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Appendix 8-A
Single and Multiple Index Models
The Single Index Model
William Sharpe, following Markowitz, developed the single-index model, which relates
returns on each security to the returns on a common index.1 A broad market index of
common stock returns is generally used for this purpose.2 Think of the S&P 500 as this
index.
The single-index model can be expressed by the following equation:
Ri = ai + iRM + ei
(8A-1)
where
Ri
= the return (TR) on security i
RM
= the return (TR) on the market index
ai
= that part of security i's return independent of market performance
i
= a constant measuring the expected change in the dependent variable, Ri,
given a change in the independent variable, RM
ei
= the random residual error
The single-index model divides a security's return into two components: a unique
part, represented by ai, and a market-related part represented by iRM. The unique part is a
micro event, affecting an individual company but not all companies in general. Examples
include the discovery of new ore reserves, a fire, a strike, or the resignation of a key
employee. The market-related part, on the other hand, is a macro event that is broad-based
and affects all (or most) firms. Examples include a Federal Reserve announcement about the
discount rate, a change in the prime rate, or an unexpected announcement about the money
supply.
UNDERSTANDING THE SIM
The error term is the difference between the left-hand side of the equation, the return on
security i, and the right-hand side of the equation, the sum of the two components of return.
Since the single-index model is, by definition, an equality, the two sides must be the same.
1
W. Sharpe, “A Simplified Model for Portfolio Analysis,” Management Science, 9 (January 1963): 277-293.
There is no requirement that the index be a stock index. It could be any variable thought to be the dominant influence on stock returns. This
means that the model cannot be based on a consistent theoretical set of assumptions.
2
EXAMPLE 8A-1
Assume the return for the market index for period t is 12 percent, the ai = 3
percent, and the bi = 1.5. The single-index model estimate for stock i is
Ri = 3% + 1.5 RM + ei
Ri = 3% + (1.5)(12%) = 21%
If the market index return is 12 percent, the likely return for stock i is 21 percent.
However, no model is going to explain security returns perfectly. The error term,
ei, captures the difference between the return that actually occurs and the return expected
to occur given a particular market index return.
EXAMPLE 8A-2
Assume in Example 8A-2 that the actual return on stock i for period t is 19
percent. The error term in this case is 19% - 21% = -2%.
This illustrates what we said earlier about the error term. For any period, it
represents the difference between the actual return and the return predicted by the
parameters of the model on the right-hand side of the equation. Figure 8A-1, which depicts
the SIM, illustrates the difference between the actual return of Example 8A-2, 19 percent,
and the predicted return of 21 percent--the error term is -2 percent.
FIGURE 8A-1: The single index model.
The β term, or beta, is important. It measures the sensitivity of a stock to market
movements. To use the single- index model, we need estimates of the beta for each stock we
are considering. Subjective estimates could be obtained from analysts, or the future beta
could be estimated from historical data. We consider the estimation of beta in more detail in
Chapter 9.
RM and ei are random variables. The single-index model assumes that the market
index is unrelated to the residual error. One way to estimate the parameters of this model is
with a time series regression. Use of this technique ensures that these two variables are
uncorrelated.
We will use ei to denote the standard deviation of the error term for stock i. The
mean of the probability distribution of this error term is zero.
CRITICAL ASSUMPTION OF THE SIM
The single-index model also assumes that securities are related only in their common
response to the return on the market. That is, the residual errors for security i are
uncorrelated with those of security j; this can be expressed as COV (ei,ej>) = 0. This is the key
assumption of the single-index model because it implies that stocks covary together only
because of their common relationship to the market index. In other words, there are no
influences on stocks beyond the market, such as industry effects. Therefore
Ri = ai + iRM + ei for stock i
and
Rj = aj + jRM + ej for stock j
It is critical to recognize that this is a simplifying assumption. If this assumption is
not a good description of reality, the model will be inaccurate.3
In the single-index model, all the covariance terms can be accounted for by stocks
being related only in their common responses to the market index; that is, the covariance
depends only on market risk. Therefore, the covariance between any two securities can be
written as
ij = i j 2M
(8A-2)
Once again, this simplification rests on the assumption about the error terms being
uncorrelated. An alternative is to consider more than one index.
SPLITTING RISK INTO TWO PARTS
In the Markowitz model, we need to consider all of the covariance terms in the variancecovariance matrix. The single-index model splits the risk of an individual security into two
components, similar to its division of security return into two components. This simplifies
the covariance and greatly simplifies the calculation of total risk for a security and for a
3
The use of regression analysis does not guarantee that this will be true. Instead, it is a specific simplifying assumption that, in fact, may or may
not be true.
portfolio. The total risk of a security, as measured by its variance, consists of two
components: market risk and unique risk.
2i = 2i[2M] + 2e
(8A-3)
= Market risk + company-specific risk
The market risk accounts for that part of a security's variance that cannot be diversified
away. This part of the variability occurs when the security responds to the market moving
up and down. The second term is the security's residual variance and accounts for that part
of the variability due to deviations from the fitted relationship between security return and
market return.
This simplification also holds for portfolios, providing an alternative expression to
use in finding the minimum variance set of portfolios.
2p
= 2p[2M]
Total
Portfolio
Portfolio
=
Variance
+
2ep
+ Portfolio
market
residual
risk
variance
(8A-4)
SOME CONCLUSIONS ABOUT THE SINGLE INDEX MODEL
The single-index model greatly simplifies the calculation of the portfolio variance, and
therefore the calculation of efficient portfolios.
Example 8A-3
In the case of the Markowitz analysis, 250 stocks require 31,125 covariances and
250 variances. Using the single-index model, we would need
3n + 2 estimates, or 752 numbers for 250 securities.1
However, this model makes a specific assumption about the process that generates
portfolio returns--the residuals for different securities are uncorrelated. Thus, the accuracy
of the estimate of the portfolio variance depends on the accuracy of the key assumption
being made by the model. For example, if the covariance between the residuals for different
securities is positive, not zero as assumed, the true residual variance of the portfolio will be
underestimated.
The end objective of the single index model is the same as that of the Markowitz
analysis, tracing the efficient frontier (set) of portfolios from which an investor would
choose an optimal portfolio. The single-index model is a valuable simplification of the full
variance-covariance matrix needed for the Markowitz model. As discussed above, this
model reduces by a large amount the number of estimates needed for a portfolio of
securities.4
An obvious question to ask is how it performs in relation to the Markowitz model.
In his original paper developing the single-index model, Sharpe found that two sets of
efficient portfolios--one using the full Markowitz model and one using his simplification-generated from a sample of stocks were very much alike.5 A later study also found that the
Sharpe model did no worse than the Markowitz model in all tests conducted, and in tests
using shorter time periods it performed better.6
Multi-Index Models
As noted in the previous section, the single-index model assumes that stock prices covary only
because of common movement with one index, specifically that of the market. Some researchers
have attempted to capture some nonmarket influences by constructing multi-index models.
Probably the most obvious example of these potential nonmarket influences is the industry
factor.7
A multi-index model is of the form
E(Ri) = ai + biRM + ciNF + ei
(8A-5)
where NF is the nonmarket factor, and all other variables are as previously defined. Equation
8A-5 could be expanded to include three, four, or more indexes.
It seems logical that a multi-index model should perform better than a single-index
model because it uses more information about the interrelationships between stock returns. In
effect, the multi-index model falls between the full variance-covariance method of Markowitz
and Sharpe's single-index model.
How well do these models perform? Given the large number of possible multi-index
models, no conclusive statement is possible. However, some studies suggest that the multi-index
model actually reproduced the historical correlations better than the single-index model.8
However, it did not perform better ex ante, which is the more important consideration, because
portfolios are built to be held for a future period of time.
4
5
6
The single-index model can be used to directly estimate the expected return and risk for a portfolio.
Sharpe, “A Simplified Model.”
G. Frankfurter, H. Phillips, and J. Seagle, “Performance of the Sharpe Portfolio Selection Model: A Comparison,” Journal of Financial and
Quantitative Analysis (June 1976): 195-204.
7
In a well-known early study, Benjamin King found a common movement between securities, beyond the market effect, associated with
industries. See B. King. “Market and Industry Factors in Stock Price Behavior,” Journal of Business 39 (January 1966): 139-190.
8
E. Elton and M. Gruber, “Estimating the Dependence Structure of Share Prices--Implications for Portfolio Selection,” Journal of Finance, 5
(December 1973): 1203-1232.