International Asset Allocation with Time Varying Risk:

An Analysis and Implementation

by

Robert Cumby *

Stephen Figlewski **

Joel Hasbrouck ***

March 1993

* Professor of Economics** Professor of Finance*** Associate Professor of Finance

Leonard N. Stern School of BusinessNew York University44 West 4th Street, Suite 9-160New York, NY 10012-1126

We would like to thank the Okasan Research Institute for funding the research and for providing data used in the analysis. We also thank Ryuzo Sato for help in setting up the project, the Interactive Data Corporation for providing additional data, Anya Khanthavit for

able and patient research assistance, and the two anonymous referees for their helpful comments.

Abstract

This paper investigates the problem of optimally allocating funds in an investment portfolio among major classes of U.S. and Japanese assets, given that the asset risk parameters vary over time. We devise an econometric specification in the ARCH/GARCH family to model the evolution of the returns covariance matrix.

We find substantial time variation in both returns variances and correlations for the asset classes we examine. To limit computational problems in what can easily become a very difficult and time consuming estimation, we fit a set of univariate models--EGARCH models for variances and modified GARCH models for correlations--and then combine them into a full covariance matrix. This procedure, which we refer to as the E/G model, offers a reasonable compromise between generality and estimability for a multivariate system.

The fitted covariance matrices are then used to analyze optimal asset allocation portfolios. The time patterns in the E/G portfolio weights and risk assessment make sense: They show the change in the character of the U.S. Treasury bond market in 1979 and the increase in stock market risk in October 1987, for example. However, while the E/G models capture some of the time variation in asset risk parameters, the improvement in portfolio performance is limited.

We also examine investment performance under different portfolio constraints, and find that allowing international diversification makes the biggest impact, especially for a U.S. investor. Prohibiting short sales and the hedging of currency risk do not seem to affect a Japanese investor or a U.S. investor following a high expected return strategy in this particular case; they do make a small difference in the U.S. Minimum Risk strategy.

Post sample performance of the E/G model deteriorates relative to the Fixed Variance model. This suggests that efforts to optimize the E/G approach for use in a forecasting mode may be worthwhile.

I. Introduction

In this paper we consider how observed data can be used to construct portfolios that take

advantage of the gains from international diversification. In constructing portfolios that are

mean-variance efficient, i.e., optimal in that they have minimum variance for a given

expected rate of return, we need an estimate of the returns covariance matrix as well as the

mean rates of return on the available securities. The key innovation of our procedure is that

we model and estimate a time-varying covariance matrix.

The assets we consider are equities, long term government bonds, and short term

borrowing and lending in the United States and Japanese markets. Both investors with the

U.S. dollar and the Japanese yen as home currencies are considered. Short-term borrowing

and lending in foreign currency are allowed in our analysis as part of the process of portfolio

selection, so investors can effectively hedge the exchange risk involved in their holdings of

foreign securities. However, the size of the hedge is chosen to optimize the portfolio's

performance, rather than being artificially constrained to be equal to the size of their foreign

security holdings.

The plan of the paper is as follows. In section II we discuss the selection of internationally

diversified portfolios and place our contribution in the context of the previous literature.

Section III discusses the data and section IV presents parameter estimates for our model of

the time-varying covariance matrix of returns. In sections V, VI and VII we use the estimates

obtained in section IV to form portfolios and examine the performance of those portfolios

both within sample and post-sample. Section VIII offers a summary and conclusion.

5

II. Choosing Internationally Diversified Portfolios

How can observed data be used to implement portfolio strategies that take advantage of

the potential gains from international diversification? The earliest studies on selecting

optimal internationally diversified portfolios are those of Grubel (1968) and Levy and Sarnat

(1970) who use sample means and the sample covariance matrix of returns on a number of

national market equity indexes to compute the mean-variance efficient frontier.1 The

portfolios they construct exhibit significant gains in within-sample performance over purely

domestic portfolios. Grauer and Hakansson (1987) show that substantial improvements in

out-of-sample performance are also possible from optimally selected international

diversified portfolios.2

An important limitation of these studies and other previous work on international asset

allocation is that the covariance matrix of returns is assumed to be constant. Evidence

presented in Maldonado and Saunders (1981) and Kaplanis (1988), however, suggests that

temporal stability of the covariance matrix of returns among different national stock indexes

can be statistically rejected at standard significance levels. This evidence is reinforced by

the findings of conditional heteroscedasticity in the excess returns on uncovered short term

foreign currency denominated assets, first reported in Cumby and Obstfeld (1984), and also

in U.S. stock returns, as reported in Giovannini and Jorion (1987), among others. Attanasio

(1988) and Hamao, Masulis, and Ng (1990) show that conditional heteroscedasticity

characterizes stock index returns in a number of countries. Subsequent research, surveyed

in Bollerslev, Ghou, Jayaraman, and Kroner (1990) suggests that the conditional 1     Both studies restrict the portfolios to include only equities, and they prohibit short sales.

2     They construct portfolios that include bonds as well as stocks. In each period, return data through the end of the previous period are used to compute the parameters of the joint distribution of returns. Portfolio weights are then chosen to maximize expected utility of an investor with isoelastic utility.

6

heteroscedasticity in asset returns can be reasonably modeled by processes that exhibit

autoregressive conditional heteroscedasticity (ARCH).

The approach we take in this paper follows this line of research and models the

covariance matrix of returns using a multivariate version of Nelson's (1990) exponential

ARCH model. The resulting time-varying covariance matrix is then used each period to

compute the efficient frontier of portfolios. That is we examine the problem,

Minimize wt'Ωt wt

subject to

E(Rp,t) = wt'E(r) = R*,

wt'1 = 1.

where wt is the time-varying vector of portfolio weights, Ωt is the covariance matrix of

returns for period t to be estimated by a multivariate exponential ARCH process, r is the

vector of excess rates of return with expected value E(r), and Rp,t is the portfolio excess

return, R* is the target expected excess return and 1 denotes a vector of 1's.

The potential gains when portfolios are diversified internationally have been questioned by

Eun and Resnick (1988) and Kaplanis and Schaefer (1990) who find that if exchange risk is

left unhedged, portfolio risk may increase. These authors and others, including Jorion (1989),

have suggested that a policy of fully hedging the exchange rate exposure of foreign assets

enhances the gains from international diversification. On the other hand, Glen (1990)

implements formal statistical tests of portfolio performance and finds no evidence that a

policy of fully hedging exchange risk increases the mean-variance efficiency of portfolios.

He attributes the findings of Eun and Resnick (1988) to their choice of 1980 - 1985 as their

7

sample, which was a period dominated by dollar appreciation in excess of the forward

discount on foreign currencies.

There is no reason to believe that a naive currency hedge (hedge ratio of one) is optimal.3

Additionally, if the covariance matrix of returns changes over time, optimal hedge ratios are

likely to be time-varying also. We therefore examine hedging by allowing short-term

borrowing and lending in foreign currency, thus including the hedging decision as part of the

problem of portfolio selection.4 This approach has the advantage of allowing hedge ratios to

change over time and takes into account the correlation between the excess returns on

short-term foreign lending and all other assets in the portfolio in choosing a hedge ratio.

III. The Data

We examine continuously compounded weekly rates of return on six broad asset classes

over a period beginning in July 1977 and ending in December 1988. The six asset classes

are equities, long term bonds, and short term borrowing and lending in both the United

States and Japanese markets. The choice of the sample period was governed by the

availability of historical data, especially for Japanese bonds and short-term Yen borrowing

and lending, and the desire to reserve part of the available data for post-sample evaluation

of the portfolios. Although we feel that a month is a more appropriate planning horizon than

a week for an asset allocation strategy, our initial experiments with monthly data suggested

that it would be better for statistical reasons to use more frequent observations. We

therefore analyze weekly data in order to provide a large data set for estimation, while

3     Glen (1990) also examines hedges computed from univariate regressions that minimize the variance of the home currency returns on each of the foreign equity markets.4     Lending one unit of foreign currency while borrowing the equivalent quantity of domestic currency is equivalent to purchasing a forward contract for one unit of foreign currency.

8

limiting the effect of "noise" from the trading process that would be present in daily prices.

U.S. stock returns were drawn from the Center for Research in Security Prices (CRSP)

market value weighted index, which includes dividends. This index measures the return on

a very broadly diversified equity portfolio that resembles the New York Stock Exchange

composite index. It is well known that such broad market value weighted U.S. stock

portfolios are very highly correlated with one another. For example the correlation of the

returns on the CRSP index with those on the S&P 500 index is over .99. We use the CRSP

index because it includes returns from dividend payout, which are an important component

of U.S. equity returns. We convert the price changes from the TOPIX index into annualized

rates of return to compute the return on Japanese equities. Japanese dividend yields were

obtained from Morgan Stanley's Capital International Perspective.

The returns on U.S. long-term government bonds are constructed from Standard and

Poor's series of long-term Treasury bond yields. We converted yields to maturity into

holding period returns by first computing the price of a hypothetical twenty year bond with a

coupon equal to the quoted yield to maturity. We then used the price change plus accrued

coupon interest to compute a continuously compounded annualized rate of return. A similar

procedure was adopted to compute the holding period return on Japanese government

bonds, with the hypothetical bond assumed to have a maturity of ten years.

The nominally riskless short-term interest rates that we use are the return on one-month

Treasury bills for the United States and the one month Gensaki rate for Japan. One

significant source of variation in asset returns over time is simply changes in the returns

available on nominally risk free securities. Unlike asset price risk, however, this component

of return variability is perfectly predictable to the investor, since the current risk free rate is

known at the time a portfolio decision is made. If portfolio performance is evaluated in

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terms of the variance of total returns, changes in the riskless rate will be incorrectly treated

as a part of the risk of the portfolio. We therefore measure each asset's return by its

premium over the return on the investor's home country riskless asset.

For the purpose of estimating the parameters of the return covariance matrix, we consider

a five asset system. The first two assets are U.S. equities and Japanese equities, with the

rate of return on each expressed as a continuously compounded annualized rate of return in

excess of the own country nominally riskless rate. The next two assets are U.S. bonds and

Japanese bonds. Again, each rate of return is expressed in excess of the own country

nominally riskless rate. The fifth asset is short-term Gensaki lending, and the return is

expressed as a continuously compounded annualized rate of return in U.S. dollar terms in

excess of the U.S. Treasury bill rate. The Japanese excess riskless rate of return measured

in dollars is thus,

100*(ln(1+R¥) - 52*ln(Et+1) + 52*ln(Et) - ln(1+R$)),

where R¥ and R$ denote the Japanese and U.S. interest rates and E is the exchange rate in

yen per dollar. When examining portfolios from the point of view of a U.S. investor, we add

the rate of return on the fifth asset to the rate of return on Japanese equities and bonds so

that all rates of return are expressed in U.S. dollar terms in excess of the U.S. Treasury bill

rate. Similarly, when examining portfolios from the point of view of a Japanese investor, we

subtract the rate of return on the fifth asset from the rate of return on U.S. equities and U.S.

bonds so that all returns are expressed in Yen terms in excess of the Gensaki rate. The

covariance matrix is adjusted accordingly in each case.

We estimate the expected excess rates of returns on equities and long-term bonds simply

as the sample averages of these excess returns over the full 599 week period. For short

term foreign lending, we assume that uncovered interest parity holds in equilibrium, so this

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expected excess rate of return is set to zero. We might doubt whether the mean realized

return correctly captures the market's ex ante expectations for Japanese equities, which

averaged an extraordinary 11.6 percent, or for the expected excess rate of return on yen

lending, which averaged nearly four percent over this time period. However, our focus in

this study is on the risk component of portfolio selection, and since mean returns are much

less amenable than variances and covariances to prediction by the statistical approaches we

are considering here, we have simply chosen a set of mean returns that are reasonable and

easily determined. It is well known that, empirically, over short time intervals like a week,

the mean return has little impact on an estimate of the returns variance. The mean

assumption does, however, significantly affect optimal portfolio choice.

IV. Time-Variation in Return Variances and Correlations:

The inputs to the portfolio selection process, the expected returns on the assets,

their return variances, and the full set of correlations between asset pairs must be estimated

from historical data. Typically, one attempts to estimate a single return variance from

historical data. Since estimates based on only a few recent periods are subject to large

errors, one usually tries to enhance the precision by using long historical samples. This

presupposes, however, that the parameters we wish to estimate are constant over time. If

these quantities are changing, then estimates formed over long historical samples will be

contaminated by obsolete data from the distant past.

The five assets under consideration are characterized by five variances and ten

correlation coefficients. While it is possible to treat all of these parameters as elements of a

variance-covariance matrix to be estimated as a single comprehensive system, the

dimensions of the problem render such an estimation procedure very difficult

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computationally. We have therefore adopted the expedient of modeling the movement in

each parameter individually. We first model the variance of each return series. Then, using

these estimates we turn to the correlation coefficients. The following discussion elaborates

on each of these steps.

Modeling variances.

The models used here belong to the autoregressive conditionally heteroscedastic

(ARCH) family of estimators. The initial variant was proposed by Engle (1982). Bollerslev

(1986) suggested the generalized ARCH (GARCH), and Nelson (1990) proposed the

exponential generalized ARCH (EGARCH).

Suppose that a security return rt is normally distributed with zero mean and variance

σ. A basic EGARCH specification for the variance is:

Use of the natural log of the variance simplifies computation because it can take on positive

or negative values. (ARCH and GARCH specifications employ the level of the variance itself

and must be examined at all stages of the estimation procedure to ensure that the fitted

variance is nonnegative in every single period.) The log variance is modeled as a

combination of a constant (a), some fraction of last period's log variance () and a third

component which is based on last period's actual return. In this last component, zt-1 is the

standardized return (rt-1/σt-1). Since it is distributed as a standard normal variate, the

expectation of its absolute value is . The deviation between these two quantities is therefore

(1)

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a measure of the extent to which the magnitude of zt-1 -- and by implication σ-1 -- exceeded

what was expected, given the estimate of σt-1). The d zt-1 term measures asymmetry. If d is

different from zero, a large positive return will not have the same effect on the variance as a

large negative return.

The specifications actually estimated here are generalizations of (1) in that they allow

for up to three lagged values of g(zt). For each return series, the full specification was

estimated. Statistically insignificant parameters were dropped, and the specification was

then reestimated. The final estimates are reported in Table 1.

The coefficients may be roughly interpreted as follows. Our prediction of next

period's risk is a combination of a constant a and b times the current risk. The c1 (and, if

present, the c2 and c3) coefficients measure the immediate impact of a large positive or

negative return. Finally, if d, the asymmetry parameter is negative, a large negative return

increases the risk prediction by more than a positive return of the same magnitude.

For a more precise example, consider the equation estimated for Japanese stocks.

The unconditional expectation for ln(σ) is a/(1-b) = .179/(1_.981) = 9.421. This roughly

implies an unconditional standard deviation of σt = [exp(9.421)]1/2 = 111%, which is close to

the value directly estimated as 98.6%.5

Now suppose we are predicting σt = 111%. This means that we expect |rt| = (σt

5     These standard deviations appear unusually large because our weekly returns are converted to annual rates by multiplying by 52. The standard deviations are therefore also 52 times the 1 week figures. However, this annualized standard deviation is not the same as volatility. In computing volatility, we assume returns are independent from one week to the next, so that when 52 weeks of returns with a given standard deviation are cumulated, the variance of that one year return is 52 times the 1 week variance. The standard deviation is therefore equal to the 1 week standard deviation multiplied by the square root of 52, in other words, it is our annualized standard deviation divided by the square root of 52. Thus the annual volatilities in this example are 111%/ = 15.4% and 98.6%/ = 13.7%. In examining portfolio performance in the next section, all standard deviations will be expressed as annual volatilities is this way.

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[2/π]1/2) = 88.6%. If the actual rt turns out to be +88.6%, then our prediction of the risk is

exactly confirmed and we will not modify next period's variance prediction.6 On the other

hand, suppose the magnitude of the return is twice as large as we predicted, i.e., |rt| =

177.2%. The predicted standard deviation for next period is raised to σt+1 = 120%.7

The parameter b has another interpretation as a measure of persistence in the

deviation from the unconditional average, with values near unity indicating high persistence.

In the case of Japanese stocks b=.98, which implies that once a higher level of uncertainty is

incorporated into stock returns, it recedes only gradually. For example, if this period's

variance were 50% higher than the overall average, the variance projected one year in the

future would be approximately (.98)52(50%) = 17.5% higher than the overall average.

In surveying the estimates for the five return series, all of the persistence parameters

are above .9, except for U.S. stocks (b = .484). This lower value implies that the effect of

large positive or negative returns dies out much more rapidly for U.S. equities. The

immediate impact coefficients (the c's) vary in sign, but the sum of these coefficients, which

captures the net immediate impact, is invariably positive. The fact that both b and the c

sums are positive is a confirmation that a period of relatively high risk is likely to be followed

by further periods of high risk. U.S. stocks exhibit the largest immediate impact coefficients

(as judged by the sum of the c coefficients). The only series which exhibited a significant

asymmetry term was Japanese bonds (d = -.149). The negative value implies that a large

negative return portends higher future risk than does a positive return of the same

6    Referring to the specification, we have g(zt) = |88.6/111| - (2/π)1/2 = 0. Substituting this into the main equation gives ln(σ-1) = .179 + .981(9.421) + .193 (0) = 9.421, which implies σt+1=111%.

7    g(zt) = |177.2/111| - (2/π)1/2 = .799 ln(σ+1) = .179 + .981(9.421) + .193 (.799) = 9.575, which implies σt+1=120%.

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magnitude.

Correlations

While the family of univariate ARCH-style models is quite well developed and varied,

extensions to joint models of two or more series are few. The best known of these is

probably the multivariate GARCH model proposed by Bollerslev (1986). If there are two

series under consideration, say rit and rjt, then the covariance portion of a multivariate

GARCH projects Cov(rit, rjt) from

Cov(ri,t-1, rj,t-1) and the product ri,t-1rj,t-1. There is a practical difficulty in implementing this

procedure, similar to the need in univariate GARCH estimation to ensure that the fitted

variance is always nonnegative. Here we must enforce the restriction that at all points

Cov(rit, rjt) < [Var(rit)Var(rjt)]1/2, i.e., that the correlation coefficient be between -1 and +1. In

the case of more than two variables, this generalizes to the requirement that the covariance

matrix be positive definite.

It was pointed out in the discussion on variance estimation that nonnegativity could

be guaranteed by using the log transform of the variance. By analogy, we employ here a

similar transform to force the correlation coefficient to remain between -1 and +1. Our

specification arises from the observation that the arctangent function has the property that

for all values of x, arctan(x) lies between -π/2 and +π/2. This suggests that the correlation

coefficient may be specified as:

where qt evolves as:8

8     An alternative approach to restricting the range of a fitted correlation coefficient is R.A.

15

Analogously to the variance models, the current value of qt is a combination of last period's

value and the extent to which the product of last period's standardized returns, zit (= rit/σit)

and zjt, diverges from last period's fitted correlation coefficient. The arctan transformation

guarantees positive definiteness of the covariance matrix for the bivariate case. This does

not extend to the multivariate case, however, since the condition that all bivariate

correlations be less than one in absolute value is not sufficient to ensure that the entire

covariance matrix is positive definite.9

In principle, this specification should be estimated jointly with the component

univariate models. To minimize the number of parameters that required estimation at each

step, however, the univariate models were held fixed, and the correlation models were then

estimated separately for each pair of assets. These estimates are reported in Table 2.

The intuitive interpretations of the coefficients are quite similar to those in the

variance estimations in Table 1. The b coefficients, which measure persistence of a deviation

of the present correlation from its long run average, are more variable than the parameters

in the variance estimations. There are five (out of ten) correlation pairs for which b was

found to be insignificantly different from zero. (In this case, the parameter was dropped from

the estimation.) For those remaining, the b's are positive, indicating persistence in the

correlations. In only two instances, however, are the values greater than .9. The overall

impression is that persistence in correlations is much lower than the persistence of the

Fisher's z-transformation. We duplicated our estimates with the z-transformation, and the results were virtually identical.9     In fact, this problem does arise in two periods immediately following the stock market crash in 1987, making it impossible to compute the returns covariance matrix for those dates. In the results presented in the next section, we drop these periods.

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variances.

The c coefficients measure the immediate impact of the unexpected component of

the correlation. These are always positive, implying that if the current correlation is higher

(more positive) than expected, the prediction of next period's correlation is revised upwards.

The model and estimation procedure just described fits a time-varying variance-

covariance matrix for a set of asset returns by estimating time-varying univariate models of

the EGARCH form for the variances and the GARCH form for the covariances. We will

therefore refer to it as the EGARCH/GARCH or E/G model.

V. Portfolio Optimization with the E/G Model

The Efficient Frontier

The investor's portfolio objective is to maximize return and minimize risk. Given the mean

returns, variances and covariances among a set of risky assets, standard optimization

methods allow us to determine the portfolios lying on the efficient frontier, where for any

expected rate of return, the frontier portfolio is the one with minimum risk. If the assets' risk

and expected return parameters are constant over time, the frontier portfolios will have fixed

weights, and the efficient frontier will be invariant as well. However, when volatilities or

correlations are time-varying, the efficient frontier also shifts from period to period.

In this section we analyze the portfolio choices resulting from the E/G model estimates

of volatilities and correlations. We determined the time-varying compositions of six different

frontier portfolios: the portfolio with overall minimum risk and the five portfolios whose

forecasted excess returns were, respectively, 0, 2, 5, 8, and 12 percent higher than the

riskless interest rate.

We will compare and contrast the performance of the E/G model efficient portfolios

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with the fixed portfolios that result from assuming the variance-covariance matrix of asset

returns to be constant over time. This "Fixed Variance" model was fitted simply by

computing the sample variances and covariances over our 599 weekly observations.

Figures 1 and 2 illustrate how the E/G model efficient frontiers change from period to

period. Each curve plots the expected excess return and volatility as an annualized percent,

with the symbols indicating the positions of the six frontier portfolios we are focusing on.

Except for the "Minimum Risk" portfolio which is the combination of assets with the lowest

volatility among all feasible portfolios, each frontier portfolio represents the lowest risk

portfolio whose expected return is at least equal to the target value. For Japanese investors,

the expected excess return on the Fixed Variance model's Minimum Risk portfolio was 2.56

percent, so this same portfolio was also the efficient portfolio for R = 0 and R = 2. This was

also true for the E/G model's R = 0 and R = 2 portfolios in most periods.

The figures each show three efficient frontiers: the Fixed Variance model's unchanging

frontier, and two frontiers based on the E/G model's efficient portfolios for two different dates.

These results display considerable variation over time in both the location of the efficient

frontier and its shape.

Figure 1 shows that on October 12, 1977, the E/G model estimated the risk for a U.S.

investor as being substantially lower than the Fixed Variance model did. On that date, if an

investor had wanted a portfolio with an expected return 2 percent above the riskless rate, the

Fixed Variance model would have indicated that the optimal portfolio's volatility was close to 9

percent, while the risk level of the optimal R = 2 portfolio according to the E/G model was only

around 5 percent. If she had been willing to accept portfolio risk of 9 percent, the E/G model

indicates that she could have constructed a portfolio with about 8 percent expected excess

return.

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In looking at comparisons such as this, it is important to understand that the different

efficient frontiers derived from these two models do not represent different choices for the

efficient portfolios that are available to an investor, but rather, different estimates of the true

efficient frontier she faces at each point in time, based on the two models' predictions of the

risks on the various asset classes. For example, if the Fixed Variance model's variance and

covariance estimates for October 12, 1977 were, in fact, the true values on that date, no asset

portfolio could really have produced the risk and return combinations indicated on the E/G

model's efficient frontier.

On October 14, 1987, i.e., the week just prior to the 1987 stock market crash, the E/G

model's volatility estimates for equities had risen well above their average values over the full

sample period, with the result that the risk levels of the high return portfolios were quite a bit

higher than the Fixed Variance model indicated. A risk averse investor would have reduced

the proportion of his portfolio held in stocks to adjust for the higher estimated risk, particularly

since the risk levels for more defensive portfolios were estimated to be lower than predicted

by the Fixed Variance model.

For Japanese investors, the estimated risk levels in all cases were lower than in the U.S.

The differences in the E/G efficient frontiers on the dates we have selected appear to be

essentially parallel shifts with little change in the estimated tradeoff between risk and return.

But, as we saw for the dollar-based investor, the E/G model evaluated the level of risk facing a

Japanese investor just before the 1987 crash to be well above the average indicated by the

Fixed Variance model.

Portfolio Risk

As these figures show, the estimated portfolio risk in the E/G model varies considerably

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over time. Figures 3A and 3B plot the forecasted volatility over time for the Minimum Risk and

the R = 5 portfolios, as viewed by a U.S. investor. We refer to these portfolios as

"Unconstrained" because we impose no constraints against short sales for any of the assets.

In the next section, we will discuss the effects of prohibiting some or all short positions.

The E/G model appears to predict lower overall risk than the Fixed Variance model for

both portfolios. Also, because the E/G model adjusts the forecasted volatility in response to

the (highly variable) asset returns observed in the most recent week or two, it varies

considerably from one week to the next. Still, there have clearly been some periods of

generally lower than average volatility and others with high estimated risk.

Estimated risk in both portfolios is substantially lower than average in the beginning of

our data sample, with a sharp jump upwards in late 1979. In October 1979, the Federal

Reserve changed its operating targets for monetary policy from stabilizing interest rates to

stabilizing money supply growth and allowing interest rates to fluctuate much more freely.

This led to a marked increase in the variability of U.S. long term interest rates, and bond

prices.

The period from 1980 through 1982 was one of about average volatility for the lowest

risk portfolio, and gradually rising risk for the R = 5 portfolio, with a few months of unusually

high volatility in the second half of 1982 for the latter. After that, risk for both portfolios fell

below average for several years. Estimated risk increased in about the beginning of 1986,

reaching a high point just before and immediately following the 1987 stock market crash.

After that, estimated risk fell abruptly, and was well below average by the end of 1988.

Volatility of the Minimum Risk portfolio also peaked in 1987, but the effect of the crash was

quite muted because the portfolio contained only a very small equity position.

Figures 4A and 4B display the somewhat different pattern of estimated risk faced by

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Japanese investors holding the Minimum Risk or R = 5 portfolios. First, the general risk level

was substantially lower for a Japanese than for a U.S. investor. The major reason for this is

that the efficient portfolios for both placed significant amounts in Japanese securities, because

of their relatively high mean returns. This introduced exchange rate risk into the U.S.

investor's position that the Japanese investor did not bear.

In Japan, the lowest volatility levels occurred in 1977 and 1978, at the very beginning

of the sample. Thereafter, risk held fairly constant for a number of years, until the middle

1980's. The end of 1985 marked the beginning of a period of increased volatility that rose for

the R = 5 portfolio through 1987. As was the experience in the U.S., after the market crash,

risk levels dropped sharply during 1988.

Portfolio Composition

As the estimated risk parameters change over time, the composition of the efficient

E/G portfolios also changes, and the week to week variability in optimal portfolio weights can

become quite extreme. This would lead to very large transactions costs and impractically

large shifts in portfolio composition for any investor who tried to implement the model's

prescriptions exactly. Translating the statistical estimates from a model like this into a viable

portfolio strategy for a real-world investor would require a smoothing procedure that balanced

the benefits of holding a portfolio that is optimized for the current market conditions against

the transactions costs imposed by frequent adjustments.

Table 3 summarizes the behavior of the portfolio weights for the Minimum Risk and the

R = 5 portfolios, for U.S. and Japanese investors. Note that the weights apply only to the

"risky" portfolio, i.e., investments in the five asset classes excluding the domestic riskless

asset, so they sum to 1.0. The full solution to an investor's optimal portfolio choice problem

21

would combine an investment in an efficient risky portfolio with the risk free asset in the

appropriate proportions to maximize the investor's expected utility. Professional portfolio

managers, however, typically leave this last decision to the investor and concentrate their

efforts on trying to choose the best risky portfolio, so we restrict our analysis to risky asset

portfolios as well.

Several features are apparent in Table 3. First, in all four cases, the mean portfolio

weights for the E/G model are quite similar to those from the Fixed Variance model. However,

the E/G weights vary considerably over time, especially for a U.S. investor. For example, a

U.S. investor assuming a fixed returns covariance matrix and seeking an optimal portfolio with

an expected return 5 percent higher than T-bills would hold fractions of about .134 of his

assets in Treasury bills and .222 in U.S. Treasury bonds. Under the E/G formulations, these

portfolio proportions would average .012 and .317, respectively. But the similarity in average

values hides a wide time variation: the optimal T-bill weight varied from -1.018 to +1.109,

with a mean difference of .333 from the Fixed Variance model value. The comparable figures

for T-bonds were a range from -0.149 to .957 and a mean discrepancy of .198.

Second, the optimal Japanese portfolios showed considerably less time variation and

smaller differences between the Fixed Variance and the E/G model weights. Only for the

Japanese government bond position did the difference between models produce an average

difference of more than 0.10 between the optimal asset weights.

In looking at the typical portfolio compositions, we see that a U.S. investor following a

minimum risk strategy would have held a large position in the Japanese riskless asset and

offset the effect of the currency fluctuations with substantial short positions in Japanese long

term bonds and stocks. The R = 5 investor, on the other hand, tended to hold all five asset

classes long, with fairly equal amounts in each.

22

The time pattern of a U.S. investor's holdings of Japanese short term lending and

Japanese government bonds for these portfolios are shown in Figures 5 and 6. In order to

reduce the extreme volatility of the weights, we have smoothed them by taking a 4 week

moving average. This is equivalent to a trading strategy of adjusting the portfolio each week

one fourth of the way from its present composition towards the currently optimal portfolio. It

is also consistent with our general view that a month (4 weeks) is an appropriate planning

horizon for asset allocation, while 1 week is too short. The resulting plots are much smoother,

though still quite choppy.

Figure 5A shows the fraction of a dollar-based investor's Minimum Risk portfolio that

would be held in the Japanese riskless asset. The effect of permitting short positions is

apparent here: the Fixed Variance model holds more than 100 percent of the total funds to be

invested in this one asset class, meaning that the combined weight on the other four assets

has to be -0.12, i.e., a net short position. The E/G model's Minimum Risk portfolio weight on

the Japanese riskless asset was above this value during most of the sample, except for the

late 1970's. In Figure 5B, we see a substantial difference in the R = 5 portfolio. The Fixed

Variance model took only a small long position in the Japanese riskless asset. The E/G model's

holdings tended to be short as often as long.

Figures 6A and 6B help to explain these results. (In comparing the graphs, note the

change in the vertical scale, which had to be expanded in order not to truncate the weights

that were over 1.6 in Figure 5A). The Minimum Risk portfolio took significant short positions in

Japanese government bonds, with a weight of -0.44 for the Fixed Variance model and values

that were mostly even more negative for the E/G model portfolio. Holding a long position of a

comparable size in the Japanese riskless asset constituted a hedge of the exchange rate risk

on the bond position. We might therefore interpret a very large long position in the Japanese

23

riskless asset as being partly an investment in that asset class and partly a currency hedge for

short positions in Japanese bonds and stocks.

The R = 5 E/G portfolio's investment in Japanese bonds was highly variable over time,

deviating widely from the Fixed Variance model's weight of 0.102. Most positions were long,

except for 1980 and a portion of 1985. At the beginning of the sample, between the middle of

1981 and the beginning of 1985 and then during two brief periods at the beginning of 1987

and 1988, the portfolio was invested heavily in Japanese bonds.

Portfolio weights on the remaining asset classes were much more stable. However, as

shown in Figure 7, the marked change in the character of U.S. Treasury bonds as an asset

class when the Federal Reserve's policy changed in October 1979 produced a significant shift

in portfolio strategy. Up to that point, both the Minimum Risk and R = 5 portfolios placed

about 80 percent of the funds in T-bonds, but this dropped to about 10 percent when the Fed

began to allow their yields to fluctuate and their price risk to rise. After that point, U.S. bonds

became much more like stocks as an asset class, from the point of view of portfolio

optimization.

The portfolio weights for the Japanese investor's Minimum Risk and R = 5 portfolios

were much more stable than for a U.S. investor. Since the mean returns for both Japanese

bonds and Japanese stocks were substantially higher than for their U.S. counterparts, and

holdings of dollar denominated assets were also exposed to exchange rate risk, the Japanese

portfolios placed very little weight on U.S. securities (mostly short). The exception was in

1986-7, when the R = 5 portfolio held a significant amount of U.S. stocks. Interestingly, this

position was not reduced immediately at the time of the 1987 crash. Rather, the portfolio

weight only dropped close to zero at the end of March 1988. U.S. Treasury bills were held as a

significant component of the optimized portfolios, however, because their yields were higher

24

than the Japanese risk free rate during much of the sample.

Figures 8, 9, and 10 show the E/G weights for U.S. T-bills, Japanese government bonds,

and Japanese stocks for the Japanese investor's optimal R = 5 portfolio.

Model Performance

An important question about the E/G model is how much improvement in risk

assessment it makes possible, relative to the Fixed Variance model. One would like to know

whether the additional effort to fit the kind of model we have been examining produces

substantially better answers than simply assuming standard deviations and correlations are

constant and computing them from past data. In this section, we will examine how well the

two models "fit" within the sample, and defer looking at post-sample prediction until later.

Unlike a linear regression, there is no "R2" statistic to look at for the E/G model. Each

model for an individual variance or correlation produced a statistically significant increase in

the log-likelihood function. But there is no direct way to transform this evidence of

"significant" explanatory power into a measure of how much forecast accuracy increases for

the specific variance or covariance being fitted, and certainly no way to combine the results of

15 individual equations to determine how much the model improves risk evaluation for a

portfolio.

Both models predict the variances and covariances for the five asset classes, from

which one can calculate the standard deviation of return for any portfolio that can be formed

from those assets. Since the two models' variance and covariance estimates differ, they will

evaluate any given portfolio's risk differently; the locations of the efficient frontiers produced

by the two models will differ; and, naturally, so will the composition of the frontier portfolios.

For example, in a given week, there will be two competing R = 5 portfolios. According

25

to the E/G model, its portfolio is on the efficient frontier while the Fixed Variance model's

selection is inefficient: its expected excess return is 5 percent but the true volatility is higher

than for the E/G model's R = 5 portfolio. According to the Fixed Variance model, on the other

hand, the positions are reversed, and the E/G model's portfolio is inefficient.

Note that an efficient portfolio produced by the true model has a lower volatility than

any other portfolio with the same expected return (regardless of what volatility that portfolio

has been estimated to have by an incorrect model). Thus, one way to gauge the difference in

performance between models is to see which one's frontier portfolios actually do achieve the

lower variance.

In order to compare the in-sample performance of the E/G model with that of the Fixed

Variance model, we computed the realized returns on the various efficient frontier portfolios

we have been considering for each week in the sample, and computed the standard

deviations around their target excess returns. Table 4 displays these results.

We see that for the E/G model, the realized standard deviation for the U.S. R = 5

portfolio, in fact, averaged 10.10 percent relative to the target rate, while the Fixed Variance

model's realized standard deviation was 10.41 percent. Thus, for this portfolio, the E/G

model's risk assessment was somewhat more correct. This would have allowed the investor to

select a more nearly optimal combination of assets, given his risk preferences and his

forecasts of expected returns on the different asset classes. Table 4 shows that for all

portfolios the E/G model produced lower realized variability around the target return than did

the Fixed Variance model.

The overall impression we get from the results so far is that the risk parameters of

these important assets do vary substantially over time and the time variation is regular

enough that it can be partly captured by the E/G model we have developed. Both dollar-based

26

and yen-based investors could have used the E/G model to construct asset allocation

portfolios with lower variance around their target returns than could have been achieved by

the Fixed Variance model. However, the improvement in portfolio performance, as measured

by the reduction in standard deviation, is relatively small, and it must be weighed against

considerable additional difficulty in obtaining parameter estimates from an E/G approach.

VI. Constrained Portfolios

In the previous section, we analyzed portfolios whose only requirement was that the

return variance should be the minimum among all possible portfolios with expected return

equal to the target value. There were no limits on the size of positions in individual asset

classes. Thus, it was permissible to sell short one asset class, take the proceeds of the short

sale and invest them in another asset class that had a higher expected return. It was equally

possible for the portfolio weight on a single asset class to be above 1.0, meaning that more

than the investor's total initial investment was held in that asset, with the additional funds

coming from short sales of the other assets.

Most institutional investors face constraints of various kinds on the portfolio allocations

they can choose. For example, it is unusual for an institution to take short positions. Most

portfolio managers view their role as allocating the investment funds at their disposal among

purchases of different securities, and they would not consider selling securities short in order

to obtain additional funds to invest. Such limitations on short selling are occasionally imposed

by government regulation, but for most U.S. investors, the constraint is self-imposed: short

selling is simply felt to be a risky strategy and inconsistent with the manager's investment

philosophy.

One exception is the practice of hedging the currency risk on foreign securities. A

27

currency hedge can take several forms, including selling forward or futures contracts based on

the foreign currency, or financing purchases of foreign risky securities (in this case, bonds and

stock) by borrowing in the foreign currency (in essence selling short the foreign riskless asset).

Thus a slightly less constrained portfolio choice would be to prohibit short positions in all but

the foreign riskless asset, but to allow a negative portfolio weight on that security up to the

size of the investor's long positions in the other foreign asset classes. Note that in practice it

is possible to achieve a hedged position essentially identical to this without short sales by

simply selling currency forward contracts.

Since one of the objects of this paper is to analyze the impact on performance of

diversifying internationally, we also consider the effects of another major constraint that many

investors impose on their portfolio choice: restricting themselves to holding only domestic

securities.

Table 5 displays the realized performance of the Minimum Risk and R = 5 portfolios

under various constraints. In each case, we show the Unconstrained portfolio analyzed above,

followed by alternative portfolios that are optimized under increasingly restrictive constraints.

The first of these is the "Currency Hedge" portfolio, in which risky securities can only be held

long, but a short position up to the total investment in foreign bonds and stocks is allowed in

the foreign riskless asset, to hedge against exchange rate changes. The "Long Only"

portfolios eliminate this option, and the last line in each section shows the performance of

portfolios that are restricted to contain only domestic assets.

Looking first at the results for a U.S. investor, we see that prohibiting short positions

makes a significant difference in portfolio risk for the Minimum Risk portfolio, but not for the R

= 5 portfolio, with the impact being greater for the E/G than the Fixed Variance model. In

general, since the E/G model portfolio changes every period in response to changes in the

28

estimated returns covariance matrix, even if a particular asset class is normally held long, the

ability to assume a short position from time to time can add an important element of flexibility

in risk management. The Fixed Variance model with its constant portfolio weights will never

try to sell short an asset that it normally holds long.

The effect of prohibiting short sales will be especially noticeable for the Minimum Risk

portfolio, because the optimal weights do not need to produce any prespecified mean return.

This gives an extra degree of freedom for the portfolio to achieve a minimum volatility. The

opposite effect occurs in the "U.S. / Japan Only" portfolios, where it may make no difference

whether short sales are permitted or not. For example, the weights in the R = 5 portfolio that

is restricted to only U.S. assets are fixed by the various constraints they must satisfy: although

there are five asset classes in the analysis, only U.S. bonds and U.S. stocks can be held in this

portfolio, the two weights must sum to 1.0 and they must also produce an expected portfolio

return of 5 percent over the riskless rate. The result is that only one combination of bonds

and stocks can satisfy all of the constraints, and risk minimization does not enter the problem

at all. In this case, the E/G and Fixed Variance model portfolios are always the same and both

domestic asset classes must be held long.

Interestingly, in this table, permitting the investor to hedge the currency risk exposure

of her foreign securities leads to virtually no difference in performance from the Long Only

positions. The most important risk reduction occurs when the investor is allowed to diversify

internationally. This is especially true for the dollar-based investor, but the Japanese investor

also can reduce risk by holding some U.S. securities.

VII. Model Performance in the Post-Sample Period

The E/G model estimation and all of the analysis presented so far has been based on

29

the 599 week sample period, from July 1977 through December 1988. Actually, of course, it

would not have been possible to choose optimal portfolios using either the E/G or Fixed

Variance model risk parameters we fitted, before the end of the period. In practice, each

week's asset allocation decision must inherently be based on estimates derived from earlier

periods, so the portfolio choice problem involves both in-sample parameter estimation and

post-sample forecasting.

This introduces additional sources of forecast error, in the form of estimation risk and

model instability. Estimation risk, or sampling error, arises because the parameters used in

deriving model forecasts are not the true values, only estimates of those values. Model

instability comes from the fact that the statistical relationships that one is trying to capture

with the model may themselves change over time. Sampling error causes parameter

estimates to differ from the true parameter values, while model instability means that those

true values, and the structure of the model as well, can change between the estimation period

and the post-sample forecasting period. As a rule, a more complex model is better able to

explain the data in in-sample estimation, but may be less robust than a simple model in post-

sample forecasting.

To examine out-of-sample performance, we extended the data series described above

from January 1989 through September 1990, yielding a total of 91 new weekly observations.

Table 6 compares the E/G and Fixed Variance models during this period in the same way as

Table 4 did for the initial sample. Both models show higher variation than in the prior period,

but the E/G model deteriorates more. It is now less accurate than the Fixed Variance model,

especially for a U.S. investor.

To see whether there was a pattern of progressive deterioration in accuracy as one got

further beyond the end of the sample period, we split the post-sample into two subperiods.

30

The results are mixed. For a U.S. investor, the E/G model shows relatively bad performance in

the first half of the post-sample period for low expected return portfolios and good

performance for high return portfolios, but in the second half, it is less accurate than the Fixed

Variance model in all cases. A Japanese investor would have experienced a considerable

increase in portfolio risk exposure between the first and second half of the post-sample period.

Although the E/G model is less accurate than the Fixed Variance model in every comparison,

its performance is substantially worse only for the lowest risk portfolios in the second half.

VIII. Conclusions

This paper has reported on an extensive investigation of the problem of optimally allocating

funds in an investment portfolio among major classes of U.S. and Japanese assets. The major

innovation that we explored was to allow the asset risk parameters to vary over time and to

construct an econometric model of this time-variation. The investigation is still in progress, as

we continue to seek ways of improving the model's performance, but we have reached a

number of conclusions based on the results reported here.

1. There appears to be substantial time variation in the risk characteristics, both variances

and correlations, for the asset classes we examined.

2. Fitting fully general ARCH, GARCH, or EGARCH models to the covariance matrix in even a

fairly modest asset allocation problem would involve many parameters and a difficult

estimation. Moreover, we may have strong prior beliefs that many of those coefficients should

be zero. Our piecemeal approach of estimating univariate models and then combining them

offers a reasonable compromise between generality and estimability. All of our submodels

31

had statistically significant and plausible estimated parameter values. We found that for this

problem, EGARCH models for variances and modified GARCH models for correlations worked

well and were the easiest to deal with in the estimation.

3. The portfolio weights that come from the E/G model are highly variable. Translating the

model's prescriptions into a viable investment strategy to be followed in practice would

require a smoothing procedure.

4. While there clearly was time variation and the E/G models did seem to capture some of it,

the improvement in portfolio performance was somewhat limited. (The real payoff in asset

allocation would appear to come from being able to forecast mean returns.) On the other

hand, the time patterns in the E/G portfolio weights and risk assessment make sense: they

showed the effect of the change in the character of the U.S. Treasury bond market in 1979

and the increase in stock market risk in October 1987, for example.

5. In comparing the performance of portfolio strategies under different constraints, we found

the biggest impact was from allowing international diversification, especially for a U.S.

investor. Prohibiting short sales and currency hedging did not seem to affect a Japanese

investor or a U.S. investor following the R = 5 strategy in this particular case; it did make a

small difference in the U.S. Minimum Risk strategy.

6. Post-sample performance of the E/G model deteriorated relative to the Fixed Variance

model. This suggests that efforts to optimize the E/G approach for use in a forecasting mode

may be worthwhile. Note that the E/G model does not actually have to more accurate than

32

other approaches in order for it to have value as an investment tool. It would enough for it to

indicate correctly at each point in time whether asset volatilities are higher or lower than

average. One step in the direction of improving forecasting performance is to restrict analysis

to parsimonious models, which we have already done. Another is to try to form an optimal

combination of the E/G and Fixed Variance forecasts. Work on these topics is currently in

progress.

33

References

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Table 1.EGARCH Estimates for all return series.

Specification:

Series(rt)

Uncond'l σ(rt)

a b c1 c2 c3 d

ForwardCurrency

78.939 .603(2.19)

.931(29.43)

.339(4.39)

-.247(-3.06)

U.S.Stocks

115.315 4.855(3.76)

.484(3.52)

.533(6.04)

JapaneseStocks

98.615 .179(2.61)

.981(130.71

)

.193(5.39)

U.S.Bonds

98.858 .312(2.92)

.966(82.78)

.340(6.82)

JapaneseBonds

38.935 .682(3.95)

.907(37.15)

.663(7.80)

-.395(-3.62)

.281(3.50)

-.149(-2.52)

Notes: Estimates are for weekly returns at annual percentage rates. T-statistics are given in parentheses. Unconditional estimates are formed over the entire sample.

Table 2.Correlation estimations for all return series.

Specification (for a pair of assets i and j):

Return Pair Uncond.Corr

a b c

ForwardCurrency

U.S. Stocks -.052

Forward Currency

Japanese Stocks

-.096 -.002(-.33)

.979(40.26)

.065(2.24)

Forward Currency

U.S. Bonds -.148

Forward Currency

Japanese Bonds

-.307 -.584(-7.49)

.143(2.26)

U.S. Stocks Japanese Stocks

.390 .027(.97)

.949(18.88)

.028(1.33)

U.S. Stocks U.S. Bonds .320U.S. Stocks Japanese

Bonds.119 .027

(1.01).886

(9.60).100

(2.15)JapaneseStocks

U.S. Bonds .100 .062(1.44)

.663(3.35)

.101(2.21)

Japanese Stocks

Japanese Bonds

.230 .115(1.47)

.775(5.29)

.052(1.25)

U.S. Bonds Japanese Bonds

.173 .345(4.77)

.163(2.33)

Notes: Estimates are for weekly returns at annual percentage rates. T-statistics are given in parentheses. Unconditional estimates are formed over the entire sample.

Table 3.Summary of Portfolio Weights for E/G and Fixed Variance Models

U.S. INVESTOR

Foreign Japanese US Japanese UsRiskless Bonds Bonds Stocks Stocks

Minimum Risk Portfolio

Fixed Variance Weights 1.114 -0.445 0.234 -0.125 0.221E/G Mean Weights 1.222 -0.627 0.335 -0.113 0.182Mean | WE/G - WFIXED | 0.304 0.245 0.181 0.111 0.095Maximum WE/G 2.318 0.392 0.932 0.189 0.562Minimum WE/G -0.211 -1.538 -0.109 -0.698 -0.031

R = 5 Portfolio

Fixed Variance Weights 0.134 0.102 0.222 0.273 0.269E/G Mean Weights 0.012 0.156 0.317 0.263 0.251Mean | WE/G - WFIXED | 0.333 0.362 0.198 0.103 0.125Maximum WE/G 1.109 1.705 0.957 0.673 0.730Minimum WE/G -1.018 -1.242 -0.149 -0.203 -0.030

JAPANESE INVESTOR

Foreign Japanese US Japanese Us Riskless Bonds Bonds Stocks Stocks

Minimum Risk Portfolio

Fixed Variance Weights 0.291 0.695 -0.011 0.062 -0.037E/G Mean Weights 0.256 0.743 -0.027 0.056 -0.028Mean | WE/G - WFIXED | 0.100 0.138 0.026 0.072 0.032Maximum WE/G 0.817 0.971 0.115 0.618 0.126Minimum WE/G -0.035 -0.071 -0.209 -0.077 -0.192

R = 5 Portfolio

Fixed Variance Weights 0.184 0.570 -0.038 0.287 -0.003E/G Mean Weights 0.157 0.611 -0.058 0.266 0.023Mean | WE/G - WFIXED | 0.097 0.126 0.040 0.050 0.050Maximum WE/G 0.588 1.004 0.263 0.618 0.521Minimum WE/G -0.165 -0.071 -0.291 0.008 -0.178

The table shows summary statistics for the portfolio weights on the five risky asset classes in

two efficient frontier portfolios, as indicated by the E/G and Fixed Variance models. The Minimum Risk portfolio was the frontier portfolio estimated to have minimum risk overall, regardless of expected return.

Table 4

Comparison of Model Performance in Assessing Portfolio Risk:

Portfolio Standard Deviation around Target Return

U.S. Investor Japanese Investor

Portfolio E/G Model Fixed Variance E/G Model Fixed Variance

Minimum Risk 7.05 7.89 3.35 4.00

R = 0 7.62 8.03 3.35 4.00

R = 2 8.35 8.68 3.36 4.00

R = 5 10.10 10.41 4.90 5.25

R = 8 12.32 12.72 8.20 8.56

R = 12 15.66 16.25 13.18 13.72

The table shows the realized standard deviations (in annualized percent) for efficient frontier portfolios selected by the E/G and Fixed Variance models during the sample period July, 1977 through December 1988. Standard deviations were computed around the target rates of excess return relative to the domestic riskless interest rate, i.e., R = 0, 2, etc. The Minimum Risk portfolio was the frontier portfolio estimated to have minimum risk overall, regardless of expected return.

Table 5

Model Performance with Constraints on Portfolio Choice:

Portfolio Standard Deviation around Target Return

U.S. Investor Japanese Investor

Portfolio E/G Model Fixed Variance E/G Model Fixed Variance

Minimum Risk

Unconstrained 7.04 7.89 3.36 4.00 Currency Hedge 8.13 8.48 3.46 4.04 Long Only 8.13 8.48 3.46 4.04 U.S./ Japan Only 11.32 11.77 4.69 5.20

R = 5

Unconstrained 10.09 10.41 4.90 5.25 Currency Hedge 10.14 10.41 5.07 5.27 Long Only 10.18 10.41 5.06 5.27 U.S./ Japan Only 14.91 14.91 5.69 5.69

The table shows the realized standard deviations (in annualized percent) for efficient frontier portfolios selected by the E/G and Fixed Variance models during the sample period July, 1977 through December 1988 under various constraints. The Minimum Risk portfolio was the frontier portfolio estimated to have minimum risk overall, regardless of expected return, while the R = 5 portfolio was required to have expected excess return of 5 percent relative to the domestic riskless interest rate.

The portfolio constraints were:

"Unconstrained" - No restrictions on portfolio weights.

"Currency Hedge" - Short positions allowed only in foreign riskless asset as a hedge against currency risk on long positions in foreign assets.

"Long Only" - Only long positions permitted.

"U.S./ Japan Only" - Portfolio restricted to domestic bonds and stocks

Table 6

Comparison of Model Performance in the Post-Sample Period

January 1989 - September 1990:

Portfolio Standard Deviation around Target Return

U.S. Investor Japanese Investor

Portfolio E/G Model Fixed Variance E/G Model Fixed Variance

Full Sample

Minimum Risk 6.93 6.38 4.66 4.12R = 0 7.55 6.61 4.66 4.12R = 2 9.17 7.93 4.66 4.12R = 5 12.77 11.15 7.26 7.06R = 8 16.96 15.01 12.47 12.21R = 12 22.92 20.51 20.02 19.55

1st Half of Sample

Minimum Risk 7.51 6.60 3.06 3.00R = 0 7.45 6.71 3.06 3.00R = 2 7.53 7.10 3.05 3.00R = 5 8.18 8.12 3.75 3.47R = 8 9.34 9.49 6.37 5.84R = 12 11.40 11.66 10.49 9.75

2nd Half of Sample

Minimum Risk 6.32 6.16 5.82 4.99R = 0 7.65 6.50 5.82 4.99R = 2 10.53 8.66 5.82 4.99R = 5 16.03 13.48 9.51 9.32R = 8 21.99 18.91 16.37 16.18R = 12 30.20 26.45 26.18 25.76

The table shows the realized standard deviations (in annualized percent) for efficient frontier portfolios selected by the E/G and Fixed Variance models during the post-sample period January 1989 through September 1990. Model parameters were fitted on data from July 1977 through December 1988. Standard deviations were computed around the target rates of excess return relative to the domestic riskless interest rate, i.e., R = 0, 2, etc. The Minimum Risk portfolio was the frontier portfolio estimated to have minimum risk overall, regardless of expected return.