PS02-solution

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Jan 9, 2024

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Problem Set 2: Solutions UCLA MQE Core Finance Prof. Pierre-Olivier Weill 1. Before starting this exercise, please check the handout on “Annualized Return” as a refresher on the computation of annualized returns. A 4-year coupon bond has face value of $ 1,000, coupons of $ 80 each, and a YTM of 8%. The coupon is paid once a year. You plan to hold the bond until maturity: (a) What is the bond’s current price? The price of the bond solves the following equation price = $80 1 . 08 + $80 1 . 08 2 + $80 1 . 08 3 + $1 , 080 1 . 08 4 = $1 , 000 It is equal to the face value of the bond because the YTM and the coupon rate are the same. (b) Suppose that the coupon can be reinvested at 8%. What is the investment’s future value at maturity? What is the investment’s annual return? FV = $80 × 1 . 08 3 + $80 × 1 . 08 2 + $80 × 1 . 08 + $1 , 080 ∼ = $1 , 360 . 49 R = $1 , 360 . 49 $1 , 000 ! 1 4 - 1 = 8% In this special case, all the coupons can be reinvested at a rate equal to the YTM, and the bond is held to maturity, so the return is exactly equal to the YTM. (c) Suppose that the first coupon can be reinvested at 8%, while the second and third coupon payments can be re-invested at a yield of 10%. What is the investment’s future value at maturity? What is the investment’s annual return? FV = $80 × 1 . 08 3 + $80 × 1 . 1 2 + $80 × 1 . 1 + $1 , 080 ∼ = $1 , 365 . 58 R = $1 , 365 . 58 $1 , 000 ! 1 4 - 1 ∼ = 8 . 1% (d) Suppose that the first coupon can be reinvested at 8%, while the second and third coupon payments can be re-invested at a yield of 4%. What is the

investment’s future value at maturity? What is the investment’s annual return? FV = $80 × 1 . 08 3 + $80 × 1 . 04 2 + $80 × 1 . 04 + $1 , 080 ∼ = $1 , 350 . 50 R = $1 , 350 . 50 $1 , 000 ! 1 4 - 1 ∼ = 7 . 8% . (e) What explains the differences in annual returns in questions (b), (c) and (d)? The higher re-investment rate in question (c) raised the return above the YTM, and the lower re-investment rate in question (d) lowered the return below the YTM. 2. Suppose you run a pension fund and you have the following liability: you will have to pay retirees $ 1,000,000 in 15 years. Suppose interest rates are equal to 1% forever and that there are only two bonds available in the market: a 2 year zero coupon bond, and a 20 year zero coupon bond. (a) What is the present value of your liability at t = 0? V L = $1 , 000 , 000 (1 + 0 . 01) 15 ∼ = $861 , 349 . 47 (1) (b) Suppose you start at t = 0 with an amount of cash equal to the present value of the liability. What portfolio of 2 year and 20 year zero-coupon bond should you buy at t = 0 in order to be immunized against change in interest rates? (that is, so that the duration of our assets and liability are the same). Let V 2 and V 20 denote the value of 2-year and 2-year zero coupon bonds I purchase. Let D 2 and D 20 denote the duration of these 2-year and 20-year zero coupon bonds. We need to satisfy the following equation system: V 2 + V 20 = V L D 2 × V 2 + D 2 × V 20 = D L × V L The first equation equates the value of our assets with the value of the liability. Together with the first equation, the second equation ensures that we are immunized, in that the duration of our assets and liability are the same. We know that D L = 15, D 2 = 2, D 20 = 20 and that V L ∼ = $861 , 349 . 47. We have to solve for V 2 and V 20 , which are the total amount in dollars that we are going to buy in each asset: V 2 + V 20 = $861 , 349 . 47

2 × V 2 + 20 × V 20 = 15 × $861 , 349 . 47 We get that V 2 ∼ = $239 , 263 . 74 and V 20 ∼ = $622 , 085 . 73 (c) Suppose that the interest rate increases from 1% to 1.25% at t = 0. Suppose that you have bought the portfolio that you found in question (b). What is the approximate change in the value of your asset and liability? We are going to calculate the approximate value change using the following formulas: Δ P ≈ - D 1 + y m Price × (Δ y ) The approximate change in the value of the liabilities is: Δ V L ≈ - 15 1 + 0 . 01 1 × $861 , 349 . 47 × 0 . 0025 ∼ = - $31 , 980 . 75 (2) The approximate change in the value of the assets is: Δ V 2 ≈ - 2 1 + 0 . 01 1 × $239 , 263 . 74 × 0 . 0025 ∼ = - $1 , 184 . 48 Δ V 20 ≈ - 20 1 + 0 . 01 1 × $622 , 085 . 73 × 0 . 0025 ∼ = - $30 , 796 . 40 Δ V 2 + Δ V 20 ∼ = - $31 , 980 . 88 (d) Re-do the calculation of question (c) assuming that, instead of the portfolio of question (b) you did not try to immunized and have bought a portfolio composed of 30 year bonds only. Explain the difference in results. Now we can only solve for one equation: V 30 = $861 , 349 . 47 The approximate change in value for this portfolio is: Δ V 30 ≈ - 30 1 + 0 . 01 1 × $861 , 349 . 47 × 0 . 0025 ∼ = - $63 , 961 . 59 Because we did not try to immunized and because the duration of our assets is twice as large as that of our liabilities, our loss is about twice larger.

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