Question

A 5-year annuity of 10 $10,000 semiannual payments will begin 9 years from now, with the first payment coming 9.5 years from now. Requirement 1: If the discount rate is 9 percent compounded semiannually, what is the value of this annuity five years and three years from now? (Enter rounded answers as directed, but do not use rounded numbers in intermediate calculations. Round your answers to 2 decimal places (e.g., 32.16).) Value of the annuity Five years $ Three years $ Requirement 2: What is the value of the annuity today? (Enter rounded answer as directed, but do not use rounded numbers in intermediate calculations. Round your answer to 2 decimal places (e.g., 32.16).) Value today $

Answer 1

Since payment is semi-annual, each semi-annual payment is of $10,000/2 = $5,000

Number of payments = 10 starting from 9.5 year

Now, rate of interest is 9% compounded semi-annually. So rate of interest per period will become 9%/2 = 4.5%

Value of annuity 5 years from now is PV of all the payments when you are sitting in year 5. the first payment happens at n = 9.5 which is 9 periods away from n=5 (5 to 5.5 is period 1, 5.5 to 6 is period 2, 6 to 6.5 is period 3 so and so forth)

Thus, PV = 5000/(1+4.5%)⁹  + 5000/(1+4.5%)¹⁰ +.....+ 5000/(1+4.5%)¹⁸

Solving the above equation, we get value of annuity five years from now = $30,533.68

Similarly,

Value of annuity 3 years from now is PV of all the payments when you are sitting in year 3. the first payment happens at n = 9.5 which is 9 periods away from n=13 (as explained in earlier part)

Thus, PV = 5000/(1+4.5%)¹³  + 5000/(1+4.5%)¹⁴ +.....+ 5000/(1+4.5%)²²

Solving the above equation, we get value of annuity five years from now = $23,329.22

Value today is the value of annuity at n=0. PV of all payments starting at 9.5 is as follows:

PV = 5000/(1+4.5%)¹⁹  + 5000/(1+4.5%)²⁰ +.....+ 5000/(1+4.5%)²⁸

Solving the above equation, we get value of annuity five years from now = $17,914.41