A 7-year annuity of fourteen $10,600 semiannual payments will begin 9 years from now, with the...

Question:

A 7-year annuity of fourteen $10,600 semiannual payments will begin 9 years from now, with the first payment coming 9.5 years from now. If the discount rate is 9% 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.)

If the discount rate is 9% 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.)

If the discount rate is 9% 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.)

Present Value of Annuity:

An annuity is a stream of equal payments, occurring at a regular time interval. The present value of an annuity is the sum of the present value of each payment in an annuity.

Answer and Explanation:

We can use the following formula to compute the present value of an annuity with periodic payments {eq}M {/eq} for {eq}T{/eq} periods, given periodic return {eq}r{/eq}:

  • {eq}\displaystyle \frac{M(1 - (1 + r)^{-T})}{r} {/eq}, when the first payment is expected in one period.

Given a 9% discount rate compounded monthly, we first compute the effective semiannual discount rate, which is:

  • {eq}(1 + 9\%/12)^6 - 1 = 4.585\% {/eq}

Applying this formula, we can first compute the value of the annuity at the end of 9 years:

  • {eq}\displaystyle \frac{10,600(1 - (1 + 4.585\%)^{-14})}{4.585\%} = 107,764.43 {/eq}

To find the value 5 years from now, we need to discount the value 9 years from now 4 years back, i.e.,

  • {eq}\displaystyle \frac{107,764.43}{(1 + 4.585\%)^{8}} = 75,285.76 {/eq}

To find the value 3 years from now, we need to discount the value 9 years from now 6 years back, i.e.,

  • {eq}\displaystyle \frac{107,764.43}{(1 + 4.585\%)^{12}} = 62,926.20 {/eq}

To find the value now, we need to discount the value 9 years from now 9 years back, i.e.,

  • {eq}\displaystyle \frac{107,764.43}{(1 + 4.585\%)^{18}} = 48,084.99 {/eq}

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How to Calculate the Present Value of an Annuity

from Business 110: Business Math

Chapter 8 / Lesson 3
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