Asset Allocation Introduction In Chap. 4 we developed a utility function for active management which separated benchmark timing from asset selection. The intervening chapters have focused on asset selection, postponing treatment of benchmark timing to Chap. 19. Before moving on to that treatment, we want to focus separately on a style of investment—asset allocation—which lies somewhere between asset selection and benchmark timing. Asset allocation comes in several varieties: strategic versus tactical, and domestic versus global. The process of selecting a target asset allocation is called strategic asset allocation. The variation in asset allocation around that target is called tactical asset allocation. There is an analogy between strategic asset allocation and the choice of a benchmark in a single equity market, and between tactical asset allocation and active management within the equity market. We do not address the important question of strategic asset allocation in this book.1 We assume that we have simply been presented with the benchmark. This chapter explicitly addresses tactical asset allocation. Domestic tactical asset allocation typically involves actively choosing allocations to at least three asset classes—stocks, bonds, and cash— 1For a discussion of strategic asset allocation issues, see Ambachtsheer (1987) or Sharpe (1987).
Page 518 and possibly more. Global asset allocation involves actively choosing allocations to typically 20 or so global equity and bond markets. The principles for actively managing asset selection in one market apply to tactical asset allocation. The main difference is in the number of assets. There are typically from 3 to 20 in tactical asset allocation and from 100 to several thousand for portfolio management in a single market. Given this important difference, the fundamental law of active management implies that tactical asset allocation will have to surmount a high skill hurdle to compete with asset selection. Asset allocation differs from asset selection in other ways as well. Asset allocation strategies often involve time series, rather than cross-sectional analysis, and rely on different types and sources of information. Currencies are more central to global asset allocation than even to international asset selection. Finally, traditional asset allocation managers have eschewed the explicit acknowledgement of a benchmark in portfolio construction. Given the relatively small number of bets, why pursue asset allocation? These strategies have several desirable features: new opportunities for enhanced returns, often beyond national boundaries; the possibility of increased diversification; and the control of the major determinant of total returns. The asset allocation decision incurs large risks and offers large potential returns. Global asset allocation is the largest source of differences in performance among global portfolios.2 This chapter will present a practical framework for researching global asset allocation strategies, and will include examples of global asset allocation to clarify the discussion. Major points will include the following: • Researching asset allocation strategies is a three-step process: forecasting returns, building portfolios, and analyzing out-of-sample performance. • The procedure for forecasting returns for asset allocation differs from that for asset selection in its focus on time series instead of cross-sectional analysis. In spite of this, 2See Solnik (1993b).
Page 519 asset allocation, like asset selection, adds value through identifying relative performance. • Traditional asset allocation ignores any explicit benchmark, simply trading off expected excess return against total risk. This causes substantial problems. We will implicitly include a benchmark in the expected returns, a procedure complicated by the presence of currencies. The Three-Step Process Researching asset allocation strategies for institutional investors is a three-step process: forecasting asset-class expected returns, building optimal portfolios, and testing their out-of-sample performance. The forecasts may come from a quantitative model, a qualitative model, or a combination of the two. We will demonstrate a quantitative model of expected returns. The quantitative framework will differ somewhat from that for individual security selection in that it will build a separate time series model for each asset class, as opposed to a cross-sectional model of differences between individual securities. There are two related reasons to focus on separate time series models rather than a cross-sectional model. First, to the extent that these asset classes exhibit low correlations as compared to asset returns (which usually include a strong market factor), a time series approach makes better sense. Second, the typical explanatory variables for asset-class returns focus on the individual time series rather than on cross-sectional comparisons, and aren't always comparable across countries and asset classes. There may still be a place for cross-sectional models, especially focusing on a smaller set of correlated asset classes (e.g., European equities), but we will not deal with such models here. Even though asset allocation forecasting uses time series analysis, asset allocation strategies add value by identifying relative performance. They bet on the relative performance of asset class A versus asset class B this month, not on the performance of asset class A this month versus last month. Underlying the reliance on time series models is the idea that accurate time series forecasts provide accurate relative valuations.
Page 520 Given the expected excess return forecasts, step 2 is to build optimal mean/variance portfolios. A problem with traditional asset allocation is its trade-off of expected excess returns against total risk, without regard to any benchmark. The resulting optimal portfolio weights are extremely sensitive to small changes in expected returns, and often vary widely from benchmark weights. A Bayesian approach (following the principles discussed in Chap. 10 and dependent on a benchmark) can control this problem. The final step is out-of-sample performance analysis on our optimal portfolios. We must test these strategies out-of-sample to determine the information content of the expected return models. Out-of-sample testing is always critical for researching investment strategies. For time series–based approaches, even the in-sample period must include explicit testing of forecasting ability. Let's say we fit a time series model over a period from T1 to T2. The model's estimates for times t < T2 will then generally include information from times all the way through T2. In particular, the estimated regression coefficients will depend on all the data through T2. Hence the in-sample testing must include estimating the model from T1 to T2 and looking at its forecasting ability beyond T2. We will measure forecasting ability using the information coefficient (IC), the correlation of forecasts and realizations. After an in-sample look at forecasting ability to choose appropriate explanatory variables, out-of-sample testing will validate the performance of the model. Step 1: Forecasting Returns We will illustrate step 1 by building a model to forecast monthly excess returns for the German, Japanese, U.K., and U.S. equity markets using data from January 1985 through December 1992. In this example, we will build a linear, regression-based model where we attempt to explain asset-class excess monthly returns r(t) using a set of explanatory variables {xj(t)}. The information used to forecast r(t), the explanatory variables {xj(t)}, are available at
Page 521 the beginning of period t. We estimate coefficients bj to relate the explanatory variables to the returns. As described before, we will estimate model coefficients over part of the in-sample data, and then test forecasting ability through the remainder of the in-sample period. In fact, our initial regression will use the first 30 months of data to forecast returns over the 31st month. Our next regression will use the first 31 months of data to forecast returns over the 32nd month. We will expand our regression window until we reach 60 months of data, after which we will always use the past 60 months of data to forecast returns over the next month. In this example, our 8 years of in-sample data will allow us to estimate information coefficients using 5.5 years of in-sample forecasts. These in-sample information coefficients help us choose appropriate explanatory variables. Later on, we will test model performance in an out-of-sample period, looking at the performance of portfolios built using the model. Building portfolios will require the information coefficient for each country, which we will estimate using the available history of forecasts and realizations (which start in month 31). We will use the same five explanatory variables for each market.3 These explanatory variables are the predicted dividend yield for the market, the short-term interest rate in the market, the difference between that short rate and the U.S. short-term interest rate, the exchange rate to U.S. dollars, and a January dummy variable. Two of these variables drop out for the U.S. model. Table 18.1 displays the statistical results for the markets, based on model estimation as of January 1993, the beginning of our out-of-sample period. Table 18.1 displays the coefficients bj and their t statistics, as well as the adjusted R2 statistics for each market. We can first make some overall observations about the performance of the model, and then focus in on some details to compare it to intuition. Overall, the explanatory power of the model varies widely from country to country over this mainly in-sample period. The 3See the references of Solnik (1993b), Emanuelli and Pearson (1994), and Ferson and Harvey (1992) for more details.
TABLE 18.1 Country Intercept Dividend Yield Short Rate Relative Short Rate Exchange Rate J Germany –0.24 19.7 2.52 –0.96 –0.58 –1.4 4.0 2.3 –1.8 –2.8 Japan –0.47 24.7 2.03 –2.28 26.10 –1.7 2.0 1.5 –2.2 0.9 United Kingdom –0.10 6.9 0.65 –1.92 –1.10 –0.9 3.1 1.5 –2.5 –1.6 United States –0.08 4.0 –0.56 –1.6 1.8 –1.4
Page 523 average adjusted R2 is roughly 9 percent, ranging from close to zero for the United States to over 20 percent for Germany. For a model of returns, these results are on average good. Looking at the estimated coefficients, we can first see that the coefficients for dividend yield are always positive. Higher dividend yields imply higher expected returns. In the United States, for example, the coefficient is 4.0. Each 1 percent increase in predicted dividend yield in the United States raises our expected monthly U.S. equity market excess return by 4.0 percent. The coefficients for the short interest rate are sometimes positive and sometimes negative. Looking again at the example of the United States, each 1 percent increase in the short rate lowers the expected monthly market excess return by 0.56 percent. The coefficients for the difference between short interest rates in the local market and in the United States are negative. In the United Kingdom, a 1 percent rise in this differential implies a decrease of 1.92 percent in the expected monthly market excess return. The intuition here is perhaps that both higher short rates and higher rate differentials to the United States should lower expected equity market returns. Our results for the short-rate effect do not entirely agree with this intuition. However a more detailed analysis of this model shows a high correlation between these two variables in some countries for the 60 months ending January 1993. Given this high correlation of exposures, coupled very often with coefficients of differing signs, the net result may well be that higher short rates do indeed imply lower market returns. If we were interested in pursuing this model further, the natural solution would be to delete one of these explanatory variables, given its high correlation with other explanatory variables. The coefficients for the local exchange rate (U.S. dollars per unit of local currency) is negative for Germany and the United Kingdom, and positive for Japan. Only the result in Germany is statistically significant. The negative coefficient means that as the local currency depreciates relative to the dollar (as the dollars per unit of local currency increases), the expected return increases. For example, if the dollars per deutsche mark decreases from $0.59 to $0.58, the expected monthly excess return to the German market will increase by –0.58 · ($0.58 – $0.59) = .0058 = 0.58 percent.
Page 524 The January effect is positive for Germany and the United Kingdom. The coefficient for Germany implies that the expected monthly excess return is higher by 2.89 percent in January than in the other months. In our implementation of this example, we have used the raw values for all the explanatory variables. This can lead to estimated coefficients of widely differing magnitudes. It is only xjbj that has the units and magnitude of returns. If the dollar value of a U.K. pound has a very different magnitude from the U.K. short interest rate, then the estimated coefficients associated with these explanatory variables will have compensatingly different magnitudes. This can obscure our insight into the relative importance of these effects (although the t statistics will help). A variant of our implementation is to linearly rescale the explanatory variables—subtract the sample mean and divide by the sample standard deviation—to put all the variables on an equal footing. This linear transformation will have no effect on explanatory power. However, after the transformation, the magnitude of the estimated coefficients will identify which variables explain most of the monthly variation in asset-class returns. This example is quantitative, yet it also provides qualitative insight and intuition into market behavior. It explicitly connects economic variables to expected returns. These connections should be intuitive. It can identify which economic variables are relatively more important for market returns, and help in predicting market direction. The above discussion shows that this simple model exhibits reasonable explanatory power, with coefficients that generally conform to our intuition. The next step is to implement these forecasts in optimal mean/variance portfolios. Before doing that, however, we should briefly discuss alternatives for maximizing the explanatory power of these models. Our simple model constrained the explanatory variables to be the same in each market. An obvious extension would be to allow different variables in different markets based on how well they forecast returns in those markets (and accounting for colinearities in some markets). We could extend the explanatory variables to include macroeconomic variables and previous or lagged monthly returns (to capture mean reversion or trends). We could add ana-
Page 525 lysts' forecasts or even forecasts of political risks. We can combine the forecasts from a quantitative model with forecasts from the more qualitative sources that are traditional in global asset allocation. We can even use the insights of the model into the size and sign of these coefficients to help the traditional analyst make better-informed qualitative forecasts. And, of course, we can extend the analysis to include bond markets as well as equity markets. Step 2: Building Optimal Portfolios Our estimated model [Eq. (18.1)] provides monthly expected excess returns to a set of equity markets. But mean/variance optimal portfolios that trade off expected excess return and total risk are extremely sensitive to these expected returns. Whereas active or residual asset returns exhibit relatively low correlations, asset-class excess returns exhibit high correlations. We could handle this problem by using expected active or residual returns relative to an asset allocation benchmark. This is the approach of the rest of the book. Instead, for this chapter, we will stick to the traditional asset allocation methodology, and describe how to avoid the sensitivity problem. To do this, we will effectively sneak the benchmark into our expected excess return forecasts.4 Our approach will be the methodology we developed in Chap. 10, refining these ''raw" expected returns The basic forecasting formula is where E{r| } is the refined expected return conditional on the forecast . This simplifies to the forecasting rule of thumb When we implement our simple model, each market will have its own information coefficient, which may vary over time depending on the model's forecasting ability in that market. Only for 4Note that the problem involving the trade-off between active or residual return and risk differs from the standard global asset allocation methodology in that it involves the aversion to active or residual risk, not to total risk.
Page 526 markets with significant information coefficients will our forecast returns differ from consensus expectations. To use Eq. (18.3) in our example model, we must analyze one further detail: Where do we find the consensus expected excess returns? Here is where we will sneak a benchmark into the problem: We will use the consensus returns implied by our benchmark portfolio. Given a benchmark, then, as we saw in Chap. 2, the consensus expected excess returns are just where β is the beta of the asset class relative to the benchmark and fB is the consensus expected return to the benchmark. This approach is more complex with multiple currencies, especially if we want consensus expected excess returns that are reasonable from many different currency perspectives.5 International Consensus Expected Returns So what is the problem with backing out consensus expected returns in the international context? If we aren't careful, the results will look very different from different perspectives. Part of the problem is the home bias. If we start by assuming, for example, that the standard asset allocation of a U.S. pension fund, with at most about 20 percent invested outside the United States, is optimal, Eq. (18.4) leads to artificially low expected excess returns for foreign assets. Perhaps more surprising, we can't solve the problem by simply reweighting the assets in the presumed efficient portfolio so that, for example, 60 percent of the holdings are outside the United States and 40 percent are within the United States, more in line with global capital market weights. At this level, Eq. (18.4) implies consensus expected excess returns for foreign assets that are too large.6 But this domestic view is only part of the problem. 5If your focus is solely on asset allocation for investors from a single country, and their benchmark is imposed externally, then you may not require "sensible" expected returns. You can use the expected returns consistent with the imposed benchmark, whether they appear sensible or not. In that case, the discussion in the next section is just a cultural diversion. 6See Grinold (1996) for details.
Page 527 An Aside: Currency Semantics We should be aware of a semantic pitfall. Currency has a double meaning: It can be either a numeraire of perspective or an asset. It is this double nature that gives a theological aspect to that old conundrum: "Is currency an asset?" We will maintain that short-term, default-free, interestbearing bills held in foreign countries are indeed assets. The return on such a bill is known in its home country; the only uncertainty in its return for a foreign investor is the exchange rate risk. Thus, when we speak loosely of currency as an asset, we actually mean a short-term, default-free, discount bill in that particular country. On the other hand, we use the notion of currency to indicate an investment perspective—i.e., how we keep score. Phrases like "for the yen-based investor" or "this is not as attractive from a sterling perspective" indicate that we are using the currency as a numeraire. We will use currency in both its meanings. We start with an example of consensus expected returns viewed from several numeraire perspectives. We will examine stocks and bonds in the four major countries treated in our expected return model: Germany, Japan, the United Kingdom, and the United States. Our presumed efficient portfolio is 60 percent stocks and 40 percent bonds in each country. The country weights are Germany, 10 percent; Japan, 30 percent; the United Kingdom, 20 percent; and the United States, 40 percent. We estimate risk from historical data for 1970 through 1995. Table 18.2 shows the consensus expected returns from each of the four different currency perspectives. The wide range of expected excess returns suggests a major inconsistency. For example, a Frankfurt-based investor holding dollars will expect 4.80 percent excess return, while a New Yorkbased investor holding deutsche marks (or Euros, post 1999) will expect a 1.93 percent excess return. While win/win is a laudable concept, this seems to carry the notion too far. In fact, as we show below, it is not possible to have numbers this high in a consistent scheme. This example suggests two questions. First, how do we consistently transform expected excess returns from one currency perspective to another? If we know the expected excess returns for a
Page 528 TABLE 18.2 Numeraire Perspective Asset Germany Japan United Kingdom United States German stocks 5.10% 5.18% 5.48% 6.27% Japan stocks 9.70% 5.36% 7.27% 7.54% U.K. stocks 10.76% 7.41% 6.49% 7.63% U.S. stocks 10.62% 6.62% 7.49% 5.54% German bonds 0.46% 1.05% 1.56% 2.73% Japan bonds 4.15% 0.36% 2.36% 2.08% U.K. bonds 4.64% 2.18% 0.91% 2.97% U.S. bonds 5.96% 2.44% 3.40% 1.87% Deutsche marks 0.00% 0.47% 0.75% 1.93% Yen 3.80% 0.00% 1.75% 1.30% Sterling 3.31% 0.97% 0.00% 1.48% Dollars 4.80% 0.83% 1.78% 0.00% London-based investor, how do we deduce the expected excess returns for a New York– or Tokyobased investor? Second, how do we determine a "reasonable" ex ante efficient portfolio? In particular, what role will currency play in that portfolio? The first question has a definite answer, as discussed in Black (1990) [also Black (1989)]. International returns, by their nature, involve products of the return in the local market and changes in the level of exchange rates. This type of relationship will hold between any two countries and for any asset. We illustrate using U.S. and U.K. investors and the Australian stock BHP: where RBHP(US/0,t) refers to the cumulative total return7 to BHP over the period from 0 to t from the U.S. dollar perspective, and we represent a 3.5 percent return as 1.035. Suppose a British investor observes a 3.5 percent return to BHP between 0 and t. Furthermore, the exchange rate at time 0 is $1.50 per pound, and the exchange 7We are assuming reinvestment of any cash flows.
Page 529 rate at time t is $1.52 per pound. Then the U.S. investor will observe a 4.88 percent return over the interval. From that simple relationship, we can derive a rule linking excess rates of return in different locales. We always measure returns in excess of the investor's risk-free alternative; i.e., the excess return on BHP for a U.S. investor is relative to the risk-free return in the U.S. market: So the excess return to BHP from the U.S. perspective equals the excess return to BHP from the U.K. perspective, less the excess return to U.S. Treasury bills from the U.K. perspective, plus the covariance between BHP and U.K. Treasury bills from the U.S. perspective (see Appendix C for details). Equation (18.6) deals with realized returns. We can also take expectations to derive a link between the expected returns: This result answers our first question about moving expected excess returns from one locale to another. It requires both expected return and risk forecasts. The second question, concerning a reasonable presumed efficient portfolio, has no totally acceptable answer in the domestic case. It will only get harder in the global case. We will have to be satisfied with a reasonable way to treat currency in the global context. Here is the procedure from Grinold (1996). How can we get rid of the exchange rates? One way is to envision a world with a single currency. Suppose we had a composite country that we will call COM. The currency for COM is a mixture of the currencies of all the countries, a composite currency. A portfolio called BSK (for basket) determines the makeup of the composite currency. For example, the basket could be 40 percent dollars, 30 percent yen, 20 percent sterling, and 10 percent deutsche marks. Recall the double nature of currency. The currency portfolio, BSK, defines the composite currency numeraire, COM. It is also an asset. It serves as the risk-free asset from the COM perspective, and German, Japanese, U.K., and U.S. investors can hold BSK. We can rewrite Eq. (18.7) using this new currency asset and numeraire:
Page 530 Now, we can't expect the consensus expected returns procedure to work perfectly from the COM perspective, since it doesn't work perfectly in a pure domestic setting. We are striving for a relative measure of satisfaction rather than an absolute measure. The best way to judge is to look at the improvement we get over using the naïve procedure. To do that, we will observe the improvement in our example. But first, we will trace the presumed efficient portfolio as we move from one locale to the other. We'll see that the presumed efficient portfolio changes. Let's start from the COM currency perspective and presume that portfolio Q is efficient. Now, from the COM perspective, portfolioQ explains all expected excess returns. For example, and for a U.S. risk-free instrument from the COM currency perspective, We can use Eqs. (18.9) and (18.10) for the expectations in Eq. (18.8). Furthermore, we can use this approach to calculate an effective efficient portfolio from other perspectives. The idea is to transform the betas from the COM perspective to the dollar perspective. Then we can solve for the portfolio, QU.S., that explains excess returns from the U.S. perspective. The technical appendix contains the details. We find that the only change required in order to move from Q to QU.S., or from QU.S. to QU.K., comes in the currency position. For example, to move from Q to QU.S., we partially hedge against changes in the value of the currency basket. Portfolio QU.S. will be long dollars and short the basket currency, relative to portfolio Q. As the technical appendix will show, to move from Q to QU.S., we subtract a fraction [1 – (σQ/SRQ)] of the portfolio that is long the basket currency (e.g., 40 percent dollars, 30 percent yen, 20 percent deutsche marks, 10 percent pounds) and short (–100 percent) dollars. That fraction is roughly 65 percent, assuming a Sharpe ratio
Page 531 of about 0.35 and a volatility of about 12 percent for the global asset allocation portfolio Q.8 Our presumed efficient portfolio from the COM perspective will have 60 percent stock and 40 percent bond allocations in each country, with country weights 40 percent in the United States, 30 percent in Japan, 20 percent in the United Kingdom, and 10 percent in Germany. It contains no explicit currency positions. Table 18.3 displays the consensus expected returns from varying perspectives, including the composite currency. Notice the very low expected excess returns for holding currency. The forecasts of expected excess returns for the currencies are more reasonable and the expected excess returns for stocks and bonds more consistent across countries than in Table 18.2. In particular, let's look at the German investor holding U.S. T-bills and the U.S. investor holding German T-bills. Using Eq. (18.7), and TABLE 18.3 Numeraire Perspective Asset Germany Japan United Kingdom United States Composite German stocks 4.72% 4.59% 4.68% 4.96% 4.51% Japan stocks 6.81% 5.70% 6.15% 6.32% 5.89% U.K. stocks 7.51% 6.62% 6.45% 6.80% 6.49% U.S. stocks 6.79% 5.76% 5.98% 5.65% 5.60% German bonds 0.65% 0.64% 0.77% 1.14% 0.61% Japan bonds 1.52% 0.53% 1.00% 1.04% 0.67% U.K. bonds 2.14% 1.45% 1.19% 1.75% 1.33% U.S. bonds 2.64% 1.71% 1.95% 1.71% 1.60% Deutsche marks 0.00% –0.04% 0.05% 0.41% –0.09% Yen 0.99% 0.00% 0.41% 0.42% 0.09% Sterling 0.85% 0.19% 0.00% 0.43% 0.06% Dollars 1.08% 0.06% 0.29% 0.00% –0.07% 8In other chapters, we discussed portfolio Q volatilities in the range of 15 to 20 percent. Those numbers assume only equity investments. The lower volatility quoted here reflects the significant exposure to fixed income.
Page 532 remembering that the excess return for a German investor on German T-bills is, by definition, zero, In Table 18.3, the 0.41 percent return for a U.S. investor holding German bills and 1.08 percent for a German investor holding U.S. bills are consistent with a 12.2 percent exchange rate volatility. By contrast, the expectations in Table 18.2—a large 1.93 percent for a U.S. investor holding German bills and an enormous 4.80 percent for a German investor holding U.S. bills—are consistent with a huge exchange rate volatility of 25.9 percent. The realized volatility of this exchange rate over any sizable period in the last 20 years has consistently been in the 11 to 13 percent range. Step 3: Analyzing Out-of-Sample Portfolio Performance The last step in our methodology involves analyzing the out-of-sample performance of our optimized portfolios. Remember that a fair test of these time series models requires this out-ofsample testing. Focusing on excess returns to the strategy, we can examine the cumulative return plots and compare portfolio performance to benchmark performance. We can calculate the Sharpe ratios for the portfolio and the benchmark. We can also look at the portfolio's active returns. We can examine the cumulative active returns, the information ratio, and the t statistics. A more detailed performance analysis would also look at turnover, performance in up markets versus down markets (defined by the benchmark or the numeraire market), the number of up and down months, and which particular markets contributed the most to the outperformance. Performance dominated by one particular market would be a worrisome sign. The goal of this out-of-sample performance analysis step is to fairly measure the value added of the strategy, and possibly identify research directions for model improvement. Note that this final step of analysis of asset allocation performance measures how well we identify relative value among asset classes, since we overweight some and underweight others. Our return forecasts use time series
Page 533 models. To add value, those time series models must forecast cross-sectional returns. Summary We have discussed a three-step procedure for researching practical asset allocation strategies. These steps include forecasting expected asset class returns, building optimal portfolios, and analyzing the out-of-sample performance of those portfolios. We saw that asset allocation strategies, unlike asset selection strategies, rely on time series analysis to forecast returns. We discussed how traditional asset allocation trades off expected excess returns against total risk in building portfolios. We showed how to retain that traditional approach, but avoid its problems, by refining the expected excess returns based on a benchmark. This process is complicated for global investors by the presence of currencies. This framework has importance far beyond quantitative strategies for global asset allocation. It can provide intuition and control for quantitative or qualitative approaches to this critical element of global investing. Notes There are two distinct strands of research concerning asset allocation. One involves building models to forecast expected returns for different asset classes. The other focuses on appropriate methods for constructing portfolios. Let's start with the research on international expected returns. Solnik has worked extensively on modeling expected returns to international equity and bond markets. In his 1993 Journal of Empirical Finance paper, he models expected returns to nine equity and nine bond markets using three fundamental variables—dividend yield, the short interest rate, and the long interest rate—plus a January dummy variable. Solnik's 1993 monograph, Predictable Time-Varying Components of International Asset Returns, extends his model by including the spread between short interest rates and the U.S. short interest rate and lagged market returns as additional explanatory variables. Emanuelli and Pearson (1994) also use IBES earnings forecast data to help explain market returns. In particular,
Page 534 they define an earnings forecast revision ratio based on the number of up and down earnings forecast revisions in the market as a predictor of future market returns. Ferson and Harvey (1992) have also built models of expected returns to 18 different international equity markets. The review paper by Solnik includes an extensive bibliography of these and other efforts to build models of expected returns. Black and Litterman's work on portfolio construction for global asset allocation focuses on an entirely different challenge of global investing. Starting with the traditional mean/variance framework, they discuss a well-known problem for global asset allocation: The optimal portfolio weights can be extremely sensitive to small changes in expected returns. The underlying cause is the high correlation of many asset classes. To an optimizer, they appear as reasonable substitutes. Because of this, optimized portfolios can often exhibit weights that vary greatly from global benchmarks. Exposures may be restricted to only a very few assets. Black and Litterman describe a Bayesian approach to portfolio construction. Instead of using raw expected returns and mean/variance optimization, they back out the consensus expected returns which will lead exactly to benchmark portfolio weights, and then move away from those consensus forecasts toward the raw expected forecasts in proportion to the information content of the model. Grinold (1996) also covers this territory, describing how to back out consistent and reasonable expected excess returns for multiple currency perspectives. This is especially important for institutional portfolios, which require not only an optimal return/risk trade-off, but also low risk relative to a benchmark. Problems 1. You are researching an asset allocation strategy among global equity markets using price/earnings and price/book ratios. What difficulties might you encounter in trying to implement this with crosssectional analysis? 2. Why might you expect to see differing January effects across countries?
Page 535 3. Suppose a U.K. investor observes a BHP return of 3.5 percent, but the pound depreciates relative to the dollar by 3.5 percent over the period (e.g., from $1.50 to $1.4475). Does a U.S. investor observe a net BHP return of zero? Why or why not? 4. Evidently (from examining Table 18.3), the sample data have produced a Q(GER) for Germany that has a high correlation with currencies and a Q(JPN) for Japan that has a low correlation with currencies. How would you explain the lack of symmetry between the Japanese relationship with other currencies and the German relationship with other currencies? References Ambachtscheer, Keith P. ''Pension Fund Asset Allocation: In Defense of a 60/40 Equity/Debt Asset Mix." Financial Analysts Journal, vol. 43, no. 5, 1987, pp. 14–24. Black, Fischer. "Capital Market Equilibrium with Restricted Borrowing." Journal of Business, vol. 45, 1972, pp. 444–455. ———. "Universal Hedging: Optimizing Currency Risk and Reward in International Equity Portfolios." Financial Analysts Journal, vol. 45, no. 4, 1989, pp. 16–22. ———. "Equilibrium Exchange Rate Hedging." Journal of Finance, vol. 65, no. 3, 1990, pp. 899– 907. Black, Fischer, and Robert Litterman. "Asset Allocation: Combining Investor Views with Market Equilibrium." Goldman Sachs Fixed Income Research Publication, September 1990. ———. "Global Asset Allocation with Equities, Bonds, and Currencies." Goldman Sachs Fixed Income Research Publication, October 1991. Emanuelli, Joseph F., and Randal G. Pearson. "Using Earnings Estimates for Global Asset Allocation." Financial Analysts Journal, vol. 50, no. 2, 1994, pp. 60–72. Ferson, Wayne E., and Campbell R. Harvey. "The Risk and Predictability of International Equity Returns," Review of Financial Studies, vol. 6, no. 3, 1992, pp. 527–566. Grinold, Richard C. "Alpha Is Volatility Times IC Times Score." Journal of Portfolio Management, vol. 20, no. 4, 1994, pp. 9–16. ———. "Domestic Grapes from Imported Wine." Journal of Portfolio Management, Special Issue December 1996, pp. 29–40. Kahn, Ronald N., Jacques Roulet, and Shahram Tajbakhsh. "Three Steps to Global Asset Allocation." Journal of Portfolio Management, vol. 23, no. 1, 1996, pp. 23–32. Sharpe, William F. "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk." Journal of Finance, vol. 19, no. 3, 1964, pp. 425–442.
Page 536 ———. "Integrated Asset Allocation." Financial Analysts Journal, vol. 43, no. 5, 1987, pp. 25–32. Singer, Brian D., Kevin Terhaar, and John Zerolis. "Maintaining Consistent Global Asset Views (with a Little Help from Euclid)." Financial Analysts Journal, vol. 54, no. 1, 1998, pp. 63–71. Solnik, Bruno. "The Performance of International Asset Allocation Strategies Using Conditioning Information." Journal of Empirical Finance, vol. 1, 1993a, pp. 33–55. Solnik, Bruno. Predictable Time-Varying Components of International Asset Returns. (Charlottesville, Va.: Research Foundation of Institute of Chartered Financial Analysts, 1993b). Technical Appendix In this appendix, we will derive the link between excess returns from different perspectives [Eq. (18.6)], and also the transformation of portfolio Q from the COM perspective to other perspectives. To derive the result connecting excess returns from different perspectives, we will continue the tradition of this chapter and focus on BHP from the U.K. and U.S. perspectives. The generalization of the approach is obvious, and by using a specific example, we avoid having to remember which subscript or superscript refers to numeraire versus local market, etc. Let's begin this treatment with Eq. (18.5) from the main text: Just to understand the logic of this starting point, imagine taking $1 at time 0 and converting it to U.K. pounds. You will have 1/[($/£)(0)] pounds. Invest that amount in BHP from t = 0 to t. You then have 1/[($/£)(0)] · RBHP(U.K.|0,t) pounds. Converting back to dollars leads to Eq. (18A.1). To account for the multiplicative relationship between local returns and currency returns, we will use a slightly unusual definition of cumulative excess return: where R$(U.S.|0,t) is the value at time t of a U.S. money market account starting with $1 at t = 0. We can similarly define excess
Page 537 returns to BHP from the U.K. perspective, and excess returns to U.K. pounds from the U.S. perspective. In fact, Now we want to look at instantaneous excess returns r: The calculation of this from Eq. (18A.3) is complicated by the ratio of two stochastic terms. We can apply Ito's lemma, to show that, in general, for Ito's lemma effectively tells us to expand dF in a Taylor series and keep all terms up to second order. Applying this to Eq. (18A.3), we find We will need one more step to derive the result in the main text. In general, for any asset n, If we set n = U.K. pounds, we can simplify this because the excess return to U.K. pounds from the U.K. perspective is zero:
Page 538 Substituting this into Eq. (18A.7) leads to This is Eq. (18.6) in the main text. Transforming Portfolio Q We saw in the main text that We also saw that Here we are using Q to refer to the efficient portfolio from the COM perspective. We are seeking a portfolio we will call QU.S.. If we apply standard (not international) CAPM analysis to this portfolio from the U.S. perspective, we will find the same consensus expected excess returns as if we had used Q in the full international context. First, we will use Eq. (18A.8) to convert the necessary covariances from Eqs. (18A.12) and (18A.11) to the U.S. perspective: Substituting Eqs. (18A.12), (18A.14), and (18A.15) into Eq. (18A.11) leads to
Page 539 If we define QU.S. such that Hence, we can define QU.S. as We can freely add or subtract h$ without influencing the covariances. We have used this property to preserve the investment level (e.g., fully invested) of Q. Note that the only adjustment in moving from Q to QU.S. involves a shift in currency exposure. We are lowering our exposure to the currency basket, i.e., hedging our international currency exposure.
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