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Chapter 10Chapter 10Chapter 10Chapter 10

Index ModelsIndex Models

10-210-2

A Single-Index Security Models

From a practical point of view Markowitz model has two shortcomings1. Problem: Computational complexity: In Markowitz Model we estimate the exp. Returns and covariance matrix. Suppose we analyze 50 stocks. Our input list;

N=50 estimate of expected returnsN=50 estimates of variances(N2-N)/2= 1125 estimates of covariancesTotal 1325 estimates

10-310-3

2. Problem: Model assumes that all the risk and return characteristics can be explained by the covariance of the return with that of the other securities. Changes in the non-financial factors such as the growth rate of the economy or the inflation rate are not accounted for directly.

10-410-4

These considerations have led to various simplications and extentions of the model. One simlication is by Sharpe (1963), who developed the Single-Index Model. This model imposes restrictions on how security returns can covary. In particular, it is assumed that all covariance arises through an index. This leads to a reduction in complexity. Sharpe’s model has since been extended to multi-index models, and leads to a more general theory called the Arbitrage Pricing Theory, developed by Ross (1976).Besides simplifying the covariance matrix, this approach is easily to take account of non-financial factors. In the multi-index models for example one of the indexes could be tha rate of inflation.

10-510-5

A Single-Index Security Models

Suppose that we summarize all relevant economic factors by one macro economic indicator and assume that it moves the security market as a whole.

We further assume that all remaining uncertainity in stock returns is firm-specific.

10-610-6

A Single-Index Security Models

Return on stock i;

ri = E(ri)+mi +ei

mi:impact of unanticipated macro events on security returns.

ei:impact of unanticipated firm-specific events.

Both have zero expected values because each represent the impact of unanticipated events and must average out to zero.

10-710-7

A Single-Index Security Models

Different firms have different sensitivities to macro economic events. If we denote the unanticipated component of the macro factor by F, the responsiveness of security i to macroevents by beta, ßi, then macro component of the security is mi = ßiF

10-810-8

ri = E(Ri) + ßiF + ei

ßi = index of a securities’ particular return to the factor

F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns.The variability of all stock returns can be completely described by one common index plus firm-specific events. Individual responsiveness to the index is captured by the weight ßi.

Single Factor Model

10-910-9

The major assumption of Sharpe's single-index model is that all the covariation of security returns can be explained by a single factor. This factor is called the index, hence the name "single-index model." One version of the model, called the market model, uses a market index such as the S&P500 as the factor, although in principle, any factor that influences security returns can serve as the index. Here, we will adopt the terminology associated with the market model.

10-1010-10

The second assumption of the single-index model is that the ei (firm specific factors) are independent across firms. This means that the covariance of ei and ej is zero, and is based on the assumption that the part of the return that is not explained by the index results from purely firm-specific events.This independence leads to a significant simplification of the covariance matrix of returns.

10-1110-11

Single Index Model

According to the index model, we can seperate the actual (realized) rate of return on a security into macro (systematic) and micro (firm-specific) components.

10-1210-12

(ri - rf) = i + ßi(rm - rf) + ei

Risk Prem Market Risk Prem or Index Risk Prem

i= the stock’s expected return if the market’s excess return is zero

ßi(rm - rf) = the component of return due to

movements in the market index

(rm - rf) = 0

ei = firm specific component, not due to market

movements

Single Index Model

Holding period excess return on the stock:

10-1310-13

Let: Ri = (ri - rf)

Rm = (rm - rf)

Risk premiumformat

Ri = i + ßi(Rm) + ei

Risk Premium Format

If we denote excess returns over the risk-free rate by capital R

10-1410-14

Security Characteristic Line

Excess Returns (i)SCL

....

........

.. ..

.. ...... ..

.... ..

.. ....

......

.. ..

.. ....

..

..

.. ..

.. ....

.. .. ....

.. .... ......

..

.. ....

..

Excess returnson market index

Ri = i + ßiRm + ei

......

10-1510-15

Jan.Feb...DecMeanStd Dev

5.41-3.44

.

.2.43-.604.97

7.24.93

.

.3.901.753.32

ExcessMkt. Ret.

ExcessGM Ret.

Using the Text Example from Table 10-1

10-1610-16

Estimated coefficientStd error of estimateVariance of residuals = 12.601Std dev of residuals = 3.550R-SQR = 0.575

ßß

-2.590(1.547)

1.1357(0.309)

rGM - rf = + ß(rm - rf)

Regression Results

10-1710-17

Market or systematic risk: risk related to the macro economic factor or market index.Unsystematic or firm specific risk: risk not related to the macro factor or market index.Total risk = Systematic + Unsystematic

Components of Risk

10-1810-18

We can break the variance of the rate of return on each stock into components:

i2 = i

2 m2 + 2(ei)

where;

i2 = total variance

i2 m

2 = systematic variance

2(ei) = unsystematic variance

Measuring Components of Risk

10-1910-19

Total Risk = Systematic Risk + Unsystematic Risk

Systematic Risk/Total Risk = 2

ßi2

m2 / 2 = 2

i2 m

2 / i2 m

2 + 2(ei) = 2

Examining Percentage of Variance

10-2010-20

Single Index Model

Input list for single index model:N estimates of E(Ri), expected excess returns.N estimates of ßi, sensitivity coefficients.

N estimates of 2(ei), firm specific variances.

1 estimate m2 , variance of macro-economic

factor.3n +1Suppose we have 50 securities;3x50 + 1= 151 estimates.It is easy to use the single index model.

10-2110-21

Reduces the number of inputs for diversification.

Easier for security analysts to specialize.

Advantages of the Single Index Model

10-2210-22

Index Model and Diversification

As the number of stocks in the portfolio rises, the unsystematic risk can be diversified away. The market risk remains.

Excess rate of return on the equally weighted portfolio. wi=1/n

10-2310-23

Index Model and Diversification

)(

1

1

1

2222

1

1

1

PMP

N

iiP

N

iiP

N

iiP

PPPP

e

eNe

N

N

eR

p

10-2410-24

Index Model and Diversification

Because these ei s are independent and all have zero expected value, as more and more stocks are added to the portfolio, the firm-specific components tend to be cancel out.

When n gets large, 2(ep) becomes negligible.

10-2510-25

Risk Reduction with Diversification

Number of Securities

St. Deviation

Market Risk

Unique Risk

2(eP)=2(e) / n

P2M

2

10-2610-26

The CAPM and the Index Model

Actual Returns vs Expected ReturnsCAPM is a statement about ex ante or expected returns whereas all we can observe are actual or realized holding period returns.

To make a leap from expected to realized returns, we can employ the index model.

The index model beta turns out to be the same beta of CAPM expected return-beta relationship, except we place an observable market index instead of the theoretical market portfolio.

10-2710-27

The CAPM and the Index Model

The Index model and the expected return-beta relationship.

CAPM:

E(ri)-rf= ßi[E(rm)- rf]If the index M is the true market portfolio, we can take the expectation of each side of the equation Ri = αi + ßi(Rm) + ei.

E(ri)-rf= αi +ßi[E(rm)- rf]

10-2810-28

The CAPM and the Index Model

If we compare it with CAPM equation the only difference is αi. CAPM predicts that αi should be zero for all assets. The alpha of a stock is expected return in excess of the fair expected return as predicted by CAPM. If the stock is fairly priced, its alpha must be zero.If we estimate the index model for several firms, using regression equation, we should find expost (realized) alphas average will be zero.

10-2910-29

The CAPM and the Index Model

CAPM states that the expected value of alpha is zero for all securities.Index model states that realized value of alpha should average zero.Jensen examine the alphas realized by mutual funds and found that frequency distribution of these alphas seem to be distributed around zero.