The standard CAPM model would provide a complete description of the behaviour of capital markets if each of the assumptions set forth held. The test of the capm is to the degree it can describe reality. But before examining such tests it is useful to develop equilibrium models based on realistic assumption. Most of the assumptions held under CAPM model are invalid in the real world. However this does not mean that the CAPM model should be completely disregarded as the differences from reality does not undermine the explanatory power of the model. On the other hand, the incorporation of alternative, more realistic assumptions in the model has several important benefits. CAPM may describe equilibrium returns on the macro level but it is not descriptive of micro behaviour. For example, most individuals and many institutions hold portfolios of risky assets that do not resemble the market portfolio. We might get better insight into investor behaviour by examining models developed under alternative and more realistic assumptions. Another reason of examining other equilibrium models is that it allows to formulate and test alternative explanations of equilibrium returns. Also because CAPM assumes several real -world influences away, it does not provide us with a mechanism for studying the impact of those influences on capital market equilibrium or on individual decision making. Only by recognizing the presence of these influences can their impact be investigated. For example if we assume personal taxes do not exist, there is no way the equilibrium model can be used to study the effect of taxes. By constructing a model that includes taxes, the study of taxes on individual behaviour and equilibrium returns in the capital market can be studied. The effect of modifying most of the assumptions of the CAPM model has been examined in the economics and finance literature. In this paper we will review much of this work and place special emphasis on two assumptions: the ability to lend at the riskless rate and the absence of personal taxes as these are important influences and also because they lead to the development of full-fledged general equilibrium models of a form that are amenable to testing. In the remainder of this paper we will discuss general equilibrium models derived under the more realistic assumption about each of the following influences:
Short sales
Riskless lending and borrowing
Personal taxes
Nonmarketable assets
Heterogeneous expectations
Non-price-taking behaviour
Multiperiod analysis
Some of them are discusses as follows
Short Sales Disallowed
One of the important assumptions made while deriving capital asset pricing model is that the investor can engage in unlimited short sales and it is defined in very broad terms as the investor was allowed to short sell any security and use the proceeds to buy any other security. This assumption was assumed for convenience as it simplified the mathematics of the derivation, but it was not a necessary assumption. Exactly the same result would have been obtained had the short sales disallowed. The economic reason behind this is in CAPM framework all the investors hold the market portfolio in equilibrium and since in equilibrium no investor short sells any security, prohibiting shot selling cannot change the equilibrium. Hence the same relationship would be derived irrespective of whether shortsales are allowed or prohibited.
Modifications of Riskless Lending and Borrowing
Another assumption under the CAPM framework is that the investor can lend and borrow at the risk free rate of interest. It is clearly not descriptive of the real world. It is much more realistic to assume that the investor can lend unlimited amount of money at the risk free rate however it cannot borrow at riskless rate. The lending assumption is based on the fact that the investor can buy government securities equal in maturity to their single period horizon. Such securities exist and are risk free and its rate is same for all the investors. It is better and convenient to examine the case where the investor can neither borrow nor lend at the risk free rate and extend the analysis to the case where they can lend but not borrow at the riskless rate.
No Riskless Lending or Borrowing Model
This is the most widely used general equilibrium model after the simple capital asset pricing model.
Simple Proof
Beta
R
A
D
C
Figure 1: Portfolio in expected return beta space
In the simple CAPM it was argued that that the systematic risk was the appropriate measure of risk and that two assets bearing the same systematic risk could not offer different rates of return. The essence of the argument was that the unsystematic risk of large diversified portfolios was essentially zero. Thus even if an individual asset had great deal of unsystematic risk, it would have a very little impact on the portfolio risk, and hence, unsystematic risk would not require a higher return. This is formalised in the diagram given below. Let's see why all assets are plotted on the straight line. It is known that the combination of two risky portfolios lie on a straight line connecting them in expected return Beta space, For example positive combination of A and D line on the straight line A-D. Thus if securities of portfolios happened to lie on a straight line in expected return Beta space, all combinations of securities ( e.g. portfolios) would lie on the same line.
Now consider the securities C and D in the figure. They both have the same systematic risk but C has a higher return. Clearly, an investor would purchase C rather than D until the price is adjusted and both the securities offer the same return. In fact a, an investor will purchase C and short D and have an asset with positive expected return and no systematic risk. Such an opportunity cannot exist in equilibrium. Therefore all portfolios and securities must plot along a straight line. One portfolio that lies along the straight line is the market portfolio. This can be understood in either of two ways. If it did not lie along the straight line, two assets would exist with the same systematic risk and different returns, and in equilibrium, equivalent assets must offer the same returns. In addition, note that all combinations of securities lie on the line and the market portfolio is the weighted average of securities.
The straight line can be described by any two points. One convenient point is the market portfolio and the second point is where the line cuts across the vertical axis (where the beta equals zero). The equation of the straight line is
Expected return = a +b (beta)
This must hold for a portfolio of zero beta. Letting Rz be the expected return on this portfolio.
Rz=a +b (0) or a=Rz
The equation must also hold for a market portfolio. If RM is the expected return on the market and Beta of the market portfolio is 1.
RM=Rz + b (1) or b= RM - Rz
Putting this together and letting Ri and Bi be the expected return and beta on an asset or portfolio, the equation for the expected return on any security or portfolio becomes
Ri=RZ + (RM - RZ) Bi
This is the so called zero beta version of the capital asset pricing model and is plotted in the figure shown below. This form of general equilibrium relationship is often referred to alternatively as two-factor model.
Slope (RM - Rz)
Rz
M
R
Beta
Figure 2: Zero beta capital asset pricing line
Riskless Lending But No Riskless Borrowing
While it is not realistic to assume that individuals can borrow at the riskless rate, it is realistic to assume that they can lend at risk less rate. Individuals can invest their funds in government securities which have time maturity equal to their time horizon and thus be guaranteed of riskless payoff at the horizon. If risk less lending is allowed then the investor's choice can be pictured as shown in the figure. All combinations of riskless of a riskless asset and risky portfolio lie on the straight line connecting the asset and the portfolio. The preferred combination lies on the straight line passing through the risk free asset and tangent to the efficient frontier. This is the RFT in the figure.
M
C
R
T
S
A
Z
Standard deviation
Figure 3: Opportunity set with riskless lending
Notice that T is drawn below and to the left of the market portfolio M and hence RZ>RF .Let us examine why this must hold true. Before the ability to lend at riskless rate all investors held portfolios along the efficient frontier SMC. With riskless lending the investor can hold portfolio of risky and riskless assets along the line RFT. If the investor chooses to hold an investment on the line RFT, he would be placing some of his assets in the risky assets denoted by T and some in riskless assets. The choice to hold any portfolio of risky assets other than T would never be made. Now why can't T and M be the same portfolio? As long as any investor has risk risk-return tradeoff such that he or she opts to hold a portfolio of investments to the right of T, the market must lie to the right of T. For example all investor but one chose to lend money and hold portfolio T. Now this one investor who does not chose T must hold a portfolio to the right of T in the efficient frontier STC. If the investor did not then he or she would be better off holding a portfolio on the line RFT and hence holding portfolio T. Since the market portfolio is the average of all the portfolios held by all investors, the market portfolio must be a combination of investor's portfolio and T. Thus it lies to the right of the T.M, being to the right of T, leads directly to RZ being larger than RF, RF is the intersection of the vertical axis and a line tangent at M. Since slope of the efficient frontier at M is less than that at T and since M lies above T, the line tangent at M must intersect the vertical axis above the line tangent at T. Thus RZ must be greater than RF .
Therefore in the case where riskless lending is allowed not all combinations of efficient portfolios are efficient. Portfolio from the line segment RFT and a portfolio from the curve TMC are dominated by a portfolio lying along the curve TMC. Portfolio T can be obtained by the combination of Portfolio Z and M. Examining the efficient frontier we can find that the investors who select the portfolio along the line segment RFT are placing some of their money in T and some in riskless asset. Those who select a portfolio on MC are selling portfolio Z short and investing all the proceeds in M. All the investors can be satisfied by holding some combination of the market portfolio, the minimum variance zero beta portfolio and the riskless asset. Having examined all the efficient portfolios in expected return standard deviation space, let us turn our attention to the location of the securities and portfolios in expected return Beta space. The market portfolio M is still an efficient portfolio. Therefore all the securities contained in M have an expected return given by
RF= RZ+ Bi (RM-Rz)
Similarly, all portfolios composed solely of risky assets have their return given by the equation above. This equation holds only for risky assets and for portfolios of risky assets. It does not describe the return on the riskless asset or the return on portfolios that contain riskless asset.
Thus , while the straight line RZM can be thought of as the security market line for all risky assets and for all portfolios composed entirely of risky assets. It does not describe the return on portfolio that contain riskless asset. Efficient portfolios have their returns given by the two line segments RFT and TC. The fact that the efficient portfolios have lower return for a given level of beta than individual assets may be startling. But the securities or portfolios on RZT have a higher standard deviation than portfolios with the same return on segment RZT.
C
R
Rm
M
RT
T
RZ
RF
Beta
Bt
1.0
Figure 4: The location of investments in expected return Beta space
Personal Taxes
The simple form of CAPM ignores the presence of taxes in arriving at an equilibrium solution. The implication of this assumption is that investors are indifferent between receiving income in the form of capital gains or dividends and that all investors hold the same portfolio of risky assets. If we recognize the fact of existence of taxes and in particular, the fact that capital gains are taxed at a rate lowers than dividends, the equilibrium prices should change. Investor should judge the return and risk on their portfolio after taxes. This implies that even with homogenous expectations about before tax return on a portfolio, the relevant efficient frontier faced by each investor will be different. However a general equilibrium relationship should still exist since, in the aggregate, markets must clear. The general equilibrium pricing equation for all assets and portfolios, given differential taxes on income and capital gains. The return on any asset or portfolio is given by
E (RI) = RF +B [E (RM) - RF) - t (qM - RF)] + t (qI- RF)
qM = dividend yield of the market portfolio
qI= the dividend yield for stock i
T=tax factor
Non-Price Taking Behaviour
Up to now we have assumed that individuals are price takers in that they ignore the impact of their buying or selling behaviour on the equilibrium price of securities and , hence, on their optimal portfolio holdings. The obvious question to ask is what happens if there are one or more investors, such as mutual funds or large pension funds, who believe that their behaviour impact prices. The method of analysis used by Lindenberg [77, 78] derives equilibrium condition under all possible demands by the price affector. The price affector selects the portfolio to maximize utility given the equilibrium prices that will result from her action. Assuming that price affector operates so as to maximize utility, we can then arrive at equilibrium condition. Lindenberg assumed that all investors, including the price taker hold some combination of the market portfolio and the riskless asset. However, the price affector would hold less of the riskless asset than would be the case if the price affector did not recognize the fact that her actions affected price. By doing so the price affector increases the utility. Beacause the price affector still holds a portfolio of riskless asset and the market portfolio; we still get a simple form of CAPM. But the market price of the risk is lower than it would be if all the investor were price takers. Lindenberg goes to analyze collective portfolio selection and efficient allocation among groups of investor. He finds that by colluding or merging, individuals or institutions can increase their utility.
Multiperiod CAPM
Upto now we assumed that all investors make investment decision based on a single period horizon. In fact, the portfolio an investor selects, at any time is really one step in a series of portfolios that he intends to hold over time to maximize his utility of life time consumption. Two questions immediately become apparent:
What are the condition under which the simple CAPM adequately describes the market equilibrium?
Is there a fully generated multi period equilibrium model?
Fama [29] and Elton and Gruber [25.26] have explored condition under which multperiod investment consumption decision can be reduced to the problem of maximize a one period utility function. These conditions are
The consumer's tastes for particular consumption goods and services are independent of future events
The consumer acts as if consumption opportunities in terms of goods and their prices are known at the beginning of the decision period
The consumer acts as if the distribution of one period returns on all assets are known at the beginning of the decision period.
Furthermore, Fama[29] has shown that if investors multiperiod utility function , expressed in terms of multiperiod consumption, exhibits both a preference of more to less and risk aversion with respect to each period's consumption , then the derived one- period utility has the same properties with respect to that period's consumption. Fama multiperiod assumptions make single-period capital asset pricing models appropriate for investors with multiperiod horizons. The particular single period model that results depends on the additional assumptions that are being made.
The consumption -oriented CAPM
A number of authors starting with Breeden [5] and Rubinstein [104] have taken a different approach in defining equilibrium in the capital markets. They start with a set of assumptions : investors maximize multiperiod utility function for lifetime consumption ;have homogenous beliefs conceringreturn characteristics of assets; there is an infinitely lived fixed population; there is a single consumption good; and there is an existence of a capital market that allows investor to reach consumption patter that they cannot jointly fare better by additional trades. They are able to shoe under these assumptions that the return on assets should be linearly related to the growth rate in aggregate consumption if the parameters of the linear relationship can be assumed constant over time. Furthermore, the residuals from the linear relationship are uncorrelated with the growth rate in aggregate consumption, have zero mean, and are uncorrelated with one another.
To be more explicit define
Ct =the growth rate in aggregate consumption per capita at time t
Rit= the rate of return on asset I in period t
Rit = ai + BiCt + e it
Where
E(eit)=0
the covariance residuals and the index is zero E(eit,Ct) =0
Bi= COV(Rit,Ct)/Var(Ct)
Given the equation above there are a number of ways to show that the equilibrium condition is
Ri=Rz + y1B1
Where
Y1 is the market price of the consumption beta
Rz is the expected return on a portfolio with zero consumption beta
This model is directly analogous to the simple form of the CAPM. The growth rate of percapita consumption has replaced the rate of return on the market portfolio as the influence affecting the time series of returns and hence equilibrium returns.
Conclusion
The simple for of CAPM is remarkable robust. Modifying some of its assumptions leaves the general model unchanged, whereas modifying other assumptions leads to the appearance of new terms in the equilibrium relationship or in some cases, to the modification of old terms. That the CAPM changes with changes in the assumption s are not unusual. What is unusual is
The robustness of the methodology in that it allows us to incorporate these changes
The fact that many of the conclusions of the original model hold, even with the changes in assumptions.
However these results seem stronger than they are as the assumptions are modified one at a time. When assumptions are modified, simultaneously, the departure from the standard CAPM may be much more serious. For example, when short sales were disallowed but lending and borrowing were allowed, the standard CAPM held. When riskless lending and borrowing were disallowed but short sales were allowed, we got a model that very much resembled the standard CAPM, except the slope and intercept were changed. Ross [100] has shown that when both riskless lending and borrowing and short sales are disallowed, one cannot derive a simple general equilibrium.
