

1
Introduction This text presents proofs of the CAP model.
Note that footnotes, e.g., [2] can be viewed by clicking them. 2 The CAPmodel This section presents a
derivation of the capital asset pricing model (CAPM). The CAPmodel 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 (k_{f}).
·
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 nonexisting (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 CAPmodel 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 (s^{2}). 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 146153]). 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 p_{i} is the probability of a
random event X_{i}, and N is the total number of events. 

Some mean and varians properties can be
derived[2]: 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] = c^{2}VAR[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: m_{p} =
E[k_{p}] = 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: s^{2}_{p} = VAR[R_{p}] = E[(k_{p}  E[k_{p}])^{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[a^{2}(X  E[X])^{2} + (1 a)^{2}(Y  E[Y])^{2} + 2a(1 a)(X  E[X])(Y  E[Y])] (using property 2)
= a^{2}E[(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)
= a^{2}VAR[X]+ (1 a)^{2}VAR[Y]+ 2a(1 a)COV[X,Y] = a^{2}VAR[X]+ (1 a)^{2}VAR[Y]+ 2a(1 a) r_{xy }s_{x}s_{y} (4) Very important the equations (1)
 (4) demonstrate the concept of portfolio diversification. In general it is
true that VAR[R_{p}] < aVAR[X] + (1 a)VAR[Y]
if 1 ≤ r_{xy} < 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 r_{xy} = 0, VAR[X] = VAR[Y], and a = 0,5 then {VAR[R_{p}] < aVAR[X] + (1 a)VAR[Y]} becomes {0,5VAR[X] < VAR[X]}.
Furthermore, if r_{xy }= 1 then {VAR[k_{p}] < 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
CAPmodel. 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 riskaverse 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: MRSs_{p}m_{p} = MRTs_{p}m_{p}. 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 (k_{f}). 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: m_{p} = aE[k_{m}] + (1  a)k_{f} (5) s_{p}^{2} = a^{2}VAR[k_{m}]+ (1 a)^{2}VAR[k_{f}]+ 2a(1 a)COV[k_{m},k_{f}] using
property 3 = a^{2}VAR[k_{m}]+ (1
a)^{2}0+ 2a(1 a)0 = a^{2}VAR[k_{m}] <=> s_{p} = as_{m} (6) This shows that the new minimum
varians opportunity set will be linear in the (m,s_{p}) 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: ¶m_{p}/¶a = E[k_{m}]  k_{f} ¶s_{p}/¶a = s_{m} Therefore the slope of the line is: ¶m_{p}/¶a /¶s_{p}/¶a = (E[k_{m}]  k_{f})/s_{m} (7) and since the intercept with the
mean axle is (s,m) = (0,k_{f}) the equation for the
minimum variance portfolio is m_{p} = k_{f} +
[(E[k_{m}]  k_{f})/s_{m}]s = R_{f} + (E[k_{m}]
 k_{f})s/ s_{m} (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 twofund 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: w_{i }=
V_{i}/SV_{i}, where V_{i}
is the market value of asset i and
SV_{i} is the market value
of all assets. Proposition 2 may now be stated: Proposition 2: With all the above assumptions
in mind (step 15) 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 CAPmodel does not requires any new assumptions only
new algebraic manipulations within the framework of the CPPmodel. 

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 (m_{p}, the CPPM). Now, imagine a portfolio
consisting of a% in a risky asset I and (1  a)% in the market portfolio M from the
CPPmodel. The mean and the variance of this portfolio is by definition E[k_{P}] = aE[k] + (1  a)E[k_{m}] (9) VAR[k_{P}] = a^{2}VAR[k]+ (1 a)^{2}VAR[k_{m}]+ 2a(1 a)COV[k,k_{m}] <=> s_{Rp} = {a^{2}VAR[k]+ (1 a)^{2}VAR[k_{m}]+ 2a(1 a)COV[k,k_{m}]}^{0,5} <=> s_{Rp} = {a^{2}VAR[k]+ (1 a)^{2}VAR[k_{m}]+ 2aCOV[k,k_{m}]  2a^{2}COV[k,k_{m}]}^{0,5} (10) Taking the derivative of (9) and (10) with
respect to a yields ¶E[k_{P}]/¶a = E[k]  E[k_{m}] (11) ¶s_{Rp}/¶a = 0,5{a^{2}VAR[k]+ (1 a)^{2}VAR[k_{m}]+ 2aCOV[k,k_{m}]  2a^{2}COV[k,k_{m}]}^{0,5} *{2aVAR[k]  2(1a)VAR[k_{m}] + 2COV[k,k_{m}]  4aCOV[k,k_{m}]} (12) The basic insight that the Nobel
laureate William Sharpe [the farther of the CAPmodel, 1963, 1964] provided,
was that he noted that in the CPPmodelequilibrium 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[k_{P}]/¶a_{a=0} = E[k]  E[k_{m}] (11) ¶s_{Rp}/¶a_{a}_{=0} = 0,5(VAR[k_{m}])^{0,5}*( 2VAR[k_{m}] + 2COV[k,k_{m}]) <=> ¶s_{Rp}/¶a_{a}_{=0} = (COV[k,k_{m}]
 VAR[k_{m}])/(VAR[k_{m}])^{0,5} <=> ¶s_{Rp}/¶a_{a}_{=0} = (COV[k,k_{m}]
 VAR[k_{m}])/s_{m} (13) Now, the slope of an equilibrium portfolio
evaluated at point M (a = 0) becomes: ¶E[k_{P}]/¶a/¶s_{Rp}/¶a_{a}_{=0} = (E[k]  E[k_{m}])/[(COV[k,k_{m}]  VAR[k_{m}])/s_{m}] (14) The final insight is to note that
this slope must be equal to the slope (7) of the CPPmodel 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 CPPmodel and under the same
assumptions. Therefore: (E[k_{m}]  k_{f})/s_{m} = (E[k]  E[k_{m}])/[(COV[k,k_{m}]  VAR[k_{m}])/s_{m}] <=> (E[k_{m}]  k_{f})/VAR[k_{m}] = (E[k]  E[k_{m}])/(COV[k,k_{m}]  VAR[k_{m}]) <=> (E[k_{m}]  k_{f})/ VAR[k_{m}]*(COV[k,k_{m}]  VAR[k_{m}]) = E[k]  E[k_{m}] <=> E[k] = E[k_{m}]
+ (E[k_{m}]  k_{f})/ VAR[k_{m}]*(COV[k,k_{m}]  VAR[k_{m}]) <=> E[k] = E[k_{m}]
+ (E[k_{m}]  k_{f})*(COV[k,k_{m}]/VAR[k_{m}])  (E[k_{m}]  k_{f}) <=> E[k] = (E[k_{m}]
 k_{f})*(COV[k,k_{m}]/VAR[k_{m}]) + k_{f} <=> _{} (15) Equation
(15) is the CAPmodel. It is also known as the security market line. See
figure III below. Q.E.D. 



Comparing the CAPmodel (15) by the
CPPmodel (8) reveals that they are almost identical. They are both linear
and they have the same measure for the price of risk (E[k_{m}]  k_{f}),
but they measure the quantity of risk differently. Where the CAPM measures
the quantity of risk by its normalised covariance (b_{m} = COV[k,k_{m}]/VAR[k_{m}])
the CPPM measures the quantity of risk by its normalised standard deviation
(s/s_{m} » VAR[k]/VAR[k_{m}]). The
reason to this difference is that investors only want to pay (E[k_{m}]
 k_{f}) 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 covaries 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: w_{i
}=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: _{} <=> <=> and lim
VAR[k] = E[s_{ij}] 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[k_{m}]  k_{f}) for an individual
risky asset in accordance with its covariance with a perfect diversified
portfolio M. The same could be said about the CPPmodel. 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: s^{2}_{m} = VAR[k_{m}] = COV[k_{m},k_{m}]
ad notam (2). In other words, the CPPmodel is a special case of the more
general CAPmodel. 

 Copyright 19972018, H. Mathiesen. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Legal notice. 