The relevant interest rates are obtained as follows. The domestic interest rate is calculated
using the rates implied by the average of the Bid and Ask prices for the U.S. Treasury Bill
expiring on the date closest to the expiration of the option. The ratio i/i* is obtained using the
currency futures contract expiring at the same time as the option, together with the covered
interest parity relation.2! The Treasury Bill discounts and futures prices are from the New York

Times (Wednesday closing prices).

4.2 Generating The Exj Variance Serie:

The binomial valuation model of section 3 requires working through a binomial "valuation
tree" just like the one presented there for the case of n=2 subintervals. In calculating the final
implicit volatilities, we set n equal to 50 (a number of subintervals that provides a "sufficiently"
close approximation to the Black-Scholes value when valuing a stock call option).22

The numerical approximation procedure used starts with an initial value for the implicit
volatility and combines it with the other determining variables in the model. After going through
the tree completely, the procedure generates a call value. This call value is then comipared to the
actual market price. At this point an adjustment is made to the initial volatility value according to
the Newton-Raphson numerical search procedure.23 The process then iterates unti. a change in
the annual spot volatility (standard deviation) of less than .0001 is encountered.”4

The resulting three series of the expected variance (of changes in the log of the exchange

21 Violations of covered interest parity represent riskless arbi 9 unities; the condition holds very well
across assets within the same jurisdiction. For evidence see Frenkel and Levich(1979).

22 See Cox and Rubinstein, Potions Markets, p. 243, for examples of the accuracy of the Binomial Approximation
to the Black-Scholes values for different values of n.

23 See Manaster and Koehler(1982) for further information regarding the search procedure.

24. Since the derivative of a call value with respect to the volatility is always positive, if an implied volatility
exists then it is unique.

18


IFDP
NUMBER

298

297

296
295

294

293

292

291

290

289

288

287

286

International Finance Discussion Papers

TITLES
1986

The International Debt Situation

The Cost Competitiveness of the Europaper
Market

The United States International Asset
and Liability Position: A Comparison of
Flow of Funds and Commerce Department
Presentation

An International Arbitrage 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

Tests of the Foreign Exchange Risk Premium
Using the Expected Second Moments Implied
by Option Pricing

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

AUTHOR(s)

Edwin M. Truman

Rodney H. Mills

Guido E. van der Ven
John E. Wilson

Ross Levine

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

Robert A. Johnson

Richard K. Lyons

Robert A. Johnson

Manuel H. Johnson
Bonnie E. Loopesko

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

Hali J. Edison
Ralph Tryon

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.

39


Wealth

The net wealth of the citizens of each of the four countries, in millions of local currency
units, end of month, is calculated as the cumulation of the government deficit plus the current
account surplus. The government deficit for the U.S. is the change in the gross public debt for
the U.S. Treasury, excluding only that held by U.S. government agencies and trust funds
(DODEBT not corrected for the two issues of ‘Carter Bonds’); for Germany and Japan it is the
change in government debt (DMDEBT and JYDEBT respectively); for the U.K. it is the Public
Sector Borrowing Requirement.

The monthly current account for each country is obtained by taking the current account for
a given quarter, subtracting the balance of trade for that quarter to obtain the balance on services
and transfers, dividing by three to get a monthly estimate, and.adding to the balance of trade,
which is available monthly. The quarterly current accounts for all four countries come from IMF
IES, line 77a.d. The monthly balance of merchandise trade for each country is taken as the
difference between exports and imports (IES, lines 70 and 71).

The benchmarks for the accumulation of wealth are at year end 1972. They are very ad
hoc, since accurate data on the level of wealth are impossible to get, and the benchmark is
essentially only a constant term in the equation anyway. For the U.S. the wealth benchmark is
400 billion dollars, taken from Dooley and Isard (1979, p. 24), and ‘estimated from end-of-1972
stocks in Federal debt, monetary base, and net claims on foreigners. For Germany, the U.K.,
and Japan the wealth benchmarks are taken as the ratio of the dollar value of own GNP (from
IES) to U.S. GNP in 1983, times U.S. wealth at 1983:2 (which includes the cumulation of U.S.
current accounts and deficits between 1972:12 and 1983:2).

Consumption Shares

When residents of all countries are assumed to consume the same basket of goods, the
share of world consumption allocated to each country's goods is computed as its 1983 GNP
divided lsy the total GNP of the four countries in dollars. The shares are: U.S. 0.596, U.K.
0.079, Germany 0.111, and Japan 0.214. When residents of different countries are allowed to
consume different baskets of goods, we use the consumption shares computed and reported in
Frankel(1982, p. 272), which are derived from import and export to GNP ratios in 1973.

35


a = max[0,uS-K]

Cq= max([0,dS-K]

Now, in order to value the option, it is necessary to design an equivalent (replicating)
portfolio--a portfolio whose value behaves exactly as the call value does. Since by construction
the call option and the equivalent portfolio will have the same value in all possible future states of
nature, arbitrage will ensure that they have the same current value; consequently, the cost of the
equivalent portfolio establishes the current option value. Let i denote one plus the domestic
interest rate and i* one plus the foreign interest rate. Suppose we form a portfolio with AS
dollars invested in foreign bonds, where A is some number as yet undetermined. At the end of

one period our investment of AS dollars will take on one of two values in dollars:

uASi*

as
Nasi

That is, both the binomial process and the foreign interest rate are affecting its dollar value.
In order to build a replicating portfolio, it is necessary to combine this with an investment
of B dollars (as yet undetermined) in domestic bonds.!5 The end of period dollar value of this

portion of the portfolio will necessarily be Bi. Thus, at the end of one period the whole portfolio

TS ‘The intuition behind this is that the dollar value of the Portion of our replicating portfolio tha: is invested in
foreign bonds will be affected one-for-one by the exc age rate change. e value of a call option on
foreign exchange, on the other hand, will not be affected one-for-one by an exchange rate change, but rather by
some fraction oi the change. If we let this fraction be denoted by A, then the proportion of our rep! cating rtfolio
that is invested in domestic bonds (the value of which is unaffectecd by exchange rate movement) wil 1-A,
exactly matching the call option's degree of insensitivity to spot changes.

12


(17)

where

Pe ip - As - HPL = Egy

HPL = [(Qp- 0.5) 0754 + Qn Spee + Q py,]

as per equation (8) with no risk premium. The OLS results for the single equation un>iasedness

test appear in Table 1 below (weekly data). The regressors are the implied variances and

covariances as described below the table.

TABLE 1

i*, - ip - As, 7 - Hy = By + By*Xy + B.*X, + B3*X3 + & 1

Bo B1 B2 $3. F R2 pw
MARK -.001 -16.9 -54.1 133 03 07 1.77
(0.20) (0.49) (1.89) (1.91)
POUND .002 19.6 -62.7 139 03 08 2.05
(0.39) (0.28) (1.93) (1.05)
YEN .000 -69.3 -20.1 49.8 7 03 1.46
(0.06) (1.19) (0.40)

(1.80)

* T-statistics using White matrix in parentheses. "F" denotes marginal significance level of the
test that B)=8,=8,=0.

22



IFDP
NUMBER
285
284

283

282

281

280

279

278

277

276

275

274

273

International Finance Discussion Papers

TITLES

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

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

40

AUTHOR(s)

B. Dianne Pauls

Michael K. Gavin
Allen B. Frankel
Catherine L. Mann

Hali J. Edison

Kathryn M. Dominguez

Garry J. Schinasi

Jacques Melitz
Michael K. Gavin
Jaime Marquez

Caryl McNeilly

Neil R. Ericsson

Robert A. Johnson

Garry Schinasi

Andrew Feltenstein
David Lebow
Anne Sibert


the proposition is easily tested within the framework of the unbiasedness tests.

We then apply some "structure" to help interpret the unbiasedness results by applying a
portfolio balance model developed by Kouri(1976,1977) and Dornbusch(1983). This
"intermediate" step naturally leads into a test of the Kouri-Dornbusch model itself. While this
model has previously been tested by Frankel(1982), his test assumes a constant return
variance-covariance matrix. Our test replicates Frankel's except for the fact that it explicitly
includes a time-varying matrix. This is an important consideration; while there is a tradition in
the finance literature of assuming a constant variance-covariance matrix of stock returns, the
measured. expectations suggest that this assumption is far less justifiable when applied to the
foreign exchange market.

The paper is in six sections. Section 2 outlines the hypothesis of forward rate
unbiasediess as well as the Kouri-Dornbusch model, and discusses some previous empirical
results. Section 3 introduces the option pricing model used to reveal the second moment
expectations. Section 4 describes the numerical procedure used to extract the implied
expectations, and presents plots of the resulting series. Section 5 presents the results of the

various tests. Finally, conclusions are in section 6.


central banks as foreign exchange reserves. Millions of dollars, end of month. These data
are available on an annual basis from the IMF Annual Report, 1985 and 1986, Table 15.
Tae numbers are multiplied by the dollar/SDR rate to get the dollar value of official
holdings of U.S. dollars at the end of 1983:12, 1984:12, and 1985:12. The monthly
numbers are arrived at by linear interpolation.

Deutsche Mark Assets

DMASST = world supply of mark assets in millions of marks, end of month. Calculated as
DMDEBT + BBINT - NDMCB.

DMDEBT = debt of the German Federal Government, end of month. Source: Monthly Report,
Deutsche Bundesbank, Table VII 9.

BBINT = cumulative Bundesbank sales of mark assets for international reserves in exchange
market intervention, calculated as GRES - GADJ.

GRES =: net external position of the Bundesbank, valued in millions of marks, end of month.
Source: Monthly Report, Deutsche Bundesbank, Table IX 6(a).

GADJ =: ‘balancing item' to the Bundesbank's external position, and adjustment by the
Bundesbank every December to reflect capital gains on foreign exchange and other
Teserves. These items must be taken back out of GRES so that only changes in reserves
due to purchases or sales of mark assets are counted. Cumulated with a benchmark of zero
in 1983:2. Source: Monthly Report, Table IX, col. 7.

NDMCB = holdings of mark assets (regardless whether government securities) by foreign central
banks as foreign exchange reserves. These data are available on an annual basis from
the IMF Annual Report, 1985 and 1986, Table 15. The numbers are converted to marks
by using the dollar/SDR exchange rate (from IFS line sa) and the mark/dollar exchange
rate (from IFS line ae). Monthly numbers are obtained by linear interpolation.

Pound S‘erling Assets
PSASST = world supply of pound assets in millions of pounds, end of month. Calculated as

PSDEBT + BEINT - NPSCB.

PSDEBT = pound sterling debt of the British government. Computed as the cumulation of the
public sector borrowing requirement (Central Statistical Office Financial Statistics, Table
2.5, col. 14) on a 1983:3 benchmark of 127,727 million pounds (CSO Financial Statistics,
Table S4).

BEINT = cumulative Bank of England sales of pound assets for international reserves in

33


rate on an annual basis) are plotted on the following page together with a measure of the historical
mean squared change in the log of the exchange rate calculated from the previous month's daily

spot movements (20 business days) according to:

tl _
(15) RVAR,= © (Rj-R)2/18
jetl9
where Rj =In(S)) - In(S;_))
_ tl
and R= 2 R/9
jet-19

Since RVAR, in (15) is a measure of the variance of daily changes in the log of the exchange

rate, we use an assumption of independence to arrive at the variance on an annual basis by

multiplying by 262. This is the series that appears as the REALIZED VARIANCE in the plots.25

25 The horizon over which the REALIZED VARIANCE is calculated is chosen so as to generate similar volatility

in the two series; a longer horizon smooths the series while a shorter one increases the volatility. Econometric

results bearing on the question of which horizon best "explains" the implied variance series are not included in
is paper.

19


The significance of the coefficient 8 is even stronger evidence against the null hypothesis
that all information is orthogonal to the left-hand-side. Additionally, assuming risk averse
agents, the negative sign of B, suggests that investors perceive the existing share of pound-
denominated assets to be lower than that share which minimizes consumption risk. The
insignificance of 8, and 8, is consistent with the perception that asset shares for the mark and
yen are more closely in balance with the minimum risk shares. The implication for @ of the
magnitudes of the coefficients is puzzling, however. Since (X - Q) is the difference of two
fractions (shares), the magnitude of 8, suggests that investors are "extremely" averse to exchange
rate risk, i.e that 6 is much larger than is commonly believed.

The analysis above naturally leads us to a test of the Kouri-Dornbusch model itself. Once

we have a measure of the asset supply shares and the consumption shares, we can estimate

the single coefficient 6 in (8):

® ig = FB AS = Hy = 0, %-Qw) + ey

As previous results point out, however, given the correlations of the variables in ©, with the
left-hand-side, the estimated coefficient @ will be very sensitive both in sign and magnitude to the
sign and magnitude of the relatively stable (X, - Qw,). This said, we choose to measure the asset

shares as is done previously in the literature by Frankel(1982), so that the effect of changing
second monients can more easily be discerned. The world portfolio which he considers includes

only outside government debt27

T See the Data Appendix for definitions.

25


5, EMPIRICAL RESULTS

Abstracting from terms arising from Jensen's inequality, forward rate unbiasedness
implies that forward rate prediction errors have zero mean and are uncorrelated with any
information available to agents at the time the forward rate is determined. In subsection 2.1, we

defined the one-period forward rate prediction error as €, =f,1-Ste1 where f,, is the lo
P 41,1 = 11 ~ Stel tl 8

of the one-period forward rate set at time t and s,,, is the log of the realized spot rate at t+1.

As is common in the literature [e.g., Hsieh(1984)], we assume that covered interest parity

holds, which implies that: °

(16) i, - i - ASu1 = Sa

where i*, and i, are the foreign and domestic nominal interest rates, As,,; is equal to the
change in the log of the spot rate between t and t+1, and Sur is the prediction error realized at

t+1 (we will suppress the one-period subscript from now on). Notice that the left-hand-side is
the same as that which appears in the Kouri-Dornbusch model. That is, the test of the
Kouri-Dornbusch model is really a test of forward rate unbiasedness in which the exact form of
the bias--the risk premium--is specified. This fact will be helpful for interpreting the coefficients
in the unbiasedness tests.

First, we test to see if expected exchange rate variances and covariances are correlated with
the prediction errors. In light of the Krugman(1981) results, however, the prediction errors
should not be orthogonal to these variables due to the Jensen's inequality terms. Under the null
hypothesis of unbiasedness, the prediction errors netted of these terms are orthogonal to all

information. For example, in the case of the pound equation we have:

21


exchange market intervention. Computed by BEINT, = BEINT,_, + ABEINT, (U.K.

Balance for Official Financing; source: CSO Financial Statistics, Table 10.1, divided by
three to get monthly numbers), on a 1983:2 benchmark of total reserves minus gold in
dollars (from IMF IES, line 11.d) times the pound/dollar exchange rate (from IFS line ag).

NCSCB = holdings of pound assets (regardless whether government securities) by foreign
central banks as foreign exchange reserves. These data are available on an annual basis
from the IMF Annual Report, 1985 and 1986, Table 15. The numbers are converted into
pounds by using the pound/dollar exchange rate (from the JES, line ae) and the dollar/SDR
exchange rate. Monthly numbers are obtained by linear interpolation.

Japanese Yen Assets

JYASST = world supply of yen-denominated assets in millions of yen, end of month.
Calculated as SYDEBT + BJINT - NJYCB.

JYDEBT = yen-denominated debt of the Japanese government not held by the governrrent itself.
Computed as the Total Debt minus Total Government Holdings in millions of yen (source:
The Bank of Japan Economic Statistics Monthly, Table 82).

BJINT = cumulative Bank of Japan sales of yen assets for international reserves in exchange
market intervention. Computed by (1) the change in foreign exchange reserves corrected
for valuation changes, equal to the yen/dollar exchange rate (from IFS, line ae) times the
change in foreign exchange reserves expressed in dollars (from IFS line 1.dd) under the
assumption that most foreign exchange reserves are held as dollars; plus (2) the change in
the IMF position and holdings of SDRs corrected for valuation changes, equal to the
yen/dollar exchange rate times the difference between this period's IMF position and SDR
holding (from IES, lines 1.cd and 1.bd, respectively) and last period's, multiplied buy the
new over old dollar/SDR exchange rate; cumulated on a February 1983 benchmark of (3)
the yen value of total international reserves, equal to the yen/dollar exchange rate times
23,930 dollars (from IFS, line 1). There were no changes in gold stocks over the sample
period.

NJYCB = holdings of yen assets (regardless whether government securities) by foreign central
banks as foreign exchange reserves. These data are available on an annual basis from the
IMF Annual Report, 1985 and 1986, Table 15. The numbers are converted to yen by
using the yen/dollar exchange rate (from IES, line ae) and the dollar/SDR exchange rate.
Monthly numbers are obtained by linear interpolation.

34


4, THE IMPLIED SECOND MOMENT EXPECTATIONS

4.1 Inputs For The ‘ion Model

American spot currency options began trading on the Philadelphia Exchange (PHLX) in
early 1983. Beginning in September 1985, the Chicago Board Options Exchange (CBOE) began
trading European currency options. In order to generate a sufficiently long time series, we use
the PHLX option prices to recover the second moment expectations. Additionally, while the
PHLX now trades options on six major currencies, the data are too sparse on all but the three
most heavily traded currency options: the British Pound, the West German Mark, and the
Japanese Yen.

A very important consideration when attempting to back implied variances out of an option
pricing model is the synchronous data problem. A spot price that does not correspond to the
timing of the option trade will distort the variances if the spot rate differs from the rate prevailing
at the time of the trade. For this reason, we use a transactions data base which provides the
underlying spot rate at the time of each option trade (the PHLX-OSU Currency Options Data
Base, compiled with the support of the Philadelphia Stock Exchange by James N. Bodurtha Jr.).
The data base runs from February 28th, 1983 through June 27, 1985.

The weekly data used in the simple efficiency regressions covers a sample period from
July, 1983 through May, 1986, for a total of 151 observations (Wednesday closing prices). The
first four months of the transactions data are not included due to the thinness of trading; there
were eight trading days between March and June, 1983, on which no call option traded for at
least one of the three currencies, while this occurred only once after June, 1983. (The closing
trade of the previous day is used). In order to lengthen the sample beyond June 27, 1985 we use
the Philadelphia Exchange closing option and spot prices as reported in the New York Times (46
of the 151 observations). The spot rate that is printed in the Times is the Philadelphia Exchange

spot price at the close of the options market (2:30 Eastern Time). Since trading volume is quite

16


different asset-holding preferences. An implication of this is that redistributions of wealth, for
example via current account imbalances, will affect asset demands. When this change is added

to the model, equation (5) takes the form:

() i, - 138, - EAs) - HB, = 80,(X, - Qu)

where Q is defined to be a matrix whose five columns indicate the consumption preferences of
residents of the five countries (the fifth being the ROW), and Ww, is a vector of the five wealth
shares. Intuitively, the demand for a given country's asset depends positively not only on its

expected relative return, but also on the wealth of those investors who have a relatively greater

preference for that country's goods and thus for its assets.

Rational expectations implies that EAs,,, = As,,; + €,, 1, yielding the regression equation:

t+)?
(8) i, - 18, - Asya - Hy = 62, (% - Quy + &y,

This is the equation that Frankel(1982) estimates under the assumption of a constant variance-
covariance matrix Q,. The null hypothesis of his test is that the coefficient of relative risk
aversion is zero, i.e., that there exists no risk premium of the form predicted by the model. He is
unable to reject the null. In section 5 we replicate Frankel's test in the sense that we include the
same assets and use the same consumption shares that he does. However, this test differs in one
very important way in that it does not constrain the variance-covariance matrix to be constant.

This introduces a new, very volatile driving force as a complement to the much slower-moving

variable (X, - Qw,); the results should shed some light on their relative importance.


tion. That is, perhaps replacing expected depreciation with realized depreciation plus a random
error term is off the mark, especially during the sample period considered (1983 - 1986).
Working with survey data on exchange rate expectations, Frankel and Froot(1986) find evidence
that systematic prediction errors are playing a significant role. A possible remedy for this would
be to replace the realized depreciation with the median survey expectation on the left-hand-side.
The results in this case are likely to be quite different from the current results in light of the fact
that most all of the variability in the dependent variable is due to this component.

A final note concerns the more general implications of rapidly varying second moment
expectations. First, they call into question the reliability of the portfolio balance effects of
sterilized intervention. Feasible policy-induced changes in relative asset supplies are dwarfed by
autonomous movements in investors' second moment expectations. And second, theoretical and
empirical work which neglects the perceived nonstationarity of exchange rate second moments is

not consistent with the actual marketplace.

31


In the dollar-share-equals-one regression the coefficient of relative risk aversion now has the
correct sign. While this case is less plausible than the others, the result underscores the
sensitivity to the measurement of the difference (X, - Q,).

As a final measure to help identify the role of various factors in producing the estimates
above, we regress the left-hand-side on the asset supply share series alone in bilateral equations.
The theory predicts that the coefficients on asset supplies should be positive: a higher asset
supply requires a higher expected return to induce investors to hold them. Table 5 presents the

results of the OLS regressions:

TABLE 5

i, - ig ASp,1 - Hy = Bg + 8,*(ASSET SHARE), + €,,,

Bo Bi. R2 _ DW. DE

MARK 083 -1.37 Al 2.30 33
(1.96) (2.05)

POUND .096 -0.94 .05 2.09 33
(1.34) (1.35)

YEN 324 -1.39 21 1.96 33

(3.02) (2.99)

* T-statistics in parentheses.

- All of the coefficients on the asset supply shares are of the wrong sign, significantly so in the
case of the yen and mark. To the extent that the measured shares represent the relevant assets,
this is further evidence against the theory. Additionally, these results shed some light on why the

measured coefficients of risk aversion are typically negative.

28



market expectation, because of the presence of a risk premium. Thus, H2 is often called the hy-

pothesis of 'no risk premium' in the forward rate.
Note that H1 and H2 are totally independent hypotheses. Traders may have rational
expectations, but still require a risk premium for forward contracts. If they are risk averse, they

may (rationally) expect a loss, in order not to have to carry any exchange risk themselves.
Also, H1 and H2 are not separately testable, because M[ St4 1 | t ] cannot be observed.

However, they jointly imply:
(3) fer = Elsi 11]

which is often called the hypothesis of forward rate unbiasedness in the literature.

Atestable implication of (3) is the following. Define the forward rate prediction error

Fu

4) Sta = fr - Ste

Then the unbiasedness hypothesis implies that ul has zero mean and is uncorrelated with any
information in I(t).

Substantial empirical evidence has been built up against the unbiasedness hypothesis.
Typically, researchers have interpreted rejection of this joint hypothesis as evidence that a risk
premium does indeed exist. However, none of the strong rejections is due to factors that theory

links to the risk premium:? evidence for rejection is generally either (1) serial correlation of

2 ‘The theoretical work bearing on the forei gxchange tisk premium includes SolnikQo74). Adler and Dumas
(1976), Kouri(1977), Fama and Farber(1979). Stulz(1981, 1982, 1984), Hodrick(1981), Dornbusch(1983),
among others.


2. RISK PREMIUM THEORY AND PAST EMPIRICAL RESULTS

2.1 The Hypothesis of Forward Rate Unbiasedness

The following exposition follows that in Hsieh(1984). Let s, denote the natural logarithm
of the spot exchange rate at date t, and f, , the natural logarithm of the forward exchange rate
contracted at date t for delivery at date t+1. E[ s,,, | I(t) ] is the expectation of s,,) conditioned

on the information set I(t), while M[ s,,, | t] is the market's expectation of s,,; at time t.

The hypothesis of rational expectations is that the market's expectation is the true

expectation:

(l) ME S41 !t] = EL Syy1 HC) ]

We will refer to this hypothesis as H1.

A second hypothesis is required to relate the forward exchange rate to expectations. The
market is assumed to set the forward rate equal to the expected spot rate on delivery. That is, H2
maintains:

(2) fy,

1 = MES41!t]
If we abstract from terms arising from Jensen's inequality,! sufficient conditions yieldir.g H2 are
(a) that all traders are risk neutral and (b) that markets are competitive. However, if traders are

~ risk averse, then it is possible to obtain an equilibrium forward rate which is different from the

T The importance of the terms arising from Jensen's inequality relative to that of the risk premium itself is
investigated in Krugman(1981). We will return to this issue in subsection 2.2.


high through the later third of the sample, the 2:30 spot price is likely to be a very good
approximation to the spot rate prevailing at the time of the final option trade. The resulting series
do not appear to be affected by the change.

For each observation we choose the call option that is closest to being at-the-money, with a
time to maturity between three and six months. There are a number of justifications for focusing
on the at-the-money options.2° First, the Black-Scholes model has a tendency to work best for
options nearest the money. Second, the partial derivative of the call price with respect to the
future volatility is higher the closer the call is to being at-the-money, suggesting that the prices of
these options are more sensitively linked to the underlying volatility. And third, these options are
more heavily traded. The choice of the time to maturity window between three and six months is
motivated by a balancing of the increasing significance of the method's approximations at shorter
matu‘ities against the increasing thinness of trading at longer maturities.

Recall that the Binomial Currency Option Pricing Model establishes a foreign exchange

option's value as a numerical approximation using seven variables:

(14)' C=FS, K, T, i, i/i*, 6, n)

For each implied volatility calculation, S$ is taken to be the spot price of the corresponding
currency from the transactions data base (PHLX spot rate at option market close after July,
1985), K to be the exercise price of the option, t to be the number of calender days until the last
trading day prior to expiration, and the number of subintervals n_ to be 50 (see subsection 4.2

for more on the significance of n).

20 Stan Beckers(1981) has demonstrated that, with implicit variances computed from daily Closing option and
stock prices, using only the Implicit variance of the option nearest the money produces as good a prediction of

future variance as other more elal selection/weighting schemes.

17


choice problem over the vector of portfolio shares generates the equilibrium relationship:”

(5) i, - 138, - EAs, - pb, = 02,(X,- Q)

) 1,12, Q, - 0/2]
where: i’, = [ iTM, iP, i%, ] isa vector of nominal returns on the bonds denominated in marks,
pounds, and yen;8 ! is acolumn vector of three ones; E is the expectation operator; A denotes

“change in"; s, is a three-element vector of the logs of the exchange rates; i, are the terms
arising from Jensen's inequality; @ is a coefficient of relative risk aversion; Q, is the
variance-covariance matrix of currency depreciation; X, is the three-element vector of asset

supply shares; and Q, is the three-element vector of consumption shares across the currencies.
Equation (5) specifies the risk premium (the right-hand-side) as a function of three differ-

ent factors: the coefficient of relative risk aversion 0, the exchange rate variance-covariance

matrix !2,, and the difference between the vector of asset supply shares and the vector of

consumption shares. The formulation is based upon the case in which the consumption bundle is
the same for all investors, i.e., the components of Q are the same across countries. In the case

where ir-vestors in different countries have different consumption patterns, they will also have

7 The result presented in equation (5) requires the additional assumption that goods prices are nonstochastic
when ex, essed 1 in the currency of the producing count That is, the only uncertainty is exchange rate uncer-
tainty. This assumption is made by Krugman 198 1), but is considered only one special case by
Doribusch(1983), Frankel(1982) makes this assumption in his tests of the model.

8 The croice of currencies for the empirical analysis is constrained by data availability; only the mark, pound,
and yen rate options provide sufficient observations.


Mark equation: X1 = variance of log changes in the dollar/mark rate (weekly basis)
X2 = covariance of log changes in the mark rate and pound rate
X3 = covariance of log changes in the mark rate and yen rate

Pound equation: X1 = covariance of log changes in the pound rate and mark rate
X2 = variance of log changes in the pound rate
X3 = covariance of log changes in the pound rate and yen rate

Yen equation: X1 = covariance of log changes in the yen rate and mark rate

X2 = covariance of log changes in the yen rate and pound rate
X3 = variance of log changes in the yen rate

These regressions are run on the full sample period from July, 1983 through May, 1986 (151
weekly observations). The interest rates are 7-day Eurocurrency deposit rates expressed on a
weekly basis. We generate the covariance terms by assuming that the correlations are constant,

and then using the relation Cov(X,Y) = px y 0x Oy where p is taken to be the sample correla-

tions, and the standard deviations are the square roots of the implied variances.26

The results provide important evidence on two fronts. First, the results of the F-tests,
together with the results for individual coefficients, provide substantial evidence against the joint
unbiasedness hypothesis, particularly in the cases of the mark and pound. While this evidence
for rejection is not as strong as that which appears elsewhere in the literature, it is qualitatively
very different in that it is due to variables which theory links to the risk premium. In this sense,
it is better evidence for the alternative hypothesis that a risk premium does indeed exist, as
opposed to the alternative of a violation of rational expectations.

The second point to be gleaned from Table 1 concerns the relative magnitude of the

Jensen's inequality terms and the risk premium. Krugman(1981) demonstrates that in theory

they are approximately the same magnitude. However, the theoretical coefficients on Xj, Xo,

and X3 arising solely from Jensen's inequality are all fractions less than one, per equation (17).

26 The sample correlations are: pm = 0-736, Ppy = 0-378, and Pmy = 0.550.

23


International Finance Discussion Papers
Number 290
December 1986

‘TESTS OF THE FOREIGN EXCHANGE RISK PREMIUM USING THE
EXPECTED SECOND MOMENTS IMPLIED BY OPTION PRICING

by

Richard K. Lyons

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.


exchange call: ul

(9) C=SN(X+ovjeMt - KN&®eit

where

In(S/K) + [i-i*-(672)]t

tal
mM

ovr

C = the market value of the call option

S = the spot exchange rate

N()= the cumulative normal distribution function
K = the exercise price

i= the domestic (riskless) interest rate

i*= the foreign (riskless) interest rate

t =the length of time until the option expires

© = the expected standard deviation of the proportional rate of change of the stock price.

Unfortunately, the currency options that have been available in this country until very
recently have been American options, !2 which permit exercise at any time up to and including
the expiration date. One could use the European valuation formula to solve for the implied
standard deviations as is done in Shastri and Tandon(1986b), but this approximation can be quite
rough for call options that are likely to be exercised early. In order to generate a more precise
measure from the American options, this paper uses a more general model from which the
Black-Scholes model can be derived as a limiting case: the Binomial Option Pricing Method.
(The model first appeared in the literature in Cox, Ross, and Rubinstein(1979); the adaptation to

. currency options appears in Cox(1985).) This method allows one to both (1) modify the

11 This model is identical to the one developed by Merton(1973) for equity options with the underlying security
paying continuous dividends. In the case of foreign currency options, the foreign interest rate is ‘akin to the
continuous dividend rate.

12 The Chicago Board Options Exchange began trading European foreign exchange options in late 1985.

10


3. THE OPTION VALUATION MODEL

The basis for analysis of the value of options was provided by Black and Scholes(1973),
who derived a closed-form equilibrium valuation formula for a European call option on a stock

that does not pay dividends.? In general terms, the formula can be written as:

C=F(S, K, t, i, 0)
where
C=the market value of the call option
S=the price of the stock on which the option is written
K=the exercise or strike price

t=the: length of time until the option expires
i=the riskless interest rate

o=the expected standard deviation of the proportional rate of change
of the stock price.

All variables appearing in this equation are directly observable except for 6, the market's
collective i:xpectation of the volatility. Thus, given the ex-post market determined value for C
(together with S, K ,t, and i ), one can use the model to solve for the implied volatility o.10

Valuing foreign currency options requires a slightly different model because the underlying
‘asset is an amount of foreign currency, not a share of stock. Biger and Hull(1983), Garman and

Kohlhagen(1983), and Grabbe(1983) developed the model for valuing a European foreign

9 A call option on a given stock gives the owner the right to purchase one share at a predetermined price--the
exercise nee. A Eurcpean i n ean be exercised onl Sn the expiration date, while an’ American option can be
exercised at any time up to and including the expiration date.

10 Examples of work on implied volatility include Schmalensee and Trippi(1978), Beckers(1981), and Shastri and
Tandon(1986b).


6. INCLUSION:

The foreign currency option market provides a unique method for measuring ex-ante
variance expectations: beliefs embedded in option prices are rooted in actual market beliavior and
reflect the "true" information set available to investors. Moreover, the model used to recover the
implied expectations is preference-free, relying wholly upon arbitrage pricing methods.28 The
result is a series which corresponds quite closely to actual variances, and provides some insight
into how far back investors look in forming their expectations.

Overall, the results of the tests support one conclusion above all others: changes in the
market's second moment expectations are systematically related to expected return differentials.
The joint unbiasedness hypothesis is rejected in a number of different tests. Rejection of this
joint hypothesis is not new, however. What is new is that the rejections are due to variables
which theory ties to the risk premium. In this sense they provide much better evidence in favor
of the alternative hypothesis that a risk premium does indeed exist, as opposed to the alternative
of a violation of rational expectations.

The test of the Kouri-Dornbusch model of the risk premium finds little support ia the data.
The absolute size of the measured coefficients of relative risk aversion are broadly consistent with
the magnitudes of the coefficients in the unbiasedness regressions, but the sign is typically
wrong. There are at least two explanations for this result which are consistent with the validity of
the theory. First, as explained in section 5, the sign and magnitude of the measured coefficient
are very sensitive to the asset and consumption share definitions, given the correlation between
the elements of the variance-covariance matrix and the dependent variable (as demonstrated in the
unbiasedness tests). This suggests the possibility of experimenting with broader asset definitions

than those used in Frankel(1982). A second explanation works from the other side of the equa-

28" This method of measuring ex-ante investor uncertainty lends itself to a number of different applications.
The author is currently using the method to investigate time-varying risk premia in the term structure of interest rates.

30


In addition to the above analysis, the volatility of expectations in itself has potential policy
implications, e.g., for the reliability of the portfolio balance effects of sterilized intervention.

Consider for illustration a two-country model of the risk premium:

(23) i*, - i, - EAsy, = 00% (X, - Qu)

A common view is that sterilized intervention (instrument) affects the exchange rate (target)
according to the above relationship with a stable, if not constant, exchange rate variance.
However, the pronounced volatility of market's second moment expectation seriously

undermines this view. The extreme noise introduced would swamp any policy induced variation
in X, due to sterilized intervention, with the sign of the effects of changes in o; depending on
the sign of (X,- Qw,). Additionally, to the extent that current account imbalances affect

exchange rates through the above relationship (by reallocating w), their role as a driving force

pales in comparison to the role of oO; .

29


forward rate prediction errors? or (2) correlation of forward rate prediction errors with known
information that is not linked theoretically to the risk premium.4 Consequently, past rejections of
unbiasedness do not offer clear evidence of a risk premium, per se.

In section 5 we test whether there exists a significant correlation between expected
variances/covariances and forward rate prediction errors. This test differs in an important way
from previous tests because theory establishes a relation between second moment changes and
the risk premium (outlined below). A rejection, then, would provide stronger evidence than

previous tests for the alternative hypothesis that a risk premium exists, as opposed to the

alternative of a violation of rational expectations.

2.2 AP lio Balance Model of the Risk Premium
The Kouri(1977) - Dornbusch(1983) model of the risk premium makes explicit a relation
between the risk premium and the second moments of exchange rates. Because the derivation of
the model appears elsewhere in the literature,© we present only a very brief description ‘tere.
Investors are assumed to maximize a function of the mean and variance of end-of-period

real wealth, where bonds of different currency denominations are the only available assets. The

3 Examples include Geweke and Feige(1979), Hansen and Hodrick(1980), Cumby and Obstfeld(1981), and
Hsieh(1984).

4 Hsieh(1984) provides the strongest evidence along these lines. His tests find a significant correlation between
forward rate prediction errors and both holding period yields and forward discounts.

5 Evidence is now appearing which points to the role of expectational errors. For example, wcrking with
survey data on exchange rate expectations, Frankel and Froot(1986) suggest that systematic prediction errors
are significant in explaining forward discount bias.

6 See Frankel(1982) for a clear exposition.


VARIANCES OF ANNUAL LOG SPOT RATE CHANGES
— — 0.12
= IMPLIED BP VARIANCE
eketeetetentel = REALIZED BP VARIANCE — 0.090
4 0.060
—| 0.030
ttt,
_ 0.060
| = IMPLIED DM VARIANCE 4 0.080
————-—-—-= REALIZED DM VARIANCE 4
i 0.040
{!
0.030
0.020
0.010
oO
_ — 0.050
= IMPLIED JY VARIANCE
------- = REALIZED JY VARIANCE 7 9.040
\
| M | 0.030
I \ h
In ln 4 0.020
hj
‘ / " 4 0.010
Ke (Vary 1 tay? ar ae aw) \/
| ~~ Pitirirritribirtitty
1984 1985

20


Garman, Mark B. and Steven W. Kohlhagen (1983), "Foreign Currency Option Values,"
Journal of International Money and Finance, 2, 231-237.

Geweke, J. and E. L. Feige (1979), "Some Joint Tests of the Efficiency of Markets For Forward
Foreign Exchange," Review of Economics and Statistics, 61, 334-341.

Giovannini, Alberto and Philippe Jorion(1985), "Interest Rates and Risk Premia in the Stock
Market and in the Foreign Exchange Market," Unpublished manuscript.

Grabbe, Orlin J. (1983), "The Pricing of Call and Put Options on Foreign Exchange," Journal of
‘International Money and Finance, 2, 239-253.

Hansen, L. and R. Hodrick (1980),:"Forward Exchange Rates as Optimal Predictors of Future
‘Spot Rates: An Econometric Analysis," Journal of Political Economy, 88, 829-853.

Hodrick, R. J. (1981), "Intertemporal Asset Pricing With Time-Varying Risk Premia," Journal
of International Economics, 11, 573-587.

Hodrick, R. J. and S. Srivastava (1984), "An Investigation of Risk and Return in Forward
Foreign Exchange," Journal of International Money and Finance, 3, 5-29.

Hsieh, David A. (1984), "Tests of Rational Expectations And No Risk Premium in Forward
‘Exchange Markets," Journal of International Economics, 17, 173-184.

Kouri, P. (1976), "The Determinants of the Forward Premium," Institute For International
Economic Studies Seminar Paper No. 62, University of Stockholm.

Kouri, P. (1977), "International Investment and Interest Rate Linkages Under Flexible Exchange
Rates," in R. Aliber, ed., The Political Economy of Monetary Reform, London:
Macmillan Press.

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

Latane, Henry A. and Richard J. Rendleman, Jr. (1976), "Standard Deviations of Stock Price
Ratios Implied in Option Prices," Journal of Finance, 31, 369-382.

Manaster, Steven and Gary Koehler (1982), "The Calculation of Implied Variances from the
-Black-Scholes Model: A Note," Journal of Finance, 37, 227-230.

Merton, Robert C. (1973), "Theory of Rational Option Pricing," Bell Journal of Economics and
Management Science, 4, 141-183.

Patell, James M. and Mark A.Wolfson (1979), "Anticipated Information Releases Reflected in
Call Option Prices," Journal of Accounting and Economics, 1, 117-140.

Rahman, Abdul and Lawrence Kryzanowski (1986), "Alternative Specifications of the Errors
in the Black-Scholes Option Pricing Model and Various Implied-Variance Formulas,"
Economics Letters, 21, 61-65.

Rogoff, Kenneth (1984), "On The Effects of Sterilized Intervention: An Analysis of Weekly
Data," Journal of Monetary Economics, 14, 133-150.

37


Schmalensee, Richard and Robert R. Trippi (1978), "Common Stock Volatility Expectations
Implied by Option Premia," Journal of Finance, 33, 129-147.

Shastri, K. and K. Tandon (1986a), "On the Use of European Models to Price American Options
on Foreign Currency," Journal of Futures Markets, 6, 93-108.

Shastri, K. and K. Tandon (1986b), "Valuation of Foreign Currency Options: Some Empirical
Tests," Journal of Financial and Quantitative Analysis, 21, 145-160.

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

Stulz, R. M. (1981), "A Model of International Asset Pricing," Journal of Financial Economics,
9, 383-406.

Stulz, R. M. (1982), "The Forward Exchange Rate and Macroeconomics," Journal of
International Economics, 12, 285-299.

Stulz, R. M. (1984), "Currency Preferences, Purchasing Power Risks, and the Deterraination of

Exchange Rates in an Optimizing Model," Journal of Money, Credit, and Benking, 16,
302-316.

38


Frankel estimates @ under the strong assumption of a constant variance-covariance matrix
Q,. The null hypothesis of the test is that the coefficient of relative risk aversion is zero, i.e., that

there is no portfolio balance risk premium of the form the model predicts. He is unab-e to reject.
The test below also tests for the null hypothesis of a zero coefficient of risk aversion. However,
this test permits the covariance matrix to vary. The elements of each month's covariance matrix
are determined by the implied expected variances from the options market (monthly basis), with
the covariances determined as above using the assumption of constant correlations.

The results of the OLS regressions are presented in Table 3. They cover monthly
observations from February, 1983 through December, 1985, for a total of 35 months. The
EQUAL CSN SHARES regression corresponds to the case in which all countries consume the
same basket of goods (see appendix); the interest rates are those for one month Eurocurrency

deposits (monthly basis); the spot rate is that used to generate the implied expected variances;

the coefficient 8, corresponds to the coefficient of relative risk aversion:

TABLE 3

Ie > p> ASpy My = By + By "2, (Ky-Qvy) + Ey

Bo Bi R2 pw

EQUAL CSN SHARES -.006 -241 07 2.09
(1.38) (2.74)

CSN SHARES VARY .007 -172 02 2.10
(1.22) (1.49)

* T-statistics in parentheses

26



commonly tested in the literature to determine whether they provide any new insights.

Using the options market to measure time-varying expectations offers two key benefits.
First, this measure of the second moment expectation is rooted in actual market behavior. It
therefore reflects the basis on which investors are assessing risk-return tradeoffs, ard is properly
forward-looking. Second, it avoids the difficult question of whether to measure variance as
dispersion about a conditional mean or a sample mean. Past work on asset market volatility often
measures time-varying second moments using either moving variance about moving mean
(MVAMM) estimators, or the ARCH model developed by Engle(1982). The option market
approach differs from these in the sense that it implicitly measures the market's method of
conditioning.

We first use the implied expectations to test the hypothesis of unbiasedness of forward
exchange rates (which implies jointly that traders have rational expectations and that there is no
foreign exchange risk premium). We do so by testing for significant correlation between
expected variances/covariances and net-of-depreciation interest differentials. The results of these
tests allow us to answer two key questions. First, can we reject unbiasedness using a variable
upon which risk premia theoretically depend? Although the joint unbiasedness hypothesis has
been strongly rejected in the literature [e. g., Hsieh(1984)], the rejections are not due to factors
that theory links to the risk premium. Consequently, they provide little evidence for the
alternative hypothesis that a risk premium does indeed exist. Because theory establishes a
relation between second moment changes and the risk premium, this test provides a stronger
basis than previous tests for interpreting rejection of the joint hypothesis as evidence for the
existence of a risk premium, as opposed to a violation of rational expectations.

The second key question the unbiasedness tests will answer is: Are the Jenser's inequality
terms which arise from the maximization of expected real returns in fact approximately the same
magnitude as the risk premium itself? This proposition is asserted theoretically by

Krugman(1981). Since these terms are functions of exchange rate second moments in his model,

2


will take on one of the two following dollar values:

a AuSi*+Bi
ASB
AdSi*+Bi

Since A and B can be set to any value one chooses, one can choose those values such that the

two possible end-of-period portfolio values exactly match the end-of-period call values C, and

Cg. This provides an equivalent portfolio, and if there are to be no riskless arbitrage

opportunities, the current value of the call, C, must equal the cost of the equivalent portfolio.

Settirig these values equal to each other yields:
(10) AuSi*+Bi=C,
qy) AdSi*+Bi=Cy
Equations (10) and (11) allow us to solve for the two values for A and B that generate

the replicating portfolio. After a little algebra:

Cu - Cd uCd - dCu
A=-—_——ee— and B = ———
Si* (u-d) i(u-d)

Since the value of the call must be the same as the value of the equivalent portfolio, we can

simply plug in these values for A and B:

13


the valuation process. We can work leftward through the tree to solve for values of C, and Cg

given our pricing relation in equation (14) (together with values of i and i*). Once C, and Cg are

established, one more step leftward using the pricing relation establishes the current call value, C.
Thus, although this is not a closed form solution as in the Black-Scholes case, in general terms

we have the formula:

(14)' C=K(S, K, t,i, i A*, 6, n)

If the assumptions underlying the model are correct, as the number of subintervals n approaches
infinity, the call value calculated should converge to the true value.18

The benefit of the binomial procedure is that it allows, at each node of the tree, for a
comparison of the value if exercised immediately with the value if kept alive (unexercised). At
those nodes where immediate exercise is optimal, the exercise value is substituted for the value
calculated using the pricing relation in equation (14). This ensures that full consideration is given

to the impact of the early exercise feature of an American option. !9

18 The assumptions underlying the model are: (1) the spot rate is log-normally distributed with constant variance,
(2) the risk-free interest rates are known and constant, and (3) capital markets ar¢ frictionless.

19 On a stock that pays no dividends, the values of American and European call options are identical; since no
“rents” (dividends) are being forgone by not exercising an American call, it will always be worth more alive
than exercised. However, in the case of foreign exchange options, rents are being forgone: the interest rate
applying to the foreign currency if the option were exercised. Thus, it's the differentials between domestic and
foreign interest rates that make the optimality of early exercise possible.

15


Tests of the Foreign Exchange Risk Premium Using the
Expected Second Moments Implied by Option Pricing

by
Richard K. Lyons*

1. INTRODUCTION

The: search for the foreign exchange risk premium and the variables which drive it has
produced a vast empirical literature, with very limited success. However, there is reason to be
cautious when interpreting many of the results, for they are generally based on an extremely
tenuous assumption: the tests generally assume that conditional second moments are constant
[e.g., Frankel(1982) and Hodrick and Srivastava(1984)]. Recent work by Cumby and
Obstfeld(1985) and Giovannini and Jorion(1985) strongly‘ rejects the hypothesis that the
conditional variance-covariance matrix of rates of return is constant. Not only do these results
suggest that many of the previous tests involve serious misspecification errors, they also have
important policy implications (e.g., for the reliability of the portfolio balance effects of sterilized
intervention).

This paper applies a new method for investigating the risk premium which explicitly
recognizes this previously neglected dimension of the risk-return tradeoff. The approach exploits
information revealed in the foreign currency options market--a market whose principal objective

. is the prediction of future exchange rate second moments. Using a currency option pricing
model, toether with realized market option prices, we solve for the market's conditional second

moment expectation. Then, we use these time-varying expectations to test some hypotheses

* The author was a summer intern in the International Finance Division. 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. Thanks go to John Cox,
Rudiger Dornbusch, Stanley Fischer, Paul Krugman, and participants in the International
Finance seminar at the Federal Reserve Board.


REFERENCES

Adler, M., and B. Dumas (1976), "Portfolio Choice and the Demand for Forward Exchange,”
American Economic Review Proceedings, 66, 332-339.

Beckers, Stan (1981), "Standard Deviations Implied in Option Prices as Predictors of Future
Stock Price Variability," Journal of Banking and Finance, 5, 363-382.

Biger, N. and J. Hull (1983), "The Valuation of Currency Options," Financial Management, 12,

Black, Fischer and Myron Scholes (1973), "The Pricing of Options and Corporate Liabilities,"
Journal of Political Economy, 81, 637-659.

Bodurtha, James N. Jr. and Georges R. Courtadon (1986), "Efficiency Tests of the Foreign
Currency Options Market," The Journal of Finance, 41, 151-162.

Cox, John C. (1985), Lecture Notes, unpublished.

Cox, John C., Stephen A. Ross, and Mark Rubinstein (1979), "Option Pricing: A Simplified
Approach," Journal of Financial Economics, 7, 229-263.

Cox, John C. and Mark Rubinstein (1985), Options Markets, Prentice Hall:Englewood Cliffs.

Cumby, R. and M. Obstfeld (1981), "Exchange Rate Expectations and Nominal Interest
Differentials: A Test of the Fisher Hypothesis." Journal of Finance, 697-703.

Domowitz, I. and C. S. Hakkio (1985), "Conditional Variance and the Risk Premium in the
Foreign Exchange Market," Journal of International Economics, 19, 47-66.

Dornbusch, Rudiger (1983), "Exchange Rate Risk and the Macroeconomics of Exchenge Rate
Determination." in R. Hawkins et. al. (eds), Research in International Business and
Finance, Vol 3, JAI Press, 3-27.

Engle, R. F. (1982), "Autoregressive Conditional Heteroscedasticity with Estimates of the
Variance of United Kingdom Inflation," Econometrica, 50, 987-1007.

Fama, Eugene F. and Andre Farber (1979), "Money, Bonds, and Foreign Exchange,"
American Economic Review, 69, 639-649.

Frankel, Jeffrey (1982), "In Search of the Exchange Risk Premium: A Six Currency Test
Assuming Mean- Variance Optimization," Journal of International Money and Finance, 1,
55-274.

Frankel, Jeffrey and Charles M. Engel (1984), "Do Asset Demand Functions Optimize Over the
Mean and Variance of Real Returns? A Six-Currency Test," Journal of International
Economics, 17, 309-323.

Frankel, Jeffrey and Kenneth Froot (1986), "Interpreting Tests of Forward Discount Bias
Using Survey Data on Exchange Rate Expectations," Unpublished manuscript.

36


The results in Table 1 demonstrate that the coefficients 8,, 8, and 8, (which are net of the

theoretical effects from Jensen's inequality due to subtracting 1, from the left-hand-side) are

generally much larger than the fractional coefficients imputed to Jensen's inequality.
Consider now an interpretation of the coefficients which derives from the portfolio balance
model of Kouri and Dornbusch. The model, described by equation (8), specifies the risk

premium (RP) on currency i as:
RP; =0.Q), (X,-Q) = OK, - Qt + OX), - QF + OK - QF

th row of the variance-covariance

where @ is the coefficient of relative risk aversion, Q; is the i
matrix of exchange rate changes, X, is the vector of asset supply shares denominated in the

different currencies, and Q, is the vector of consumption shares in the different currencies. If one

assumes that the difference between the asset supply shares and the consumption shares (X, - Q,)
is constant, then each of the coefficients in Table 1 should be the same across the three equa-
tions. That is, 8,=@ & - Q)), 8, =9 (X,,-Q,,), and 8,=6 (Ky - Qy). Results of

OLS estimation of the pooled regression are reported in Table 2 below.

TABLE 2

i, > iy ASy.y - Hy = Bo + By*Xy + By*X, + B3*X3 +e)

Bo Bi B2 $3. R2 DW.

- POOLED 005 9.92 -22.6 -19.2 03 1.84
(2.24) (0.99) (2.70) (1.69)

* T-statistics using White matrix in parentheses.

24



ABSTRACT

This paper applies a new method to investigate the foreign exchange risk premium. The
method is new in the sense that it utilizes the time-varying second moment expectations implied
by foreign currency option pricing. The vast empirical literature on the risk premium generally
neglects the role of time-varying second moments, in spite of their importance in assessing
risk-return tradeoffs. In fact, this importance is borne out in the data: time-varying expectations
generate valuable new evidence regarding both unbiasedness in the forward rate and portfolio
balance models. Moreover, the results suggest that previous tests which assume constant second
moments involve serious misspecification errors. The results also highlight the unreliability of
the portfolio balance effects of sterilized intervention, in spite of the quantitative importance of

expected return differentials.


DATA APPENDIX

Rates of Retum
The rates of return used for the regressions reported are the seven-day and one month

Eurocurrency interest rates as reported in the London Financial Times. Wednesday closing rates
are used for the weekly data, while rates for the last business day of the month are used for the
monthly data. The depreciation of the spot rate is measured as the change in the logarithm of the
spot rate that is used to calculate the implied option volatility each week/month (Philadelphia
Exchange). This corresponds to the transactions data described in the text from July, 1983
through June, 1985, and to the Philadelphia option market closing spot reported in the New York
Times for July, 1985 through May, 1986.

US. Dollar Assets

DOASST = world supply of dollar assets in billions of dollars, end of month. Calculated as
DODEBT + FEDINT - NDOLCB.

DODEBT = gross public debt of the U.S. Treasury and other U.S. government: agencies,
excluding that held by U.S. government agencies and trust funds, i.e. held by the Federal
Reserve, private domestic investors, and foreigners, at end of month (source: Treasury
Bulletin, Table FD-1); minus two issues of 'Carter Bonds,’ which are denominated in
foreign currency: $1595.2 million dating from December 1978 and another $1,351.5
million from March 1979 (source: Federal Reserve press release, June 1979).

FEDINT = dollars supplied by the Fed in cumulative foreign exchange intervention. computed
by FEDINT, = FEDINT,_, + AFEDINT,, on a benchmark of the dollar value of all U.S.

international reserve assets (gold, foreign exchange, SDRs, and IMF position) in
February, 1983 (source: Federal Reserve Bulletin, Table 3.12).

AFEDINT = intervention, equal to increases in reserves, corrected for valuation changes.
Computed as the change in gold holdings (there were no valuation changes during the
sample period), plus the change in foreign exchange holdings in dollars minus valuation
change (last period's foreign exchange times the change in the dollar/mark rate; most of
the holdings have been in marks), plus change in SDRs and IMF position in dollars minus
valuation change (last period's SDRs and IMF position times the change in the dollar/SDR
tate). (Source for reserve holding: Federal Reserve Bulletin, Table 3.12. Source for
dollar/SDR rates: IMF Internation] Financial Statistics, line sa).

NDOLCB = holding of dollar assets (regardless whether government securities) by foreign

32


The results are both discouraging and interesting. The coefficient of relative risk aversion

(8) is both negative and much larger in absolute value than one would expect. Thus, the test

results provide no support for the model. The existence of a "pole" at the point (X, - Qw,) = 0 is
a possible explanation of the wrong sign, however. That is, given the demonstrated relationship
between the relatively volatile Q, and the left-hand-side, measuring the relatively stable (X, - Q,)

with a sign that is opposite that perceived by the market would generate a negative coefficient
rather than a positive one if the model were in fact true. The extremely significant coefficient in
the equzl-consumption-shares case suggests that a relationship exists. At the very least, it
constitutes a very strong rejection of the unbiasedness hypothesis.

In order to gauge the sensitivity of the results to the consumption shares, we examine two
alternative cases: (1) investors measure consumption risk against dollar-denominated
consumption, so that the dollar share in consumption is one and (2) investors measure
consumption risk wholly against home-denominated consumption, so that the home share equals

one. Table 4 presents the results of the OLS regressions:

TABLE 4

i, - ig- ASg. 1 - Hy = By + By*Q, (K,-Q) + Ey

Bo Bi R2 DW.

DOLLAR SHARE = 1 -.015 44.6 03 2.19
(1.69) (1.87)

HOME SHARE = 1 .009 -209 02 2.10
(1.23) (1.44)

* T-statistics in parentheses

27



valuation procedure to take into account the different nature of the underlying asset and (2)
properly include the value of early exercise. Drawing from Cox et al(1979), we begin by

making three assumptions:!3

(1) the exchange rate follows a multiplicative binomial process over discrete periods
(2) the nominal interest rate is constant and positive and

(3) there are no taxes, transaction costs, or margin requirements.

Consider the case in which only one period remains until expiration. Let S now denote the
current spot exchange rate in dollars per unit of foreign currency. After one period, the ex-
change rate will be either uS or dS where u is the factor generated by the binomial process if
the excliange rate moves upward and d the factor if it moves downward. Diagramatically:
ss
Ys

Letting C denote the current value of a call, C,, its value at the end of the period if the exchange

s

Tate gors to uS (which represents a depreciation--i.e., more dollars to purchase a unit of foreign

currency), and Cy its value at the end of the period if the spot rate goes to dS, we can diagram

the call value at expiration as:!4

13 ‘The final version of the model assumes that the exchange, rate is distributed lognormally; this is the limiting
distribution of the multiplicative binomial process we start with here.

14 A call value can never be negative, since exercise is always the option of the owner. If uS-K were negative,
the opticn would expire unexercised.

11


(12) C= AS+B
(Cu - Ca) S uCa- dCu
(13) = —— + ——
Si* u-d) i@-d)
i i
(i#-d)Cy + (u- i*) Cg
(14) =

i(u - d)

Equation (14) establishes the one period pricing relation given values for u. d, i, i*, Cy,

and Cg, all of which are known at the current time.!6 If we now consider a case in which the

number of periods to expiration (henceforth denoted n) is greater than one, the logic behind the

valuation of a multiperiod call begins to assert itself. For the case of n=2 we have:

Cyy=max(0,uuS-K]
ae
Cc —— c _Cau=maxl0,duS-K]

da
~c yg=max{0,ddS-K]

The values of u ,d ,S, and K (known at the current time) establish the three possible call values

of Cyyys Cqy» and Cyg that could be realized at the date of expiration.!7 This is the starting point in

16 It can be shown that the choice of u=exp(oVi/n) and d=1/u provide the final model with the proper limiting

properties, where o is the (annualized) standard deviation of the proportional changes in the spot rate; t is the time to
expiration in years, and n is the number of subintervals used in the approximation. See Cox and Rubinstein (1985)

Yr At this point we are still assuming that the option cannot be exercised until the expiration date.

14