Question

1.The spot rate of interest is defined by s(t) = .1(.9)t for t = 1, 2, 3, 4, 5. Find the present value of a 5-year annuity-due in which the first payment is equal to $1000, and each subsequent payment increases by 5% of the immediately preceding payment.

2.You are given the following term structure of spot interest rates:

Term (in years) Spot interest rate

1 5%

2 5.75%

3 6.25%

4 6.50%

A three-year annuity-immediate will be issued a year from now with annual payments of 5000. Using the forward rates, calculate the present value of this annuity a year from now.

Answer 1

1)

s(t) = .1(.9)t

s1 = 0.1*0.9*1 = 0.09

s2 = 0.1*0.9*2 = 0.18

s3 = 0.1*0.9*3 = 0.27

s4 = 0.1*0.9*4 = 0.36

s5 = 0.1*0.9*5 = 0.45

payment at the begining of year 2 = p1 = 1000*1.05 = 1050

payment at the begining of year 3 = p2 = 1050*1.05 = 1102.5

payment at the begining of year 4 = p3 = 1102.5*1.05 = 1157.625

payment at the begining of year 5 = p4 = 1157.625*1.05 = 1215.50625

present value = 1000 + p1/(1+s1) + p2/(1+s2)² + p3/(1+s3)³ + p4/(1+s4)⁴ = 1000 + (1050/1.09) + (1102.5/(1.18)²) + (1157.625/(1.27)³) + (1215.50625/(1.36)⁴)

= 1000 + 963.3027 + 791.7983 + 565.1408 + 355.3051 = $3675.5471 or $3675.56 ( after rounding off to 2 decimal places)