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Beta Arbitrage as an Alpha Opportunity
Tuomo Vuolteenaho, Ph.D.
Partner, Arrowstreet Capital, L.P.
January 2006
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Introduction
This document describes the alpha opportunity created by mispricing of high-beta and low-beta stocks
relative to the overall stock and bond markets. We call market-neutral strategies that seek to exploit
such pricing errors “beta arbitrage.”
The idea of beta arbitrage
The beta is a measure of systematic risk of a stock. Mathematically, it is defined as the covariance of a
stock’s return with the market’s return divided by the market’s return variance. In other words, the
stock’s beta is the regression coefficient of the stock’s return on the market’s return. According to the
Sharpe-Lintner capital asset pricing model (CAPM), a stock’s beta with the market is its sole relevant
risk measure (Sharpe 1964, Lintner 1965).
The importance of beta as a measure of systematic risk can be illustrated with the following
hypothetical examples. Suppose that the market returns 3% over the risk-free rate in a given month.
Absent any significant purely company-specific news, a stock with beta of 0.5 is expected to have a
return of 1.5%, a stock with beta of 1.0 a return of 3%, and a stock with beta of 1.5 a return of 4.5%
over the risk free rate in that month.
For an investor who can borrow and/or short, there are two ways to obtain a unit of equity-risk
exposure. The first way to obtain equity risk is to buy the broad stock market and to sell short bonds
(i.e., to borrow). The second way to obtain equity risk is to buy high-beta stocks and sell short low-
beta stocks. If high beta is taken to mean 1.5 and low beta is taken to mean 0.5, then $1 long
position in high-beta stocks and $1 short position in low-beta stocks will give an equity risk exposure
equal to $1 long the broad stock market and $1 short bonds.
The idea of beta arbitrage is to exploit the fact that the compensation for equity risk may be different
depending on how the risk is obtained. If these two ways to obtain essentially the same risk will earn
different compensation, there is an arbitrage opportunity. (In reality, any stock strategy always has
some risk, so the “arbitrage opportunity” should be taken to mean an attractive risk-return tradeoff.)
For example, if high-beta stocks are overpriced and low-beta stocks underpriced relative to the equity
premium, one can buy low-beta stocks and sell short a lesser amount of high-beta stocks, while
neutralizing the systematic equity risk and earning a positive premium.
Strategic beta arbitrage
In the absence of investor irrationalities that manifest themselves in mispricing, the Sharpe-Lintner
CAPM predicts that the risk compensation for one unit of beta among stocks (also called the slope of
the security market line or the cross-sectional beta premium) is always equal to the expected premium
of the market portfolio of stocks over short-term bills (also called the equity premium). For example, if
a risky stock has a beta of 1.5 and the equity premium is four percent, then that stock should have an
expected return equal to the risk-free rate plus six percent. Conversely, a safe stock with a beta of 0.5
should only earn a two percent premium over the risk-free rate, and the risky stock will therefore
return a premium of four percent over the relatively safe stock.
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Historically, and contradicting the CAPM theory, the market equity premium has been consistently
higher than the cross-sectional beta premium. This feature of the historical data was first discovered
by Black, Jensen, and Scholes (1972). Figure 1 plots the historical average returns on beta-sorted
portfolios using the 1927-2001 US sample, both for the full period and subsamples (early sample =
1927-1963; late sample = 1963-2001). The slope of the solid line in Figure 1 is the equity premium for
the corresponding period, or the price of risk implied by the pricing of the overall stock market relative
to short-term bills. In a world described by the Sharpe-Lintner CAPM, the beta-sorted portfolios should
fall approximately on the line. In reality, the compensation for the unit of beta in the cross-section has
been much less than the equity premium. A line drawn through the points would be flatter than the
solid line in Figure 1. The flatness is especially pronounced among high-beta stocks and during the late
sample. Stocks with betas higher than one seem to earn the same return as the overall market,
despite being much riskier.
Figure 1
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
14.00%
16.00%
18.00%
20.00%
0.00
0.50
1.00
1.50
2.00
Beta
E
xce
ss
r
e
t
u
r
n
o
v
er
T
-
b
i
l
l
s
Full sample
Early sample
Late sample
Full sample predicted
Early sample predicted
Late sample predicted
Figure 1: Returns to beta deciles in the long-term US data, 1927-2001. Source: Data set analyzed by
Polk, Thompson, and Vuolteenaho (2005)
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A strategic beta-arbitrage strategy, first suggested by Black (1993), builds into the portfolio a
systematic tilt towards low-beta stocks and away from high-beta stocks. This tilt is held constant
irrespective of the market environment. For example, holding a $1 long position in low-beta stocks
(assumed beta 0.5) and holding a $0.33 short position in high-beta stocks (assumed beta 1.5) will
result in a strategic, market-neutral beta-arbitrage portfolio. If the realized return over the risk-free
rate for high-beta stocks is lower than three times the return on the low-beta stocks, then this beta-
arbitrage strategy makes a profit. Conversely, if the realized return over the risk-free rate for high-
beta stocks is higher than three times the return on the low-beta stocks, then this beta-arbitrage
strategy makes a loss.
What creates the persistent beta-arbitrage opportunity, or equivalently, the persistent overpricing of
high-beta stocks relative to low-beta stocks that strategic beta-arbitrage seeks to exploit? First, and
perhaps the most obvious explanation, originally suggested by Black, Jensen, and Scholes (1972) and
Black (1973), is that some investors want to hold a high-risk high-return portfolio, yet are reluctant to
take leverage. This demand for risky, high-beta stocks pushes up their valuations.
Second, Karceski (2002) suggests the following agency-based explanation for the absence of cross-
sectional beta premium, which related to above-mentioned borrowing constraints. Traditional long-
only managers who manage retail mutual funds prefer high-beta and dislike low-beta stocks for the
following reason. The majority of new inflows to mutual funds take place in bull markets.
Furthermore, the inflows go to the funds with the best recent absolute performance (irrespective of the
portfolio’s beta). Asymmetrically, money in mutual funds is very sticky, and bad performance does not
typically result in significant outflows. This set of incentives makes it imperative for the mutual fund
managers to perform well in up markets. The only generally available way to systematically do so
without leverage is to buy high-beta stocks.
Third, it is possible that irrational trader activity causes both overvaluation and high beta. For
example, suppose that irrational investor sentiment affects the market as a whole. Suppose further
that irrational investors are face constraints or are otherwise reluctant to short, and thus only hold long
positions. In such situation, stocks that are held by irrational investors simultaneously exhibit
overvaluation and excessive comovement with the market.
Is the market-neutral beta-arbitrage strategy simply a value strategy in disguise, especially given that
during the recent years value (i.e., low P/B) stocks have had lower betas than growth (i.e., high P/B)
stocks? The answer to this question, given by Daniel and Titman (1997) and Davis, Fama, and French
(2000), is that the market-neutral beta-arbitrage strategy is not simply a repackaged version of the
long-value-short-growth strategy. Even if one forces the market-neutral beta-arbitrage strategy to be
neutral with respect to the price-to-book characteristics, the historical premium on the strategy is
positive as the below Figure 2 shows:
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Figure 2
Low-B/P
2
3
4
High-B/P
Low-beta
2
3
4
High-beta
-6.00
-4.00
-2.00
0.00
2.00
4.00
6.00
8.00
α% p.a.
Figure 2: Jensen’s alphas for beta and B/P sorted portfolios, 1927 - 2002. Source: Data set analyzed
by Cohen, Polk, and Vuolteenaho (2005)
Tactical beta-arbitrage
In contrast to strategic beta arbitrage, which takes a permanent tilt towards low-beta and away from
high-beta stocks, tactical beta-arbitrage adjusts these positions with market conditions. If we have
reliable forecasts of the overall market’s equity premium and on the returns of high-beta and low-beta
stocks, then tactical beta arbitrage may add more value than strategic beta arbitrage. In particular,
tactical beta-arbitrage involves taking aggressive positions if and when the forecast equity premium
and the forecast cross-sectional beta premium diverge significantly.
Polk, Thompson, and Vuolteenaho (2005) and Cohen, Polk, and Vuolteenaho (2005) have studied such
divergences between the expected equity premium and cross-sectional beta premium in the 1927-2001
U.S. sample. In this sample, one event stands out. In 1982, the expected equity premium and the
cross-sectional beta premium strongly diverged. At that point of time, a beta-arbitrage portfolio that is
long low-beta stocks (which had an expected return much higher than justified by their risk) and short
a lesser amount of high-beta stocks (which had an expected return much lower than justified by their
risk) had a very high expected return relative to its risk. Polk, Thompson, and Vuolteenaho estimate
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that, in the beginning of 1982, the expected Sharpe ratio on such a beta-arbitrage portfolio was over
one.
One possible reason for the divergence in 1982 is money illusion combined with high inflation.
Suppose that in 1982 the high rate of inflation lead the money-illusioned stock market investors
undervalue the stock market, incorrectly perceiving the market being priced to yield only a low equity
premium. Consistent with this incorrect belief, the stock market investors priced high-beta stocks to
yield only a low premium over low-beta stocks. In reality, the equity premium was high in 1982, as
evident from the very high earnings and dividend yields. The simultaneous high equity premium and
low cross-sectional beta premium thus offered an excellent opportunity for beta arbitrage.
Recent tactical opportunities
Although the above-cited academic studies support the idea that beta-arbitrage has offered an
intriguing investment opportunity, they do not prove that the opportunity presents itself on a forward-
looking basis. To examine current and recent opportunities in the context of global investing, we
compare simple equity-premium forecasts to cross-sectional beta-premium forecasts constructed using
Arrowstreet Capital’s models. If our equity-premium measure is indeed a good predictor of the
market’s excess return over bonds and if our cross-sectional beta premium measure forecasts the
return on high-beta stocks relative to that of low-beta stocks, the divergences imply trading
opportunities.
Figure 3
-6.00
-4.00
-2.00
0.00
2.00
4.00
6.00
1996m
1
2
1
997m
3
1
9
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1
997m9
1997m1
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9
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8m9
1998
m1
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999m
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9m6
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999m9
1999m1
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000m3
2
000m6
2
00
0m9
2000
m1
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001m3
2
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1m6
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001m9
2001m1
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002m6
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m9
2002m
1
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m3
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003m6
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003m9
2003m1
2
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004m3
2
004m6
2
004m9
200
4m1
2
2
005m
3
Forecast equity premium
Forecast beta premium
Smoothed forecast beta premium
Figure 3: Equity premium forecasts and cross-sectional beta-premium forecasts produced by
Arrowstreet Capital’s models (not currently used in production)
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The solid red line in Figure 3 graphs the prospective equity premium estimate for the MSCI developed
world index, computed using a derivation of the classic Gordon growth model. According to this
model, the prospective equity premium is computed as E/P * b + ROE*(1-b) - y, where E is normalized
earnings, P is price, b is the dividend payout ratio set to 50%, ROE is the long-term real accounting
return on equity set to 6%, and y is the real interest rate. We use the average of US and UK ten-year
inflation-protected bond yields as the proxy for the prospective real interest rate. It is interesting to
note that the significant decline in real interest rates over the last five years, combined with a
contemporaneous increase in equity yields, has brought the prospective equity premium back to the
3.5% level.
The dashed line in Figure 3 plots the cross-sectional beta premium computed using Arrowstreet’s stock-
level return and beta forecasts. Each month, we group stocks into five portfolios based on the forecast
beta, compute the value-weight return and beta forecasts for all portfolios, and use these portfolio
level forecasts to compute the cross-sectional beta forecast.
Figure 3 shows occasionally large divergences between the world market equity premium and the
cross-sectional beta premium. For example, near the end of our sample in March 2005, our equity-
premium calculation indicates a relatively high equity premium of three percent, while the cross-
sectional beta premium is at its lowest value in this sample period at negative four percent. Contrary
to the economic theory, our estimates suggest that high-beta stocks will earn a negative premium over
low-beta stocks, making short positions in high-beta stocks a particularly attractive way to hedge
equity risk.
There are also periods during which the cross-sectional beta premium appears higher than the equity
premium, such as December 1999 when high-beta stocks had positive momentum and attractive
forecasts relative to those of low-beta stocks yet the market as a whole was already trading at a
relatively high level yielding a low equity premium. During that period, the tactical beta-arbitrage
strategy might have taken (conservative) long positions in high-beta stocks and short positions in low-
beta stocks.
Disclosures
All information presented above relating to Arrowstreet models refers to proprietary investment models
that are not currently used in production to manage actual client assets. These models were designed
with the benefit of hindsight in order to demonstrate the relevant tactical opportunities. The data
presented as it relates to Arrowstreet models does not reflect the impact that any significant market or
economic factors might have had on our use of the model or on actual investment results, had this
model been used during the period to actually manage client assets.
It is important to note that the potential for profit is accompanied with the possibility of loss and the
data or results as presented are not a guarantee of future results and actual results could differ
materially from those presented above.
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Bibliography
Black (1993). "Beta and return." Journal of Portfolio Management, Fall/1993.
Black, Jensen, and Scholes (1972). "The capital asset pricing model: some empirical tests." In Studies
in the Theory of Capital Markets, Michael C. Jensen ed., Praeger, Inc.
Cohen, Polk, and Vuolteenaho (2005). “Money illusion in the stock market: the Modigliani-Cohn
hypothesis.” Forthcoming in the Quarterly Journal of Economics.
Daniel and Titman (1997). “Evidence on the characteristics of cross-sectional variation in stock
returns.'' Journal of Finance 52.
Davis, Fama, French (2000). "Characteristics, covariances, and average returns: 1929 to 1997."
Journal of Finance 55.
Karceski (2002). “Returns-chasing behavior, mutual funds, and beta's death.” Journal of Financial and
Quantitative Analysis 37.
Lintner (1965). “The valuation of risk assets and the selection of risky investments in stock portfolios
and capital budgets.” Review of Economics and Statistics 47.
Polk, Thomposon, and Vuolteenaho (2005). “New forecasts of the equity premium.” Forthcoming in
the Journal of Financial Economics.
Sharpe (1964). “Capital asset prices: A theory of market equilibrium under conditions of risk.” Journal
of Finance 19.
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