Download Web Note 7.1: Mean Reversion

Web Note 7.1: Mean Reversion
A. The Concept
Just as the name implies, a mean reverting economic time series is one that periodically
returns to its long run mean value. It is the idea that although the time series may stochastically
exceed its mean value for a prolonged period of time, or similarly fall short of it, it eventually
and repeatedly returns to its mean value.1 In the finance literature, this concept is most typically
applied to equity rate of return series. When authors speak of mean reversion in stock prices, it is
typically in the context of a “normalized” price series such as the price-earnings (P/E) ratio.
Equity price series are not level-stationary (as the economy grows, there is no upper bound to the
value the aggregate stock market index can assume), and thus the “long run mean value” is not
defined.2 Our discussion will focus exclusively on mean reversion as regards equity return series.
In order to make this notion a bit more intuitive, let us consider two (stationary) models
of equity return evolution which display this property:
1. rt 1  Er    rt  Er    t 1
(1)
where Er is the long term, unconditional mean return, σ is the return volatility and  t  is a
sequence of i.i.d. N(0, 1) random variables. The parameter ρ is understood to satisfy 0 < ρ < 1.
With ρ < 1, if rt  Er , then in period t + 1, there is the tendency for rt+1 to exceed Er by a lesser
amount, thereby pulling the return series back towards its mean; if rt  Er , the like tendency is
for rt 1  Er by a lesser amount.
2. rt 1  rt    Er  rt 1    t , 0    1
(2)
where the notation is the same as in our first example. A return process governed by (2) is
referred to as a “random walk with mean reversion,” and it suggests an even stronger quality of
mean reversion: if rt 1  Er -- the series is above its unconditional means – then rt 1 relative to rt
By the expression “returns to its mean value” we do not intend to suggest that the process exactly assumes its long
run mean value, but that it assumes values in a small neighborhood of it.
2
Within the class of theoretical models we consider in Chapter 10 and Web Chapter A, there are good theoretical
reasons to argue that the (P/E) ratio follows a stationary stochastic that is mean reverting.
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will be the sum of a mean zero random component  t and a negative term, the latter tending to
reduce the extent that period rt 1 ’s return exceeds rt and thus also its mean, and vice versa.
The classic AR-1 processes (those of the form  rt  0  1rt 1   t  are both stationary
and mean reverting provided 1  0 . A stationary process, however, is not guaranteed to be
mean reverting, with an i.i.d. process being a case in point.
So far our discussion of “mean reversion” has been in the context of specific return
processes, but, as yet, we have no “specific process free,” abstract definition. In fact, the
literature offers three distinct definitions (we phrase them in the language of rate of return
series):
(i) A rate of return series rt  is said to be mean reverting if and only if:
corr  rt , rt 1   0 ;
(ii) A rate of return series rt  is said to be mean reverting if and only if
var  rt  rt 1  . . .  rt  k 
k
 var  rt 
for any k  1 . This definition was first proposed in Poterba and Summers (1988), and employed
by, e.g., Mukherji (2011). It stands in specific contrast to the property of a random walk where
var  rt  rt 1  . . .  rt k   k var  rt  .
(iii) A rate of return series rt  is mean reverting if and only if
cov  rv  ru , rx  rw   0
for 0  u  v  w  x . This definition appears to be applied in the insurance industry writings (see
Exley et al. (2004)).
In general, there does not appear to be any model-free direct relationships between these
definitions. Not surprisingly, definition (i) appears to be the most widespread in the finance
literature.
What is the empirical evidence as regards mean reversion? It is first important to specify
the time horizon over which mean reversion is alleged to be observed. If we speak of mean
reversion over days (a topic of interest to traders) or weeks, the discussion is more fruitfully
undertaken in the general context of momentum phenomena which we consider in Chapter 14.
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For the case of long-term mean reversion, the principal case for its existence was made in
two papers, both of which were published in 1988: Poterba and Summers (1988), and Fama and
French (1988). Each paper models mean reversion slightly differently but ends up with the same
conclusion: between 1926 – 1985, long term mean reversion in (aggregate) stock market returns
was present at 3 – 5 year frequencies. In particular, three year lagged returns displayed a
correlation of around .25, with 5 year returns displaying a lagged correlation of .4. Without
getting into the precise statistical issues involved, suffice it to say that these results have been
questioned even by Fama and French themselves (1988), and subsequently Fama (1991).
Basically, there are not enough independent statistical observations for the relevant tests to be
robust. Furthermore, if the stock returns for the period 1926 – 1940 are excluded from the data
set, then any statistical evidence for mean reversion disappears. Using simulations, Richardson
(1993) further argues that large autocorrelations for the 3 – 5 year horizon do not, in and of
themselves, disprove the random walk hypothesis because within this range of horizons,
autocorrelations are subject to large sampling variation. Evidence against mean reversion from
an alternative perspective is found in Kim et al. (1991). See Mayost (2012) for an excellent
literature review. In summary, it is probably fair to say that the empirical evidence for long term
mean reversion in equity returns is modest at best. The same can be said of mean reversion as
regards equity prices.3
This (tentative) conclusion is a welcome one for efficient markets theory. In particular,
mean reversion at 3 – 5 year frequencies suggests predictability (see Chapter 10, footnote 14) at
these same frequencies which is a violation of weak form efficient market theory.
B. Mean Reversion and Long-Run Portfolio Risk.
As discussed in Chapter 7, Pastor and Stambaugh (2012) provide one perspective on the
ability of mean reversion to reduce long run portfolio risk. Spierdijk and Bikker (2012) provide
another in a somewhat simpler and less stylized context. Rather than including issues of mean
estimation, they assume the relevant mean returns are precisely estimated and try to measure the
3
In general, negative autocorrelation in equity prices (defn. (i) for mean reversion) implies the same for returns, but
the converse is not true (see Spierdijk and Bikker (2012) for an illustration).
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effects solely of mean reversion on long run portfolio composition and risk. We give some
details of their underlying model of portfolio return evolution and their results in what follows.
The model they employ is taken from Summers (1986) and Fama and French (1988). Let
qte represent the price of the equity security or portfolio and qˆte its rational expectations price.4
Their model presumes that the actual equity price experiences random deviations from the
rational expectations price and that these deviations are mean reverting:
ln qte  ln qˆte  zt where
(1)
zt     zt 1  nt , and
(2)
ln qˆte  ln qˆte1  t .
(3)


In the above system equations, nt is i.i.d. N(0, σ2) and  t is i.i.d. N 0,  2 ; nt and  t are also
independent of one another.
Substituting (1) into (2) yields
ln qte    ln qˆte   zt 1  nt .
If we note that  
ln qte 
(4)
ln qˆte  ln qˆte




and ln qˆte 
, expression (4) can be rewritten as
1
1
1 1

 
ln qˆte
ln qˆte  


   zt 1  

   nt .

1 1
1


1





Accordingly, ln qte can be reviewed as mean reverting around
(5)
ln qˆte


. Spierdijk and
1 1
Bikker (2012) parameterized this model as follows (means and standard deviations being
measured in terms of monthly returns):   .9%,   .975,   3.2%,   3.2% , these latter two
values yielding a monthly standard deviation of 4.5%, the average for the U.S. aggregate stock
market index from 1.1.1982 through 8.31.2010. For the bond return they assume Erb = .7% and
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 

ds
where E denotes the expectation conditional on all information,


s t 
t
t
 s t 1 1  r  
e
In this setting, qˆt  E
available to the market in time period t, Ωt .
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 rb = 1.4% as reflecting the Citibank U.S. Overall Bond Investment Grade Total Return Index
for the same historical period, with no mean reversion in bond returns. They further assume


corr rte , rtb  .2 , which is the historical contemporaneous correlation between the




aforementioned stock and bond indices, and corr rte, t 1 , rtb, t 1  .18 , corr rte, t 5 , rtb, t 5  .17 ,




corr rte, t 10 , rtb, t 10  .17 , and corr rte, t  20 , rtb, t 20  .17 . Note that the two risky assets in this
exercise are the equity market portfolio and a risky bond portfolio.
With this information, the authors compute the mean and variance of the minimum risk
portfolio and the mean variance efficient tangency portfolio for the indicated four investment
horizons, as well as the associated portfolio proportions.5 In the latter case the assumed risk free
rate is the average over the investment horizon from which the data was taken. The exercise was
repeated for various variance ratios, where in their model, this quantity is defined by
2
2

for   .975 , a ratio that signifies the relative risk contributions of the
 2 /  2    2
permanent and transitory components, respectively. The results of this exercise are presented in
Table 1 below (Table 1 of Spierdijk and Bikker (2012)).
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Recalling the discussion in Chapter 6, the portfolios of interest are the ones presented below.
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Table 1: Optimal Portfolio Weights (in Percentage) With and Without Mean Reversion in
Stock Prices (1)
This table reports the optimal portfolio weights for the global minimum variance portfolio
(GMVP) and the tangency portfolio (TP). The investment categories are stocks (Datastream U.S.
Aggregate Stock Market Index) and bonds (Citigroup U.S. Overall Bond Investment Grade Total
Return Index) for different values of the variance ratio. The variance ratio is defined as the return
variance of the permanent price component divided by the return variance of the transitory
component. The risk-free rate is based on the nominal interest rate term structure as compiled by
the Dutch Central Bank. The last two columns in each panel display the expected portfolio return
  p  and the portfolio volatility  p 
(1)
Table 1 in Spierdijk and Bikker (2012).
In the above table, the absence of mean reversion requires   0 , and the relevant
comparison concerns a comparison of the like entry in the top and bottom halves of the table
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(e.g., for a variance ratio of 1:1, and a time horizon of 20 years, we compare  p  21.34% (a 20
year monthly standard deviation) versus  p  21.47% ). A quick survey of the table suggests that
for the indicated parameterizations neither portfolio returns and variances nor portfolio
proportions are much affected by the presence or absence of mean reversion for this datamotivated parameterization.
References
Exley, J., Mehta, S., and Smith, A. (2004), “Mean Reversion,” Finance and Investment
Conference, Brussels, mimeo.
Fama, E., and French, K. (1988), “Dividend Yields and Expected Stock Returns,” Journal of
Financial Economics, 22, 3-25.
Kim, M., Nelson, C., and Startz, R. (1991), “Mean Reversion in Stock Prices? A Reappraisal of
the Empirical Evidence,” Review of Economic Studies, 58, 515-528.
Mayost, D (2012), “Evidence of Mean Reversion in Equity Prices,” OSFI-BSIF Canada mimeo.
Mukherji, S. (2011), “Are Stock Returns Mean Reverting?”, Review of Financial Economics, 20,
22-27.
Poterba, J., and Summers, L. (1988), “Mean Reversion in Stock Prices,” Journal of Financial
Economics, 22, 27-59
Richardson, M. (1993), “Temporary Components of Stock Prices: A Skeptic’s View,” Journal of
Business and Economic Statistics, 11, 199-207.
Spierdijk, L., and Bikker, J. (2012), “Mean Reversion in Stock Prices: Implications for LongTerm Investors,” De Nederlandsche Bank NV Working Paper #343.
Summers, L. (1986), “Does the Stock Market Rationally Reflect Fundamental Values?”, The
Journal of Finance, 41, 591-601.
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