Roger G. Clarke Analytic/TSA Global Asset Management R. Matthew Tullis, C.F.A.
Liquid Asset Management, Zions Bank
July 1998
Abstract
How Much International Exposure is Advantageous in a Domestic Portfolio?
The question of how much international exposure is advantageous in a domestic portfolio is dependent on what the investor assumes about the long-run risk and expected return of the foreign assets and currency exposure and on the investor’s risk/return penalty. The analysis begins with the investor holding a core position in foreign assets to minimize the risk of the portfolio. Using estimates of volatility and correlation from market history, we suggest that a long-run allocation of 20% to 30% in foreign equity would not be unreasonable. However, an investor may be enticed to deviate from this core allocation based on the expected relative returns of domestic and foreign equity and on the expected currency return.
The authors thank Steven Thorley from Brigham Young University and Harin de Silva from Analytic/TSA Global Asset Management for their comments.
HOW MUCH INTERNATIONAL EXPOSURE IS ADVANTAGEOUS IN A DOMESTIC PORTFOLIO?
In recent years many U.S. investors have added international exposure to their domestic portfolios. The question naturally arises—how much international exposure is advantageous to add?
The answer to such a question depends in part on what the investor means by advantageous. In this analysis we frame the answer in the context of the impact that foreign asset exposure has on the portfolio’s expected return and risk. Within this framework, we suggest that from 20% to 30% of an equity portfolio might be allocated to a core foreign equity position using typical market parameters. The investor may be enticed to deviate from this core position as expected relative returns change in the short run.
Portfolio Risk and Return
The addition of foreign asset exposure affects both the return and risk of the portfolio. This can be seen by writing the return of a simple portfolio containing both domestic and foreign asset positions as:
where
wd = portfolio proportion in domestic assets wl = portfolio proportion in foreign assets
H = proportion of foreign currency exposure in the portfolio Rd = return on domestic assets
Rl = local return on foreign assets
Rc = currency return
f = currency forward premium or discount
The variance of the portfolio return can be written as:
The framework for analyzing how much foreign asset exposure is advantageous can be structured in two ways as illustrated in Exhibit 1. The first is to consider the impact on both risk and return from adding foreign asset exposure. The second considers only the impact on risk. Within this framework, the investor can choose to jointly optimize the asset and currency positions or to allocate the asset exposures given a currency policy of maintaining currency exposure at a predetermined proportion of the foreign asset exposure. The two extremes of this policy would be to maintain either a fully hedged or a fully unhedged currency position.
Forming Optimal Allocations: Joint Determination of Asset and Currency Exposure
To illustrate the effects of international assets we begin with a simple portfolio allocated to domestic and foreign assets. If the investor chooses the proportions of asset and currency exposure in order to jointly optimize the portfolio’s risk-adjusted expected return,
the optimal allocations to foreign assets, domestic assets, and currency exposure are
Equation (3) indicates that the optimal allocation to foreign assets is a complex expression related to the expected return on both the assets and the currency in addition to the respective variances and covariances. The tradeoff between expected return and risk is governed by the investor’s risk/return penalty . Investors with extreme risk aversion will have large values of while investors with low aversion to risk will have small values of .
In the event that the investor wants to only minimize the variance of the portfolio return without regard to the impact on return, equations (3) and (5) reduce to:
Notice that equations (3a) and (5a) contain only risk-related terms. Expected returns drop out of the equations because the investor is interested in only minimizing the risk of the portfolio without regard to the impact on return3. The minimum-variance perspective is often interesting as a starting point because measures of risk are usually more stable over time and easier to estimate than expected returns.
Forming Optimal Allocations: Predetermined Currency Strategy
The other cases of interest relate to the optimal allocations of asset exposure if currency exposure is maintained as a fixed proportion of foreign asset exposure. If both risk and return are taken into account in the optimization, the optimal asset allocation when currency exposure is fully hedged is:
Notice that the expected currency return is not a factor in equation (6) because the currency exposure is fully hedged. The forward premium captures the cost of the hedge.
When currency exposure is unhedged, the optimal asset allocation becomes:
When currency exposure is unhedged, the expected currency return becomes important in
determining the optimal asset allocation while the forward premium drops out.
The minimum-variance solutions to the two extreme cases are given in equations (6a) and (7a). If currency exposure is fully hedged, the minimum-variance allocation to foreign assets is:
The minimum-variance allocation to foreign assets if currency exposure is unhedged is:
Notice again that the minimum-variance solutions in equations (6a) and (7a) contain only risk-related terms since the investor is disregarding the impact on expected return.
Historical Market Experience
In order to use the allocation framework developed in the previous section, the investor must supply estimates of expected returns and risk. Historical market experience may provide some insights as to what estimates might be reasonable to use as long-run forecasts. As a broad proxy for international exposure in an equity portfolio we use data from the Morgan Stanley EAFE index. We recognize that many investors may want to do a more complex analysis using individual country indices while incorporating both stocks and bonds in the analysis. However, to illustrate the general principle without adding additional complexity, we restrict our analysis to the S&P 500 for U.S. equities and to the EAFE index as a commonly used proxy for international equities.
Exhibit 2 captures the historical experience of the risk of the respective equity indexes and the currency exposure in the EAFE index from a U.S. investor perspective. The standard deviations are calculated using rolling 3 years of monthly data beginning in 1988. Notice that the respective equity markets have generally had higher volatility than the composite currency exposure in the EAFE index. Underlying equity market volatility for the S&P 500 and EAFE (currency hedged) has averaged 11.4% and 15.4% respectively over this period, while currency volatility has averaged 8.8%.
We also plot the volatility of the combined asset and currency exposure embodied in the EAFE index (currency unhedged). Note that in general the volatility of the underlying foreign equity markets is increased by adding currency exposure with the combination averaging 17.4% over the period. This is because the diversification benefits of adding currency exposure are generally not large enough to overcome the volatility added by the underlying currency exposure. Some have suggested that currency exposure provides a natural diversification benefit in an international portfolio. For currency exposure to decrease the volatility of the underlying foreign assets, it would require that;
This condition would be met if the correlation between asset returns and currency returns ( lc) were negative and less than one-half the ratio of the currency and asset standard deviations.
Correlations have generally not been this negative in the past suggesting that not hedging currency exposure will generally increase the volatility of international assets.
Exhibit 3 shows the relationship in equation (9) using rolling 3-year standard deviations and the correlation between EAFE-Hedged and EAFE-Currency returns. Over this period of time, the threshold correlation has usually been between –0.2 to –0.3. Notice that the foreign hedged equity correlation with currency returns has generally not been sufficiently negative for the diversification benefits of currency exposure to overcome its own volatility. For only a brief period in recent years has the correlation been negative enough for an unhedged EAFE portfolio to have comparable volatility to a fully hedged EAFE portfolio.
Also of interest are the correlations between domestic and international equity markets and EAFE-Currency returns. Rolling 3-year correlations are plotted in Exhibit 4 for each pair. The correlation between the equity markets has averaged 0.51 while the correlation between each equity index and the combined currency return has averaged -0.04 and -0.06 for the S&P 500 and EAFE-Hedged respectively.
Examination of the historical data from Exhibit 4 shows that the correlation between equity markets has been positive and has fluctuated around 0.50. The
correlation of currency returns with U.S. equity returns has fluctuated around zero while the correlation of currency returns with international equity market returns has also been close to zero with a downward trend pulling it below zero in recent years. The standard deviations of returns for the equity markets, shown in Exhibit 2, have been trending downward though we have seen a movement back above 10.0% in recent years. The standard deviation of currency returns has been stable at around 8.0% for several years.
The cumulative return from each of the respective indices is shown in Exhibit 5. Over this period of time the return on the U.S. equity market has far outstripped the foreign equity markets as captured by the EAFE index while the cumulative relative return on the U.S. dollar (net of the forward premium) has been close to zero. However, notice that there have been periods where domestic assets performed differently than foreign assets and the U.S. dollar performed differently than the combination of currencies in the EAFE index. As with most markets the historical data over this period of time suggest that there is usually more stability in the relative risk between markets than there is between the relative returns.
Minimum-Variance Allocations
We might consider the minimum-variance allocation as a core allocation to foreign equity exposure in a portfolio. In theory, departure from the minimum-variance position would be expected to add additional risk to the portfolio. As a result one might think of the minimum-variance position as a starting point for an allocation. The investor could be enticed to move away from this allocation and accept higher risk upon expectation of higher returns.
Given the historical pattern of standard deviations and correlations it is interesting to look at the minimum-variance foreign equity allocation to foreign equity based on the rolling three-year statistics reported in Exhibits 2 and 4. We first use these historical patterns as estimates of the corresponding risk parameters in equations (3a), (4), and (5a) to track the minimum-variance allocations as shown in Exhibit 6. Notice that based on three-year rolling estimates the allocation to foreign equity averaged about 20% for many years with currency exposure ranging mostly between –20% and 20%. These minimum- variance allocations are derived by jointly optimizing the asset and currency positions. In most periods, less currency exposure than foreign equity exposure is recommended suggesting that at least some currency exposure should be hedged in the minimum- variance allocation.
The minimum-variance foreign equity exposures implied from equations (6a) and (7a) are given in Exhibit 7 along with the joint optimization exposures. One line shows the minimum-variance foreign equity allocation when currency exposure is fully hedged while a second line shows the minimum-variance allocation when currency exposure is unhedged. Notice the difference between the allocations. In general, the minimum- variance allocation to foreign equity when currency exposure is unhedged is smaller than when currency exposure is completely hedged or when currency exposure is jointly
optimized. This occurs because forcing foreign assets to maintain complete currency exposure usually adds to their volatility requiring the optimizer to allocate less to foreign assets4. In periods when joint optimization recommends currency exposure close to zero, the minimum-variance allocation to foreign stocks is close to the completely hedged allocation. Otherwise, the fully hedged currency allocation to foreign stocks is smaller than when the currency can be optimized jointly with the asset positions. The jointly optimized minimum-variance solution and the completely hedged minimum-variance solution track closely over much of this period because the correlations of both domestic and foreign equity returns with currency returns have been close to zero.
If we make assumptions about the long-run risk characteristics of the equity and currency markets, we can calculate the long-run minimum-variance allocation to foreign equity using the equations derived in the first section. Investors may make a variety of assumptions about the parameters, but suppose we choose the following scenario consistent with recent market experience; suppose that foreign equity markets are assumed to have 20% greater volatility than the domestic market and that the currency volatility is assumed to be 20% less than that of the domestic equity market in the long run. Furthermore, assume that the correlation of the equity markets is assumed to be 0.50 while the currency market is assumed to have zero correlation with the equity markets. Under these assumptions the long-run minimum-variance allocation to domestic and foreign equity is given in Exhibit 8 for the three cases we have highlighted.
Notice that the long-run minimum-variance allocation to foreign equity exposure using our assumptions varies between approximately 20% to 30%. If currency exposure is fully hedged, the minimum-variance foreign equity exposure moves to the higher end of the range while if currency exposure is unhedged, foreign equity exposure moves to the lower end of the range. This occurs because coupling currency exposure to the foreign equity exposure adds additional volatility and reduces the desire to invest internationally when minimizing risk. The jointly optimized allocation produces the same recommendations as the fully hedged position because we have assumed that currency returns are uncorrelated on average with asset returns. When this is the case, the minimum-variance allocation to currency exposure in the portfolio is zero. For a typical portfolio with positive asset exposures we note in equation (5a) that only if currency returns are negatively correlated with asset returns will positive currency exposure be optimal in a minimum-variance framework.
Long-Run Risk/Return Tradeoffs
An investor might be enticed away from the minimum-variance allocation by expectations of increased return from one equity market or the other or because of expected currency movements. In such a case equations (3), (5), (6), and (7) allow the investor to trade off increased risk against increased expected return. The investor’s tradeoff penalty is expressed as .
To illustrate how the investor might be enticed away from the minimum-variance allocation to foreign equity by expected asset and currency returns, suppose that we assume the same long-run risk assumptions as in the previous section. In addition, we assume that because of the higher perceived risk of foreign equity, the expected return on foreign equity is 2.0% more than that for domestic equity. Furthermore, if the expected
5 E(R ) c F Rreturn on currencies is equal to the forward premium in the long run , i.e. c , we
can show the impact on portfolio allocations for various levels of investor risk aversion using equations (3)-(7). Exhibit 9 shows the recommended portfolio allocations and the impact on expected return and risk for the simple cases we have outlined. In Exhibit 10 we plot the tradeoff between risk and return for the respective cases. Notice that the curve for the unhedged EAFE exposure lies below that of the hedged and jointly optimized cases because the currency exposure adds risk without adding any extra return in the long-run.
Finally, in Exhibit 11 we examine the impact of the expected forward currency
surprise E(Rc ) f on the portfolio allocations. In the previous example we used the
long-run assumption that E(Rc ) f 0 . Some have observed that when the forward
premium is negative, Rc f tends to be positive and when f is positive Rc f tends to be negative. In Exhibit 11 we show the impact on the asset and currency allocations
under the assumption that E(Rc ) f is non-zero. Notice that under our assumptions the jointly optimized currency exposure is either positive or negative depending on whether the forward premium is either negative or positive. A negative forward premium (higher foreign interest rates) entices the investor to add currency exposure and a positive forward premium (higher domestic interest rates) entices the investor to reduce currency exposure. One way to think about the allocations in Exhibit 11 is to consider them to be tactical changes in the long-run allocation caused by changes in the forward currency premium6. Depending on the investor’s hedging policy, an increase in the expected currency surprise could induce the investor to either increase or decrease the foreign asset exposure. If the currency exposure is fully hedged, a negative forward premium would tend to decrease the allocation to hedged foreign equity because of the increased cost of hedging. However, if a negative forward premium is associated with positive expected currency returns, unhedged foreign equity exposure would tend to be increased in order to take advantage of the extra return provided by the currency exposure.
Conclusions
The question of how much international exposure is advantageous in a domestic portfolio is dependent on what the investor assumes about the long-run risk and expected return of the foreign assets and currency exposure and on the investor’s risk/return penalty. The analysis began with the investor holding a core position in foreign assets to minimize the risk of the portfolio. Using estimates of volatility and correlation from market history, we suggest that a long-run allocation of 20% to 30% in foreign equity would not be unreasonable. However, an investor may be enticed to deviate from this core allocation based on the expected relative returns of domestic and foreign equity and
on the expected currency return. One example of a tactical deviation from this long-run allocation is the presence of positive or negative forward currency premiums. A positive forward currency premium may entice the fully hedged investor to decrease currency exposure and increase foreign equity exposure while a negative forward premium would entice the fully hedged investor to increase currency exposure and decrease foreign equity exposure.
Footnotes
- For computational ease we have chosen to approximate the foreign asset return by separating it into two components—the fully hedged local asset return and the currency exposure. This structure allows us to isolate the foreign asset effects from the foreign currency effects.
- See Clarke and Kritzman (1996) for a more detailed treatment of the derivation of these relationships.
- Notice that the expectational terms also drop out if the expected domestic asset return is equal to the expected hedged local foreign asset return, i.e. E(Rd) = E(Rl) + f,
and if the expected currency return is equal to the forward currency premium,
i.e. E(Rc) = f. - The diversification benefits of currency exposure when both domestic and foreign assets are included in the portfolio are more complex because the variance of the portfolio return depends on the currency correlation with both domestic and foreign assets. Using equations (6a) and (7a) it can be shown that the minimum-variance allocation to foreign assets when currency exposure is fully hedged is greater than the allocation when currency exposure is unhedged if the correlation between foreign asset and currency returns is less than:
If the domestic and foreign asset return variances are equal to each other, this condition reduces to equation (9) in the text. That is, unless the correlation of foreign asset and currency returns is sufficiently negative, unhedged currency exposure usually adds more volatility than its diversification benefit would eliminate. The minimum-variance allocation to foreign assets is subsequently reduced compared to a fully-hedged allocation.
- On average this seems to be a good assumption though there is some evidence that on a conditional basis the expected currency surprise [E(Rc)-f] tends to be positive when the forward premium f is negative and [E(Rc)-f] tends to be negative when f is positive. See Green (1992), Hazuka and Huberts (1994), and Kritzman (1993).
- For purposes of illustration in Exhibit 11, we have assumed that an increase in expected currency surprise is split between an increase in expected currency return and a decrease in the forward premium. If the change in expected currency surprise is caused totally by a change in the forward premium, there would be relatively little impact on the allocation to unhedged foreign equity because expected currency returns would remain unchanged.
References
Clarke, R. and M. Kritzman. 1996. Currency Management: Concepts and Practices Charlottesville, Va.: The Research Foundation of the Institute of Chartered Financial Analysts.
Green, P. 1992. “Is Currency Trading Profitable? Exploiting Deviations from Uncovered Interest Parity.” Financial Analysts Journal (July/August):82-86.
Hazuka, T., and L. Huberts. 1994. “A Valuation Approach to Currency Hedging.” Financial Analysts Journal (March/April):55-59.
Kritzman, M. 1993. “The Optimal Currency Hedging Policy with Biased Forward Rates.” The Journal of Portfolio Management (Summer):94-100.
