Investments Ch. 5-7/11-13 (final exam) Practice Problems

A pension plan is obligated to make disbursements of $1 million, $2 million, and $1 million at the end of each of the next three years, respectively. Find the duration of the plan's obligations if the interest rate is 10% annually.

(Yrs) - Paymnt - Paymnt Discount@10% - Weight - Column (1×4) 1 1 0.9091 0.2744 0.2744 2 2 1.6529 0.4989 0.9977 3 1 0.7513 0.2267 0.6803 Column Sum: 3.3133 1.0000 1.9524 Duration = 1.9524 years

Assume that you manage a risky portfolio with an expected rate of return of 17% and a standard deviation of 27%. The T-bill rate is 7% 1) Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. a. What are the expected return and standard deviation of your client's portfolio? b. Suppose your risky portfolio includes the following investments in the given proportions. What are the investment proportions of each stock in your client's overall portfolio, including the position in T-bills? Stock A: 27% - Stock B: 33% - Stock C: 40% c. What is the Sharpe ratio (S) of your risky portfolio and your client's overall portfolio? d. Draw the CAL of your portfolio on an expected return/standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund's CAL. 2) Suppose the same client in the previous problem decides to invest in your risky portfolio a proportion (y) of his total investment budget so that his overall portfolio will have an expected rate of return of 15%. a. What is the proportion y? b. What are your client's investment proportions in your three stocks and in T-bills? c. What is the standard deviation of the rate of return on your client's portfolio? 3) Suppose the same client as in the previous problem prefers to invest in your portfolio a proportion (y) that maximizes the expected return on the overall portfolio subject to the constraint that the overall portfolio's standard deviation will not exceed 20%. a. What is the investment proportion, y? b. What is the expected rate of return on the overall portfolio? 4) You estimate that a passive portfolio invested to mimic the S&P 500 stock index provides an expected rate of return of 13% with a standard deviation of 25% a. Draw the CML and your fund's CAL on an expected return/standard deviation diagram. b. What is the slope of the CML? c. Characterize in one short paragraph the advantage of your fund over the index fund.

1) a. E(rC) = 0.14 or 14% per year Oc = 0.189 or 18.9% per year b. Security Investment Proportions T-Bills 30.0% Stock A 0.7 x 27% = 18.9% Stock B 0.7 x 33% = 23.1% Stock C 0.7 x 40% = 28.0% c. For the risky portfolio: = 0.3704 For the client's overall portfolio: = 0.3704 d. ________ 2) a. y = 0.8 b. Security Investment Proportions T-Bills 20.0% Stock A 0.8 x 27% = 21.6% Stock B 0.8 x 33% = 26.4% Stock C 0.8 x 40% = 32.0% c. Oc = y x P = 0.8 x 0.27 = 0.216 or 21.6% per year 3) a. Standard deviation of the complete portfolio = OC = y x 0.27 If the client wants the standard deviation to be equal or less than 20%, then: y = (0.20/0.27) = 0.7407 = 74.07% He should invest, at most, 74.07% in the risky fund. b. E(rC) = rf + y x [E(rP) - rf] = 0.07 + 0.7407 x 0.10 = 0.1441 or 14.41% 4) a. __________ b. Slope of the CML = 0.24 c. our fund allows an investor to achieve a higher expected rate of return for any given standard deviation than would a passive strategy, i.e., a higher expected return for any given level of risk.

The standard deviation of return on investment A is 24%, while the standard deviation of return on investment B is 19%. If the correlation coefficient between the returns on A and B is −0.263, the covariance of returns on A and B is __________

Covariance = −0.263(0.24)(0.19) = −0.0120

Here are data on two companies. The T-bill rate is 4% and the market risk premium is 6% Company $1 Discount Store Everything $5 Forecast return 12% 11% Standard deviation of returns 8% 10% Beta 1.5 1.0 What should be the expected rate of return for each company, according to the capital asset pricing model (CAPM)?

E(r$1 Discount Store) = .04 + 1.5 x 0.6 = 0.13 = 13% E(rEverything $5) = .04 + 1.0 x .06 = .11 = 10%

Suppose there are two independent economic factors, M1 and M2. The risk-free rate is 7%, and all stocks have independent firm-specific components with a standard deviation of 50%. Portfolios A and B are both well diversified. Portfolio Beta on M1 Beta on M2 Expected Return (%) A 1.8 2.1 40 B 2.0 −0.5 10 What is the expected return-beta relationship in this economy?

E(rP) = 7% + 4.47BP1 + 11.86BP2

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.5%. The probability distributions of the risky funds are: Expected Return Standard deviation Stock fund (S) 15% 32% Bond fund (B) 9% 23% The correlation between the fund returns is 0.15 a. Tabulate and draw the investment opportunity set of the two risky funds. Use investment proportions for the stock fund of 0% to 100% in increments of 20%. What expected return and standard deviation does your graph show for the minimum-variance portfolio? c. What is the Sharpe ratio of the best feasible CAL? d. Suppose now that your portfolio must yield an expected return of 12% and be efficient, that is, on the best feasible CAL. What is the standard deviation of your portfolio? What is the proportion invested in the T-bill fund and each of the two risky funds?

E(rS) = 15%, E(rB) = 9%, OS = 32%, OB = 23%, Corr = 0.15, rf = 5.5% The minimum-variance portfolio proportions are: wMin(S) = OB2 － Cov(rS, rB)/OS2 + OB2 － 2Cov(rS, rB) = 0.0529 － 0.01100/.1024 + 0.0529 － (2 ×0.0110) = .3142 wMin(B) = 1 - .3142 = .6858 E(rMin) = ( .3142 15%) + ( .6858 9%) = 10.89% OMin = [wS2S2 + wB2B2 + 2 wS wB Cov(rS, rB)]1/2 = [( .31422 x 0.1024) + ( .68582 0.0529) + (2 x .3142 x .6858 x 0.0110)]^1/2 = 19.94% c. (E(rP) － rf)/(OP) = (12.88% － 5.50%)/(23.34%) = .3162 d. The equation for the CAL is: E(rC) = rf + (E(rP) － rf)/(OP)OC = 5.50% + .3162OC Setting E(rC) equal to 12% yields a standard deviation of 20.56%. Setting E(rC) = 12% --> y = .8808 (88.08% in the risky portfolio) 1 - y = .1192 (11.92% in T-bills)

Currently, the term structure is as follows: One-year bonds yield 7%, two-year zero-coupon bonds yield 8%, three-year- and longer-maturity zero-coupon bonds all yield 9%. You are choosing between one-, two-, and three-year maturity bonds all paying annual coupons of 8%. Which bond will provide the highest rate of return if at year-end the yield curve will be flat at 9%?

Maturity One year Two years Three years YTM@beginning of yr 7.00% 8.00% 9.00% Beginning of yr price $1,009.35 $1,000.00 $974.69 End of yr price(@9% YTM) $1,000.00 $990.83 $982.41 Capital gain −$ 9.35 −$ 9.17 $7.72 Coupon $80.00 $80.00 $80.00 One year total $ return $70.65 $70.83. $87.72 One yr total rate of return 7.00% 7.08% 9.00%

A bond has a par value of $1,000, a time to maturity of 10 years, and a coupon rate of 8% with interest paid annually. If the current market price is $800, what will be the percentage capital gain of this bond over the next year if its yield to maturity remains unchanged?

PV = -800, FV = 1,000, n = 10, PMT = 80. The YTM is 11.46%. The new price will be 811.70. Thus, the capital gain is $11.70. the percentage capital gain: 11.70 / 800

A coupon bond paying semiannual interest is reported as having an ask price of 117% of its $1,000 par value. If the last interest payment was made one month ago and the coupon rate is 6%, what is the invoice price of the bond. The coupon period has 182 days.

Semi-annual coupon = $1,000 x 6% x 0.5 = $30. Accrued Interest = (Annual Coupon Payment / 2) x (Days since Last Coupon Payment / Days Separating Coupon Payment) = $30 x (30/182) = $4.945 At a price of 117, the invoice price is: $1,170 + $4.945 = $1,174.95

The stock of Business Adventures sells for $40 a share. Its likely dividend payout and end-of-year price depend on the state of the economy by the end of the year as follows: Dividend Stock Price Boom $2.00 $50 Normal Economy 1.00 43 Recession 0.50 34 Calculate the expected holding-period return and standard deviation of the holding-period return. All three scenarios are equally likely.

The holding period returns for the three scenarios are: Boom: 30% Normal: 10% Recession: -13.75% E(HPR) = 0.0875 or 8.75% Var(HPR) = 0.031979 SD(r) = √0.031979 = 0.1788 or 17.88%

A nine-year bond paying coupons annually has a yield of 10% and a duration of 7.194 years. If the bond's yield changes by 50 basis points, what is the percentage change in the bond's price?

The percentage change in the bond price is: 𝛥𝑃/𝑃 = - Duration x (𝛥𝑦/1+𝑦) =−7.194 x (0.0050/1.10) =−0.0327 or a 3.27% decline

A newly issued 20-year-maturity, zero-coupon bond is issued with a yield to maturity of 8% and face value $1,000. Find the imputed interest income in the first, second, and last years of the bond's life.

The price schedule is as follows, (assuming annual compounding): Yr-Remaining Maturity-Constant Yield Value-Imputed Interest 0 20 $214.55 1 19 231.71 231.71 - 214.55 = 17.16 2 18 250.25 250.25 - 231.71 = 18.54 19 1 925.93 20 0 1,000 1,000 - 925.93 = 74.07 *constant yield value = (1,000/1.08^T)

A bond with a coupon rate of 7% makes semiannual coupon payments on January 15 and July 15 of each year. The Wall Street Journal reports the ask price for the bond on January 30 at 100.125. What is the invoice price of the bond? The coupon period has 182 days.

The reported bond price is $1,001.25 15 days have passed since the last semiannual coupon was paid, so there is an accrued interest, which can be calculated as: Accrued Interest = (Annual Coupon Payment/2) x (Days since Last Coupon Payment/Days Separating Coupon Payment) = $35 x (15/182) = $2.8846 The invoice price is the reported price plus accrued interest: $1,001.25 + $2.8846 = $1,004.13

The standard deviation of the market index portfolio is 20%. Stock A has a beta of 1.5 and a residual standard deviation of 30%. a. What would make for a larger increase in the stock's variance: an increase of 0.15 in its beta or an increase of 3% (from 30% to 33%) in its residual standard deviation?

Total variance = Systematic variance + Residual variance = β2×Var(rM) + Var(e) When β = 1.5 and σ(e) = .3, variance = 1.52 × .22 + .32 = .18. In the other scenarios: sM s(e) b Total Variance. Correlation Coefficient 0.2 0.3. 1.65 0.1989 0.7399 0.2 0.33 1.5 0.1989 0.6727 Both will have the same impact. Total variance will increase from .18 to .1989.

Find the duration of a 6% coupon bond making annual coupon payments if it has three years until maturity and a yield to maturity of 6%. What is the duration if the yield to maturity is 10%?

When YTM = 6%, the duration is 2.8334. (Years)-Paymnt-Paymnt Discount@6%-Weight-Column (1)×(4) 1 60 56.60 0.0566 0.0566 2 60 53.40 0.0534 0.1068 3 1060 890.00 0.8900 2.6700 Column Sum: 1000.00 1.0000 2.8334 When YTM = 10%, the duration is 2.8238 (Years)-Paymnt-Paymnt Discount@10%-Weight-Column (1)x(4) 1 60 54.55 0.0606 0.0606 2 60 49.59 0.0551 0.1101 3 1060 796.39 0.8844 2.6531 Column Sum: 900.53 1.0000 2.8238 When the yield to maturity increases, the duration decreases.

The following figure shows plots of monthly rates of return and the stock market for two stocks.

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Stock XYZ has an expected return of 12% and β = 1. Stock ABC is expected to return 13% with a beta of 1.5. The market's expected return is 11% and rf = 5%. According to the CAPM, which stock is a better buy? What is the alpha of each stock? Plot the SML and the two stocks. Show the alphas of each on the graph.

a(XYZ) = 12% - {5% + 1.0(11% - 5%)} =1% a(ABC) = 13% - {5% + 1.511% - 5%)} =1%

Consider the two (excess return) index-model regression results for stocks A and B. The risk-free rate over the period was 6%, and the market's average return was 14%. Performance is measured using an index model regression on excess returns. Stock A Stock B Index model regression estimates 1% + 1.2(rM - rf) 2% + 0.8(rM - rf) R-square 0.576 0.436 Residual standard deviation, σ(e) 10.3% 19.1% Standard deviation of excess returns 21.6% 24.9% a. Calculate the following statistics for each stock: ▪ Alpha ▪ Information ratio ▪ Sharpe ratio ▪ Treynor's measure b. Which stock is the best choice under the following circumstances? ▪ This is the only risky asset to be held by the investor. ▪ This stock will be mixed with the rest of the investor's portfolio, currently composed solely of holdings in the market-index fund. ▪ This is one of many stocks that the investor is analyzing to form an actively managed stock portfolio

a. Stock A Stock B Alpha = regression intercept 1.0% 2.0% Information ratio = [αp / σ(ep)] 0.0971 0.1047 *Sharpe measure = (rp-rf / σp) 0.4907 0.3373 †Treynor measure = (rp-rf / βp) 8.833 10.500 b. (i) If this is the only risky asset held by the investor, then Sharpe's measure is the appropriate measure. Since the Sharpe measure is higher for Stock A, then A is the best choice. (ii) If the stock is mixed with the market index fund, then the contribution to the overall Sharpe measure is determined by the appraisal ratio; therefore, Stock B is preferred. (iii) If the stock is one of many stocks, then Treynor's measure is the appropriate measure, and Stock B is preferred.

A 30-year maturity bond has a 7% coupon rate, paid annually. It sells today for $867.42. A 20-year maturity bond has a 6.5% coupon rate, also paid annually. It sells today for $879.50. A bond market analyst forecasts that in five years, 25-year maturity bonds will sell at yields to maturity of 8% and that 15-year maturity bonds will sell at yields of 7.5%. Because the yield curve is upward-sloping, the analyst believes that coupons can be invested in short-term securities at a rate of 6%. a. Calculate the expected rate of return of the 30-year bond over the five-year period. b. What is the expected return of the 20-year bond?

a. The maturity of the 30-year bond will fall to 25 years, and the yield is forecast to be 8%. Therefore, the price forecast for the bond in five years is: $893.25 [N = 25; I/Y = 8; FV = 1,000; PMT = 70] At a 6% interest rate, the five coupon payments will accumulate to $394.60 (FV) after five years. [N =5; I/Y = 6; PV = 0; PMT = 70] Therefore, total proceeds will be: $394.60 + $893.25 = $1,287.85 The five-year return is therefore: ($1,287.85/867.42) - 1 = 1.48469 - 1 = 48.469% The annual rate of return is: (1.48469)(1/5) -1 = 0.0822 = 8.22% b. The maturity of the 20-year bond will fall to 15 years, and its yield is forecast to be 7.5%. Therefore, the price forecast for the bond in five years is: $911.73 [N = 15; I/Y = 7.5; FV = 1000; PMT = 65] At a 6% interest rate, the five coupon payments will accumulate to $366.41 after five years. [N = 15; I/Y = 6; PV = 0; PMT = 65] Therefore, total proceeds will be: $366.41 + $911.73 = $1,278.14 The five-year return is therefore: ($1,278.14/$879.50) - 1 = 1.45326 - 1 = 45.326% The annual rate of return is: 1.45326(1/5) - 1 = 0.0776 = 7.76% Conclusion: The 30-year bond offers the higher expected return

Abigail Grace has a $900,000 fully diversified portfolio. She subsequently inherits ABC Company common stock worth $100,000. Her financial adviser provided her with the following estimates: Risk and Return Characteristics Expected Monthly Returns Standard Dev. of Monthly Returns Original Portfolio 0.67% 2.37% ABC Company 1.25 2.95 The correlation coefficient of ABC stock returns with the original portfolio returns is .40. a. The inheritance changes Grace's overall portfolio and she is deciding whether to keep the ABC stock. Assuming Grace keeps the ABC stock, calculate the: 1) Expected return of her new portfolio, which includes the ABC stock. 2) Covariance of ABC stock returns with the original portfolio returns. 3) Standard deviation of her new portfolio, which includes the ABC stock. b. If Grace sells the ABC stock, she will invest the proceeds in risk-free government securities yielding 0.42% monthly. Assuming Grace sells the ABC stock and replaces it with the government securities, calculate the: 1)Expected return of her new portfolio, which includes the government securities. 2) Covariance of the government security returns with the original portfolio returns. 3) Standard deviation of her new portfolio, which includes the government securities. c. Determine whether the beta of her new portfolio, which includes the government securities, will be higher or lower than the beta of her original portfolio.

a. 1) E(rNP) = wOP E(rOP ) + wABC E(rABC ) = ( .9 x .67%) + ( .1 x 1.25%) = .7280% 2) CovOP , ABC = CorrOP , ABC x OOP x OABC = .40 x 2.37% x 2.95% = 0.00027966 3) ONP = [wOP 2 x OP 2 + wABC 2 OABC 2 + 2 wOP wABC (CovOP , ABC)]^1/2 = [( .92 x 0. 02372) + ( .12 x 0.02952) + (2 x .9 x .1 x 0.00027966)]^1/2 = 2.2672% b. 1) E(rNP) = wOP E(rOP ) + wGS E(rGS ) = ( .9 x .67) + ( .1 x .42) = .6450% 2) CovOP , GS = CorrOP , GS x OOP x OGS = 0 x 2.37 x 0 = 0 3) ONP = [wOP 2 OOP 2 + wGS 2 OGS 2 + 2 wOP wGS (CovOP , GS)]^1/2 = [( .92 x 2.372) + ( .12 x 0) + (2 x .9 x .1 x 0)]^1/2 = 2.1330% c. Adding the risk-free government securities would result in a lower beta for the new portfolio. The new portfolio beta will be a weighted average of the individual security betas in the portfolio; the presence of the risk-free securities would lower that weighted average.

Kelli Blakely is a portfolio manager for the Miranda Fund, a core large-cap equity fund. The benchmark for performance measurement purposes is the S&P 500. Although the Miranda portfolio generally mirrors the asset class and sector weightings of the S&P, Blakely is allowed a significant amount of leeway in managing the fund. Blakely was able to produce exceptional returns last year (as outlined in the table below) through her market timing and security selection skills. At the outset of the year, she became extremely concerned that the combination of a weak economy and geopolitical uncertainties would negatively impact the market. Taking a bold step, she changed her market allocation. For the entire year her asset class exposures averaged 50% in stocks and 50% in cash. The S&P's allocation between stocks and cash during the period was a constant 97% and 3%, respectively. The risk-free rate of return was 2%. One-Year Trailing Return Miranda Fund S&P 500 Return 10.2% -22.5% Standard deviation 37% 44% Beta 1.10 1.00 a. What are the Sharpe ratios for the Miranda Fund and the S&P 500? b. What are the M2 measures for Miranda and the S&P 500? c. What is the Treynor's measure for the Miranda Fund and the S&P 500? d. What is the Jensen measure for the Miranda Fund?

a. (rp-rf / σp) --> Smiranda = (.102-.02 / .37) =.2216 Ss&p = (-.225-.02/.44)=-.5568 b. 𝑀^2 =𝑅̄𝑃∗ −𝑅̄𝑀 =𝜎𝑀(𝑆𝑃 −𝑆𝑀) = 0.44(0.2216- (-0.5568)) =0.3425 c. (rp-rf / βp) --> Tmiranda = (.102-.02 / 1.10) =.0745 Ts&p = (-.225-.02 / 1.00) = -.245 d. αp = rp - [rf + βp(rm - rf)] = 0.102 - [0.02 + 1.10x (-0.225 - 0.02)] = .3515 = 35.15%

Suppose the yield on short-term government securities (perceived to be risk-free) is about 4%. Suppose also that the expected return required by the market for a portfolio with a beta of 1 is 12%. According to the capital asset pricing model: a. What is the expected return on the market portfolio? b. What would be the expected return on a zero-beta stock? c. Suppose you consider buying a share of stock at a price of $40. The stock is expected to pay a dividend of $3 next year and to sell then for $41. The stock risk has been evaluated at β = −.5. Is the stock overpriced or underpriced?

a. 12% b. 4% c. 10% > 0%: the expected return exceeds the fair return, the stock must be under-priced.

XYZ stock price and dividend history are as follows: YEAR - BEGINNING-OF-YEAR - PRICE DIVIDEND PAID AT YEAR-END 2018 - $100 - $4 2019 - 110 - 4 2020 - 90 - 4 2021 - 95 - 4 An investor buys three shares of XYZ at the beginning of 2018, buys another two shares at the beginning of 2019, sells one share at the beginning of 2020, and sells all four remaining shares at the beginning of 2021. a. What are the arithmetic and geometric average time-weighted rates of return for the investor? b. What is the dollar-weighted rate of return? (Hint: Carefully prepare a chart of cash flows for the four dates corresponding to the turns of the year for January 1, 2018, to January 1, 2021. If your calculator cannot calculate internal rate of return, you will have to use a spreadsheet)

a. Arithmetic mean: [0.14 + (-0.1455) + 0.10]/3] = 0.0315 or 3.15% Geometric mean: √(1 + 0.14) [1 + (-0.1455)] (1 + 0.10)3 - 1 = 0.0233 or 2.33% b. 0 = -$300 + -$2081+ IRR + $110(1+ IRR)2 + $396(1+ IRR)3 Dollar-weighted return = Internal rate of return = -0.1661%

Consider the following table, which gives a security analyst's expected return on two stocks and the market index in two scenarios: Scenario Probability Market Return Aggressive Stock Defensive Stock 1 0.5 5% 2% 3.5% 2 0.5 20 32 14 a. What are the betas of the two stocks? b. What is the expected rate of return on each stock? c. If the T-bill rate is 8%, draw the SML for this economy. d. Plot the two securities on the SML graph. What are the alphas of each? e. What hurdle rate should be used by the management of the aggressive firm for a project with the risk characteristics of the defensive firm's stock?

a. Ba = (2-32) / (5-20) = 2.00 Bd = (3.5-14) / (5-20) = 0.70 b. E(rA) = 0.5 x (2% + 32%) = 17% E(rD) = 0.5 x (3.5% + 14%) = 8.75% c. ______________ d. -2.4% e. The hurdle rate is determined by the project beta (i.e., 0.7), not by the firm's beta. The correct discount rate is therefore 11.15%, the fair rate of return on stock D.

Which security has a higher effective annual interest rate? a. A three-month T-bill with face value of $100,000 currently selling at $97,645. b. A coupon bond selling at par and paying a 10% coupon semiannually.

a. Effective annual rate on a three-month T-bill: ( $100,000 / $97,645 )^4 - 1 = (1.02412)^4 - 1 = 0.1000 = 10% b. Effective annual interest rate on coupon bond paying 5% semiannually: (1 + 0.05)^2 - 1 = 0.1025 = 10.25% Therefore, the coupon bond has the higher effective annual interest rate.

A 20-year maturity bond with par value $1,000 makes semiannual coupon payments at a coupon rate of 8%. Find the bond equivalent and effective annual yield to maturity of the bond if the bond price is: a. $950 b. $1,000 c. $1,050

a. Effective annual yield to maturity = (1.0426)2 - 1 = 0.0870 = 8.70% b. Effective annual yield to maturity = (1.04)2 - 1 = 0.0816 = 8.16% c. Effective annual yield to maturity = (1.0376)2 - 1 = 0.0766 = 7.66%

You manage an equity fund with an expected risk premium of 10% and a standard deviation of 14%. The rate on Treasury bills is 6%. Your client chooses to invest $60,000 of her portfolio in your equity fund and $40,000 in a T-bill money market fund. a. What are the expected return and standard deviation of your client's portfolio? b. What is the reward-to-volatility (Sharpe) ratio for the equity fund in the previous problem?

a. Expected return of client's overall portfolio = (0.6 x 16%) + (0.4 x 6%) = 12% Standard deviation of client's overall portfolio = 0.6 x 14% = 8.4% b. 0.7143

The yield curve for default-free zero-coupon bonds is currently as follows: Maturity(years) YTM 1 10% 2 11% 3 12% a. What are the implied one-year forward rates? b. Assume that the pure expectations hypothesis of the term structure is correct. If market expectations are accurate, what will be the yield to maturity on one-year zero-coupon bonds next year? c. What will be the yield to maturity on two-year zeros? d. If you purchase a two-year zero-coupon bond now, what is the expected total rate of return over the next year? (Hint: Compute the current and expected future prices.) Ignore taxes. e. What if you purchase a three-year zero-coupon bond?

a. Maturity (years) YTM Forward rate 1 10.0% 2 11.0% 12.01% 3 12.0% 14.03% Year 2 Forward Rate: (1 + 11%)^2/(1 + 10%) - 1 = 0.1201 = 12.01% Year 3 Forward Rate: (1+12%)^3/(1+11%)^2 - 1 = 0.1403 = 14.03% b. ------- c. forward rates derived in part (a): Maturity(years) Price YTM 1 $892.78 [ = 1,000/1.1201] 12.01% 2 $782.93 [ = 1,000/(1.1201 1.1403)]. 13.02% d. ------- e. Maturity(years) YTM Forward rate Price(for part c) 1 10.0% $909.09 2 11.0% 12.01% $811.62 3 12.0% 14.03% $711.78 Year 1 Price: 1,000/(1 + 10%) = 909.09 Year 2 Price: 1,000/(1 + 11%)2 = 811.62 Year 3 Price: 1000/(1 + 12%)3 = 711.78 Maturity(years) Price YTM 1 $892.78 [ = 1,000/1.1201] 12.01% 2 $782.93 [ = 1,000/(1.1201 1.1403)] 13.02% Next year, the two-year zero will be a one-year zero, and it will therefore sell at: $1000/1.1201 = $892.78 Similarly, the current three-year zero will be a two-year zero, and it will sell for: $782.93 Expected total rate of return: Two-Year Bond: ($892.78 / $811.62) - 1 = 0.1000 = 10.00% Three-Year Bond: ($782.93 / $711.78) - 1 = 0.1000 = 10.00%

Assume the risk-free rate is 8% and the expected rate of return on the market is 18%. Use the SML of the simple (one-factor) CAPM to answer these questions. a. A share of stock is now selling for $100. It will pay a dividend of $9 per share at the end of the year. Its beta is 1. What must investors expect the stock to sell for at the end of the year? b. I am buying a firm with an expected perpetual cash flow of $1,000 but am unsure of its risk. If I think the beta of the firm is 0, when the beta is really 1, how much more will I offer for the firm than it is truly worth? c. A stock has an expected return of 6%. What is its beta?

a. P1 = $109 b. If beta is zero, the cash flow should be discounted at the risk-free rate, 8%: PV = $1,000/0.08 = $12,500 If, however, beta is actually equal to 1, the investment should yield 18%, and the price paid for the firm should be: PV = $1,000/0.18 = $5,555.56 The difference ($6,944.44) is the amount you will overpay if you erroneously assume that beta is zero rather than 1. c. -0.2

A 30-year maturity, 6% coupon bond paying coupons semiannually is callable in five years at a call price of $1,100. The bond currently sells at a yield to maturity of 5% (2.5% per half-year). a. What is the yield to call? b. What is the yield to call if the call price is only $1,050? c. What is the yield to call if the call price is $1,100 but the bond can be called in two years instead of five years?

a. PV = $1,154.5433 n = 10; PV = -1,154.54; FV = 1,100; PMT = 30 Therefore, yield to call is 2.1703% semiannually, 4.3407% annually. b. n = 10; PV = -1,154.54; FV = 1,050; PMT = 30 Therefore, yield to call is 1.7625% semiannually, 3.5249% annually. c. n = 4; PV = -1,154.54; FV = 1,100; PMT = 30 Therefore, yield to call is 1.4426% semiannually, 2.8852% annually.

Consider a bond paying a coupon rate of 10% per year semiannually when the market interest rate is only 4% per half-year. The bond has three years until maturity. a. Find the bond's price today and six months from now after the next coupon is paid. b. What is the total rate of return on the bond?

a. The bond pays $50 every six months. Use the following inputs: n = 6, FV = 1,000, I/Y= 4, PMT = 50, Solve for PV Current price: [$50 x Annuity factor(4%, 6)] + [$1000 x PV factor(4%, 6)] = $1,052.42 Assuming the market interest rate remains 4% per half year, price six months from now. Use the following inputs: n = 5, FV = 1,000, I/Y= 4, PMT = 50, Solve for PV [$50 x Annuity factor(4%, 5)] + [$1000 x PV factor(4%, 5)] = $1,044.52 b. Rate of Return = [$50 + ($1,044.52－ $1,052.42)] / ($1,052.42) = ($50 － $7.90) / ($1,052.42) = 0.0400 = 4.00% per six months.

Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $50,000 or $150,000, with equal probabilities of 0.5. The alternative riskless investment in T-bills pays 5%. a. If you require a risk premium of 10%, how much will you be willing to pay for the portfolio? b. Suppose the portfolio can be purchased for the amount you found in (a). What will the expected rate of return on the portfolio be? c. Now suppose you require a risk premium of 15%. What is the price you will be willing to pay now?

a. The expected cash flow is: (0.5 x $50,000) + (0.5 x $150,000) = $100,000 X x (1 + 0.15) = $100,000, we get X = $86,957 b. E(r): 15% The portfolio price is set to equate the expected return with the required rate of return. c. X x (1 + 0.20) = $100, 000 ----> X = $83,333

The yield to maturity on one-year zero-coupon bonds is 8%. The yield to maturity on two-year zero-coupon bonds is 9%. a. What is the forward rate of interest for the second year? b. If you believe in the expectations hypothesis, what is your best guess as to the expected value of the short-term interest rate next year? c. If you believe in the liquidity preference theory, is your best guess as to next year's short-term interest rate higher or lower than in (b)? The yield to maturity on the zero-coupon bonds assumes annual compounding.

a. The forward rate (f2) is the rate that makes the return from rolling over one-year bonds the same as the return from investing in the two-year maturity bond and holding to maturity: (1 + 8%) x (1 + f2) = (1 + 9%)^2 --> f2 = 0.1001 = 10.01% b. According to the expectations hypothesis, the forward rate equals the expected value of the short-term interest rate next year, so the best guess would be 10.01%. c. According to the liquidity preference hypothesis, the forward rate exceeds the expected short-term interest rate next year, so the best guess would be less than 10.01%.

You will be paying $10,000 a year in tuition expenses at the end of the next two years. Bonds currently yield 8% a. What are the present value and duration of your obligation? b. What maturity zero-coupon bond would immunize your obligation? c. Suppose you buy a zero-coupon bond with value and duration equal to your obligation. Now suppose that rates immediately increase to 9%. What happens to your net position, that is, to the difference between the value of the bond and that of your tuition obligation? d. What if rates fall to 7%

a. The present value of the obligation is $17,832.65 and the duration is 1.4808 years, as shown in the following table: Computation of duration, interest rate = 8% Yrs - Paymnt - Paymnt Discount@8% - Weight - Column (1x4) 1 10,000 9,259.26 0.5192 0.51923 2 10,000 8,573.39 0.4808 0.96154 Column Sum: 17,832.65 1.0000 1.48077 b. To immunize the obligation, invest in a zero-coupon bond maturing in 1.4808 years. Since the present value of the zero-coupon bond must be $17,832.65, the face value (i.e., the future redemption value) must be: $17,832.65 x (1.08)^1.4808 = $19,985.26 c. If the interest rate increases to 9%, the zero-coupon bond would fall in value to [($19,985.26) / (1.09^1.4808)] =$17,590.92 The present value of the tuition obligation would fall to $17,591.11, so that the net position changes by $0.19. d. If the interest rate falls to 7%, the zero-coupon bond would rise in value to: [($19,985.26) / (1.07^1.4808)]=$18,079.99 The present value of the tuition obligation would increase to $18,080.18, so that the net position changes by $0.19. The reason the net position changes at all is that, as the interest rate changes, so does the duration of the stream of tuition payments.

A 30-year maturity bond making annual coupon payments with a coupon rate of 12% has duration of 11.54 years and convexity of 192.4. The bond currently sells at a yield to maturity of 8%. a. Use a financial calculator or spreadsheet to find the price of the bond if its yield to maturity falls to 7%. b. What price would be predicted by the duration rule? c. What price would be predicted by the duration-with-convexity rule?

a. Using a financial calculator, we find that the price of the bond is: For y = 7%: N = 30; I/Y = 7; FV = 1000; PMT = 120 → PV = $1,620.45 For y = 8%: N = 30; I/Y = 7; FV = 1000; PMT = 120 → PV = $1,450.31 b. Using the duration rule, assuming yield to maturity falls from 8% to 7%: Predicted price change: = - Duration × (𝛥𝑦 /1+𝑦) ×𝑃0 = -11.54 × (−0.01 / 1.08) x $1,450.31 =$154.97 Therefore: Predicted price = $154.97 + $1,450.31 = $1,605.28 c. Using the duration-with-convexity rule, if yield to maturity falls from 8% to 7%: Predicted price change = [−Duration × (∆y/1 + y) + (0.5 × Convexity × (∆y)^2)]x P0 = [(−11.54(－0.01/1.08) )+ (0.5 × 192.4 × (−0.01)^2)] ́ $1,450.31 = $168.92 Predicted price = $168.92 + $1,450.31 = $1,619.23

Two investment advisers are comparing performance. One averaged a 19% return and the other a 16% return. However, the beta of the first adviser was 1.5, while that of the second was 1. a. Can you tell which adviser was a better selector of individual stocks (aside from the issue of general movements in the market)? b. If the T-bill rate were 6% and the market return during the period were 14%, which adviser would appear to be the superior stock selector? c. What if the T-bill rate were 3% and the market return 15%?

a. Without information about the parameters of this equation (i.e., the risk-free rate and the market rate of return), we cannot determine which investment adviser is the better selector of individual stocks. b. α1 = 19% - [6% + 1.5 x (14% - 6%)] = 19% - 18% = 1% α2 = 16% - [6% + 1.0 x (14% - 6%)] = 16% - 14% = 2% Here, the second investment adviser has the larger abnormal return and thus appears to be the better selector of individual stocks. c. α1 =19% - [3% + 1.5 x (15% - 3%)] = 19% - 21% = -2% α2 = 16% - [3%+ 1.0 x (15% - 3%)] = 16% - 15% = 1%

A two-year bond with par value $1,000 making annual coupon payments of $100 is priced at $1,000. What is the yield to maturity of the bond? What will be the realized compound yield to maturity if the one-year interest rate next year turns out to be (a) 8%, (b) 10%, (c) 12%?

r Total proceeds Realized YTM =√Proceeds/1,000 = 1 8% $1,208 √1,208/1,000 - 1 = 0.0991 = 9.91% 10% $1,210 √1,210/1,000 - 1 = 0.1000 = 10.00% 12% $1,212 √1,210/1,000 - 1 = 0.1009 = 10.09%

The risk-free rate is 8% and the expected return on the market portfolio is 16%. A firm considers a project with an estimated beta of 1.3. What is the required rate of return on the project? If the IRR of the project is 19%, what is the project alpha?

rf + Bi(E(rm)-rf) =8 +1.3(16-8) =18.4%

Assume both portfolios A and B are well diversified, that E(rA) = 14% and E(rB) = 14.8%. If the economy has only one factor, and βA = 1 while βB = 1.1, what must be the risk-free rate?

rf = 6%

A newly issued 10-year maturity, 4% coupon bond making annual coupon payments is sold to the public at a price of $800. What will be an investor's taxable income from the bond over the coming year? The bond will not be sold at the end of the year. The bond is treated as an original-issue discount bond.

we can compute that its price in one year will be (at an unchanged yield) $814.60, representing an increase of $14.60. Total taxable income is: $40 + $14.60 = $54.60.

A portfolio's expected return is 12%, its standard deviation is 20%, and the risk-free rate is 4%. Which of the following would make for the greatest increase in the portfolio's Sharpe ratio? a. An increase of 1% in expected return. b. A decrease of 1% in the risk-free rate. c. A decrease of 1% in its standard deviation

𝑆𝑃 =(𝑟𝑃 −𝑟𝑓)/(𝜎𝑃) =(.12−.04)/(.20) =.40 𝑎. 𝑆𝑃 =(.13−.04)/(.20) =.45 𝑏. 𝑆𝑃 =(.12−.03)/(.20) =.45 𝑐. 𝑆𝑃 =(.12−.04)/(.19) =.42

You own a fixed-income asset with a duration of five years. If the level of interest rates, which is currently 8%, goes down by 10 basis points, how much do you expect the price of the asset to go up (in percentage terms)?

𝛥𝑃/𝑃 = - D x (𝛥𝑦/1+𝑦) =−5.0 x (−0.0010/1.08) =0.00463 or a 0.463% increase.