18

premium without assuming the state variables are independent) !!

dc
s Oo]
a7) a - of = (R)) aaa,

2
yo

where Ry represents investor j's coefficient of relative risk aversion,
dc h
a0.c is the partial elasticity of consumption with respect to the sth state
Jj
variable, and z is the variance of factor s.

The left hand side of equation (17) is the same as the right hand side of
equation (9). Both equations identify the risk factor premium. The equili-
brium risk factor premiums are seen to depend positively on the individual's
level of relative risk aversion, and on the factor's own variance. The sign
of the risk premium can be positive or negative depending on the Sign of the
partial elasticity of wealth with respect to the relevant state variable.

Since the sign of the risk premium, A, can be reversed by reversing the sign

of the state variable, I will assume that A is always positive.

V. The IAPM's Dynamic Nature and Its Empirical Applicability

This section is divided into two parts. The first subsection describes
the intertemporal nature of the international arbitrage pricing model derived
in Section III. The second subsection briefly outlines how the IAPM can be

empirically implemented.

lthis section borrows heavily from Roll and Ross (1980) and Cox,
Ingersoll, and Ross (1985). They derive different expressions for A’. This
is because they make different assumptions regarding the independence of the
state variables. In the text, I follow Roll and Ross while, in Appendix C I
follow Cox, Ingersoll and Ross (1985). Both of these papers derive the factor
risk premiums in a domestic setting but, the familiar reader will recognize
that the extension to the international setting is trivial.


IFDP
NUMBER

294

293

292

291

290

289

288

287

286

285

284

283

37

International Finance Discussion Papers

TITLES
1986

An International Arbtrage Pricing Model
with PPP Deviations

The Structure and Properties of the FRB
Multicountry Model

Short-term and Long-term Expectations of
the Yen/Dollar Exchange Rate: Evidence
from Survey Data

Anticipated Fiscal Contraction: The
Economie Consequences of the Announcement.
of Gramm-Rudman-Hollings

Deposit Risk Pooling, Irreversible
Investment, and Financial Intermediation

The Yen-Dollar Relationship: A Recent
Historical Perspective

Should Fixed Coefficients be Reestimated
Every Period for Extrapolation?

An Empirical Analysis of Policy
Coordination in the U.S., Japan and
Europe

Comovements in Aggregate and Relative
Prices: Some Evidence on Neutrality

Labor Market Rigidities and Unemployment:
The Case of Severance Costs

A Framework for Analyzing the Process
of Financial Innovation

AUTHOR(s)

Ross Levine

Hali J. Edison
Jaime R. Marquez
Ralph W. Tryon
Jeffrey A. Frankel
Kenneth A. Froot

Robert A. Johnson

Robert A. Johnson
Manuel H. Johnson
Bonnie E. Loopesko

P.A.V.B. Swamy
Garry J. Schinasi

Hali J. Edison
Ralph Tryon
B. Dianne Pauls

Michael K. Gavin

Allen B. Frankel
Catherine L. Mann

Please address requests for copies to International Finance Discussion
Papers, Division of International Finance, Stop 24, Board of Governors of

the Federal Reserve System, Washington, D.C.

20551.


12

expected return vector, o> must be a linear combination of the constant

vector and the BJ vectors. Thus, there exist K +1 weights such that
J Jg3 Jpi
(7) a, = ze + DBL + eee + Lar

with the understanding that if a riskless real asset exists for residents of
country j it would have real return a, with an expected value of af = ae

For asset i in particular:

a) dpisd jpisj
(8) a of + adpyed +... + abae

Intuitively, equations (6) and (8) are easily explained. Equation (6)
expresses the notion that the real return on asset i evaluated in country j is
equal to its expected value plus the effect of random shocks, The magnitude
of these unanticipated shocks is measured by 6 = (Sy 5000s 8) and the
sensitivity of the real return of asset i in j is captured by Bioj =
(ated, ...,abed), Due to stochastic PPP deviations these sensitivities depend
on the asset's home country and the country in which the real return of that
asset is being evaluated. For example, unanticipated changes in the U.S.
money supply may affect the real return on U.S. securities evaluated in
Germany differently from the real return evaluated in Italy. Equation (8)
states that the expected real return of an asset will depend upon the
sensitivity of that asset's real return to shocks. That is, a risky asset —-
one whose real return is expected to be very sensitive to unexpected changes
in fundamental state variables —- would also be expected to ‘have a
compensating higher real rate of return than a less risky asset.

The effect on the expected real return of an asset due to that asset's
exposure to unanticipated shocks is captured by the A's. These A's are
called factor risk premiums. They indicate the market price of any asset's

sensitivity to various shocks. An asset's sensitivity to these common shocks


24

Similarly, the difference between the expected real return of asset i
evaluated in country i and the expected real return of asset j evaluated in

country i can be expressed as either

i
(22") ot — a) = Cy -u) ) - Cp thy up ) + Var(P,)
17% a, “9, ", "ay "P, 1
- Var(P,) + Cov(P, Dyy)>
or
3") t_ga. ptgttpity
ot 1 s° Ss s .

These two sets of equations (8qhations (22)-(23) and equations (22')-(23"'))
are needed to derive the two versions of the risk premium.
Substituting equation (22) [(22')] into (21), rearranging, and using

equation (19) we obtain

(24) RP = Cov(Py,P4) + Cov(Py,Dy4) - Cov(Py ,Dy4) ~ Var(Py) - De? (aj-a})

(24") RP = Cov(Py 5P4) + Cov(P, »D44) - Cov(P; 5Dy4) - Var(P,) + (ated),

Now we can express the risk premium using the IAPM by substituting (23)

[(23')] into (24) [(24')] to get

(25) RP = Cov(Py,P3) + Cov(Py,Dy4) - Cov(Py ,Dy5) - Var(Py) - Up
ij
K
o_ 0° i,i,t _ \J_)-3
+ (apa) + oy {a,B, 3S |
(25") RP = Cov(Ps Py) + Cov(Py »Dy4) ad Cov( Ps ,D44) - Var(Py)
K
+ £ xf fptetigi et,
8 8 s
s=1

Equations (24), (24'), (25), (25') are all expressions for the forward
foreign exchange market's risk premium between countries i and j. ‘%quation
(24) states that the risk premium is a function of the expected real return

differential evaluated in the home country, the expected change in the real


This paper further extends Solnik's (1974) model by allowing stochastic
inflation rates, stochastic PPP deviations, and correlation between PPP
deviations and real asset returns. In order to accomplish this objective, I
follow Ross and Walsh (1983) in extending Ross’ (1976, 1977) arbitrage pricing
theory (APT) to an international environment with PPP deviations.2 This new
technique is necessary because, given this paper's extensions, international
capital asset pricing models (ICAPM) cannot price assets without severe
restrictions on utility functions and correlation matrices; and because, even
granting those restrictions, the ICAPMs are empirically intractable?

The arbitrage pricing approach alone, however, is insufficient because it
does not yield an optimal portfolio equation, which is necessary for many
extensions of the model. The arbitrage asset pricing approach must be applied
in conjunction with the representative investor utility maximization approach
to obtain both equilibrium asset prices and optimal portfolio shares.

The model's structural equations are discussed in Section II, Section
III uses these equations to derive an international arbitrage pricing model
(IAPM);, Section IV rigorously analyzes the price of systematic risk derived
in Section III. In the process, we obtain the optimal portfolio for a
representative individual. This is important for extensions of the model.
Section V highlights the dynamic nature of the IAPM and describes how to

implement empirically the model. The forward exchange market's risk premium

2soinitk (1983) extends Ross' APT to an international setting where PPP
holds. This model is estimated by Cho, Eun, and Senbet (1986).

3these models are empirically suspect because (1) it would be difficult
to construct portfolios which mimick the world state variable; (2) Roll's
(1977) critique of the CAPM applies to the ICAPM; and (3) even if a world
market portfolio could be constructed and state variable portfolios assembled,
Stulz (1981b) shows that in a world with differential tax structures the world
market portfolio is inefficient for all individuals.


35

Roll, R., “A Critique of the Asset Pricing Theory's Test; Part 1: On Past and

Potential Testability of the Theory,” Journal of Financial Economics 4

(March 1977), 129-76.
» "Violations of Purchasing Power Parity and Their Implications for

Efficient International Commodity Markets,” in M. Sarnat and G, Szego,

eds., International Finance and Trade, vol. 1, Cambridge, MA: Ballinger,
1976,
____ and S. Ross, “An Empirical Investigation of the Arbitrage Pricing
Theory,” Journal of Finance 35 (December 1980), 1073-103.
Ross, S.A., “The Arbitrage Theory of Capital Asset Pricing,” Journal of
Ecoromic Theory 13 (December 1976), 341-60.

» “Return, Risk, and Arbitrage,” in Risk and Return in Finance, eds.

I. Friend and J. Bicksler. Cambridge, MA: Ballinger, 1976.
» and M.W. Walsh, “A Simple Approach to the Pricing of Risky Assets

with Uncertain Exchange Rates," Research in International Business and

Finence, 3 (1983), 39-54.
Sercu, P., “A Generalization of the International Asset Pricing Model,” Reveu

de_1'Association Francaise de Finance 1 (June 1980), 91-135.

Shanker, J., “The Arbitrage Pricing Theory: Is It Testable,” Journal of
Finance 37 (December 1982), 1129-140.
Solnik, B., “An Equilibrium Model of the International Capital Market,”

Journal of Economic Theory 8 (August 1974), 500-24.

» “International Arbitrage Pricing Theory,” Journal of Finance 38
(May 1983), 449-57.
Stockman, A.C., “Risk, Information, and Forward Exchange Rates,” in The

Economics of Exchange Rates, eds., J.A. Frenkel and H.G. Johnson. Reading,

MA: Addison-Wesley, 1978).


19

V.A. The Dynamic Nature of the TAPM

Up until now, I have been lax in distinguishing ex ante from ex post
perceptions. In order to apply the international arbitrage pricing equation,
however, this distinction is important. Therefore a little backtracking and
clarification is necessary before explaining how the LAPM can be used.

Given the model specified in Section IL and assuming rational
expeci-ations, equation (6) describes both the ex ante perceptions of the return
generating process and the ex post realization, The real return on an asset is
its expected value plus the effect of shocks on the asset's real return.

Equation (8), on the other hand, is an equation describing an asset's
expected real rate of return. The B's in equation (8) are expected values;
they zre the expected sensitivity of the asset's real return to unanticipated
movements in the state variables. The expectation of Bidcty is formed
conditional on the expected state of the world at time t. (Recall that

i,j _ s _ +8 _ +8
BS bg, (oCt) TD bp, (Ot) 7) D, C2)?»

L
A more rigorous characterization of “expected state of the world” can be

obtained by integrating equation (1)

qs) a(t) = (0) + J* un(e(v),v)av + Jt 5 (e(v),v)az(v),
0 0 8

where equation (18) represents the actual state of the world at time t. It
obviously includes the unanticipated movements happening the instant before
time t,

The expectation of O(t) at time t-h is

W

= [o(t)/1, 1]

9(0) + f* ug(0¢v),woav + f°" 5 Carvy,v) azcw)
t-h 0 8

0
where h is positive, and Tih represents information available at time

t-h. The expectation of Bid(t) is formed conditional on an information set


27

This paper applies the model to the forward exchange market's risk
premium. The lack of an empirically tractable model of the time-varying risk
premiun has hindered empirical inquiry into the sources of intertemporal
variation between forward and corresponding future spot exchange rates.
Section VI derives an equation of the risk premium that in future work will

allow Investigators to discover the risk premium's economic determinants.


21

factors are prespecified. !2

In order to determine which economic state variables —- systematic risk
factors -— influence real returns, and in order to construct estimated real
returns using the IAPM, it is necessary to:

(1) choose a set of assets and systematic risk factors, 6's;

(2) estimate each asset's real return sensitivity to the chosen systematic
risk factors, i.e., collect a time series of B's for each asset from
equation (6);

(3) estimate equation (8) in each time period, i.e., using the estimated
systematic risk coefficients, the B's, from step (2) as predetermined
variables and realized real rates as proxies for expected real rates as
the dependent variables, equation (8) may be estimated and the risk
factor premia, the i's, retrieved.

The results are a time series of systematic risk estimates, B's, a time
series of risk factor premia, A's, and hence a time series of estimated
expected real rates of return, a's.

The most difficult econometric step in the above procedure is step 2.
Estimating the exposure of each asset's return to the chosen systematic risk
factors involves the collection of a time series of B's from equation (6).
Econometrically this entails recovering a series of time-varying parameters,
and using this series as the explanatory variables in equation (8). Equation

(8) is then estimated in order to determine which systematic risk factors are

127+ should be emphasized that use of the IAPM is a separate issue from
the debate surrounding the APT's testability. The seminal article in this
area is Roll and Ross (1980), where they conclude that the APT is testable.
After using factor analysis, they find that about five risk factors seem to
explain stock price movements in the U.S. For criticisms of the APT and its
testability see Shanken (1982) and Dhrymes, Friend, and Gultekin (1984). For
a response to these papers, see Dybvig and Ross (1985).


36

Stulz, R.M., "A Model of International Asset Pricing,” Journal of Financial

Economics 9 (December 1981), 383-406.

» “On the Effects of Barriers to International Investment,” Journal
of Finance, 36 (September 1981), 923-34.
» “The Forward Exchange Rate and Macroeconomics,” Journal of

International Economics, 12 (May 1982), 285-99.

» “Currency Preferences, Purchasing Power Risks and the
Determination of Exchange Rates in an Optimizing Model,” Journal of Money,

Credit and Banking, 16 (1984), 302-16.


16

Let J(W,6) represent the value function, iee., J is the solution to
equation (11) for given levels of real wealth and state variables. The
fundamental principle of stochastic dynamic programming lets us transform the

problem given in equation (11) to

0 = (ns, [U(C) - py + JylWCq'a) = C] + 1/2 JN Eq! oq]
Cyq

+ SygWla'] + G[1- q'.1]
where G is the Lagrange multiplier. 9 is the covariance matrix of real
asset returns. is the covariance matrix of real asset returns with the K
state variables. & and are country specific and implicitly have j
subscripts. They are defined more rigorously in Appendix B. Jy and Saw
are the first and second partial derivatives of the value function with res-
pect to real wealth. Jy8 is a K x1 vector of second partial derivatives
of the value function with respect to real wealth and the K state variables.

The first order conditions for the problem described above are:
(13) U.- J, = 0
Zogk - =
(14) JWa + J Wagk + J Wo - GL = 0

Equation (13) states that at an optimum consumption will be withdrawn
until its marginal utility equals the marginal utility of wealth.
Equation (14) can be solved for the real asset return vector on optimally

invested wealth.
(15) a= Roq* + OA + G/WI . 1

where: R= Wig! Iyy5 the coefficient of relative risk aversion
A=-J_./J 3; a k x1 vector
wow
iL=N x1 = vector of ones

and, q* = vector of optimally invested shares of real wealth.


It is not necessary, however, to model explicitly the microeconomic
components of risk for the framework to be consistent with a general equilib-
rium model. In order to be consistent with general equilibrium, prices must
be endogenously determined through the equilibrium of supply and demand.
Since all random shocks are captured as elements of the state vector, 06, and
assuming asset supply and demand schedules are functions of the state
variables which follow Ito processes, then the resulting equilibrium prices
will also follow Ito processes.’ Thus, the economic model presented is

consistent with endogenously determined prices.

III. An International Arbitrage Pricing Model with PPP Deviations

This section follows Ross and Walsh (1983) and applies the APT developed
by Ross (1976, 1977) to the model presented in Section II in order to price
internationally traded assets in a world with PPP deviations.

This section is divided into three subsections. The first subsection
demonstrates that the framework presented in Section II is consistent with
real asset returns following a linear return generating process, which is a
necessary component in any APT approach. The second subsection derives an
intertemporal, international asset pricing equation using the arbitrage
pricing approach. This equation specifies that the real return on asset i
evaluated in country j, for example, is a linear combination of asset i's
sensitivity to unanticipated movements in the state variables. The third sub-
section gives an intuitive explanation behind the “pricing” of an asset's

sensitivity to specific shocks.

7This section borrows heavily from Breeden (1979) and Richard (1979).


33

Dhrymes, P., I. Friend, and B. Gultekin, “A Critical Reexamination of the
Empirical Evidence on the Arbitrage Pricing Theory,” Journal of Finance 39
(June 1984), 323-46.

Domowitz, I., and C.S. Hakkio, “Conditional Variance Exchange Market," Journal

of International Economics, 19 (1985), 47-66.

Dybvig, P., and S. Ross, "Yes, the APT is Testable,” Journal of Finance 15

(September 1985), 1173-188.

Fama, E.F., “Forward and Spot Exchange Rates,” Journal of Monetary Economics

(November 1984), 319-38,

» and A. Farber, “Money, Bonds, and Foreign Exchange,” American
Economic Review 67 (September 1979), 639-49.

Fleming, W.H., and R.W. Rishel, Deterministic and Stochastic Optimal Control,

New York: Springer-Verlag, 1975.

Graver, F., R. Litzenberger, and R. Stehle, "Sharing Rules and Equilibrium in
an International Capital Market under Uncertainty,” Journal of Financial
Economics 3 (June 1976), 233-56.

Hansen, L.P., and R.J. Hodrick, “Risk Averse Speculation in the Forward
Foreign Exchange Market: An Econometric Analysis of Linear Models,” in

Exchange Rates and International Macroeconomics, ed. J.A. Frenkel.

Chicago: University of Chicago Press, 1983.
___» and R.J. Hodrick, “Forward Exchange Rates as Optimal Predictors of
Future Spot Ratees: An Econometric Analysis,” Journal of Political
Economy, 88 (October 1980), 829-53.

Hodrick, R., “International Asset Pricing with Time-Varying Risk Premia,”

Journal of International Economics 11 (November 1981), 573-87.

» and S. Srivastava, “An Investigation of Risk and Return in Forward

Foreign Exchange,” Journal of International Money and Finance 3 (April



30

After maximizing utility subject to the given constraints, the following
matrices are constructed:
Q, is an NxN matrix of real asset return covariances according to
residents of j.
$, is an Nx K matrix of covariances between real asset returns and the K
state variables,
The (2,m)th element of &, is the covariance between the real return

J

on asset & and the real return on asset m, and is equal to

K Lyj K m,j
= oJ >
25 (2m) [= BS 0,42.) cn B, 9,42].
s=1 s=1

The (a,x) element of 6, is the covariance between the real return on

j

asset a and the state variable x, and is represented by

K
= a yj *
9,(a,x) tr Be’ 9,d2,] * o dz.

Appendix C

This Appendix derives the factor risk premium for a representative
individual from country j, equation (16), assuming that the state variables
follow (1) independent diffusion processes, and (2) correlated diffusion
paths.

Consider first the simple case when the diffusion processes are

independent. Equation (15a) is rewritten:

- = *
(C1) a a el Ra, a4 + 9,4;-

Now, form a (hypothetical) portfolio whose expected real rate of return,
as depends only on state variable s, and has unit systematic risk with state
variable s. Multiply (C1) by a> which represents this hypothetical

portfolio's vector of asset shares, and replace A, to obtain:


time.4 Thus, the unanticipated change in state variable s at time t is
equal to 9,(0(t) ,t)dz.(t).

The intertemporal development of the state variables determines the
production and credit opportunities available to the economy. They include
such variables as technological growth, the money supply, institutional
arrangements, tax and tariff structures, the price of intermediate inputs,
etc. The state variables provide the dynamic nature of the model. More
specifically, the probability distribution of important variables such as
aggregate price levels, exchange rates, and deviations from PPP, depends on
the current level of the state variables, ©, which are themselves changing
randomly over time.

Remembering the notational convention described above the remaining
structural equations may be specified.

Each country's inflation rate is assumed to follow an Ito-type continuous
time Markov process.

aPy K 8
(2) Po 7 Hp (Ot) t)dt + £ by (0(t),t) 9,(0(t),t)d2(t) 1 = lesen
i i s=1 i
where P, is country i's domestic price level, and where throughout
vi @(t),t) quantifies the sensitivity of variable 2's rate of change to
unanticipated movements in state variable s at time t, when the state of the
world is 6(t). Note that the expected change in country i's domestic price

level at time t, oy (@(t),t), is made conditional on all information
i

4a veal valued function z on [t,t'] is a Wiener process if: (i) 2z
is a continuous process with independent increments, and (ii) z(t") - z(t)
has a normal distribution with mean zero and variance t' - t. For a more
rigorous definition of Wiener processes and their applications in economics
and finance see Malliaris and Brock (1982).


is derived in Section VI. A brief summary is presented in Section VII.

Il. The Model

The world economy consists of n countries each with its own currency.
Every investor is able to issue and purchase assets in any country through
freely floating international exchange markets. It is assumed that each

n
country, for example country i, has N,; assets, where n << = N, =N.
isl

Although all goods are available in each country, residents of country i
can consume only goods purchased in country i. The existence of differential
transactions costs, storage costs, tariffs, tastes, etc., make the relative
price of representative consumption baskets differ internationally. Given
these deviations from Purchasing Power Parity, one cannot trade a basket of
goods in country i for an identical basket in country j at a 1:1 ratio.
Consequently, the real return on any asset will depend upon the investor's
nationality.

It is assumed that there exists a K x1 vector of state variables, 06,
which describes the state of the world and provides the fundamental dynamic
relations of the model. I assume that the state vector, 0, follows a

continuous time vector Markov process of the Ito type,
(1) dé = Wal Ct) ,tdt + Ep (O(t),t)dz(t)

where throughout this paper, uy @(t),t), represents the expected rate of
change in variable 2 at time t, given that the state vector is O(t).

DING @(t),t) is a K xK diagonal matrix of instantaneous standard deviations.
The main diagonal consists of: 9, (O(t) ,t) seoe (OC) ,t)5 where o,( a(t) ,t)
is state variable s's instantaneous standard deviation at time t when the
state of the world is described by 6(t). The 4dZ(t) variables are

correlated Wiener Processes having zero mean and unit variance per unit of


V7

Now, form a minimum variance portfolio with shares q°, such that it has
zero covariance with optimally invested wealth, and zero covariance with the
state variables, The real rate of return on this portfolio is

, 1 ' 1
a = a a= WRq? Qqt + 9° OAq* + c/w q° el

°
a G/WI,
Substitute a into equation (15) and reintroduce the j-subscripts to
obtain:

- =
(15a) a as 1 R945 + OA,

Solving for optimal portfolio shares, > yields

1 9°) — -1
(16) a Cay a5) LR, a oA,

Equation (16) expresses the optimal portfolio shares for a representative
investor from country j. The expected rates of return in equation (16) can be

obtained from an IAPM of Section III, equation (8) .10

IV.B The Factor Risk Premiums

The factor risk premiums may now be obtained from the existing equations.
Let q® be the vector of portfolio weights with unit systematic risk on the
sth factor, iee., q® is the vector of portfolio shares whose expected rate
of return, a, depends only on state variable s. Assuming, with some loss
of generality but with worthwhile gains in clarity, that the state variables
follow independent diffusion processes, the following expression can be

derivec. (Appendix C derives equation (17) and also solves for the factor risk

assuming asset returns follow a linear return generating process,
market equilibrium implies that the no-arbitrage condition holds. Therefore,
the IAPM and the optimal representative investor model are consistent, and the
expected real returns generated from the IAPM can be used in the optimal
portfolio equation.


Note that dDy 3 /Day = 0 by definition, and each country has a different
stochastic PPP relationship with each other country. The expected change in
PPP deviations is not necessarily zero. Thus, while a martingale is not
inconsistent with the above formulation, I do not restrict PPP deviations to
be a random walk,

Assuming identical consumption bundles, Diy can be interpreted as the
real exchange rate —- by which I mean the relative cost of the same bundle of
goods in two different countries, e.g., let i = Germany and j = U.S., so
that S43 = D.M./$, Py = D.M./bundle(Germany), Py = $/bundle(U.S) then
Daj = bundle(Germany)/bundle(U.S.). The real exchange rate may be different
if a different representative bundle of goods is chosen.®

This model is partial equilibrium in the sense that unspecified state
variables provide the fundamental dynamic relations of the model. It is not
general equilibrium in the sense of Arrow-Debreu because technological sources
of uncertainty are not explicitly related to the equilibrium prices. Cox,
Ingersoll, and Roll (1985) (CIR), and Brock (1982) construct general equilib-
rium asset pricing models in a domestic setting. Production possibilities are
explicitly modelled as a set of linear stochastic activities where it is these
direct technological shocks that ultimately induce stochastic contingent claim
prices. They, however, construct their models in a completely real setting
with no aggregate price level. Since the model used in this paper incorpor-
ates stochastic inflation rates in each of n countries, a money market and
aggregate price level would have to be added to the CIR and Brock models: a

non-trivial task left for future work.

the assumption of identical consumption bundles is made for expositional
purposes. It plays no role in this paper.


25

exchange rate, and four terms which capture the explicitly stochastic nature
of the model employed. Equation (24') is the same as equation (24) except
that the real return differential is specified using the IAPM developed above.
In equation (25), the expected change in the real exchange rate is embodied in
the expected real return differential. The expected real return differential
in equation (25) is evaluated solely in country i terms, and thus considers
expected changes in the real exchange rate. The difference between equation
(25) and equation (25') is that the expected real return differential is
specified using the IAPM.

Korrajczyk (1985) estimates a model that closely resembles equation (24).
He, however, does not construct an explicitly stochastic model, and thus, he
does not. derive an expression for the risk premium that contains the first
four terms on the right hand side of equation (24). Korajczyk (1985) assumes
that ex ante PPP deviations follow a martingale process. This assumption has
two important implications: (1) real asset returns are independent of
residence, and (2) the expected change in the real exchange rate is zero.
Given these assumptions, Korajczyk (1985) estimates an equation of the form
RP = a + b( a-al) + u. Using three stage least squares, he finds that the
expected real return differential is useful in predicting the forward forecast
error. This represents an encouraging step toward the conclusion that the
forward bias is due to a risk premium rather than market inefficiency,

Although Korajczyk's (1985) paper is a seminal contribution, it can be
improved upon in at least two ways. First, given the IAPM developed above, it
is possible to test for the risk premium's existence within the context of a
specific asset pricing model. Using the IAPM to analyze the risk premium's
existence is an improvement over past work because now the risk premium's

determinants are fully articulated by a theoretical paradigm explicitly


International Finance Discussion Papers
Number 294

December 1986

AN INTERNATIONAL ARBITRAGE PRICING MODEL
WITH
PPP DEVIATIONS

by

Ross Levine

NOTE: International Finance Discussion Papers are preliminary materials
circulated to stimulate discussion and critical comment. References in
publications to International Finance Discussion Papers (other than an
acknowledgment that the writer has had access to unpublished material)
should be cleared with the author or authors.


10

asset i's home country's price level to changes in state variable s. These
two components are the same regardless of the country in which final. evalua-—

tion is to occur. The third term, b5 indicates the sensitivity of

1j

purchasing power parity deviations between asset i's home country ard country
j, the country in which the real return of asset i is being evaluated. This
term is what causes the real return of assets to be evaluated differently
internationally. Models in which PPP is assumed to hold are special cases of
the above formulation (see Hodrick (1981), Kouri (1979)).

The real return on any asset computed by any country is thus seen to
follow a linear return generating model. This satisfies the quintessential
assumption of the APT. The other assumptions necessary to use the arbitrage
pricing approach are perfectly competitive capital markets, frictionless asset
markets, and that the number of stochastic state variables affecting asset
returns is relatively small, i.e., K << n.8 In the parlance of the APT
literature, there are only a few systematic components of risk existing in

nature, 8) se 00s Ss these are called systematic risk factors.

III.B The IAPM

Given these assumptions, it is possible to derive an international
arbitrage pricing model with PPP deviations. Following Ross (1976) and Roll
and Ross (1980), consider an individual in country j who is considering a
change in his portfolio. Let % (i = 1,...,N) equal the amount purchased
or sold of asset i as a fraction of total real wealth, where “real” is defined

in country j terms. Since purchases of assets must be financed by sales of

8there is an entire literature on the APT's “necessary” conditions. See
Ross (1976) and especially Ross (1977) for the original presentation, and see
Dybvig and Ross (1985) for a recent list of relevant citations.


ILI.A ‘The Linear Return Generating Process

To facilitate exposition, let:

6 = 9, ((t) ,t)dz (t) BS = 1,000,K

o§ = byCa(e) ,t) for all s and j

and

4 uy (Ct) ,t)

It shoul.d be remembered, however, that (1) bf is a time varying, state
dependent parameter, and (2) 6, is the unanticipated movement in state
variable s, and is distributed normally with mean zero, and variance
9,( Qt),t)dt.

Equation (3) then becomes
3") we, dt + t % 8,

1 L s=1 oa
For. simplicity, assume that asset i's home country is country i. Then,

the inst:antaneous real return on asset i in country j is (after using Ito's

Lemma and collecting terms)

k
ae i,j
(6) 4, a, + £ By 8

s=1
where Es is the real return of asset i in country j, 5 is the expected

real ret.urn of asset i in country j, Bid(= b& - pS - b= ) is the
8 Q P. D.
4 i 14
sensitivity of the real return of asset in country j to an unanticipated
movement. in the state variable s. (See Appendix A for the derivation of

equation (6)).

Note that pied consists of three components. The first component,

65 » signifies the sensitivity of the nominal return of asset i to changes in
i
state variable s. The second component, b® represents the sensitivity of

Pi


32

BIBLIOGRAPHY

Adler, M., and B. Dumas, "International Portfolio Choice and Corporation
Finance: A Synthesis,” Journal of Finance 38 (June 1983), 925-84,

Bilson, J.F.0., "The Speculative Efficiency Hypothesis," Journal of Business
54 (July 1981), 435-51.

Breeden, D.T., “An Intertemporal Asset. Pricing Model with Stochastic

Consumption and Investment Opportunities,” Journal of Financial Economics 7

(September 1979), 265-96.
Brock, W.eA., “Asset Prices in a Production Economy,” in The Economies of

Information and Uncertainty, ed. J.J. McCall. Chicago: University of

Chicago Press, 1982,
Chen, N., R. Roll, and S.A. Ross, "Economic Forces and the Stock Market,”
University of Chicago Working Paper (1985).
Cho, D., “On Testing the Arbitrage Pricing Theory,” Journal of Finanze 39
(December 1984), 1485-1502.
» C. Eun, and L.W. Senbet, “International Arbitrage Pricing Theory:
An Empirical Investigation,” Journal of Finance 41 (June 1986), 313-30.

Cornell, B., “Spot Rates, Forward Rates and Exchange Market Efficiency,"

Journal of Financial Economics 5 (August 1977), 55-65.

Cox, J.C., J. Ingersoll, and S. Ross, “An Intertemporal General Equiiibrium
Model of Asset Prices," Econometrica 53 (March 1985), 363-84.

Cumby, R., and M. Obstfeld, “A Note on Exchange-Rate Expectations and Nominal
Interest Differentials," Journal of Finance 36 (June 1981), 697-704.

Darby, M.R., “Movements in Purchasing Power Parity: The Short and Long Runs,”

in M.R. Darby et al., The International Transmission of Inflation, NBER

Monograph Series, Chicago: University of Chicago Press, 1983,


31

(C2) a& - oo = Wo @,

where & =q!> 6

4 5 because this portfolio only varies with the s state

variable.

Using the implicit function theorem on equation (13), we know that

oer) = THe AC/48,
J, Me
- (tee*®) dc
uy qo, (cy

From Bellman's Principle of Optimality, we know that ES = () =
c w

investors' coefficient of relative risk aversion. Thus, equation (C2) can be
rewritten as

s ° ac
(3) of - of = (R) Sco o, Q.E.D.

The second part of this Appendix derives the factor risk premium in the
case of covaring state variables. This section follows directly from Cox,
Ingersoil, and Ross (1985), and is merely included here for completeness,

The hypothetical portfolio, a5 is now more accurately interpreted as a
portfol:.o whose excess expected real return has only the risk of factor s but,
the st® factor covaries with the other state variables. The factor risk
premium then becomes

ww.

K
) (cov(W*, 6.) + £ ¢

*) (cov 9, ,6.))1,
w x=l

where W* is optimally invested wealth. Note that if the conditions of the

first part of this Appendix are imposed, the first term in brackets disappers,
-J,
wd

and of the K following terms only s cov( 6,9.) would remain. For a
Ww

more detailed explanation see Cox, Ingersoll, and Ross (1985) and Roll and

Ross (1980).


available up until time t.

The instantaneous nominal return on the N nominally risky assets is?
dQ, K 8
(3) = (C(t), tidt + E b. (O(t),t) o (O(t),t)dz_(t) i =1,...,N
qe sl & s s

where Q; is the nominal price of asset i in home currency terms.
The nominal exchange rate is assumed to reflect not only relative price

differences but PPP deviations as well:
(4) $13 = py?

where S44 is the amount of currency i exchangeable for a unit of currency j
on the spot exchange market, and Diy represents PPP deviations between
countries i and j.

There are three important stylized facts about PPP deviations which
should be incorporated into an international model. First, PPP deviations are
the rule rather than the exception. Second, PPP deviations between, for
example, the U.S. and France are different from PPP deviations between the
U.S. and Japan. Third, although Darby (1983) finds a significant MA term,
much empirical work (Adler and Dumas (1983)) suggests that PPP deviations are
not statistically very different from a martingale.

This paper recognizes the empirical evidence and models PPP deviations
accordingly. PPP deviations are modeled as a sub-system of stochastic
differential equations.

aD
(5) ih

=u (0(t),t) +
Pay Pay

K
z b5 (0(t),t) o (O(t),t)da_(t)
=1 14 8 8

S5Note that a nominally risk free bond is also representable within
equation (3). Simply let the uncertain component of dQ/Q equal zero for the
nominally risk free return, i.e., set the b's equal to zero.


20

that does not include the realization of shocks between t-h and t, i.ee.,
the expectation of Bid(t) is formed conditional on En £8(t)/T,_) not
a(t).

These expectations of systematic risk are then weighted by the factor
risk premiums to yield the expected real rate of return (equation (8)). As
the expectations of systematic risk, Bid, vary intertemporally the expected
real return on assets will also vary.

In conclusion to this clarifying subsection on the intertemporal behavior
of the IAPM, I pose a basic question: Why do expected real returns vary with-
in the context of an intertemporal international arbitrage pricing model? By
looking at equation (8) and using Section IV, the answer is clear. The
expected real rate of return on an asset will vary through time as the asset's
expected exposure to systematic risk varies. For example, as the expected
values of Bry jee. Bie change, the expected real return on asset 1 in
country j will change. Additionally, changes in attitudes toward risk, and
perceived alterations in probability distributions governing unanticipated
movements in the state variables, i.e., changes in factor risk premlums will
also induce changes in the expected real returns on assets. These dynamic
considerations must be accounted for in any meaningful econometric work

utilizing the IAPM.

V.B. Empirical Application of International APT

In order to estimate an asset's expected real return within the context
of the IAPM, one can either specify the systematic risk factors bel-.eved to
induce risk, or use factor analysis. This subsection will briefly outline the

econometric methodology required for using the IAPM when systematic risk


28

Appendix A

This Appendix provides additional steps in moving from equation (3') to
equation (6).
Recall that equation (3'), the nominal return on asset i in country j

(assuming i is its home country), is

dQ, K s
(Al) ay = Ya, dt + a a, 8,
The instantaneous real return on asset i in country j is AQ, /PyDy51/
Q/ PDs ye After using Ito's Lemma this becomes

41Q; /P, Dy 5] _ dQ; dP, 4,5 dQ, dPy

(A2) -
IPPs A Pp Py Py
2 2
_ aD Pay . ae, . apy
aD P,D 2 2
Py 144 DOE
Now use equations (2), (3) and (5) to obtain:
4[Q,/P D341 Yo, Ye, Hp
(43) a < is | ma - = - = ~ cov(Q, ,P,)
QPP ay 1 i 14

+ Cov(P; 5D; 5) + Var(D, 5) + Var(P,)]dt

K
+EE 42,9, (64 - bp - >) I
s=1 ‘i 1 ij

The following multiplication rules have been used:
(dt)? = 0
(dt)(dz,) = 0

(dz,)(dz,) = correlation between state variable s and r.

So that:


Kouri (1979), Fama and Farber (1979), and Grauer, Litzenberger, and Stehle
(1976)), nations are defined as economic units holding the same currency as a
means of payment.

This paper follows Solnik (1974) and defines a country as a place where a
subset of agents uses the same price index to deflate nominal returns.
Nations are delineated by deviations from PPP which cause residents to evalu-
ate differently the real return on the same asset. PPP deviations may exist
because of differences in consumption tastes, or from differences in the price
of individual commodities arising from transactions costs, storage costs,
tariffs, etc. “This heterogeneity in individuals' evaluation of returns plays
havoc with the standard Separation, Aggregation, and Asset Pricing results of

Portfolio Theory." 1

The majority of this paper will be devoted to resolving
the problem of portfolio choice when investors’ real returns differ.
Resolving the problem of portfolio choice and asset pricing when
investors' evaluations of real returns differ internationally is a necessary
first step towards a truly international theory of finance. Solnik (1974)
takes this first step. In a model with PPP deviations, no inflation, and no
correlation between PPP deviations and local real asset returns, Solnik
derives an international capital asset pricing model. This model is extended
by Sercu (1980) to permit correlation between exchange rates and local real
asset returns, and by Kouri and de Macedo (1978) to allow for non-stochastic
inflation. Each of these models is an extension of Merton's (1973) inter-

temporal CAPM.

paler and Dumas (1983, pe 926). Much of this introduction owes its
origin to their survey of the international finance literature.


13

is called the asset's systematic risk, and is signified by Bd .6.,Bd The

me
relevant shocks in the economy are By seen dy and are called systematic risk

factors,

III.C Factor Risk Premia: Intuition

To see why the factor risk premiums are the market price of systematic
risk consider a world with only one factor. The international arbitrage

pricing model derived above tells us that
i _ 0 - J plod
474
The expected excess real return on asset i in country j above the country j

risk-free rate depends upon the exposure of asset i in j to the lone risk

factor times a constant, a -- constant in the sense of not changing across
assets evaluated in country j. The factor risk premiums must be equal across
assets to rule out riskless arbitrage profits. This constant, M, is the

market orice of a unit of systematic risk. That is, a unit increment in asset

i's systematic risk increases the expected real return of asset i in j by
4.
This unifactor example can be extended by forming portfolios with unit

systemaic risk on each factor and no risk on other factors. Each may

then be interpreted as
J 2 8 - ° =
(9) 8 a, a, s 1, 000K
which is the excess real return, or factor risk premium, on portfolios with
only systematic risk factor s. Equation (8) can then be rewritten as

iL jo. 1_.0)pi,j K_ 0) 1,4
(10) a7 Ca, By + sre # Cay PBK .

This section derived an intertemporal, international arbitrage pricing

model with PPP deviations. Equation (8) is the intertemporal arbitrage


15

(12a) dWy = Way aydt - C,dt + Way 4,42

(12b) 44 -l-1i,

where: Wy is the real wealth held by the representative individual of
country j.
44 is the vector of representative investor j's invested wealth
shares, i.e., the first component of qj is 45 and it signifies
the optimal proportion of j's wealth invested in asset l.
cy is j's instantaneous real consumption
¥5 is the matrix containing country j's real asset return sensitivi-
ties to shocks. It is derived in Appendix B.
is the expectation operator evaluated at t = 0.
p is a discount factor.
1 isan N x1 vector of ones.

The first constraint, equation (12a), specifies the intertemporal budget
constraint. The accumulation of real wealth occurs when real capital gains on
assets exceed instantaneous real consumption, Cy. It is assumed that no
other source of income exists. The second constraint, equation (12b), merely
states that the summation of portfolio shares over assets is equal to one.

This utility maximization problem can be solved using stochastic optimal
control theory.? For simplicity, I will drop the j subscripts, indicating
the hom2 country of the representative individual. The subscript will

reappear at important times.

Por a mathematical explanation of stochastic optimal control see Fleming
and Rishel (1975), Deterministic and Stochastic Optimal Control. For a more
heuristic presentation with economic applications see Malliaris and Brock
(1982), Stochastic Methods in Economics and Finance.



AN INTERNATIONAL ARBITRAGE PRICING MODEL
WITH
PPP DEVIATIONS

by
Ross Levine*

I. Introduction

This paper develops an intertemporal, international asset pricing model
for use in applied theoretical and empirical research, An important feature
of the model is that it incorporates both stochastic inflation rates and
stochastic Purchasing Power Parity (PPP) deviations. The model derives the
equilibrium real return on assets, and obtains empirically tractable reduced
form equations which can be used to examine such issues as capital market
segmentation, currency substitution, exchange rate volatility, and the risk
premium in forward foreign exchange markets. The model is consistent with the
domesti.c-general equilibrium asset pricing models of Cox, Ingersoll, and Ross
(1985) and Brock (1982). This paper's international emphasis, however,
focuses attention on different issues.

In order to distinguish domestic from international asset pricing
paradigms, an economic concept of statehood is necessary. The Ricardian model
identifies nations by their technologies, while the Heckscher-Ohlin theory
defines countries as areas within which physical factors of production are

confined. In monetary models and some asset pricing models (Hodrick (1981),

*the author is a student intern in the International Finance Division of
the Board of Governors, and a graduate student at the University of
California, Los Angeles. This paper represents the views of the author and
should not be interpreted as reflecting those of the Board of Governors of the
Federal Reserve System or other members of its staff. I have benefited from
valuable discussions with Maria Carkovic, Michael Darby, Sebastian Edwards,
Mark Grinblatt, Arnold Harberger, Giovanna Mossetti, and Guido Tabellini.


IFDP
NUMBER

282

281

280

279

278

277

276

275

274

273

272

271

38

International Finance Discussion Papers

TITLES
Is the ECU an Optimal Currency Basket?

Are Foreign Exchange Forecasts Rational?
New Evidence from Survey Data

Taxation of Capital Gains on Foreign
Exchange Transactions and the Non-neutrality
of Changes in Anticipated Inflation

The Prospect of a Depreciating Dollar
and Possible Tension Inside the EMS

The Stock Market and Exchange Rate Dynamics

Can Debtor Countries Service Their Debts?
Income and Price Elasticities for Exports
of Developing Countries

Post-simulation Analysis of Monte Carlo
Experiments: Interpreting Pesaran's (1974)
Study of Non-nested Hypothesis Test Statistics

A Method for Solving Systems of First Order
Linear Homogeneous Differential Equations
When the Elements of the Forcing Vector
are Modelled as Step Functions

International Comparisons of Fiscal Policy:
The OECD and the IMF Measures of Fiscal
Impulse

An Analysis of the Welfare Implications of
Alternative Exchange Rate Regimes: An
Intertemporal Model with an Application

1985
(partial listing)

Expected Fiscal Policy and the Recession
of 1982

Elections and Macroeconomic Policy Cycles

AUTHOR (3).
Hali J. Edison

Kathryn M. Dominguez

Garry J. Schinasi

Jacques Mel:.tz

Michael K. Gavin
Jaime Marquez

Caryl McNei:ly

Neil R. Ericsson

Robert A, Johnson

Garry Schinasi

Andrew Feltenstein
David Lebow
Anne Sibert

William H. Branson
Arminio Fraga
Robert A. Johnson

Kenneth Rogoff
Anne Sibert


26

designed to characterize risk. The second improvement which can be made is to
allow for ex ante real exchange rate changes. This extension is presented in
Levine (1986),

Estimation of equation (24') or (25') will consider each of these
improvements, Using equation (24') has the advantage of being able to
distinguish the relative explanatory powers of ex ante real return differences
and ex ante real exchange rate movements. It has the disadvantage that real
exchange rate movements would then have to be explicitly modeled. 'stimation
of equation (25') has the advantage that it can be estimated completely within
the context of the IAPM. The next step is clear: estimate equation (25')

using the empirical techniques outlined in Section ve

VII. Conclusions

This paper has applied techniques developed by Merton (1973) and Ross
(1976) to the problem of pricing assets and deriving optimal portfolio shares
in a model where investors’ real returns differ internationally. I derive an
intertemporal, international arbitrage pricing model incorporating both
stochastic PPP deviations and stochastic inflation rates. Imposing (1) the
no-arbitrage condition; and (2) the condition that the supply and demand for
assets are always equal, this paper obtains equilibrium asset prices and
optimal portfolio shares.

The IAPM relaxes more assumptions than previous international asset
pricing models, e.g., stochastic PPP deviations, stochastic, imperfectly
correlated inflation rates, and time-varying covariances are permitted without
concomitant restrictions on agents utility functions. Furthermore, he IAPM
is empirically tractable. Because of these features, the model can orm the

basis of applied theoretical and empirical research,


ABSTRACT

This paper develops an intertemporal, international asset pricing model
for use in applied theoretical and empirical research. An important feature
of the model is that it incorporates both stochastic inflation rates and
stochastic Purchasing Power Parity deviations (PPP). The model derives the
equilitrium real return on assets, and obtains empirically tractable reduced
form equations which can be used to examine such issues as capital market
segmentation, currency substitution, exchange rate volatility, and the forward
exchange market's risk premium. Mechanically, the model begins as a system of
stochastic differential equations which describe the dynamic paths of a vector
of state variables, prices, and PPP deviations. The state variables' inter-
temporal development determines the production and credit opportunities, and
provides the model's fundamental dynamic nature. The model is shown to be
consistent with the domestic-general equilibrium asset pricing models of Cox,
Ingersoll, and Ross (1985) and Brock (1982). The model is applied to pricing
forward exchange, and an empirically tractable equation of the risk premium is
derived which will allow researchers to uncover the risk premium's economic

determinants,


34

1984), 5-29,
Korajezyk, R.A., “The Pricing of Forward Contracts for Foreign Exchange,”

Journal of Political Economy 93 (April 1985), 346-68.

Kouri, P., “International Investment and Interest Rate Linkages under Flexible

Exchange Rates,” in The Political Economy of Monetary Reform, ed. R.

Aliber. London: Macmillan & Co., 1977.

and J.B. de Macedo, “Exchange Rates and the International

Adjustment Process," Brookings Papers (1:1978), 111-50.
Levine, R., “The Pricing of Forwward Foreign Exchange Rates,” unpublished ms.,
UCLA (October 1986).
Los, L.A., “Econometrics of Models with Evolutionary Parameter Structures,”
unpublished Ph.D. dissertation, Columbia University (1984).
» "Regression Analysis, and Diagnosis by an Evolutionary Parameter

Estimator,” Journal of the American Statistical Association (forthcoming

1986).

and C. McC. Kell, “Computer Programs for the Estimation of Economic

Models with Evolutionary Parameter Structures," Federal Reserve Bank of New
York Version 4.0 (December 1985).
Lucas, R.E., “Interest Rates and Currency Prices in a Two-Country World,”

Journal of Monetary Economics 10 (1982), 335-60.

Malliaris, A.G., and W.A. Brock, Stochastic Methods in Economics and Finance,

New York: North-Holland Press, 1982.
Merton, R.C., “An Intertemporal Capital Asset Pricing Model,” Econometrica 41
(September 1973), 867-87.

Richard, S.F., "A Generalized Capital Asset Pricing Model," TIMS Studies in

the Management Sciences 11 (1979), 215-32,


29

k k
8s s
Cov(Q;,P;) =[ 2 b 9,42] * TE bp g, dz],

s=1 % s=1 Py

and the remaining covariances and variances are similarly defined.

The term in the first pair of brackets in equation (A3) is the expected

real rei:urn of asset i evaluated in country j. Let this term equal

i,j s s s
Let. Buy = be - be - b ’
5 % Py Pay
and 6 =odZ.
8 8's
Equation (A3) then becomes
4[Q,/P;D, 5] kK
(As) na pl - a + zr Bldg,
Lo iaj s=1
ix B

This Appendix more rigorously defines 4 » 2, and 6

j

ot

is

Q.E.D.

Since capital gains are the only sources of income, the instantaneous

change in real wealth of a country j resident is

N K 4
i,t i,j
= = dt + = B? dZ -cC
(B1) aw; Qo gy Cay wn 9,42.) j
Let
i,j i,j
By o eee By %
¥ = : : ; .
N,j >
By o eee Be
(B1) then becomes
= Wat - 1 y.
(B2) aw, W545 aydt Cydt + W345 yjaz

dt.

where ¥ is the N xK matrix containing the sensitivity of each asset's

real return to changes in the fundamental state variables according to a

resident of country j.


lt

other assets, the new portfolio must have the same real value as the old

portfolio in equilibrium. Thus,

Portfolios that use no wealth such as X = (xg seve oxy) are called arbitrage
portfolios.

Consider an arbitrage portfolio, Xp chosen to have no systematic risk,
i.e., unanticipated changes in the state variables will not affect the real
return to residents of country j holding Xye Formally this is written:
toatdxt so for al s
ir 8 4

This arbitrage portfolio, Xj» will have real return to residents of

country j,
Xia, =X! a,+x! Bl 6 +... +x! Bl 6 = xt
joj jj joi. j KK ji’
where
a ay Bed
a=]: |, a=]: |, Ble] :
j N J N phe
“y % 8

Notice that Ky is a special arbitrage portfolio: it has no risk and
uses no wealth. Therefore, the real return on Xj must equal zero in
equilibrium or, a riskless and costless pump of consumption goods would exist!

Using simple linear algebra we can now solve for the equation of expected
real asset returns from the vantage point of a country j resident. Any
vector, Xy> which is orthogonal to the constant vector and to each of the
coefficient vectors, al,...,a), must also be orthogonal to the vector of

expected returns, The algebraic consequence of this statement is that the


14

pricing equation. It indicates that the expected real return on asset i
evaluated in country j is a linear function of asset i's exposure to a set of
systematic risk factors. These exposures, or sensitivities, are then weighted

by the factor risk premiums to yield the expected real return of asset i in je

IV. The Optimal Portfolio and the Risk Factor Premiums

This section has two related objectives. The first is to derive the
factor risk premium's economic determinants. This bolsters the economic
foundation of the IAPM, and enhances its empirical usefulness. The section's
second objective is to solve for the optimal portfolio of country j's
representative individual. This is important because it allows the IAPM to be
extended to such issues as currency substitution and exchange rate volatility.

It is important to note that the arbitrage approach alone cannot be used
to obtain an optimal portfolio equation. Therefore, we must hypothesize a
representative individual and solve for his optimal portfolio. This utility
maximization approach, however, is also insufficient because assets cannot
easily be priced within the context of the above model. The two approaches

used together are necessary to derive a meaningful optimal portfolio equation.

IV.A The Individual Investor Choice Problem

It is assumed that each investor chooses that consumption and portfolio
strategy that maximizes the expected value of utility over an infinite time
horizon. The instantaneous utility function is assumed to be a strictly
concave function of consumption. The choice problem for the representative

investor of country j is

(1) Max E [f® e 7 ydee.(t)at]
{c,.4,} °0 j

Sete