BUSN 20400 Problem Sets Winter 2024 (6)

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BUSN 20400 Problem Set 2 Professor Quentin Vandeweyer Fixed Income Solutions You may work on this assignment in groups of up to five people. Hand in one solution per group. You may discuss the problems only with members of your group. Answers should be typed and should be uploaded as a single PDF to Canvas. This problem set is due on Canvas at 11:59pm on January 21, 2024 . Please note that late submissions will NOT be accepted . Question 1 Many institutions have fixed future liabilities to meet (such as pension payments) and they fund these future liabilities using default-free fixed-income securities. When discount bonds of all maturities are available, these institutions can simply buy discount bonds to fund their liabilities. For example, if there is a fixed liability equal to 1 million dollars five years from now, Geico can buy a discount bond maturing in five years with a face value of 1 million dollars. Unfortunately, there may not be the “right” discount bonds for a fixed future liability and coupon bonds must be used. Then Geico faces reinvestment risk on the coupons. For example, suppose that the yield curve is flat at 10% and we have the following coupon bonds (paying annual coupons): Bond Prices Principal Coupon Years to Maturity A ??? 100 15 5 B ??? 100 25 10 and we have a 1 million liability five years from now. Part a Suppose that the yield curve will remain unchanged for the following five years and you have decided to use bond A to fund the liability. That is, you want to invest in bond A and invest the coupons at the prevailing interest rates to produce a future value at the end of year five of 1 million. How much should you invest in bond A ? Solution If you think in terms of year 5 future value you can directly compute the number of bonds. Here b is the number of bonds purchased. $1 , 000 , 000 = 15 b (1 . 1 4 ) + 15 b (1 . 1 3 ) + 15 b (1 . 1 2 ) + 15 b (1 . 1) + 115 b b = 5219 . 83 Note that the implied price is the PV of $ 1,000,000 divided by the number of bonds ≈ $118 . 95 Part b Now suppose that right after you invested in bond A, the yield curve makes a parallel move down by 2% to 8%. What is the future value five years from now of your investment? What is the future value if the yield curve moves up by 2% to 12% ? Please explain why the future value changes differently depending on the direction of the change in the yield curve. Solution FV = 15 b (1 + r ) 4 + 15 b (1 + r ) 3 + 15 b (1 + r ) 2 + 15 b (1 + r ) + 115 b FV r =8% = $981 , 322 . 90 = 1 , 000 , 000 − 18 , 677 FV r =12% = $1 , 019 , 394 . 75 = 1 , 000 , 000 + 19 , 394 1

BUSN 20400 Problem Set 2 Professor Quentin Vandeweyer Part c Part (b) shows that the future value of your investment is sensitive to interest rate fluctuations and you face the risk that your future liabilities may not be met. You should try to ”immunize” this interest rate risk. But how? Do the following: Part i Construct a portfolio of the two coupon bonds so that the future value of the portfolio is 1 million and the duration of this portfolio is equal to five years, assuming that the yield curve will remain flat at 10%. Solution The price of the 5 year bond is $ 118.95. The price of the 10 year bond is $ 192.17. Using this information we can apply the duration formula: Duration = PV (CF 1 ) P × 1 + PV (CF 2 ) P × 2 + · · · + PV (CF T ) P × T The duration of the 5 year bond is 3.953. The duration of the 10 year bond is 5.784. To get our weights we need: w · 3 . 953 + (1 − w )5 . 784 = 5 ⇒ w = 0 . 428 Here w was the weight on the 5 year bond. To translate weights to quantities we start with the present value of our desired investment and divide by price. q 5 y = w · PV (1 , 000 , 000) P 5 y = 0 . 428 · 620921 . 32 118 . 95 = 2234 . 17 q 10 y = (1 − w ) · PV (1 , 000 , 000) P 10 y = (1 − 0 . 428) · 620921 . 32 192 . 17 = 1848 . 19 Part ii Show that if immediately after you purchased this portfolio the yield curve makes a permanent parallel downward or upward move of 2%, the future value of this portfolio at the end of year 5 will still be approximately 1 million. You have immunized the portfolio of the risk associated with parallel movements of the yield curve by buying a portfolio of coupon bonds so that the duration of the portfolio matches the number of years to the payment of the fixed liability. Solution CF 5y CF 10y CF total Y5 value 10% Y5 value 8% Y5 value 12% $ 33,511.42 $ 46,205.15 $ 79,716.57 $ 116,713.03 $ 108,453.51 $ 125,435.56 $ 33,511.42 $ 46,205.15 $ 79,716.57 $ 106,102.75 $ 100,419.92 $ 111,996.04 $ 33,511.42 $ 46,205.15 $ 79,716.57 $ 96,457.05 $ 92,981.41 $ 99,996.46 $ 33,511.42 $ 46,205.15 $ 79,716.57 $ 87,688.23 $ 86,093.89 $ 89,282.56 $ 256,920.86 $ 46,205.15 $ 303,126.02 $ 303,126.02 $ 303,126.02 $ 303,126.02 $ - $ 46,205.15 $ 46,205.15 $ 42,004.68 $ 42,782.55 $ 41,254.60 $ - $ 46,205.15 $ 46,205.15 $ 38,186.08 $ 39,613.47 $ 36,834.46 $ - $ 46,205.15 $ 46,205.15 $ 34,714.61 $ 36,679.14 $ 32,887.91 $ - $ 46,205.15 $ 46,205.15 $ 31,558.74 $ 33,962.17 $ 29,364.21 $ - $ 231,025.76 $ 231,025.76 $ 143,448.82 $ 157,232.25 $ 131,090.22 Y5 PV $ 1,000,000.00 $ 1,001,344.31 $ 1,001,268.04 2

BUSN 20400 Problem Set 2 Professor Quentin Vandeweyer Part iii Suppose now that you have held your portfolio for one year after a 2% decrease in the yield curve to 8% which occurred immediately after you constructed your initial portfolio with a duration of five. There are now four years to the payment of the fixed liability. Use the money at your disposal (the market value of your investment at the end of year one) to construct a portfolio of coupon bonds with duration equal to four years and a future value in four years hence equal to approximately 1 million, given the new flat yield curve at 8%. You are essentially supposed to re-balance your immunization portfolio at this point. Show that if the yield curve then makes a parallel upward or downward move of 2%, the future value of your portfolio four years from now will be unchanged. You have approximately funded your liability of 1 million at the end of the fifth year. (Compare the difference between the future value of your portfolio and your fixed liability here and in part (b).) This technique is called ”duration matching”: if you adjust your portfolio over time so that its duration always matches the years to the payment date of your fixed liabilities, you will approximately immunize the risk of parallel shifts in the yield curve. Solution Note the rate change happened immediately after the initial investment so we can use the 8% column from part ii. We see that the PV of the portfolio is $1 , 001 , 344 . 31 1 . 08 4 = $ 736,017.95. That is the 5year FV discounted to period 1 at 8% The duration of the 4y bond is 3.356 and the duration of the 9y bond is 5.585. Weights are calculated w · 3 . 356 + (1 − w )5 . 585 = 4 ⇒ w = 0 . 711 These weights imply q 4 y = w · PV (1 , 000 , 000) P 4 y = 0 . 711 · 736 , 017 . 95 123 . 18 = 4248 . 33 q 9 y = (1 − w ) · PV (1 , 000 , 000) P 9 y = (1 − 0 . 711) · 736 , 017 . 95 206 . 20 = 1031 . 57 CF 4y CF 9y CF total Y4 value 8% Y4 value 6% Y4 value 10% $ 63,722.36 $ 25,789.55 $ 89,511.90 $ 112,759.22 $ 106,610.11 $ 119,140.34 $ 63,722.36 $ 25,789.55 $ 89,511.90 $ 104,406.69 $ 100,575.58 $ 108,309.40 $ 63,722.36 $ 25,789.55 $ 89,511.90 $ 96,672.86 $ 94,882.62 $ 98,463.09 $ 488,538.07 $ 25,789.55 $ 514,327.62 $ 514,327.62 $ 514,327.62 $ 514,327.62 $ 25,789.55 $ 25,789.55 $ 23,879.21 $ 24,329.76 $ 23,445.04 $ 25,789.55 $ 25,789.55 $ 22,110.38 $ 22,952.61 $ 21,313.68 $ 25,789.55 $ 25,789.55 $ 20,472.57 $ 21,653.40 $ 19,376.07 $ 25,789.55 $ 25,789.55 $ 18,956.09 $ 20,427.74 $ 17,614.61 $ 128,947.74 $ 128,947.74 $ 87,759.66 $ 96,357.25 $ 80,066.40 $ 1,001,344.30 $ 1,002,116.68 $ 1,002,056.26 3

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