Chapter 14: Nonstandard Forms of Capital Asset Pricing Models

Short Sales Disallowed Modifications of Riskless Lending and Borrowing Personal Taxes Nonmarketable Assets Heterogeneous Expectations Non-Price-Taking Behavior Multiperiod CAPM The Consumption-Oriented CAPM Inflation Risk and Equilibrium The Multi-Beta CAPM Conclusion

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Short Sales Disallowed

Short selling is illegal or severely restricted in many countries. This assumption was not necessary and has no bearing on CAPM derivation.

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Modifications of Riskless Lending and Borrowing

Second assumption of the CAPM is the ability of investors to lend or borrow unlimited funds at the riskless rate of interest.

This assumption is not descriptive of the real world. It is more realistic to assume that investors can lend unlimited sums of money at riskless rate but cannot borrow at riskless rate.

Investors can buy government securities equal in maturity to their single period horizon.

Only the government can borrow at the T-bill rate. No other investors in the world can borrow money at these rates unless they have some special concession.

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No Riskless Lending or Borrowing

• Riskless Lending but No Riskless Borrowing

• Other Lending and Borrowing Assumptions

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Figure 14.1Portfolios in expected return Beta space.

No Riskless Lending or Borrowing

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Figure 14.2

The zero Beta capital asset pricing line.

If no riskless asset exists investors can use a portfolio of risky assets which is uncorrelated with the market portfolio instead as the riskless asset. This portfolio is called the zero-beta portfolio.

)( ZMiZi RRRR

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Figure 14.5

The minimum variance frontier.

Properties of the Zero-beta portfolio(1) Of all the zero-beta portfolios this has the minimum variance(2) The separation principle applies here with the two portfolios, the market portfolio and the zero-beta portfolio, i.e. all investors hold combinations of the two.(3) The expected return on any security can be expressed as a linear function of its beta.

where E(Rz) is the expected return on a zero beta portfolio

)( ZMiZi RRRR

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Riskless Lending But No Riskless Borrowing

(a) There is a piecewise linear relationship between expected return and beta for efficient portfolios.(b) Efficient portfolios with the risk-free asset lie along the segment RFT and those containing only risky assets lies along the segment TMC.

Figure 14.6

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Figure 14.6

The opportunity set with riskless lending.

In the graph, the efficient frontier with riskless lending but no riskless

borrowing is the ray extending from RF

to the tangent portfolio T and then along the minimum-variance curve

through the market portfolio M and out toward infinity (assuming unlimited

short sales). All investors who wish to lend will hold tangent portfolio T in

some combination with the riskless asset, since no other portfolio offers a higher slope. Furthermore, unless all

investors lend or invest solely in portfolio T, the market portfolio M will

be along the minimum-variance curve to the right of portfolio T, since the

market portfolio is a wealth-weighted average of all the efficient risky-asset

portfolios held by investors, and no rational investor would hold a risky-

asset portfolio along the curve to the left of T.

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The expected return on a zero-beta asset is the intercept of a line

tangent to the market portfolio, and the zero-beta portfolio on the

minimum-variance frontier must be below the global minimum

variance portfolio of risky assets by the geometry of the graph.

Furthermore, since the risk-free lending rate is the intercept of the

line tangent to portfolio T, and since T is to the left of M on the

minimum-variance curve, the risk-free lending rate must be below the expected return on a zero-

beta asset.

Figure 14.6The opportunity set with riskless lending.

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Figure 14.7

The location of investments in expected return Beta space.

There is a piecewise linear relationship between expected return and beta for efficient portfolios.

Efficient portfolios with the risk-free asset lie along the segment RFT and those containing only risky assets lies along the segment TMC.

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In Summary

The zero-beta security market line is the line extend from the expected return on a zero-beta asset through the market portfolio and out toward infinity (assuming unlimited short sales). The expected return-beta relationships of all risky securities risky-asset portfolios (including the market portfolio M and portfolio T) are described by that line. The other line from the risk-free lending rate to portfolio T only describes the expected return-beta relationships of combination portfolios of the risk-free asset and portfolio T; those combination portfolios are not described by the zero-beta security market line.

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Figure 14.8

Other Lending and Borrowing Assumptions

The opportunity set with a differential lending and borrowing rate.

)( ZMiZi RRRR

)( LMiLi RRRR

)( BMiBi RRRR

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Personal Taxes

FiiFmFmFi RRRRRR

The model considers differential taxes on dividends and capital gains in a one-period context where investors maximize their one-period returns. The final model for expected return has a dividend component.

When dividends are on average taxed at a higher rate than capital gains t is positive and expected return is an increasing function of dividend yield.Above equation is no longer sufficient to describe the equilibrium relationship. The expected return is now affected by both Beta and Dividend yield factors. This means equilibrium relationship must be described in 3-dimensional plane.

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Personal Taxes

It should be noted that the optimal portfolio for any investor is a function of tax rate paid on capital gains and dividends.Investors in low tax bracket will favor to hold high-dividend stocks in their portfolio and vice versa.

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Nonmarketable Assets

Hi

m

Hmi

Hmm

Hm

Fm

Fi RRP

PRR

RRPP

RRRR cov cov

cov 2

The effect of leaving out bonds depends on two factors:1.)

Whether or not the returns on the aggregate of all bonds are negatively or

positively correlated with the returns on the aggregate of all stocks;2.)

The correlation between the returns on a particular stock and the returns on

the aggregate of all bonds.From the above equation, if returns on stocks and bonds are generally positively

correlated (as empirical evidence shows), then the denominator in the second term of the equation will tend to lower the expected return on any stock. If the return on a

particular stock is negatively correlated with bonds, that will further lower the stock’s expected return. However, if the stock is positively correlated with bonds, this will offset

the effect of positive correlation between all stocks and bonds and may actually result in a higher expected return for the stock.

Existence of Non-Marketable Assets (such as Human Capital)The separation principle still holds but,(a) Investors hold different portfolios of risky assets depending upon the portfolios of non-marketable assets that they possess.(b) The market price of risk includes the variance of the market and the covariance between the market portfolio and the portfolio of non-marketable assets. The equation also holds for deleted assets, with the subscript H now standing for those assets that were left out:

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Heterogeneous Expectations

Existence of Heterogeneous Expectations and Information

To get strong conclusions we have to assume that all investors have a certain class of utility functions (Constant Absolute risk aversion) and complete markets (At least as many independent securities as states).

To derive the model Lintner assumed a negative exponential utility function and was able to derive the simple CAPM .

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Non-Price-Taking BehaviorThis relates to assumption that individuals act as price takers and the impact of their buying is ignored.

What happens if some investors such as mutual funds or large pension funds believe that their trading will impact the prices.

Lindenberg finds that all investors, including the price takers, hold some combination of riskless asset and risk portfolio.

However, the price affector will hold less riskless asset than the price takers.

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The Consumption-Oriented CAPM

This model starts with the following assumptions:1. Investors are assumed to maximize a multi-period utility function of lifetime consumption. 2.Have homogenous beliefs concerning characteristics of assets3.There is an infinity fixed population4.There is a single consumption good5.There exists a capital market that allows investors to reach a consumption pattern such that they cannot jointly fare better by additional trade.With these assumptions, Breeden and Rubinstein showed that the expected rate of return on asset should be linearly related to the growth rate in aggregate consumption if the parameters of the linear relationship can be assume constant over time. Let Ci = the growth rate in aggregate consumption per capita at time tRit = the rate of return on asset I in period tE(eit )=0 and E(eit ,Ci)=0

itiit CR ii )(

),(

i

iiti CVar

CRCov

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Inflation Risk and Equilibrium

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The Multi-Beta CAPMMerton has constructed a generalized intertemporal CAPM which the sources of uncertainty would be priced. Merton assumes investors as solving lifetime consumption decisions when faced with multiple sources of uncertainty.

Sources of uncertainty are:Future laborFuture prices of goods,Future investment opportunities, and so on.The inflation is the simplest form of a multi-Beta CAPM where expected return is based on two sensitivities;

)()R-(RmR Fi FIiIiMF RRR Other risk which might be important areDefault riskTerm Structure RiskDeflation RiskProfit risk

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