Kidd wants to purchase an annuity. The first annuity, an ordinary annuity, will

Question

Kidd wants to purchase an annuity. The first annuity, an ordinary annuity, will pay $1,000 at the end of each quarter for 20 years. The second annuity, an annuity due, will pay $1,000 at the beginning of each quarter for 20 years. Which of the following statements is correct regarding these annuities?

A. An annuity due has both a higher present value and a higher future value than an ordinary annuity.

B. The present value of an ordinary annuity is equal to the present value of an annuity due.

C. An ordinary annuity has a higher future value but a lower present value than an annuity due.

D. The future value of an ordinary annuity is greater than the future value of an annuity due.

E. An annuity due will pay one more payment than an ordinary annuity will.

Explanation / Answer

Option A

1

Ordinary annuity, PV of 1st annuity=1000/(1+r/4)+1000/(1+r/4)^2................1000/(1+r/4)^80

=1000/(1+r/4)*(1-1/(1+r/4)^80)/(1-1/(1+r/4))

=1000*4/r*(1-1/(1+r/4)^80)

Annuity due, PV of 2nd annuity=1000+1000/(1+r/4)................1000/(1+r/4)^79

=1000*(1-1/(1+r/4)^80)/(1-1/(1+r/4))

=1000*4/(r*(1+r/4))*(1-1/(1+r/4)^80)

So, PV of 2nd annuity/PV of 1st annuity=1+r/4

Hence, PV of 1st annuity is less than 2nd annuity or PV of annuity due is more than PV of ordinary annuity

2

Ordinary annuity, FV of 1st annuity=1000*(1+r/4)^79+1000/(1+r/4)^78................1000/(1+r/4)+1000

=1000*(1-(1+r/4)^80)/(1-(1+r/4))

=1000*4/r*((1+r/4)^80-1)

Annuity due, FV of 2nd annuity=1000*(1+r/4)^80+1000*(1+r/4)^79................1000*(1+r/4)

=1000*(1+r/4)*(1-(1+r/4)^80)/(1-(1+r/4))

=1000*4/r*(1+r/4)*((1+r/4)^80-1)

So, FV of 2nd annuity/FV of 1st annuity=(1+r/4)

Hence, future value of 2nd annuity is more than FV of 1st annuity. Future value of annuity due is more than FV of ordinary annuity

3

Both will have same number of payments

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