A3_sol - ActSc 371 Assignment#3 Solutions Question 1(a By...

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ActSc 371: Assignment #3 Solutions Question 1 (a) By definition, the spot rates satisfy 92 . 5926 = 100 1 + r 1 84 . 1680 = 100 (1 + r 2 ) 2 75 . 1315 = 100 (1 + r 3 ) 3 The spot rates are therefore 8%, 9% and 10%. (b) The rate over the second year is f 2 = (1 . 09) 2 1 . 08 - 1 = 0 . 10 and the rate over the third year is f 3 = (1 . 10) 3 (1 . 09) 2 - 1 = 0 . 12 The rates are therefore 10% and 12%, respectively. (c) Discounting all cash flows from this bond with the appropriate spot rate, we see that its price should be P = 100 1 . 08 + 100 (1 . 09) 2 + 1100 (1 . 1) 3 = 1003 . 21 Why do we say that this is what the coupon bond’s price should be? Note that $1003.21 is the amount it would cost to purchase a portfolio consisting of one 1-year zero, one 2-year zero and eleven 3-year zeros. Note also that the coupon bond is identical (in terms of future cash flows) to this portfolio. This means that the coupon bond and the portfolio are essentially identical investments, and they should there- fore have the same price. In part (d), we will see what can happen if they do not have the same price. 1
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(d) The price of the coupon bond is $1020, while the price of the portfolio described in (c) is equal to $1003.21. Therefore the coupon bond is overpriced (remember that they are essentially identical, and should therefore have the same price), and we can profit from this by short selling the coupon bond and using the proceeds to purchase the port- folio (“buy low, sell high”). The cash flows arising from this strategy would be as follows. Remember that if we short sell the coupon bond, we must reimburse the lender for coupon payments and the payment of face value. Bond Today 1 Year 2 Years 3 Years Coupon (Sell One) 1020 -100 -100 -1100 1-Yr (Buy One) -92.5926 100 0 0 2-Yr (Buy One) -84.1680 0 100 0 3-Yr (Buy Eleven) -826.4465 0 0 1100 Net 16.79 0 0 0 This strategy puts $16.79 in our pocket today, with future net cash flows of zero. This is an example of an arbitrage opportunity , which means an opportunity for a risk-less profit.
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