Asset pricing, higher moments and the market risk premium : a note

FACULTY WORKING
PAPER NO. 1014

Asset Pricing, Higher Moments and the
Market Risk Premium: A Note

R. Stephen Sears
John Wei

College of Commerce and Business Administration
Bureau of Economic and Business Research
University of Illinois, U'bana-Champaign

BEBR

FACULTY WORKING PAPER NO. 1014
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
February 1984

Asset Pricing, Higher Moments
and the Market Risk Premium: A Note

R. Stephen Sears, Professor
Department of Finance

John Wei, Professor
Department of Finance

Digitized by the Internet Archive

in 2011 with funding from

University of Illinois Urbana-Champaign

http://www.archive.org/details/assetpricinghigh1014sear

Abstract

The purpose of this note is to examine, theoretically, why the market
risk premium (R^ _ g\ raay influence tests of asset pricing models with
higher moments. When moments of higher order than the variance are added
to a pricing model developed within the usual two-fund separation assump-
tions, the market risk premium enters the pricing equation in a nonlinear
fashion and is implicit in the estimation of each moment's coefficient.
Unless this nonlinearity is recognized, incorrect conclusions regarding
the tests of such models may result.

Asset Pricing, Higher Moments and the Market Risk Premium: A Note

I. Introduction

Following the work of Markowitz [15], Sharpe [22], Lintner [14] and
Mossin [17] developed the first formulations of the mean-variance capi-
tal asset pricing model (CAPM). Subsequent modifications to the theory
were made by Fama [5], Brennan [4] and Black [2] as well as others.
Proponents of the CAPM note its simplicity and potential for testability;
however, the model has not been empirically validated in the tests of
Black, Jensen and Scholes [3], Miller and Scholes [16], Fama and MacBeth
[6] and many others. Furthermore, Roll [18] has warned us of the ambi-
guous nature of such tests because of a number of measurement difficul-
ties and joint hypotheses present in the model.

Efforts to respecify the pricing equation have gone in several
directions. The direction that is of interest in this note is the re-
search that has expanded the utility function beyond the second moment
to examine the importance of higher moments. There has been recent
interest in the importance of higher moments as evidenced in a paper by
Scott and Horvath [20] which develops a utility theory of preference for
all moments under rather general conditions. The third moment (skewness)
has already received some attention in the literature [1, 8, 9, 10, 11,
12, 20]. Following the work of Rubinstein [19], Kraus and Litzenberger
(KL) [13] derived and tested a linear three moment pricing model,
finding the additional variable (co-skewness) to explain the empirical
anomalies of the two moment CAPM. The three moment model was re-examined
by Friend and Westerfield (FW) [7] with mixed results. The FW study
found some, but not conclusive evidence of the importance of skewness in

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the pricing of assets. In particular, FW found empirical tests of the
three moment model to be "...especially sensitive to the relationship
between the market rate of returns (O and the risk-free rate (R_)..."
[7, p. 899] and concluded "...there is no obvious reason to expect the
sign of the co-skewness coefficient to depend on the relationship
between R and R " [7, p. 908].

The purpose of this note is to examine, theoretically, why the market
risk premium (R^ _ d ) may influence tests of asset pricing models with
higher moments. When moments of higher order than the variance are added
to a pricing model developed within the usual two-fund separation assump-
tions, the market risk premium enters the pricing equation in a nonlinear
fashion and is implicit in the estimation of each moment' s coefficient.
Unless this nonlinearity is recognized, incorrect conclusions regarding
the tests of such models may result.

In Section II, the three moment model is re-examined to demonstrate
the presence of the market risk premium in each moment's coefficient.
Because the market risk premium introduces non-linearities in the model,
empirical tests should be redesigned to distinguish between the effects
of (R^^ - R.) and skewness. Furthermore, expressing the model in this
manner provides a clearer understanding of the conditions which are
necessary if skewness is to be useful in explaining the two moment CAPM
empirical results. A brief summary is contained in Section III.

II. Higher Moments and the Market Risk Premium
The Three Moment Model

Using the framework and notation developed in KL [13], the theore-
tical market equilibrium relationship between security excess returns

-3-

(R. - Rf ) , the market risk premium (R - R ), systematic risk (3 ) and
systematic skewness (y.) is:

R. "

1

Rf - [(^ " Rf)/(l+K3)JSi + [K3<8^ " Rf)/(1+K3)]y1 CD

where: K. = [(dW/din )/(dW/da ) ] (m /a ) , the market's marginal rate"
of substitution between skewness and risk times the
risk-adjusted skewness of the market portfolio
a , m^ = second and third central moments about the market port-
folio's return
W, a , m^ = first, second and third central moments about end of
period wealth.

The KL version of the model is given by (KL 3):

R - Rf - [(dW/do )<j ]3± + [(dW/dmw)mMJyi (KL 3)

Kraus and Litzenberger recognize that, in equilibrium, (KL 3) also im-
plies the following condition:

^ - Rf = UdW/daw)aM] + [(dW/dm^] (2)

since 3M ■ YM = 1« Thus, (2) produces one empirical hypothesis of the
model — that the sum of the estimated coefficients should equal (R^ - R-)
However, the (KL 3) specification does not reveal all of the informa-
tion in the theoretical model since it does not specify the effects of
the market risk premium on the individual coefficients of 3. and Y-»
That is, the development of (KL 3) and (2) also impose restrictions
between (R - R ) and each of the coefficients individually. These

-4-

effects can be seen by dividing (KL 3) by (2) which produces equation
(1). Equation (1) indicates that (It - R ) is implicit in each of the
model's coefficients on $. and y.. Although (KL 3) and (1) both come

from the same theoretical model, the derivation of (1) is consistent

2
with the risk premium formulation of the two moment CAPM and brings to

light some insights regarding the three moment model and why empirical

tests of the model can be sensitive to the market risk premium.

Consider the linear empirical version of (1):

h - Rf ■ bo + Vi + Vi (3)

where: b = intercept, hypothesized to equal 0

bl = C(*M " V/(1+K3)]
b2 = [K3(\ " Rf>/(1+V]

Previous studies have focused upon the entire coefficients of 3. and

y., b, and b_ , in the examination of risk and skewness. Such examina-
i 1 2

tions, however, measure the joint effects of (R^ - Rf) and K~. Failure
to separate (R^ - R ) from K may result in incorrect inferences
regarding the importance as well as the sign of risk and skewness.

Consistent with the two moment model (a special case of (1)), the
importance of risk is more properly measured by (R^ - R_) = b + b. ,
rather than b ; likewise, the importance of skewness, K^., is gauged by

a . ^

b./b1 , rather than b^. Thus, skewness is evaluated on a relative basis

3

as measured by the market's tradeoff between skewness and risk. Since

the nonlinear parameters of (1) are identified in terms of the linear
parameters of (3), time series (non-stationary), cross-sectional tests
such as those performed in [7, 13] can still be used to test separate
hypotheses about K. and (R^ - R ) .

-5-

In addition, because of the interaction between (IL. - R ) and K in
the determination of b and b , under certain conditions b and b can
also give misleading signals regarding the signs of risk and skewness in
the model. When the empirical model estimates a negative market premium
(b1 + b? < 0) , b_ attaches the wrong sign to skewness. When Of - R )
< 0 and L < 0, b > 0. Similarly, when (IL - Rf) < 0 and iL > 0,
b < 0. Since the linear model focuses on b9 rather than K_, its use
will lead to incorrect conclusions regarding the sign of skewness when
(R^ - Rf) < 0. This is especially troublesome in a study such as FW in
which 26 of the 68 regressions result in (R^ - Rf) < 0. In 24 of these

cases, the use of b rather than K- leads to incorrect inferences

5
regarding the sign of skewness. In similar fashion, when K_ < -1, the

use of b , rather than b + b , results an incorrect inconclusion

regarding the sign of risk.

Skewness Preference and the Two Moment CAPM

Empirical tests of the two moment CAPM have found a positive inter-
cept and a slope value less than its theoretical value, (R^ - Rf ) . If
the three moment model is the correct pricing mechanism, then the omis-
sion of y. from the two moment model should explain, empirically, the
two moment model's results. Explicit consideration of (R^ - R,) in each
coefficient in the three moment model (1) provides a linkage between the
two models and enables an examination of the theoretical conditions under
which the omission of y. is consistent with the two moment empirical
results.

The two moment CAPM is given by (4):

R. - R - b * + b *0. (4)

i f 0 1 i

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Under the hypothesis that the three moment model is correct:

b * =» cov[("r. - R.),3.]/var(3.)
1 1 f 1 i

= cov[(bQ + b10i + b2Yi),Si]/var(Bi)

= (\- Rf)[(l + aK3)/(l+K3)] (5)

where a = cov( 3 . ,Y. )/var(8 . ) , the slope of the regression of
y. against 3.

Equation (5) provides theoretical support for KL's "heuristic rationale"
[13, p. 1098] and their empirical results since if a > 1 when K < 0
(ni > 0) , b- < (R,. - Rf ) and b^ > 0. The empirical evidence provided
by KL and FW indicates considerable correlation between 3. and y. when
in > 0 as well as when ni < 0. Furthermore, it seems reasonable that
var(Y.) > var(3.). Together, these imply that a > 1 and the empirical
results of the two moment CAPM are consistent with a market preference
for positive skewness when m^ > 0. However, note that b < (R^ - R )
and bn > 0 when IC > 0 (n^ < 0) only if a < 1. Thus, a preference for
positive skewness when n^ < 0 requires higher 3.'s to be associated
with proportionately smaller Y.'s.

Extension of the Pricing Model to N Moments

With the recent interests in asset prices and in higher moments,
some researchers may be tempted to expand the asset pricing model beyond
three moments. The interaction of (r~ - R ) with higher moments becomes
compounded when moments higher than skewness are included. The theoreti-
cal N moment pricing model is:

-7-

Ri"Rf * (RM-V V(Vi)/E?Kn! <«

n=2 n=2

where: K = [(dW/dm TT)/(d¥/dm0 ..) ] (m /m. )

a n,W 2,W n,M 2,M

m w * the ntn central moment about the market portfolio's
n,M

rate of return, where m0 w = a and m_ w = m as in (1)

2 , M M 3 , M M

m = the ntn central moment about the investor's end of

n,W

period wealth, where ra9 „ = a and m» = m^ as in (1)

Y. = the systematic portion of the nth moment for asset i,

2 3

where Y . - B . and y. = Y . as in (1)

The two (N=2) and three (N=3) moment models are simply special cases of
(6). As seen in (6), (IL - Rf) appears in each of the N moments' coef-
ficients and the importance of the ntn moment is an assessment of the
preference tradeoffs in the market between the ntn moment and the second
moment (risk) .

Conclusion

Recent research has examined the importance of skewness in the
pricing of risky assets, finding the results of such tests to be influ-
enced by the market risk premium. The purpose of this note has been to
explore a not so obvious theoretical relationship within such models —
namely, that such models are intrinsically non-linear in the market
risk premium. Failure to account for this interaction may lead to
erroneous conclusions regarding the empirical results of such models.

-8-

Footnotes

In this paper, the word "skewness" refers to the third moment of the
return distribution. Many authors use the term "skewness" as the third
moment divided by the standard deviation cubed.

2
In the two-moment _vers ion of the model, the investor's problem is to

maximize: E[U(W)j = U[W,a] subject to: Zq^ + q = Wq. The equilibrium

conditions are: x

(¥. - R.) - [(d¥/daTT)c IB, (a)

1 r W M i

(\ ' V = [(d¥/daW)0M] (b)

Dividing (a) by (b) p_roduces the_familiar two-moment CAPM in terms of the
market risk premium, R-j_ - Rf = (R^ - Rf)3i« In both the two-moment model
and equation (1), the market ' s_marginal rate of substitution between return
and risk times market risk, (dW/doy)^, is not present in the final equa-
tion. However, the contribution of skewness to the model is evaluated by
the relative importance of the third moment vis a vis the second moment
(K3).

3
When jK.3 | < 1 (> 1) , risk is more (less) important than skewness in

the pricing of assets. When JK3 j = 1, the market views risk and skewness

as equally important. When K3 = 1, equation (1) becomes:

*i-Rf-T<W(f!i + V

However, when K3 = -1, the theoretical model specification becomes
ambiguous since as K3 -*■ -1 # (R^ - Rf) ♦ 0 and:

lim
h ~ Rf =j^yoj [(\ " V/(1+K3)]3i + [V*m - V/(1+K3)]Yi

= [dW/daTJ)aJ(B. - Y.) or
W M i 1

■ [dW/dmw)mM](Yi - S.)

It is only in this case, when K3 = -1, where (R^ - Rf) is not theoreti-
cally implicit in both coefficients on 3i and Yi»

4 . .

This assumes that [K3 j < 1 when K3 < 0. Exceptions to this in FW

[7] correspond to Table IV: 1972-1976 and Table VII: 1952-1976.

The two exceptions to this in FW [7] correspond to instances where
K3 < -1 (see footnote 4). The most dramatic illustration of the effects of

(R^ - Rf ) on b2 can be seen in the FW study where the periods are divided
into cases where % > Rf and where % < Rf (e.g., Table VI). Since (R^ - Rf)
is implicit in b£ , the sign of b£ will be influenced by the sign of (R^ - Rf)

For example, see [7, p. 902, fn. 15] and [13, p. 1098, Table III].

-9-

Bibliography

1. F. Arditti and H. Levy. "Distribution Moments and Equilibrium:
A Comment . " Journal of Financial and Quantitative Analysis 7
(January, 1972), pp. 1429-1433.

2. F. Black. "Capital Market Equilibrium with Restricted Borrowing."
Journal of Business 45 (July, 1972), pp. 444-454.

3. F. Black; M. Jensen; and M. Scholes. "The Capital Asset Pricing
Model: Some Empirical Tests." In Studies in the Theory of Capital
Markets , Michael C. Jensen, ed., NY: Praeger Publishers (1972),
pp. 79-121.

4. M. Brennan. "Capital Market Equilibrium with Divergent Borrowing
and Lending Rates." Journal of Financial and Quantitative Analysis
6 (December, 1971), pp. 1197-1206.

5. E. Fama. "Risk, Return, and Equilibrium: Some Clarifying Comments.
Journal of Finance 23 (March, 1968), pp. 29-40.

6. and J. MacBeth. "Risk, Return and Equilibrium: Empirical

Tests." Journal of Political Economy 81 (May/June, 1973), pp. 607-
636.

7. I. Friend and R. Westerfield. "Co-Skewness and Capital Asset
Pricing." Journal of Finance 35 (September, 1980), pp. 897-914.

8. J. Ingersoll. "Multidimensional Security Pricing." Journal of
Financial and Quantitative Analysis 10 (December, 1975), pp. 785-
798.

9. W. Jean. "The Extension of Portfolio Analysis to Three or More
Parameters." Journal of Financial and Quantitative Analysis 6
(January, 1971), pp. 505-515.

10. . "Distribution Moments and Equilibrium: Reply." Journal

of Financial and Quantitative Analysis 7 (January, 1972), pp. 1435-
1437.

11. . "More on Multidimensional Portfolio Analysis." Journal

of Financial and Quantitative Analysis 8 (June, 1973), pp. 475-490.

12. A. Kane. "Skewness Preference and Portfolio Choice." Journal of
Financial and Quantitative Analysis 17 (March, 1982), pp. 15-26.

13. A. Kraus and R. Litzenberger. "Skewness Preference and the Valua-
tion of Risk Assets." Journal of Finance 31 (September, 1976),

pp. 1085-1100.

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14. J. Lintner. "The Valuation of Risk Assets and the Selection of
Risky Investments in Stock Portfolios and Capital Budgets." Review
of Economics and Statistics 47 (February, 1965), pp. 13-67.

15. H. Markowitz. Portfolio Selection: Efficient Diversification of
Investments. NY: John Wiley and Sons, Inc. (1959).

16. M. Miller and M. Scholes. "Rates of Return in Relation to Risk:
A Re-examination of Some Recent Findings." In Studies in the
Theory of Capital Markets, Michael C. Jensen, ed., NY: Praeger
Publishers (1972), pp. 47-77.

17. J. Mossin. "Equilibrium in a Capital Asset Market." Econometrica

34 (October, 1966), pp. 768-783.

18. R. Roll. "A Critique of the Asset Pricing Theory's Tests."
Journal of Financial Economics 4 (May, 1977), pp. 129-176.

19. M. Rubinstein. "The Fundamental Theorem of Parameter-Preference
Security Valuation." Journal of Financial and Quantitative Analy-
sis 8 (January, 1973), pp. 61-69.

20. R. Scott and P. Horvath, "On the Direction of Preference for
Moments of Higher Order than the Variance." Journal of Finance

35 (September, 1980), pp. 915-920.

21. C. Schweser. "Multidimensional Security Pricing: A Correction."
Journal of Financial and Quantitative Analysis 13 (March, 1978),
pp. 177-183.

22. W. Sharpe. "Capital Asset Prices: A Theory of Market Equilibrium
under Conditions of Risk." Journal of Finance 19 (September, 1964),
pp. 425-442.

D/164

Notes for the Reviewer

Derivation of the N Moment
Capital Asset Pricing Model

Extension of the KL framework to an N moment pricing model implies
that the investor seeks to:

maximize:

E[U(W)] = U[W, m2jW, m^, ..., m^] (1)

subject to: Iq. + qf = WQ (2)

i

where:

E[U(W)] = expected value of the utility of terminal wealth W

W = E(W) = Eq^. + qfRf (3)

i

m2 = [E(W-W)2]1/2 = [ZZq.q m ]1/2 (4)

ij

m3 = [E(W-W)3i1/3 - [BS, q %« J1/3

ijk J J

-N.W" tE(W-W)N]1/N= [^..-Zqq ...V ]

1 ij N J J

1/N

q.,qf = amount (in dollars) of initial wealth (W~) invested

in asset i and the riskless asset f
R. ,R. = expected holding period return on i and the holding

period return on f

mij = EI(Ri ■ Ri)(Rj ■ V1 (5)

mijk = E[(Ri- h"*i-TiHR*-\.)]

At the end of the period, the investor receives W = Sq.R. + qfRf (6)

i

For the investor's portfolio, define the following terms:

R_ = E(R ) = Z(qi/W0)¥i + (qf/WQ)Rf (7)

i

y2 - m. /m* - E[(R. -*R.)(R -"r)]/E(R - T )2 (8)

lp ip2,p l ip p P p

2

= S(q./W-)m. ./EECq.q./VJ )m. . = the systematic risk of
j J 0 ij Minj 0 ij

asset i with the investor's portfolio p

Y3 = m. /ml = E[(R. - R.)(R - R )2J/E(R - R )3
ip ipp 3,p l i P p P P

2 3

■ 2Z(q q^/W0 )m /EZE(qiq qk/WQ )rn = the systeraati«

skewness of asset i with the investor's portfolio p

YN - m. /taJJ = E[(R. - R.)(R -R)N"1]/E(R - R )N

'ip ip**»p N,p i i p p P P

E — Z(q i---qN/W0N 1)niii...N/22---2(qiq1---qN/W0N)m N

j N J ij N

the systematic portion of the ntn moment for asset i
with portfolio p

The Lagrangian and first order conditions are:

L = U(W,m n,3 .... % ) - X[Iq + q - W J (9)

1

dL/dqi = (dU/dW,)(d"w/dqi) + (dU/dm2 w)(dm2 w/dq±) + (dU/dm3 ^(dn^ w/dqi)

+ ... + (du7dmKT TT)(dnLT „/dq.) - X = 0 for all i
N,W N,W ^1

(10)

dL/dqf = (dU/dW)(dW/dq ) - X =0 (11)

dL/dX = Iq + q - WQ =0 (12)

In solving for the investor's portfolio equilibrium conditions, note
that:

m2,W = ^qiYipm2,P (13)

3
m- rT = Eq . Yj m-
3,W ,4i'ip 3,p

"N.W = ?qiYipmN,p

Conditions (3) and (13) imply:

dW/dq - R. (14)

dW/dq = R (15)

dm2)W/dq. = YJpm2>p (16)

dm3)W/dq. = Y^pm3>p

dmN,W/dqi = Ylp"N,p

dm rT/dq£ = 0 for n = 2, ..., N
n,W ^f '

Conditions (11) and (15) imply:

X = (dU/dW)Rf (17)

Substituting (14), (16) and (17) into (10):

(dU/dWKR. - Rf) - - (dU/dm2(W)Y2ipm2(p - (dU/dm3(W)Y3pm3ip

- ... - (dU/dmN)W)Y^pmN>p for all 1 (18)

Moving from the investor's equilibrium condition (18) to a market equi-
librium requires that (18) holds for all individuals and that markets
clear. For markets to clear, all assets have to be held which requires
the value weighted average of all individual's portfolios equal the
market portfolio m. Summing (18) across all individuals gives:

(dU/dW)(Ri - Rf) = - (dU/dm2)W)Y2n2>M - (dU/d^^T^^

- ... - (dU/dn^y-^m^ for all i (19)

Since (19) holds for any security or portfolio, it also holds for the
market portfolio:

(dU/dW)^ - Rf> - - (dU/dm2)W)m2>M - (dU/d^^ M

- ... - (dU/d^y*^ (20)

Dividing (19) by (20) gives the capital asset pricing model in terms of
the N moments and the market risk premium (IL - R )

2 N 3 N

R. - R = (R,. - R.)[(K9Y,/ I K ) + (K.Y,/ 2 K )
l r M r 2 l „ n ii„n

n= 2 n= 2

N N
+ ... + (KMY / S K )]

N x n=2 n

_ N N

R. - R = (P - R ) Z [(KL)/ £ K ] (21)

i r n r n n I n

n=2 n=2

where:

K = [(dW/dm Tj)/(dW/dm. ) ] (m /m_ M)
n n,W 2,W n,M 2,M

In words, equation (21) says that in equilibrium the excess return on
security i,(R. - Rf ) , is a function of the excess return on the market
(R^ - Rf ) , the market-related systematic risks of variance and the
higher moments (y.)» and the preference tradeoffs in the market
between risk and all higher moments. This is equation (6) in the text.

Special Cases: Mean-Variance and
Three Moment Pricing Models

An investor who makes investment decisions solely upon the mean and

variance of wealth seeks to maximize E[U(W)] = U[W,ra9 „] . Similarly,

an investor who considers only the first three moments will maximize

E[U(W)] = U[W,m„ TT,m0 rT] . These two versions are special cases of (21)

I , w j , w

where N = 2 and N = 3. When N = 2, we have the two moment CAPM
model:

Ri " Rf = (RM " VYi (22)

and when N = 3, we obtain the three moment version (equation (1) in
text) :

¥. - Rf = [(H^ - Rf)/(1+K3)]Y? + [(\ " Rf)K3/(l+K3)JY^ (23)