Financial_Markets_CFT2020_Problem_Set_2

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Corporate Finance Track Financial Markets Problem Set 2 Bond Markets Problem 1 [YTM given, calculate price] (a) Bond A is a 5-year bond that pays 8% coupon per year at the end of each year. The face value of the bond is $1,000. The bond’s yield-to-maturity is 8%. Calculate the bond price. (b) Bond B is a 5-year bond that pays 8% coupon per year at the end of each year. The face value of the bond is $1,000. The bond’s yield-to-maturity is 7%. Calculate the bond price. (c) Bond C is a 5-year bond that pays 8% coupon per year at the end of each year. The face value of the bond is $1,000. The bond’s yield-to-maturity is 9%. Calculate the bond price. (d) Compare the YTMs and prices of the three bonds. Solution Please do it yourself. Use Excel for calculations. Problem 2 [Price given, calculate YTM] (a) Bond A is a 5-year bond that pays 8% coupon per year at the end of each year. The face value of the bond is $1,000. The bond’s price is $1,000. Calculate the yield-to-maturity of the bond. (b) Bond B is a 5-year bond that pays 8% coupon per year at the end of each year. The face value of the bond is $1,000. The bond’s price is $1,050. Calculate the yield-to-maturity of the bond. (c) Bond C is a 5-year bond that pays 8% coupon per year at the end of each year. The face value of the bond is $1,000. The bond’s price is $950. Calculate the yield-to-maturity of the bond. (d) Compare the prices and YTMs of the three bonds. Solution Please do it yourself. Use Excel for calculations. Problem 4.11. [Hull] Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero coupon rates are 4%, 4.2%, 4.4%, 4.6%, and 4.8% per annum with continuous compounding respectively. (a) Calculate the price of a bond with a face value of 100 that will mature in 30 months and pays a coupon of 4% per annum semiannually. (b) Calculate the yield-to-maturity of the bond. Solution (a) The bond pays $2 in 6, 12, 18, and 24 months, and $102 in 30 months. The bond price is (b) The bond YTM is the value of that solves 0 04 0 5 0 042 1 0 0 044 1 5 0 046 2 0 048 2 5 2 2 2 2 102 98 04 e e e e e - . ´ . - . ´ . - . ´ . - . ´ - . ´ . + + + + = . y

2 2e -y*0.5 + 2e -y*1 + 2e -y*1.5 + 2e -y*2 + 102e -y*2.5 = 98.04 => y = 4.78% with continuous compounding. Problem 4.11. [Hull] (modified) Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero coupon rates are 4.08%, 4.29%, 4.50%, 4.71%, and 4.92% per annum with annual compounding respectively. (a) Calculate the price of a bond with a face value of 100 that will mature in 30 months and pays a coupon of 4% per annum semiannually. (b) Calculate the yield-to-maturity of the bond. Solution (a) P = 2/(1+4.08%) 0.5 +2/(1+4.08%) 1 +2/(1+4.08%) 1.5 +2/(1+4.08%) 2 +102/(1+4.08%) 2.5 = 98.04 (b) 2/(1+y) 0.5 +2/(1+y) 1 +2/(1+y) 1.5 +2/(1+y) 2 +102/(1+y) 2.5 = 98.04 => YTM = 4.90% with annual compounding. Problem 4.11. [Hull] (modified) Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero coupon rates are 4.04%, 4.24%, 4.45%, 4.65%, and 4.86% per annum with semi-annual compounding respectively. (a) Calculate the price of a bond with a face value of 100 that will mature in 30 months and pays a coupon of 4% per annum semiannually. (b) Calculate the yield-to-maturity of the bond. Solution (a) P = 2/(1+4.04%/2) 1 +2/(1+4.24%/2) 2 +2/(1+4.45%/2) 3 +2/(1+4.65%/2) 4 +102/(1+4.86%/2) 5 = 98.04 (b) 2/(1+y/2) 1 +2/(1+y/2) 2 +2/(1+y/2) 3 +2/(1+y/2) 4 +102/(1+y/2) 5 = 98.04 => YTM = 4.84% with semi-annual compounding. Clarification: Why did we write power 1 (and not 0.5) in the first term? Note that (1+4.04%/2) 1 = [(1+4.04%/2) 2 ] 0.5 . So, the power is in fact 0.5 but after simplification of the two powers (2*0.5=1) the power becomes 1. Annual: (1+ra) 0.5 Semi-annual: [(1+rs/2) 2 ] 0.5 Quarterly: [(1+rq/4) 4 ] 0.5 Monthly: [(1+rm/12) 12 ] 0.5 Daily: [(1+rs/365) 365 ] 0.5 Continuously: As n tends to infinity, [(1+r/n) n ] 0.5 = e r*0.5

3 Problem 4.8. [Hull] What does duration tell you about the sensitivity of a bond portfolio to interest rates? What are the limitations of the duration measure? Solution Duration provides information about the effect of a small parallel shift in the yield curve on the value of a bond portfolio. The percentage decrease in the value of the portfolio equals the duration of the portfolio multiplied by the amount by which interest rates are increased in the small parallel shift. The duration measure has the following limitation. It applies only to parallel shifts in the yield curve that are small. Problem 4.35. [Hull] Portfolio A consists of a one-year zero-coupon bond with a face value of $2,000 and a 10-year zero- coupon bond with a face value of $6,000. Portfolio B consists of a 6.04-year zero-coupon bond with a face value of $5,000. The current yield on all bonds is 10% per annum. (a) Show that both portfolios have the same duration. (b) Calculate the percentage changes in the values of the two portfolios for a 0.5% per annum increase in yields using the duration formula. (c) Calculate the percentage changes in the values of the two portfolios for a 0.5% per annum increase in yields using the bond price formula. Solution (a) The duration of Portfolio A is 1 ∗ 2000 (1 + 10%) ! + 10 ∗ 6000 (1 + 10%) !" 2000 (1 + 10%) ! + 6000 (1 + 10%) !" = 6.04 Portfolio B is a zero-coupon bond with only one cash flow at the end of 6.04 years. Therefore, its duration is 6.04 years. Therefore, the two portfolios have the same duration. (b) Using duration formula: Portfolio A: Percentage decrease in price = -D*(change in y)/(1+y) = -6.04*0.5%/(1+10%) = -2.75% Portfolio B: Percentage decrease in price = -D*(change in y)/(1+y) = -6.04*0.5%/(1+10%) = -2.75% Since both portfolios have the same duration, so the percentage change in price using duration formula will be same for the two portfolios. (c) Using bond price formula: Portfolio A original price (at y=10%): P = 2000/(1+10%) 1 +6000/(1+10%) 10 = 4131.44

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4 Portfolio A new price (at y=10.5%): P = 2000/(1+10.5%) 1 +6000/(1+10.5%) 10 = 4020.65 Percentage change in price = (4020.65-4131.44)/4131.44 = -2.68% Portfolio B original price (at y=10%): P = 5000/(1+10%) 6.04 = 2811.63 Portfolio B new price (at y=10.5%): P = 5000/(1+10.5%) 6.04 = 2735.66 Percentage change in price = (2735.66-2811.63)/ 2811.63= -2.70% The real percentage decline in value of Portfolio A is less than that of Portfolio B. Please also note that duration is not completely accurate, it overestimates the percentage decrease in price for both portfolios. This error can be fixed by calculating bond convexity (we will not discuss bond convexity in this course).

5 Stock Markets Problem 1 Consider the following limit-order book for a share of stock. The last trade in the stock occurred at a price of $50. Limit Buy Orders Limit Sell Orders Price ($) Shares Price ($) Shares 49.75 500 50.25 100 49.50 800 51.50 100 49.25 500 54.75 300 49.00 200 58.25 100 48.50 600 (a) If a market buy order for 100 shares comes in, at what price will it be filled? (b) The next market buy order is for 120 shares. At what prices would this buy order be filled? (c) If a market sell order for 100 shares comes in, at what price will it be filled? (d) At what price would the next market sell order for 120 shares be filled? Solution (a) The buy order will be filled at the best limit-sell order price: $50.25 (b) First 100 shares at $51.50, remaining 20 shares at $54.75 (c) The sell order will be filled at the best limit-buy order price: $49.75 (d) Next market sell order for 120 shares will be filled at $49.75 Problem 2 You are bullish on Telecom stock. The current market price is $50 per share, and you have $5,000 of your own to invest. You borrow an additional $5,000 from your broker at an interest rate of 8% per year and invest $10,000 in the stock. (a) What will be your rate of return if the price of Telecom stock goes up by 10% during the next year? The stock currently pays no dividends. (b) What will be your rate of return if the price of Telecom stock goes down by 10% during the next year? The stock currently pays no dividends. (c) How far does the price of Telecom stock have to fall for you to get a margin call if the maintenance margin is 30%? Assume the price fall happens immediately. Solution (a) You buy 200 shares of Deutsche Telekom for $10,000. These shares increase in value by 10%, or $1,000. You pay interest of: 0.08 ´ $5,000 = $400. Your net profit/loss is $1,000-$400 = $600, and your initial investment is $5,000. The rate of return will be: $1,000 $400 0.12 12% $5,000 - = =

6 (b) You buy 200 shares of Deutsche Telekom for $10,000. These shares decrease in value by 10%, or $1,000. You pay interest of: 0.08 ´ $5,000 = $400. Your net profit/loss is -$1,000-$400 = - $1600, and your initial investment is $5,000. The rate of return will be: (c) The value of the 200 shares is 200P. Equity (i.e. value of your investment) is (200P – $5,000). You will receive a margin call when the percentage value of your investment is less than 30%: = 0.30 Þ when P = $35.71 or lower Problem 3 Here is some price information on FinCorp stock. Suppose that FinCorp trades in a dealer market. Bid: $55.25 Ask: $55.50 (a) Suppose you have submitted an order to your broker to buy at market. At what price will your trade be executed? (b) Suppose you have submitted an order to sell at market. At what price will your trade be executed? (c) Suppose you have submitted a limit order to sell at $55.62. What will happen? (d) Suppose you have submitted a limit order to buy at $55.37. What will happen? Solution (a) $55.50 (b) $55.25 (c) The trade will not be executed because the bid price is lower than the price specified in the limit sell order. (d) The trade will not be executed because the asked price is greater than the price specified in the limit buy order. Problem 4 You bought a stock one year ago for $50 per share. It paid a $1 per share dividend today. You sold it today for $55 per share. (a) What was your realized return? (b) How much of the return came from dividend yield and how much came from capital gain? Solution a. R = !#(%%&%") %" = 0.12 = 12% b. R div = ! %" = 2% $1,000 $400 0.28 28% $5,000 - - = - = - P 200 000 , 5 $ P 200 -

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7 R capital gain = %%&%" %" = 10% Problem 5 Suppose Pepsico’s stock has a beta of 0.57. If the risk-free rate is 3% and the expected return of the market portfolio is 8%, what is Pepsico’s equity cost of capital? Solution 3% + 0.57 ´ (8%-3%) = 5.85% Problem 6 Suppose the market portfolio has an expected return of 10% and a volatility of 20%, while Microsoft’s stock has a volatility of 30%. Given its higher volatility, should we expect Microsoft to have an expected return that is higher than 10% (according to CAPM)? Solution No, higher volatility does not mean higher expected returns. The expected return depends on beta which is not known in this case. Problem 7 Suppose the risk-free return is 4% and the market portfolio has an expected return of 10% and a volatility of 16%. Johnson and Johnson Corporation (Ticker: JNJ) stock has a 20% volatility and a correlation with the market of 0.06. (a) What is Johnson and Johnson’s beta with respect to the market? (b) Under the CAPM assumptions, what is its expected return? Solution (a) β (( = 0.06 × ".*" ".!+ = 0.075 (b) E[R JJ ] = 0.04 + 0.075(0.1 − 0.04) = 4.45% Problem 8 Suppose Intel stock has a beta of 2.16, whereas Boeing stock has a beta of 0.69. If the risk-free interest rate is 4% and the expected return of the market portfolio is 10%, what is the expected return of a portfolio that consists of 60% Intel stock and 40% Boeing stock, according to the CAPM? Solution β = (0.6)(2.16) + (0.4)(0.69) = 1.572 E[R] = 4 + (1.572)(10 − 4) = 13.432%