Chapter 5: Introduction to Risk, Return, and the Historical Record (Review Questions)

18. Consider these long-term investment data: ∙ The price of a 10-year $100 face value zero-coupon inflation-indexed bond is $84.49. ∙ A real-estate property is expected to yield 2% per quarter (nominal) with a SD of the (effective) quarterly rate of 10%. b. Compute the continuously compounded annual risk premium on the real-estate investment. With a per quarter yield of 2%, the annual yield is _____

(1 + .02)^4 = 1.0824, or 8.24%.

3. The Narnian stock market had a rate of return of 45% last year, but the inflation rate was 30%. What was the real rate of return to Narnian investors?

1 + rREAL = 1 + rNominal / 1 + i = 1 + .45 / 1 + .30 = 1.1154 ---> 11.54%

1. The Fisher equation tells us that the real interest rate approximately equals the nominal rate minus the inflation rate. Suppose the inflation rate increases from 3% to 5%. Does the Fisher equation imply that this increase will result in a fall in the real rate of interest? Explain. Hence, if the inflation rate increases from 3% to 5% while there is no change in the real rate, then the nominal rate will increase by ____%.

2%

11. Using historical risk premiums from Table 5.5 over the 1927-2018 period as your guide, what would be your estimate of the expected annual HPR on the Big/Value portfolio if the current risk-free interest rate is 3%? Adding 11.69% to the 3% risk-free interest rate, the expected annual HPR for the Big/Value portfolio is: _____

3.00% + 11.69% = 14.69%.

13. During a period of severe inflation, a bond offered a nominal HPR of 80% per year. The inflation rate was 70% per year. a. What was the real HPR on the bond over the year?

= 0.80 - 0.70 / 1.70 = 0.0588 or 5.88%

8. Derive the probability distribution of the 1-year HPR on a 30-year U.S. Treasury bond with an 8% coupon if it is currently selling at par and the probability distribution of its yield to maturity a year from now is as follows: State of the Economy Probability YTM Boom 0.20 11.0% Normal growth 0.50 8.0 Recession 0.30 7.0 For simplicity, assume the entire 8% coupon is paid at the end of the year rather than every 6 months. Capital Gain: Boom = ____ Normal Growth = ____ Recession = _____

Capital Gain -$25.95 0.00 12.28

9. Determine the standard deviation of a random variable q with the following probability distribution: Value of q Probability 0 0.25 1 0.25 2 0.50 E(q) = _____

E(q) = (0 × 0.25) + (1 × 0.25) + (2 × 0.50) = 1.25

Use the following scenario analysis for Stocks X and Y to answer CFA Problems 3 through 5 (round to the nearest percent). Bear Market Normal Market Bull Market Probability 0.2 0.5 0.3 Stock X −20% 18% 50% Stock Y −15% 20% 10% 5. Assume that of your $10,000 portfolio, you invest $9,000 in Stock X and $1,000 in Stock Y. What is the expected return on your portfolio?

E(r) = (0.9 × 20%) + (0.1 × 10%) =19% à $1,900 in returns

CFA Problems 2. Based on the scenarios below, what is the expected return for a portfolio with the following return profile? Bear Market Normal Market Bull Market Probability 0.2 0.3 0.5 Rate of return −25% 10% 24%

E(r) = [0.2 × (−25%)] + [0.3 × 10%] + [0.5 × 24%] =10%

8. Derive the probability distribution of the 1-year HPR on a 30-year U.S. Treasury bond with an 8% coupon if it is currently selling at par and the probability distribution of its yield to maturity a year from now is as follows: State of the Economy Probability YTM Boom 0.20 11.0% Normal growth 0.50 8.0 Recession 0.30 7.0 For simplicity, assume the entire 8% coupon is paid at the end of the year rather than every 6 months. HPR: Boom = ____ Normal Growth = ____ Recession = _____

HPR -17.95% 8.00 20.28

6. You are considering the choice between investing $50,000 in a conventional 1-year bank CD offering an interest rate of 5% and a 1-year "Inflation-Plus" CD offering 1.5% per year plus the rate of inflation. d. If we observe a risk-free nominal interest rate of 5% per year and a risk-free real rate of 1.5% on inflation-indexed bonds, can we infer that the market's expected rate of inflation is 3.5% per year?

No.

5. Use Figure 5.1 in the text to analyze the effect of the following on the level of real interest rates: c. The Federal Reserve Board undertakes open-market purchases of U.S. Treasury securities in order to increase the supply of money

Open market purchases of U.S. Treasury securities by the Federal Reserve Board are equivalent to an increase in the supply of funds (a shift of the supply curve to the right). The FED buys treasuries with cash from its own account or it issues certificates which trade like cash. As a result, there is an increase in the money supply, and the equilibrium real rate of interest will fall.

16. You are faced with the probability distribution of the HPR on the stock market index fund given in Spreadsheet 5.1 of the text. Suppose the price of a put option on a share of the index fund with exercise price of $110 and time to expiration of 1 year is $12. a. What is the probability distribution of the HPR on the put option?

STOCK PUT Excellent 0.25 $ 131.00 31.00% $ 0.00 100% Good 0.45 114.00 14.00 $ 0.00 100 Poor 0.25 93.25 −6.75 $ 20.25 68.75 Crash 0.05 48.00 52.00 $ 64.00 433.33 Remember that the cost of the index fund is $100 per share, and the cost of the put option is $12.

14. Suppose that the inflation rate is expected to be 3% in the near future. Using the historical data provided in this chapter, what would be your predictions for: a. The T-bill rate?

T-bills: 0.46% real rate + 3% inflation = 3.46%

CFA Problems 1. Given $100,000 to invest, what is the expected risk premium in dollars of investing in equities versus risk-free T-bills (U.S. Treasury bills) based on the following table? Action Probability Expected Return Invest in equities 0.6 $50,000 0.4 −$30,000 Invest in risk-free T-bill 1.0 $ 5,000

The expected dollar return on the investment in equities is $18,000 (0.6 × $50,000 + 0.4 × −$30,000) compared to the $5,000 expected return for T-bills. Therefore, the expected risk premium is $13,000.

14. Suppose that the inflation rate is expected to be 3% in the near future. Using the historical data provided in this chapter, what would be your predictions for: c. The risk premium on the stock market?

The risk premium on stocks remains unchanged. A premium, the difference between two rates, is a real value, unaffected by inflation.

9. Determine the standard deviation of a random variable q with the following probability distribution: Value of q Probability 0 0.25 1 0.25 2 0.50 σq = _____

[0.25 × (0 - 1.25)^2 + 0.25 × (1 - 1.25)^2 + 0.50 × (2 - 1.25)2]^1/2 = 0.8292

6. You are considering the choice between investing $50,000 in a conventional 1-year bank CD offering an interest rate of 5% and a 1-year "Inflation-Plus" CD offering 1.5% per year plus the rate of inflation. b. Can you tell which offers the higher expected return? The expected return depends on the ____

expected rate of inflation over the next year.

13. During a period of severe inflation, a bond offered a nominal HPR of 80% per year. The inflation rate was 70% per year. b. Compare this real HPR to the approximation rreal ≈ rnom − i

r_"nominal" -i=.80-.70=.10≈r_real Clearly, the approximation gives a real HPR that is too high.

18. Consider these long-term investment data: ∙ The price of a 10-year $100 face value zero-coupon inflation-indexed bond is $84.49. ∙ A real-estate property is expected to yield 2% per quarter (nominal) with a SD of the (effective) quarterly rate of 10%. a. Compute the annual rate of return on the real (i.e., inflation-indexed) bond. Total return of the bond is _____

(100/84.49)-1 = 0.1836.

10. The continuously compounded annual return on a stock is normally distributed with a mean of 20% and standard deviation of 30%. With 95.44% confidence, we should expect its actual return in any particular year to be between which pair of values? (Hint: Look again at Figure 5.3.) a. −40.0% and 80.0% b. −30.0% and 80.0% c. −20.6% and 60.6% d. −10.4% and 50.4%

(a) With probability 0.9544, the value of a normally distributed variable will fall within 2 standard deviations of the mean; that is, between -40% and 80%. Simply add and subtract 2 standard deviations to and from the mean.

18. Consider these long-term investment data: ∙ The price of a 10-year $100 face value zero-coupon inflation-indexed bond is $84.49. ∙ A real-estate property is expected to yield 2% per quarter (nominal) with a SD of the (effective) quarterly rate of 10%. b. Compute the continuously compounded annual risk premium on the real-estate investment. The risk-free rate is 3.55% with a cc rate of ln(1+.0355) = ____

.0349, or 3.49%.

18. Consider these long-term investment data: ∙ The price of a 10-year $100 face value zero-coupon inflation-indexed bond is $84.49. ∙ A real-estate property is expected to yield 2% per quarter (nominal) with a SD of the (effective) quarterly rate of 10%. b. Compute the continuously compounded annual risk premium on the real-estate investment. The cc risk premium will equal ____

.0792 - .0349 = .0443, or 4.433%.

11. Using historical risk premiums from Table 5.5 over the 1927-2018 period as your guide, what would be your estimate of the expected annual HPR on the Big/Value portfolio if the current risk-free interest rate is 3%? From Table 5.4, the average risk premium Big/Value for the period 1927-2018 was: ____

11.69% per year.

18. Consider these long-term investment data: ∙ The price of a 10-year $100 face value zero-coupon inflation-indexed bond is $84.49. ∙ A real-estate property is expected to yield 2% per quarter (nominal) with a SD of the (effective) quarterly rate of 10%. a. Compute the annual rate of return on the real (i.e., inflation-indexed) bond. With t = 10, the annual rate on the real bond is (1 + EAR) = ____

= 1.1836^1/10 = 1.69%

7. Suppose your expectations regarding the stock price are as follows: State of the Market Probability Ending Price HPR Boom 0.35 $140 44.5% Normal growth 0.30 110 14.0 Recession 0.35 80 −16.5 Use Equations 5.11 and 5.12 to compute the mean and standard deviation of the HPR on stocks E(r) = _____

= [0.35 × 44.5%] + [0.30 × 14.0%] + [0.35 × (-16.5%)] = 14% s2 = [0.35 × (44.5 - 14)2] + [0.30 × (14 - 14)2] + [0.35 × (-16.5 - 14)2] = 651.175 s = 25.52%

16. You are faced with the probability distribution of the HPR on the stock market index fund given in Spreadsheet 5.1 of the text. Suppose the price of a put option on a share of the index fund with exercise price of $110 and time to expiration of 1 year is $12. c. In what sense does buying the put option constitute a purchase of insurance in this case?

Buying the put option guarantees the investor a minimum HPR of 0.0% regardless of what happens to the stock's price. Thus, it offers insurance against a price decline.

8. Derive the probability distribution of the 1-year HPR on a 30-year U.S. Treasury bond with an 8% coupon if it is currently selling at par and the probability distribution of its yield to maturity a year from now is as follows: State of the Economy Probability YTM Boom 0.20 11.0% Normal growth 0.50 8.0 Recession 0.30 7.0 For simplicity, assume the entire 8% coupon is paid at the end of the year rather than every 6 months. Coupon Interest: Boom = ____ Normal Growth = ____ Recession = _____

Coupon Interest $8.00 8.00 8.00

7. An analyst estimates that a stock has the following probabilities of return depending on the state of the economy: State of Economy Probability Return Good 0.1 15% Normal 0.6 13 Poor 0.3 7 What is the expected return of the stock?

E(r) = (0.1 × 15%) + (0.6 × 13%) + (0.3 × 7%) = 11.4%

Use the following scenario analysis for Stocks X and Y to answer CFA Problems 3 through 5 (round to the nearest percent). Bear Market Normal Market Bull Market Probability 0.2 0.5 0.3 Stock X −20% 18% 50% Stock Y −15% 20% 10% 3. What are the expected rates of return for Stocks X and Y?

E(rx) = [0.2 × (−20%)] + [0.5 × 18%] + [0.3 × 50%] =20% E(ry) = [0.2 × (−15%)] + [0.5 × 20%] + [0.3 × 10%] =10%

17. Suppose the risk-free interest rate is 6% per year. You are contemplating investing $107.55 in a 1-year CD and simultaneously buying a call option on the stock market index fund with an exercise price of $110 and expiration of 1 year. Using the scenario analysis of Spreadsheet 5.1, what is the probability distribution of your dollar return at the end of the year?

Excellent 0.25 $ 114.00 $16.50 $130.50 Good 0.45 114.00 0.00 114.00 Poor 0.25 114.00 0.00 114.00 Crash 0.05 114.00 0.00 114.00

16. You are faced with the probability distribution of the HPR on the stock market index fund given in Spreadsheet 5.1 of the text. Suppose the price of a put option on a share of the index fund with exercise price of $110 and time to expiration of 1 year is $12. b. What is the probability distribution of the HPR on a portfolio consisting of one share of the index fund and a put option?

Excellent 0.25 $ 131.00 17.0% = (131 112)/112 Good 0.45 114.00 1.8 = (114 112)/112 Poor 0.25 113.50 1.3 = (113.50 112)/112 Crash 0.05 112.00 0.0 = (112 112)/112

14. Suppose that the inflation rate is expected to be 3% in the near future. Using the historical data provided in this chapter, what would be your predictions for: b. The expected rate of return on the Big/Value portfolio?

Expected return on Big/Value: 3.46% T-bill rate + 11.69% historical risk premium = 15.15%

5. Use Figure 5.1 in the text to analyze the effect of the following on the level of real interest rates: a. Businesses become more pessimistic about future demand for their products and decide to reduce their capital spending.

If businesses reduce their capital spending, then they are likely to decrease their demand for funds. This will shift the demand curve in Figure 5.1 to the left and reduce the equilibrium real rate of interest.

5. Use Figure 5.1 in the text to analyze the effect of the following on the level of real interest rates: b. Households are induced to save more because of increased uncertainty about their future Social Security benefits.

Increased household saving will shift the supply of funds curve to the right and cause real interest rates to fall.

8. Derive the probability distribution of the 1-year HPR on a 30-year U.S. Treasury bond with an 8% coupon if it is currently selling at par and the probability distribution of its yield to maturity a year from now is as follows: State of the Economy Probability YTM Boom 0.20 11.0% Normal growth 0.50 8.0 Recession 0.30 7.0 For simplicity, assume the entire 8% coupon is paid at the end of the year rather than every 6 months. Price: Boom = ____ Normal Growth = ____ Recession = _____

Price $ 74.05 100.00 112.28

15. An economy is making a rapid recovery from steep recession, and businesses foresee a need for large amounts of capital investment. Why would this development affect real interest rates?

Real interest rates are expected to rise. The investment activity will shift the demand for funds curve (in Figure 5.1) to the right. Therefore the equilibrium real interest rate will increase.

6. Probabilities for three states of the economy and probabilities for the returns on a particular stock in each state are shown in the table below. What is the probability that the economy will be neutral and the stock will experience poor performance?

The probability that the economy will be neutral is 0.50, or 50%. Given a neutral economy, the stock will experience poor performance 30% of the time. The probability of both poor stock performance and a neutral economy is therefore: 0.30 × 0.50 = 0.15 = 15%

1. The Fisher equation tells us that the real interest rate approximately equals the nominal rate minus the inflation rate. Suppose the inflation rate increases from 3% to 5%. Does the Fisher equation imply that this increase will result in a fall in the real rate of interest? Explain. On the other hand, it is possible that an increase in the expected inflation rate would be accompanied by ____

a change in the real rate of interest.

2. You've just stumbled on a new dataset that enables you to compute historical rates of return on U.S. stocks all the way back to 1880. What are the advantages and disadvantages in using these data to help estimate the expected rate of return on U.S. stocks over the coming year? If we assume that the distribution of returns remains reasonably stable over the entire history, then _____ increases the precision of the estimate of the expected rate of return; this is a consequence of the fact that the standard error decreases as the sample size increases

a longer sample period (i.e., a larger sample)

6. You are considering the choice between investing $50,000 in a conventional 1-year bank CD offering an interest rate of 5% and a 1-year "Inflation-Plus" CD offering 1.5% per year plus the rate of inflation. c. If you expect the rate of inflation to be 3% over the next year, which is the better investment? Why? If you expect the rate of inflation to be 3% over the next year, then the _____ offers you an expected real rate of return of 2%, which is 0.5% higher than the real rate on the inflation-protected CD.

conventional CD

6. You are considering the choice between investing $50,000 in a conventional 1-year bank CD offering an interest rate of 5% and a 1-year "Inflation-Plus" CD offering 1.5% per year plus the rate of inflation. d. If we observe a risk-free nominal interest rate of 5% per year and a risk-free real rate of 1.5% on inflation-indexed bonds, can we infer that the market's expected rate of inflation is 3.5% per year? We cannot assume that the entire difference between the risk-free nominal rate (on conventional CDs) of 5% and the real risk-free rate (on inflation-protected CDs) of 1.5% is the ____

expected rate of inflation.

2. You've just stumbled on a new dataset that enables you to compute historical rates of return on U.S. stocks all the way back to 1880. What are the advantages and disadvantages in using these data to help estimate the expected rate of return on U.S. stocks over the coming year? However, if we assume that the mean of the distribution of returns is changing over time but we are not in a position to determine the nature of this change, then the _____ must be estimated from a more recent part of the historical period.

expected return

6. You are considering the choice between investing $50,000 in a conventional 1-year bank CD offering an interest rate of 5% and a 1-year "Inflation-Plus" CD offering 1.5% per year plus the rate of inflation. a. Which is the safer investment? The "Inflation-Plus" CD is the safer investment because ____

it guarantees the purchasing power of the investment.

18. Consider these long-term investment data: ∙ The price of a 10-year $100 face value zero-coupon inflation-indexed bond is $84.49. ∙ A real-estate property is expected to yield 2% per quarter (nominal) with a SD of the (effective) quarterly rate of 10%. b. Compute the continuously compounded annual risk premium on the real-estate investment. The equivalent continuously compounding (cc) rate is _____

ln(1+.0824) = .0792, or 7.92%.

1. The Fisher equation tells us that the real interest rate approximately equals the nominal rate minus the inflation rate. Suppose the inflation rate increases from 3% to 5%. Does the Fisher equation imply that this increase will result in a fall in the real rate of interest? Explain. While it is conceivable that the nominal interest rate could remain constant as the inflation rate increased, implying that the real rate decreased as inflation increased, this is ____

not a likely scenario

7. Suppose your expectations regarding the stock price are as follows: State of the Market Probability Ending Price HPR Boom 0.35 $140 44.5% Normal growth 0.30 110 14.0 Recession 0.35 80 −16.5 Use Equations 5.11 and 5.12 to compute the mean and standard deviation of the HPR on stocks The mean is unchanged, but the standard deviation has increased, as the _____ have increased

probabilities of the high and low returns

6. You are considering the choice between investing $50,000 in a conventional 1-year bank CD offering an interest rate of 5% and a 1-year "Inflation-Plus" CD offering 1.5% per year plus the rate of inflation. a. Which is the safer investment? Using the approximation that the real rate equals the nominal rate minus the inflation rate, the CD provides a _____ regardless of the inflation rate.

real rate of 1.5%

6. You are considering the choice between investing $50,000 in a conventional 1-year bank CD offering an interest rate of 5% and a 1-year "Inflation-Plus" CD offering 1.5% per year plus the rate of inflation. d. If we observe a risk-free nominal interest rate of 5% per year and a risk-free real rate of 1.5% on inflation-indexed bonds, can we infer that the market's expected rate of inflation is 3.5% per year?' Part of the difference is probably a ____ associated with the uncertainty surrounding the real rate of return on the conventional CDs.

risk premium

6. You are considering the choice between investing $50,000 in a conventional 1-year bank CD offering an interest rate of 5% and a 1-year "Inflation-Plus" CD offering 1.5% per year plus the rate of inflation. c. If you expect the rate of inflation to be 3% over the next year, which is the better investment? Why? But unless you know that inflation will be 3% with certainty, the conventional CD is also ____

riskier.

Use the following scenario analysis for Stocks X and Y to answer CFA Problems 3 through 5 (round to the nearest percent). Bear Market Normal Market Bull Market Probability 0.2 0.5 0.3 Stock X −20% 18% 50% Stock Y −15% 20% 10% 4. What are the standard deviations of returns on Stocks X and Y?

sX 2 = [0.2 × (- 20 - 20)2] + [0.5 × (18 - 20)2] + [0.3 × (50 - 20)2] = 592 sX = 24.33% sY 2 = [0.2 × (- 15 - 10)2] + [0.5 × (20 - 10)2] + [0.3 × (10 - 10)2] = 175 sY = 13.23%

6. You are considering the choice between investing $50,000 in a conventional 1-year bank CD offering an interest rate of 5% and a 1-year "Inflation-Plus" CD offering 1.5% per year plus the rate of inflation. d. If we observe a risk-free nominal interest rate of 5% per year and a risk-free real rate of 1.5% on inflation-indexed bonds, can we infer that the market's expected rate of inflation is 3.5% per year? This implies ____

that the expected rate of inflation is less than 3.5% per year.

1. The Fisher equation tells us that the real interest rate approximately equals the nominal rate minus the inflation rate. Suppose the inflation rate increases from 3% to 5%. Does the Fisher equation imply that this increase will result in a fall in the real rate of interest? Explain. The Fisher equation predicts ____

that the nominal rate will equal the equilibrium real rate plus the expected inflation rate.

6. You are considering the choice between investing $50,000 in a conventional 1-year bank CD offering an interest rate of 5% and a 1-year "Inflation-Plus" CD offering 1.5% per year plus the rate of inflation. b. Can you tell which offers the higher expected return? If the expected rate of inflation is less than 3.5% then ____

the conventional CD offers a higher real return than the inflation-plus CD

2. You've just stumbled on a new dataset that enables you to compute historical rates of return on U.S. stocks all the way back to 1880. What are the advantages and disadvantages in using these data to help estimate the expected rate of return on U.S. stocks over the coming year? In this scenario, we must determine how far back, historically, to go in selecting the relevant sample. Here, it is likely to be disadvantageous to use ____

the entire data set back to 1880.

6. You are considering the choice between investing $50,000 in a conventional 1-year bank CD offering an interest rate of 5% and a 1-year "Inflation-Plus" CD offering 1.5% per year plus the rate of inflation. b. Can you tell which offers the higher expected return? if the expected rate of inflation is greater than 3.5%, then ____

the opposite is true.

6. You are considering the choice between investing $50,000 in a conventional 1-year bank CD offering an interest rate of 5% and a 1-year "Inflation-Plus" CD offering 1.5% per year plus the rate of inflation. c. If you expect the rate of inflation to be 3% over the next year, which is the better investment? Why? The question of which is the better investment then depends on _____

your attitude towards risk versus return.