A manager should allocate capital to an investment project when the present value of the net cash flows generated by the project exceeds the current investment outlay. Applying this net present value principle requires a discount rate. It is one of the hallmarks of modern finance that this discount rate - the cost of equity capital - is set by investors in the capital markets. When investors finance a firm by purchasing its equity shares, they forgo the opportunity to invest in the equities of many other firms. Therefore, investors demand to be compensated for the opportunity cost of their investment with an appropriate expected rate of return. Consequently, the manager of a firm in a capital budgeting situation should set the discount rate for a project to be the expected return for the firm’s investors as if they were investing directly in that project.

It has been suggested that the international bond market sets the cost of a company’s debt equal to the risk-free (government) interest rate on bonds plus a risk premium to compensate for the possibility that the company may default on the debt. The appropriate rate for discounting the equity cash flows of any project similarly depends on how risky the investors in the firm view the cash flows from that particular project to be. However, thinking about risk in increasingly global equity markets is difficult because there are many more factors involved.

How, then, do investors determine the riskiness of an investment, and how do managers know the required rate of return on a risky investment? Unfortunately, there are no easy answers to these questions, and there are competing theories. This section develops the theories necessary to determine the cost of equity capital. It then demonstrates how these theories apply in an international context. Since investors set the cost of equity capital, we start with a detour through the fascinating world of international investing and the theory of optimal portfolio choice. The idea of portfolio diversification figures prominently, and we will argue that international diversification is highly desirable.

The old saying “Don’t put all your eggs in one basket” should entice investors to explore foreign stocks, perhaps in exotic places. As has been noted, global stock markets offer investors an incredible menu of choices, offering potentially higher rates of return and different types of risks. To understand the benefits and pitfalls of international investments, we must fully understand what determines risk and return in international markets. This necessitates that we understand how currency fluctuations affect international investments.

When a U.S. investor is bullish about the South African stock market, she must realize that investing in South African equity market also implies an exposure to the South African rand. Let us analytically derive the dollar return on a South African equity investment. Let \(S_t\) be the $/R exchange rate, and let \(s_{t+1}=[S_{t+1}-S_{t}]/S_{t}\) indicate the rate of appreciation of the rand relative to the dollar. We are interested in the dollar rate of return on South African equity, which we denote by \(r_{t+1,\$}\). This return will have two components: the rand rate of return on South African equity, denoted by \(r_{t+1,\mathsf{R}}\), and the rate of change in the value of the rand, \(s_{t+1}\). This reasoning is identical to the derivation of the return on a foreign money market investment. In this case, however, we replace the foreign interest rate with the foreign equity rate of return. We first convert from dollars to rands to get \(1/S_t\) rands, which we will invest. Each rand earns the rand return \(1+r_{t+1, \mathsf{R}}\) in the equity market. Subsequently, the total rand return is sold for dollars at \(S_{t+1}\) . Thus, the dollar return on a South African equity investment is

\[\begin{eqnarray*} 1+r_{t+1,\$} =[1/S_t] \times [1+r_{t+1,\mathsf{R}}] \times S_{t+1} \end{eqnarray*}\]Subtracting 1 from each side and using \(\frac{S_{t+1}}{S_{t}} = 1+s_{t+1}\) gives

\[\begin{eqnarray*} r_{t+1,\$} = \big[1+r_{t+1,\mathrm{R}} \big] \times \big[1+s_{t+1}\big]-1 \end{eqnarray*}\]or

\[\begin{eqnarray*} r_{t+1,\$} = r_{t+1,\mathsf{R}} + s_{t+1} + r_{t+1,\mathsf{R}} \times s_{t+1} \end{eqnarray*}\]We see that the dollar rate of return on a foreign investment depends on the local equity rate of return plus the currency return plus a cross-product term (the product of the two rates of return). The cross-product term is often small relative to the other two terms because it is percentages of percentages, and it is thus often ignored.

Table 1 lists several characteristics of the equity markets of the G7 countries. The data are from Morgan Stanley Capital International (MSCI) for the period from January 1980 to August 2010. We first focus on the three volatility columns. Remember that volatility, \(\mathsf{vol}[r],\) is defined to be the standard deviation, which is the square root of the variance, \(\mathsf{var}[r]\); it indicates how much returns vary around the mean or average return.

For an investor in a highly developed country, such as the U.S., international investments appear to have two problems. First, the volatilities of equity returns in foreign countries may exceed the volatility of equity returns in countries. Here we note that the U.S. market appears to be the least volatile market, with a volatility of only 15.6%. The second-least-volatile market is the United Kingdom, with a volatility of 18.9%. The other three European markets have volatilities exceeding 20%.

Second, Table 1 (second to last column) shows that currency changes are pretty variable themselves, with volatilities around 11%, for the most part. The only exception is the substantially lower volatility of the Canadian dollar, which is driven by the close economic ties between the United States and Canada and episodes during which Canadian monetary policy focused on exchange rate stability.

*Notes*: The original data are monthly total equity returns (including capital gains and dividends) taken from Morgan Stanley Capital International (MSCI) for the period January 1980 to August 2010. Means and volatilities are expressed as annualized percentage rates by multiplying monthly means by 12 and monthly volatilities by \(\sqrt{12}\). The market return is in foreign currency; the currency return is the change in the value of the foreign currency relative to the dollar.

We know that the volatility of the exchange rate affects the volatility of the dollar return on a foreign equity. But the volatility of the dollar return on foreign equity is generally much less than the sum of the exchange rate volatility and local equity return volatility. That is, the return to a foreign investment is well approximated by the sum of a local equity return and the currency return, \(r_{t+1,\$} = r_{t+1,\mathsf{FC}} + s_{t+1}\), with FC denoting foreign currency. Volatility is not additive because it is the square root of the variance, and the variance of the sum of two variables involves their covariance. Thus,

\[\begin{eqnarray*} \mathsf{var} \big[r_{t+1, \mathsf{FC}} + s_{t+1} \big] &=& \mathsf{var} \big[r_{t+1, \mathsf{FC}} \big] + \mathsf{var} \big[s_{t+1} \big] \ldots \\ && +2 \mathsf{cov} \big[r_{t+1, \mathsf{FC}}, s_{t+1} \big] \end{eqnarray*}\]Recall that the covariance of two variables equals the correlation between the variables multiplied by the product of the two volatilities, and the correlation is a number between -1 and 1 that indicates how closely related the variations are in the two variables. Rewriting the variance as a function of the correlation, \(\rho\), is informative:

\[\begin{eqnarray*} \mathsf{var}[r_{t+1, \mathsf{FC}} + s_{t+1}] &=& \mathsf{var}[r_{t+1, \mathsf{FC}}] + \mathsf{var}[s_{t+1}] \ldots \\ && +2 \rho \mathsf{vol}[r_{t+1, \mathsf{FC}}]\mathsf{vol}[s_{t+1}] \end{eqnarray*}\]Suppose the correlation is 1. Then, because the variance is the square of the volatility and using \((A+B)^{2}=A^{2}+B^{2}+2AB\), we see that

\[\begin{eqnarray*} \mathsf{var}[r_{t+1, \mathsf{FC}} + s_{t+1}] &=& \mathsf{vol}[r_{t+1, \$}]^2 \\ &=& \Big\{ \mathsf{vol} [r_{t+1, \mathsf{FC}}] + \mathsf{vol}[s_{t+1}] \Big\}^2 \end{eqnarray*}\]Hence, if \(\rho=1\), the volatility of the dollar return on foreign equity is indeed the sum of the foreign equity volatility and currency return volatility. Because of the perfect correlation, there is no natural diversification advantage to having exposure to two sources of risk. However, as long as \(\rho<1\), the total dollar volatility will be less than the sum of the two volatilities.

*Notes*: The original monthly data are taken from MSCI and cover the period January 1980 to August 2010.

Table 1 shows that the volatilities of dollar-denominated foreign equity returns are often not much above the original volatility in the local currency. This indicates that the correlation between exchange rate changes and local equity market returns is low. It is sometimes argued that it should be negative, appealing to the competitiveness ideas of that were discussed previously. When countries experience real depreciations (usually brought about by nominal exchange rate depreciations), exporting firms and import competing firms in that country experience a boost to their competitiveness and profitability, which might increase local stock market values. Under this scenario, the exchange rate and the stock market move in opposite directions. As shown in Table 2, in most countries, the correlation between exchange rate changes and local stock market returns is indeed slightly negative.

In Canada, the correlation is positive. For such a country, the primary forces may be foreign capital flows that appreciate both the foreign currency and the stock market as investors enter the capital markets and depreciate both markets when foreign investors repatriate capital. Nevertheless, the main conclusion of Table 2 is that dollar currency returns and foreign currency-denominated equity returns show little correlation.

In efficient markets, risky securities should earn returns higher than the risk-free rate. In Table 1, we also report the average (mean) returns earned in the various markets over the 31 years as a measure of the expected return, \(\mathbb{E}[r]\). If these returns are representative of true expected returns, they do not indicate that volatility is rewarded in the international marketplace. Whereas the most volatile market (Italy) does have the highest average local currency return (over 14%) and in dollars (over 12.5%), the two low-volatility markets (the United States and the United Kingdom) have relatively high average returns as well. Moreover, although Japan is a comparatively high-volatility country, it has low average stock market returns. Something else must drive average returns. We explore this issue later in this section.

Table 1 splits up the average dollar return into the average equity return in the foreign currency and the average currency return. The currency returns range between -1.5% (Italy) and 4.1% (Japan). It should not be a surprise that countries such as Japan and Germany feature positive currency returns and that countries such as France, Italy, and the United Kingdom feature negative currency returns. In the long run, currency changes reflect nominal interest rate differentials (recall our discussion of uncovered interest rate parity), and these interest rate differentials partially reflect inflation differentials. For example, Japan and Germany are both countries with historically low inflation and interest rates. In contrast, prior to the adoption of the euro, France and Italy historically experienced relatively high inflation and high nominal interest rates. The United Kingdom similarly witnessed high inflation in the first part of the sample, but reformed its monetary policy in the 1990s to focus on inflation targeting.

Figure 1 updates a classic study by Solnik (1974) who was one of the first to demonstrate the benefits of international diversification. The horizontal axis in Figure 1 depicts the number of stocks in a particular portfolio, and the vertical axis shows the typical variance of a portfolio. For the top line, we consider a universe of only U.S. stock and compute the average variance of a typical individual U.S. stock, which is normalized to 1. Then, we consider equally weighted portfolios of two stocks (one-half each), find the average variance of this portfolio expressed as a fraction of the average variance of one stock to produce a second point on the graph, and so on.

Since the correlation between stocks is imperfect, the relative portfolio variances decline with the addition of stocks. The graph shows that the portfolio variance falls quickly as more stocks are added, but after including around 30 stocks, it becomes difficult to reduce the variance further. The curve finally settles at a level of about 29% of the beginning variance. In other words, more than 70% of the variance of a typical stock can be eliminated through diversification. The part of the variance that can be diversified away is called nonsystematic variance. The lower line in Figure 1 repeats the exercise, but now stocks can be added from the United States and the major developed stock markets. Since there is even less correlation between U.S. and foreign stocks, the variance of the equally weighted portfolios goes down much more quickly as more stocks are added. The variance of the portfolio falls to barely 10% of the variance of a typical U.S. stock.

*Note*: The sample period is 1999 to 2008.

Recall that the variance of a large, equally weighted portfolio equals the average covariance among the stocks in the portfolio. Consequently, the variance of U.S. portfolios cannot be reduced further because there are systematic sources of variation that affect all stocks in the United States in the same way. The macroeconomic forces driving stock returns are factors that affect the cash flow prospects of firms and the discount rates used by investors to value these cash flows. We know that stock returns are sensitive to interest rates, which, in turn, depend on monetary policies and business cycles. Business cycles of course affect cash flow prospects, but they may also affect discount rates, as investors may become more risk averse in recessions and less risk averse during booms. These risks cannot be diversified away in a single domestic portfolio.

Notice, though, that when foreign stocks are added to the portfolio, these risks can, to some extent, be diversified away because U.S. monetary policies and business cycles are not perfectly correlated with those of the rest of the world. However, for the most part, stocks are positively correlated, so you cannot diversify away all of a portfolio’s variance, no matter how many international stocks you add to the portfolio. Since the average covariance is positive, even a large portfolio of international stocks will have a positive variance. We call the variance that cannot be diversified away the systematic variance or market variance. The important insight here is that when an investor holds a diversified portfolio, a stock’s contribution to the variance of the portfolio depends on its covariance relative to the other stocks in the portfolio.

The variance of a firm’s return can be split up into an *idiosyncratic* component and a *systematic* component, with the latter variance being the source of risk. For most firms, the idiosyncratic variance constitutes between 60% and 75% of the total variance of the firm’s return. This may sound like a lot, but Figure 1 shows that this idiosyncratic variance disappears relatively quickly when a portfolio is constructed with securities that are less than perfectly correlated.

Recent research by Bekaert, Hodrick, and Zhang (2012) demonstrates that idiosyncratic volatility, both in the United States and other G7 countries, seems to go through low- and high-volatility regimes. These findings provide a different interpretation of the results in Campbell et al. (2001), who argued that the general level of idiosyncratic risk in the U.S. market substantially increased from the early 1960s to 1997, whereas the level of long-run systematic risk roughly remained constant. In periods of high idiosyncratic volatility, more stocks are needed to achieve full diversification than the 30 that Figure 1 suggests.

Table 4 reports a full correlation matrix of the stock market returns of 23 developed countries. The sample period starts in 1980 for most countries. The correlations range from 0.23 for Japan and Greece to 0.79 for Germany and the Netherlands. It is striking that the stock returns of countries that are in close geographic proximity to one another and have significant exports and imports to one another correlate more highly. This is true for Canada and the United States, and it is also true for European Union countries (in particular, Belgium, France, Germany, and the Netherlands). Ireland and the United Kingdom are also highly correlated, at 0.71; New Zealand and Australia returns have a correlation of 0.73. This suggests that trade increases correlations, presumably because importing and exporting firms are affected by the economic factors in the other countries.

*Notes*: The countries are Australia (AU), Austria (AT), Belgium (BE), Canada (CA), Denmark (DK), France (FR), Germany (DE), Hong Kong (HK), Italy (IT), Japan (JP), the Netherlands (NL), Norway (NO), Singapore (SG), Spain (SP), Sweden (SE), Switzerland (CH), the United Kingdom (UK), the United States (US), Greece (GR), Portugal (PT), Ireland (IE), Finland (FI), and New Zealand (NZ). The data are monthly dollar returns from MSCI for the period from January 1980 to August 2010, although for some countries, the sample starts later.

The lowest correlations are observed for Japan and Greece. Greece has a correlation of less than 0.30 with Japan and Hong Kong. The correlations with non-European countries are invariably below 40%. Even within Europe, Greece does not correlate very highly with most other markets, although the correlations are always higher than 40%. Interestingly, the highest correlation Greece has with any other country is with Portugal, another ex-emerging market. Portugal naturally correlates most closely with its neighbor and trading partner Spain.

Apart from trade patterns, what drives the different return co-movements we observe in Table 4? To analyze this, it is best to first think of pure fundamental factors. Think of a country as a set of firms. Then figure that each firm is priced rationally, using a discounted cash flow analysis. In such a world, common variations in discount rates and common variations in expected cash flow growth rates will lead to correlations among the firms.

The first fundamental factor that may drive the correlations of stock returns in different countries is their industrial structures. Firms in the same industry are likely to be buffeted by the same shocks affecting cash flows and profitability. Moreover, it is likely that their systematic risks also move together, so both their discount rates and expected cash flow variations are closely related. Both Canada and Australia have many firms operating in the mining industry, for example. This might explain why Australia is highly correlated with Canada but not with Germany.

A long debate has ensued about the importance of industry factors when it comes to return correlations across countries. Some researchers, such as Brooks and Del Negro (2004), have found that industry factors are starting to dominate country factors. It used to be the case that country factors clearly dominated when markets were less integrated and discount rates were not highly correlated across countries. Moreover, limited trade across countries and relatively independent monetary policies implied that business cycles showed little correlation across countries, resulting in low correlations among cash flows in different countries. Consequently, policies affecting the degree of integration and the independence of business cycles appear to be important determinants of cross-country correlations. For example, the adoption of a common currency has helped synchronize business cycles in Europe. In contrast, emerging markets typically act more independently of integrated countries. This may explain why Greek stock market returns have historically not been highly correlated with the returns of other countries. If Greece continues to integrate into the European Union, we would expect these correlations to increase, but Greece’s recent sovereign debt crisis obviously jeopardizes the integration process.

Finally, irrational investor behavior may induce excess correlations across equity markets, especially during crisis periods.

Since the correlations overall are so far from unity, there are ample opportunities for investors to internationally diversify their portfolios. Some investors may be less impressed and argue that they really only care about diversification when their home market is going down. Longin and Solnik (2001) confirm what casual observations may have led you to suspect: International diversification benefits evaporate when you need them the most - that is, in bear markets. To demonstrate this rather annoying fact, Longin and Solnik computed “bear market correlations” (correlations using returns below the average for both of the stock markets) and “bull market correlations” (correlations using returns above the average) for various developed markets.

The results are striking: The bear market return correlations are much higher than the bull market correlations. This finding does not justify staying at home with your equity portfolio, however. Research by Ang and Bekaert (2002) shows that these asymmetric correlations do not negate the benefits of international diversification because bear markets remain imperfectly correlated.

Table 3 shows the U.S. Sharpe ratio to be historically higher than the Sharpe ratios for the other G7 countries. Even so, international diversification makes perfect sense for U.S. investors. This is because it is not the Sharpe ratio of the foreign asset that the U.S. investor should care about but the Sharpe ratio of the portfolio that results from international diversification. Intuitively, because equity markets in other countries are not perfectly correlated with the U.S. market, part of their volatility disappears through portfolio diversification.

Let’s consider formally how international diversification affects Sharpe ratios. Imagine putting a fraction \(w\) of your all-U.S. portfolio in international equity. Let’s denote the U.S. return by \(r\) and the foreign return (in dollars) by \(r^{*}\). The expected return of the new portfolio is the weighted average of the expected returns on the individual assets with the weights equal to the fractions of wealth invested in each asset, \((1-w)\mathbb{E}[r]+w\mathbb{E}[r^{*}]\). Expected returns aggregate linearly. As we already know, volatility does not aggregate linearly. The volatility of the new portfolio equals

\[\begin{eqnarray*} \Big\{(1-w)^{2} \mathsf{var}[r] + w^{2}\mathsf{var}[r^{*}] + 2w(1-w) \mathsf{cov}[r, r^{*}]\Big\}^{1/2} \end{eqnarray*}\]Since the covariance is a function of the correlation, correlations really matter.

Suppose you start with an all-U.S. portfolio. The U.S. Sharpe ratio is \(\mathbb{E}[r-r_{f}]/\mathsf{vol}[r]\), and the Sharpe ratio on the foreign equity is \(\mathbb{E}[r^{*}-r_{f}]/\mathsf{vol}[r^{*}]\). We denote the correlation between the U.S. and foreign returns as \(\rho\). From a zero investment in foreign equities, the Sharpe ratio goes up when you add a little bit of foreign equity exposure, if the following condition holds:

\[\begin{eqnarray} \frac{\mathbb{E}[r^{*}]-r_{f}}{\mathsf{vol}[r^{*}]}>\rho\frac{\mathbb{E}[r]-r_{f}}{\mathsf{vol}[r]} \tag{2.1} \end{eqnarray}\]The appendix contains a formal proof of this statement. Equation (2.1) states that your Sharpe ratio improves when you add a little bit of the foreign asset to your portfolio if the Sharpe ratio of the new asset is higher than the Sharpe ratio of the U.S. portfolio multiplied by the correlation between the U.S. return and the international return. In other words, the lower the correlation with the U.S. market, the lower the Sharpe ratio of the foreign market needs to be for it to become an investment that increases your Sharpe ratio. This is because markets that have low correlation with the U.S. market are the best diversifiers of a U.S. portfolio. Another way to see this is to bring \(\rho\) to the other side and notice that it is not the foreign asset’s volatility that matters when computing the return to risk ratio but, rather, volatility adjusted for correlation \((\rho \; \mathsf{vol}[r^{*}])\) . The lower \(\rho\) is, the lower this adjusted risk number becomes, and the easier it is to exceed the U.S. Sharpe ratio.

Given the correlations and volatilities provided earlier, we can compute hurdle rates on international investments for U.S.-based investors. The *hurdle rate* is the lowest possible expected foreign return that must be earned for investors with purely domestic assets to improve their Sharpe ratio when they invest in that foreign market and when the expected return on the domestic market takes a specific value.

To find the hurdle rates, we fill in \(\mathbb{E}[r]\) in Equation (2.1) with a reasonable number (for instance, 10%), and we use the data to estimate correlations and volatilities, leaving \(\mathbb{E}[r^{*}]\) as an unknown variable. The minimum \(\mathbb{E}[r^{*}]\) we need for the Sharpe ratio with some foreign investment to be at least as large as the domestic Sharpe ratio is the one that equates the two sides of the equation. That is,

\[\begin{eqnarray*} \mathsf{Hurdle} \; \mathsf{rate} = \rho \; \frac{\mathbb{E}[r]-r_{f}}{\mathsf{vol}[r]}\mathsf{vol}[r^{*}]+r_{f} \end{eqnarray*}\]The hurdle rate is higher when the domestic market has a high Sharpe ratio, the foreign market is more volatile, or there is high correlation between foreign and U.S. stock returns.

Whereas Table 1 reports the dollar volatilities of the various international equity market returns, and Table 3 reports their Sharpe ratios, Table 5 reports their correlations with the U.S. market. The market returns of Canada and the United Kingdom have the highest correlations with U.S. returns, whereas Japanese and Italian market returns have the lowest correlations. For France and Germany, the correlations are about 60%.

*Notes*: All returns have been converted to U.S. dollars. The original monthly data are taken from MSCI.

The hurdle rates for the countries with low correlations will be low. Let’s illustrate the computation of the hurdle rate for Japan, when the expected return for the United States is 10% (\(\mathbb{E}[r]\) =0.10). The number is

\[\begin{eqnarray*} 0.05 +0.37 \times\frac{0.10-0.05}{0.156}\times 0.225=0.0767 \;\; \mathsf{or} \;\; 7.67\% \end{eqnarray*}\]The risk-free rate is 0.05, and the correlation between Japanese and U.S. equity returns is 0.37, the U.S. Sharpe ratio is (0.10-0.05)/0.156, and the volatility of the Japanese equity return is 0.225. Hence, a U.S. investor should put some money in Japanese equity even if he believes the expected dollar return on Japanese equity is only 7.67%.

Hurdle rates appear in Table 6. The correct conclusion is that international diversification can easily improve performance for U.S. investors because the hurdle rates for expected dollar returns on foreign investments are low. In fact, they are lower than the expected return on the U.S. equity market in every case. It is difficult to imagine that foreign equity markets have such dramatically lower expected returns relative to the U.S. market. Italy and Japan have the lowest correlation with the United States and therefore offer the easiest performance enhancement.

*Notes*: The hurdle rate equals \(r_{f}+ \rho \; \frac{\mathbb{E}[r]-r_{f}}{\mathsf{vol}[r]}\mathsf{vol}[r^{*}]\). The correlation number is taken from Table 5; the volatility numbers (in dollars) are taken from Table 1 (both for the United States and the foreign country); \(r_{f}\) is set at 5%; and \(\mathbb{E}[r]\) is the U.S. expected return specified on top of the two columns. Data are from MSCI, and the sample is from January 1980 to August 2010.

**How to Diversify at Home**

Retail investors do not necessarily need to call a foreign broker to invest in far-flung places. Many investment vehicles can be used to accomplish international diversification. First of all, would Richemont not constitute an ideal international investment for a South African investor? After all, Richemont sells its luxury goods in many countries around the world. Hence, its cash flows must be influenced by the local economies of all those countries. It was long thought that a portfolio of multinational companies would capture the benefits of international diversification. While the recent literature does indicate that the stock returns of multinational companies behave quite internationally (see, for example, Diermeier and Solnik 2001), Rowland and Tesar (2004) find that restricting oneself to domestically traded multinational companies remains a flawed diversification strategy. The best diversification opportunities may be exactly the companies for which local factors remain important drivers of their returns.

It is noted that many companies cross-list in the United States using American depositary receipts (ADRs). Why not simply buy these companies? Again, the problem is one of representation: The ADR companies tend to be the larger, more internationally focused companies, and they may not give full exposure to foreign stock markets.

Another possibility is to invest in closed-end funds, or investment trusts, which trade on the local equity market.

These funds represent a fixed portfolio that may invest in the world markets, sometimes restricted to a region (Asia, for instance) or a particular country, in which case they are called country funds. The only way to buy into this portfolio is for the investor to buy the fund from another investor selling it. Therefore, closed-end funds can trade at prices that are different from the value of the portfolio, especially when they invest in emerging markets. Hence, it is conceivable that closed-end fund returns fail to offer the same diversification benefits as the underlying portfolio. This is not a problem with open-end funds, where the portfolio grows with new investments and contracts with redemptions, and the fund is not traded on an exchange. These represent the bulk of the international funds available to retail investors.

Finally, a hybrid alternative that is rapidly gaining popularity is the exchange-traded fund (ETF), which trades on an exchange but where prices are kept close to the value of the underlying portfolio through arbitrage activities by a few institutional investors. Both diversified funds and funds focusing on one country, mimicking the performance of the corresponding MSCI indices, are now available. As the availability of these vehicles expands, an internationally diversified portfolio is only a phone call away.

We have established that diversifying internationally is likely to reduce risk and improve your Sharpe ratio. But how much should you invest internationally? This is a portfolio choice problem - one of the most fundamental finance problems, and one that brings us very close to a formula for the cost of equity capital.

To solve for the optimal portfolio, we must first specify feasible portfolios, which are all portfolios that use up all wealth. Let’s consider the G7 example. An investor can invest in the risk-free asset or in seven different equity markets. We can represent the investor’s feasible portfolios by a series of wealth fractions - the proportions of wealth devoted to each asset - and these proportions must add to 1. For example, putting 50% of your portfolio in the risk-free asset and 50% in the U.S. equity market is a feasible portfolio. The combination of all feasible portfolios constitutes the investor’s menu. Of course, there is an infinite number of possible portfolios, so to figure out which portfolio is best for any investor seems like a daunting task.

Luckily, finance theory has come up with some rather simple answers. We start by defining investors’ preferences regarding risk and return, and then we consider a simplified set of ingredients: one risky asset and one riskless security. After we extend the ingredients to multiple risky assets, we can solve the portfolio problem. For example, we will find that no smart investor should ever choose the 50-50 portfolio we proposed.

In economics, preferences are typically represented by utility functions. Typically, a utility function mathematically links the consumption of units of real goods to a level of satisfaction. Here, we specify a utility function for the individual investor in terms of the statistical properties of the portfolio that the investor holds - that is, expected returns and portfolio variance. We assume that investors would like to generate the highest possible expected return with as little variance as possible, but each investor may have a different risk tolerance. A simple function that captures the trade-off the investors face is

\[\begin{eqnarray*} U =\mathbb{E}[r_{p}]-\frac{A}{2}\sigma_{p}^{2} \end{eqnarray*}\]where the subscript \(p\) indicates the portfolio, \(\mathbb{E}[r_{p}]\) is the expected return on the portfolio, and \(\sigma_{p}\) is the volatility of the portfolio. The parameter \(A\) in this mean-variance preference function indicates the penalty the investor assigns to the variance of the portfolio. The higher \(A\) is, the more the investor dislikes variance or risk; in other words, \(A\) characterizes the investor’s risk aversion.

**The Investor’s Utility Calculation**

Suppose the expected portfolio return is 9.87%, and its standard deviation is 7.84%. For an investor with \(A=4\), utility equals

\[\begin{eqnarray*} 9.87\% - \frac{1}{2} \times 4 \times (7.84\%)^{2}= 9.87\% - 1.23\% = 8.64\% \end{eqnarray*}\]One interpretation of this number is that the investor in this portfolio achieves the same utility as he would by investing in a completely risk-free portfolio with a return of 8.64%.

The portfolio problem is considerably simplified and much intuition is gained if we begin by restricting the set of ingredients to one single risky asset and the risk-free asset. Let’s introduce some notation. Let the risk-free return be \(r_{f}\), let the risky return be \(r\), and let the weight on the risky asset be \(w\).

If the proportion \(w\) of the portfolio is invested in the risky asset, then \(1-w\) is invested in the risk-free asset. Hence, the return on a portfolio is

\[\begin{eqnarray*} r_{p}=w\times r+(1-w)\times r_{f}=r_{f}+w\times(r-r_{f}) \end{eqnarray*}\]The variable \(r-r_{f}\) is the excess return. Therefore, the portfolio’s expected return is \(\mathbb{E}[r_{p}]=r_{f}+w\times \mathbb{E}[r-r_{f}]\), which increases linearly with the weight in the risky asset when the expected excess return is positive. To find the variance of the portfolio return, note that the risk-free rate is known with certainty. Therefore, we simply have \(\sigma_{p}^{2}=w^{2}\sigma^{2}\), where \(\sigma^{2}\) is the variance of the risky return, \(r\). Hence, the volatility of the portfolio is \(\sigma_{p}=w\sigma\), and the risk of the portfolio is also linear in \(w\). Now, use this volatility expression to substitute for \(w\) in the expected return expression, and find

\[\begin{eqnarray} \mathbb{E}[r_{p}]=r_{f}+ \frac{\mathbb{E}[r]-r_{f}}{\sigma}\sigma_{p} \tag{3.1} \end{eqnarray}\]This expression describes the relationship between the expected return on the portfolio and its standard deviation. Consequently, Equation (3.1) fully describes the “menu”, or the possible risk-return combinations, for this simple case. Also, note that the relationship is of the form \(y= \mu +\beta x\), with \(y=\mathbb{E}[r_{p}]\) and \(x=\sigma_{p}\), which is the equation for a straight line.

We call the line describing the risk-return trade-off in the single risky asset case the capital allocation line (CAL) because it describes the ways capital can be allocated in the single risky asset case. The CAL is graphed in Figure 2.

*Notes*: The vertical axis shows the expected return, and the horizontal axis is the standard deviation of the portfolio. The line is the capital allocation line of feasible risk-expected return patterns. It emanates at the risk-free rate (5% in this example) and slopes upward with the Sharpe ratio of the risky asset, \(\frac{E(r)-r_{f}}{\sigma}\), as its slope.

**The Capital Allocation Line**

`r figr("f13_1", type="table")`

), and let \(r_{f}=5\%\). Then, the CAL is given by \(\mathbb{E}[r_{p}]=0.05+SR \times \sigma_{p}\), with \(SR= \frac{\mathbb{E}[r]-r_{f}}{\sigma}=\frac{0.1152-0.05}{0.1558}=0.42\), where we recognize the Sharpe ratio, \(SR\), as the return premium per unit of risk.
To find the optimal portfolio, we must combine the CAL menu with the investor’s preferences. The mathematical problem can be written as

\[\begin{eqnarray*} \max_{w}U=\max_{w} \Big( \mathbb{E}[r_{p}]-\frac{1}{2}A\sigma_{p}^{2} \Big) \end{eqnarray*}\]In words, we try to find the weight on the risky asset (\(w\)) that maximizes the utility function. We can substitute the expressions for \(\mathbb{E}[r_{p}]\) and \(\sigma_{p}^{2}\) to obtain

\[\begin{eqnarray*} \max_{w} \Big[r_{f}+w \big( \mathbb{E}[r]-r_{f} \big) -\frac{1}{2}Aw^{2}\sigma^{2} \Big] \end{eqnarray*}\]To solve for the optimal \(w\), denoted \(w^{*}\), we must take the derivative of this function with respect to \(w\) and set it equal to zero, in which case we find

\[\begin{eqnarray*} \mathbb{E}[r]-r_{f}-Aw^{*}\sigma^{2}=0 \end{eqnarray*}\]Solving for the optimal portfolio gives a very intuitive solution:

\[\begin{eqnarray} w^{*}= \frac{\mathbb{E}[r]-r_{f}}{A\sigma^{2}} \tag{3.2} \end{eqnarray}\]The allocation to the risky asset is increasing in the expected return on the asset, decreasing in its variance, and decreasing in the investor’s risk aversion.

**Calculations of Optimal Portfolios**

Let’s apply the formula to investors who have different levels of risk aversion:

To fill in the numbers of the table, we use the formula for \(w^{*}\), and then the expected return is \(\mathbb{E}[r_{p}]=r_{f}+w^{*}\mathbb{E}[r-r_{f}]\) and the volatility is \(\sigma_{p}=|w^{*}|\sigma\).

Note that \(w^{*}=1\) implies that 100% of wealth is invested in the risky asset. As risk aversion increases, the weight on the risky asset decreases, which decreases the expected return and the standard deviation. Since we stay along the CAL, the risk-return trade-off (Sharpe ratio) of the portfolio, \([\mathbb{E}(r_{p})-r_{f}]/\sigma_{p}=0.42\), remains the same because it is the slope of the line.

Figure 3 demonstrates the above point graphically.

Note that investors with different preferences toward risk and return invest in different portfolios, represented by different points on the capital allocation line in Figure 3.

For low \(A\), we are at a point such as \(L\). The investor is more than 100% invested in the risky asset \((w>1)\), and the investor finances this position by borrowing. For example, for \(A=1\), the investor borrows $1.69 for every dollar of his own wealth invested, and he invests the $2.69 in the stock market. For high \(A\), the investor combines stock investing with an investment in the T-bill. Hence in this case, \(w<1\). For example, for \(A=4.0\), the investor places 67% of her wealth in the risky asset and 33% in the risk-free asset.

What if there are multiple risky assets? Consider Figure 4. The circles represent the expected returns and standard deviations of various assets. Even with just two risky assets, many different capital allocation lines are available. After all, we could consider all feasible risky portfolios as “the risky asset”. What is the optimal risky portfolio? Economist Harry Markowitz (1952) won the Nobel Prize in 1990 for showing us how to proceed.

First, we must get rid of a large number of “inefficient” portfolios by creating the mean-standard deviation frontier, which is the locus of the portfolios in expected return-standard deviation space that have the minimum variance for each expected return. It is therefore also often referred to as the minimum-variance frontier. For two assets, the frontier would have a shape similar to the one graphed in Figure 4. Imagine combining a low expected return-low variance asset (say asset \(X\)) with a high expected return-high variance asset (say asset \(Y\)). Starting from a portfolio 100% in asset \(X\), adding some of asset \(Y\) to the portfolio increases the expected return of the portfolio in a linear fashion. However, unless assets \(X\) and \(Y\) have perfectly correlated returns, the standard deviation will not change in a linear fashion. In fact, it may even decrease at first, but in any case, when it starts to increase, imperfect correlation makes the standard deviation of the portfolio increase at a rate lower than linear, giving rise to the curved shape also seen in Figure 4.

Creating the frontier for multiple assets as in Figure 4 is the solution to a complex mathematical problem. We want to minimize the return variance for a portfolio of \(N\) securities, for each possible expected return:

\[\begin{eqnarray*} \min_{\{w_{1},\ldots,w_{N}\}} \left[ \sum_{i=1}^{N}w_{i}^{2}\sigma_{i}^{2}+\sum_{i=1j}^{N}\sum_{\neq 1}^{N}w_{i}w_{j} \mathsf{cov} [r_{i},r_{j}] \right] \Rightarrow \mathsf{Minimum} \; \mathsf{variance} \end{eqnarray*}\]such that

\[\begin{eqnarray*} \sum_{i=1}^{N}w_{i} = 1 \Rightarrow \mathsf{Feasible} \; \mathsf{portfolio}, \; \sum_{i=1}^{N}w_{i} \mathbb{E}[r_{i}] = \overline{r}\Rightarrow \mathsf{Target} \; \mathsf{return} \end{eqnarray*}\]By varying \(\overline{r}\), we trace out the frontier. Although analytical solutions are possible, using one of the many optimization packages in **R**, such as `lpSolve`

or the Excel Solver is another popular way of finding minimum-variance portfolios.

Interestingly, when this problem is solved for two target returns, we are done. This is called two-fund separation: The minimum-variance frontier is said to be spanned (or generated) by any two minimum-variance frontier portfolios. That is, if we find two portfolios - say, portfolio \(X\) with weights \([w_{1}^{X}, w_{2}^{X}, \ldots , w_{N}^{X}]\) and portfolio \(Y\) with weights \([w_{1}^{Y}, w_{2}^{Y}, \ldots , w_{N}^{Y}]\) - that are on the frontier, we can generate the whole frontier by taking combinations of these two portfolios. If there are only two assets, then the mean-standard deviation frontier can be found by simply mixing the two assets in all possible combinations with weights adding up to 1. Two-fund separation says that with multiple assets, all portfolios on the frontier can be viewed as a mix of any two frontier portfolios.

Once we have determined the mean-standard deviation frontier, we can focus on a rather limited set of possible portfolios. Clearly, no one will want to invest in a portfolio on the inside of the frontier: You can either lower risk at the same expected return or increase the expected return at the same risk. Also, no one will invest in a portfolio on the portion of the frontier below the global minimum-variance portfolio, which is indicated on Figure 4. The global minimum-variance portfolio is the portfolio with the least variance among all possible portfolios. If you are below that portfolio, you can increase expected return without increasing volatility.

What remains is the upper portion of the frontier, starting at the global minimum-variance portfolio. This set of risky portfolios is called the efficient frontier. It yields a large number of “efficient” risky portfolios that could be combined with a risk-free asset to form a capital allocation line.

Starting from the risk-free rate on the vertical axis of 5%, we can consider any portfolio on the mean-standard deviation frontier as a potential risky asset. We can draw a potential capital allocation line (CAL) from the risk-free rate to the risky portfolio’s point on the graph. As before, the slope of the CAL is the Sharpe ratio. People with utility functions that depend positively on the expected return and negatively on the variance of the portfolio would naturally prefer higher Sharpe ratios. Once we have a CAL, we know how to optimally combine the risky portfolio with the risk-free asset from our previous analysis.

For example, consider Figure 5. It graphs the mean-standard deviation frontier for two assets, the U.S. and Japanese equity markets, using the expected return and volatility properties reported in Table 1 and the correlation reported in Table 5. Clearly, the “best” CAL has the steepest slope, or highest Sharpe ratio. This is the line emanating from the risk-free return to the point where the line is tangent to the mean-standard deviation frontier. This portfolio is called the mean-variance-efficient (MVE) portfolio, and it represents the risky portfolio that maximizes the Sharpe ratio.

The theory is surprisingly powerful. It states that there is a superior risky portfolio that all investors will prefer: Of course, preferences toward risk still differ, and investors can combine the MVE portfolio with the risk-free asset in different ways. Portfolios to the left (right) of the tangency represent the MVE portfolio for the more (less) risk-averse investors. Notice how the risky efficient frontier is completely below the CAL going through the MVE portfolio. By borrowing at the risk-free rate and investing more than 100% in the MVE portfolio, investors use leverage and can achieve a much higher expected return for the same risk than if they only considered risky assets. The actual weight on the MVE portfolio versus the risk-free asset can be determined using Equation (3.2).

*Notes*: We form the mean-standard deviation frontier from two assets. The U.S. portfolio has a mean return of 11.52% and a standard deviation of 15.58%. The Japanese portfolio has a mean return of 9.28% and a standard deviation of 22.51%. The correlation between the two returns is 0.37. The mean-variance-efficient portfolio dominates either individual portfolio.

To determine the international cost of equity capital, we must first determine how investors view risk in a global investments context. When investing abroad, an investor must assess both the returns of the international asset in its local currency and variations in the value of the foreign currency relative to the investor’s home currency. In this regard, it is worth noting that the volatility of an international equity investment is mostly determined by the volatility of the local equity market. Although exchange rate changes are quite variable, they are nearly uncorrelated with local stock returns.

International diversification results in portfolios with risk levels much lower than what can be achieved with domestic diversification alone. The main reason is that the stock market returns of different countries are not very highly correlated with one another, despite the fact that correlations among them tend to increase during bear markets. Using available data on the volatilities of different markets and the correlation among them, investors can compute a “hurdle rate” of return for foreign investments. The hurdle rate is the expected return for which a small investment in the foreign equity market, starting from an all-domestic portfolio, increases the Sharpe ratio for the portfolio.

A mean-variance investor likes high expected returns but dislikes portfolio variance. If only a risk-free asset and just one risky asset are available, she will invest more in the risky asset the lower her risk aversion, the higher the expected excess return, and the lower the variance of the risky asset. The mean-standard deviation frontier collects portfolios that minimize the portfolio variance for each possible expected return. The mean-variance- efficient (MVE) portfolio is the one portfolio on the frontier that maximizes the Sharpe ratio and is hence optimal. This portfolio defines the capital allocation line, which determines how the investor mixes the risk-free asset with the optimal risky portfolio, depending on her preferences.

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