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**Chapter 5 Portfolio Risk and Return: Part I**

Presenter Venue Date 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. 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 reading. DISCLAIMER: Candidates should understand this presentation is NOT a substitute for a thorough understanding of the CFA Program curriculum. This presentation is NOT necessarily a reflection of all of the knowledge and skills needed for candidates to successfully complete questions regarding this topic area on the CFA exam.

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**Return on Financial Assets**

Total Return Periodic Income Capital Gain or Loss LOS: Calculate and interpret major return measures and describe their applicability. Page 176 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 mortgages make income payments as long as you live.

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Holding Period Return A holding period return is the return from holding an asset for a single specified period of time. LOS: Calculate and interpret major return measures and describe their applicability. Page 177 The following two observations are important. • A capital gain of 5 percent and a dividend yield of 2 percent is computed in the 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.

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**Holding Period Returns**

What is the 3-year holding period return if the annual returns are 7%, 9%, and –5%? LOS: Calculate and interpret major return measures and describe their applicability. Pages

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**Average Returns Average returns Arithmetic or mean return**

Geometric mean return Money-weighted return LOS: Calculate and interpret major return measures and describe their applicability. Page 178 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. Most holding period returns are reported as daily, monthly, or annual returns. Different approaches: Arithmetic or mean return Geometric mean return Money-weighted return

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**Arithmetic or Mean Return**

The arithmetic or mean return is the simple average of all holding period returns. LOS: Calculate and interpret major return measures and describe their applicability. Page 178 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.

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Geometric Mean Return The geometric mean return accounts for the compounding of returns. LOS: Calculate and interpret major return measures and describe their applicability. Page 179 The geometric mean return per year is –5.0 percent, compared with an arithmetic mean return of 4.0 percent (see previous slide). 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.

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**Money-Weighted Return**

LOS: Calculate and interpret major return measures and describe their applicability. Pages The money-weighted return accounts for the money invested and provides the investor with information on the return on actual investment. The money-weighted return and its calculation are similar to the internal rate of return. 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 internal rate of return is the discount rate at which the sum of present values of these cash flows will equal zero. 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 the table. IRR can be computed using the formula and a trial and error process to identify a discount rate that produces a sum of present values of cash flows equal to zero. The computation of IRR can be more easily done using a business calculator or spreadsheet.

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**Annualized Return Weekly return of 0.20%: 18-month return of 20%:**

LOS: Calculate and interpret major return measures and describe their applicability. Pages 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.

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**Gross and Net Returns Gross returns Net returns Expenses**

LOS: Calculate and interpret major return measures and describe their applicability. Page 186 Gross return-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. Net return-a return that 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.

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**Pre-Tax and After-Tax Nominal Return**

Pre-tax nominal return Taxes After-tax nominal return LOS: Calculate and interpret major return measures and describe their applicability. Page 186 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. In general, all returns are pre-tax nominal returns unless they are otherwise designated. The after-tax nominal return is computed as the total return minus any allowance for taxes on realized gains. 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.

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**Nominal Returns and Real Returns**

LOS: Calculate and interpret major return measures and describe their applicability. Page 187 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). 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.

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**Variance and Standard Deviation of a Single Asset**

Population Sample LOS: Calculate and interpret the mean, variance, and covariance (or correlation) of asset returns based on historical data. Page 189 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 equation to the left below, 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, then the equation to the left below underestimates the variance. The correction for sample variance is made by replacing the denominator with (T – 1), as shown in the equation to the right below, where R “bar” is the mean return of the sample observations and s2 is the sample variance The standard deviation of returns of an asset is the square root of the variance of returns. Standard deviation is another measure of the risk of an asset, which may also be referred to as its volatility.

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**Variance of a Portfolio of Assets**

Variance can be determined for N securities in a portfolio using the formulas below. Cov(Ri, Rj) is the covariance of returns between security i and security j and can be expressed as the product of the correlation between the two returns (ρi,j) and the standard deviations of the two assets, Cov(Ri, Rj) = ρi,j σiσj. LOS: Calculate and interpret portfolio standard deviation. Pages The right side of the first equation is the variance of the weighted average returns of individual securities. Weight is a constant, but the returns are variables whose variance isVar(Ri). We can rewrite the equation using the covariances as shown in the second equation. Because the covariance of an asset with itself is the variance of the asset, we can separate the variances from the covariances in the third equation.

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**EXAMPLE 5-4 Return and Risk of a Two-Asset Portfolio**

Assume that as a U.S. investor, you decide to hold a portfolio with 80 percent invested in the S&P 500 U.S. stock index and the remaining 20 percent in the MSCI Emerging Markets index. The expected return is percent for the S&P 500 and percent for the Emerging Markets index. The risk (standard deviation) is percent for the S&P 500 and percent for the Emerging Markets index. What will be the portfolio’s expected return and risk given that the covariance between the S&P 500 and the Emerging Markets index is ? LOS: Calculate and interpret portfolio standard deviation. Pages

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**EXAMPLE 5-4 Return and Risk of a Two-Asset Portfolio (Continued)**

LOS: Calculate and interpret portfolio standard deviation. Pages The portfolio’s expected return is percent and the portfolio’s risk is percent. The example shows that we can take the portfolio of a U.S. investor invested only in the S&P 500, combine it with a riskier portfolio consisting of emerging markets securities, and the return of the U.S. investor increases from 9.93 percent to percent while the risk of the portfolio actually falls from percent to percent.

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**EXAMPLE 5-4 Return and Risk of a Two-Asset Portfolio (Continued)**

LOS: Calculate and interpret portfolio standard deviation. Pages The exhibit depicts how the combination of the two assets results in a superior risk–return trade-off. Not only does the investor get a higher return, but he also gets it at a lower risk.

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

LOS: Describe the characteristics of the major asset classes that investors would consider in forming portfolios according to mean–variance portfolio theory. Pages 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 based on the real risk-free interest rate, expected inflation, and an expected risk premium for the risk of the asset. 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. Three major asset categories are the focus of 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 percent in Inflation rates have fallen since then to a negligible level of 0.09 percent in Overall, the inflation rate was 3.0 percent for the 83-year period.

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

LOS: Describe the characteristics of the major asset classes that investors would consider in forming portfolios according to mean–variance portfolio theory. Pages Along with U.S. returns, returns of major asset classes for a 17-country world and the world excluding the United States are 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 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 last 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. The risk for major asset classes for the United States, the world, and the world excluding the United States are also reported for 1900–2008 in Exhibit 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. 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.

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**Important Assumptions of Mean-Variance Analysis**

Markets are informationally and operationally efficient Returns are normally distributed LOS: Describe the characteristics of the major asset classes that investors would consider in forming portfolios according to mean–variance portfolio theory. Page 197 Two important assumptions of mean-variance analysis: Returns are normally distributed and the return distribution can therefore be fully characterized by its mean and variance. Markets are informationally and operationally efficient. If these two assumptions are violated, then there is a need to consider additional investment characteristics.

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**EXHIBIT 5-9 Histogram of U.S. Large Company Stock Returns, 1926-2008**

Violations of the normality assumption: skewness and kurtosis. LOS: Describe the characteristics of the major asset classes that investors would consider in forming portfolios according to mean–variance portfolio theory. Pages 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, and 99 percent of the observations within ±3σ of the mean. 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. Kurtosis refers to “fat tails” or higher than normal probabilities for extreme returns. 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. Exhibit 5-9 demonstrates the negative skewness of stock returns by plotting a histogram of U.S. large company stock returns for 1926– 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. The higher probability of extreme negative outcomes among stock returns can also be observed in Exhibit 5-9

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**Utility Theory Expected return Variance or risk**

Measure of risk tolerance or risk aversion Utility of an investment LOS: Explain risk aversion and its implications for portfolio selection. Pages Utility is a measure of relative satisfaction that an investor derives from different portfolios. 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. Several conclusions can be drawn 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. 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.

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Indifference Curves An indifference curve plots the combination of risk-return pairs that an investor would accept to maintain a given level of utility. LOS: Discuss the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line. Pages 203 and 204 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.

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**Portfolio Expected Return and Risk Assuming a Risk-Free Asset**

Assume a portfolio of two assets, a risk-free asset and a risky asset. Expected return and risk for that portfolio can be determined using the following formulas: LOS: Discuss the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line. Page 206 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. In the 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.

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**The Capital Allocation Line (CAL)**

E(Rp) CAL E(Ri) Rf LOS: Discuss the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line. Pages From the previous slide, note that w1 = 1 – (σP ÷ σi). Substituting the right-hand side of this equation for w1 in the E(RP) formula from the previous slide produces the formula for the capital allocation line shown on this slide. The term “capital allocation line” is typically only used when the risky asset is assumed to be a market portfolio of risky assets (the “market”). 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 comprised of an unlimited number of risk–return pairs or portfolios. σp σi

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**EXHIBIT 5-15 Portfolio Selection for Two Investors with Various Levels of Risk Aversion**

LOS: Discuss the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line. Pages 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. 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, i.e. 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.

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**Correlation and Portfolio Risk**

Correlation between assets in the portfolio Portfolio risk LOS: Describe the effect on a portfolio’s risk of investing in assets that are less than perfectly correlated. Pages 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. 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. Portfolio risk falls when the two assets are not perfectly correlated, ρ12 < +1. The fact that portfolio risk falls when the two assets are not perfectly correlated is evident from examining the formulas for portfolio risk.

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**EXHIBIT 5-16 Relationship between Risk and Return**

LOS: Describe the effect on a portfolio’s risk of investing in assets that are less than perfectly correlated. Pages Exhibit 5-16 shows how portfolio risk and return vary with different portfolio weights and different correlations. 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 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 the correlation coefficient is -1.

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**EXHIBIT 5-17 Relationship between Risk and Return**

LOS: Describe the effect on a portfolio’s risk of investing in assets that are less than perfectly correlated. Pages The graph in Exhibit 5-17 makes use of the results from Exhibit It shows that the risk–return relationship is a straight line when 12 = +1. As the correlation falls, the risk becomes smaller and smaller as in the table presented in Exhibit The curvilinear nature of a portfolio of assets is recognizable in all investment opportunity sets (except at the extremes where 12 = –1 or +1).

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**Avenues for Diversification**

Diversify with asset classes Diversify with index funds Diversify among countries Evaluate assets Buy insurance LOS: Describe the effect on a portfolio’s risk of investing in assets that are less than perfectly correlated. Pages Correlations among major asset classes are not usually high, as can be observed from the few U.S. asset classes listed in Exhibit 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. Using exchange-traded funds or mutual funds that track the respective indices could bring down the costs associated with building a well-diversified portfolio. Countries are different because of industry focus, economic policy, and political climate. 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. Employees are already heavily invested in their employer via their future earnings. Additional investments in your employer will concentrate your wealth in one asset even more so and make you less diversified. Every time you add a security or an asset class to the portfolio, recognize that there is a cost associated with diversification. The securities or assets may have different names but belong to an asset class in which you already have sufficient exposure. 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.

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**EXHIBIT 5-22 Minimum-Variance Frontier**

LOS: Describe and interpret the minimum-variance and efficient frontiers of risky assets and the global minimum-variance portfolio. Pages 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 above 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 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. 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. 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 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.

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**EXHIBIT 5-23 Capital Allocation Line and Optimal Risky Portfolio**

CAL(P) is the optimal capital allocation line and portfolio P is the optimal risky portfolio. LOS: Discuss the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line. Page 225 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, 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.

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**The Two-Fund Separation Theorem**

Investment Decision Financing Decision Optimal Investor Portfolio LOS: Discuss the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line. Pages The two-fund separation theorem states that all investors will hold a combination of two portfolios or funds: a risk-free asset and an optimal portfolio of risky assets. The separation theorem allows us to divide an investor’s portfolio choice problem into two distinct steps: Investment decision: the investor identifies the optimal risky portfolio. Financing decision: the investor identifies the allocation to the risk-free asset (lending or borrowing) and to the optimal risky portfolio. 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 capital allocation line 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).

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**EXHIBIT 5-25 Optimal Investor Portfolio**

Given the investor’s indifference curve, portfolio C on CAL(P) is the optimal portfolio. LOS: Discuss the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line. Pages 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.

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**Summary Different approaches for determining return**

Risk measures for individual assets and portfolios Market evidence on the risk-return tradeoff Correlation and portfolio risk The risk-free asset and the optimal risky portfolio Utility theory and the optimal investor portfolio

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Efficient Diversification

Efficient Diversification

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