Problem 6-52 Calculating Present Values [LO1] A 5-year annuity of ten $9,400 sem

Question

Problem 6-52 Calculating Present Values [LO1] A 5-year annuity of ten $9,400 semiannual payments will begin 9 years from now, with the first payment coming 9.5 years from now. If the discount rate is 6 percent compounded monthly, what is the value of this annuity five years from now? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) Value of the annuity $ If the discount rate is 6 percent compounded monthly, what is the value three years from now? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) Value of the annuity $ If the discount rate is 6 percent compounded monthly, what is the current value of the annuity? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) Value of the annuity $

Explanation / Answer

Answer: The cash flows in this problem are semiannual, so we need the effective semiannual rate. The interest rate given is the APR, so the monthly interest rate is:

Monthly rate = .06/ 12 = .005   

To get the semiannual interest rate, we can use the EAR equation, but instead of using 12 months as the exponent, we will use 6 months. The effective semiannual rate is:

Semiannual rate = (1.005)6– 1 = .03037 or 3.04%

Now, we can use the present value of an annuity equation. Doing so, we get:

PVA=C ({1-[1/(1+r)]^t}/r)

=$9400{[1-(1/1.0304)^10]/0.0304}

=$80020.47

This is the present value one period before the first payment.The first payment occurs nine and one half years from now, so this is the value of the annuity nine years from now.Since the interest rate is semiannual .

The value of  this annuity five years from now is:

PV=FV/(1+r)t

=$80020.47/(1+0.0304)^8

=$62972.99

And the value of  the annuity three years from now is:

PV=FV/(1+r)t

=$80020.47/(1+0.0304)^12

=$55863.85

And the value of the annuity today is:

PV=FV/(1+r)t

=$80020.47/(1+0.0304)^18

=$46676.23

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