single index model and market model - jose marin

Market and Index Models Session 08

Copyright © José M. Marín Portfolio Management 1

Outline 08: Market and Index Models 1. CAPM and Multifactor Models 2. Bloomberg Screen 3. Components of Variance 4. Diversification Effects

5. Precision of Historical Betas 6. Adjusting Historical Betas 7. Estimating Expected Returns 8. Market-Neutral Strategies (Problem Set)

Mandatory Reading: Hansell´s article.

CAPM and Multifactor Models

• In the previous section we saw that in a CAPM framework the only variable that matters for asset pricing is the return of the market portfolio.

• Consider the following statistic representation of asset returns:

( )

new informationexpected component

innovation in 's returnj jR E R j= +14444244443123

• We can also distinguish two broad types of innovations: systematic (generated by an arbitrarily large set

of factors, K) and idiosyncratic.

( ){1 1 2 2

idiosyncratic innovationsystematic innovation ( )( )

....j j j j jK K j

unique riskaggregate risk factors

R E R F F Fβ β β ε= + + + + +1444442444443

• We end up with the following linear factor model of asset returns:

1 1 2 2 ....j j j j jK K jR a F F Fβ β β ε= + + + + +



• Now we can interpret the CAPM as a model in which returns are generated by a single factor:

1 1 2 2 ....

j M

j j j j jK K j

R

R a F F Fβ

β β β ε= + + + + +1444442444443

• More specifically, the Market Model1 is defined as:

j j j M jR a Rβ ε= + + , j = 1, …, N 0,j M jE R E jε ε= = ∀ 0i jEε ε =

• This is, for instance, the model used by Bloomberg.

1 The label market model is often used for a model in which only (1) and (2) hold and the term single index model for the model in which (3) also holds. We will just use the latter.



Market Model in the industry: Bloomberg screen



Components of Variance • Consider the Market Model:

j j j M jR a Rβ ε= + + (1) • The market model provides a decomposition of the variance of an asset’s return into two components

o market-related o non-market

• Using the regression, we obtain

434214434421iancemarket var-non

,

variancerelated-market

,2

,,,,

,,,

}{var }{ var

} ,{cov },{cov}{var

titmi

titmiititmii

tititi

R

RRRRR

εβ

εβαεβα

+=

++++=

=

• The R2 (“R-squared”) gives the fraction of the asset’s total variance that can be attributed to its market-

related variance:



}{var}{var

1

}{var}{var

,

,

,

,2

2

ti

ti

ti

tmi

R

RR

R

ε

β

−=

=

(2)

• In the Bloomberg example:

o 865.0=β o std. dev.{εCB,t} = 4.476% o R2=0.324 o Using (2):

324.0)(Var

)04476.0(12

2 =−=CBR

R

o std. dev.{RCB,t} = 5.43%

• Beta is not the same as R2: high-beta stocks can actually have lower correlations with the market.



Diversification Effects • A market-model regression also holds for a portfolio of assets. To see how:

o Begin with a market-model regression for each of N assets in the portfolio o Multiply both sides of the regression equation for asset i by the portfolio’s weight in asset i, xi

o Sum the resulting equations across the N assets

,

)(

,,,

1,,

1,

tptmpptp

N

ititmiii

N

ntii

RR

RxRx

εβα

εβα

++=

∑ ++=∑==

where

∑ ∑=== =

N

i

N

iiipiip xx

1 1, , ββαα

.1

,, ∑==

N

itiitp x εε

• The single-index model implies that the variance of returns on well-diversified portfolios will consist primarily of the market-related component.



• For example, with an equally weighted portfolio ( xi = 1/N ),

,}{var1

1var}{var

,

1,,

ti

N

ititp

N

N

ε

εε

=

⎭⎬⎫

⎩⎨⎧∑==

which approaches zero as N grows large. Therefore, the variance of the total portfolio return is

},{var

}var{1}{var }{var

,2

,,2

,

tmp

titmptp

RN

RR

β

εβ

→

+=

and the market-model R2 for the portfolio approaches 1. • The single-index model also implies that well diversified portfolios will be highly correlated with each

other.



Precision of Historical Betas • The OLS regression estimator of βi, denoted iβ̂ , is often referred to as the “historical beta.” A historical beta is a

random variable, since it is computed from realizations of random returns. The standard error of iβ̂ is given (approximately) by

( )m

ii T

Rsσσβˆˆ1ˆ

2−= (3)

• Thus, (3) suggests that the precision of an OLS beta estimate can be increased by increasing the number of

observations (T), or using portfolios instead of individual securities, thereby raising R2 and lowering iσ̂ . • For example, the equally weighted portfolio of the 30 Dow Jones Industrial stocks gives the following results

(1993-99, 84 months): Beta (raw) 1.05 R2 0.93 Std. Error of Beta 0.03

• Some experiments suggest that, in estimating historical betas using monthly returns, a sample period of five to

seven years gives the optimal tradeoff between greater accuracy arising from a larger number of observations and lower accuracy arising from changes over time in the true underlying beta.

• Precision in historical betas can, in principle, be improved by increasing the frequency of return observations.



Adjusting Historical Betas for “Regression to the Mean” • Denote the historical betas on asset i estimates in two adjustment five-year sample period as )1(,

îβ and )2(,

îβ .

• For a number of assets (i’s), plot )1(,

îβ versus )2(,

îβ .

• As the plot suggests, the assets with extreme values (large or small) of )1(,ˆ

iβ tend to have values of )2(,ˆ

iβ closer to 1.

• This “regression” tendency can be represented by the fitted line in the above plot,

,ˆˆ)1(,10)2(, ii cc ββ +=

where it must be the case that



c0 + c1 = 1.

• In practice, the “raw” historical betas are often adjusted toward 1 to account for this regression tendency. The

“adjusted” beta estimator, , is formed by

.ˆ)1(1ˆ, iiADJ ww ββ ⋅−+⋅=

• A typical specification (e.g., Bloomberg) sets w = 1/3. Estimating Expected Returns: Why Bother with Beta? • Based on the past 84 months of data, you estimate the mean and volatility of the return on a given stock, Ri,t, and

the return on the S&P in excess of the riskless (T-Bill) rate, tftSPm RRr ,, −= .

Mean Std. Dev. (s)

Ri 1.5% 13.3% rm 0.5 4.8

• Using the simple historical average return, you would estimate the stock’s expected return as 1.5%.



• Assuming that the current T-Bill return is 0.3% monthly, and the stock’s estimated beta equals 1.41, the CAPM-based estimate is

%01.1)5.0(41.13.0* =+=iE

• The CAPM approach combines estimates of two unknown quantities: the expected excess market return and

beta, and the CAPM could be wrong. Is it worth the trouble? • Compare the standard errors of the two estimates:

greater precision ⇔ lower standard error

• For historical mean:

%45.184

3.13error std. ===Ts

• For CAPM-based estimates:

( ) %76.0ˆ dev. std. std.error =⋅= mrβ

• Why? – Greater precision in the estimated market risk premium, mr , outweighs the effect of additional estimation error in β̂ .



• Allows substantial error in the CAPM pricing relation before the CAPM-based estimate of expected return

becomes less precise. • Even greater differences arise between approaches if the market equity premium is based on a longer sample

period than used for the individual stock’s average return.



Market Neutral Strategies (Read Articles in the reading list) Market-Neutral Strategies. The following questions are based on the assigned reading, “The Other Side of Zero,” by Saul Hansell, which appeared in the April 1992 issue of International Investor.

1. Explain why the “market-neutral” strategy is described as “two-alpha, no beta”. 2. On page 59 (middle column, bottom), Salomon’s Sorensen is quoted as saying “The volatility of the market-

neutral funds isn’t 16 percent like the S&P, but 5 to 10 percent.” The last paragraph on page 60 begins with the statement, “Of course, any strategy that doubles up on its alpha has, by definition, twice the risk. ‘The two alphas represent a form of leverage,’ says Unisys’ Service.”

a) What prediction can one make, if any, about the volatility of a market –neutral strategy? Upon what will the

volatility depend? b) If a market-neutral position is constructed by combining (1) a long position in a portfolio of oil stocks with a

beta of 1.2 and an annual volatility (standard deviation) of 30% with (2) a short position in S&P 500 index futures, what is the volatility of the return on the market-neutral position? Assume that the annual rate of return on the S&P 500 index has a volatility of 16%, and ignore margin deposits on the futures position (assume there are none).

3. On page 62 (middle column, bottom), the article reasons that “though the managers are investing in stocks, this

can hardly be considered equity, because the product is hedged. With a legitimate benchmark of the T-bill rate, a long-short equity strategy is in fact most appropriate as an alternative to other investments.”



a) Do you agree that the market neutral strategies should best be viewed as alternatives to cash? b) What would justify the statement that the T-bill rate is a “legitimate benchmark” for market-neutral

strategies? c) Would you recommend a market neutral strategy to a client with “a negative short-term view of the equity

market”? (see the quote from Mr. Borneman of Consolidated Gas) 4. Suppose the true expected excess return on a market-neutral strategy is zero, and the annual volatility of the

strategy is 10%.

a) What is the probability of observing a (simple arithmetic) average excess annual return over two years greater than 19 percent?

b) What is the probability of observing a five-year average excess annual return greater than 10%? c) If there are 24 such market neutral strategies, and the returns on the strategies are independent of each other,

what is the probability that at least one of the strategies will achieve the performance in part a.? In part b.?

(see page 59, middle column, middle paragraph)

single index model and market model - jose marin

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