Estimation of Portfolio-Balance Functions that are Mean-Variance Optimizing: The Mark and the Dollar

International Finance Discussion Papers Number 188

September 1981

ESTIMATION OF PORTFOLIO-BALANCE FUNCTIONS THAT ARE MEAN-VARIANCE OPTIMIZING: THE MARK AND THE DOLLAR

by

Jeffrey A. Frankel

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to International Finance Discussion Papers (other than an acknowledgment by a writer that he has had access to unpublished material) should be cleared with the author or authors.

Sept. 1, 1981

ESTIMATION OF PORTFOLIO-BALANCE FUNCTIONS THAT ARE MEAN-VARIANCE OPTIMIZING:

THE MARK AND THE DOLLAR by Jeffrey A. Frankel* ABSTRACT

This paper offers a way of efficiently estimating the parameters in demand functions for mark and dollar assets. The technique imposes the constraint that the parameters, rather than being determined arbitrarily, are based on investors’ optimizing behavior regarding expected returns and variances, It dominates some previous empirical applications of finance theory in the respect that the expected returns are allowed to vary from period =o period, which is a necessary feature of any macro model. The constraint that the parameters are based on mean-variance optimization is

also tested, and not rejected.

*Acting Associate Professor, University of California, Berkeley. This paper was completed while I was a Visiting Scholar at the Federal Reserve Board. I would like to thank Charles Engel, Brian Newton and Tony. © Rodrigues, for research assistance; the Institute for Business and Economic Research at U.C. Berkeley and the National Science Foundation under Grant No. SES--8007162, for support of parts of this research; and Stanley Black, Barry Eichengreen, Donald Hester, Richard Marston, Richard Meese and Ken Rogoff or comments. This paper represents the view of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or members of its staff.

1. Introduction

One of the most rapidly-progressing sub-areas of international finance theory recently has been the application of the principles of expected utility maximization to the problem of international asset demand functions. Investors balance their portfolios between domestic and foreign assets as a function of the expected relative rate of return, that is, the interest differential in excess of expected exchange rate depreciation. The function itself is shown to depend on parameters such as the variance of the exchange rate and the degree of risk-aversion.

Substantive theoretical results include the following. (1) (Cmly the supply of "outside" assets matters. For example, if residents of different countries consume a common basket of goods, then a current account imbalance that redistributes wealth among countries has no effect on aggregate asset demand or supply, and thus no effect on the relative price of domestic and foreign assets. (2) The portfolio share that is optimally allocated to a given country's assets can be expressed, with suitable assumptions, in a simple linear form: as the sum of a "minimum-variance" portfolio share that depends on the share of the consumption basket allocated to that country's goods and a "speculative" portfolio share that depends on the expected rates of return and the degree of risk aversion. (3) A necessary condition for domestic residents to have a greaterpropénsity to hold domestic assets than the foreign residents, and thus for a current account surplus to increase the relative price of domestic assets, is that domestic residents have a greater propensity to consume domestic goods. However an additional neces-

sary condition is that the coefficient of relative risk-aversion be greater

than unity. Some specific references on the three results are (1) Frankel (1979), (2) Dornbusch (1980), and (3) Krugman (1980). However these points were made earlier, implicitly or explicitly, by Kouri (1976, 1977) and Kouri and de Macedo (1978).2

The empirical literature in this area has lagged behind the theoretical. Several people have taken the tests of the Capital Asset Pricing Model (CAPM) that have been developed for other financial markets and have extended them to foreign currencies.“ These finance studies vary according to whether they take the relevant "market basket" to include real assets (equities) or nominal assets (bonds). They also vary according to what numeraire the investor is assumed to consider as riskless-domestic currency, domestic goods, or a basket of domestic and foreign goods--and according to whether investors who live in different countries

are assumed to consider different numeraires as riskless.

But the finance studies all make the assumption that the expected

currency returns perceived by investors are constant over time, and that the variances and covariances are constant as wel]. This assumption is

made (usually implicitly) in order to be able to estimate the parameters

from ex post sample data. In the case of the variances and covariances,

bother contributors to the literature include Adler and Dumas (1978), Grauer ,Litzenberger and Stehle (1976), Fama and Farber (1979), Solnik (1973), Garman and Kohlhagen (1980), Stulz (1980) and Hodrick (1981).

“Examples are Roll and. Solnik (1977), Cornell and Dietrich (1978), Kouri and Macedo (1978), Macedo (1980), and Dornbusch (1980b).

the stationarity assumption is perfectly appropriate. It is necessary if the parameters of the asset-demand functions are to be considered unchanging over time. However in the case of the expected returns, the stationarity assumption, while it may be appropriate from a micro CAPM perspective, is notappropriate for a macro model. It would imply ‘hat the arguments and values of the asset-demand functions, as opposed to the parameters of the functions themselves, are constant over time. I is an essential element of most macro models that expected returns, and thus asset demands, be allowed to vary over time.

This point is given extra practical relevance by the striking reversals in trend which the most important exchange rates have undergone in recent years. In 1977 and 1978 the dollar was depreciating steadily against the mark and other currencies. This fact partly explains why previous estimates of the optimal portfolios have given a

_larger-than-expected weight to the mark and a smaller-than-expected weight

3 to the dollar. But, as of 1981, the mark is down sharply and the

3 Dornbusch (1980b, p. 165) estimates that an optimal portfolio of

these two currencies would be 56% in marks, of which 50% represents a minimum-variance portfolio and 6% represents speculation to exploit the higher return on the mark over the sample period of 1976 1 to 1979 2. Kouri and Macedo (1978, p. 129) find that an optimal five-currency portfolio would include a 37% share in marks, of which 33% represents a minimum-variance portfolio and 4% represents speculation. In their study the dollar also benefits from speculation. The big loser is the found, which depreciated sharply over their earlier sample period of 1973-77. (The negative share calculated in the optimal portfolio for a currency like the pound does not jibe with the positive supply of pounds known to exist in the world market. Thus we know that either the actual portfolio held by investors is not in facet equal to the optimal one, or else the method of calculation of the optimal portfolio is incorrect.)

dollar up. If the estimates were redone on more recent data, they would certainly give a larger weight to the dollar and a smaller weight to the mark. The important point is that expected returns do vary over ‘time, and any estimates that neglect this are suspect.

Thus we require a measure of expected returns that can vary over time, rather than relying on the sample mean, and we require a measure of variances and covariances that computes squared deviations around this varying expected value, rather than computing deviations around the sample mean. At first this might appear to be asking the impossible. But this paper offers a strategy for constructing exactly such measures. The strategy involves imposing the constraint that actual asset-demand functions are in fact based on mean-variance optimization on the part of investors. It thus produces more efficient estimates of the parameters than earlier studies that do not impose any constraints regarding from where the asset-demand functions are derived.» The strategy also allows us to test formally the proposition that asset-demand functions are based on mean-variance optimization. We simply compare the likelihood with the constraint imposed, to the likelihood unconstrained.

The next section of this paper shows how asset-demand functions can be estimated, in a world in which investors have different preferences

depending on their country of residence, without imposing the constraint

bne study of the optimal portfolio, by von Furstenberg (1981), does allow investors' expected returns to change each period; they are computed from the data observed up until the period in question.

. . ‘

Sthis refers to macro studies in which asset supplies are present as explanatory variables. In most portfolio-balance studies, the dependent variable is the exchange rate; examples include Branson, Halttunen and Masson [1977, 1979]. But the dependent variable is the relative rate of return (i.e. exchange rate depreciation in excess of the interest = differencial) in Dooley and Isard [1979] and Frankel [1981], as it is in the present paper.

of mean-variance optimization. Section 3 derives theoretically the optimizing form of the functions. Section 4 estimates the asset-demand functions subject to the constraint that they are indeed of this form. In a comparison of the unconstrained likelihood from Section 2 with the constrained likelihood from Section 4, one is statistically unable to reject the constraint. This evidence would tend to support the hypothesis that actual asset-demand functions are indeed optimizing. However, the power of the test is probably very low. Thus the contribution of the paper may lie primarily in the estimation framework, which is of general. macroeconomic applicability. Section 5 briefly discusses extensions of the framework, in particular relaxations of two simplifying assumptions made in the paper: (a) the limitation of the portfolio to two assets - dollar bonds and mark bonds, and (b) the assumption that exchange rate variability is the

only source of uncertainty, for example because prices are "sticky" in the

short run,

2. Estimation of Unconstrained Asset~Demand Functions

In this paper we assume that investors allocate their portfolio

between mark bonds and dollar bonds only. Let Xa be the share allocated

to marks, by residents of country i attime t. The asset—demand

function By gives us the demand as a function of the interest rate on

_ marks a » the interest rate on dollars i° » and the expected depreciation of the mark (from time t to time t+1l) Ast :

M ,$ x > i,

D e it = BF (4, > ds. ),

More specifically, assume that B; is linear in the expected relative

e return Ze :

e = > > *y a, + biz.) > a; 0, »b 0 . (1)

$

e —. ,DM * J24i0 - 1 where ae i, t

- dst - This-functional form is assumed for two reasons: (1) some form must be assumed for estimation, and (2) meanvariance opt imizat ion implies such a linear form, as will be seen in the next section. But the important point is that we are not constraining

“the parameters a and b to be anything in particular. They could be based on investors' arbitrary "tastes" for assets as easily as on meanvariance optimization. Of course we have already restricted the function somewhat; for example many macroeconomic models include real income levels,

. 6 representing a transactions demand for the assets.

If all investors in the market had the same preferences, then we could

I tested various ‘other possibilities such as including income in Frankel [1981], on which this section is based. ;

.

estimate equation (1) by itsel£.7 But, in general, asset-demand functions certainly vary, residents of each country having a relatively greater preference for their own currency. For the purposes of this paper

we distinguish among residents of Germany (1=6G), the United States

(i= US) and the rest of the world (L=R). The mark shares of the three

countries' portfolios are given by

Mo, /Mo. = a, t b(z-) “us, “us, ~ ust b(a) YR /YR = ap + b(z-), t where M, = holdings of marks by residents of country i, and Mie = total wealth (narks and dollars) held by residents of country i, expressed in

8

marks. Presumably ag? ap > aye.

If we had data on mark assets broken down by country of holder, we could estimate each of these equations separately. But given adequate data on only the aggregate supply of marks Mes we must aggregate the

three equations. We do this by defining the countries' shares in

aggregate world wealth Weiwe = We Me» Wyg = Wys MM. and wp = We MM - . t t t t t t We multiply each equation by that country's Wy and.add them up. We

t

"two common kinds of models satisfy this description. Some, like Branson, Haltunnen and Masson [1977], assume that domestic residents are the only Ones who hold domestic assets, and thus are the only investors who count .in the market. Others, like Frankel [1979] and Dornbusch {1980a], assume that all investors, domestic and foreign alike, share the same preferences. In Frankel [1980 ] I call the former "small-country" portfolio-balance

models and the latter "uniform-preference" portfolio-balance models, and give further references:

We are assuming that the responsiveness with respect to expected returns, b , is the same for all investors. In the next section this will be seen to hold if all have the same degree of risk-aversion, (It is also

necessary if we are to have % 7 > Sg for all 2.)

' marks x, is high, they must pay a high expected relative return z

thus obtain an expression for the share of the world portfolio

occupied by marks:

= = _ _w e, FMM, = ag b, fus"ts, * xO “%e ~ “ts + b(ze)> (2) where we have used the fact that w, + Ws + a1. t t t Equation (2) is simply a weighted average of equation (1) over the

three countries, where the weights are their shares in world wealth. ?

So far we have not said anything about measuring z©

t? the expected

relative rate of returns on mark assets. This is not a trivial task, in light of the nonobservability of expected depreciation, and our unwillingness to assume it constant as in the previous finance studies. The solution adopted here is to invert equat ion (2), so that the expected relative return is expressed as a function of the asset supplies and the distribution of wealth: -a

a a,-a. a e_ _(7R, _ (7G 3R R” “us z= -G) - ¢ dw. + ¢ b

a | db “Gt ys. FOX 3)

Let us check the economic relationships in (3). If the relative supply of

. t

in order to be willingly held. Also if world wealth is redistributed toward U.S. residents (e.g. by a current account surplus with the rest of the world), then the relative return on marks ze must rise, in order for

them to be willingly held, because U.S. residents have a lower preference

for them (ap - ays > 0) . On the other hand if world wealth fs redistributed

toward German residents (e.g. by a current account surplus with the rest

’ of the world), then ze must fall, because German residents have a higher

Irhis trick is borrowed from Dornbusch [1980a, appendix]. Notice that in

the case a, = a, = a). , equation (2) reduces to the "uniform-pref erence"

model mentioned in footnote 7.

.

preference for marks (a, - ap > 0). To deal with the unobservability of expectations, we make the assumption

that investors form them rationally. The ex post relative return

Cray = im - i° - As,) » which is observable, is assumed equal to the

e expected return 2z” plus a random error ¢€

t - By “random", we mean

ttl uncorrelated with all information available at the beginning of the

period over which the return is measured:

e een 7b t Seay > Eley | rT.) = 0.

Substituting into (3),

a a,7-a a,-a R G7 3R R~ 4us 1 Zz = -~(—— - (—_—— —— ee — xX cH G? — KEG, + yg FE et Ee

The parameters of equation (4) can now be estimated by regression. All

the variables (Zap oe? WUS,» and x.) are observable. And the regression

error is simply the expectational error » which we know to be

ea

uncorrelated with the right-hand-side variables by the assumption of rational expectations. Indeed, the reason we inverted equation (2) to

begin with was so that E.4, Would enter the left-hand-side variable

rather than one of the right-hand-side variables, allowing us to use

regression estimation.10 Table 1 reports regressions of equation (4) for the period Jatuary

1974 to October 1978. Data are discussed in appendix 2. Unfortunately,

1Wote the importance of the strong assumption that the asset-—demand .function (1) is correctly specified, so that the only source of regression error is the expectational error. If the asset stocks are measured

with error, or if any other determinants of asset demands have been omitted, then a regression of equation (4) will produce estimates that are biased and inconsistent. But the defense of this procedure is that, as a way

of estimating asset-demand functions, it is a step forward from the typical finance studies, which would require not only an absence of error terms other than the expectational error, but also the assumptions that (a) the actual functions are based on mean-variance optimization, «and

(b) the expected returns are constant. In this paper we rule out the latter and consider the former open to testing.

10

.

even the few implications of our hypothesis so far ~- a negative constant term and coefficient on German wealth, and positive coefficients on U.S. wealth and the supply of marks - are not borne out. The coefficients

are generally wrong in sign, and always statistically insignificant.

In light of the high standard errors, there seems no purpose in unscrambling the point estimates to obtain estimates of the original parameters

Gg 4yg > 4p and b , even though they are fully identified.++

a

However, one assumption that we have already made is borne out.

The absence of serial correlation in the error term supports the hypothesis of rational expectations.

Perhaps the main lesson to be drawn from Table 1 is the very low degree of precision that plagues estimation of general portfolio-balance equations, and the need to bring additional information to bear. This provides the motivation for considering the constraints placed on the parameters by the hypothesis, developed in the following section, that they are derived from mean-variance optimization by investors. If one

believes this hypothesis, then the resulting estimates will be more

precise.

llwe should not be concerned with the very high sums ‘of squared | residuals-and consequent very low R2s - in Table 1 (and later, in Table

2). The empirical literature on the forward exchange market has shown that deviations of the forward discount (or, equivalently,the interest differential) from ex post spot depreciation are enormou's, regardless whether they are random. Thus we would expect a high SSR and low R

even if the explained sum of squares were significantly greater than zero.

ll

Table 1: Unconstrained Asset—Demand Functions OLS

- Dependent Variable: 2, , relative return on marks

24d measured as: German-U.S, interest differential minus depreciation of mark Sample: Jan. 1974 - Oct. 19738 (58 obs.)

Asset supplies . Coefficients

. log - 2 a . x

measured as: Constant “Ce “Use x. D.W. R SSR VC@)* Likelihood Total assets -246 .138 8 8-.133 -.654 1.82 .04 .05617 .3009684 118.96

(.256) (.731) (.188) (1.122) [.100104} [116.88] Bonds only .152 -152. -.064 -.774 1.82 .05 .05598 .0009651 119.05

(.149) (.346) (.098) (.909) . [.00104] [116.98] Monetary base -.083 -.144 .028 .354 , 1.87 .03 .05703 .0009832 118.52 only (.191). (.249) (.033) (.948) [-00106] [116.45] Za measured as: forward discount minus depreciation of mark

Sample: March 1974 - October 1978 (56 obs.)

Asset Supplies Coefficients 3 log measured as: - Constant wg Wys x D.W. R SSR V(€)* Likelihood* EE ee ‘Total assets "313 .581 -.210 -1.207 1.92 .04° .05014 .0008953 117.05 (.985) (.758) (.194) (1.156) {0009643} [114.97]

(Standard errors are reported in parentheses.) * Maximum likelihood estimates of the variance and log likelihood are reported.

{Unbiased estimates are reported in brackets.]

12

.

3. Derivation of Asset-Demand Functions from Mean-Variance Optimization

In this section we derive the correct form for the asset~—demand of an investor who maximizes a function of the mean and variance of his end-of-period real wealth.!2The analysis is lifted wholesale from Dornbusch [1980a]. The reader familiar with that paper or with the general approach, which is standard in the CAPM literature, is urged to skip to the next

section, and thus to conserve his patience for the rest of the journey

ahead.

let W, be tne real wealth of investors of country i. Their decision i , t variable is Xs 5 the fraction of the portfolio that they put into mark

t .. «DM assets. The real rate of return on the mark assets is i, - 1 - As $ ttl

so og .DM . and the real rate of return on dollar assets is i” - 1 » Where i, is +1 the current one-period interest rate on mark assets, i, is the current one-p2riod interest rate on dollar assets, 1 is the dollar-denominated t+1 inflation rate in the goods consumed by residents of country i, and AS ay is

the © depreciation rate of the mark between times t and t + 1. Thus end-of-

period wealth depends on Xs and the realized rates of return: t

DM $ .$ $ We =W, {x, (i, - 7 - As_.,) + (1 - x, Ge - Tap +1} tet te ot, Ct deny ttl tt ~ .DM .8 7) $ - =W. {x. (i. - i” - As Y+i7-T + 1}. (5) i, i t t t+1 t qe

If the relative return on marks is positive, end-of-period wealth is an

increasing function of Xp Notice that the inflation rate drops out of the t relative return.

12a he assumption that returns are normally distributed is sufficient to

imply that investors look only at the mean and variance. The normality assumption might be justified by an appeal to Brownian motion observed at discrete intervals, and is necessary for the maximum likelihood estimation in arly case.

13

“We define the dollar-denominated inflation index to be a weighted average of dollar-denominated inflation in German-produced goods and dollar-denominated

inflation in U.S.-produced goods, with the weights a, and (1 - a.) equal to shares in consumption:. — $ _ DM _ _ $ "1 = O15 (% As.) + (1 04) (mys).

Now we make a major simplifying assumption: goods prices are non-stochastic when denominated in the currency of the producing country.

Only the exchange tate is uncertain .23/

Then the mean and variance of end-cf-period wealth (5) are as follows:

DM .$ .$ DM S . . E(W )=w, {x, (i, - if - E(As )) + it - [a, (9 - E(As )) + (1 - a.) (0 yy +1 * etl i, ai, t t+1 t i Gea t+1 i US nay

vai, ) = 2x, + IPCs, ))

‘i 7 4 ne Conca Shan t+1 t t The hypothesis is that investors maximize a function of the mean and variance

ule, ), VW, dd). “ttl “ttl

We differentiate with respect to x,

~ ~ t

dE(W, ) qvw, )

| t+1 ttl _

ax, By dx, + Vs dx, _ 0. t t t

~ , DM SS 2

)=0 t ttl

iM _ 48 _ gus...) = wei, 2)V(As

t t ttl U. i ~

(x, - a,).

ite t+1 i. i 13phis assumption is made by Krugman, but is considered only one special case by Kouri [1976] and Dornbusch {1980a}. Assuming that prices are sticky, at least in the short run - and in this case we are talking about one month - is of course standard in Keynesian macroeconomics.

.

14

The expectation and variance of the exchange rate change are conditional on all information available at time t. We use the notation of section 2, in which Zig Was defined to be the relative return on marks and € was defined to be the expectational error (As i445 - EAs. 4.): Also we define — to be the W 14/ measure of relative risk-aversion: p = 2U5W, /U,- Thus our expression

t for the expected relative return on marks is

Bey) = PVC) Dy = 4) | (6) This expression for the expected relative return is analogous to

the equation that was estimated in the previous section, except that

we have not yet aggregated over the countries. But let us invert to

get the form analogous to the earlier asset-demand function:

- 1, | 4 =O, + oye; FC44) (7)

Compare this equation to our general unconstrained asset-demand function : . = - 1 7

(1). The two are the same, with a, =a, and b= pve) * Following

Kouri and Macedo, Dornbusch [1980a] calls the first term the minimun-

variance portfolio. If the investor is highly risk-averse (p=) or

14

The Arrow-Pratt measure of risk aversion is defined as 9 = -u"W/u', where u(W) is the utility function, the expectation of which is to be maximized. One can take a Taylor-series approximation to Eu(W) and differentiate it

with respect to E(W) and V(W) to show that the two definitions of Pp are equivalent.

The utility function will have a constant coefficient of relative riskaversion if it is exponential in form:

u(W) = = WY, where p =. 1 - y..

(The solution to the one-period maximization problem considered here will

be the correct solution to the general intertemporal maximization problen, if the utility function is further restricted to the logarithmic form, the limiting case as y goes to zero, which implies p = 1, or if events occurring during the period are independent of the expected returns that prevail in the following period.)

. 15

the returns are very uncertain (V(e)= ~), he will hold marks and dollars in the same proportions as he consumes German and U.S. goods. Of course, he would have to give up some expected return to hold the minimum--variance

portfolio. They call the second term the speculative portfolio. A higher expected relative return on marks induces investors to hold more of them than the minimum-variance portfolio, to an extent limited only by the variance of the exchange rate and the degree of risk-aversion. For example, under risk-neutrality (p=0) , the two assets become perfect subst itutes and arbitrage insures that E(z 4.) =O.

Before we proceed to the est imat ion of the parameters of equation (6) or (7), which is the main point of this paper, we must add a footnote to the foregoing derivation, pointed out by Krugman: equations (5) to (7) involve some sleight-of—hand. Is the expected rate of mark depreciation, which enters EZ 47> defined as the percentage increase in the mark cost of

dollars E(S /$,)-12 Or is it the percentage decrease in the dollar cost of

ttl

marks -[E(S,/s )-1] 2? The two are not equivalent, by Jensen's inequality.

t+1 The latterdefinition is correct only if a = 0. For example, under risk-neutrality (p=0) , dollar assets and mark assets will have the

same expected purchasing powers in this case, i.e. in terms of U.S. goods;

then E Z41 8° defined is zero. But to the extent that investors

consume German goods (a > 0) , the variance of the mark/dollar exchange rate will have a positive effect on the expected purchasing power of dollar assets, due to Jensen's inequality. Thus the variance should enter.

z -neutrality. EZ 4d even under risk-neutrality

16

Similarly, the former definition is correct only if a =1. For example, under risk-neutrality dollar and mark assets will have the same expected purchasing powers, in this case in terms of German goods; then E Z4, 80 defined is zero. But to the extent that investors consume U.S. gocds (a <1) , the variance of the dollar/mark exchange rate will have a positive effect on the expected purchasing power of mark assets, due to Jensen's inequality. Thus, again, the variance should enter

E 44, even under risk-neutrality.

This rather counter~intuitive, but important, point of Krugman's is discussed, and the estimation technique is modified accordingly, in Appendix 1. We stick with the Dornbusch formulation in the text, as it is more intuitive. The ultimate finding of this paper, a statistical inability to reject the hypothesis that asset-demand functions are based

on mean-variance optimization, is the same in either formulation.

17

.

4. Estimation of Asset-Demand Functions Constrained to be Optimizing

In the last section we found that. the linear form assumed in

equation (1) is mean-variance optimizing, provided

a a, , the share of consumption occupied by German goods,

i i and b = siGy , where p is the constant of relative risk-aversior. and V(e) is the variance of exchange rate prediction errors. To aggregate across residents of the three countries, Germany, the United States, and the rest of the world, we can substitute directly into

equation (4), the pre-constrained aggregate form:

Zaz 7 7 PV(E)OR - PV(E) (OQ - oe, * evce) Can = °us’“us, + pvVCedx, + egy (8)

We could simply estimate equation (8) by OLS. ‘If we use actual import and consumption averages for Ops Oo and Ais » then the coefficients are overidentified (by three). This is what we want; overidentifying restrictions are necessary for hypothesis-testing.

Table 2 presents regressions of equation (8) using the same data sanple

as Table 1. The constraints are imposed by constructing the variable

= - - - x Ye = “Op — Og OR) Mey + OR“ A%ys) Mys + 7

and then running the regression

o|F

741 Ye + Seay -

The shares of consumption allocated to German, as opposed to U.S., goods, were taken to be Oo = 0.986 , Op = 0.469 and 5 = 0.005 , as

explained in appendix 2.

18 Table 2: Constrained Asset ~ Demand Functions (Dornbusch version) OLS

Dependent Variable: » relative return on marks

241

‘Independent Variable: y, = -a, - (a, -O,)wG, + (a, - %y5) "ys +x, re)

24d measured as: German-U.S. interest differential minus depreciation of mark

Sample: Jan. 1974 - Oct. 1978 (58 obs.)

- Coefficient

Asset supplies —-—x""- log _Measured as: _ PV(e) D.W. v(€) * likelihood * Total assets .035 1.88 -.00 .05877 001013 117.64

(.033) — [ .001031] (117.14) Bonds only -015 1.88 -.00 .05873 -001013 117.66

(.014) - [001030] {117.16} 241 measured as: forward discount minus depreciation of mark

Sample: Mar. 1974 - Oct. 1978 (56 obs.)

Asset supplies Soefficient 2: — log measured 2s: Pv(e) D.W. R SSR v(& * likelihood * Total asset:s .019 1.98 -.00 .05216 .0009314 - 115.94

(.033) [ .0009480) {115.44}

(Standard errors are reported in parentheses.)

* Maximun likelihood estimates of the variance and log likelihood are reported. [Unbiased estimates are reported in brackets. }.

19

.

First consider a formal test of the constraint. To illustrate, we take the last regression, in which a is calculated using the forward exchange rate rather than the interest differential. The constrained log likelihood is 115.94 . The unconstrained log likelihood in Table 1 was 117.05 . Twice the difference is distributed x? with three degrees of freedom. We easily fail to reject the hypothesis that actual asset-demand functions are in fact based on mean-variance op=imization. (Intuitively, the imposition of the three constraints worsened the fit, i.e. the sum of squared residuals, very little.)

If we accept the optimization hypothesis either on a priori grounds or on the basis of the likelihood ratio test, then we can use it 1 6? 0.019 , is presumably more efficient than its estimate in Table 1 (the

to get efficient estimates of the parameters. The estimate of

coefficient of x ). Indeed, unlike before, it is now of the correct positive sign. For example, an increase in the expected relative return on marks z of 100 basis points would raise the demand for marks by .(119Z of thé aggregate portfolio. If asset supplies were unchanged, this point estimate would imply (as of May 1980, when the share of the portfolio in marks x happened to be .19038) an appreciation of the mark of .023%. Unfortunately the standard error is still too high for the estimates to be statistically significant.

Under the optimization hypothesis, i. pV{e) . Let us say we are

b willing to ignore the high standard error and use the point estimate

“a

L for + 0.019 ; then how can we separate out p from V(¢) 2? The

in.troduction promised a measure of the expected relative return that was

20

allowed to change over time and an estimate of the variance around the changing expected value, rather than around the sample mean. Under the

hypotheses we have adopted (rational expectations and optimization), an

e efficient estimate of the expected relative return 2, is simply the fitted value from the regression in Table 2, Zl "3 D%e° An efficient

“estimate of the expectational error is simply the residual ead - And an efficient estimate of the variance is simply the variance of the residual.

In Table 2 the variance is 0.0009314 . We also get an estimate of

the coefficient of relative risk aversion? 0.0009314

= 20.08 .

‘ The high standard errors attached to the coefficients are a consequence of (1) the small numbers of overidentifying restrictions imposed by the hypothesis, relative to the number of observations, and (2) the high sum of squared residuals (SSR). Given the high SSR in the unconstrained case, a high SSR in the. constrained case was inevitable. To obtain more efficient estimates, we would need additional overidentifying restrictions,

the prospect of which arises in the next and final section.

21

5. Future Research

In the foregoing analysis two assumptions have been made that it would be desirable to relax in the interest of greater realism: (a) the limitation of the portfolio to two assets, and (b) the non-stochastic nature of goods prices.

Relaxing the first of these assumptions, i.e. extending the number of assets, not only is more realistic, but turns out to have the additional advantage that the hypothesis of optimization then imposes addit:onal constraints. This should allow a more powerful likelihood ratio test of the hypothesis than in the present paper. And, assuming we cont:inue to accept the optimization hypothesis, imposing the constraints should earn us more precise estimates than in the present paper.

The outcome of the optimization problem in the multiasset case,

analogous to equation (6), turns out to be simply

ey = PAEx, — a) + EL yy

>» X& , @ and € are now vectors of dimension n-l =

where Z44 t t+l

the number of assets (not counting the dollar). The testable constraint is that 2 (an n-l x n-l matrix) is the variance-covariance matrix of

€ , which is the vector of expectational errors pertaining to the relative returns on the various assets. If the constraints are to be imposed,

the parameters cannot be estimated by OLS, but instead must be estimated by maximum likelihood. Nor isthe programming problem as easy as it is

in the M.L.E. of the Krugman version of the one-dimensional equation, “discussed in Appendix 1. But results on the multicountry case should be ready soon.

To relax the second assumption, the nonstochastic nature of goods

22

.

prices, is a matter of explicitly deflating rates of return

by inflation rates as measured by price indices. Up until now even the standard efficiency tests, for example tests of serial correlation in

our Z > have not used price data. Some of them recognize the relevance of Jensen's inequality (or "the Siegal Paradox"). But nevertheless

they all simply test that the forward rate equals the expected future spot rate. Doing it right would mean testing that the forward rate equals

the ratio of the expected future purchasing power of foreign currency

to the expected future purchasing power of domestic currency 2> The culmination of this line of research would be to repeat all the estimation in the present paper allowing uncertainty in the price levels as well

as the exchange rate.

mn

13 56e Engel {1981]. Frenkel and Razin, (1980) offer the first test to “recognize explicitly that the equation for the expected future spot

rate should include not only the forward rate but also a term representing

the covariance of the spot rate with the purchasing power of domestic

currency, even under thenull] hypothesis of market efficiency and risk-

neutrality. However they calculate the covariance term with ex post

data, implicitly assuming that the expected purchasing powersof the cur-

rencies are constant, which is inconsistent with the intrinsically

present fact that the expected exchange rate varies. In other words,

their approach to the efficiency hypothesis is subject to the same

limitation as the literdture on the mean/variance-optimization hypothesis cited in footnote 2.-

23 APPENDIX 1: The Krugman Version

As explained at the end of Section 3, Krugman points out that Jensen's inequality is not merely a mathematical annoyance that can be swept away by an appeal to approximation, but is substantive to the question of how the parameters of the asset-demand functions depend on 9 and v(4s,) . His equation (18), translated from his continuous~t ime model to our

discrete-time notation, is

EZ, .,7 pv(as.) x, - (p-1) Vv (As,) o, (6")

“1 . t

where depreciation As. is defined as the percentage change in 3

(as opposed to the percentage change in 1/S ). It differs from our equation (6) by the addition of a V(AS, Ja, term. An increase in the variance raises the expected purchasing power of dollar assets over German goods, due to Jensen's inequality. and thus (even under riskneutrality) raises the necessary expected relative nominal return that must be paid on mark assets. When (6")' is aggregated across the three

countries of residence, we get a discrete-time version of Krugman's equation (21):

Ez 4, = pv(As.)X, + (o-1) v(As,)[-ap - (a,-a2) “c+ (Ap -%M5) Wy 1) (8")

Unlike the Dornbusch form, our equation (8), equation (8') is not homogeneous in pv(As,) . Thus we must take advantage of the constraint between the coefficients and the variance of the regression error as part of the estimation process. Imposing a constraint between coef ficients and the variance of the error is unusual in econometrics. It cannot be done by OLS, but requires maximum likelihood estimation. Fortunately in the simple two-asset case, it is very easy to write down the likelihood

function and to compute the values of 9 and vids.) that maximize it.

24

The results are given in Table 3. As in the Dornbusch formulation, the reduction in the likelihood that results from imposing the constraint is very small; we are again unable to reject the hypothesis that the parameters of the asset-demand functions are based on mean-variance optimization. The likelihoods under the Dornbusch and: Krugman formulations are sc extremely close as to permit no possible inference as to which fits the data better. The point estimate of the variance is essentially

the same as it was in the Dornbusch formulation. The point estimate of

the constant of relative risk-aversion is somewhat larger than before.

25 Table 3: Constrained Asset—Demand Functions

(Krugman version) Maximum Likelihood Estimation

z =

at pv (As) x + (p-1) V(As)[-a, - (og - OR) ¥ Gy + Cap - ys) yg] + Eeal tel measured as: forward discount minus depreciation of mark Asset supplies measured as: total assets Sample: March 1974 - Oct. 1978 (56 obs.) A log

6 _ V(As) likelihood 27.53 . 000931 115.59 (1.06) (.000926)

(Standard errors reported in parentheses.)

26

APPENDIX 2: Data

The total net supply of assets denominated in the currency of a particular country (Germany or the United States) was calculated as the stock of federal debt outstanding (whether monetized by the Central Bank or not) plus the Central Bank's cumulative sales of domestic assets in foreign exchange intervention (measured as its international reserve holdings corrected for valuation changes) minus a measure of the holdings by foreign central banks of the country's assets in the form of foreign exchange reserves. The net supply of bonds to the private market was calculated as the total net supply of assets minus the monetary base.

For purposes of distinguishing German-held wealth and U.S.-held wealth, in each country the current account surplus and federal debt were cumulated to arrive at the private sector's total claims on outside assets.

The data sample was ended at October 1978 because the calculation of reserve holdings becomes especially difficult after that date due to the “issuing of mark-denominated Carter notes by the U.S. Treasury, the holding of foreign exchange reserves valued at current exchange rates by the Federal Reserve, and the turning over of reserves to the European Monetary System by the Bundesbank.

Further details on the above data calculations, and sources, are given in the appendix to Frankel {1981}.

The tests were run once using the interest differential and once

‘using using the forward discount. The two are theoretically the same, and in practice very close, by covered interest parity. In the former case, one-month Eurocurrency interest rates were used: the mean of the bid-ask

spread on the last day of each month, as reported by the Financial Times

27

of London the first day of the following month. The exchange rate was also end+of-month, as reported in the IMF's International Financial Statistics. In the latter case, the 30-day forward rate and spot: rate were both taken from the Wall Street Journal.

Comput. ing ore > Qs and Op» the consumption shares allocated by residents of the various countries to German rather than U.S. goods,

involves a number of arbitrary assumptions. The numbers used in this

paper were computed as follows:

Qs = (U.S. imports from Germany)/(U.S. income) = 0.005 a = 1 - (German imports from U.S.)/(German income) = 0.986 Op, = (R.O.W. imports from Germany)/(R.O.W, imports from Germany

and U.S.) = 0.469 .

The arbitrary assumptions are that all goods are denominated in the currency of the producer, and that R.O.W. currency values are uncorrelated with the mark/dollar exchange rate. The data are for 1974, taken from the IMF's

Direction of Trade Annual 1969-75.

28 References

Adler, M. and Dumas, Bernard, "Portfolio Choice and the Demand for Forward Exchange," American Economic Review 66, 2 (1976), 332-39, ae can Economic Review

Branson, William, Hannu Halttunen and Paul Masson, "Exchange Rates in

the Short Run: The Dollar—Deutschemark Rate," European Economic Review 10 (1977), 303-24.

» "Exchange Rates in ‘the Short Run: Some Pin he So ; . the Short Run: Some Further Results," European Economic Review 12, 4 (October, 1979), 395-402.

Cornell, Bradford and J. kK. Dietrich, "The Efficiency of the Market for Foreign Exchange under Floating Exchange Rates," Review of Economics and Statistics 60, 1 (February 1978).

Dooley, Michael and Peter Isard, "The Portfolio-Balance Model of Exchange

Rates," International Finance Discussion Pa er No. 141, Federal a _ Sane nce Viscussion Paper Reserve Board (May 1979).

Dornbusch, Rudiger, "Exchange Risk and the Macroeconomics of Exchange Rate Determination," M,I.T. (April 1980).

» "Exchange Rate Economics: Where Do We Stand?"

Brookings Papers on Economic Activity 1 (1980), 143-94.

Engel, Charles, "Testing for Risk-Neutrality and Rationality," International Finance Division, Federal Reserve Board (June 1981).

_—_

Fama, Eugene and Andre Farber, "Money, Bonds and Foreign Exchange," American Economic Review 69, 4 (September, 1979), 639-49.

Frankel, Jeffrey, "The Diversifibility of Exchange Risk," Journal of International Economics 9 (August, 1979), 379-93.

; , "Monetary and Pertfolio-Balance Models of Exchange Rate Determination,” NBER Summer Institute Paper No. 80-7 (December 1980), forthcoming in Economic Interdependence and Flexible Exchange Rates, edited by J. Bhandari and B. Putnam, Cambridge, Massachusetts: M.1.T. Press. .

, "A Test of Perfect Substitutability in the Foreign Exchange Market," U.C. Berkeley (July 1981), revised version of

International Finance Discussion Paper 149, Federal Reserve Board, onbernattona’ rinance Discussion Paper

Frenkel, Jacob and Razin, Assaf, "Stochastic Prices and Tests of Efficiency 7

of Foreign Exchange Markets." Economic Letters 6, No. 2 (1980), 165-70.

29

Garman, Mark and Kohlhagen, Steven, ‘Inflation and Foreign Exchange Rates under Production and Monetary Uncertainty," U.C. Berkeley (June 1980).

Grauer, F. L. A., R. M. Litzenberg and R. E. Stehle, "Sharing Rules and Equilibrium in an International Capital Market under Uncertainty, Journal of Financial Economics 3, 3 (1976), 233-56.

Hodrick, Robert, "International Asset Pricing with Time-Varying Risk Premia." Carnegie-Mellon V., Working Paper # 29-80-81 (January 1981).

Kouri, Pentti, "The Determinants of theForward Premium," IIES Seminar Paper 62, University of Stockholm (August 1976),

“International Investment and interest Rate Linkages under

-- Flexible Exchange Rates,'' in The Political Economy of Monetary Reform, edited by Robert Aliber, London: Macmillan (1977).

and Jorge Braga de Macedo, "Exchange Rates and the International

Adjustment Process," Brookings Papers on Economic Activity 1 (1978), 111-150.

Krugman, Paul, "Consumption Preferences, Asset Demands, and Distribution Effects in International Financial Markets," NBER Working Paper No. 651 (March 1981).

Macedo, Jorge Braga de, "Portfolio Diversification Across Countries,"

International Finance Section Working Paper, Princeton University (November 1980).

Roll, Richard and Bruno Solnik, "A Pure Foreign Exchange Asset Fricing Model," Journal of International Economics 7, 2 (1977), 161-180.

Solnik, Bruno, "An Equilibrium Model of the International Capital Market,” Journal of Economic Theory 8, 4 (1974), 500-524.

Stulz, Rene, "A Model of International Asset Pricing," M.1.T. (August 1980).

Von Furstenberg, George, "Incentives for International Currency Diversification by U.S. Financial Investors,"' International Monetary Fund DM/81/37 (April, 1981).