1.01 Rates and Returns - Answers

1. An analyst compares portfolios in which the amount invested changes each year. Which of the following is the most appropriate measure for the analyst to use in comparing the portfolios' returns over a multiyear period?  A. Holding period return B. Arithmetic mean return C. Geometric mean return Explanation A portfolio's value can change during a period due to contributions, withdrawals, and/or income earned. Changes in portfolio value should be included in the beginning value in each subsequent period so that those changes are compounded. The geometric mean return (G in the image above) provides a better representation of a portfolio's return since it allows for the compounding of returns and the varying amounts invested. The geometric mean return produces an average periodic return from actual periodic returns (eg, three-year average annual return from annual returns over three years). Analysts prefer the geometric mean return since it allows for better performance comparisons between investments with different amounts invested each year. (Choice A) A holding period return is an asset's return earned over a specific period of time such as a day, week, or year. However, this return will be inaccurate if the amounts invested change during the period. It is best used as a periodic return when the amount invested remains the same. (Choice B) An arithmetic mean return (ie, simple average return) assumes an equal base amount invested at the beginning of each period. Things to remember: The geometric mean return converts performance data from multiple periods into a comparable format and produces a constant periodic return. It provides a better representation of an portfolio's return since it allows for the compounding of returns and the varying base amounts invested. Calculate and interpret major return measures and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 2. At year end, a bondholder reviews performance records of a INR 1,000 par value bond. The holding period return of the security is 3.0%. If the beginning price of the bond was INR 998 and the ending price of the bond was INR 1,008, the bond's interest income for the period was closest to:  A. INR 19.94 B. INR 29.94 C. INR 39.94 Explanation A holding period return (HPR) is a single-period return, where the single period can represent any timeframe. There are two sources of investment returns to the HPR: capital changes and income. The capital changes reflect the security's market value difference from the beginning to the end of the period. Income during the period is either the result of dividends from stocks or of interest earned from bonds. For example, a stock or bond's price may move up or down during the period, which creates a capital gain or loss. In addition, most bondholders earn interest income during the period, and some stockholders earn dividend income. HPR compares these combined sources (gain or loss plus income) to the value the security had at the beginning of the period. To find the interest income of a bond, use the HPR formula and input all but the missing interest income. (Choice B) This incorrect answer of INR 29.94 is obtained by calculating interest as the HPR multiplied by the beginning price. This ignores the capital gain part of the return and assumes all return is generated from income. (Choice C) This answer of INR 39.94 incorrectly uses the ending price, rather than the beginning price, in the dominator to find the interest. This is flawed since the interest found is greater than the holding period return. Things to remember: A holding period return compares the change in investment value plus interest to the beginning value of the investment to determine performance. Calculate and interpret major return measures and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 3. A hedge fund manager invested in three investments, A, B, and C, with differing holding periods. The manager collected the following information: Investment performance A B C Holding period return 0.5% 50.0% 350.0% Holding period return 5 days 23 months 6.7 years Which of the three investments has the lowest annualized rate of return?  A. Investment A B. Investment B C. Investment C Explanation The annualized rate of return is an application of the geometric mean return. It converts a holding period return to an equivalent annualized return. To find the annualized rate of return, determine the time unit (eg, days, months, or years). Identify how many time units occur in one year (eg, days = 365, months = 12, year = 1). Express the time in years. Apply this standardized unit of time to the geometric mean return formula. When the annualized returns of all investments are compared, Investment B has the lowest return, 23.56%. The annualized returns of Investment A (Choice A) and Investment C (Choice C) both show higher annualized returns than Investment B's annualized return. Things to remember: In all performance calculations, match the time unit of the return (eg. annual, monthly or daily), r, with the corresponding time, t (ie, how many periods in the time unit). Apply this standardized unit of time to the geometric mean return formula. Calculate and interpret major return measures and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 4. An analyst records the price of an asset over a two-year period: Time (years) Price (AUD) 0 99 1 105 2 113 Based only on price, the asset's Year 1 continuously compounded return is closest to:  A. 5.88% B. 6.06% C. 6.41% Explanation An infinite number of infinitesimally small compounding periods during a holding period is known as continuous compounding. The continuously compounded return is related to the holding period return; it is the natural log of the ratio of the ending value to the beginning value of the investment. In this scenario, solve to determine the Year 1 continuously compounded return using the price at the beginning (St) and price at the end (St+1) of the one-year holding period, which are both given. Note that the continuously compounded return in this scenario is lower than the holding period return (6.06%). This makes sense due to the fact that compounding at a lower rate more frequently results in the same return as compounding at a higher rate less frequently. (Choice B) 6.06% is the Year 1 holding period return. (Choice C) 6.41% is the geometric mean of the two-year, continuously compounded return. Things to remember: An infinite number of infinitesimally small compounding periods during a holding period is known as continuous compounding. The continuously compounded return is related to the holding period return; it is the natural log of the ratio of the ending value to the beginning value of the investment. Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 5. A lender gathers data on three different loans in its portfolio: Stated Annual Loan Interest Rate ( rs ) Compounding Frequency X 10.40% weekly Y 10.45% monthly Z 10.50% quarterly Based on the data above, the loan with the highest effective annual rate is:  A. Loan X. B. Loan Y. C. Loan Z. Explanation With interest-bearing investments, lenders (investors) earn interest periodically (eg, monthly, quarterly) based on a specific periodic rate over time. When interest is compounded, interest income is reinvested each period at the periodic rate, so additional interest is earned on previous interest. Therefore, the total value of the investment (including principal and interest) grows at the periodic rate. The investment's percentage growth in annual terms, accounting for periodic compounding, is known as the effective annual rate (EAR). The EAR is derived from the periodic rate. Since the periodic rate is not given, it must be derived from the stated annual rate (rs). The rs is an annualized quotation convention (used by financial institutions) that equals the periodic rate multiplied by the number of compounding periods per year (m). To derive the periodic rate, divide the rs by m. Calculate EAR for the three loans: The EAR for Loan Y is 10.97%. An EAR example illustrates how $1 invested today in Loan Y will result in an annual return of 10.97%. Things to remember: An investment that generates compounded interest will grow in value at the periodic rate each period until maturity. The investment's percentage growth, in annual terms, is known as the effective annual rate (EAR), which accounts for the effect of periodic compounding. Greater compounding frequency or periodic rates can lead to higher EAR. Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 6. For a manager reporting historical performance of an open-end mutual fund, the most appropriate measure of return is the:  A. arithmetic mean return. B. time-weighted rate of return. C. money-weighted rate of return. Explanation Money-weighted rate of return (MWR) Time-weighted rate of return (TWR) Internal rate of return (IRR) of a portfolio; accounts for all Compound growth rate (geometric mean) of a portfolio; excludes contributions and withdrawals contributions and withdrawals Appropriate for measuring an investor's change in wealth Preferred return measure for evaluating investment performance Affected by amount and timing of cash flows Unaffected by amount and timing of cash flows The time-weighted rate of return (TWR) is used to calculate the compound growth rate for each dollar (or other currency unit) invested in the portfolio. It defines subperiods based on cash flows and compounds the return of each subperiod as the geometric mean return. As a result, the size of cash additions and withdrawals and of the portfolio do not figure into the calculated return. Open-end mutual fund managers generally do not control the amount and timing of their clients' additions and withdrawals, so TWR provides the best view of investment performance without any distorting effects from cash flows; it is the preferred way for managers to present historical performance. Therefore, the most appropriate choice in this case is the TWR. (Choice A) The arithmetic mean return is not suitable since it ignores compounding. It is generally used for forecasting or modeling purposes, not for reporting actual performance. (Choice C) The money-weighted rate of return is inappropriate in this scenario since its calculation is affected by the amount and timing of cash flows. Things to remember: The time-weighted rate of return (ie, geometric mean return) is not influenced by timing and amount of investor cash flows, making it the best choice for fund managers reporting on their performance. compare the money-weighted and time-weighted rates of return and evaluate the performance of portfolios based on these measures LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 7. The following table shows a company's sales growth over a five-year period: Year Sales Growth Rate 2008 6.8% 2009 12.2% 2010 4.3% 2011 −6.9% 2012 14.7% The company's geometric mean for sales from 2008 to 2012 is closest to:  A. 5.9% B. 6.2% C. 6.8% Explanation Calculating an average growth rate for an asset over multiple periods requires the use of compounding, since the growth rate each year is affected by the growth rates in previous years. In Year 2, the asset will have already grown by the rate in Year 1. In Year 3, the asset will have already grown by the rate in both Years 1 and 2. This is true for asset returns and financial figures like sales growth and earnings per share growth. The geometric mean calculates the average growth rate over multiple periods of time by taking compounding into account. Because of this, the geometric mean is used for the time series for the company's sales growth since sales each year are affected by the growth in previous years. The geometric mean helps to calculate the terminal value of an asset or, in this case, sales. The company's five-year compounding annual growth rate rounds to 5.9%. (Choice B) The arithmetic mean, or simple average, of the company's sales growth is 6.2%. The arithmetic mean does not take compounding into consideration. (Choice C) The median of the company's sales growth is 6.8%. Since there is an odd number of years being calculated, the median is the middle data point. Things to remember: The geometric mean calculates the average growth rate over multiple periods of time by taking compounding into account. Calculate and interpret different approaches to return measurement over time and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 8. The rate on a US Treasury bill (T-bill) is least likely to include a(n):  A. risk-free rate. B. inflation premium. C. liquidity premium. Explanation An interest rate (r) can be viewed as the risk-adjusted rate of return required by investors. It reflects the real risk-free rate adjusted for different risk premiums. A completely risk-free security with no expected inflation has a real risk-free rate of return. The return of a riskier security includes additional risk premiums to compensate investors for accepting various risks, which include: Inflation: expected inflation over the term of the investment Default risk: potential failure on behalf of the borrower to make promised timely payments Liquidity risk: potential losses related to the sale of securities at below fair value due to illiquidity (eg, thinly traded securities) Maturity risk: greater sensitivity of a longer-termed security's value to changes in market interest rates Treasury bills (T-bills) are issued and guaranteed by the US government. They are the most liquid of securities, so investors do not require a liquidity premium. A T-bill rate captures the nominal risk-free rate, which includes the real risk-free rate and the inflation premium (Choices A and B). The nominal risk-free rate can be thought of as the return expected for the least risky investment. Things to remember: The interest rate is the risk-free rate plus the inflation risk premium, default risk premium, liquidity premium, and maturity risk premium. A US Treasury bill captures the nominal risk-free rate (real risk-free rate plus inflation premium). Treasury bills are the most liquid of securites, and their rates do not require a liquidity premium. Interpret interest rates as required rates of return, discount rates, or opportunity costs and explain an interest rate as the sum of a real risk-free rate and premiums that compensate investors for bearing distinct types of risk LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 9. An analyst reviews the performance of three exchange-traded funds (ETFs): ETF Annual Returns Large-cap Mid-cap Small-cap Year 1 13.0% 17.0% 10.0% Year 2 14.0% −1.5% 18.5% Year 3 −0.1% 9.6% −5.3% Which ETF had the highest three-year annual geometric mean return?  A. Large-cap ETF B. Mid-cap ETF C. Small-cap ETF Explanation The geometric mean return (GMR) formula is used to convert actual performance data over multiple periods into a comparable format. The GMR produces a constant periodic return, usually annual, from actual periodic returns. The actual periodic returns, which are geometrically linked, produce an actual total return. The GMR breaks this multiperiod total return into a conforming time period. Advantages of the GMR include allowing for varying investment levels and comparability. The simple arithmetic mean assumes that each period's level of investment is constant. The GMR is independent of levels of investment, which allows for better performance comparisons between securities such as large-, mid-, and small-cap exchange-traded funds (ETFs). To calculate the GMR, link the annual returns to produce a three-year total return. Link the returns geometrically (by multiplying), not arithmetically (by adding), then convert the three-year total return to an annualized one-year return. Use the total number of years (3) as the root number when reconstructing the GMR. The large-cap ETF has the highest annual GMR at 8.8%. Add 1 to the ETF return to represent the initial investment. Subtracting 1 after calculating the GMR removes the effect of the initial investment. Scaling all initial investments of any dollar amount to a base of 1 allows for comparable performance figures. (Choice B) This option incorrectly employs the arithmetic mean to find each of the ETFs' returns. The arithmetic mean is dependent on the level of investment. It assumes a constant investment. (Choice C) In this calculation, +1.5% instead of −1.5% is used in Year 2 for the mid-cap ETF. Omitting the minus sign for negative returns is a common computational error. Things to remember: The geometric mean return (GMR) is used to convert a security's total return over multiple periods to a more comparable constant return. Use the total number of years, which can be whole or fractional, when reconstructing the GMR. A common mistake is to enter negative returns incorrectly. Calculate and interpret major return measures and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 10. An asset's present value is £1,150 and its annual continuously compounded rate of return is 5.9%. Assuming no income, if the asset is sold in 4 years, its value (in £) is closest to:  A. 1,220 B. 1,446 C. 1,456 Explanation The return an investor receives on an investment depends on the rate of return and number of compounding periods. All else being equal, for a given investment holding period many shorter compounding periods (ie, more frequent compounding) result in a greater return than a smaller number of longer compounding periods (ie, less frequent compounding). An infinite number of infinitesimally small compounding periods during a holding period is known as continuous compounding. In this scenario, the number of years (T) and the asset's present value (S0) and annual continuously compounded return (r0,1) are given, so the future price after 4 years can be determined by substituting as follows: (Choice A) £1,219.89 results from incorrectly treating 5.9% as the continuously compounded return for the entire period, not the annual continuously compounded return. (Choice B) £1446.38 results from incorrectly treating 5.9% as an annually compounded return and compounding for 4 years. Things to remember: An infinite number of infinitesimally small compounding periods during a holding period is known as continuous compounding. With continuous compounding, an ending value is equal to its beginning value multiplied by erT, where r is the annual continuously compounded return and T is the number of years. Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 11. A mutual fund has the following annual rates of return: Year 20X1 20X2 20X3 Returns −10% 5% 16% The mutual fund's annual geometric mean return (%) is closest to:  A. 3.1 B. 3.7 C. 9.6 Explanation The geometric mean return captures the impact of compounding over multiple periods. Compounding involves multiplying (ie, "chaining") the returns for each of n periods and then taking the nth root of the result. The calculation convention is to add 1 to each return before multiplying the series and then subtract 1 from the calculated nth root. In this scenario, the geometric mean is calculated as follows: (Choice B) 3.7 is the arithmetic mean (ie, simple average) and does not take into account multiperiod compounding. (Choice C) 9.6 is the total holding period return, not an annual return. Things to remember: The geometric mean return measures a compounded return over multiple periods. Compounding involves multiplying the returns for each of n periods and then taking the nth root of the result. The calculation convention is to add 1 to each return before multiplying the series and then subtract 1 from the calculated nth root. compare the money-weighted and time-weighted rates of return and evaluate the performance of portfolios based on these measures LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 12. An analyst gathers monthly investment returns for a 9-month period: Monthly Portfolio Returns Month Return January 5% February 6% March −4% April 16% May 8% June 6% July 8% August 5% September 13% The analyst notes that the data set has outliers and decides to calculate the average considering all nine observations while limiting the range of returns to be between the 20th and 80th percentile. Using these parameters, the mean is closest to:  A. 7.00% B. 7.29% C. 7.67% Explanation Outliers can significantly distort the accuracy of statistical measures used to summarize a data set of observations. They can be treated in three different ways when calculating the mean: The arithmetic mean reflects the entire data set, including outliers. The trimmed mean eliminates all outliers from the data set. The winsorized mean includes outliers but "corrects" their values to fall within an acceptable range. The winsorized mean thus reflects the total number of observations without allowing distortion by outliers. In this scenario, the analyst specifies that the acceptable range of values is between the 20th and 80th percentiles. The winsorized mean is calculated as follows: Step 1: Find the range of acceptable values by locating the observations at the 20th and 80th percentiles: L20 = (9 + 1) × (20 / 100) = 2: the lower limit is the 2nd observed value L80 = (9 + 1) × (80 / 100) = 8: the upper limit is the 8th observed value Step 2: With the data in ascending order, find the values of the 2nd and 8th observations, which are 5% and 13%, respectively: Step 3: Adjust the values of observations below the specified range (ie, 1st observation) to the lower bound of 5%, and observations above (ie, 10th observation) to the upper bound, 13%. Step 4: Calculate the simple average of the adjusted data set: (5% + 6% + 5% + 13% + 8% + 6% + 8% + 5% + 13%) / 9 = 7.67% (Choice A) 7.00% is the arithmetic mean, which accounts in its calculation for the values of all observations. By including outliers, it may be misleading as a measure of central tendency. (Choice B) 7.29% is the trimmed mean that is calculated as the average of the middle seven observations and excludes outliers. Things to remember: The winsorized mean is a measure of central tendency that "corrects" outlier values to fall within an acceptable outcome range, so that the actual number of observations is reflected in the mean without allowing distortion by outliers. Calculate and interpret different approaches to return measurement over time and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 13. If an asset's two-year continuously compounded return is 13.42%, then its holding period return for the same period is closest to:  A. 13.00% B. 13.87% C. 14.36% Explanation In a discretely compounded return, an investment's return compounds periodically (eg, monthly, quarterly) for a defined number of periods within a certain time frame. In a continuously compounded return, returns compound over infinite periods within a certain time frame. The continuously compounded return is the natural log of (1 + Holding period return). In this scenario, the holding period return is calculated as follows: (Choice A) 13.00% results from incorrectly multiplying the annual geometric return by 2. (Choice B) 13.87% results from incorrectly dividing the two-year continuously compounded return by 2, and then annually compounding it for 2 years. Things to remember: With a continuously compounded return, returns compound over infinite periods within a certain time frame. The continuously compounded return equals the natural log of (1 + Holding period return (HPR). The HPR is the change in price (ie, Ending price − Beginning price + Income) as a percentage of the beginning price. Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 14. The following table shows an investment's closing price on each day: Date Closing Price March 1 50 April 1 74 May 1 65 Based only on this information, the continuously compounded return of the investment from March 1 to May 1 is closest to:  A. 23.1% B. 26.2% C. 30.0% Explanation A continuously compounded rate of return is one way to measure an asset's change in value over a holding period, assuming that changes in value are compounded continuously. This differs from discrete compounding, which assumes that compounding occurs at evenly spaced intervals, such as weekly or annually. The asset's change in value has two elements: Change in price over the period (ie, gain, loss) Income received from the asset over the period These two elements are used to calculate the asset's total return. The formula for calculating total return is: The total return is used to calculate the holding period return (HPR), which is (Total return − 1). In this scenario, the investment does not pay interest or a dividend, so its HPR from March 1 to May 1 is (65 / 50) − 1 = 0.3, or 30%. The continuously compounded return over this period is the natural logarithm of (1 + HPR) = the natural logarithm of (Total return), or ln(1 + 0.3) ≈ 26.2%. (Choice A) 23.1% is the approximate result incorrectly calculating the HPR as the average of the three prices divided by the beginning price, and then calculating the natural logarithm of (1 + result), as shown above. (Choice C) 30.0% is the HPR, not the continuously compounded return, for the asset over the time period. Things to remember: An asset's continuously compounded return is the natural logarithm of [1 + Holding period return (HPR)]. The HPR is the asset's (Total return − 1). When an asset does not pay income (eg, interest, dividends) over the period, total return is the ending price divided by the beginning price. Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 15. An investment fund holds investments X, Y, and Z. An analyst reviews the following information: The geometric mean return for the fund over two years is closest to:  A. 12.5% B. 13.0% C. 14.0% Explanation To find the two-year geometric mean return, first determine the portfolio returns for 20X8 and 20X9. The portfolio return is the weighted average return of the holdings of the portfolio. It is based on each holding's weight and return. Using the portfolio return formula, find the fund's annual return for 20X8 and 20X9. Link these annual returns to determine the two-year geometric mean return using the geometric mean formula. Time-weighted returns nullify the influences created from cash flows and improve comparability of returns. The geometric mean return for the fund over two years is 12.51%. (Choice B) This choice correctly finds the portfolio return for 20X8 and 20X9 but uses the arithmetic mean instead of the geometric mean to find the two-year mean return. (Choice C) This option correctly uses the geometric mean formula but mistakenly applies a simple arithmetic return (arithmetic mean), not a weighted average return, for the components of the portfolio return. Things to remember: The geometric mean return, derived from a total or holding period return, creates a periodic performance measure. Unlike to other performance measures, it provides comparability of different portfolios. Calculate and interpret major return measures and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 16. A bank offers a certificate of deposit (CD) with a stated annual rate of 7% compounded continuously. The CD's effective annual rate is closest to:  A. 6.77% B. 7.00% C. 7.25% Explanation If an investment's returns are compounded more than once per year—such that the number of compounding periods in a year is greater than 1— then the stated annual rate of return does not accurately reflect the investor's effective annual rate (EAR) of return: the investor's EAR is greater than the stated annual rate. A rate of return is considered continuously compounded when the number of discrete compounding periods is infinite and each compounding period is infinitesimally small. In this scenario, the stated rate is 7%, which is continuously compounded. Solve for the EAR using the continuously compounded rate of return: Note that this question could be answered without doing the math since 7.25% is the only choice greater than the APR of 7% and so must be the correct answer. (Choice A) 6.77% is the result of substituting the continuously compounded rate of return as the EAR. (Choice B) 7.00% is the result of incorrectly assuming the EAR is equal to the annual continuously compounded rate of return. Things to remember: The effect of compounding over a certain number of discrete periods during the year causes the investor to realize a higher effective annual rate (EAR) of return than the stated rate. A rate of return is considered continuously compounded when the number of discrete compounding periods is infinite, and each is infinitesimally small. EAR = er − 1, where the stated annual rate r is continuously compounded. Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 17. An investor purchases stock for $20 and sells it for $14. On the date of sale, the stock paid a $1 dividend. The investor's continuously compounded rate of return for the investment is closest to:  A. −25.0% B. −28.8% C. −35.7% Explanation Continuously compounded rate of return r0,t = ln(1 + Holding period return) A continuously compounded rate of return measures an asset's change in value over a holding period. The assumption is that the changes are compounded continuously over the period. This differs from discrete compounding, which assumes that compounding occurs at evenly spaced intervals, such as weekly or annually. The change in value has two elements: Change in price over the period (ie, gain, loss); and/or Income received from the asset over the period The combination of these two elements is known as the total return. The formula for calculating total return is given as: The total return for an investment is also its holding period return (HPR). In this question, the ending price is less than the beginning price, so the investor incurs a loss; even after adding in the dividend income, the HPR is negative: The investor's continuously compounded rate of return is the natural logarithm (ln) of (1 + HPR), or ln(0.75), which is approximately −0.2877. (Choice A) −25.0% is the holding period loss for the investment. (Choice C) −35.7% is the approximate continuously compounded return when neglecting to account for the dividend. Although the investment generated an overall loss, the dividend reduced the loss. Things to remember: The holding period return (HPR) for an investment is its total return. Total return is the sum of price appreciation plus income received on the investment. The continuously compounded return is the natural logarithm of (1 + HPR). Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 18. In portfolio management, it is assumed that the expected return is:  A. never equal to the historical mean return. B. equal to the historical mean return over the next 30 years. C. equal to the historical mean return over a future period of unknown length. Explanation The historical mean return is the actual return that an investor earned in the past. Since assets are risky, it is unlikely that this exact return will be earned again over a specific period in the future. However, in portfolio management it is assumed that given a long enough, although unspecified, period, it is reasonable to expect that a future return will be equal to the historical mean return. Although this assumption may not prove accurate, in practice the historical mean return is often used as a reasonable estimate of an asset's expected return (Choice A). (Choice B) It is incorrect to say that historical mean return is equal to expected return over any specific period, be it 30, 50, or 100 years, since the period may not be long enough. Things to remember: Given a long enough, although unspecified, period, it is reasonable to expect that a future return will be equal to the historical mean return. Although this assumption may not prove accurate, in practice the historical mean return is often used as a reasonable estimate of an asset's expected return. Calculate and interpret major return measures and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 19. An investor is reviewing investment performance and collects the following information: (AUD millions) Year 1 Year 2 Year 3 0 1,150 1,285 Beginning of year contributions 1,000 500 0 Total investment at beginning of year 1,000 1,650 1,285 15% −10% 20% 150 (165) 257 0 (200) 0 1,150 1,285 1,542 Beginning value of investment Portfolio performance (%) Portfolio performance Portfolio withdrawals Ending value of investment If withdrawals only occur at year-end, the investor's money-weighted return over the three years is closest to:  A. 6.03% B. 7.49% C. 8.28% Explanation The money-weighted return (MWR), an application of the internal rate of return (IRR), measures investment performance based on an investor's cash flows. The net present value (NPV) represents the sum of all discounted investor cash flows. When cash flows are discounted by the MWR, the NPV = 0 and the initial investment is equal to the future discounted contributions and withdrawals. The relevant cash flows for calculating MWR are the: investor's contributions (ie, investor's outflows) investor's withdrawals (ie, investor's inflows), and investment's value at the end of the measuring period. The investment's ending value is an inflow; as if the investor liquidated the entire investment at the end of the measuring period. Discount the investor's cash flows over the appropriate time period. The initial investment (AUD 1,000) occurs at time zero, so it is not discounted. Consider the beginning of the year contributions as occurring at the end of the previous period. When withdrawals occur at the end of the year, as in this question, they are discounted over that year as well as all prior periods. After the cash flows have been identified, solve for the MWR using a financial calculator. Enter positive values for cash inflows and negative values for cash outflows. (Choice B) 7.49% is the time-weighted return, which uses the geometric mean and is based on investment performance, not cash flows. MWR uses cash flows to determine the return. (Choice C) 8.28% results from discounting both the initial contribution and the beginning of Year 2 contribution over one year. Things to remember: The money-weighted return is an application of the internal rate of return. It measures an investor's return based on the amount of contributions to and withdrawals from the investment, as well as when those cash flows occur. Calculate and interpret major return measures and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 20. An analyst gathers the following information on three investments with a two-year holding period: Which investment most likely has the highest continuously compounded return over the two-year period?  A. Investment X B. Investment Y C. Investment Z Explanation For a given holding period, different frequencies of compounding affect an investment's returns. Many shorter compounding periods (ie, greater frequecy) result in a greater return than a smaller number of longer compounding periods for the same investment period, all else being equal. An infinite number of infinitesimally small compounding periods during an investment holding period is known as continuous compounding. A holding period return, Rt,t+1, can be converted into an equivalent continuously compounded return using the formula Continuously compounded return = ln(1 + holding period return): Convert Investment X's two-year holding period return to a continuously compounded return. Investment Y is stated in terms of a one-year holding period return and is held for two years. Calculate the two-year holding period return of Investment Y by compounding the one-year holding period return for two years using R0,2 = (1 + R0,1)2 − 1. Then use the formula for continuously compounded return to convert the two-year holding period return to a continuously compounded return. All else being equal, Investment Z has the highest continuously compounded return (Choices A and B). Things to remember: Continuously compounded return involves compounding with infinitesimally small compounding periods over an investment period. Continuously compounded return is equal to the natural log of (1 + holding period return): rt, t+1 = ln(1 + Rt, t+1). Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 21. A student chooses to spend $100 today on a concert ticket instead of investing that money in a savings account that earns interest. The interest that could have been earned can be best interpreted as this student's:  A. sunk cost. B. opportunity cost. C. required rate of return. Explanation An interest rate can be used to express the percentage growth in money between two different points in time. For example, $100 invested today at an annual interest rate of 5% will grow to $105 in a year. Depending on the circumstances, an interest rate can be interpreted in many ways: as a required rate of return, as a discount rate, or as an opportunity cost. The student forgoes the future investment return (ie, the interest rate) by choosing to spend the money today. The foregone return is the opportunity cost of current consumption. (Choice A) A sunk cost is an actual cost that has already been incurred. In this question, the $100 spent on the purchase, not the foregone interest, is the sunk cost. (Choice C) The required rate of return is the minimum return percentage that an investor requires for accepting an investment. Since the student did not accept the investment, the interest rate offered on the account is not the student's required rate of return. Things to remember: Opportunity cost in terms of an interest rate relates to a return that an investor forgoes by choosing consumption today over investing for the future. Sunk cost is an actual expense that has already been incurred. Interpret interest rates as required rates of return, discount rates, or opportunity costs and explain an interest rate as the sum of a real risk-free rate and premiums that compensate investors for bearing distinct types of risk LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 22. An investor deposits $5,000 into an account with a stated interest rate of 10%. The bank will compound interest using annual compounding for the first year and continuous compounding thereafter. If all interest is reinvested and there are no withdrawals from the account, the number of years needed for the account to grow from $5,000 to $10,000 is closest to:  A. 6.0 B. 7.0 C. 7.3 Explanation This scenario requires calculating the total time in months needed for an initial deposit of $5,000 to grow to $10,000 given a stated interest rate of 10%. The compounding periods change during the investment's time horizon, although the stated interest rate does not. All things equal, more frequent compounding results in faster growth of a lump sum. First, calculate the amount on deposit at the end of Year 1, when the funds are compounded annually. The initial $5,000 deposit grows to $5,500 in one year. Then, calculate the number of years required for $5,500 to grow to $10,000 assuming continuous compounding. The formula for calculating the future value of a lump sum using continuous compounding is as follows: The solution is shown below: Things to remember: A lump sum grows more quickly with more frequent compounding periods. Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld Copyright CFA Institute All rights reserved 23. A university receives contributions to its endowment fund at the beginning of each year. The fund generates the returns displayed in the chart below: Return during Year of Contributions investment ($ millions) 1 20 12% 2 50 16% 3 70 18% 4 200 −2% year of investment Over the four-year period, the measure that shows the highest return is the:  A. internal rate of return. B. time-weighted rate of return. C. money-weighted rate of return. Explanation The time-weighted rate of return (TWR) and the money-weighted rate of return (MWR) are both portfolio return measures. The TWR measures the compound growth rate of a portfolio and excludes impact from contributions and withdrawals. The MWR is the internal rate of return on a portfolio and accounts for the impact of contributions and withdrawals. The MWR equates the present value of the cash outflows to the present value of the investor's cash inflows. The outflows (contributions) are provided in the question. The value of the fund at the end of Year 4 should be treated as the only inflow for the purpose of calculating MWR. Solve for MWR: Determine cash inflows: Calculate MWR: The TWR is the geometric growth of the portfolio without considering the impact of contributions and withdrawals. To solve for TWR, find the annual geometric mean of annual returns over all four years. Solve for TWR: The TWR of 10.71% is greater than the MWR of 5.77%. Note that the MWR being less than the TWR is attributed to the timing and size of cash flows. Since Year 4 had a negative return, the significant addition of $200 million at the beginning of Year 4 significantly reduces the overall MWR. (Choice A) The internal rate of return is the MWR. (Choice C) The MWR is lower than the TWR in this question. Things to remember: The time weighted-rate of return (TWR) and the money-weighted rate of return (MWR) measure portfolio returns over time. The TWR measures the compound growth rate of a portfolio and excludes the impact from contributions and withdrawals. The MWR is the internal rate of return on a portfolio and accounts for impact of contributions and withdrawals. compare the money-weighted and time-weighted rates of return and evaluate the performance of portfolios based on these measures LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 24. An analyst is comparing three investments: Frequency of Stated Compounding Annual Rate X Semiannually 6.1% Y Quarterly 6.0% Z Continuously 5.9% Investment The investment with the greatest effective annual rate is:  A. Investment X B. Investment Y C. Investment Z Explanation If an investment's return is compounded more than once per year, then a stated annual rate of return does not accurately reflect the investor's effective annual rate (EAR) of return. The effect of compounding over a certain number of discrete periods during the year causes the investor to receive more than the stated rate. All else equal, a greater number of compounding periods means a greater return for the investor. The EAR can be used to compare investments with different compounding frequencies and therefore different numbers of compounding periods per year. The periodic rate is the stated annual rate per compounding period. In this scenario, the EAR of each investment is calculated as follows: Things to remember: The effective annual rate (EAR) reflects the actual return that an investor receives. The EAR can be used to compare investments with different numbers of compounding periods per year. It can be calculated as EAR = (1 + periodic interest rate)m − 1 or, if the stated annual return is continuously compounded, as EAR = er − 1. Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 25. A credit union offers an investment that pays interest quarterly. If the investment's effective annual rate is 5.93%, then its stated annual rate (rs) quoted by the credit union is closest to:  A. 5.76% B. 5.80% C. 6.06% Explanation With interest-bearing investments, investors earn interest periodically (eg, monthly, quarterly) over time based on a specific periodic rate. When interest is compounded, interest income is reinvested each period at the periodic rate, so additional interest is earned on previous interest. Therefore, the total value of the investment (including principal and interest) grows at the periodic rate. The effective annual rate (EAR), which reflects periodic compounding, is the investment's effective annual return. For the sake of simplicity, financial institutions usually quote a stated annual interest rate (rs) instead of an EAR. The rs equals the periodic rate multiplied by the number of compounding periods per year (m). This rate does not reflect periodic compounding. The rs is always less than the EAR when interest is paid more than once a year. The candidate can derive rs from EAR and m. (Choice A) This results from finding the natural log of 1 + EAR. This is the continuously compounded rate that would generate the same return as an EAR of 5.93%. (Choice C) This results from mistakenly swapping the placement of rs and EAR in the original equation. Things to remember: The stated annual rate (rs) equals the periodic rate multiplied by the number of compounding periods per year. The rs can be found from the effective annual rate (EAR) and the number of compounding periods per year. The stated annual rate does not capture the effect of compounded interest; EAR does. The rs is always less than the EAR when interest is paid more than once a year. Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 26. The price of silver is $14.00 per ounce at the beginning of the year and $14.50 at the end of the year. The continuously compounded rate of return for silver is closest to:  A. 3.51% B. 3.57% C. 3.64% Explanation In a discretely compounded return, an investment's return compounds periodically (eg, monthly, quarterly, semiannually) for a defined number of periods within a certain timeframe. In a continuously compounded return, an investment's return compounds over infinite periods within a certain time frame. The continuously compounded return equals the natural log of ending price over beginning price. Note that ST/S0 is the same as (1 + holding period return) The beginning price of $14.00 grown continuously over one year at 0.0351 equals $14.50 at year-end. (Choice B) 3.57% is the holding period return (14.5/14 − 1), not the continuously compounded rate of return. (Choice C) 3.64% results from dividing the holding period return by 365 and compounding that daily. Things to remember: With continuously compounded return, an investment's return compounds over infinite periods within a certain time frame. The continuously compounded return equals the natural log of the ending price over the beginning price (or the natural log of [1 + holding period return]). Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 27. An analyst compares the performance of portfolios that receive annual contributions in addition to earning returns. The analyst most appropriately uses a(n):  A. internal rate of return. B. time-weighted rate of return C. money-weighted rate of return. Explanation A time-weighted rate of return (TWR) is more appropriate for comparing portfolios since its calculation neutralizes the impact on performance when the portfolio's market value changes due to investor contributions or withdrawals rather than the returns that the investment itself generates. Contributions increase the market value of the investment but should not be included as part of the investment's performance. The TWR neutralizes contributions by splitting a performance period into smaller sub-periods each time contributions are made. A TWR is found by geometrically linking each sub-period's holding period return. This results in a compounded growth rate. (Choices A and C) A money-weighted rate of return (MWR) is an application of the internal rate of return. The MWR includes all the cash flows, including contributions to and withdrawals from a portfolio, by equating the present value of the investor's cash outflows to the present value of the investor's cash inflows. Therefore, it does not calculate just the return from the portfolio's actual earnings. Things to remember: Analysts prefer the time-weighted rate of return to compare the performance of portfolios since its calculation neutralizes the impact on performance when there are changes in the portfolio's market value due to contributions rather than returns. It neutralizes the impact of contributions by splitting a performance period into smaller sub-periods each time there is a contribution. compare the money-weighted and time-weighted rates of return and evaluate the performance of portfolios based on these measures LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 28. An initial investment of $9,520 grows to $15,000 in five years with quarterly compounding. If there are no withdrawals or deposits during those five years, the investment's effective annual rate is closest to:  A. 9.2% B. 9.5% C. 11.5% Explanation The effective annual rate (EAR) is the annual compounded growth rate. Since the EAR accounts for compounding, it connects the present value of a lump sum to its future value for a given time period. Using the formula shown, the compounded annual growth rate needed to grow $9,520 to $15,000 within five years is: If the stated interest rate and the number of compounding periods are known, those variables can be used to derive the EAR. The EAR relates to the stated interest rate and the number of compounding periods per year (m) as follows: In this scenario, the stated rate is not known but the calculation can be solved using the future value relationship. (Choice A) Since the stated interest rate is not given, the compounding convention is irrelevant. A common error is to use the financial calculator to generate the periodic rate by using 20 as the number of periods (ie, 5 years × 4 periods per year). This results in a quarterly rate of approximately 2.3%, which when multiplied by 4 equals 9.2%. 9.2% is not the effective rate; it is the stated rate that produces an effective rate of 9.52%: (Choice C) 11.5% results from first calculating [(15,000/9,520) − 1], which generates the cumulative growth for the five-year period, and then dividing that by 5 (years). This is not the EAR. Things to remember: The effective annual rate (EAR) is the investment's compounded annual growth rate. When no stated interest rate is provided, the EAR is calculated as the quotient of the ending value divided by the beginning value, raised to the power of (1 ÷ number of years of the investment), minus 1. Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 29. A charity foundation invests donations it has received at the beginning of each year. The foundation has had the following yearly returns: Year of Contributions Return During Investment ($ millions) Year of Investment 1 40 10% 2 45 4% 3 40 12% 4 60 5% Over the four years, the fund most likely had an annual:  A. time-weighted return of 7.75%. B. money-weighted return of 7.27%. C. money-weighted return of 7.70%. Explanation Money-weighted rate of return (MWR) Time-weighted rate of return (TWR) Internal rate of return (IRR) of a portfolio; accounts for all Compound growth rate (geometric mean) of a portfolio; excludes contributions and withdrawals contributions and withdrawals Appropriate for measuring an investor's change in wealth Preferred return measure for evaluating investment performance Affected by amount and timing of cash flows Unaffected by amount and timing of cash flows The time-weighted rate of return (TWR) and money-weighted rate of return (MWR) are measures of portfolio return. TWR is: the geometric mean return (a compound growth rate) and is not impacted by interim cash flows, such as deposits and withdrawals. MWR is: the IRR and is impacted by interim cash flows. Therefore, MWR is affected by the timing and size of cash flows. MWR equates the present value of cash outflows to the present value of cash inflows. Deposits into the fund are treated as outflows from the investor (eg, charity). In this scenario, the value of the fund at the end of Year 4 should be treated as the only inflow (since it could be withdrawn by the charity) for the purpose of calculating MWR. To solve for MWR: 1. Determine fund value at end of Year 4: 2. Calculate MWR (r): (Choice A) 7.75% is the simple average of returns over four years. (Choice C) 7.70% is the time-weighted return and excludes the impact of contributions and withdrawals. Things to remember: The time-weighted rate of return (TWR) and money-weighted rate of return (MWR) measure portfolio returns over time. TWR measures a portfolio's compound growth rate and excludes the impact from contributions and withdrawals. MWR is a portfolio's IRR and accounts for the impact of contributions and withdrawals. compare the money-weighted and time-weighted rates of return and evaluate the performance of portfolios based on these measures LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 30. Two AUD 50,000 investments have a 3-year term. One compounds quarterly at a stated annual rate of 2.00%. The other compounds at a continuous rate of 1.75%. The difference in their future values (in AUD) at the end of 3 years is closest to:  A. 364 B. 389 C. 540 Explanation The future value of the first investment can be solved using the formula for multiple compounding periods per year. The present value is AUD 50,000 and the stated annual rate is 2.00%. Quarterly compounding implies 4 per year × 3 years = 12 compounding periods (N = 3 and m = 4). The other investment's future value can be solved using the continuous compounding formula. The stated annual rate is 1.75% and all other values are the same as those of the first investment. The two values and the difference between them are calculated as follows: The difference between the future values is 388.76, which can be rounded to 389. (Choice A) 364 is the result of solving for the future value of the first investment without adjusting for the number of compounding periods—in effect, assuming annual compounding. (Choice C) 540 results from mistakenly assuming continuous compounding has discrete compounding periods and then using the same frequency of compounding for both investments. Things to remember: The future value of an initial investment depends on whether the rate of compounding is continuous or discrete. In the case of discrete compounding, adjust for the frequency of compounding. Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 31. A manager buys 500 shares of a stock at $15.00 per share at the beginning of the year. At the end of the year, the stock's price appreciates to $18.50. If this stock pays its annual dividend of $0.50 at the end of the year, then the manager's holding period return is closest to:  A. 21.6% B. 23.3% C. 26.7% Explanation A holding period return (HPR) is a single-period measurement of return. The period can represent a day, a year, or any other timeframe. The difference between the security's ending price and its beginning price is its capital gain or loss during the holding period. Also affecting the HPR are company distributions to bondholders (interest) or shareholders (dividends). These distributions are the income part of a security's return and increase HPR. (Choice A) This answer incorrectly applies the HPR using the ending price, rather than the beginning price, to find the HPR. (Choice B) This incorrect choice does not include the stock's dividend and therefore results in an understatement of HPR. Things to remember: A holding period return is a performance measurement that compares the beginning and ending values of an investment and includes any income earned. Calculate and interpret major return measures and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 32. Two investors each initially invest ¥6 million in the same fund, which earns 5% during Year 1. At the beginning of Year 2: Investor X invests an additional ¥8 million to the fund, and Investor Y invests an additional ¥9 million to the fund. The fund's return is −2% in Year 2. After two years, which of the following best describes the investors' returns?  A. Investor Y will have a greater money-weighted return B. Both investors will have equal geometric mean returns C. Both investors will have negative time-weighted returns Explanation Money-weighted rate of return (MWR) Time-weighted rate of return (TWR) Internal rate of return (IRR) of a portfolio; accounts for all Compound growth rate (geometric mean) of a portfolio; excludes contributions and withdrawals contributions and withdrawals Appropriate for measuring an investor's change in wealth Preferred return measure for evaluating investment performance Affected by amount and timing of cash flows Unaffected by amount and timing of cash flows A time-weighted rate of return (TWR) is calculated by linking periodic returns without consideration for the investment's size. This way, the impact of cash additions or withdrawals does not skew reported returns. This method of linking returns sets subperiods based on cash flows and then links them using the same calculation as the geometric mean return. In this case, since the investors invest in the same fund and thus have the same returns in each period, the two investors have TWRs (ie, geometric mean returns) that are equal, and the returns are positive (Choice C): A money-weighted return (MWR) accounts for all cash flows, in effect weighting each periodic return by the amount invested during that period. In this scenario, the investors have different MWRs since they invest different amounts in Year 2. Investor Y's MWR will be less due to the larger investment during a period of lower returns (Choice A). Things to remember: A time-weighted rate of return excludes effects from multiple cash additions or withdrawals; a money-weighted return accounts for all cash flows. Therefore, investors with the same periodic returns but unequal cash flows will have equal time-weighted returns but different money-weighted returns. compare the money-weighted and time-weighted rates of return and evaluate the performance of portfolios based on these measures LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 33. A manager is concerned about the current inflation rate and gathers the following information: Expected annual return (%) Certificates of deposit 1.8 Treasury bills 2.3 Commercial paper 2.8 Equities 8.0 If the equity risk premium is 3.5%, what is the current inflation rate?  A. 1.5% B. 2.0% C. 2.5% Explanation Nominal return is the expected return on an asset. Three factors influence nominal return: the risk-free rate, the inflation rate, and the risk premium. Most investors estimate these factors, resulting in varying nominal return figures. The risk-free rate is the return received by investors who take no risk. The risk premium is the additional return an investor receives for investing in a risky asset. The inflation rate is the percent change in the cost of a basket of goods over time. The nominal return (expected annual return) of equities is given at 8%. Since the return on short-term government bonds is the optimal proxy for the risk-free rate, it can be used here to find the current inflation rate. Using the other given information, rearrange the nominal return formula to solve for the missing variable. (Choice A) This option incorrectly calculates inflation using the expected return on certificates of deposit as a proxy for the risk-free rate. Certificates of deposit are bank-issued investments; they do not have the full faith and credit of the United States. This is not the best proxy for the risk-free rate. (Choice C) This choice calculates the inflation rate using the expected return on commercial paper. Commercial paper is a company-issued, short-term fixed income investment. This investment is subject to more risks than affect government securities. The expected return of commercial paper is not the best proxy for the risk-free rate and overstates the inflation rate. Things to remember: To calculate the nominal return, an investor must approximate the inflation rate, risk-free rate, and risk premium. Most often, government-based securities are used as a proxy for the risk-free rate. Calculate and interpret major return measures and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 34. An investment has an effective annual rate of 7.7% with monthly compounding. Its stated annual rate is closest to:  A. 7.44% B. 7.70% C. 7.97% Explanation If an investment's return is compounded more than once per year, then the investor's effective annual rate (EAR) of return is greater than the stated annual rate of return. All else being equal, the more frequently a return is compounded, the greater the return for the investor, as measured by the EAR. In this scenario, since 7.44% is less than the EAR, it is the only possible correct answer. To calculate the stated annual rate associated with an EAR of 7.7% with monthly compounding, rearrange the EAR formula and substitute as follows: To convert a monthly periodic rate to a stated annual rate, multiply the monthly rate by 12. Therefore, the annual stated rate is 12 × 0.62% = 7.44% (Choice B) 7.70% is the EAR. If the periodic rate is compounded more than once per year, the EAR must be greater than, not equal to, the stated annual rate. (Choice C) 7.97% results from mistakenly swapping the EAR and the periodic rate in the formula. Things to remember: If an investment's return is compounded more than once per year, then the investor's effective annual rate of return is greater than the stated annual rate of return. All else being equal, a greater number of compounding periods means a greater return for the investor, based on a given stated annual rate. Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 35. An investor buys a share of stock for €50 today. At the end of the Year 1, the investor buys another share of the same stock for €60 and receives a dividend of €2. At the end of Year 2, the investor receives total dividends of €4 and then sells both shares for €130. The moneyweighted rate of return is closest to:  A. 12.05% B. 13.05% C. 15.68% Explanation The money-weighted rate of return (MWR) and the time weighted-rated of return (TWR) are two methods used to calculate portfolio return over time. The TWR is often used to measure an investment manager's return performance. The MWR is often used to measure an investor's change in wealth when an investor has control over cash flows. The MWR is the internal rate of return on a portfolio. The MWR is the rate that equates the present value of the investor's cash outflows (contributions) to the investor's cash inflows (withdrawals). Due to this relationship, MWR is determined by the timing and amount of cash flows. To solve for the MWR, first determine net cash flow for each period: Then, input net cash flows into a financial calculator: (Choice A) This results from excluding dividends when solving for MWR. (Choice B) This results from incorrectly treating the dividend as a cash outflow to the investor at T = 1. Things to remember: The money-weighted rate of return (MWR) is the internal rate of return on a portfolio. It equates the present value of the investor's cash outflows (contributions) to the investor's cash inflows (withdrawals). MWR is heavily influenced by the timing and amount of cash flows. MWR is appropriate when an investor has control over withdrawals and contributions. compare the money-weighted and time-weighted rates of return and evaluate the performance of portfolios based on these measures LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 36. An analyst has gathered the following data to calculate the expected return of an asset: The asset's expected return is closest to:  A. 7.99% B. 8.80% C. 9.05% Explanation An investment's return is the investor's compensation for delaying consumption and bearing risk. Therefore, expected return is a function of: Real risk-free rate (ie, compensation for delaying consumption) Expected inflation rate (ie, compensation for the loss of purchasing power) Expected risk premium (ie, compensation for bearing the risks of an investment) The expected return in this scenario is calculated as: (Choice A) 7.99% is the result of incorrectly using the economic growth rate instead of the real rate of return to calculate expected inflation. (Choice B) 8.80% is the result of adding the real risk-free rate, expected risk premium, and expected inflation. Although that is one way to approximate expected return, it does not result in the exact expected return. Things to remember: The expected return of an asset is the investor's compensation for postponing consumption and bearing risks. Expected return = [(1 + real riskfree rate) ⨯ (1 + expected inflation) ⨯ (1 + expected risk premium)] − 1. Calculate and interpret major return measures and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 37. An asset's one-year continuously compounded return is 9.2%. Assuming no income from the asset, if its beginning price is BRL 136.50, then its price at the end of one year (in BRL) is closest to:  A. 149.06 B. 149.65 C. 150.33 Explanation In a discretely compounded return, an investment's return compounds periodically (eg, monthly, quarterly) for a defined number of periods within a certain time frame. In a continuously compounded return, an investment's return compounds over infinite periods within a certain time frame. The continuously compounded return equals the natural log of the ratio of ending price to beginning price. It should be noted that the ratio of the ending price to the beginning price is equivalent to 1 + Holding period return. In this scenario, the asset's ending price is calculated through the following steps: (Choice A) 149.06 is the result of treating 9.2% as the asset's holding period return. The incorrect setup would have been rt,t+1 = In(1 + 0.092). (Choice C) 150.33 is the result of treating 9.2% as the asset's discount yield. The incorrect setup would have been: (Face value − Purchase price) / Face value = 0.092. Things to remember: In a continuously compounded return, an investment's return compounds over infinite periods within a certain time frame. The continuously compounded return is the natural log of the ratio of the ending price to the beginning price. This ratio is equivalent to (1 + Holding period return). Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 38. A portfolio manager has compiled an investment's annual returns for three years: Time Period Return (%) Year 1 −3% Year 2 11% Year 3 −5% The average annual compounded return on the investment over the three-year holding period is closest to:  A. 0.76% B. 1.00% C. 2.29% Explanation The compounded periodic rate of return for an asset is the geometric mean return. It is used to measure the periodic average of investment performance over multiple consecutive periods. Compounding assumes that one period's ending investment value is the next period's beginning value (ie, that returns accumulate). The geometric mean is the average periodic rate of return that accounts for compounding and includes accumulated returns in subsequent periods. In this instance, there are three annual periods (ie, N = 3), so the geometric mean is: (Choice B) 1.00% is the arithmetic mean, which is calculated as the simple average of multiple periodic returns and does not reflect compounding. (Choice C) 2.29% is the holding period return, which is the total cumulative return received over the three-year period, not the annual return. Things to remember: The geometric mean is the average periodic rate of return that accounts for compounding and includes accumulated returns in subsequent periods. It is used to measure investment performance over multiple consecutive periods. Calculate and interpret different approaches to return measurement over time and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 39. An investor deposits ¥100 million into a fund at the beginning of Year 1 and deposits another ¥50 million at the beginning of Year 2. At the beginning of Year 3, the investor withdraws ¥15 million. The fund has the following returns: Year Return (%) 1 20 2 10 3 −20 The investor's money-weighted annual rate of return over the three-year period is closest to:  A. 0.7% B. 1.0% C. 1.8% Explanation The money-weighted return (MWR), an application of the internal rate of return (IRR), measures investment performance based on an investor's cash flows (ie, deposits and withdrawals). By contrast, the arithmetic mean and time-weighted return do not account for cash flows into and out of the portfolio. As an application of the IRR, the MWR discounts cash flows so that their net present value (NPV) is equal to zero and the initial investment is equal to the future discounted deposits and withdrawals. For the investor, deposits are outflows and withdrawals are inflows. When calculating MWR, the investment's ending value is assumed to be the final cash inflow (if positive) or outflow (if negative). Therefore, to calculate the MWR, the ending balance must be calculated first, using the investor's deposits and withdrawals and the return on the investment over each period. (¥ millions) Balance at beginning of year Year 1 Year 2 Year 3 0 120 187 Investor's deposits (withdrawals) 100 50 −15 Net balance after deposits (withdrawals) 100 170 172 Return for the year 20% 10% −20% Investment gain (loss) 20 17 −34.4 Balance at end of year [= Net balance + Investment gain (loss)] 120 187 137.6 After the cash flows have been identified, solve for the MWR using a financial calculator. Enter positive values for cash inflows and negative values for cash outflows. Things to remember: The money-weighted return is an application of the internal rate of return. It measures the actual return on an investment based on the amount of contributions and withdrawals and the periodic returns on the investment. Calculate and interpret major return measures and describe their appropriate uses LOS 40. An individual makes the following deposits, one every six months with the first deposit beginning today: If the stated interest rate is 5.4% compounded monthly, the value of the account (in SGD) six months after the last deposit is made is closest to:  A. 46,540 B. 47,779 C. 47,811 Explanation This time value of money question requires the candidate to calculate the future value of a bank account that is funded by a series of three unequal cash flows. The first deposit is made today, and subsequent deposits occur every six months. Each deposit is a different amount, so the future value cannot be calculated as an annuity. Instead, each deposit's future value must be calculated separately as the amount of the deposit (ie, its present value) multiplied by a future value factor: The stated rate of 5.4% is compounded monthly, so there are 12 compounding periods per year. To account for monthly compounding, the stated rate must be converted to a monthly rate: 0.054 ÷ 12 = 0.0045. (Choice A) SGD 46,540 results from forgetting to compound for the 6-month period following the last deposit or from incorrectly assuming that all deposits occur at the end of the period rather than at the beginning. (Choice B) SGD 47,779 results from assuming semiannual, rather than monthly, compounding. It may be useful to remember that if two series are identical with respect to the stated rate, amount, and timing of cash flows, the series with the more frequent compounding always produces a higher future value. Things to remember: The future value of a series of uneven cash flows is the sum of the cash flows' future values. The series cannot be calculated as an annuity since each deposit is a different amount. Calculate and interpret annualized return measures and continuously compounded returns, and describe their appropriate uses LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 41. Country A and Country B are both large issuers of government debt, and each country issues approximately the same amount of debt. The interest rate for one-year government debt is 3% in Country A and 3.5% in Country B. With respect to variables such as credit rating and issuance in multiple maturities, the characteristics of each country's government debt are similar. Based only on the information given, which of the following risk premiums most likely accounts for the difference in the interest rates?  A. Default premium B. Inflation premium C. Liquidity premium Explanation Interest rates are based on a real risk-free rate, which is the return that investors demand for a bond with no default risk or inflation expectations. The other premiums shown above compensate investors who hold riskier bonds. Interest rates are influenced by supply and demand for borrowed funds; therefore, each premium quantifies the additional compensation investors demand for lending when specific risks are present. In this scenario, both countries issue approximately the same amount of debt, so both countries' bonds should be similarly liquid. A liquidity premium most likely applies when a country issues fewer bonds since smaller issues tend to trade less frequently (Choice C). Less trading means less liquidity since the supply and demand forces that contribute to market prices are absent. (Choice A) The countries' similar credit ratings indicate a similar default risk; however, a default risk premium does not apply to this scenario. This leaves only the possibility that lenders anticipate higher inflation in Country B and demand an appropriate premium to compensate for the risk. Things to remember: Interest rates are based on a real risk-free rate, which is the return that investors demand for a bond with no default risk or inflation expectations. Premiums for specific risks such as liquidity, default, and inflation are added to the real risk-free rate to compensate investors for assuming those risks. Interpret interest rates as required rates of return, discount rates, or opportunity costs and explain an interest rate as the sum of a real risk-free rate and premiums that compensate investors for bearing distinct types of risk LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved. 42. An investor creates a portfolio by buying one share of stock for CAD 100 today. At the end of the Year 1, the investor buys another share of the same stock for CAD 105. At the end of Year 2, the investor sells both shares for CAD 115 each. The stock paid a dividend of CAD 10 at the end of Year 1 and paid total dividends of CAD 20 at the end of Year 2. The dividend at the end of Year 1 was reinvested in the portfolio. The timeweighted rate of return is closest to:  A. 9.23% B. 14.32% C. 17.01% Explanation The time-weighted rate of return (TWR) and the money-weighted rate of return (MWR) are two methods used to calculate portfolio returns over time. The MWR is often used to measure an investor's increase in wealth. The TWR is often used to measure an investment return performance; it is the preferred measure of the two. The TWR measures a portfolio's compounded rate of growth over time. The main advantage of the TWR is that unlike the MWR, the TWR is not affected by the investor's own contributions or withdrawals—both of which may alter portfolio returns. To calculate the TWR, divide the portfolio's overall time horizon into sub-periods based on the dates of contributions or withdrawals. Value the portfolio right before these cash flows. Determine the holding period return (HPR) for each sub-period. Finally, link the HPR of each holding subperiod. Determine activities and calculate HPR (amounts in CAD): Input HPR: A TWR of 14.32% means that a dollar initially invested into this portfolio grew at a compound annual rate of 14.32% over the two years. (Choice A) 9.23% results from excluding dividends received of $10 from the HPR calculation for the first sub-period. (Choice C) 17.01% results from not including the reinvested dividends in P1 when calculating the HPR for the second sub-period. Things to remember: The time-weighted rate of return (TWR) is the preferred method for calculating portfolio returns. The TWR measures a portfolio's compound rate of growth over time. Its main advantage is that it is not impacted by contributions and withdrawals. compare the money-weighted and time-weighted rates of return and evaluate the performance of portfolios based on these measures LOS Copyright © UWorld. Copyright CFA Institute. All rights reserved.