ALPHAS AND THE CHOICE OF RATE OF RETURN IN REGRESSIONS

Transcription

1

2 ALPHAS AND THE CHOICE OF RATE OF RETURN IN REGRESSIONS March 25, 2015 Michael Edesess, PhD Principal and Chief Strategist, Compendium Finance Senior Research Fellow, City University of Hong Kong Research Associate, EDHEC-Risk Institute

3 ALPHAS AND THE CHOICE OF RATE OF RETURN IN REGRESSIONS Spoiler / Abstract / Executive Summary: The standard method of calculating alphas from rate of return series may be causing alphas to be found where they don t exist, especially in highly volatile portfolios.

4 Alpha: the Holy Grail Alpha the «Holy Grail» of investing: Most seek it; many claim it; all desire it A top investment web site is called «Seeking Alpha» The Financial Times s financial markets commentary is called «Alphaville» Journalist Sebastian Mallaby, in his 2010 book «More Money Than God», which sings the praises of hedge funds, rests his claim of their superiority on a single journal article that found they had a positive alpha, in spite of having on average underperformed the Standard & Poors 500 index

5 But Alpha is Elusive Evidence can only be found in past returns Past records of outperformance are unreliable predictors of future alpha one out of twenty empirical alphas mined from random data will spuriously be found significant in a t-test at the.05 level Those who wish to find alphas in a strategy s past performance can often try variations on the strategy and different time periods and data series until they find a significant alpha

6 «Alpha Schmalpha?» Schwert Rule: After they are documented and analyzed in the academic literature, anomalies often seem to disappear, reverse, or attenuate. * Economists Nouriel Roubini and Stephen Mihm, in their 2010 book, Crisis Economics, ridiculed alpha by calling it Schmalpha. *G. William Schwert, Anomalies and Market Efficiency. Chapter 15 in Handbook of the Economics of Finance, eds. George Constantinides, Milton Harris, and Rene M. Stulz

7 And there is a problem with the method A finance professor at a prominent university told me That s all we do, run regressions, regressions When I asked him about the problem I am about to speak on, he said, Everybody knows it but nobody talks about it. Perhaps they all know about it possibly many do but certainly not everybody.

8 Portfolio returns Portfolio returns How regressions are performed in finance A brief tutorial: 36 monthly returns Market returns Find a line that minimizes the sum of the squared deviations from the line Alpha 36 monthly returns 0 Market returns

9 Probability Here is the problem Returns have a skewed distribution: Rate of Return The probability distribution is more like a lognormal distribution than the normal distribution assumed in the mathematical derivation that defines regression analysis When I was an undergraduate at MIT, I was told in a science class that when you perform regression analysis, if a data series clearly does not adhere approximately to the normal distribution assumption, you should transform it before performing a regression analysis and I have taught the same myself in statistics classes. THIS STILL HOLDS TRUE.

10 Here is the problem Returns have a skewed distribution: The skew is greater for longer time periods, but even for only one-month periods the distribution is still skewed.

11 Illustrative Example Here s the data Meaningful fit? Regression coefficients of the trendline meaningful? To correct the problem, returns should be transformed by taking their logarithms

12 Log scale More Familiar Example Which makes more sense: A Trendline? or B Trendline

13 My realization of the problem It bothered me for years that this wasn t being done, but like most people in finance, I assumed the regression analysis was «robust» enough that it didn t matter, and that the returns for short periods were close enough to their logarithmic returns that the results would come out about the same either way so like everyone else, I never took the time to check But then, the publisher of an online journal that I write for called my attention to an article in a journal that is little-known (and perhaps little respected) among finance academics, Financial Advisor Magazine

14 A serious discrepancy Background: In 1981 a peer-reviewed article by Rolf Banz appeared in the Journal of Financial Economics showing that small stocks had realized a significantly positive alpha over the period The authors of the Financial Advisor Magazine article, Gary Miller and Scott MacKillop, reran the regression but found no alpha I investigated; it turned out that Miller & MacKillop had logarithmically transformed the monthly returns data used by Banz before running their regression I asked for the data and confirmed their findings

15 A brief digressive tutorial on rates of return Rates of return (ROR) can be stated in many ways For example they can be stated as annualized rates Or they can be stated as the rate over whatever time period they apply, such as a month The definition in the simplest case is ROR = ending value divided by beginning value minus one (it is more complicated if there are cash flows in between) We ll call this (as did finance professor Zvi Bodie in his widely-used textbook on investments), the Holding Period Rate, or HPR The HPR is the one usually used in regression analyses in finance

16 A brief tutorial on rates of return (continued) The annual rate can even be stated as an APR (Annual Percentage Rate) which is 12 times the monthly rate (and it is required to be stated this way for mortgage quotes in the U.S.) The APR is not the HPR but the monthly-compounded rate; the HPR over a year s time comes out different for example if the APR is 6% then the HPR is (1+.06/12)^12-1, or % Compounding could also be done weekly, daily, hourly, or by the minute or second; each method of compounding yields a slightly different annual HPR The limit of more and more frequently compounded rates is the continuously-compounded rate, or CCR; this is also sometimes called the log-return; that is, it is the logarithmic transformation of the HPR, because CCR=log(1+HPR)

17 A brief tutorial on rates of return (continued) Rates of return can be stated in any of these ways There is no inherent reason why one way of stating them is necessarily better than another It depends on how they will be used As pointed out earlier, for use in regression analyses there is a strong reason to use the CCR, the continuously-compounded return, which is created by taking the logarithm of one plus the HPR

18 Period studied by Banz Results of doing it both ways Annualized alphas averaged about 2% lower when continuouslycompounded rates (CCRs) were used in the regressions than when the usual holding period returns (HPRs) were used When CCRs were used, the alpha for the time period that Banz wrote about in 1981 was negative, not positive Banz s results created an impression of risk-adjusted superiority of small-stock returns that has endured for more than 30 years

19 Could this just be due to the particular dataset a fluke? How to check? Randomly generate many series of rates of return from the lognormal distribution (the usual assumption of Monte Carlo simulations in finance) For each rate of return series, express the return both as a CCR and as a HPR For both the CCR series and the HPR series, run regressions to find alpha For this study, CCR series were generated for the market and a market sector (e.g. small stocks) designed to have zero alpha The result (next slide) was that while the CCR regressions of course resulted in zero alphas, the HPR regressions yielded significantly positive alphas

20 Spuriously high %age of significant alphas Spurious 3% alpha with HPR Simulation Results But falls to near-zero when volatility is low

21 Illustration helps explain Blue is CCR, red is HPR (graph is not exact only illustrative) Red HPRs tend to be more outliers on the upside than CCRs, and CCRs tend more to be outliers on the downside This drags the regression line and the alpha up when HPRs used

22 Conclusions Results of regression analyses to find alpha are unreliable, and may be mostly wrong when the dependent variable is the rate of return (represented by the HPR) on a highly volatile asset Beware of claims of alpha Findings that a three-factor (or four-factor) regression model «span» the data (i.e., that the factors completely «explain» the returns because the residual alpha is zero) may be incorrect And there are other reasons why the results of regression analyses in investment finance may be unreliable as well

23 Merci Thank You