Which of the following available portfolios most likely falls below the efficient frontier?

aversion). An indifference curve is a tool from economics that, in this application, plots combinations of risk (standard deviation) and expected return among which an investor is indifferent. In constructing indifference curves for portfolios based on only their expected return and standard deviation of returns, we are assuming that these are the only portfolio characteristics that investors care about. In Figure 52.4, we show three indifference curves for an investor. The investor’s expected utility is the same for all points along a single indifference curve. Indifference curve I1 represents the most preferred portfolios in Figure 52.4; our investor will prefer any portfolio along I1 to any portfolio on either I2 or I3. Figure 52.4: Risk-Averse Investor’s Indifference Curves Indifference curves slope upward for risk-averse investors because they will only take on more risk (standard deviation of returns) if they are compensated with greater expected returns. An investor who is relatively more risk averse requires a relatively greater increase in expected return to compensate for a given increase in risk. In other words, a more risk-averse investor will have steeper indifference curves, reflecting a higher risk aversion coefficient. In our previous illustration of efficient portfolios available in the market, we included only risky assets. Now we will introduce a risk-free asset into our universe of available assets, and we will consider the risk and return characteristics of a portfolio that combines a portfolio of risky assets and the risk-free asset. Recall from Quantitative Methods that we can calculate the expected return and standard deviation of a portfolio with weight WA allocated to risky Asset A and weight WB allocated to risky Asset B using the following formulas: E(Rportfolio) = WAE(RA) + WBE(RB) σportfolio =√W2Aσ2A + W2Bσ2B + 2WAWBρABσAσB Allow Asset B to be the risk-free asset and Asset A to be the risky asset portfolio. Because a risk-free asset has zero standard deviation and zero correlation of returns with those of a risky portfolio, this results in the reduced equation: σportfolio =√W2Aσ2A = WAσA The intuition of this result is quite simple: If we put X% of our portfolio into the risky asset portfolio, the resulting portfolio will have standard deviation of returns equal to X% of the standard deviation of the risky asset portfolio. The relationship between portfolio risk and return for various portfolio allocations is linear, as illustrated in Figure 52.5. Combining a risky portfolio with a risk-free asset is the process that supports the two-fund separation theorem, which states that all investors’ optimum portfolios will be made up of some combination of an optimal portfolio of risky assets and the risk-free asset. The line representing these possible combinations of risk-free assets and the optimal risky asset portfolio is referred to as the capital allocation line. Point X on the capital allocation line in Figure 52.5 represents a portfolio that is 40% invested in the risky asset portfolio and 60% invested in the risk-free asset. Its expected return will be 0.40[E(Rrisky asset portfolio)] + 0.60(Rf), and its standard deviation will be 0.40(σrisky asset portfolio). Figure 52.5: Capital Allocation Line and Risky Asset Weights Now that we have constructed a set of the possible efficient portfolios (the capital allocation line), we can combine this with indifference curves representing an individual’s preferences for risk and return to illustrate the logic of selecting an optimal portfolio (i.e., one that maximizes the investor’s expected utility). In Figure 52.6, we can see that Investor A, with preferences represented by indifference curves I1, I2, and I3, can reach the level of expected utility on I2 by selecting portfolio X. This is the optimal portfolio for this investor, as any portfolio that lies on I2 is preferred to all portfolios that lie on I3 (and in fact to any portfolios that lie between I2 and I3). Portfolios on I1 are preferred to those on I2, but none of the portfolios that lie on I1 are available in the market. Figure 52.6: Risk-Averse Investor’s Indifference Curves The final result of our analysis here is not surprising; investors who are less risk averse will select portfolios that are more risky. Recall that the less an investor’s risk aversion, the flatter his indifference curves. As illustrated in Figure 52.7, the flatter indifference curve for Investor B (IB) results in an optimal (tangency) portfolio that lies to the right of the one that results from a steeper indifference curve, such as that for Investor A (IA). An investor who is less risk averse should optimally choose a portfolio with more invested in the risky asset portfolio and less invested in the risk-free asset. Figure 52.7: Portfolio Choices Based on Investor’s Indifference Curves MODULE QUIZ 52.3 To best evaluate your performance, enter your quiz answers online. Use the following data to answer Questions 1 and 2. A portfolio was created by investing 25% of the funds in Asset A (standard deviation = 15%) and the balance of the funds in Asset B (standard deviation = 10%). 1. If the correlation coefficient is 0.75, what is the portfolio’s standard deviation? A. 10.6%. B. 12.4%. C. 15.0%. 2. If the correlation coefficient is –0.75, what is the portfolio’s standard deviation? A. 2.8%. B. 4.2%. C. 5.3%. 3. Which of the following statements about covariance and correlation is least accurate? A. A zero covariance implies there is no linear relationship between the returns on two assets. B. If two assets have perfect negative correlation, the variance of returns for a portfolio that consists of these two assets will equal zero. C. The covariance of a 2-stock portfolio is equal to the correlation coefficient times the standard deviation of one stock’s returns times the standard deviation of the other stock’s returns. 4. Which of the following available portfolios most likely falls below the efficient frontier? Portfolio Expected return Expected standarddeviation A. A 7% 14% B. B 9% 26% C. C 12% 22% 5. The capital allocation line is a straight line from the risk-free asset through: A. the global maximum-return portfolio. B. the optimal risky portfolio. C. the global minimum-variance portfolio. KEY CONCEPTS LOS 52.a Holding period return is used to measure an investment’s return over a specific period. Arithmetic mean return is the simple average of a series of periodic returns. Geometric mean return is a compound annual rate. Gross return is total return after deducting commissions on trades and other costs necessary to generate the returns, but before deducting fees for the management and administration of the investment account. Net return is the return after management and administration fees have been deducted. Pretax nominal return is the numerical percentage return of an investment, without considering the effects of taxes and inflation. After-tax nominal return is the numerical return after the tax liability is deducted, without adjusting for inflation. Real return is the increase in an investor’s purchasing power, roughly equal to nominal return minus inflation. Leveraged return is the gain or loss on an investment as a percentage of an investor’s cash investment. LOS 52.b The money-weighted rate of return is the IRR calculated using periodic cash flows into and out of an account and is the discount rate that makes the PV of cash inflows equal to the PV of cash outflows. The time-weighted rate of return measures compound growth. It is the rate at which $1 compounds over a specified performance horizon. If funds are added to a portfolio just before a period of poor performance, the money- weighted return will be lower than the time-weighted return. If funds are added just prior to a period of high returns, the money-weighted return will be higher than the time-weighted return. The time-weighted return is the preferred measure of a manager’s ability

frontier. Risk-averse investors would only choose a portfolio that lies on the efficient frontier. LOS 43.h An indifference curve plots combinations of risk and expected return that an investor finds equally acceptable. Indifference curves generally slope upward because risk-averse investors will only take on more risk if they are compensated with greater expected returns. A more risk-averse investor will have steeper indifference curves. Flatter indifference curves (less risk aversion) result in an optimal portfolio with higher risk and higher expected return. An investor who is less risk averse will optimally choose a portfolio with more invested in the risky asset portfolio and less invested in the risk­ free asset. ©2012 Kaplan, Inc. Study Session 12 Cross-Reference to CFA Institute Assigned Reading #43 - Portfolio Risk and Return: Part I CONCEPT CHECKERS 1 . An investor buys a share of stock for $40 at time t = 0, buys another share of the same stock for $50 at t = 1 , and sells both shares for $60 each at t = 2. The stock paid a dividend of $ 1 per share at t = 1 and at t = 2 . The periodic money­ weighted rate of return on the investment is closest to: A. 22.2%. B. 23.0%. c. 23.8%. 2 . Which of the following asset classes has historically had the highest returns and standard deviation? A. Small-cap stocks. B. Large-cap stocks. C. Long-term corporate bonds. 3 . In a 5-year period, the annual returns on an investment are 5%, -3%, -4%, 2%, and 6%. The standard deviation of annual returns on this investment is closest to: A. 4.0%. B. 4.5%. c. 20.7%. 4. A measure of how the returns of two risky assets move in relation to each other is the: A. range. B. covanance. C. standard deviation. 5 . Which of the following statements about correlation is least accurate? A. Diversification reduces risk when correlation is less than + 1 . B. If the correlation coefficient is 0, a zero variance portfolio can be constructed. C. The lower the correlation coefficient, the greater the potential benefits from diversification. 6. The standard deviation of returns is 0.30 for Stock A and 0.20 for Stock B. The covariance between the returns of A and B is 0.006. The correlation of returns between A and B is: A. 0 . 10 . B. 0.20. c. 0.30. 7. Which of the following statements about risk-averse investors is most accurate? A risk-averse investor: A. seeks out the investment with minimum risk, while return is not a major consideration. B . will take additional investment risk if sufficiently compensated for this risk. C. avoids participating in global equity markets. ©20 12 Kaplan, Inc. Page 155 Study Session 12 Cross-Reference to CFA Institute Assigned Reading #43 - Portfolio Risk and Return: Part I Page 156 Use the following data to answer Questions 8 and 9. A portfolio was created by investing 25% of the funds in Asset A (standard deviation = 15%) and the balance of the funds in Asset B (standard deviation = 10%). 8. If the correlation coefficient is 0.75, what is the portfolio's standard deviation? A. 10.6%. B. 12.4%. c. 15 .0%. 9. If the correlation coefficient is -0.75, what is the portfolio's standard deviation? A. 2.8%. B. 4.2%. c. 5.3%. 10 . Which of the following statements about covariance and correlation is least accurate? A. A zero covariance implies there is no linear relationship between the returns on two assets. B. If two assets have perfect negative correlation, the variance of returns for a portfolio that consists of these two assets will equal zero. C. The covariance of a 2-stock portfolio is equal to the correlation coefficient times the standard deviation of one stock's returns times the standard deviation of the other stock's returns. 1 1 . Which of the following available portfolios most likely falls below the Markowitz efficient frontier? 12. Expected Expected Portfolio return standard deviation A. A 7% 14% B. B c. c 9% 12% 26% 22% The capital allocation line is a straight line from the risk-free asset through the: A. global maximum-return portfolio. B. optimal risky portfolio. C. global minimum-variance portfolio. ©2012 Kaplan, Inc. Study Session 12 Cross-Reference to CFA Institute Assigned Reading #43 - Portfolio Risk and Return: Part I ANSWERS - CONCEPT CHECKERS 1 . C Using the cash flow functions on your financial calculator, enter CFO = -40; CF 1 = -50 + 1 = -49; CF2 = 60 X 2 + 2 = 122; CPT IRR = 23.82%. 2. A Small-cap stocks have had the highest annual return and standard deviation of return over time. Large-cap stocks and bonds have historically had lower risk and return than small-cap stocks. 3. B Mean annual return = (5o/o - 3o/o - 4o/o + 2o/o + 6o/o) I 5 = 1 .2o/o Squared deviations from the mean: 5o/o - 1 .2o/o = 3.8o/o -3o/o -1.2% = -4.2% -4o/o -1 .2% = -5.2% 2o/o -1 .2% = 0.8o/o 6o/o -1 .2% = 4.8o/o 3.82 = 14.44 -4.22 = 17.64 -5.22 = 27.04 0.82 = 0.64 4.82 = 23.04 Sum of squared deviations = 14.44 + 17.64 + 27.04 + 0.64 + 23.04 = 82.8 Sample variance = 82.8 I (5 - 1) = 20.7 Sample standard deviation = 20.7 112 = 4.55% 4. B The covariance is defined as the co-movement of the returns of two assets or how well the returns of two risky assets move together. Range and standard deviation are measures of dispersion and measure risk, not how assets move together. 5 . B A zero-variance portfolio can only be constructed if the correlation coefficient between assets is -1 . Diversification benefits can be had when correlation is less than + 1 , and the lower the correlation, the greater the potential benefit. 6. A Correlation = 0.006 I [(0.30)(0.20)] = 0 . 10 7. B Risk-averse investors are generally willing to invest in risky investments, if the return of the investment is sufficient to reward the investor for taking on this risk. Participants in securities markets are generally assumed to be risk-averse investors. 8. A �(0.25)2 (0.15)2 + (0.75)2 (0. 1 0)2 + 2(0.25)(0.75)(0. 15)(0. 10)(0.75) = .Jo.oo1406 + o.oo5625 + o.oo4219 = .Jo.o 1 1 25 = o.1o6 = 1o.6o/o 9. c �(0.25)2 (0.15)2 + (0.75)2 (0.10)2 + 2(0.25)(0.75)(0.15)(0.10)( -0.75) = .)0.001406 + 0.005625- 0.004219 = �0.002812 = 0.053 = 5.3o/o 10. B If the correlation of returns between the two assets is -1 , the set of possible portfolio risk/return combinations becomes two straight lines (see Figure 2). A portfolio of these two assets will have a positive returns variance unless the portfolio weights are those that minimize the portfolio variance. Covariance is equal to the correlation coefficient multiplied by the product of the standard deviations of the returns of the two stocks in a 2-stock portfolio. If covariance is zero, then correlation is also zero, which implies that there is no linear relationship between the two stocks' returns. ©20 12 Kaplan, Inc. Page 157 Study Session 12 Cross-Reference to CFA Institute Assigned Reading #43 - Portfolio Risk and Return: Part I 1 1 . B Portfolio B must be the portfolio that falls below the Markowitz efficient frontier because there is a portfolio (Portfolio C) that offers a higher return and lower risk. 12. B An investor's optimal portfolio will lie somewhere on the capital allocation line, which begins at the risk-free asset and runs through the optimal risky portfolio. Page 158 ©2012 Kaplan, Inc. The fo llo wing is a review o f the Portfo lio M anagement principles designed to address the learning o utcome statements set fo rth by CFA Institute. This to pic is also co vered in: PORTFOLIO RISK AND RETURN: PART II Study Session 12 EXAM FOCUS The concepts developed here are very important to finance theory and are also used extensively in practice. You must know this material completely-not only the formulas and definitions, but the ideas that underlie their use. A model assumption that diversification is costless leads to the conclusion that