The capital asset pricing model (CAPM) underlies all modern financial theory. It was derived by Sharpe (1964), Lintner (1965), and Mossin (1966), using principles of diversification, with simplified assumptions building on the original mean-variance optimization analytics developed by Markowitz (1952). Markowitz and Sharpe won the 1990 Nobel Prize in economics for their efforts.

This section describes the application of the CAPM, which may be used to compute the international cost of capital. We describe its origins, provide a formal derivation and interpretation, and discuss the difference between domestic and international CAPMs.

# 1 The International Cost of Capital

## 1.1 Assumptions and Origins

The CAPM requires a long list of rather strong assumptions:

• There is a single-period investment horizon.
• Individual investors are price takers.
• Investments are limited to traded financial assets.
• There are no taxes and transaction costs.
• Information is costless and available to all investors.
• Investors are rational mean-variance optimizers.
• Expectations are homogeneous; that is, all investors agree on the expected returns, standard deviations, and covariances between security returns.

The CAPM then derives the optimal asset demands of all investors and derives restrictions on expected returns by imposing that markets have to clear (that is, supply must equal demand), implying that all assets must be willingly held. Given these assumptions, it is not surprising that the CAPM yields strong predictions:

• All investors hold the same portfolio of risky assets (i.e the market portfolio).
• The market portfolio contains all securities, and the proportion of each security is its market value as a percentage of total market value.
• The risk premium on the market depends on the average risk aversion of all market participants.
• The risk premium on an individual security is a function of its covariance with the market portfolio.

Although no one literally believes that the assumptions underlying the CAPM hold in the real world, the CAPM is one of the most useful models in finance. For example, it serves as a benchmark for evaluating portfolio managers, and it provided an impetus for the development of index funds. Index funds are open-end funds that passively track a stock index such as the FTSE/JSE Top 40 without trying to outperform it.1 Finally, the CAPM is the basis for cost-of-capital computations; it is this application of the CAPM that is most useful for this book. The next section provides a technical introduction to the main CAPM equation. The following sections help interpret it and illustrate its practical use in a global context, where exchange rate movements may complicate the model’s application.

## 1.2 A Derivation of the CAPM (Advanced)

To derive the CAPM, recall the results of the diversification problem. We argued that adding a little bit of the new asset to a portfolio improves the investor’s Sharpe ratio when when

$\begin{eqnarray*} SR_{\mathsf{NEW}}\geq\rho\times SR_{p} \end{eqnarray*}$

where $$\rho$$ is the correlation between the portfolio, $$p$$, and the new asset; $$SR_{\mathsf{NEW}}$$ is the Sharpe ratio of the new asset; and $$SR_{p}$$ is the Sharpe ratio of the present portfolio. The correlation of the new asset return with $$r_{p}$$, which now contains some of the new asset, increases as we add more of the new asset, making the condition harder to satisfy. We should keep adding the asset until

$\begin{eqnarray} SR_{\mathsf{NEW}}=\rho\times SR_{p} \tag{1.1} \end{eqnarray}$

At that point, further additions no longer increase the Sharpe ratio; that is, we have reached the portfolio that maximizes the Sharpe ratio, implying that we have found the MVE portfolio. Thus, $$r_{p}$$ should now be interpreted as the return on the MVE portfolio. Rewriting Equation (1.1) using the definition of the Sharpe ratio and bringing $$\rho$$ to the other side gives

$\begin{eqnarray*} \frac{\mathbb{E}(r_{\mathsf{NEW}})-r_{f}}{\rho\times\sigma_{\mathsf{NEW}}}=\frac{\mathbb{E}(r_{p})-r_{f}}{\sigma_{p}} \end{eqnarray*}$

Substituting $$\rho=\frac{\mathsf{cov}(r_{\mathsf{NEW}},r_{p})}{\sigma_{\mathsf{NEW}}\sigma_{p}}$$ gives

$\begin{eqnarray} \frac{\mathbb{E}(r_{\mathsf{NEW}})-r_{f}}{\mathsf{cov}(r_{\mathsf{NEW}},r_{p})}=\frac{\mathbb{E}(r_{p})-r_{f}}{\sigma_{p}^{2}} \tag{1.2} \end{eqnarray}$

This relationship holds for any security $$i$$. Equation (1.2) implies that expected excess returns per unit of covariance risk are the same for all assets and are equal to

$\begin{eqnarray*} \frac{\mathbb{E}(r_{p})-r_{f}}{\sigma_{p}^2} \end{eqnarray*}$

The relevant risk for a security is its covariance with the MVE portfolio. Rewriting Equation (1.2) for security $$i$$ gives

$\begin{eqnarray} \mathbb{E}(r_{i})-r_{f}= \frac{\mathsf{cov}(r_{i},r_{p})}{\sigma_{p}^{2}}\times[\mathbb{E}(r_{p})-r_{f}] \tag{1.3} \end{eqnarray}$

Equation (1.3) establishes a relationship between the expected excess return on an individual asset and the expected return on the MVE portfolio.

We are almost finished. Let’s review the major findings of the previous section on optimal asset allocation:

1. The efficient frontier is a set of “dominant” portfolios in risk-return space. Non-efficient portfolios would not be held by any mean-variance investor.
2. If a risk-free asset exists, one portfolio of risky securities offers the best risk-return trade-off: the MVE portfolio.

Now, if everybody is a mean-variance investor facing the same frontier, what must the MVE portfolio be for there to be no excess demand or supply for any security? It must be the market portfolio - and that is what the CAPM says! The implication is

$\begin{eqnarray*} \mathbb{E}(r_{i})-r_{f}=\frac{\mathsf{cov}(r_{i},r_{m})}{\sigma_{m}^{2}}\times[\mathbb{E}(r_{m})-r_{f}] \end{eqnarray*}$

where the subscript $$m$$ represents the market portfolio. The relationship between the expected return on an individual security and the expected return on the market portfolio depends on the statistical construct $$\frac{\mathsf{cov}(r_{i},r_{m})}{\sigma_{m}^{2}}$$, which is called the beta $$(\beta )$$ of security $$i$$.

## 1.3 Interpreting the CAPM

The CAPM is often used as a benchmark to determine the required rate of return on risky equity capital. The CAPM provides a formula for the required rate of return on an equity investment, which is its expected rate of return, $$\mathbb{E}(r_{e})$$.

### 1.3.1 The CAPM Equilibrium

Equity investors require compensation for the time value of money based on the risk-free rate, $$r_{f}$$. In addition, they require compensation for the systematic, or non-diversifiable, risk of the investment. Systematic risk is measured by the beta of the equity, $$\beta_{e}$$, multiplied by the risk premium on the market, $$[\mathbb{E}(r_{m})-r_{f}]$$. An equity’s beta is the covariance of the rate of return on the equity with the rate of return on the market portfolio divided by the variance of the rate of return on the market portfolio:

$\begin{eqnarray*} \beta_{e}=\frac{\mathsf{cov}(r_{e},r_{m})}{\mathsf{var}(r_{m})} \end{eqnarray*}$

Hence, the CAPM states that

$\begin{eqnarray} \mathbb{E}(r_{e})=r_{f}+\beta_{e}[\mathbb{E}(r_{m})-r_{f}] \tag{1.4} \end{eqnarray}$

The logic of the CAPM begins with the assumptions that investors prefer higher expected returns but are averse to risk. From the investor’s perspective, risk is measured by the variance of the return on the investor’s overall portfolio. Given the expected future cash flows of the assets, changes in the market prices change the assets’ expected returns and their variances and covariances. In equilibrium, the market prices of assets adjust such that the expected returns on the different assets and their variances and covariances allow the market portfolio to be willingly held by investors. This will happen when the expected excess returns per unit of covariance risk are equalized across assets and are equal to the expected excess return on the market divided by its variance, as in Equation (1.2). In equilibrium, all investors are thought to be holding the market portfolio because they are assumed to have the same expectations and the same investment opportunities. The market portfolio is the MVE portfolio.

### 1.3.2 The Risk Premium on the Market

The risk premium on the market portfolio is the amount by which the expected return on the market exceeds the risk-free rate. The CAPM actually predicts that this risk premium will depend on the average risk aversion of investors and the variance of the market portfolio return. To see this, consider an expression for the optimal portfolio, $$w^{*}$$, when applied to the market portfolio. Since every investor chooses to combine the market portfolio with the risk-free asset according to her preferences, someone with average risk aversion, say $$\bar{A}$$, will hold exactly the market portfolio.

Consequently,

$\begin{eqnarray*} w^{*}=1= \frac{1}{\bar{A}}\frac{\mathbb{E}(r_{m})-r_{f}}{\sigma_{m}^{2}}, \end{eqnarray*}$

or

$\begin{eqnarray} \mathbb{E}(r_{m})-r_{f}=\bar{A}\sigma_{m}^{2} \tag{1.5} \end{eqnarray}$

Hence, the market risk premium balances the variance of the market portfolio to reflect the average risk aversion of the investors in the market.

### 1.3.3 Individual Expected Returns and the Role of Beta

In the CAPM equilibrium, if an equity return is not correlated with the return on the market portfolio, that equity’s expected return is equal to the risk-free rate because investors do not need to be compensated for bearing the uncertainty associated with that particular return. In Equation (1.4), if $$\beta_{e}=0$$, then $$\mathbb{E}(r_{e})=r_{f}$$. If an asset does not covary with the market portfolio, it becomes effectively riskless when it is held in a large, diversified portfolio that mirrors the market portfolio.

Equity returns that covary positively with the return on the market portfolio contribute to the variance of the return on the market portfolio. Consequently, these positive beta assets require an expected rate of return that is greater than the risk-free rate. On the other hand, an asset with a negative beta, whose return covaries negatively with the return on the market portfolio, actually reduces the overall variance of the portfolio. Investors willingly hold this asset even though its expected return is driven below the return on the risk-free interest rate in the competitive equilibrium. Most equities have positive betas, however, because the market environment tends to affect all stocks the same way.

Notice that an asset’s beta measures its relative risk because the beta is the covariance of the asset’s return with the return on the market portfolio divided by the variance of the return on the market portfolio. For example, if the beta is 1, the covariance of the asset’s return with the return on the market portfolio equals the variance of the return on the market, and the asset’s expected return is the same as the market’s expected return.

## 1.4 Domestic Versus World CAPMs

In a domestic CAPM, the market portfolio is defined as the aggregate asset holdings of all investors in a particular country. Many real-world applications of the CAPM use domestic CAPMs. For example, the beta for a U.K. firm that is listed on the London Stock Exchange (LSE) would be calculated relative to the LSE value-weighted market return, and the beta for a South African firm that is listed on the FTSE/JSE would be calculated relative to the FTSE/JSE value-weighted market return.

What are the implications of this assumption? The domestic CAPM assumes that assets of a country are held only by investors who reside in that country. In such a case, there would be no international diversification of risk, and countries’ capital markets would be completely internationally segmented. We discuss the concept of a segmented and integrated market more fully in a later section. When the CAPM was first developed in the 1960s, international segmentation seemed reasonable because capital flows and portfolio investments were limited. Today, in an increasingly globalized world, it makes more sense to use an internationally diversified portfolio of securities as the market portfolio. This CAPM is called the world CAPM.

### 1.4.1 The Role of Exchange Rates

One major theoretical problem with using the world CAPM is that the development of the theory assumes that investors share the same expectations about the real returns on different assets. Given the observed deviations from purchasing power parity and fluctuations in real exchange rates discussed previously, there is a substantial amount of evidence contrary to this premise. When real exchange rates fluctuate, investors in different countries have different perceptions about the real returns on different assets. Let’s illustrate this with an example.

Let $$r_{e}$$ be the real equity return on a U.S. security for a U.S.-based investor, and let $$r_{f}$$ be the real risk-free rate in the United States. The world CAPM states

$\begin{eqnarray} \mathbb{E}(r_{e})-r_{f}=\beta_{e}[\mathbb{E}(r_{m})-r_{f}] \tag{1.6} \end{eqnarray}$

where $$r_{m}$$ is the real rate of return on the world market portfolio. Since we are defining real returns for a U.S.-based investor, they are computed relative to the U.S. consumption basket, using the U.S. price level. For example, the real rate of return on equity, $$r_{e}$$, can be computed by subtracting 1 from 1 plus the nominal rate of return divided by 1 plus the U.S. rate of inflation: $$\frac{1+r_{e, \}}{1+\pi_{\}}-1$$. Similarly, we know that $$r_{f}$$, the ex ante real interest rate, is the expected value of the ex post real interest rate:

$\begin{eqnarray*} r_{f}=r_{f, \mathsf{US}} = \mathbb{E} \left[\frac{1+i_{\}}{1+\pi_{\}}-1 \right] \end{eqnarray*}$

where $$i_\$$ is the nominal interest rate.

Now, what is the expected real return on the same U.S. security for a South African investor? The South African investor cares about real South African returns, hence

$\begin{eqnarray*} \frac{1+r_{e, \mathsf{R}}}{1+\pi_{\mathsf{R}}}=\frac{[1+r_{e,\}](1+s)}{1+\pi_{\mathsf{R}}} \end{eqnarray*}$

with $$s$$ representing the percentage change in the rand-dollar exchange rate. But the expression for the dollar-based version of the CAPM contains the real return for the U.S. investor, $$\frac{1+r_{e,\}}{1+\pi_{\}}$$. This only equals the real return for the South African investor when $$\frac{1+s}{1+\pi_{\mathsf{R}}}=\frac{1}{1+\pi_{\}}$$, or $$1+s= \frac{1+\pi_{\mathsf{R}}}{1+\pi_{\}}$$. In other words, the real returns for the U.S.-based and South African-based investors are identical only when purchasing power parity (PPP) holds.

What about the risk-free rate? For the South African-based investor, it should be defined relative to her consumption basket. Consequently, the ex ante South African risk-free rate is $$r_{f, \mathsf{R}}=\mathbb{E} \left[ \frac{1+i_{\mathsf{R}}}{1+\pi_{\mathsf{R}}} \right]-1$$. If we assume that PPP holds, we find that

$\begin{eqnarray*} r_{f, \mathsf{R}}= \mathbb{E} \left[ \frac{1+i_{\mathsf{R}}}{(1+s)(1+\pi_{\})}-1 \right]. \end{eqnarray*}$

Of course, $$\mathbb{E} \left[ \frac{1+i_{\mathsf{R}}}{1+s} \right]$$ is the dollar return on an investment in the rand money market. For the real interest rates to be equalized across countries, we need more than just PPP to hold. We also need the real expected returns on money market investments to be equal across countries - that is, we need a real version of uncovered interest rate parity to hold.2 We conclude that translating the world CAPM to the other country’s perspectives works only when all the international parity conditions hold.

So far, we have focused on real returns as the theory demands. However, in practice, CAPMs are mostly applied to nominal returns. Let the nominal equity return be denoted by $$r_{e,\}$$, and let $$i_{\}$$ represent the money market interest rate in the United States. The world CAPM for the U.S.-based investor is then formulated as follows:

$\begin{eqnarray} \mathbb{E} \left[ r_{e, \} - i_{\} \right] = \beta_{e} \mathbb{E} [r_{m, \} - i_{\}] \tag{1.7} \end{eqnarray}$

where the equity return is earned over a short interval such as 1 month, and the interest rate is the 1-month T-bill rate known at the beginning of the month. For such small intervals of time, Equations (1.6) and (1.7) are indeed nearly equivalent. This is because, by definition, $$r_{e} = \frac{1+r_{e, \}}{1+\pi_{\}}-1 \approx r_{e, \} - \pi_{\}$$. Moreover, $$r_{f} = \mathbb{E} \left[ \frac{1+i_{\}}{1+\pi_{\}}-1 \right] \approx \mathbb{E}[i_{\} - \pi_{\}]$$. It is easy to see that the inflation rates cancel out of the equation.

Of course, the beta computation in the two equations is different, involving real returns in Equation (1.6) and nominal excess returns in Equation (1.7). Since equity returns are much more variable than inflation and interest rates, these differences are immaterial from a practical perspective.

The conditions for the world CAPM to apply to all countries are rather stringent. With deviations from the parity conditions, theory suggests more complex models where inflation and exchange rate risks enter the expected return computation. Many models of international capital market equilibrium have been developed, but none has attained a dominant status.3 Most models allow for currency risk premiums in one form or another.

An example of the most popular model in this class builds on the theories of Solnik (1974) and Sercu (1980) and forms the counterpart to the nominal returns model in Equation (1.7):

$\begin{eqnarray} \mathbb{E}[r_{j, \} - i_{\}] = \beta_{j} \mathbb{E}[r_{w, \} - i_{\} ]+ \sum_{k=1}^{K} \gamma_{j,k} \mathbb{E}[s_{k, t+1} - fp_{k,t}] \tag{1.8} \end{eqnarray}$

We assume that the dollar is the numeraire and that risk is measured for a U.S. investor.4 The first term represents the standard world market risk; the other terms represent exchange rate risk, with $$s_{k}$$ representing the rate of foreign currency appreciation and $$fp_{k}$$ representing the forward premium on currency $$k$$. Exchange rates are thus measured as \$ per currency $$k$$. Recall that

$\begin{eqnarray*} \mathbb{E}[s_{k, t+1}-fp_{k, t}]= \mathbb{E} \left[ \frac{S_{k,t+1} - F_{k, t}}{S_{k, t}} \right], \end{eqnarray*}$

which is the expected excess dollar return to a long forward market position in currency $$k$$. The $$\gamma_{j,k}$$’s in Equation (1.8) measures the exposures of the $$j$$-th firm’s returns to the various exchange rate risks. For example, an exporter with many unhedged foreign currency receivables may exhibit positive $$\gamma$$. That is, if these currencies appreciate substantially, the firm’s return will be high as well. Of course, if uncovered interest rate parity holds, this model collapses to the world CAPM. To compute the cost of capital in such a setting, we must run a multivariate regression of excess returns for security $$j$$ onto the world market return and various relevant currency returns. In practice, people use only a few major currencies or even a currency basket.

It is not clear whether the international CAPM is a better model than the world CAPM. Research by Dumas and Solnik (1995) and Zhang (2006) suggests that exchange rate risk is priced and that adding exchange rate factors to cost of capital computations is important. Other studies, such as that by Griffin and Stulz (2001), cast doubt on this conclusion. Due to the continuing academic controversy and the scant use of such models in practical capital budgeting situations, we will not discuss them further.

# 2 The CAPM in Practice

Firms need expected returns on their equity to get appropriate discount rates when doing capital budgeting. These expected returns represent what investors demand as compensation for giving capital to the firm. The CAPM delivers such discount rates. Let’s be very concrete about how to compute the cost of equity capital.

## 2.1 A Recipe for the Cost of Equity Capital

Recall the CAPM equation for security $$j$$:

$\begin{eqnarray} \mathbb{E}(r_{j})=r_{f}+\beta_{jm}[\mathbb{E}(r_{m})-r_{f}] \tag{2.1} \end{eqnarray}$

where $$\beta_{jm}=\frac{\mathsf{Cov}(r_{j},r_{m})}{\mathsf{Var}(r_{m})}$$. You find the expected nominal return on security $$j$$ by taking these steps:

Step 1. Get data on the market portfolio return, the equity returns on security $$j$$, and the T-bill interest rate, $$r_{f}$$.

Step 2. Determine the market risk premium, $$[\mathbb{E}(r_{m})-r_{f}]$$. The market risk premium is the expected excess return on a portfolio that approximates the market portfolio.

Step 3. Obtain an estimate of $$\beta_{jm}$$.

Step 4. Compute the expected return on security $$j$$ from Equation (2.1).

This recipe reveals three problems in applying the CAPM to a practical capital budgeting situation: the choice of a benchmark (how to measure the market portfolio), the estimation of beta, and the determination of the risk premium on the market portfolio. We discuss each in turn.

## 2.2 The Benchmark Problem

### 2.2.1 The Market Portfolio

One problem that has plagued the CAPM since its early development is what portfolio to use as the market portfolio.5 The theoretically correct value of the return on the market portfolio is the value-weighted return on all assets that are available for investors to purchase. If the return on the market portfolio is measured in rands, it would consequently include the rand-denominated returns on the equities of all the corporations of the different countries of the world, the rand-denominated returns on the bonds of all the corporations and the governments of these countries, and the rand-denominated returns on real estate and assets such as gold and land.

No one has ever attempted to use this version of the theory because its data requirements are too stringent. We simply do not have all the data. More importantly, though, financial markets are too imperfect to allow us to think that highly illiquid assets, such as real estate, would be bought and sold like stocks and bonds. Since data on the returns on corporate and government bonds in many countries are also difficult to obtain, in practice, people use the CAPM as if it were a theory that relates individual equity rates of return to a market portfolio composed of only equities.

### 2.2.2 World Market Proxies

When the CAPM is applied for a particular company’s project, the proxy for the market portfolio should in theory represent the well-diversified portfolio that the firm’s investors are holding. In practice, many U.S. companies use the U.S. stock market index as the market portfolio. With the increasing globalization of investors’ portfolios, a world market index is becoming more and more appropriate. Although the availability of data on a world market index is imperfect, there are reasonable proxies available, such as the Morgan Stanley Capital International (MSCI) Index and the Financial Times Actuaries (FTA) Index.

### 2.2.3 Getting the Benchmark Wrong

We would like to know how large a mistake is made quantitatively if we use a domestic, country-specific CAPM when the assets of the country are actually priced by investors with a world CAPM. If the assets of this country are actually priced internationally, the expected return on asset $$j, \mathbb{E}[r_{j}]$$, satisfies the world CAPM in Equation (2.1), where $$r_{m}$$ is the return on the world market portfolio and $$\beta_{jm}$$ is the beta of the return on asset $$j$$ with respect to the world market return. We denote this “true” expected return or cost of equity capital by $$\mathsf{COE}_{j}^{TR}$$. Now, suppose we postulate incorrectly that the expected return on asset $$j$$ is determined by the covariance of the return on asset $$j$$ with the return on the home market portfolio, $$r_{h}$$, as in the following version of a domestic CAPM:

$\begin{eqnarray} \mathbb{E}(r_{j})=r_{f}+\beta_{jh}[\mathbb{E}(r_{h})-r_{f}] \tag{2.2} \end{eqnarray}$

Denote the cost of equity capital number resulting from this computation by $$\mathsf{COE}_{j}^{FA}$$.

To compute the error in using Equation (2.2) rather than Equation (2.1), we first compute the correct expected return on the home market portfolio. The return on the home-country market portfolio is the value-weighted return on the individual assets in the country, and hence, it will also satisfy the world CAPM, as in Equation (2.1):

$\begin{eqnarray} \mathbb{E}(r_{h})=r_{f}+\beta_{hm}[\mathbb{E}(r_{m})-r_{f}] \tag{2.3} \end{eqnarray}$

Using Equations (2.1) to (2.3), we can investigate the difference between the two costs of equity capital:

$\begin{eqnarray*} \mathsf{COE}_{j}^{FA}-\mathsf{COE}_{j}^{TR} &=& \beta_{jh}[\mathbb{E}(r_{h})-r_{f}]-\beta_{jm}[\mathbb{E}(r_{m})-r_{f}] \\ &=& (\beta_{jh}\beta_{hm}-\beta_{jm})[\mathbb{E}(r_{m})-r_{f}] \end{eqnarray*}$

Thus, the expected return on asset $$j$$ will be correct if $$\beta_{jm}=\beta_{jh}\beta_{hm}$$. The following example provides some insight into when this expression is likely to be right and how badly things go if it is wrong.

Example: The Nestlé Cost of Equity Capital

R. M. Stulz (1995) applies the previous analysis to derive two estimates of the expected return for the Swiss company Nestlé. Stulz estimates the beta of the Swiss franc return on Nestlé with respect to the Swiss franc return on the Swiss market portfolio $$(\beta_{jh})$$ to be 0.885. The beta of the Swiss franc return on Nestlé with respect to the Swiss franc return on the world market portfolio $$(\beta_{jm})$$ using the FTA world market index is 0.585. The beta of the Swiss franc return on the Swiss market portfolio with respect to the Swiss franc return on the world market portfolio $$(\beta_{hm})$$ is 0.737. Hence, the pricing error in beta from using the domestic CAPM rather than the world CAPM is

$\begin{eqnarray*} \beta_{jh}\beta_{hm}-\beta_{jm}=(0.885\times 0.737)-0.585=0.067 \end{eqnarray*}$

Stulz uses an expected excess return on the world market portfolio $$[\mathbb{E}(r_{m})-r_{f}]$$ of 6.22%, in which case the error for Nestlé from using a domestic CAPM instead of the global CAPM is $$0.067 \times 6.22\% = 0.42\%$$.

Thus, using local pricing instead of global pricing implies an expected return for Nestlé that is 0.42% higher than it should be. If Nestlé is priced in the world market and not the local market, its required expected return should be the risk-free return on Swiss franc bonds plus a risk premium equal to the beta with the world market portfolio multiplied by the excess return on the world market portfolio, $$0.585 \times 6.22\% = 3.64\%$$. If Nestlé is priced in the local market, its required expected return would be the risk-free return on Swiss franc bonds plus a risk premium equal to $$3.64\% + 0.42\% = 4.06\%$$. This example demonstrates that, at least for Nestlé, the error from using a domestic CAPM when the world CAPM is appropriate does not seem to be too big. Estimation error in the betas and the mean return on the world market portfolio could easily lead one to consider discount rates that are in this range when doing sensitivity analysis. In a similar exercise, Harris et al. (2003) show that the world CAPM and the domestic CAPM led to similar cost-of-capital estimates for S& P 500 firms.

Figure 1: Estimating Beta

Note: The CAPM implies $$\hat{\alpha}_{\iota}=0$$. The $$x$$’s represent a combination of the excess return on the j-th asset and the excess return on the market portfolio.

## 2.3 Beta Estimation

Recall that the beta for security $$j$$ is given by $$\beta_{j}=\frac{\mathsf{Cov}[r_{j},r_{m}]}{\mathsf{Var}[r_{m}]}$$. Astute readers will recognize that $$\beta_{j}$$ is the regression coefficient from regressing $$r_{j}-r_{f}$$ onto $$r_{m}-r_{f}$$. Suppose you have data on excess returns for security $$j$$, $$r_{t,j}^{e}$$, and for the market, $$r_{t,m}^{e}$$. You obtain $$\beta_{j}$$ by running a regression:

$\begin{eqnarray*} r_{t,j}^{e}=\alpha_{j}+\beta_{t,j}r_{m}^{e}+\epsilon_{t,j} \end{eqnarray*}$

where $$\epsilon_{t,j}$$ is the error term in the regression. Figure 1 demonstrates graphically what we would find in a regression framework.

Many firms use the CAPM in their capital budgeting analyses. They can estimate the beta of a firm directly by choosing a portfolio to represent the market portfolio that is held by their investors and run the regression just described. Firms such as Barra and Value Line do the regressions and sell the information. Typically, the regression analysis uses only 60 months of data to accommodate the possibility that the risk profiles of companies change over time.

Estimating a beta using a regression is often imprecise because a firm’s returns exhibit considerable idiosyncratic volatility. That is, much of the variation in a firm’s return is driven by firm-specific events. This idiosyncratic volatility reduces the fit of the regression and increases the standard errors of the estimates. Therefore, some beta providers (such as Bank of America-Merrill Lynch) shrink the estimates toward 1, which is the value we would expect without other information. Another approach is to use industry portfolios. If firms in the same industry have about the same systematic risk, their betas will be about the same as well. A portfolio of firms diversifies away a lot of idiosyncratic risk and is consequently much less variable than an individual firm’s stock returns. Therefore, beta estimates from industry portfolios are more precise.

Example: Comparing Firm and Industry Betas

Yahoo’s financial website http://www.finance.yahoo.com provides estimates of betas for free. Let’s compare beta estimates obtained from there on March 21, 2011, with beta estimates obtained from Aswath Damodaran’s Web site at New York University http://pages.stern.nyu.edu/~adamodar/ for industry portfolios. The Yahoo estimates use 5 years of individual stock returns on a monthly basis, whereas the industry estimates also use 5 years of data, but at a weekly frequency:

The individual stock betas vary between 0.40 (McDonald’s) and 2.52 (Ford), whereas the industry estimates are much closer to 1.0.

There are good reasons for some companies to have betas that deviate from the industry average. For instance, they may have more or less financial leverage (debt value relative to equity value). If equity holders have to pay off bondholders before laying claim to the firm’s assets, their claims are riskier. Nevertheless, betas of only 0.40 for McDonald’s and 2.52 for Ford are almost surely due to unusual idiosyncratic movements of the firm’s stock prices over the sample period and are unlikely to give rise to reliable cost-of-capital estimates. A firm’s beta also changes over time as its business changes. Microsoft used to be a growth company with a very high beta. As it has become more mature with a more steady cash flow, its beta has also converged to 1.

## 2.4 The Risk Premium on the Market

### 2.4.1 Historical Estimates

It is surprising how little consensus there is about the magnitude of the equity risk premium.6 To estimate the risk premium, the first logical step is to look at history. Since stock returns are so volatile, it is important to take a long-run perspective. Dimson, Marsh, and Staunton (2007) collected 106 years’ worth of data, and Table 1 reproduces the historical risk premiums for 17 countries. These equity premiums vary between 4.51% for Denmark and 10.46% for Italy. The estimate for South Africa is 8.25%.

### 2.4.2 Caveats

Historical estimates, even for long samples, are still prone to large sampling errors, and different subperiods give very different answers. The recent global financial crisis illustrates how sensitive risk premium estimates can be. Many stock markets decreased by 40% or more in 2008. Even with 100 years of data, such dramatic outcome would lower the average by approximately 40 basis points. When shorter time periods are relied on, the effect would be even more dramatic.

Table 1: Equity Risk Premiums Around the World

Notes: Data from Dimson, Marsh, and Staunton (2007). The mean column reports the average return on equity in percentage per annum over and above a risk-free return for the period 1900 to 2005. The standard deviation column reports the annual standard deviation of these excess returns. [Germany values omit 1922 to 1923]

Research has suggested that these premiums have declined over more recent periods of time, even before the crisis. Brown, Goetzmann, and Ross (1995) note that the equity markets of various countries have periodically closed or failed outright. If investors thought that the market might actually fail, but it did not, then the average return over a long period would be abnormally high and not a good estimate of the expected ex ante return. In addition, Fama and French (2002) suggest that the high average realized equity returns post-World War II are greater than what was expected over the past 50 years because the ex post returns include “large unexpected capital gains” caused by a decline in discount rates. Claus and Thomas (2001) use analysts’ forecasts to argue that the equity premium should be 3%, which is less than half the historical average.

It is certainly possible that risk premiums have permanently declined. Investing in the stock market was traditionally difficult, costly, and limited to a select few, but now better technology, improved communication, an efficient collective investment scheme industry, and improved retirement fund legislation have increased stock market participation. Broadening the base of equity holders spreads risks and should decrease the risk premium. A decline in the risk premium produces a capital gain in stocks, but these high past returns signal future lower expected returns. Lettau, Ludvigson, and Wachter (2008) ascribe a decrease in discount rates to a reduction in macroeconomic risk, as measured by the volatility of consumption and output growth, witnessed in the 1980s and 1990s (the so-called Great Moderation). However, although the 2007 to 2010 economic crisis surely implied much lower returns on equities, it also signaled the end of the Great Moderation. Given all of this, substantial uncertainty about a correct value for the risk premium remains. Most practitioners would propose making use of an equity premium of between 4% and 7%.

### 2.4.3 The Need for Sensitivity Analysis

The imprecision in estimates of the equity premium combined with imprecision in the estimates of betas means that costs of equity capital are difficult to measure. In light of these difficulties, conducting sensitivity analysis when estimating the cost of equity capital is a good idea. Considering a range of values that are $$\pm$$ 2% around the estimates of the cost of capital seems appropriate.

# 3 Integrated versus Segmented Markets

In this section, we first discuss investing in emerging markets and the critical role investment barriers play. We then discuss how integrated versus segmented markets affect a company’s cost of capital. We end the section by describing the phenomenon of home bias.

## 3.1 Investing in Emerging Markets

Table 2 reports characteristics of annualized emerging market equity returns in dollars for the period from 1988 to 2010. The average returns vary between 5.78% for Jordan to a stellar 34.00% for Brazil. However, emerging market returns are very volatile, with most of the volatilities exceeding 30%. Turkey’s volatility is a whopping 59%. Nevertheless, the volatility of an index of emerging market returns measured in dollars is only 24%, which is about the same magnitude as that experienced by a developed country such as Japan.

The reduced volatility of the index reflects the low correlations across the emerging markets and the substantial benefits of diversification. The last four columns of Table 2 report the correlations of emerging market returns with the stock returns of the United States, Japan, the United Kingdom, and Germany. The correlations are generally lower than the correlations among developed countries, but there is lots of variation. The correlations vary between 0.08 for Argentina and Japan, and 0.61 for Hungary with the United Kingdom and Germany. The lowest correlations are typically observed with Japan, with the exception of Korea, which is more highly correlated with its close neighbour than with the other developed markets.

Table 2: Average Returns and Volatilities in Emerging Markets

Notes: For most emerging markets, the monthly data run from January 1988 to August 2010. All returns are in U.S. dollars. The last line reports characteristics for returns on the Emerging Market Index, a value-weighted average of all 26 country indexes.

Such low correlations should make it possible to construct low-risk portfolios. Therefore, it is not surprising that early studies showed significant diversification benefits for emerging market investments. However, these studies used market indexes compiled by the International Finance Corporation (IFC) that generally ignored the high transaction costs, low liquidity, and investment constraints associated with emerging market investments. More generally, older data may no longer be relevant given that many emerging markets imposed severe investment restrictions on foreign investors in the early 1990s. For example, in Korea, most stocks were subject to strict foreign ownership restrictions (foreign ownership was limited to 10% of market capitalization for most stocks).

Research by Bekaert (1996) and Bekaert (1999) shows that the returns cited in the early diversification studies using market index data could not actually be realized by foreign investors. To do so, they examined the diversification benefits U.S. investors enjoyed through investing in a variety of actually available investment vehicles for emerging markets, such as closed-end funds, ADRs, and open-end funds. These assets are easily accessible to retail investors, and investment costs are comparable to the investment costs for U.S.-traded stocks. Bekaert and Urias found that investors give up a substantial part of the diversification benefits by holding these investment vehicles relative to holding the indices.7

These results suggest that investment barriers may prevent the diversification benefits of emerging markets from being fully realized. They also make it unlikely that emerging markets satisfy the strong assumptions underlying the CAPM. In particular, emerging markets may not be completely integrated with world capital markets, making the world CAPM the wrong model to use. We now clarify the crucial distinction between integrated and segmented markets.

## 3.2 The Cost of Capital in Integrated and Segmented Markets

Markets are integrated when assets of identical risk command the same expected return, irrespective of their domicile. The governmental interferences with free capital markets in emerging markets can prevent market integration and effectively segment the capital markets of a country from the world capital market. If foreign investors are taxed or otherwise prohibited from holding the equities of a country, then that country’s assets are not part of the world market portfolio, and that country is said to be segmented from international capital markets.

The implications of segmentation for determining the cost of capital are important. Suppose we want to figure out the expected return on the Pakistani stock market. If the Pakistani stock market is integrated with world capital markets, we can simply use the world CAPM and the world market return as the benchmark portfolio. However, such an exercise would yield a very low expected return for Pakistan because the low correlation Pakistan displays with the world market translates into a low beta. Whereas this is the right computation to make for a foreign multinational corporation (MNC) investing in Pakistan, it yields a poor estimate of the true expected market return for local investors when the market is segmented. Harvey (1995) shows that the world CAPM provides a poor description of emerging market returns in general and that the domestic CAPM fares much better. Since the Pakistani market is segmented, all securities will be priced according to their correlation with the Pakistani market portfolio, but Pakistani investors will not be able to diversify the risk of the Pakistani market. Therefore, the expected return on the Pakistani market will be a function of its own volatility. This follows from aggregating the CAPM to the market level, as in Equation (1.4):

$\begin{eqnarray} \mathbb{E}[r_{j}]=r_{f}+\beta_{j}\mathbb{E}[r_{\mathsf{pak}}-r_{f}] \tag{3.1} \end{eqnarray}$

for every $$j$$ security in Pakistan, where $$r_{\mathsf{pak}}$$ is the return on the Pakistani market. We know that the $$\beta_{j}$$ is the covariance of security $$j$$ with the market portfolio; hence, we can rewrite Equation (3.1) as

$\begin{eqnarray*} \mathbb{E}[r_{j}]=r_{f}+\mathsf{Cov}(r_{j}, r_{\mathsf{pak}})\frac{\mathbb{E}[r_{\mathsf{pak}}-r_{f}]}{\mathsf{Var}(r_{\mathsf{pak}})} \end{eqnarray*}$

The expected excess return on the market portfolio divided by its variance is called the price of covariance risk. If investors hold only equities, Equation (1.5) shows that this price of risk equals the average risk aversion of the investors in Pakistan. Let’s denote this by $$A_{\mathsf{pak}}$$. Consequently, $$\mathbb{E}[r_{j}]=r_{f}+A_{\mathsf{pak}}$$ Cov $$(r_{j},\ r_{\mathsf{pak}})$$, and aggregating over all securities in Pakistan,

$\begin{eqnarray*} \mathbb{E}[r_{\mathsf{pak}}]=r_{f}+A_{\mathsf{pak}} \mathsf{Var}(r_{\mathsf{pak}}) \end{eqnarray*}$

Therefore, in segmented markets, expected and average returns should be related to the variance of returns rather than to the covariance with the world market return.

Example: The Expected Return in Pakistan

From data since 2000 on Pakistani stock returns, we determine that its world market beta is 0.4265. Given a risk-free rate of 5% and a world market equity premium of 5%, full integration dictates an expected return for the Pakistani market of

$\begin{eqnarray*} 5\% +0.4265 \times 5\%= 7.13\% \end{eqnarray*}$

While some foreign investors may find this cost-of-capital estimate low, most of the risk associated with investing in Pakistan may indeed be political in nature and idiosyncratic to Pakistan. Thus, it would not represent systematic risk.

However, if Pakistan is truly segmented, the local expected return depends on the local market volatility, which stands at 39.32% in dollar terms (see the above table). Suppose the average risk aversion in Pakistan is 2.0. Under a domestic CAPM for Pakistan, the expected return on the Pakistani market is

$\begin{eqnarray*} \mathbb{E}[r_{\mathsf{pak}}]=5\% +2.0(0.3932)^{2}= 35.92\% \end{eqnarray*}$ Clearly, the cost-of-capital estimates from the domestic CAPM and the world CAPM are very different. The fact that the domestic CAPM expected return is so unrealistically high may suggest that the Pakistani market is not fully segmented and that part of its variability is diversifiable.

### 3.2.1 Equity Market Liberalizations

Equity market liberalizations allow inward and outward foreign equity investment. The equity market liberalizations that took place in the late 1980s and early 1990s in many emerging markets form a nice laboratory to investigate the effects of potential integration into global capital markets.

If liberalization brings about integration with the global capital market, and if the world CAPM holds, what do we expect to happen? Suppose that the country is completely segmented from world capital markets before the liberalization. In this case, it is possible for the real interest rate in the country to be quite a bit higher than the world real interest rate. Also, the risk premiums associated with the equities in that country will be dictated by the variance of the return on that country’s market portfolio. As we saw in the above example, these risk premiums may be quite high.

Now, suppose the country unexpectedly opens its capital markets to the world economy. Two things will happen: First, the real interest rate in the country should fall dramatically because the country’s residents are now free to borrow and lend internationally, and there is additional foreign supply of capital.8 Second, the equities of the country will now be priced based on their covariances with the return on the world market portfolio, which are likely to be much smaller than the variance of the local market. Both of these effects will reduce the discount rate on the country’s assets.

A big reduction in the discount rate, of course, causes the price of an asset to rise dramatically, which provides a big rate of return to the investors holding these assets. Simply put, foreign investors will bid up the prices of local stocks in an effort to diversify their portfolios, while all investors will shun inefficient sectors.9 Thus, equity prices should rise substantially (as expected returns decrease) when a market moves from a segmented to an integrated state. When a market is opened to international investors, though, the country’s assets may become more sensitive to world events. In other words, their covariances with the rest of the world’s assets may increase. Even with this effect, it is likely that these covariances will remain much smaller than the variance of the local market. The data bear out the theory. Studies by Kim and Singal (2000), Henry (2000), Bekaert and Harvey (2000), and others show that equity market liberalizations were accompanied by positive returns to integration as foreign investors bid up local prices. Post-liberalization returns, in contrast, were lower on average, as the theory predicts. While the exact estimates differ somewhat, liberalization causes the cost of capital to decline by about 1%.

An interesting parallel occurs with respect to the price of a firm’s shares following the issuance of an ADR. An ADR issued by a company headquartered in a country with investment restrictions can be viewed as a sort of liberalization of investment. For example, when Chile had repatriation restrictions in place, it lifted the restrictions for those companies listing their shares overseas to allow cross-market arbitrage. When an ADR is announced, we therefore expect positive announcement returns (e.g., relative to a similar firm not introducing an ADR) and lower expected returns after the liberalization. Several studies demonstrate that this effect is typically larger than 1%, and the studies find lower costs of capital after the ADR issuance. Of course, there are many reasons, apart from liberalization, that ADR issues may result in a positive effect on the price of equity shares.

Many studies, as surveyed in Bekaert and Harvey (2003), have investigated the effects of liberalizations on other return characteristics. First, there is no significant impact on the volatility of market returns. Indeed, it is not obvious from finance theory that volatility should increase or decrease when markets are opened to foreign investment. On the one hand, markets may become informationally more efficient, leading to higher volatility as prices quickly react to relevant information, or hot speculative capital may induce excess volatility. On the other hand, in the preliberalized market, there may be large swings from fundamental values, leading to higher volatility. In the long run, the gradual development and diversification of the market should lead to lower volatility. Second, the correlation of the return and its beta with the world market increases after equity market liberalizations, and for some countries, the increase is dramatic. This is also consistent with these liberalizing emerging markets becoming more integrated with world capital markets.

## 3.3 Segmentation and Integration over Time

Although the empirical studies on the financial effects of equity market liberalizations confirm the intuition predicted by the simple CAPM, this does not mean that we are now living in a globally integrated capital market. In fact, using official regulatory reforms to measure liberalization is fraught with difficulties because it is difficult to know what effectively segments a market from the global capital market.

There are three different kinds of barriers. The first are legal barriers, such as foreign ownership restrictions and taxes on foreign investments. An additional complication here is that the liberalization process is typically a complex and gradual one. It took Korea almost 10 years between 1991 and 2000 to gradually remove its foreign ownership restrictions. The second are indirect barriers arising from differences in available information, accounting standards, and investor protection. The third are emerging-market-specific risks (EMSRs) that discourage foreign investment. EMSRs include liquidity risk, political risk, economic policy risk, and perhaps currency risk. In general, indirect barriers and EMSRs may make institutional investors in developed countries reluctant to invest in emerging markets and segment them from the world market.

Finally, regulatory restrictions might not have posed a barrier prior to liberalization because canny investors often find ways to circumvent them. Alternatively, there may be legal, indirect ways to access local equity markets, such as through country funds or ADRs. The Korea Fund, trading on the NYSE, is a good example; it was launched in 1986, well before the liberalization of the Korean equity market. In short, determining whether a market is segmented, integrated, or something in between is far from easy.

### 3.3.1 A Model of Time-Varying Market Integration

Given the imperfections posed by official regulatory reform dates, researchers have come up with a variety of models to determine when and to what extent markets are integrated. For example, Bekaert and Harvey (1995) build on the CAPM model to measure the degree of market integration. In integrated markets, the covariance with the world market should determine the expected return on the domestic market. However, if the market is truly segmented, the variance of the return on the domestic market should affect the domestic expected return. Bekaert and Harvey apply an econometric framework, which allows the degree of a country’s integration with the world market to vary over time, directly to equity return data. They find that the degree of equity market integration seems to vary for all countries in the sample, but variation in the integration measure does not always coincide with capital market reforms. For example, consider the market rate of return in Greece, which is completely open to foreign investors. The market return was more sensitive to the variance of the return on the Greek market in some periods than to the covariance between the return on the Greek market and the return on the world market portfolio. In contrast, Mexico has had rather strong legal restrictions on foreign investment, which would lead us to think that the variance of Mexico’s stock market ought to be important when it comes to determining its expected return. But the analysis implies that Mexico is actually quite integrated with the world market. Consistent with this analysis, Table 2 shows that Mexican equity returns have a 57% correlation with U.S. returns, whereas we already discussed the low correlation of Greek returns with other developed markets.

Bekaert (2011) follow a different approach. They compare the valuation of industry portfolios in different countries with the valuation of the same industry globally by computing earnings yields (total earnings divided by market capitalization). Under some assumptions, industry earnings yields in different countries converge toward the global earnings yields when markets are economically and financially integrated. They take the market capitalization weighted average of these earnings yields differentials for various industry portfolios to arrive at a “segmentation measure” for each country, which essentially measures the absolute difference in earnings yields with the global yield. For developed countries, these average yield differentials are 2% for 2001 to 2005, which could be generated through noise and measurement error in a fully integrated market. However, for emerging markets, these differentials were, on average, 4.3%, suggesting segmentation. Bekaert et al. also document considerable convergence of earnings yields over time and demonstrate that, apart from the regulatory liberalization process, indirect barriers (such as the quality of the regulatory and legal framework) and emerging-market-specific risks (such as the liquidity in the stock markets) play an important role in explaining variation across countries and across time in the degree of segmentation. For examples that describe the practical implications of segmentation and time-varying integration see Bodnar, Dumas, and Marston (2003).

## 3.4 Home Bias and Its Implications

Unlike what the CAPM predicts, investors in different countries are generally not very well internationally diversified. In other words, most of their portfolios have a strong home bias. Home bias means that British investors, for example, hold a disproportionately large share of British assets compared to the world market portfolio. Table 3 documents home bias for equity portfolios using data from the International Monetary Fund (IMF).

The home bias in Table 3 is measured in a “raw” and “normalized” form for 6 years between 1997 and 2005 and averaged, following Bekaert, Harvey, and Lundblad (2010). Raw home bias measures the difference between the portfolio share that each country invests in its own market (home market share) and the share of the country’s market in the world market (world market benchmark). By this measure, the United States is by far the least home-biased market. However, this is largely true because the U.S. market represents a large fraction of the world market.

Table 3: Characterizing Home Bias

Note: Reproduced from Table 2 in Bekaert, Harvey, and Lundblad (2010).

The normalized home bias measure divides the raw measure by 1 - world market benchmark weight, which is nothing but the maximum bias that can occur. A fully home-biased country has a normalized measure of 1, whereas a country that invests in its own market consistent with its share in the world market has a home bias measure of zero. Table 3 delivers a few stark results. First, all around the world, people hold far less foreign securities than the world CAPM would dictate. Investors do not seem to take full advantage of the considerable benefits of international diversification. Second, the biases are large. Of 27 countries, only the Netherlands has a bias less than 50%. Third, the bias is much larger for emerging markets than for developed markets. This is particularly striking because the benefits of portfolio diversification are presumably larger for emerging market residents than for developed market residents, given how volatile their domestic stock markets tend to be.

Finally, it is generally known that the degree of home bias has substantially decreased over time. Cai and Warnock (2006) claim that the degree of home bias is overstated because institutional investors tend to overweight their domestic investments toward multinationals that have international exposure through their foreign operations and cash flows. Yet, even adjusting the numbers for this additional foreign exposure, home bias remains significant for most countries in the world, and it is something that is not well understood by financial economists.

### 3.4.1 Implications for Pricing

If investors are not fully internationally diversified, should we discard the world CAPM as the benchmark model? This is a difficult issue. However, it might not be necessary for every individual in the world to be fully internationally diversified for asset returns to be well described by a world CAPM. In fact, whereas it is true that emerging market returns do not look at all consistent with a world CAPM, the evidence against other stock markets is not strong. Harvey (1991) and R. Hodrick, Ng, and Sengmueller (1999) show that a version of the CAPM in fact works well for most developed stock markets most of the time.

### 3.4.2 Time-Varying Correlations

The trend toward less home bias, and the move toward ever-increased integration, as investment barriers, both direct and indirect, are dismantled, should also increase the correlations across countries, making international diversification less viable. Table 4 sheds some light on this issue. It reports correlations for Japan, Canada, the United Kingdom, France, and Italy with the United States for every decade since 1970 and for the past decade (until August 2010). Until 1999, the correlations increase steadily for all countries except Japan. However, for all countries, the correlations are substantially higher during 2000 to 2010 than they were previously.

Table 4: Correlations with U.S. Equity Returns

Note: The data are from MSCI.

Whether the increases in correlations are due to increased market integration and, therefore, represent a permanent change is an important question. Since temporarily higher volatility in equity markets also tends to temporarily increase the correlations between markets, it is difficult to separate temporary from permanent correlation changes. The intuition for this fact is best understood if we consider two countries satisfying the world CAPM. As a consequence, part of the return variation in both countries is driven by the returns on the world market, and this joint exposure likely induces positive correlation between the returns on the two stock markets. Intuitively, if the world market movements became extremely variable, they would dominate all return variation in the two stocks, and the correlation would converge to 1. This is relevant for the numbers produced in Table 4, as the world market volatility at the end of the 1990s, in the early 2000s, and again during the 2007 to 2010 financial crisis was indeed relatively high. A study by Bekaert, Hodrick, and Zhang (2009) concludes that return correlations within Europe have permanently increased, but their tests do not reject the hypothesis that return correlations elsewhere have remained unchanged, once account is taken of temporary changes in volatility.

# 4 Conclusion

To determine the cost of equity capital in global financial markets we can make use of the capital asset pricing model (CAPM). It states that under some simplifying assumptions, the MVE portfolio ought to be the market portfolio, which contains all securities in proportion to their market capitalization. The mechanisms of the model imply that the expected return of any security equals the risk-free rate plus the beta of the security multiplied by the market risk premium. The beta of the security is the covariance of its return with the return on the market portfolio divided by the variance of the market portfolio return. In an international setting, the relevant benchmark for the market portfolio should be the world market portfolio, giving rise to the world CAPM, which ignores exchange rate risk.

In an international setting, investors in different countries evaluate real returns using different consumption baskets and view money market investments in other countries as risky because of exchange rate risk. Although it is possible to adjust the CAPM for these considerations, the resulting international CAPMs are rarely used in practice. To use the CAPM to obtain a cost of capital we must determine the betas, the market risk premium, and a risk-free rate. The risk-free rate is usually the Treasury bill rate. The beta is estimated from a regression of excess returns on the security in question onto excess returns on the world market portfolio. Sometimes, industry portfolios are used to reduce the sampling error in estimating the betas. The risk premium on the market portfolio is the subject of much controversy. An estimate of 4% to 7% is reasonable. In any case, any cost-of-capital estimation and project evaluation should be accompanied by a sensitivity analysis.

Emerging equity markets display relatively low correlations with the stock markets of developed countries. Many of the emerging markets underwent a liberalization process in the 1990s that made their stock markets fully or partially accessible to foreign investors. Equity markets are integrated when assets of identical risk command the same expected return, irrespective of their domicile. The many investment barriers in place in emerging markets have effectively segmented them from the global capital market. The liberalization process, however, has led to increased asset prices, higher correlations with the world market, and lower expected returns in emerging markets.

The benchmark used in the cost-of-capital computation should reflect the composition of the portfolio of the investors in the company, even when the project takes place in a potentially segmented emerging market. Historical data in these emerging markets may not be very useful for a cost-of-capital analysis if the market is truly segmented or if it underwent a liberalization process that caused a structural break in the return data.

Even in the developed world, investors have not fully internationally diversified. Instead, their portfolios are heavily invested in their home markets. This phenomenon is known as home bias. Evidence suggests that there has been a gradual increase in the correlations between the G7 countries, potentially reflecting increased economic and financial integration.

# 5 Appendix: The Mathematics of International Diversification

Here, we formally prove two results that we used.

## 5.1 Risk Reduction

### 5.1.1 Statement:

As long as the correlation between two assets is less than 1, the standard deviation of a portfolio of the two assets will be less than the weighted average of the two individual standard deviations.

Proof: Let $$w$$ and $$1-w$$ denote the investment proportions in the two assets. Let $$\sigma_{1}$$ and $$\sigma_{2}$$ denote the two standard deviations of the two assets. We use two statistical properties:

1. The variance of a sum of two random variables equals the sum of the variances plus twice the covariance between the variables.
2. The correlation, $$\rho$$, between two variables is their covariance divided by the product of their standard deviations.

Hence, the variance of the portfolio with weights $$\{w, 1-w\}$$ is:

$\begin{eqnarray*} w^{2}\sigma_{1}^{2}+(1-w)^{2}\sigma_{2}^{2}+2w(1-w)\rho\sigma_{1}\sigma_{2} \end{eqnarray*}$

We want to show

$\begin{eqnarray*} \{w^{2}\sigma_{1}^{2}+(1-w)^{2}\sigma_{2}^{2}+2w(1-w)\rho\sigma_{1}\sigma_{2}\}^{0.5} < w\sigma_{1}+(1-w)\sigma_{2} \end{eqnarray*}$

Squaring both sides gives

$\begin{eqnarray*} w^{2}\sigma_{1}^{2}+(1-w)^{2}\sigma_{2}^{2}+2w(1-w)\rho\sigma_{1}\sigma_{2} < w^{2}\sigma_{1}^{2}+(1-w)^{2}\sigma_{2}^{2}+2w(1-w)\sigma_{1}\sigma_{2} \end{eqnarray*}$

Strict inequality follows from $$\rho<1$$. Hence, when $$\rho$$ is smaller than 1, the variance of the portfolio is always smaller than the variance of either asset. As a special case, if $$\sigma_{1}=\sigma_{2}=\sigma$$, the variance is minimized by setting $$w=0.5$$, and the portfolio variance is 0.5 $$[1+\rho]\sigma^{2}.$$

## 5.2 Improving the Sharpe Ratio

### 5.2.1 Statement:

If $$\frac{\mathbb{E}[r^{*}]-r_{f}}{\mathrm{V}\mathrm{o}1[r^{*}]}>\rho\frac{\mathbb{E}[r]-r_{f}}{\mathrm{V}\mathrm{o}1[r]}$$, the Sharpe ratio improves when an asset with return $$r^{*}$$ is added (marginally) to the portfolio with return $$r$$. Without loss of generality, we set the return on the risk-free asset equal to $$0$$ in our proof.

Proof: The Sharpe ratio of the portfolio with $$w$$ invested in the foreign asset is

$\begin{eqnarray*} SR=\frac{(1-w)\mathbb{E}(r)+w\mathbb{E}(r^{*})}{\mathsf{Var}(P)} \end{eqnarray*}$

where $$\mathsf{Var}(P)=(1-w)^{2}\mathsf{Var}(r)+w^{2}\mathsf{Var}(r^{*})+2w(1-w) \mathsf{Cov}(r,\ r^{*})$$ .

We want to show that if the statement holds, then $$\partial SR -> 0$$ evaluated at $$w\equiv 0$$. Taking the derivative and $$\partial w$$ leaving out the (positive) denominator, we obtain:

$\begin{eqnarray*} \frac{\partial SR}{\partial w}>0\Leftrightarrow(\mathbb{E}[r^{*}]-\mathbb{E}[r])\mathsf{Var}[P]^{0.5}-\mathbb{E}[P] \times\frac{1}{2}\mathsf{Var}[P]^{\frac{1}{2}}\times[-2\mathsf{Var}[r]+2\mathsf{Cov}[r,r^{*}]]>0 \end{eqnarray*}$

Evaluating this at $$w=0$$ means that $$P$$ equals the U.S. portfolio. Multiplying through with Var $$[P]^{0.5}$$ and simplifying, we obtain

$\begin{eqnarray*} \mathbb{E}[r^{*}] \mathsf{Var} [r]-\mathbb{E}[r] \mathsf{Cov} [r,r^{*}]>0 \end{eqnarray*}$

or

$\begin{eqnarray*} \frac{\mathbb{E}[r^{*}]}{\mathsf{Var}[r^{*}]^{0.5}}>\frac{\mathbb{E}[r]}{\mathsf{Var}[r]^{0.5}}\times\frac{\mathsf{Cov}[r,r^{*}]}{\mathsf{Var}[r]^{0.5}\mathsf{Var}[r^{*}]^{0.5}} \end{eqnarray*}$

where the terms refer to the Foreign Sharpe Ratio, Domestic Sharpe Ratio, and Correlation CORR $$(r, r^{d})$$, respectively.

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1. There are about fifty index tracking funds that are listed in South Africa.

2. We derived that real interest rates are equalized across countries when PPP, uncovered interest rate parity, and the Fisher hypothesis hold.

3. See Adler and Dumas (1983) for an early model.

4. One problem with the many variants of the international CAPM, including the one presented here, is that the exact outcome of the cost-of-capital computation may depend on the numeraire currency.

5. This issue is often called the “Roll critique” because Roll (1977) was the first to write about the problems involved in testing the CAPM. Roll argued that statistical rejections of the theory could be incorrect if a statistician did not observe the true market portfolio.

6. A direct perspective on this issue can be gleaned from Ivo Welch’s survey of the opinions of professional economists. Welch’s 2009 survey puts the average estimate at 6%.

7. The reduction in benefits is only partially due to investment barriers being priced in. For open-end funds, active investment management may cause a reduction in diversification benefits. Didier, Rigobon, and Schmukler (2013) demonstrate that mutual fund managers tend to hold concentrated portfolios that hamper full international diversification.

8. It is conceivable that before the liberalization, the government may have kept interest rates artificially low - for instance, through interest rate ceilings - in which case, the interest rate may rise upon liberalization.

9. A more formal analysis can be found in Bekaert and Harvey (2003), which builds on work by Errunza and Losq (1985).