,

CHAPTER 5

PORTFOLIO RISK AND RETURN: PART I

Vijay Singal, CFA

Blacksburg, VA, U.S.A.

LEARNING OUTCOMES

After completing this chapter, you will be able to do the following:

• Calculate and interpret major return measures and describe their applicability.
• Describe the characteristics of the major asset classes that investors would consider in forming portfolios according to mean–variance portfolio theory.
• Calculate and interpret the mean, variance, and covariance (or correlation) of asset returns based on historical data.
• Explain risk aversion and its implications for portfolio selection.
• Calculate and interpret portfolio standard deviation.
• Describe the effect on a portfolio’s risk of investing in assets that are less than perfectly correlated.
• Describe and interpret the minimum-variance and efficient frontiers of risky assets and the global minimum-variance portfolio.
• Discuss the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line.

1. INTRODUCTION

Construction of an optimal portfolio is an important objective for an investor. In this chapter, we will explore the process of examining the risk and return characteristics of individual assets, creating all possible portfolios, selecting the most efficient portfolios, and ultimately choosing the optimal portfolio tailored to the individual in question.

During the process of constructing the optimal portfolio, several factors and investment characteristics are considered. The most important of those factors are risk and return of the individual assets under consideration. Correlations among individual assets along with risk and return are important determinants of portfolio risk. Creating a portfolio for an investor requires an understanding of the risk profile of the investor. Although we will not discuss the process of determining risk aversion for individuals or institutional investors, it is necessary to obtain such information for making an informed decision. In this chapter, we will explain the broad types of investors and how their risk–return preferences can be formalized to select the optimal portfolio from among the infinite portfolios contained in the investment opportunity set.

The chapter is organized as follows: Section 2 discusses the investment characteristics of assets. In particular, we show the various types of returns and risks, their computation and their applicability to the selection of appropriate assets for inclusion in a portfolio. Section 3 discusses risk aversion and how indifference curves, which incorporate individual preferences, can be constructed. The indifference curves are then applied to the selection of an optimal portfolio using two risky assets. Section 4 provides an understanding and computation of portfolio risk. The role of correlation and diversification of portfolio risk are examined in detail. Section 5 begins with the risky assets available to investors and constructs a large number of risky portfolios. It illustrates the process of narrowing the choices to an efficient set of risky portfolios before identifying the optimal risky portfolio. The risky portfolio is combined with investor risk preferences to generate the optimal risky portfolio. Section 6 summarizes the concepts discussed in this chapter.

2. INVESTMENT CHARACTERISTICS OF ASSETS

Financial assets are generally defined by their risk and return characteristics. Comparison along these two dimensions simplifies the process of selecting from millions of assets and makes financial assets substitutable. These characteristics distinguish financial assets from physical assets, which can be defined along multiple dimensions. For example, wine is characterized by its grapes, aroma, sweetness, alcohol content, and age, among other factors. The price of a television depends on picture quality, manufacturer, screen size, number and quality of speakers, and so on, none of which are similar to the characteristics for wine. Therein lies one of the biggest differences between financial and physical assets. Although financial assets are generally claims on real assets, their commonality across two dimensions (risk and return) simplifies the issue and makes them easier to value than real assets. In this section, we will compute, evaluate, and compare various measures of return and risk.

2.1. Return

Financial assets normally generate two types of return for investors. First, they may provide periodic income through cash dividends or interest payments. Second, the price of a financial asset can increase or decrease, leading to a capital gain or loss.

Certain financial assets, through design or choice, provide return through only one of these mechanisms. For example, investors in non-dividend-paying stocks, such as Google or Baidu, obtain their return from capital appreciation only. Similarly, you could also own or have a claim to assets that only generate periodic income. For example, defined benefit pension plans, retirement annuities, and reverse mortgages1 make income payments as long as you live.

You should be aware that returns reported for stock indices are sometimes misleading because most index levels only capture price appreciation and do not adjust for cash dividends unless the stock index is labeled “total return” or “net dividends reinvested.” For example, as reported by Yahoo! Finance, the S&P 500 Index of U.S. stocks was at 903.25 on 31 December 2008. Similarly, Yahoo! Finance reported that the index closed on 30 July 2002 at 902.78 implying a return of close to 0 percent over the approximately six-and-a-half-year period. The results are very different, however, if the total return S&P 500 Index is considered. The index was at 1283.62 on 30 July 2002 and had risen 13.2 percent to 1452.98 on 31 December 2008, giving an annual return of 1.9 percent. The difference in the two calculations arises from the fact that index levels reported by Yahoo! Finance and other reporting agencies do not include cash dividends, which are an important part of the total return. Thus, it is important to recognize and account for income from investments.

In the following subsection, we consider various types of returns, their computation, and their application.

2.1.1. Holding Period Return

Returns can be measured over a single period or over multiple periods. Single period returns are straightforward because there is only one way to calculate them. Multiple period returns, however, can be calculated in various ways and it is important to be aware of these differences to avoid confusion.

A holding period return is the return earned from holding an asset for a single specified period of time. The period may be 1 day, 1 week, 1 month, 5 years, or any specified period. If the asset (bond, stock, etc.) is bought now, time (t – 1), at a price of 100 and sold later, say at time t, at a price of 105 with no dividends or other income, then the holding period return is 5 percent [105 – (100/100)]. If the asset also pays an income of 2 units at time t, then the total return is 7 percent. This return can be generalized and shown as a mathematical expression:

In the preceding expression, P is the price and D is the dividend. The subscript indicates the time of that price or dividend: t – 1 is the beginning of the period and t is the end of the period. The following two observations are important.

• We computed a capital gain of 5 percent and a dividend yield of 2 percent in the previous example. For ease of illustration, we assumed that the dividend is paid at time t. If the dividend was received any time before t, our holding period return would have been higher because we would have earned a return by putting the dividend in the bank for the remainder of the period.
• Return can be expressed in decimals (0.07), fractions (7/100), or as a percent (7%). They are all equivalent.

The holding period return can be computed for a period longer than one year. For example, you may need to compute a three-year holding period return from three annual returns. In that case, the holding period return is computed by compounding the three annual returns: R=[(1+R1)×(1+R2)×(1+R3)] – 1, where R1, R2, and R3 are the three annual returns.

In this and succeeding parts of Section 2.1, we consider the aggregation of several single period returns.

2.1.2. Arithmetic or Mean Return

When assets have returns for multiple holding periods, it is necessary to aggregate those returns into one overall return for ease of comparison and understanding. It is also possible to compute the return for a long or an unusual holding period. Such returns, however, may be difficult to interpret. For example, a return of 455 percent earned by AstraZeneca PLC over the last 16 years (1993 to 2008) may not be meaningful unless all other returns are computed for the same period. Therefore, most holding period returns are reported as daily, monthly, or annual returns.

Aggregating returns across several holding periods becomes a challenge and can lead to different conclusions depending on the method of aggregation. The remainder of this section is designed to present various ways of computing average returns as well as discussing their applicability.

The simplest way to compute the return is to take the simple average of all holding period returns. Thus, three annual returns of −50 percent, 35 percent, and 27 percent will give us an average of 4 percent per year= . The arithmetic return is easy to compute and has known statistical properties, such as standard deviation. We can calculate the arithmetic return and its standard deviation to determine how dispersed the observations are around the mean or if the mean return is statistically different from zero.

In general, the arithmetic or mean return is denoted by and given by the following equation for asset i, where Rit is the return in period t and T is the total number of periods.

2.1.3. Geometric Mean Return

The arithmetic mean return is the average of the returns earned on a unit of investment at the beginning of each holding period. It assumes that the amount invested at the beginning of each period is the same, similar to the concept of calculating simple interest. However, because the base amount changes each year (the previous year’s earnings needs to be added to or “compounded” to the beginning value of the investment), a holding or geometric period return may be quite different from the return implied by the arithmetic return. The geometric mean return assumes that the investment amount is not reset at the beginning of each year and, in effect, accounts for the compounding of returns. Basically, the geometric mean reflects a “buy-and-hold” strategy whereas arithmetic reflects a constant dollar investment at the beginning of each time period.2

A geometric mean return provides a more accurate representation of the return that an investor will earn than an arithmetic mean return assuming that the investor holds the investment for the entire time. In general, the geometric mean return is denoted by and given by the following equation for asset i,

where Rit is the return in period t and T is the total number of periods.

In the example in Section 2.1.2, we calculated the arithmetic mean to be 4 percent. Exhibit 5-1 shows the actual return for each year and the actual amount at the end of each year using actual returns. Beginning with an initial investment of €1.0000, we will have €0.8573 at the end of the three-year period as shown in the third column. Note that we compounded the returns because, unless otherwise stated, we receive return on the amount at the end of the prior year. That is, we will receive a return of 35 percent in the second year on the amount at the end of the first year, which is only €0.5000, not the initial amount of €1.0000. Let us compare the actual amount at the end of the three-year period, €0.8573, with the amount we get using an annual arithmetic mean return of 4 percent calculated previously. The year-end amounts are shown in the fourth column using the arithmetic return of 4 percent. At the end of the three-year period, €1 will be worth €1.1249 (= 1.0000 × 1.043). This ending amount of €1.1249 is much larger than the actual amount of €0.8573. Clearly, the calculated arithmetic return is greater than the actual return. In general, the arithmetic return is biased upward unless the actual holding period returns are equal. The bias in arithmetic mean returns is particularly severe if holding period returns are a mix of both positive and negative returns, as in the example.

EXHIBIT 5-1

For our example and using the previous formula, the geometric mean return per year is −5.0 percent, compared with an arithmetic mean return of 4.0 percent. The last column of Exhibit 5-1 shows that using the geometric return of −5.0 percent generates a value of €0.8574 at the end of the three-year period, which is very close to the actual value of €0.8573. The small difference in ending values is the result of a slight approximation used in computing the geometric return of −5.0 percent. Because of the effect of compounding, the geometric mean return is always less than or equal to the arithmetic mean return, , unless there is no variation in returns, in which case they are equal.

2.1.4. Money-Weighted Return or Internal Rate of Return

The preceding return computations do not account for the amount of money invested in different periods. It matters to an investor how much money was invested in each of the three years. If she had invested €10,000 in the first year, €1,000 in the second year, and €1,000 in the third year, then the return of −50 percent in the first year significantly hurts her. On the other hand, if she had invested only €100 in the first year, the effect of the −50 percent return is drastically reduced.

The money-weighted return accounts for the money invested and provides the investor with information on the return she earns on her actual investment. The money-weighted return and its calculation are similar to the internal rate of return and the yield to maturity. Just like the internal rate of return, amounts invested are cash outflows from the investor’s perspective and amounts returned or withdrawn by the investor, or the money that remains at the end of an investment cycle, is a cash inflow for the investor.

The money-weighted return can be illustrated most effectively with an example. In this example, we use the actual returns from the previous example. Assume that the investor invests €100 in a mutual fund at the beginning of the first year, adds another €950 at the beginning of the second year, and withdraws €350 at the end of the second year. The cash flows are shown in Exhibit 5-2.

EXHIBIT 5-2

The internal rate of return is the discount rate at which the sum of present values of these cash flows will equal zero. In general, the equation may be expressed as follows, where T is the number of periods, CFt is the cash flow at time t, and IRR is the internal rate of return or the money-weighted rate of return:

A cash flow can be positive or negative; a positive cash flow is an inflow where money flows to the investor whereas a negative cash flow is an outflow where money flows away from the investor. We can compute the internal rate of return by using the preceding equation. The flows are expressed as follows, where each cash inflow or outflow occurs at the end of each year. Thus, CF0 refers to the cash flow at the end of year 0 or beginning of year 1, and CF3 refers to the cash flow at end of year 3 or beginning of year 4. Because cash flows are being discounted to the present—that is, end of year 0 or beginning of year 1—the period of discounting CF0 is zero.

IRR = 26.11% is the internal rate of return, or the money-weighted rate of return, which tells the investor what she earned on the actual euros invested for the entire period. This return is much greater than the arithmetic and geometric mean returns because only a small amount was invested when the mutual fund’s return was −50 percent.

Although the money-weighted return is an accurate measure of what the investor actually earned on the money invested, it is limited in its applicability to other situations. For example, it does not allow for return comparison between different individuals or different investment opportunities. Two investors in the same mutual fund may have different money-weighted returns because they invested different amounts in different years.

2.1.5. Comparison of Returns

The previous subsections have introduced a number of return measures. The following example illustrates the computation, comparison, and applicability of each measure.

2.1.6. Annualized Return

The period during which a return is earned or computed can vary and often we have to annualize a return that was calculated for a period that is shorter (or longer) than one year. You might buy a short-term treasury bill with a maturity of 3 months, or you might take a position in a futures contract that expires at the end of the next quarter. How can we compare these returns? In many cases, it is most convenient to annualize all available returns. Thus, daily, weekly, monthly, and quarterly returns are converted to an annual return. In addition, many formulas used for calculating certain values or prices may require all returns and periods to be expressed as annualized rates of return. For example, the most common version of the Black–Scholes option-pricing model requires annualized returns and periods to be in years.

To annualize any return for a period shorter than one year, the return for the period must be compounded by the number of periods in a year. A monthly return is compounded 12 times, a weekly return is compounded 52 times, and a quarterly return is compounded 4 times. Daily returns are normally compounded 365 times. For an uncommon number of days, we compound by the ratio of 365 to the number of days.

If the weekly return is 0.2 percent, then the compound annual return is computed as shown because there are 52 weeks in a year:

If the return for 15 days is 0.4 percent, the annualized return is computed assuming 365 days in a year. Thus,

A general equation to annualize returns is given, where c is the number of periods in a year. For a quarter, c = 4 and for a month, c = 12:

How can we annualize a return when the holding period return is more than one year? For example, how do we annualize an 18-month holding period return? Because one year contains two-thirds of 18-month periods, c = 2/3 in the preceding equation. An 18-month return of 20 percent can be annualized, as shown:

Similar expressions can be constructed when quarterly or weekly returns are needed for comparison instead of annual returns. In such cases, c is equal to the number of holding periods in a quarter or in a week. For example, assume that you want to convert daily returns to weekly returns or annual returns to weekly returns for comparison between weekly returns. For converting daily returns to weekly returns, c = 5, assuming that there are five trading days in a week. For converting annual returns to weekly returns, c = 1/52. The expressions for annual returns can then be rewritten as expressions for weekly returns, as shown:

One major limitation of annualizing returns is the implicit assumption that returns can be repeated precisely; that is, money can be reinvested repeatedly while earning a similar return. This type of return is not always possible. An investor may earn a return of 5 percent during a week because the market went up that week or he got lucky with his stock, but it is highly unlikely that he will earn a return of 5 percent every week for the next 51 weeks, resulting in an annualized return of 1,164.3 percent (= 1.0552 – 1). Therefore, it is important to annualize short-term returns with this limitation in mind. Annualizing returns, however, allows for comparison among different assets and over different time periods.

2.1.7. Portfolio Return

When several individual assets are combined into a portfolio, we can compute the portfolio return as a weighted average of the returns in the portfolio. The portfolio return is simply a weighted average of the returns of the individual investments, or assets. If asset 1 has a return of 20 percent and constitutes 25 percent of the portfolio’s investment, then the contribution to the portfolio return is 5 percent (= 25% of 20%). In general, if asset i has a return of Ri and has a weight of wi in the portfolio, then the portfolio return, RP, is given as:

Note that the weights must add up to 1 because the assets in a portfolio, including cash, must account for 100 percent of the investment. Also, note that these are single period returns, so there are no cash flows during the period and the weights remain constant.

A two-asset portfolio is easier to work with, so we will use only two assets to illustrate most concepts. Extending the analysis to multiple assets, however, is easily achieved and covered in later sections. With only two assets in the portfolio, the portfolio return can be written as shown, where w1 and w2 are weights in assets 1 and 2.

Because the portfolio consists of only two assets, the sum of the two weights should equal 100 percent. Therefore, w1 + w2 = 1 or w2 = (1 – w1). By substituting, we can rewrite the preceding equation as follows:

2.2. Other Major Return Measures and Their Applications

The statistical measures of return discussed in the previous section are generally applicable across a wide range of assets and time periods. Special assets, however, such as mutual funds, and other considerations, such as taxes or inflation, may require return measures that are specific to a particular application.

Although it is not possible to consider all types of special applications, we will discuss the effect of fees (gross versus net returns), taxes (pre-tax and after-tax returns), inflation (nominal and real returns), and leverage. Many investors use mutual funds or other external entities (i.e., investment vehicles) for investment. In those cases, funds charge management fees and expenses to the investors. Consequently, gross and net-of-fund-expense returns should also be considered. Of course, an investor may be interested in the net-of-expenses after-tax real return, which is in fact what an investor truly receives. We consider these additional return measures in the following sections.

2.2.1. Gross and Net Return

A gross return is the return earned by an asset manager prior to deductions for management expenses, custodial fees, taxes, or any other expenses that are not directly related to the generation of returns but rather related to the management and administration of an investment. These expenses are not deducted from the gross return because they may vary with the amount of assets under management or may vary because of the tax status of the investor. Trading expenses, however, such as commissions, are accounted for in (i.e., deducted from) the computation of gross return because trading expenses contribute directly to the return earned by the manager. Thus, gross return is an appropriate measure for evaluating and comparing the investment skill of asset managers because it does not include any fees related to the management and administration of an investment.

Net return is a measure of what the investment vehicle (mutual fund, etc.) has earned for the investor. Net return accounts for (i.e., deducts) all managerial and administrative expenses that reduce an investor’s return. Because individual investors are most concerned about the net return (i.e., what they actually receive), small mutual funds with a limited amount of assets under management are at a disadvantage compared with the larger funds that can spread their largely fixed administrative expenses over a larger asset base. As a result, many small-sized mutual funds waive part of the expenses to keep the funds competitive.

2.2.2. Pre-Tax and After-Tax Nominal Return

All return measures discussed previously are pre-tax nominal returns—that is, no adjustment has been made for taxes or inflation. In general, all returns are pre-tax nominal returns unless they are otherwise designated.

Investors are concerned about the tax liability of their returns because taxes reduce the actual return that they receive. The two types of returns, capital gains (change in price) and income (such as dividends or interest), are usually taxed differently. Capital gains come in two forms: short-term capital gains and long-term capital gains. Long-term capital gains typically receive preferential tax treatment in a number of countries. Interest income is taxed as ordinary income in most countries. Dividend income may be taxed as ordinary income, may have a lower tax rate, or may be exempt from taxes depending on the country and the type of investor. The after-tax nominal return is computed as the total return minus any allowance for taxes on realized gains.3

Because taxes are paid on realized capital gains and income, the investment manager can minimize the tax liability by selecting appropriate securities (e.g., those subject to more favorable taxation, all other investment considerations equal) and reducing trading turnover. Therefore, many investors evaluate investment managers based on the after-tax nominal return.

2.2.3. Real Returns

A nominal return (r) consists of three components: a real risk-free return as compensation for postponing consumption (rrF), inflation as compensation for loss of purchasing power (π), and a risk premium for assuming risk (RP). Thus, nominal return and real return can be expressed as:

Often the real risk-free return and the risk premium are combined to arrive at the real “risky” rate as given in the second equation above, simply referred to as the real return. Real returns are particularly useful in comparing returns across time periods because inflation rates may vary over time. Real returns are also useful in comparing returns among countries when returns are expressed in local currencies instead of a constant investor currency in which inflation rates vary between countries (which are usually the case). Finally, the after-tax real return is what the investor receives as compensation for postponing consumption and assuming risk after paying taxes on investment returns. As a result, the after-tax real return becomes a reliable benchmark for making investment decisions. Although it is a measure of an investor’s benchmark return, it is not commonly calculated by asset managers because it is difficult to estimate a general tax component applicable to all investors. For example, the tax component depends on an investor’s specific taxation rate (marginal tax rate), how long the investor holds an investment (long-term versus short-term), and the type of account the asset is held in (tax-exempt, tax-deferred, or normal).

2.2.4. Leveraged Return

In the previous calculations, we have assumed that the investor’s position in an asset is equal to the total investment made by an investor using his or her own money. This section differs in that the investor creates a leveraged position. There are two ways of creating a claim on asset returns that are greater than the investment of one’s own money. First, an investor may trade futures contracts in which the money required to take a position may be as little as 10 percent of the notional value of the asset. In this case, the leveraged return, the return on the investor’s own money, is 10 times the actual return of the underlying security. Note that both the gains and losses are amplified by a factor of 10.

Investors can also invest more than their own money by borrowing money to purchase the asset. This approach is easily done in stocks and bonds, and very common when investing in real estate. If half (50 percent) of the money invested is borrowed, then the asset return to the investor is doubled but the investor must account for interest to be paid on borrowed money.

2.3. Variance and Covariance of Returns

Having discussed the various kinds of returns in considerable detail, we now turn to measures of riskiness of those returns. Just like return, there are various kinds of risk. For now, we will consider the total risk of an asset or a portfolio of assets as measured by its standard deviation, which is the square root of variance.

2.3.1. Variance of a Single Asset

Variance, or risk, is a measure of the volatility or the dispersion of returns. Variance is measured as the average squared deviation from the mean. Higher variance suggests less predictable returns and therefore a more risky investment. The variance (σ2) of asset returns is given by the following equation,

where Rt is the return for period t, T is the total number of periods, and μ is the mean of T returns, assuming T is the population of returns.

If only a sample of returns is available instead of the population of returns (as is usually the case in the investment world), then the previous expression underestimates the variance. The correction for sample variance is made by replacing the denominator with (T – 1), as shown next, where is the mean return of the sample observations and s2 is the sample variance:

2.3.2. Standard Deviation of an Asset

The standard deviation of returns of an asset is the square root of the variance of returns. The population standard deviation (σ) and the sample standard deviation (s) are given next.

Standard deviation is another measure of the risk of an asset, which may also be referred to as its volatility. In a later section, we will decompose this risk measure into its separate components.

2.3.3. Variance of a Portfolio of Assets

Like a portfolio’s return, we can calculate a portfolio’s variance. When computing the variance of portfolio returns, standard statistical methodology can be used by finding the variance of the full expression of portfolio return. Although the return of a portfolio is simply a weighted average of the returns of each security, this is not the case with the standard deviation of a portfolio (unless all securities are perfectly correlated—that is, correlation equals one). Variance can be expressed more generally for N securities in a portfolio using the notation from Section 2.1.7 of this chapter:

The right side of the equation is the variance of the weighted average returns of individual securities. Weight is a constant, but the returns are variables whose variance is shown by Var (Ri). We can rewrite the equation as shown next. Because the covariance of an asset with itself is the variance of the asset, we can separate the variances from the covariances in the second equation:

Cov(Ri, Rj) is the covariance of returns, Ri and Rj, and can be expressed as the product of the correlation between the two returns (ρ1,2) and the standard deviations of the two assets. Thus, Cov(Ri,Rj) = ρijσiσj.

For a two-asset portfolio, the expression for portfolio variance simplifies to the following using covariance and then using correlation:

The standard deviation of a two-asset portfolio is given by the square root of the portfolio’s variance:

2.4. Historical Return and Risk

At this time, it is beneficial to look at historical risk and returns for the three main asset categories: stocks, bonds, and Treasury bills. Stocks refer to corporate ownership, bonds refer to long-term fixed-income securities, and Treasury bills refer to short-term government debt securities. Although there is generally no expectation of default on government securities, long-term government bond prices are volatile (risky) because of possible future changes in interest rates. In addition, bondholders also face the risk that inflation will reduce the purchasing power of their cash flows.

2.4.1. Historical Mean Return and Expected Return

Before examining historical data, it is useful to distinguish between the historical mean return and expected return, which are very different concepts but easy to confuse. Historical return is what was actually earned in the past, whereas expected return is what an investor anticipates to earn in the future.

Expected return is the nominal return that would cause the marginal investor to invest in an asset based on the real risk-free interest rate (rrF), expected inflation [E(π)], and expected risk premium for the risk of the asset [E(RP)]. The real risk-free interest rate is expected to be positive as compensation for postponing consumption. Similarly, the risk premium is expected to be positive in most cases.4 The expected inflation rate is generally positive, except when the economy is in a deflationary state and prices are falling. Thus, expected return is generally positive. The relationship between the expected return and the real risk-free interest rate, inflation rate, and risk premium can be expressed by the following equation:

The historical mean return for investment in a particular asset, however, is obtained from the actual return that was earned by an investor. Because the investment is risky, there is no guarantee that the actual return will be equal to the expected return. In fact, it is very unlikely that the two returns are equal for a specific time period being considered. Given a long enough period of time, we can expect that the future (expected) return will equal the average historical return. Unfortunately, we do not know how long that period is—10 years, 50 years, or 100 years. As a practical matter, we often assume that the historical mean return is an adequate representation of the expected return, although this assumption may not be accurate. For example, Exhibit 5-5 shows that the historical equity returns in the past nine years (2000–2008) for large U.S. company stocks were negative whereas the expected return was nearly always positive. Nonetheless, longer-term returns (1926–2008) were positive and could be consistent with expected return. Though it is unknown if the historical mean returns accurately represent expected returns, it is an assumption that is commonly made.

Going forward, be sure to distinguish between expected return and historical mean return. We will alert the reader whenever historical returns are used to estimate expected returns.

EXHIBIT 5-5 Risk and Return for U.S. Asset Classes by Decade (%)

Source: 2009 Ibbotson SBBI Classic Yearbook (Tables 2-1, 6-1, C-1 to C-7).

2.4.2. Nominal Returns of Major U.S. Asset Classes

We focus on three major asset categories in Exhibit 5-5: stocks, bonds, and T-bills. The mean nominal returns for U.S. asset classes are reported decade by decade since the 1930s. The total for the 1926–2008 period is in the last column. All returns are annual geometric mean returns. Large company stocks had an overall annual return of 9.6 percent during the 83-year period. The return was negative in the 1930s and 2000s, and positive in all remaining decades. The 1950s and 1990s were the best decades for large company stocks. Small company stocks fared even better. The nominal return was never negative for any decade, and had double-digit growth in all decades except two, leading to an overall 83-year annual return of 11.7 percent.

Long-term corporate bonds and long-term government bonds earned overall returns of 5.9 percent and 5.7 percent respectively. The corporate bonds did not have a single negative decade, although government bonds recorded a negative return in the 1950s when stocks were doing extremely well. Bonds also had some excellent decades, earning double-digit returns in the 1980s and 2000s.

Treasury bills (short-term government securities) did not earn a negative return in any decade. In fact, Treasury bills earned a negative return only in 1938 (–0.02 percent) when the inflation rate was −2.78 percent. Consistently positive returns for Treasury bills are not surprising because nominal interest rates are almost never negative and the Treasury bills suffer from little interest rate or inflation risk. Since the Great Depression, there has been no deflation in any decade, although inflation rates were highly negative in 1930 (–6.03 percent), 1931 (–9.52 percent), and 1932 (–10.30 percent). Conversely, inflation rates were very high in the late 1970s and early 1980s, reaching 13.31 percent in 1979. Inflation rates have fallen since then to a negligible level of 0.09 percent in 2008. Overall, the inflation rate was 3.0 percent for the 83-year period.

2.4.3. Real Returns of Major U.S. Asset Classes

Because inflation rates can vary greatly, from −10.30 percent to +13.31 percent in the past 83 years, comparisons across various time periods is difficult and misleading using nominal returns. Therefore, it is more effective to rely on real returns. Real returns on stocks, bonds, and T-bills are reported from 1900 in Exhibits 5-6 (page 195) and 5-7 (page 196).

EXHIBIT 5-7 Nominal Returns, Real Returns, and Risk Premiums for Asset Classes (1900–2008)

T-bills and inflation rates are not available for the world and world excluding the United States. All returns are in percent per annum measured in US$. GM = geometric mean, AM = arithmetic mean, SD = standard deviation.“World” consists of 17 developed countries: Australia, Belgium, Canada, Denmark, France, Germany, Ireland, Italy, Japan, the Netherlands, Norway, South Africa, Spain, Sweden, Switzerland, United Kingdom, and the United States. Weighting is by each country’s relative market capitalization size.

Source: Credit Suisse Global Investment Returns Yearbook, 2009. Compiled from tables 62, 65, and 68.

Exhibit 5-6 shows that $1 would have grown to $582 if invested in stocks, to only $9.90 if invested in bonds, and to $2.90 if invested in T-bills. The difference in growth among the three asset categories is huge, although the difference in real returns does not seem that large: 6.0 percent per year for equities compared with 2.2 percent per year for bonds. This difference represents the effect of compounding over a 109-year period. The graph tracks the growth of money through major economic and political events, such as the world wars, the Great Depression, oil shocks, and other economic crashes and booms. Another interesting statistic to note is that most of the total return in stocks came from dividends, not from capital gains. If an investor relied only on capital gains, his investment would have grown to only $6 instead of $582 over the 109-year period.

Exhibit 5-7 reports both the nominal and real rates of return. As we discussed earlier and as shown in the table, geometric mean is never greater than the arithmetic mean. Our analysis of returns focuses on the geometric mean because it is a more accurate representation of returns for multiple holding periods than the arithmetic mean. We observe that the real returns for U.S. stocks are higher than the real returns for U.S. bonds, and that the real returns for bonds are higher than the real returns for U.S. T-bills.

2.4.4. Nominal and Real Returns of Asset Classes in Major Countries

Along with U.S. returns, returns of major asset classes for a 17-country world and the world excluding the United States are also presented in Exhibit 5-7. Equity returns are weighted by each country’s GDP before 1968 because of a lack of reliable market capitalization data. Returns are weighted by a country’s market capitalization beginning with 1968. Similarly, bond returns are defined by a 17-country bond index except GDP is used to create the weights because equity market capitalization weighting is inappropriate for a bond index and bond market capitalizations were not readily available.

The nominal mean return for the world stock index over the past 109 years was 8.4 percent, and bonds had a nominal geometric mean return of 4.8 percent. The nominal geometric mean returns for the world excluding the United States are 7.9 percent for stocks and 4.2 percent for bonds. For both stocks and bonds, the United States has earned higher returns than the world excluding the U.S. Similarly, real returns for stocks and bonds in the United States are higher than the real returns for rest of the world. No separate information is available for Treasury bills for non-U.S. countries.

2.4.5. Risk of Major Asset Classes

Risk for major asset classes in the United States is reported for 1926–2008 in Exhibit 5-5, and the risk for major asset classes for the United States, the world, and the world excluding the United States are reported for 1900–2008 in Exhibit 5-7. Exhibit 5-5 shows that U.S. small company stocks had the highest risk, 33.0 percent, followed by U.S. large company stocks, 20.6 percent. Long-term government bonds and long-term corporate bonds had lower risk at 9.4 percent and 8.4 percent, with Treasury bills having the lowest risk at about 3.1 percent.

Exhibit 5-7 shows that the risk for world stocks is 17.3 percent and for world bonds is 8.6 percent. The world excluding the United States has risks of 20.1 percent for stocks and 13.0 percent for bonds. The effect of diversification is apparent when world risk is compared with U.S. risk and world excluding U.S. risk. Although the risk of U.S. stocks is 20.2 percent and the risk of world excluding U.S. stocks is 20.1 percent, the combination gives a risk of only 17.3 percent for world stocks. We can see a similar impact for world bonds when compared with U.S. bonds and world bonds excluding U.S. bonds. We observe a similar pattern in the risk levels of real returns.

2.4.6. Risk–Return Trade-Off

The expression “risk–return trade-off” refers to the positive relationship between expected risk and return. In other words, a higher return is not possible to attain in efficient markets and over long periods of time without accepting higher risk. Expected returns should be greater for assets with greater risk.

The historical data presented previously show the risk–return trade-off. Exhibit 5-5 shows for the United States that small company stocks had higher risk and higher return than large company stocks. Large company stocks had higher returns and higher risk than both long-term corporate bonds and government bonds. Bonds had higher returns and higher risk than Treasury bills. Uncharacteristically, however, long-term government bonds had higher total risk than long-term corporate bonds, although the returns of corporate bonds were slightly higher. These factors do not mean that long-term government bonds had greater default risk, just that they were more variable than corporate bonds during this historic period.

Turning to real returns, we find the same pattern: Higher returns were earned by assets with higher risk. Exhibit 5-7 reveals that the risk and return for stocks were the highest of the asset classes, and the risk and return for bonds were lower than stocks for the United States, the world, and the world excluding the United States. U.S. Treasury bills had the lowest return and lowest risk among T-bills, bonds, and stocks.

Another way of looking at the risk–return trade-off is to focus on the risk premium, which is the extra return investors can expect for assuming additional risk, after accounting for the nominal risk-free interest rate (includes both compensation for expected inflation and the real risk-free interest rate). Worldwide equity risk premiums reported at the bottom of Exhibit 5-7 show that equities outperformed bonds and bonds outperformed T-bills. Investors in equities earned a higher return than investors in T-bills because of the higher risk in stocks. Conversely, investors in T-bills cannot expect to earn as high a return as equity investors because the risk of their holdings is much lower.

A more dramatic representation of the risk–return trade-off was shown in Exhibit 5-6, which shows the cumulative returns of U.S. asset classes in real terms. The dashed line representing T-bills is much less volatile than the other lines. Adjusted for inflation, the average real return on T-bills was 1.0 percent per year. The grey line representing bonds is more volatile than the line for T-bills but less volatile than the lines representing stocks. The total return for equities including dividends and capital gains is represented by the dark line where $1 invested at the beginning of 1900 grows to $582, generating an annualized return of 6.0 percent in real terms.

Over long periods of time, we observe that higher risk does result in higher mean returns. Thus, it is reasonable to claim that, over the long term, market prices reward higher risk with higher returns, which is a characteristic of a risk-averse investor, a topic that we discuss in Section 3.

2.5. Other Investment Characteristics

In evaluating investments using mean (expected return) and variance (risk), we make two important assumptions. First, we assume that the returns are normally distributed because a normal distribution can be fully characterized by its mean and variance. Second, we assume that markets are not only informationally efficient but that they are also operationally efficient. To the extent that these assumptions are violated, we need to consider additional investment characteristics. These are discussed next.

2.5.1. Distributional Characteristics

A normal distribution has three main characteristics: its mean and median are equal; it is completely defined by two parameters, mean and variance; and it is symmetric around its mean with:

• 68 percent of the observations within ±1σ of the mean.
• 95 percent of the observations within ±2σ of the mean.
• 99 percent of the observations within ±3σ of the mean.

Using only mean and variance would be appropriate to evaluate investments if returns were distributed normally. Returns, however, are not normally distributed; deviations from normality occur both because the returns are skewed, which means they are not symmetric around the mean, and because the probability of extreme events is significantly greater than what a normal distribution would suggest. The latter deviation is referred to as kurtosis or fat tails in a return distribution. The next sections discuss these deviations more in-depth.

2.5.5.1. Skewness

Skewness refers to asymmetry of the return distribution, that is, returns are not symmetric around the mean. A distribution is said to be left skewed or negatively skewed if most of the distribution is concentrated to the right, and right skewed or positively skewed if most is concentrated to the left. Exhibit 5-8 shows a typical representation of negative and positive skewness, whereas Exhibit 5-9 demonstrates the negative skewness of stock returns by plotting a histogram of U.S. large company stock returns for 1926–2008. Stock returns are usually negatively skewed because there is a higher frequency of negative deviations from the mean, which also has the effect of overestimating standard deviation.

2.5.1.2. Kurtosis

Kurtosis refers to fat tails or higher than normal probabilities for extreme returns and has the effect of increasing an asset’s risk that is not captured in a mean–variance framework, as illustrated in Exhibit 5-10. Investors try to evaluate the effect of kurtosis by using such statistical techniques as value at risk (VAR) and conditional tail expectations.5 Several market participants note that the probability and the magnitude of extreme events is underappreciated and was a primary contributing factor to the financial crisis of 2008.6 The higher probability of extreme negative outcomes among stock returns can also be observed in Exhibit 5-9.

2.5.2. Market Characteristics

In the previous analysis, we implicitly assumed that markets are both informationally and operationally efficient. Although informational efficiency of markets is a topic beyond the purview of this chapter, we should highlight certain operational limitations of the market that affect the choice of investments. One such limitation is liquidity.

The cost of trading has three main components—brokerage commission, bid–ask spread, and price impact—of which liquidity affects the latter two. Brokerage commission is usually negotiable and does not constitute a large fraction of the total cost of trading except in small-sized trades. Stocks with low liquidity can have wide bid–ask spreads. The bid–ask spread, which is the difference between the buying price and the selling price, is incurred as a cost of trading a security. The larger the bid–ask spread, the higher is the cost of trading. If a $100 stock has a spread of 10 cents, the bid–ask spread is only 0.1 percent ($0.10/$100). On the other hand, if a $10 stock has a spread of 10 cents, the bid–ask spread is 1 percent. Clearly, the $10 stock is more expensive to trade and an investor will need to earn 0.9 percent extra to make up the higher cost of trading relative to the $100 stock.

Liquidity also has implications for the price impact of trade. Price impact refers to how the price moves in response to an order in the market. Small orders usually have little impact, especially for liquid stocks. For example, an order to buy 100 shares of a $100 stock with a spread of 1 cent may have no effect on the price. On the other hand, an order to buy 100,000 shares may have a significant impact on the price as the buyer has to induce more and more stockholders to tender their shares. The extent of the price impact depends on the liquidity of the stock. A stock that trades millions of shares a day may be less affected than a stock that trades only a few hundred thousand shares a day. Investors, especially institutional investors managing large sums of money, must keep the liquidity of a stock in mind when making investment decisions.

Liquidity is a bigger concern in emerging markets than in developed markets because of the smaller volume of trading in those markets. Similarly, liquidity is a more important concern in corporate bond markets and especially for bonds of lower credit quality than in equity markets because an individual corporate bond issue may not trade for several days or weeks. This certainly became apparent during the global financial crisis.

There are other market-related characteristics that affect investment decisions because they might instill greater confidence in the security or might affect the costs of doing business. These include analyst coverage, availability of information, firm size, and so on. These characteristics about companies and financial markets are essential components of investment decision making.

3. RISK AVERSION AND PORTFOLIO SELECTION

As we have seen, stocks, bonds, and T-bills provide different levels of returns and have different levels of risk. Although investment in equities may be appropriate for one investor, another investor may not be inclined to accept the risk that accompanies a share of stock and may prefer to hold more cash. In the last section, we considered investment characteristics of assets in understanding their risk and return. In this section, we consider the characteristics of investors, both individual and institutional, in an attempt to pair the right kind of investors with the right kind of investments.

First, we discuss risk aversion and utility theory. Later we discuss their implications for portfolio selection.

3.1. The Concept of Risk Aversion

The concept of risk aversion is related to the behavior of individuals under uncertainty. Assume that an individual is offered two alternatives: one where he will get £50 for sure and the other is a gamble with a 50 percent chance that he gets £100 and 50 percent chance that he gets nothing. The expected value in both cases is £50, one with certainty and the other with uncertainty. What will an investor choose? There are three possibilities: an investor chooses the gamble, the investor chooses £50 with certainty, or the investor is indifferent. Let us consider each in turn. However, please understand that this is only a representative example, and a single choice does not determine the risk aversion of an investor.

3.1.1. Risk Seeking

If an investor chooses the gamble, then the investor is said to be risk loving or risk seeking. The gamble has an uncertain outcome, but with the same expected value as the guaranteed outcome. Thus, an investor choosing the gamble means that the investor gets extra “utility” from the uncertainty associated with the gamble. How much is that extra utility worth? Would the investor be willing to accept a smaller expected value because he gets extra utility from risk? Indeed, risk seekers will accept less return because of the risk that accompanies the gamble. For example, a risk seeker may choose a gamble with an expected value of £45 in preference to a guaranteed outcome of £50.

There is a little bit of gambling instinct in many of us. People buy lottery tickets although the expected value is less than the money they pay to buy it. Or people gamble at casinos in Macau or Las Vegas with the full knowledge that the expected return is negative, a characteristic of risk seekers. These or any other isolated actions, however, cannot be taken at face value except for compulsive gamblers.

3.1.2. Risk Neutral

If an investor is indifferent about the gamble or the guaranteed outcome, then the investor may be risk neutral. Risk neutrality means that the investor cares only about return and not about risk, so higher return investments are more desirable even if they come with higher risk. Many investors may exhibit characteristics of risk neutrality when the investment at stake is an insignificant part of their wealth. For example, a billionaire may be indifferent about choosing the gamble or a £50 guaranteed outcome.

3.1.3. Risk Averse

If an investor chooses the guaranteed outcome, he/she is said to be risk averse because the investor does not want to take the chance of not getting anything at all. Depending on the level of aversion to risk, an investor may be willing to accept a guaranteed outcome of £45 instead of a gamble with an expected value of £50.

In general, investors are likely to shy away from risky investments for a lower, but guaranteed return. That is why they want to minimize their risk for the same amount of return, and maximize their return for the same amount of risk. The risk–return trade-off discussed earlier is an indicator of risk aversion. A risk-neutral investor would maximize return irrespective of risk and a risk-seeking investor would maximize both risk and return.

Data presented in the last section illustrate the historically positive relationship between risk and return, which demonstrates that market prices were based on transactions and investments by risk-averse investors and reflect risk aversion. Therefore, for all practical purposes and for our future discussion, we will assume that the representative investor is a risk-averse investor. This assumption is the standard approach taken in the investment industry globally.

3.1.4. Risk Tolerance

Risk tolerance refers to the amount of risk an investor is willing to tolerate to achieve an investment goal. The higher the risk tolerance, the greater is the willingness to take risk. Thus, risk tolerance is negatively related to risk aversion.

3.2. Utility Theory and Indifference Curves

Continuing with our previous example, a risk-averse investor would rank the guaranteed outcome of £50 higher than the uncertain outcome with an expected value of £50. We can say that the utility that an investor or an individual derives from the guaranteed outcome of £50 is greater than the utility or satisfaction or happiness he/she derives from the alternative. In general terms, utility is a measure of relative satisfaction from consumption of various goods and services or, in the case of investments, the satisfaction that an investor derives from different portfolios.

Because individuals are different in their preferences, all risk-averse individuals may not rank investment alternatives in the same manner. Consider the £50 gamble again. All risk-averse individuals will rank the guaranteed outcome of £50 higher than the gamble. What if the guaranteed outcome is only £40? Some risk-averse investors might consider £40 inadequate, others might accept it, and still others may now be indifferent about the uncertain £50 and the certain £40.

A simple implementation of utility theory allows us to quantify the rankings of investment choices using risk and return. There are several assumptions about individual behavior that we make in the definition of utility given in the following equation. We assume that investors are risk averse. They always prefer more to less (greater return to lesser return). They are able to rank different portfolios in the order of their preference and that the rankings are internally consistent. If an individual prefers X to Y and Y to Z, then he/she must prefer X to Z. This property implies that the indifference curves (see Exhibit 5-11) for the same individual can never touch or intersect. The following is an example of a utility function:

where, U is the utility of an investment, E(r) is the expected return, and σ2 is the variance of the investment.

In the preceding equation, A is a measure of risk aversion, which is measured as the marginal reward that an investor requires to accept additional risk. More risk-averse investors require greater compensation for accepting additional risk. Thus, A is higher for more risk-averse individuals. As was mentioned previously, a risk-neutral investor would maximize return irrespective of risk and a risk-seeking investor would maximize both risk and return.

We can draw several conclusions from the utility function. First, utility is unbounded on both sides. It can be highly positive or highly negative. Second, higher return contributes to higher utility. Third, higher variance reduces the utility but the reduction in utility gets amplified by the risk aversion coefficient, A. Utility can always be increased, albeit marginally, by getting higher return or lower risk. Fourth, utility does not indicate or measure satisfaction. It can be useful only in ranking various investments. For example, a portfolio with a utility of 4 is not necessarily two times better than a portfolio with a utility of 2. The portfolio with a utility of 4 could increase our happiness 10 times or just marginally. But we do prefer a portfolio with a utility of 4 to a portfolio with a utility of 2. Utility cannot be compared among individuals or investors because it is a very personal concept. From a societal point of view, by the same argument, utility cannot be summed among individuals.

Let us explore the utility function further. The risk aversion coefficient, A, is greater than zero for a risk-averse investor. So any increase in risk reduces his/her utility. The risk aversion coefficient for a risk-neutral investor is 0, and changes in risk do not affect his/her utility. For a risk lover, the risk aversion coefficient is negative, creating an inverse situation so that additional risk contributes to an increase in his/her utility. Note that a risk-free asset (σ2=0) generates the same utility for all individuals.

3.2.1. Indifference Curves

An indifference curve plots the combinations of risk–return pairs that an investor would accept to maintain a given level of utility (i.e., the investor is indifferent about the combinations on any one curve because they would provide the same level of overall utility). Indifference curves are thus defined in terms of a trade-off between expected rate of return and variance of the rate of return. Because an infinite number of combinations of risk and return can generate the same utility for the same investor, indifference curves are continuous at all points.

A set of indifference curves is plotted in Exhibit 5-11. By definition, all points on any one of the three curves have the same utility. An investor does not care whether he/she is at point a or point b on indifference curve 1. Point a has lower risk and lower return than point b, but the utility of both points is the same because the higher return at point b is offset by the higher risk.

Like curve 1, all points on curve 2 have the same utility and an investor is indifferent about where he/she is on curve 2. Now compare point c with point b. Point c has the same risk but significantly lower return than point b, which means that the utility at point c is less than the utility at point b. Given that all points on curve 1 have the same utility and all points on curve 2 have the same utility and point b has higher utility than point c, curve 1 has higher utility than curve 2. Therefore, risk-averse investors with indifference curves 1 and 2 will prefer curve 1 to curve 2. The utility of risk-averse investors always increases as you move northwest—higher return with lower risk. Because all investors prefer more utility to less, investors want to move northwest to the indifference curve with the highest utility.

The indifference curve for risk-averse investors runs from the southwest to the northeast because of the risk–return trade-off. If risk increases (going east) then it must be compensated by higher return (going north) to generate the same utility. The indifference curves are convex because of diminishing marginal utility of return (or wealth). As risk increases, an investor needs greater return to compensate for higher risk at an increasing rate (i.e., the curve gets steeper). The upward-sloping convex indifference curve has a slope coefficient closely related to the risk-aversion coefficient. The greater the slope, the higher is the risk aversion of the investor as a greater increment in return is required to accept a given increase in risk.

Indifference curves for investors with different levels of risk aversion are plotted in Exhibit 5-12. The most risk-averse investor has an indifference curve with the greatest slope. As volatility increases, this investor demands increasingly higher returns to compensate for risk. The least risk-averse investor has an indifference curve with the least slope and so the demand for higher return as risk increases is not as acute as for the more risk-averse investor. The risk-loving investor’s indifference curve, however, exhibits a negative slope, implying that the risk-lover is happy to substitute risk for return. For a risk lover, the utility increases both with higher risk and higher return. Finally, the indifference curves of risk-neutral investors are horizontal because the utility is invariant with risk.

In the remaining parts of this chapter, all investors are assumed to be risk averse unless stated otherwise.

3.3. Application of Utility Theory to Portfolio Selection

The simplest application of utility theory and risk aversion is to a portfolio of two assets, a risk-free asset and a risky asset. The risk-free asset has zero risk and a return of Rf. The risky asset has a risk of σi (>0) and an expected return of E(Ri). Because the risky asset has risk that is greater than that of the risk-free asset, the expected return from the risky asset will be greater than the return from the risk-free asset, that is, E(Ri)>Rf.

We can construct a portfolio of these two assets with a portfolio expected return, E(Rp), and portfolio risk, σp, based on sections 2.1.7 and 2.3.3. In the following equations, w1 is the weight in the risk-free asset and (1 – w1) is the weight in the risky asset. Because σf=0 for the risk-free asset, the first and third terms in the formula for variance are zero leaving only the second term. We arrive at the last equation by taking the square root of both sides, which shows the expression for standard deviation for a portfolio of two assets when one asset is the risk-free asset:

The two-asset portfolio is drawn in Exhibit 5-13 by varying w1 from 0 percent to 100 percent. The portfolio standard deviation is on the horizontal axis and the portfolio return is on the vertical axis. If only these two assets are available in the economy and the risky asset represents the market, the line in Exhibit 5-13 is called the capital allocation line. The capital allocation line represents the portfolios available to an investor. The equation for this line can be derived from the previous two equations by rewriting the second equation as:

Substituting the value of w1 in the equation for expected return, we get the following for the capital allocation line:

This equation can be rewritten in a more usable form:

The capital allocation line has an intercept of Rf, and a slope of

which is the additional required return for every increment in risk, and is sometimes referred to as the market price of risk.

Because the equation is linear, the plot of the capital allocation line is a straight line. The line begins with the risk-free asset as the leftmost point with zero risk and a risk-free return, Rf. At that point, the portfolio consists of only the risk-free asset. If 100 percent is invested in the portfolio of all risky assets, however, we have a return of E(Ri) with a risk of σi.

We can move further along the line in pursuit of higher returns by borrowing at the risk-free rate and investing the borrowed money in the portfolio of all risky assets. If 50 percent is borrowed at the risk-free rate, then w1 = −0.50 and 150 percent is placed in the risky asset, giving a return = 1.50E(Ri) – 0.50Rf, which is > E(Ri) because E(Ri) > Rf.

The line plotted in Exhibit 5-13 is composed of an unlimited number of risk–return pairs or portfolios. Which one of these portfolios should be chosen by an investor? The answer lies in combining indifference curves from utility theory with the capital allocation line from portfolio theory. Utility theory gives us the utility function or the indifference curves for an individual, as in Exhibit 5-11, and the capital allocation line gives us the set of feasible investments. Overlaying each individual’s indifference curves on the capital allocation line will provide us with the optimal portfolio for that investor. Exhibit 5-14 illustrates this process of portfolio selection.

The capital allocation line consists of the set of feasible portfolios. Points under the capital allocation line may be attainable but are not preferred by any investor because the investor can get a higher return for the same risk by moving up to the capital allocation line. Points above the capital allocation line are desirable but not achievable with available assets.

Three indifference curves for the same individual are also shown in Exhibit 5-14. Curve 1 is above the capital allocation line, curve 2 is tangential to the line, and curve 3 intersects the line at two points. Curve 1 has the highest utility and curve 3 has the lowest utility. Because curve 1 lies completely above the capital allocation line, points on curve 1 are not achievable with the available assets on the capital allocation line. Curve 3 intersects the capital allocation line at two points, a and b. The investor is able to invest at either point a or b to derive the risk–return trade-off and utility associated with curve 3. Comparing points with the same risk, observe that point n on curve 3 has the same risk as point m on curve 2, yet point m has the higher expected return. Therefore, all investors will choose curve 2 instead of curve 3. Curve 2 is tangential to the capital allocation line at point m. Point m is on the capital allocation line and investable. Point m and the utility associated with curve 2 is the best that the investor can do because he/she cannot move to a higher utility indifference curve. Thus, we have been able to select the optimal portfolio for the investor with indifference curves 1, 2, and 3. Point m, the optimal portfolio for one investor, may not be optimal for another investor. We can follow the same process, however, for finding the optimal portfolio for other investors: the optimal portfolio is the point of tangency between the capital allocation line and the indifference curve for that investor. In other words, the optimal portfolio maximizes the return per unit of risk (as it is on the capital allocation line) and it simultaneously supplies the investor with the most satisfaction (utility).

As an illustration, Exhibit 5-15 shows two indifference curves for two different investors: Kelly with a risk aversion coefficient of 2 and Jane with a risk aversion coefficient of 4. The indifference curve for Kelly is to the right of the indifference curve for Jane because Kelly is less risk averse than Jane and can accept a higher amount of risk, that is, has a higher tolerance for risk. Accordingly, their optimal portfolios are different: point k is the optimal portfolio for Kelly and point j is the optimal portfolio for Jane. In addition, for the same return, the slope of Jane’s curve is higher than Kelly’s suggesting that Jane needs greater incremental return as compensation for accepting an additional amount of risk compared with Kelly.

4. PORTFOLIO RISK

We have seen before that investors are risk averse and demand a higher return for a riskier investment. Therefore, ways of controlling portfolio risk without affecting return are valuable. As a precursor to managing risk, this section explains and analyzes the components of portfolio risk. In particular, it examines and describes how a portfolio consisting of assets with low correlations has the potential of reducing risk without necessarily reducing return.

4.1. Portfolio of Two Risky Assets

The return and risk of a portfolio of two assets was introduced in Section 2 of this chapter. In this section, we briefly review the computation of return and extend the concept of portfolio risk and its components.

4.1.1. Portfolio Return

When two individual assets are combined in a portfolio, we can compute the portfolio return as a weighted average of the returns of the two assets. Consider assets 1 and 2 with weights of 25 percent and 75 percent in a portfolio. If their returns are 20 percent and 5 percent, the weighted average return = (0.25×20%) + (0.75×5%) = 8.75%. More generally, the portfolio return can be written as in the following, where Rp is return of the portfolio and R1, R2 are returns on the two assets:

4.1.2. Portfolio Risk

Portfolio risk or variance measures the amount of uncertainty in portfolio returns. Portfolio variance can be calculated by taking the variance of both sides of the return equation as follows, where Cov(R1, R2) is the covariance of returns, R1 and R2, w1 is the weight in asset 1, w2 ( = 1 – w1) is the weight in asset 2, and σ12, σ22 are the variances of the two assets:

The standard deviation, or risk, of a portfolio of two assets is given by the square root of the portfolio’s variance:

4.1.3. Covariance and Correlation

The covariance in the formula for portfolio standard deviation can be expanded as (R1, R2) = ρ12σ1σ2 where ρ12 is the correlation between returns, R1, R2. Although covariance is important, it is difficult to interpret because it is unbounded on both sides. It is easier to understand the correlation coefficient (ρ12), which is bounded but provides similar information.

Correlation is a measure of the consistency or tendency for two investments to act in a similar way. The correlation coefficient, ρ12, can be positive or negative and ranges from −1 to + 1. Consider three different values of the correlation coefficient:

• ρ12 = + 1: Returns of the two assets are perfectly positively correlated. Assets 1 and 2 move together 100 percent of the time.
• ρ12 = −1: Returns of the two assets are perfectly negatively correlated. Assets 1 and 2 move in opposite directions 100 percent of the time.
• ρ12 = 0: Returns of the two assets are uncorrelated. Movement of asset 1 provides no prediction regarding the movement of asset 2.

The correlation coefficient between two assets determines the effect on portfolio risk when the two assets are combined. To see how this works, consider two different values of ρ12. You will find that portfolio risk is unaffected when the two assets are perfectly correlated (ρ12 = + 1). In other words, the portfolio’s standard deviation is simply a weighted average of the standard deviations of the two assets and as such a portfolio’s risk is unchanged with the addition of assets with the same risk parameters. Portfolio risk falls, however, when the two assets are not perfectly correlated (ρ12< + 1). Sufficiently low values of the correlation coefficient can make the portfolio riskless under certain conditions.

First, let ρ12 = + 1

The first set of terms on the right side of the first equation contain the usual terms for portfolio variance. Because the correlation coefficient is equal to + 1, the right side can be rewritten as a perfect square. The final row shows that portfolio risk is a weighted average of the risks of the individual assets’ risks. In subsection 4.1.1., the portfolio return was shown always to be a weighted average of returns. Because both risk and return are just weighted averages of the two assets in the portfolio, there is no reduction in risk when ρ12 = + 1.

Now let ρ12 < + 1

The preceding analysis showed that portfolio risk is a weighted average of asset risks when ρ12 = + 1. When ρ12 < + 1, the portfolio risk is less than the weighted average of the individual assets’ risks.

To show this, we begin by reproducing the general formula for portfolio risk, which is expressed by the terms to the left of the “<” sign in the following. The term to the right of “<” shows the portfolio risk when ρ12 = + 1:

The left side is smaller than the right side because the correlation coefficient on the left side for the new portfolio is <1. Thus, the portfolio risk is less than the weighted average of risks while the portfolio return is still a weighted average of returns.

As you can see, we have achieved diversification by combining two assets that are not perfectly correlated. For an extreme case in which ρ12 = −1 (that is, the two asset returns move in opposite directions), the portfolio can be made risk free.

4.1.4. Relationship between Portfolio Risk and Return

The previous example illustrated the effect of correlation on portfolio risk while keeping the weights in the two assets equal and unchanged. In this section, we consider how portfolio risk and return vary with different portfolio weights and different correlations. Formulas for computation are in Subsections 4.1.1 and 4.1.2.

Asset 1 has an annual return of 7 percent and annualized risk of 12 percent, whereas asset 2 has an annual return of 15 percent and annualized risk of 25 percent. The relationship is tabulated in Exhibit 5-16 for the two assets and graphically represented in Exhibit 5-17.

EXHIBIT 5-16 Relationship between Risk and Return

The table shows the portfolio return and risk for four correlation coefficients ranging from + 1.0 to −1.0 and 11 weights ranging from 0 percent to 100 percent. The portfolio return and risk are 15 percent and 25 percent respectively when 0 percent is invested in asset 1, versus 7 percent and 12 percent when 100 percent is invested in asset 1. The portfolio return varies with weights but is unaffected by the correlation coefficient.

Portfolio risk becomes smaller with each successive decrease in the correlation coefficient, with the smallest risk when ρ12  =  –1. The graph in Exhibit 5-17 shows that the risk–return relationship is a straight line when ρ12  =   + 1. As the correlation falls, the risk becomes smaller and smaller as in the table. The curvilinear nature of a portfolio of assets is recognizable in all investment opportunity sets (except at the extremes where ρ12 = −1 or + 1).

4.2. Portfolio of Many Risky Assets

In the previous section, we discussed how the correlation between two assets can affect the risk of a portfolio and the smaller the correlation the lower is the risk. The previous analysis can be extended to a portfolio with many risky assets (N). Recall the following equations from Sections 2.1.7 and 2.3.3 for portfolio return and variance:

To examine how a portfolio with many risky assets works and the ways in which we can reduce the risk of a portfolio, assume that the portfolio has equal weights (1/N) for all N assets. In addition, assume that and are the average variance and average covariance. Given equal weights and average variance/covariance, we can rewrite the portfolio variance as the following. Intermediate steps are omitted to focus on the main result:

The second equation shows that as N becomes large, the first term on the right side with the denominator of N becomes smaller and smaller, implying that the contribution of one asset’s variance to portfolio variance gradually becomes negligible. The second term, however, approaches the average covariance as N increases. It is reasonable to say that for portfolios with a large number of assets, covariance among the assets accounts for almost all of the portfolio’s risk.

4.2.1. Importance of Correlation in a Portfolio of Many Assets

The analysis becomes more instructive and interesting if we assume that all assets in the portfolio have the same variance and the same correlation among assets. In that case, the portfolio risk can then be rewritten as:

The first term under the root sign becomes negligible as the number of assets in the portfolio increases leaving the second term (correlation) as the main determining factor for portfolio risk. If the assets are unrelated to one another, the portfolio can have close to zero risk. In the next section, we review these concepts to learn how portfolios can be diversified.

4.3. The Power of Diversification

Diversification is one of the most important and powerful concepts in investments. Because investors are risk averse, they are interested in reducing risk preferably without reducing return. In other cases, investors may accept a lower return if it will reduce the chance of catastrophic losses. In previous sections of this chapter, you learned the importance of correlation and covariance in managing risk. This section applies those concepts to explore ways for risk diversification. We begin with a simple but intuitive example.

4.3.1. Correlation and Risk Diversification

Correlation is the key in diversification of risk. Notice that the returns from Beachwear and DVDrental always go in the opposite direction. If one of them does well, the other does not. Therefore, adding assets that do not behave like other assets in your portfolio is good and can reduce risk. The two companies in the previous example have a correlation of −1.0.

Even when we expand the portfolio to many assets, correlation among assets remains the primary determinant of portfolio risk. Lower correlations are associated with lower risk. Unfortunately, most assets have high positive correlations. The challenge in diversifying risk is to find assets that have a correlation that is much lower than + 1.0.

4.3.2. Historical Risk and Correlation

When we discussed asset returns in Section 2.4.1, we were careful to distinguish between historical or past returns and expected or future returns because historical returns may not be a good indicator of future returns. Returns may be highly positive in one period and highly negative in another period depending on the risk of that asset. Exhibit 5-5 showed that returns for large U.S. company stocks were high in the 1990s but have been very low in the 2000s.

Risk for an asset class, however, does not usually change dramatically from one period to the next. Stocks have been risky even in periods of low returns. T-bills are always less risky even when they earn high returns. From Exhibit 5-5, we can see that risk has typically not varied much from one decade to the next, except that risk for bonds has been much higher in recent decades when compared with earlier decades. Therefore, it is not unreasonable to assume that historical risk can work as a good proxy for future risk.

As with risk, correlations are quite stable among assets of the same country. Intercountry correlations, however, have been on the rise in the last few decades as a result of globalization and the liberalization of many economies. A correlation above 0.90 is considered high because the assets do not provide much opportunity for diversification of risk, such as the correlations that exist among large U.S. company stocks on the NYSE, NASDAQ, S&P 500 Index, and Dow Jones Industrial Average. Correlations below 0.30 are considered attractive for portfolio diversification.

4.3.3. Historical Correlation among Asset Classes

Correlations among major U.S. asset classes and international stocks are reported in Exhibit 5-20 for 1970–2008. The highest correlation is between U.S. large company stocks and U.S. small company stocks at about 70 percent, whereas the correlation between U.S. large company stocks and international stocks is approximately 66 percent. Although these are the highest correlations, they still provide diversification benefits because the correlations are less than 100 percent. The correlation between international stocks and U.S. small company stocks is lower, at 49 percent. The lowest correlations are between stocks and bonds, with some correlations being negative, such as that between U.S. small company stocks and U.S. long-term government bonds. Similarly, the correlation between T-bills and stocks is close to zero and is marginally negative for international stocks.7

EXHIBIT 5-20 Correlation among U.S. Assets and International Stocks (1970–2008)

Source: 2009 Ibbotson SBBI Classic Yearbook (Table 13-5).

The low correlations between stocks and bonds are attractive for portfolio diversification. Similarly, including international securities in a portfolio can also control portfolio risk. It is not surprising that most diversified portfolios of investors contain domestic stocks, domestic bonds, foreign stocks, foreign bonds, real estate, cash, and other asset classes.

4.3.4. Avenues for Diversification

The reason for diversification is simple. By constructing a portfolio with assets that do not move together, you create a portfolio that reduces the ups and downs in the short-term but continues to grow steadily in the long term. Diversification thus makes a portfolio more resilient to gyrations in financial markets.

We describe a number of approaches for diversification, some of which have been discussed previously and some of which might seem too obvious. Diversification, however, is such an important part of investing that it cannot be emphasized enough, especially when we continue to meet and see many investors who are not properly diversified.

• Diversify with asset classes. Correlations among major asset classes8 are not usually high, as can be observed from the few U.S. asset classes listed in Exhibit 5-20. Correlations for other asset classes and other countries are also typically low, which provides investors the opportunity to benefit from diversifying among many asset classes to achieve the biggest benefit from diversification. A partial list of asset classes includes domestic large caps, domestic small caps, growth stocks, value stocks, domestic corporate bonds, long-term domestic government bonds, domestic Treasury bills (cash), emerging market stocks, emerging market bonds, developed market stocks (i.e., developed markets excluding domestic market), developed market bonds, real estate, and gold and other commodities. In addition, industries and sectors are used to diversify portfolios. For example, energy stocks may not be well correlated with health care stocks. The exact proportions in which these assets should be included in a portfolio depend on the risk, return, and correlation characteristics of each and the home country of the investor.
• Diversify with index funds. Diversifying among asset classes can become costly for small portfolios because of the number of securities required. For example, creating exposure to a single category, such as a domestic large company asset class, may require a group of at least 30 stocks. Exposure to 10 asset classes may require 300 securities, which can be expensive to trade and track. Instead, it may be effective to use exchange-traded funds or mutual funds that track the respective indices, which could bring down the costs associated with building a well-diversified portfolio. Therefore, many investors should seriously consider index mutual funds as an investment vehicle as opposed to individual securities.
• Diversification among countries. Countries are different because of industry focus, economic policy, and political climate. The U.S. economy produces many financial and technical services and invests a significant amount in innovative research. The Chinese and Indian economies, however, are focused on manufacturing. Countries in the European Union are vibrant democracies whereas East Asian countries are experimenting with democracy. Thus, financial returns in one country over time are not likely to be highly correlated with returns in another country. Country returns may also be different because of different currencies. In other words, the return on a foreign investment may be different when translated to the home country’s currency. Because currency returns are uncorrelated with stock returns, they may help reduce the risk of investing in a foreign country even when that country, in isolation, is a very risky emerging market from an equity investment point of view. Investment in foreign countries is an essential part of a well-diversified portfolio.
• Diversify by not owning your employer’s stock. Companies encourage their employees to invest in company stock through employee stock plans and retirement plans. You should evaluate investing in your company, however, just as you would evaluate any other investment. In addition, you should consider the nonfinancial investments that you have made, especially the human capital you have invested in your company. Because you work for your employer, you are already heavily invested in it because your earnings depend on your employer. The level of your earnings, whether your compensation improves or whether you get a promotion, depends on how well your employer performs. If a competitor drives your employer out of the market, you will be out of a job. Additional investments in your employer will concentrate your wealth in one asset even more so and make you less diversified.
• Evaluate each asset before adding to a portfolio. Every time you add a security or an asset class to the portfolio, recognize that there is a cost associated with diversification. There is the cost of trading an asset as well as the cost of tracking a larger portfolio. In some cases, the securities or assets may have different names but belong to an asset class in which you already have sufficient exposure. A general rule to evaluate whether a new asset should be included to an existing portfolio is based on the following risk–return trade-off relationship:

where E(R) is the return from the asset, Rf is the return on the risk-free asset, σ is the standard deviation, ρ is the correlation coefficient, and the subscripts new and p refer to the new stock and existing portfolio. If the new asset’s risk-adjusted return benefits the portfolio, then the asset should be included. The condition can be rewritten using the Sharpe ratio on both sides of the equation as:

If the Sharpe ratio of the new asset is greater than the Sharpe ratio of the current portfolio times the correlation coefficient, it is beneficial to add the new asset.

• Buy insurance for risky portfolios. It may come as a surprise, but insurance is an investment asset—just a different kind of asset. Insurance has a negative correlation with your assets and is thus very valuable. Insurance gives you a positive return when your assets lose value, but pays nothing if your assets maintain their value. Over time, insurance generates a negative average return. Many individuals, however, are willing to accept a small negative return because insurance reduces their exposure to an extreme loss. In general, it is reasonable to add an investment with a negative return if that investment significantly reduces risk (an example of a classic case of the risk–return trade-off).

Alternatively, investments with negative correlations also exist. Historically, gold has a negative correlation with stocks; however, the expected return is usually small and sometimes even negative. Investors often include gold and other commodities in their portfolios as a way of reducing their overall portfolio risk, including currency risk and inflation risk.

Buying put options is another way of reducing risk. Because put options pay when the underlying asset falls in value (negative correlation), they can protect an investor’s portfolio against catastrophic losses. Of course, put options cost money and the expected return is zero or marginally negative.

5. EFFICIENT FRONTIER AND INVESTOR’S OPTIMAL PORTFOLIO

In this section, we formalize the effect of diversification and expand the set of investments to include all available risky assets in a mean–variance framework. The addition of a risk-free asset generates an optimal risky portfolio and the capital allocation line. We can then derive an investor’s optimal portfolio by overlaying the capital allocation line with the indifference curves of investors.

5.1. Investment Opportunity Set

If two assets are perfectly correlated, the risk–return opportunity set is represented by a straight line connecting those two assets. The line contains portfolios formed by changing the weight of each asset invested in the portfolio. This correlation was depicted by the straight line (with ρ = 1) in Exhibit 5-17. If the two assets are not perfectly correlated, the portfolio’s risk is less than the weighted average risk of the components and the portfolio formed from the two assets bulges on the left as shown by curves with the correlation coefficient (ρ) less than 1.0 in Exhibit 5-17. All of the points connecting the two assets are achievable (or feasible). The addition of new assets to this portfolio creates more and more portfolios that are either a linear combination of the existing portfolio and the new asset or a curvilinear combination depending on the correlation between the existing portfolio and the new asset.

As the number of available assets increases, the number of possible combinations increases rapidly. When all investable assets are considered, and there are hundreds and thousands of them, we can construct an opportunity set of investments. The opportunity set will ordinarily span all points within a frontier because it is also possible to reach every possible point within that curve by judiciously creating a portfolio from the investable assets.

We begin with individual investable assets and gradually form portfolios that can be plotted to form a curve as shown in Exhibit 5-21. All points on the curve and points to the right of the curve are attainable by a combination of one or more of the investable assets. This set of points is called the investment opportunity set. Initially, the opportunity set consists of domestic assets only and is labeled as such in Exhibit 5-21.

5.1.1. Addition of Asset Classes

Exhibit 5-21 shows the effect of adding a new asset class, such as international assets. As long as the new asset class is not perfectly correlated with the existing asset class, the investment opportunity set will expand out further to the northwest providing a superior risk–return trade-off.

The investment opportunity set with international assets dominates the opportunity set that includes only domestic assets. Adding other asset classes will have the same impact on the opportunity set. Thus, we should continue to add asset classes until they do not further improve the risk–return trade-off. The benefits of diversification can be fully captured in this way in the construction of the investment opportunity set, and eventually in the selection of the optimal portfolio.

In the discussion that follows in this section, we will assume that all investable assets available to an investor are included in the investment opportunity set and no special attention needs to be paid to new asset classes or new investment opportunities.

5.2. Minimum-Variance Portfolios

The investment opportunity set consisting of all available investable sets is shown in Exhibit 5-22. There are a large number of portfolios available for investment, but we must choose a single optimal portfolio. In this subsection, we begin the selection process by narrowing the choice to fewer portfolios.

5.2.1. Minimum-Variance Frontier

Risk-averse investors seek to minimize risk for a given return. Consider points A, B, and X in Exhibit 5-22 and assume that they are on the same horizontal line by construction. Thus, the three points have the same expected return, E(R1), as do all other points on the imaginary line connecting A, B, and X. Given a choice, an investor will choose the point with the minimum risk, which is point X. Point X, however, is unattainable because it does not lie within the investment opportunity set. Thus, the minimum risk that we can attain for E(R1) is at point A. Point B and all points to the right of point A are feasible but they have higher risk. Therefore, a risk-averse investor will choose only point A in preference to any other portfolio with the same return.

Similarly, point C is the minimum variance point for the return earned at C. Points to the right of C have higher risk. We can extend the preceding analysis to all possible returns. In all cases, we find that the minimum variance portfolio is the one that lies on the solid curve drawn in Exhibit 5-22. The entire collection of these minimum-variance portfolios is referred to as the minimum-variance frontier. The minimum-variance frontier defines the smaller set of portfolios in which investors would want to invest. Note that no risk-averse investor will choose to invest in a portfolio to the right of the minimum-variance frontier because a portfolio on the minimum-variance frontier can give the same return but at a lower risk.

5.2.2. Global Minimum-Variance Portfolio

The left-most point on the minimum-variance frontier is the portfolio with the minimum variance among all portfolios of risky assets, and is referred to as the global minimum-variance portfolio. An investor cannot hold a portfolio consisting of risky assets that has less risk than that of the global minimum-variance portfolio. Note the emphasis on “risky” assets. Later, the introduction of a risk-free asset will allow us to relax this constraint.

5.2.3. Efficient Frontier of Risky Assets

The minimum-variance frontier gives us portfolios with the minimum variance for a given return. However, investors also want to maximize return for a given risk. Observe points A and C on the minimum-variance frontier shown in Exhibit 5-22. Both of them have the same risk. Given a choice, an investor will choose portfolio A because it has a higher return. No one will choose portfolio C. The same analysis applies to all points on the minimum-variance frontier that lie below the global minimum-variance portfolio. Thus, portfolios on the curve below the global minimum-variance portfolio and to the right of the global minimum-variance portfolio are not beneficial and are inefficient portfolios for an investor.

The curve that lies above and to the right of the global minimum-variance portfolio is referred to as the Markowitz efficient frontier because it contains all portfolios of risky assets that rational, risk-averse investors will choose.

An important observation that is often ignored is the slope at various points on the efficient frontier. As we move right from the global minimum-variance portfolio (point Z) in Exhibit 5-22, there is an increase in risk with a concurrent increase in return. The increase in return with every unit increase in risk, however, keeps decreasing as we move from left to the right because the slope continues to decrease. The slope at point D is less than the slope at point A, which is less than the slope at point Z. The increase in return by moving from point Z to point A is the same as the increase in return by moving from point A to point D. It can be seen that the additional risk in moving from point A to point D is three to four times more than the additional risk in moving from point Z to point A. Thus, investors obtain decreasing increases in returns as they assume more risk.

5.3. A Risk-Free Asset and Many Risky Assets

Until now, we have only considered risky assets in which the return is risky or uncertain. Most investors, however, have access to a risk-free asset, most notably from securities issued by the government. The addition of a risk-free asset makes the investment opportunity set much richer than the investment opportunity set consisting only of risky assets.

5.3.1. Capital Allocation Line and Optimal Risky Portfolio

By definition, a risk-free asset has zero risk so it must lie on the y-axis in a mean-variance graph. A risk-free asset with a return of Rf is plotted in Exhibit 5-23. This asset can now be combined with a portfolio of risky assets. The combination of a risk-free asset with a portfolio of risky assets is a straight line, such as in Section 3.3 (see Exhibit 5-13). Unlike in Section 3.3, however, we have many risky portfolios to choose from instead of a single risky portfolio.

All portfolios on the efficient frontier are candidates for being combined with the risk-free asset. Two combinations are shown in Exhibit 5-23: one between the risk-free asset and efficient portfolio A and the other between the risk-free asset and efficient portfolio P. Comparing capital allocation line A and capital allocation line P reveals that there is a point on CAL(P) with a higher return and same risk for each point on CAL(A). In other words, the portfolios on CAL(P) dominate the portfolios on CAL(A). Therefore, an investor will choose CAL(P) over CAL(A). We would like to move further northwest to achieve even better portfolios. None of those portfolios, however, is attainable because they are above the efficient frontier.

What about other points on the efficient frontier? For example, point X is on the efficient frontier and has the highest return of all risky portfolios for its risk. However, point Y on CAL(P), achievable by leveraging portfolio P as seen in Section 3.3, lies above point X and has the same risk but higher return. In the same way, we can observe that not only does CAL(P) dominate CAL(A) but it also dominates the Markowitz efficient frontier of risky assets.

CAL(P) is the optimal capital allocation line and portfolio P is the optimal risky portfolio. Thus, with the addition of the risk-free asset, we are able to narrow our selection of risky portfolios to a single optimal risky portfolio, P, which is at the tangent of CAL(P) and the efficient frontier of risky assets.

5.3.2. The Two-Fund Separation Theorem

The two-fund separation theorem states that all investors, regardless of taste, risk preferences, and initial wealth, will hold a combination of two portfolios or funds: a risk-free asset and an optimal portfolio of risky assets.9

The separation theorem allows us to divide an investor’s investment problem into two distinct steps: the investment decision and the financing decision. In the first step, as in the previous analysis, the investor identifies the optimal risky portfolio. The optimal risky portfolio is selected from numerous risky portfolios without considering the investor’s preferences. The investment decision at this step is based on the optimal risky portfolio’s (a single portfolio) return, risk, and correlations.

The capital allocation line connects the optimal risky portfolio and the risk-free asset. All optimal investor portfolios must be on this line. Each investor’s optimal portfolio on the CAL(P) is determined in the second step. Considering each individual investor’s risk preference, using indifference curves, determines the investor’s allocation to the risk-free asset (lending) and to the optimal risky portfolio. Portfolios beyond the optimal risky portfolio are obtained by borrowing at the risk-free rate (i.e., buying on margin). Therefore, the individual investor’s risk preference determines the amount of financing (i.e., lending to the government instead of investing in the optimal risky portfolio or borrowing to purchase additional amounts of the optimal risky portfolio).

5.4. Optimal Investor Portfolio

The CAL(P) in Exhibits 5-23 and 5-25 contains the best possible portfolios available to investors. Each of those portfolios is a linear combination of the risk-free asset and the optimal risky portfolio. Among the available portfolios, the selection of each investor’s optimal portfolio depends on the risk preferences of an investor. In section 3, we discussed that the individual investor’s risk preferences are incorporated into their indifference curves. These can be used to select the optimal portfolio.

Exhibit 5-25 shows an indifference curve that is tangent to the capital allocation line, CAL(P). Indifference curves with higher utility than this one lie above the capital allocation line, so their portfolios are not achievable. Indifference curves that lie below this one are not preferred because they have lower utility. Thus, the optimal portfolio for the investor with this indifference curve is portfolio C on CAL(P), which is tangent to the indifference curve.

5.4.1. Investor Preferences and Optimal Portfolios

The location of an optimal investor portfolio depends on the investor’s risk preferences. A highly risk-averse investor may invest a large proportion, even 100 percent, of his/her assets in the risk-free asset. The optimal portfolio in this investor’s case will be located close to the y-axis. A less risk-averse investor, however, may invest a large portion of his/her wealth in the optimal risky asset. The optimal portfolio in this investor’s case will lie closer to point P in Exhibit 5-25.

Some less risk-averse investors (i.e., with a high risk tolerance) may wish to accept even more risk because of the chance of higher return. Such an investor may borrow money to invest more in the risky portfolio. If the investor borrows 25 percent of his wealth, he/she can invest 125 percent in the optimal risky portfolio. The optimal investor portfolio for such an investor will lie to the right of point P on the capital allocation line.

Thus, moving from the risk-free asset along the capital allocation line, we encounter investors who are willing to accept more risk. At point P, the investor is 100 percent invested in the optimal risky portfolio. Beyond point P, the investor accepts even more risk by borrowing money and investing in the optimal risky portfolio.

Note that we are able to accommodate all types of investors with just two portfolios: the risk-free asset and the optimal risky portfolio. Exhibit 5-25 is also an illustration of the two-fund separation theorem. Portfolio P is the optimal risky portfolio that is selected without regard to investor preferences. The optimal investor portfolio is selected on the capital allocation line by overlaying the indifference curves that incorporate investor preferences.

6. SUMMARY

This chapter provides a description and computation of investment characteristics, such as risk and return, that investors use in evaluating assets for investment. This was followed by sections about portfolio construction, selection of an optimal risky portfolio, and an understanding of risk aversion and indifference curves. Finally, the tangency point of the indifference curves with the capital allocation line allows identification of the optimal investor portfolio. Key concepts covered in the chapter include the following:

• Holding period return is most appropriate for a single, predefined holding period.
• Multiperiod returns can be aggregated in many ways. Each return computation has special applications for evaluating investments.
• Risk-averse investors make investment decisions based on the risk–return trade-off, maximizing return for the same risk, and minimizing risk for the same return. They may be concerned, however, by deviations from a normal return distribution and from assumptions of financial markets’ operational efficiency.
• Investors are risk averse, and historical data confirm that financial markets price assets for risk-averse investors.
• The risk of a two-asset portfolio is dependent on the proportions of each asset, their standard deviations, and the correlation (or covariance) between the asset’s returns. As the number of assets in a portfolio increases, the correlation among asset risks becomes a more important determinant of portfolio risk.
• Combining assets with low correlations reduces portfolio risk.
• The two-fund separation theorem allows us to separate decision making into two steps. In the first step, the optimal risky portfolio and the capital allocation line are identified, which are the same for all investors. In the second step, investor risk preferences enable us to find a unique optimal investor portfolio for each investor.
• The addition of a risk-free asset creates portfolios that are dominant to portfolios of risky assets in all cases except for the optimal risky portfolio.

By successfully understanding the content of this chapter, you should be comfortable calculating an investor’s optimal portfolio given the investor’s risk preferences and universe of investable assets available.

PROBLEMS11

1. An investor purchased 100 shares of a stock for $34.50 per share at the beginning of the quarter. If the investor sold all of the shares for $30.50 per share after receiving a $51.55 dividend payment at the end of the quarter, the holding period return is closest to:

A. −13.0%.

B. −11.6%.

C. −10.1%.

2. An analyst obtains the following annual rates of return for a mutual fund:

Year	Return
2008	14%
2009	−10%
2010	−2%

The fund’s holding period return over the three-year period is closest to:

A. 0.18%.

B. 0.55%.

C. 0.67%.

3. An analyst observes the following annual rates of return for a hedge fund:

Year	Return
2008	22%
2009	−25%
2010	11%

The hedge fund’s annual geometric mean return is closest to:

A. 0.52%.

B. 1.02%.

C. 2.67%.

4. Which of the following return calculating methods is best for evaluating the annualized returns of a buy-and-hold strategy of an investor who has made annual deposits to an account for each of the last five years?

A. Geometric mean return.

B. Arithmetic mean return.

C. Money-weighted return.

5. An investor evaluating the returns of three recently formed exchange-traded funds gathers the following information:

ETF	Time since Inception	Return since Inception
1	146 days	4.61%
2	5 weeks	1.10%
3	15 months	14.35%

The ETF with the highest annualized rate of return is:

A. ETF 1.

B. ETF 2.

C. ETF 3.

6. With respect to capital market theory, which of the following asset characteristics is least likely to impact the variance of an investor’s equally weighted portfolio?

A. Return on the asset.

B. Standard deviation of the asset.

C. Covariances of the asset with the other assets in the portfolio.

7. A portfolio manager creates the following portfolio:

Security	Security Weight	Expected Standard Deviation
1	30%	20%
2	70%	12%

If the correlation of returns between the two securities is 0.40, the expected standard deviation of the portfolio is closest to:

A. 10.7%.

B. 11.3%.

C. 12.1%.

8. A portfolio manager creates the following portfolio:

Security	Security Weight	Expected Standard Deviation
1	30%	20%
2	70%	12%

If the covariance of returns between the two securities is −0.0240, the expected standard deviation of the portfolio is closest to:

A. 2.4%.

B. 7.5%.

C. 9.2%.

Use the following data to answer Questions 9 and 10.

A portfolio manager creates the following portfolio:

Security	Security Weight	Expected Standard Deviation
1	30%	20%
2	70%	12%

9. If the standard deviation of the portfolio is 14.40%, the correlation between the two securities is equal to:

A. −1.0.

B. 0.0.

C. 1.0.

10. If the standard deviation of the portfolio is 14.40%, the covariance between the two securities is equal to:

A. 0.0006.

B. 0.0240.

C. 1.0000.

Use the following data to answer Questions 11 through 14.

An analyst observes the following historic geometric returns:

Asset Class	Geometric Return
Equities	8.0%
Corporate Bonds	6.5%
Treasury Bills	2.5%
Inflation	2.1%

11. The real rate of return for equities is closest to:

A. 5.4%.

B. 5.8%.

C. 5.9%.

12. The real rate of return for corporate bonds is closest to:

A. 4.3%.

B. 4.4%.

C. 4.5%.

13. The risk premium for equities is closest to:

A. 5.4%.

B. 5.5%.

C. 5.6%.

14. The risk premium for corporate bonds is closest to:

A. 3.5%.

B. 3.9%.

C. 4.0%.

15. With respect to trading costs, liquidity is least likely to impact the:

A. Stock price.

B. Bid-ask spreads.

C. Brokerage commissions.

16. Evidence of risk aversion is best illustrated by a risk-return relationship that is:

A. Negative.

B. Neutral.

C. Positive.

17. With respect to risk-averse investors, a risk-free asset will generate a numerical utility that is:

A. The same for all individuals.

B. Positive for risk-averse investors.

C. Equal to zero for risk-seeking investors.

18. With respect to utility theory, the most risk-averse investor will have an indifference curve with the:

A. Most convexity.

B. Smallest intercept value.

C. Greatest slope coefficient.

19. With respect to an investor’s utility function expressed as , which of the following values for the measure for risk aversion has the least amount of risk-aversion?

A. −4

B. 0

C. 4

Use the following data to answer Questions 20 through 23.

A financial planner has created the following data to illustrate the application of utility theory to portfolio selection:

Investment	Expected Return	Expected Standard Deviation
1	18%	2%
2	19%	8%
3	20%	15%
4	18%	30%

20. A risk-neutral investor is most likely to choose:

A. Investment 1.

B. Investment 2.

C. Investment 3.

21. If an investor’s utility function is expressed as and the measure for risk aversion has a value of −2, the risk-seeking investor is most likely to choose:

A. Investment 2.

B. Investment 3.

C. Investment 4.

22. If an investor’s utility function is expressed as and the measure for risk aversion has a value of 2, the risk-averse investor is most likely to choose:

A. Investment 1.

B. Investment 2.

C. Investment 3.

23. If an investor’s utility function is expressed as and the measure for risk aversion has a value of 4, the risk-averse investor is most likely to choose:

A. Investment 1.

B. Investment 2.

C. Investment 3.

24. With respect to the mean-variance portfolio theory, the capital allocation line, CAL, is the combination of the risk-free asset and a portfolio of all:

A. Risky assets.

B. Equity securities.

C. Feasible investments.

25. Two individual investors with different levels of risk aversion will have optimal portfolios that are:

A. Below the capital allocation line.

B. On the capital allocation line.

C. Above the capital allocation line.

Use the following data to answer Questions 26 through 28.

A portfolio manager creates the following portfolio:

Security	Expected Annual Return	Expected Standard Deviation
1	16%	20%
2	12%	20%

26. If the portfolio of the two securities has an expected return of 15%, the proportion invested in security 1 is:

A. 25%.

B. 50%.

C. 75%.

27. If the correlation of returns between the two securities is −0.15, the expected standard deviation of an equal-weighted portfolio is closest to:

A. 13.04%.

B. 13.60%.

C. 13.87%.

28. If the two securities are uncorrelated, the expected standard deviation of an equal-weighted portfolio is closest to:

A. 14.00%.

B. 14.14%.

C. 20.00%.

29. As the number of assets in an equally weighted portfolio increases, the contribution of each individual asset’s variance to the volatility of the portfolio:

A. Increases.

B. Decreases.

C. Remains the same.

30. With respect to an equally weighted portfolio made up of a large number of assets, which of the following contributes the most to the volatility of the portfolio?

A. Average variance of the individual assets.

B. Standard deviation of the individual assets.

C. Average covariance between all pairs of assets.

31. The correlation between assets in a two-asset portfolio increases during a market decline. If there is no change in the proportion of each asset held in the portfolio or the expected standard deviation of the individual assets, the volatility of the portfolio is most likely to:

A. Increase.

B. Decrease.

C. Remain the same.

Use the following data to answer Questions 32 through 34.

An analyst has made the following return projections for each of three possible outcomes with an equal likelihood of occurrence:

32. Which pair of assets is perfectly negatively correlated?

A. Asset 1 and Asset 2.

B. Asset 1 and Asset 3.

C. Asset 2 and Asset 3.

33. If the analyst constructs two-asset portfolios that are equally weighted, which pair of assets has the lowest expected standard deviation?

A. Asset 1 and Asset 2.

B. Asset 1 and Asset 3.

C. Asset 2 and Asset 3.

34. If the analyst constructs two-asset portfolios that are equally weighted, which pair of assets provides the least amount of risk reduction?

A. Asset 1 and Asset 2.

B. Asset 1 and Asset 3.

C. Asset 2 and Asset 3.

35. Which of the following statements is least accurate? The efficient frontier is the set of all attainable risky assets with the:

A. Highest expected return for a given level of risk.

B. Lowest amount of risk for a given level of return.

C. Highest expected return relative to the risk-free rate.

36. The portfolio on the minimum-variance frontier with the lowest standard deviation is:

A. Unattainable.

B. The optimal risky portfolio.

C. The global minimum-variance portfolio.

37. The set of portfolios on the minimum-variance frontier that dominates all sets of portfolios below the global minimum-variance portfolio is the:

A. Capital allocation line.

B. Markowitz efficient frontier.

C. Set of optimal risky portfolios.

38. The dominant capital allocation line is the combination of the risk-free asset and the:

A. Optimal risky portfolio.

B. Levered portfolio of risky assets.

C. Global minimum-variance portfolio.

39. Compared to the efficient frontier of risky assets, the dominant capital allocation line has higher rates of return for levels of risk greater than the optimal risky portfolio because of the investor’s ability to:

A. Lend at the risk-free rate.

B. Borrow at the risk-free rate.

C. Purchase the risk-free asset.

40. With respect to the mean-variance theory, the optimal portfolio is determined by each individual investor’s:

A. Risk-free rate.

B. Borrowing rate.

C. Risk preference.

1A reverse mortgage is a type of loan that allows individuals to convert part of their home equity into cash. The loan is usually disbursed in a stream of payments made to the homeowner by the lender. As long as the homeowner lives in the home, they need not be repaid during the lifetime of the homeowner. The loan, however, can be paid off at any time by the borrower not necessarily by selling the home.

2A buy-and-hold strategy assumes that the money invested initially grows or declines with time depending on whether a particular period’s return is positive or negative. On the one hand, a geometric return compounds the returns and captures changes in values of the initial amount invested. On the other hand, arithmetic return assumes that we start with the same amount of money every period without compounding the return earned in a prior period.

3Bonds issued at a discount to the par value may be taxed based on accrued gains instead of realized gains.

4There are exceptions when an asset reduces overall risk of a portfolio. We will consider those exceptions in Section 4.3.

5Value at risk (VAR) is a money measure of the minimum losses expected on a portfolio during a specified time period at a given level of probability. It is commonly used to measure the losses a portfolio can suffer under normal market conditions. For example, if a portfolio’s one-day 10 percent VAR is £200,000, it implies that there is a 10 percent probability that the value of the portfolio will decrease by more than £200,000 over a single one-day period (under normal market conditions). This probability implies that the portfolio will experience a loss of at least £200,000 on one out of every ten days.

6For example, see Bogle (2008) and Taleb (2007).

7In any short period, T-bills are riskless and uncorrelated with other asset classes. For example, a three-month U.S. Treasury bill is redeemable at its face value upon maturity irrespective of what happens to other assets. When we consider multiple periods, however, returns on T-bills may be related to other asset classes because short-term interest rates vary depending on the strength of the economy and outlook for inflation.

8Major asset classes are distinguished from subclasses, such as U.S. value stocks and U.S. growth stocks.

9In the next chapter, you will learn that the optimal portfolio of risky assets is the market portfolio.

10You can maximize by taking the first derivative of the slope with respect to wA and setting it to 0.

11These practice questions were developed by Stephen P. Huffman, CFA (University of Wisconsin, Oshkosh).