next period is: Et[Pt+1 - Pt] = Et[Et+1 V* - Et V*] = 0 because It is contained in It+1 , so Et[Et+1 V*] = Et V* by the law of iterated expectations.

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Discussion The second sentence of Malkiel’s definition expands Fama’s definition and suggests a test for efficiency useful in a laboratory. The third sentence suggests a way to judge efficiency that can be used in empirical work. This is what is concentrated on in the finance literature. Examples: mutual fund managers profits if they are true economic profits then prices are not efficient with respect to their information. Difficult to test for good reasons we will discuss.

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**Versions of Efficiency**

Weak Form Information set is the set of historical prices (and sometimes volumes). Semi-strong Form Information set is the set of all publicly available information. Strong Form Information set includes all knowable information.

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**Violations of Efficiency**

That technical traders can make money violates which form? Reading the Wall Street Journal and devising a profitable trading strategy violates which form? Corporate insiders making profitable trades violates which form? Question: Can markets really be strong-form efficient?

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**What Does “Profitable” Mean?**

We’re talking about economic profits, adjusting for risk and costs. Need models of such things, particularly the risk adjustment. One way of thinking of the tests of efficiency is that they are joint tests of efficiency and some asset pricing model, or benchmark. For example, many benchmarks typically assume constant “normal” returns. This is easier to implement, but doesn’t have to be right. Hence rejections of efficiency could be due to rejections of the benchmark. Time varying returns or time varying risk premia for example. Just the wrong measure of risk, etc.

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The Tests Most tests suggest that if the security return (beyond the mean) is unforecastable, then market efficiency is not rejected. With the wrong asset pricing model, we can wind up rejecting efficiency. It would be easy to find (de-meaned) returns to be forecastable if we had the wrong mean.

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