Performance in Normal and Turbulent Markets...VZ WMT XOM However, if mean reversion is present, then...

1 Michael Tan, Ph.D., CFA, AIC Capital LLC © Copyright (2008) All Rights Reserved Confidential Working Draft – Do Not Distribute

Performance in Normal and Turbulent Markets An Explanation of the Accordant Strategy

The Accordant Strategy is a price-based quantitative equity program operated by AIC Capital LLC that exploits short-term mean reversion in stock prices. Recognizing that expected returns and return predictability for stocks vary over time, the Strategy employs a trading strategy that is more opportunistic than many of its peers. With bets on the reversion of temporary stock price displacements as well as short-term market direction that vary in number and size as prospects change, the Strategy seeks to perform well in times of market turmoil without giving up the return naturally due it during normal times.


1 Predictable Returns Not only are stock returns predictable, they can be much more predictable at certain times than others. This fact should be exploited for profit in a well-crafted strategy.

Stock returns are predictable. This is true even when they are modeled endogenously based on past returns and not on external factors such as earnings or dividend yield1. They can also be more predictable at certain times, and less so at others, as shown by academic studies of time-varying risk premia (mean excess returns over risk-free rate) and model and parameter specification risks2,3,4,5. For example, the value effect is regarded as a risk premium that can be more reliably captured via exposure to systematic risks during times of economic uncertainty or financial distress. While the time horizons in these studies are too long to be relevant to a competitive hedge fund strategy, and the conclusions therein already operating principles for most practitioners, the possibility of capturing a fast time-varying risk premium from a non-macroeconomic source by means of a mechanical trading strategy is a tantalizing one. This is because the strategy will have a high Sharpe ratio and is likely to work for a long time. Here we argue that the Accordant Strategy operated by AIC Capital LLC is just such a strategy. The return of the Strategy comes from exposure to the well-known risks of liquidity provision, such as adverse selection or information asymmetry (trading against better informed traders or insiders) and noise trading (trading against uninformed but well-endowed traders). The Strategy is distinguished by its ability to perform well in times of market turmoil without giving up the returns naturally due it during normal times. It does this by allowing


Table 1

A study of what happens to the Accordant Strategy during and following market declines greater than a given percentage: The Strategy falls much less as the Index falls and rises much more when it rebounds. The period for the study is between January 3, 1990 and July 31, 2008.

Return series S&P 500 Accordant (simulated)

p -3%

Average 5-day return when Index 5-

day return is less than p

-4.5% -0.4%

Average ensuing 5-day return 0.9% 1.3%

Standard Deviation of ensuing 5-day

returns 3.3% 3.2%

% of positive ensuing 5-day

returns 62% 72%

number of observations

317

The Strategy rises sharply in the aftermath of severe market declines with a high degree of certainty and is profitable overall even when the market is not.


p -7%

Average 5-day return when Index 5-

day return is less than p

-8.4% -2.5%



returns 2.7% 4.3%


returns 62% 88%


22

its bets to vary according to return predictability, i.e. the uncertainty of the parameters that govern the probability distribution of returns. It bets heavily when there are many stocks in its universe whose expected excess returns are high and of the same sign (this is a harbinger of greater return predictability) and keeps a partial reserve of capital when they are not. We will argue later that the Strategy is also favored from the point of view of hedging demand despite its tendency to take on greater risk when stocks are perceived to be doing very poorly or very well. To draw the reader to the Strategy, we showcase its salient features in Table I. This is a simple statistical analysis of consecutive 5-day returns of a simulation of the historical performance of the Strategy and the S&P 500 Index between January 3, 1990 and July 31, 2008. In the analysis, all consecutive 5 trading day periods during which the aggregate S&P 500 Index return is less than p, where p is set in turn to -3% and -7%, are identified. Then for each value of p the average returns of the Index and the simulated Strategy for the period and the ensuing 5-day period are computed. The standard deviation of the returns and the percentage of positive returns in the ensuing 5-day period are also computed. It is clear from the statistics presented that there is mean reversion in S&P 500 Index returns. If we were to buy the Index every time it falls at least 3% percent, we would make about 6% annually in the 19-year sample period. Comparing the Index and simulated Strategy returns, we see that the simulated Strategy falls much less on average when the Index falls at lot and yet recovers in the aftermath much more than what it lost. As the Index rebounds, the Strategy generates a 1.0% return on average in the ensuing period when the market declines 3% or more in the preceding period; it generates a 3.3% return on average when the market declines 7% or more in the preceding period. Because the Strategy maintains approximate market neutrality at all times, it is not surprising that it should decline much less than the Index. However, what is striking is that when the market declines sharply (7% or more), the Strategy rises sharply in the aftermath of the decline with a high degree of certainty (88%). Moreover, the average total return of the Strategy for the combined decline and aftermath episode is positive, even though the corresponding average


Table 2

A study analogous to the one presented in Table 1 showing that the Accordant Strategy has a more consistent albeit lower return than the market during and following sharp rallies.


p 7%

Average 5-day return when Index

5-day return is more than p

8.5% 3.2%



returns 3.3% 2.4%


returns 62% 65%


20

Figure 1

The Strategy builds positions slowly as the market declines, always keeping some capital in reserve to trade at the extremes. This investment posture, coupled with approximate market neutrality and high alpha, turns sharp market declines and rallies into great trading opportunities.

Index

Strategy

reserve capital

build positions

liquidate positions

return for the Index is negative. Thus the Strategy is likely to do well in times of market turmoil, which are invariably precipitated by sharp market declines. This behavior is achieved by design through restraining capital usage in quiet times so that enough is left to be deployed in tumultuous times when expected excess returns are high.

2 Mean Reversion One cannot trade for profit unless one’s counterparty also thinks he has a chance to do the same. This is why mean reversion will always be present in the market. If all market participants were to agree on the value of a stock, then there would be no incentive to trade the stock. For how could a trade occur unless both parties to the exchange think that each could profit from the other’s miscalculation. To induce one’s counterparty to trade, the price must be attractive enough for him to think that he too has a chance to make money. Thus the fact that trading occurs at all imply that prices will fluctuate more than warranted by news or changes in market or industry factors and will revert with time to warranted levels. We believe the above to be the most fundamental truth about markets. Lest such a self-evident truth be lost on those who require mathematical proof, we cite in the bibliography several studies (including one of ours) that show that mean reversion in stock prices is persistent, ubiquitous and present on many time scales1,6. One statistical test for the presence of mean reversion which is also intuitively appealing is the following: If stock returns has a stationary probability distribution and are independent over time (this means there is no mean reversion and no return predictability), then both the mean return and variance of returns will increase linearly with the investment horizon. In this case, the variance of multiperiod returns is just the variance of one-period returns times the number of periods. This is a distinguishing feature of purely diffusive processes, i.e. random walks.


Figure 2

The term structure of variance of returns of stocks from the Dow Jones Industrial Index: The graph below shows the normalized centered variance ratio of the returns of the given stock versus the aggregation period (time horizon). The formula for the variance ratio is

111122

13

M

qM

qq

NqR

where

qN

iiqiq

qN

iiqiq

yyqN

qyy

Nq

qNqqM

/

0

2/

0

loglog1

1

loglog

11)(

1

and N is length of the sample, yi the price on day i, the mean return and q the aggregation period. The date range for the price data used in computing the variance ratios is from January 1, 1990 to August 31, 2008. The variance ratio is basically the ratio of long-term variance to short-term variance after adjusting for time horizon. The fact that many stocks exhibit a monotonically decreasing variance term structure means that they are comparatively more mean reverting the longer the time horizon, relative to the excursions they could have had given the time horizon. Similar studies at even longer horizons point to the practical aphorism: Stocks are less risky the longer they are held.

0 10 20 30 40 50-5

-4

-3

-2

-1

0

aggregation period (days)

norm

aliz

ed c

ente

red

varia

nce

ratio

s

PGTUTXVZWMTXOM

However, if mean reversion is present, then the variance will increase slower with horizon than the rate implied by a random walk process. In other words, short-term volatility will be greater than long-term volatility after adjusting for horizon, which is a state of affairs that aligns well with our casual belief that stocks are comparatively less risky over the long run because prices tend to come back after moving too much in the short run. To test for mean reversion, we simply compare long-term return variance to short-term return variance after normalizing for time horizon and finite sample size effects7. Mathematically, we compute the variance ratio, i.e. the variance of q period returns divided by q times the variance of one-period returns, from price data, where q is the aggregation period or time horizon of interest. We subtract one from this value and divide by a normalization factor (given in Figure 2) to get the normalized centered variance ratio for ease of comparison to its asymptotic (i.e. large-sample) probability distribution, which is a standard normal distribution. Figure 2 shows the variance ratios as a function of time horizon for a selection of stocks from the Dow Jones Industrial Index. Since values of the variance ratio less than zero indicate mean reversion, we see that the returns of these stocks are indeed mean reverting – and in many cases significantly so. In a separate study, we find that for time horizons up to 50 trading days the proportion of stocks in the S&P 500 Index with an average negative variance ratio is 80%. It appears that the returns of a vast majority of stocks are indeed mean reverting to some extent. If we average the negative of the variance ratios at a chosen time horizon for all stocks, then we create a mean reversion index that depicts the amount of mean reversion over that horizon in the market. The negative is used because the resulting index would be easier to interpret. It would increase with increasing mean reversion and would be greater than zero if mean reversion is present. Figure 4 depicts a mean reversion index constructed from the returns of the largest 350 stocks by market capitalization in the U.S. between June 1997 and July 2008. The graph shows that mean reversion has almost always been present in the market. Moreover the amount of mean reversion (magnitude of the index values) has remained at historical levels.


Figure 3

The variance ratio test: The graph below shows the minimum variance ratios (vertical streaks) observed for time horizons up to 50 trading days for stocks from the Dow Jones Industrial Average plotted together with the asymptotic distribution corresponding to no mean reversion. For normalized centered variance ratios, the asymptotic distribution is a standard normal. Hence testing for mean reversion simply involves choosing a significance level, e.g. 10% or equivalently a -1.31 standard deviation cut-off, and checking that the observed variance ratio lies to the left of the cut-off.

-6 -4 -2 0 2 4 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

normalized centered variance ratio

pro

ba

bili

ty d

istr

ibu

tion

Figure 4

The historical levels of mean reversion and return dispersion in the market: The graph below depicts a mean reversion index based on an aggregation period of 20 trading days juxtaposed against a return dispersion index calculated from the returns of the largest 350 stocks by market cap in the U.S. between June 1997 and July 2008. The mean reversion index is almost always greater than zero (indicating presence of mean reversion). Return dispersion was higher between 1998 and 2002, trended lower between 2003 and 2007, and then rose considerably in 2008.

1997 1998 2000 2001 2002 2004 2005 2006 2008 2009-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

mea

n re

vers

ion

inde

x

mean reversion index

return dispersion

Figure 4 also depicts the historical levels of return dispersion, a related quantity which is the cross-sectional variance or contemporaneous variation of returns among stocks. It shows that return dispersion has waxed and waned considerably over the past decade. Mathematically, return dispersion is the same as one minus the average correlation between stocks6. In other words, at any given instant, return dispersion is greater if the average correlation is lower and vice versa. Return dispersion is relevant to stock pairs trading strategies because they require both mean reversion and some degree of negative correlation between stocks to work. Clearly if all stocks are perfectly correlated, then the return spread between any two stocks will be static and no money can be made from it. Pairs trading strategies may not work when return dispersion is low. In contrast, the Accordant Strategy is primarily a bet on mean reversion and does not require the presence of return dispersion to generate excess returns.

3 Enduring Risk Premia By itself the fact that price displacements are mean reverting does not imply that money can be made. What is also needed is that betting against such displacements involves risk. The presence of mean reversion in stock returns is but one of many “biases” and “anomalies” that permeate the post information efficiency view of the markets that has gained currency in the past two decades. It is a view that says convergence to efficient pricing is no longer instantaneous and guaranteed, but is rather a result of trading that involves risk. The fact that price displacements are mean reverting does not by itself imply that money can be made. What is also needed, apart from market friction being surmountable, is that betting against such displacements is a risky undertaking. Indeed the Accordant Strategy does not capture a market inefficiency. Rather it captures a risk premium by placing risky bets on contrarian opportunities that arise when prices overshoot a


measured trend. In our view, the advantages of such a strategy are greater precisely because its returns come from exposure to real risks. If the return had come from a market inefficiency involving little or no risk exposure, then it may be arbitraged away quickly. But because the risks of liquidity provision, viz. adverse selection and noise trading, are well-known and widely shared, the risk premium will persist because a large class of investors will choose to avoid the opportunity, even if it is readily accessible, for reason of sensitivity to the risk that comes with it. To illustrate the argument for why mean reversion trades will generate high excess returns and will do so for a long time to come, we consider the following two scenarios: In the first scenario, suppose the price of a stock is falling quickly and sizably. Liquidity demand manifested as “selling pressure” arose in the stock from investors who must reduce their portfolio holdings of this stock – perhaps because their portfolios are not diversified enough or because as agents they are occupationally unable to tolerate the risk of the stock doing badly. The Accordant Strategy would step in to help satisfy this liquidity demand, but only at a low enough price that assures it a reasonable chance of making money on the purchase. The investors who must sell the stock quickly will likely have to accept a price that is lower than justified by the public information currently available on the stock (for fear that the price will go down even more when new information comes out). Since the Strategy would only purchase a small position in the stock relative to its entire portfolio, it is able to wait for the liquidity demand to fade and the price to recover. Thus risk transference works in this case because the Strategy is systematically positioned to bear this risk by allowing time and diversification to work in its favor. In the second scenario, suppose there is a stock whose fundamental value should not change, because no news specific to the company has come out. Suppose the price of the stock does in fact change because an uninformed (or “noise”) trader has been buying it. Such a trader is defined as one who trades not on information (perhaps because he has liquidity needs) or one who trades on noise as if it were material information. Many market microstructure studies indicate that the noise trader is a good paradigm to model how markets work8. One study has argued, on the basis of comparison between days

Table 3

Lead-lag relationship between the returns of the Accordant Fund and the market: The coefficients and confidence intervals of linear regressions of the simulated 5-day returns of the Strategy on the lagged 5-day returns of the market between January 3, 1990 and July 31, 2008: On all market returns:

051.0,091.00039.0,0030.0

070.00035.0 5,105,

tttt Rr

On the subset of market returns which are negative:

264.0,364.0000.0,002.0

314.0001.0 5,105,

tttt Rr

Here rt,t-5 is the Strategy return from day t to day t-5,

and Rt-10,t-5 is the market return from day t -10 to day t - 5. The numbers in square parentheses are the 90% confidence intervals for the regression coefficients. The second regression is that of the subset of Strategy returns on lagged market returns which are negative. The loading of the Strategy return on the market return, i.e. the slope of the regression equation, is significantly negative, indicating that negative market returns presage positive Strategy returns and vice versa.

Figure 5

Optimal allocation between the Accordant Strategy and a hurdle rate asset as a function of investment horizon: Assuming that the investor’s preferences for wealth W at the end of the investment is described by the following power utility function

A

WWU

A

1)(

1

where A is a constant that characterizes the degree of risk aversion, we maximize U(W) as a function of the weight allocated to the Strategy versus a competitive “risk-free” asset acting as a hurdle rate. We seek the utility maximizing weight as a function of the investment horizon h under two different assumed probability distributions of Strategy returns: (1) a normal distribution whose mean and variance correspond to those of the historical returns of the Strategy; and (2) a distribution generated by regressing the Strategy returns on the lagged market returns. For (2), we randomly select segments of size h from the time series of market returns between January 3, 1990 and July 31, 2008, and then regress the Strategy returns on the lagged market returns in these segments.


on which company-specific news was released and those on which it was not, that more than 70% of price changes cannot be explained by news or changes in market and industry factors9. Any price change caused by noise traders ought to be small and brief, for surely arbitrageurs will provide the necessary liquidity to revert the price to its intrinsic valuation. However, the provision of liquidity is risky because noise traders may be large in numbers and well-funded. They may move the price even further from its intrinsic value and prevent it from reverting for a long time. Additionally, the intrinsic value is never certain or precise, for it involves discounting uncertain estimates of the future cash flows of the company. Noise trading therefore creates a risk which is not easily arbitraged away. Hence mean reversion works because (1) providing liquidity involves the real risks of adverse selection and noise trading; and (2) the compensation for carrying these risks away from those who must lay them off quickly can be repeatedly earned by a well-designed trading strategy.

4 Hedging Demand A strategy that has high expected returns after stocks have fallen a lot is a good hedge against reinvestment risk, i.e. the risk of poorer prospects in the future. One argument in favor of the Accordant Strategy when we look at the historical returns presented in Section 6 is that it has a decent Sharpe ratio and low correlation to the S&P 500 Index. Hence it provides diversification benefit (reduces risk without reducing return) to a traditional stock portfolio. A less conventional argument in its favor is the following: The Strategy tends to perform well during turbulent times from increased expected stock returns, increased model certainty and increased leverage. In particular, it tends to rally substantially during the aftermath of sharp market declines as we showed in Section 1 (even if the market does not

Figure 5 (continued)

We then iterate the following equation:

Rr

where R is the market return, and the coefficients of the regression calculated for the segment in question, a random number drawn from a normal distribution with the same mean and variance as the residuals from the regression. We derive a sample for the terminal W by generating many samples of size h for the Strategy return as explained above, and setting W, for each sample of the Strategy returns ri, to the following:

h

iif rhrW

1

1)(

Here is the weight allocated to the Strategy and rf the hurdle rate over one time period. We then compute the expected utility by averaging U(W) over many values of W. In this way we obtain the expected utility U(,h) as a function of and h. To see how the optimal (i.e. expected utility maximizing) allocation changes with horizon, we plotted the allocation max that maximizes U(,h) as a function of h below. We set A to 10, limit to between 0 and 1 and set rf to 10% per annum so that it could represent a strongly competitive asset (else the Strategy would always be preferred, i.e. equal 1, for all but the shortest horizons). The pink line in the graph corresponds to the allocation under distribution (1), i.e. independent and unpredictable Strategy returns. The blue line corresponds to the allocation under distribution (2), i.e. Strategy returns linked to market returns via the regression equation and therefore predictable by the latter. The optimal allocation is higher at practically all horizons when Strategy returns are predictable.

0 2 4 6 8 100.6

0.7

0.8

0.9

1

Time Horizon (Years)

Opt

imal

Allo

catio

n to

Fun

d (

max

)

Independent Returns

Predictable Returns


much). This means large negative returns in the market will presage large positive returns in the Strategy. In Table 3, we linearly regress the Strategy’s cumulative returns over all 5-day periods on the lagged 5-day period returns of the S&P 500 Index. The loading of the Strategy’s return on the lagged “market factor” is –0.07 within a 95% confidence interval of -0.091 to –0.051. If we regress the subset of the Strategy’s returns that correspond to lagged market returns that are negative, we get an even more negative loading of -0.31 within a 95% confidence interval of -0.36 to -0.26. Hence the market return is a good predictor of the Strategy return, especially when it is negative. Now the concept of hedging demand addresses the question of whether we should change our overall allocation to the Strategy in view of the predictability of its returns10. In other words, we wish to know whether we should increase our allocation if we believed that the Strategy return can be predicted by the market return relative to the allocation we would have had if we believed it cannot be predicted. To answer this question, we study a simple model of investor preferences for the Strategy versus a hypothetical “risk-free” asset paying a given “hurdle rate” of return. The model presented in Figure 5 describes an risk averse investor who prefers more wealth to less but whose utility saturates quickly with increasing wealth. The investor has to allocate between the Strategy and the competitive “risk-free” asset under two different assumptions about the distribution of the Strategy returns, viz. one when they are independent (hence unpredictable) and the other when they are predictable by the lagged market return. In Figure 5, we calculate the optimal (i.e. expected utility maximizing) allocation to the Strategy as a function of investment horizon under the two distributional assumptions. The results show that the allocation to the Strategy should be greater the longer the time horizon. More importantly, the allocation is also greater when returns are predictable than when they are not. This conclusion can be understood intuitively as follows: The allocation to the Strategy should be

higher because the Strategy has high expected returns just when stocks in general have fallen a lot. Hence it provides a good hedge against their reinvestment risk, i.e. the risk of poorer return prospects in the future. Noting that the Strategy is slightly positively correlated to the market return but very negatively correlated to the lagged market return, the hedging demand, which is the need for protection against future return shortfall, on balance increased the investor’s optimal allocation to the Strategy because of the future risk-reducing benefit of the hedge, even though the Strategy does not provide a good current hedge against sharp market declines.


Trading induces price fluctuations much larger than warranted by news or changes in market or industry factors

Fundamental Premises

Betting against unwarranted price fluctuations is risky because of adverse selection and noise trading, but the opportunity is well-known and widely accessible

Risk premium due to liquidity provision is time-varying and increases during and after sharp market declines

Mean reversion is a widespread and permanent feature of markets

Implications

Risk premium endures

Opportunistic strategy with varying numbers and sizes of bets

Increased model certainty and high expected returns during turbulent times

Good hedge against reinvestment risk or risk of poor future prospects for stocks

Market timing bets

Implementation

Guiding principles are parsimony and statistical robustness

Purely mechanical strategy dependent only on constrained information set, i.e. prices.

Provides liquidity only at extremes of trading range for margin of safety

Trades only global large market cap stocks

Advanced real-time trend estimation techniques

Dynamic stock and index hedges to protect against directional market risk

5 Trading Strategy Concept Summary


6 Bibliography

1 Lo, A.W., and C. MacKinlay, Stock Market Prices Do Not Follow Random Walks: Evidence From a Simple Specification Test. Review of Financial Studies 1 (1988), 41-66.

2 Campbell, John., and Luis Viceira, Consumption and Portfolio Decisions when Expected Returns are Time Varying, Quarterly Journal of Economics 114 (1999), 433–495.

3 Nicholas Barberis, Investing for the Long Run when Returns Are Predictable. The Journal of Finance LV, 1 (2000), 225-264.

4 Doron Avramov, Stock Return Predictability and Model Uncertainty, Journal of Financial Economics 64 (2002), 423–458.

5 Pascal J Maenhout, Robust Portfolio Rules and Detection-Error Probabilities for a Mean-Reverting Risk Premium, Journal of Economic Theory 128 (2006) 136–163.

6 Tan, M. and Greyserman, A., Return Dispersion and Statistical Arbitrage, 2004. Available for download at: http://weebly-file/4/0/3/9/40396163/returndispersionandstatisticalarbitrage_2004.pdf

7 Lo, A.W., and C. MacKinlay, The Size and Power of the Variance Ratio Test in Finite Samples: A Monte Carlo Investigation. Journal of Econometrics 40 (1989), 203-238.

8 DeLong, J.B., Shleifer, A., Summers, L. and Waldmann, R.J., Noise trader risk in financial markets, Journal of Political Economy 98 (1990), 703–738.

9 Roll, R.W., R2, Journal of Finance 43 (1988), 541–566.

10 Cochrane, J. H., Portfolio Advice for a Multifactor World, Economic Perspectives Federal Reserve Bank of Chicago 23(3), 59-78, 1999.

Performance in Normal and Turbulent Markets...VZ WMT XOM However, if mean reversion is present, then...

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