 Page info:  *Author: Mathiesen, H.  *Document version: 2.2. *Copyright 1997-2019, H. Mathiesen. Legal notice.

Model: The CAP model (CAPM)

1                   Introduction

This text presents proofs of the CAP model. Note that footnotes, e.g.,  can be viewed by clicking them.

2       The CAP-model

This section presents a derivation of the capital asset pricing model (CAPM). The CAP-model is a ceteris paribus model. It is only valid within a special set of assumptions. They are:

·      Investors are risk averse individuals who maximize the expected utility of their end of period wealth. Implication: The model is a one period model.

·      Investors have homogenous expectations (beliefs) about asset returns. Implication: all investors perceive identical opportunity sets. This is, everyone have the same information at the same time.

·      Asset returns are distributed by the normal distribution.

·      There exists a risk free asset and investors may borrow or lend unlimited amounts of this asset at a constant rate: the risk free rate (kf).

·      There is a definite number of assets and their quantities are fixed within the one period world.

·      All assets are perfectly divisible and priced in a perfectly competitive marked. Implication: e.g. human capital is non-existing (it is not divisible and it can’t be owned as an asset).

·      Asset markets are frictionless and information is costless and simultaneously available to all investors. Implication: the borrowing rate equals the lending rate.

·      There are no market imperfections such as taxes, regulations, or restrictions on short selling.

Step 1. The derivation of the CAP-model starts by assuming that all assets are stochastic and follow a normal distribution. This distribution is described completely by its two parameters: mean value (m) and variance (s2). The mean value is a measure of location among many such as median and mode. Likewise, the variance value is a measure of dispersion among many such as range, semiinterquartile range, semivariance, mean absolute deviation (For definitions of the mentioned measures of location and dispersion, see Copeland and Weston [1988, pages 146-153]). In the hypothetical world of the CAPM theory all that the investor bothers about is the values of the normal distribution. In the real world asset return are not normally distributed and investors do find other measures of location and dispersion relevant. However, the assumption may be seen as a reasonable approximation and it is needed in order to simplify matters.

As a result the mean and the variance of an asset X is defined as: (1) (2)

and the covariance and the correlation coefficient between two assets X and Y are:  where pi is the probability of a random event Xi, and N is the total number of events.

Some mean and varians properties can be derived:

Property 1: E[X + c] = E[X] + c

Property 2: E[cX] = cE[X]

Property 3: VAR[X + c] = VAR[X]

Property 4: VAR[cX] = c2VAR[X]

where c is a constant.

Consider a portfolio of two risky assets, X and Y with a % in asset X and (1- a) % in asset Y. They are both normally distributed. The return on this portfolio (using property 1 and 2) is:

mp  = E[kp] = E[aX + (1- a)Y] = E[aX] + E[(1- a)Y] = aE[X] + (1- a)E[Y]                        (3)

and the varians on this portfolio is:

s2p = VAR[Rp] = E[(kp - E[kp])2] = E[({aX + (1- a)Y} - E[aX + (1- a)Y])2]

(using property 2)

= E[({aX + (1- a)Y} - {aE[X] + (1- a)E[Y]})2]

= E[({aX - aE[X]} + {(1- a)Y - (1- a)E[Y]})2]

= E[(a{X - E[X]} + (1- a){Y - E[Y]})2]

= E[a2(X - E[X])2 + (1- a)2(Y - E[Y])2 + 2a(1- a)(X - E[X])(Y - E[Y])]

(using property 2)

= a2E[(X - E[X])2]+ (1- a)2 E[(Y - E[Y])2]+ 2a(1- a)E[(X - E[X])(Y - E[Y])]

(using property 4)

= a2VAR[X]+ (1- a)2VAR[Y]+ 2a(1- a)COV[X,Y]

= a2VAR[X]+ (1- a)2VAR[Y]+ 2a(1- a) rxy sxsy                                                          (4)

Very important the equations (1) - (4) demonstrate the concept of portfolio diversification. In general it is true that VAR[Rp] < aVAR[X] + (1- a)VAR[Y]  if -1 ≤ rxy < 1. In words, the variance of a portfolio is less than the simple average of variances of the assets in the portfolio if the assets are not perfectly correlated. This will not be demonstrated rigorously but set rxy = 0, VAR[X] = VAR[Y], and a = 0,5 then {VAR[Rp] < aVAR[X] + (1- a)VAR[Y]} becomes {0,5VAR[X] < VAR[X]}. Furthermore, if rxy = -1 then {VAR[kp] < aVAR[X] + (1- a)VAR[Y]} becomes {0 = VAR[X] < VAR[X] } a perfect hedge (the resulting portfolio is riskless)! The diversification property implies that the minimum variance opportunity set will be convex, and this is a necessary condition for the existence of unique and efficient portfolio equilibrium. As will be seen this property is used for the derivation of the CAP-model. The minimum variance opportunity set is the locus of mean and variance combinations offered by portfolios of risky assets that yield the minimum variance for a given return. The locus is illustrated as the fat curve in figure I below. The convexity property holds for two risky assets or more. The area on and behind the locus (the oval) is sometimes refereed to as the portfolio production possibility area. Each point in this region represent the return and risk from some single asset available in the market, or some portfolio made on those assets. Step 2. The next assumption is that investors are risk averse and maximize expected utility. They perceive variance as a bad and mean as a good. This is also illustrated in figure I where tree risk-averse indifference curves are drawn. Now, the first conclusion is.

Proposition 1: An individual investor will maximize expected utility of his end of period wealth where his subjective marginal rate of substitution between risk and return represented by his indifference curves is equal to the objective marginal rate of transformation offered by the minimum variance opportunity set: MRSspmp = MRTspmp.

Step 3. Assume now that there in addition to the many risky assets exist a risk free asset and that investors may borrow or lend unlimited amounts of this asset at a constant rate: the risk free rate (kf). Furthermore, capital markets are assumed to be frictionless. The effect on the shape of the portfolio production possibility area is profound as illustrated in figure II below. The reason for this dramatic change is simple. With the existence of the risk free asset the mean and the variance for a portfolio consisting of the risk free asset and the portfolio M (see figure) will be:

mp  = aE[km] + (1 - a)kf                                                                                                           (5)

sp2 = a2VAR[km]+ (1- a)2VAR[kf]+ 2a(1- a)COV[km,kf]

using property 3

= a2VAR[km]+ (1- a)20+ 2a(1- a)0 = a2VAR[km]                                                                                                                                                                                                             <=>

sp = asm                                                                                                                               (6)

This shows that the new minimum varians opportunity set will be linear in the (m,sp) space and consists of portfolios with some fraction a of portfolio M and (1 - a) of the risk free asset. In the following an equation for the linear minimum variance opportunity set is developed. Taking the derivative of (5) and (6) yields:

mp/a = E[km] - kf

sp/a = sm

Therefore the slope of the line is:

mp/a /sp/a = (E[km] - kf)/sm                                                                                             (7)

and since the intercept with the mean axle is (s,m) = (0,kf) the equation for the minimum variance portfolio is

mp = kf + [(E[km] - kf)/sm]s = Rf  + (E[km] - kf)s/ sm                                                              (8)

This equation has come to be known as the capital market line (CML). It is the fat line in figure II. This formula is referred to as the capital portfolio pricing model (CPPM), because it prices efficient portfolios. The following explains why.

Step 4. Assume that all investors have homogeneous beliefs about the expected distribution of returns offered by all assets. Also, capital markets are frictionless and information is costless and simultaneously available to all investors. Furthermore, there are no market imperfections. Taken together this implies that all investors calculate the same equation for the market capital line and that the borrowing rate equals the lending rate.

Within broad degrees of risk aversion each investor will maximize their utility by holding some combination of the risk free asset and the portfolio M. This property is known as the two-fund separation principle. It is illustrated in figure II by the tangency of the indifference curves on the CML for different degrees of risk.

Step 5. Assume further that all assets are perfectly divisible and priced in a perfectly competitive marked. Furthermore, there is a definite number of assets and their quantities are fixed within the one period world. Then the portfolio M turns out to be the market portfolio of all risky assets. The reason is that equilibrium requires all prices to be adjusted so that the excess demand for any asset is zero. That is, each asset is equally attractive to investors. Theoretically the reduction of variance from diversification increases as the number of risky assets included in the portfolio M rise. Therefore, all assets will be hold in the portfolio M in accordance to their market value weight: wi = Vi/SVi, where Vi is the market value of asset i and SVi is the market value of all assets. Proposition 2 may now be stated:

Proposition 2: With all the above assumptions in mind (step 1-5) the capital market line (8) shows the relation between mean and variance of portfolios (consisting of the risk free asset and the market portfolio) that are efficiently priced and perfectly diversified.

The capital market line equation could rightly be called the capital portfolio pricing model (CPPM) since it prices efficient portfolios. What is more interesting is to develop an equation for pricing of individual assets. This is exactly what the capital asset pricing model (CAPM) does. The CAP-model does not requires any new assumptions only new algebraic manipulations within the framework of the CPP-model.

Step 6. From CPPM to CAPM. What is wanted is a model for efficient pricing of capital for individual assets (E[k], the CAPM), not one for efficient cost of capital for portfolios (mp, the CPPM). Now, imagine a portfolio consisting of a% in a risky asset I and (1 - a)% in the market portfolio M from the CPP-model. The mean and the variance of this portfolio is by definition

E[kP] = aE[k] + (1 - a)E[km]                                                                                                   (9)

VAR[kP] = a2VAR[k]+ (1- a)2VAR[km]+ 2a(1- a)COV[k,km]

<=>

sRp = {a2VAR[k]+ (1- a)2VAR[km]+ 2a(1- a)COV[k,km]}-0,5

<=>

sRp = {a2VAR[k]+ (1- a)2VAR[km]+ 2aCOV[k,km] - 2a2COV[k,km]}-0,5                   (10)

Taking the derivative of (9) and (10) with respect to a yields

E[kP]/a = E[k] - E[km]                                                                                                       (11)

sRp/a = 0,5{a2VAR[k]+ (1- a)2VAR[km]+ 2aCOV[k,km] - 2a2COV[k,km]}-0,5

*{2aVAR[k] - 2(1-a)VAR[km] + 2COV[k,km] - 4aCOV[k,km]} (12)

The basic insight that the Nobel laureate William Sharpe [the farther of the CAP-model, 1963, 1964] provided, was that he noted that in the CPP-model-equilibrium the market portfolio M already contains the risky asset I. If the risky asset I is added to the market portfolio M in any positive quantities it creates an excess demand for asset I by aI. Therefore, equations (11) and (12) must be evaluated at a = 0 for the equations to describe an equilibrium portfolio. This is done below:

E[kP]/a|a=0 = E[k] - E[km]                                                                                                  (11)

sRp/a|a=0 = 0,5(VAR[km])-0,5*(- 2VAR[km] + 2COV[k,km])

<=>

sRp/a|a=0 = (COV[k,km] - VAR[km])/(VAR[km])0,5

<=>

sRp/a|a=0 = (COV[k,km] - VAR[km])/sm                                                                          (13)

Now, the slope of an equilibrium portfolio evaluated at point M (a = 0) becomes:

E[kP]/a/sRp/a|a=0 = (E[k] - E[km])/[(COV[k,km] - VAR[km])/sm]                        (14)

The final insight is to note that this slope must be equal to the slope (7) of the CPP-model since the capital market line is tangent to the market portfolio M and the slope (14) is evaluated at M identical to M in the CPP-model and under the same assumptions. Therefore:

(E[km] - kf)/sm = (E[k] - E[km])/[(COV[k,km] - VAR[km])/sm]

<=>

(E[km] - kf)/VAR[km] = (E[k] - E[km])/(COV[k,km] - VAR[km])

<=>

(E[km] - kf)/ VAR[km]*(COV[k,km] - VAR[km]) = E[k] - E[km]

<=>

E[k] = E[km] + (E[km] - kf)/ VAR[km]*(COV[k,km] - VAR[km])

<=>

E[k] = E[km] + (E[km] - kf)*(COV[k,km]/VAR[km]) - (E[km] - kf)

<=>

E[k] = (E[km] - kf)*(COV[k,km]/VAR[km]) + kf

<=> (15)

Equation (15) is the CAP-model. It is also known as the security market line. See figure III below.

Q.E.D. Comparing the CAP-model (15) by the CPP-model (8) reveals that they are almost identical. They are both linear and they have the same measure for the price of risk (E[km] - kf), but they measure the quantity of risk differently. Where the CAPM measures the quantity of risk by its normalised covariance (bm = COV[k,km]/VAR[km]) the CPPM measures the quantity of risk by its normalised standard deviation (s/sm » VAR[k]/VAR[km]). The reason to this difference is that investors only want to pay (E[km] - kf) for undiversifiable risk. The CPPM prices portfolios that are perfectly diversified. Therefore, the appropriate measure for risk is the variance of that portfolio. Contrary, the CAPM prices an individual asset that will be diversified. Therefore, only the part of the variance that co-varies with a perfect diversified portfolio is relevant to pay for. The following argument helps making this clearer.

The variance of an equally weighted portfolio of N risky assets (weight: wi =1/N, for all iÎ[1,N]) is <=> <=> As an approximation we may replace the individual variances and covariance’s with their mean values. This implies: <=>
VAR[k] = 1/N2   N*E[sii] + 1/N2*(N2 - N)E[sij]

<=>
VAR[k] = 1/N*E[sii] + (1 - 1/N)E[sij]

and

lim VAR[k] = E[sij]

N ® ¥

This demonstrates that as the portfolio becomes more diversified by letting the number of risky asset (N) in the portfolio rise, the covariance term becomes relatively more important. Indeed, in the limit it is the only thing that matters. Therefore, investors capable of creating perfect diversified portfolios will only be willing to pay the price of risk (E[km] - kf) for an individual risky asset in accordance with its covariance with a perfect diversified portfolio M. The same could be said about the CPP-model. However, this model is pricing assets (portfolios) that are already perfectly diversified and they will by definition have the same characteristics as the market portfolio M. This implies that the covariance is equal to the variance: s2m = VAR[km] = COV[km,km] ad notam (2). In other words, the CPP-model is a special case of the more general CAP-model.

- Copyright 1997-2019, H. Mathiesen. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Legal notice.

     This derivation draw on the derivation given in Copeland and Weston [1988, pages 194-198].

     For derivation confer Copeland and Weston [1988, pages 147-152].