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homework4-1

# homework4-1 - 4.1(a Let the continuously compounded rate be...

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4.1. (a) Let the continuously compounded rate be r . Then we must have e r ° 0 : 25 = 1 + 0 : 14 = 4 ; r = 0 : 1376 : (b) Let the annually compounded rate be r . Then we must have 1 + r = (1 + 0 : 14 = 4) 4 ; r = 0 : 1475 : 4.3. The bond price is related to its yield via the following: p = 4 1 + 0 : 104 = 2 + 4 (1 + 0 : 104 = 2) 2 + 104 (1 + 0 : 104 = 2) 3 = 96 : 744 : Now let°s use the zero rates to price the bond. Assuming that the zero rate at 18 months is R per annum with semiannual compounding, we must have 96 : 744 = 4 1 + 0 : 10 = 2 + 4 (1 + 0 : 10 = 2) 2 + 104 (1 + R= 2) 3 The solutions is R = 0 : 1042 : 4.5. The forward rate for the second quarter (between end of ±rst and end of second quarters) is: f (2) = 8 : 2 ° 2 ± 8 : 0 ° 1 2 ± 1 = 8 : 4% : The rest of the forward rates are: 8.8%, 8.8%, 9.0%, and 9.2%. 4.6. Using continuously compounded zero rates and forward rates, the value of the FRA is: 1000000 ° ° 0 : 25 ° 0 : 095 ± ° e 0 : 09 ° 3 12 ± 1 ±± ° e ± 0 : 086 ° 15 12 = 893 : 56 : Note that the 9.5% is quoted with quarterly compounding, but the forward rate of 9% (between 4th and 5th quarter) is continuously compounded. There- fore ° e 0 : 09 ° 3 12 ± 1 ± allows us to convert the continuously compounded forward rate into a forward interest rate that would apply to a one-quarter period. The

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