Question

Amount of annuity-$32,000

Interest rate-9%

Period (years)-11

a. Calculate the present value of the annuity assuming that it is ​(1) An ordinary annuity. ​(2) An annuity due.

b. Compare your findings in parts a​(1) and a​(2). All else being​ identical, which type of annuity—ordinary or annuity due—is ​preferable? Explain why.

Answer 1

Solution:

a (1). The formula for calculating the present value of annuity assuming that it is an ordinary annuity :

PV = A * [ ( 1 - ( 1 / ( 1 + r ) ⁿ )) / r ]

Where

PV = Present value of annuity   ;   A = Amount of annuity   ;   r = rate of interest ; n = no. of years

As per the information given in the question we have

r = 9 % = 0.09   ; n = 11 years   ; A = $ 32,000

Applying the above information in the formula we have

= $ 32,000 * [ ( 1 - ( 1 / ( 1 + 0.09 ) ¹¹ )) / 0.09 ]

= $ 32,000 * [ ( 1 - ( 1 / ( 1.09 ) ¹¹ )) / 0.09 ]

= $ 32,000 * [ ( 1 - ( 1 / 2.580426 )) / 0.09 ]

= $ 32,000 * [ ( 1 – 0.387533 ) / 0.09 ]

= $ 32,000 * [ 0.612467 / 0.09 ]

= $ 32,000 * 6.805191

= $ 217,766.0976

= $ 217,766.10    ( When rounded off to two decimal places )

The present value of the annuity assuming that it is an ordinary annuity = $ 217,766.10

Note: The value of ( 1.09 ) ¹¹   is calculated using the Excel formula =POWER(Number,Power)

=POWER(1.09,11) = 2.580426

a(2). The formula for calculating the present value of annuity assuming that it is an annuity due :

PV = A + [ A* [ ( 1 - ( 1 / ( 1 + r ) ^{( n - 1 )} )) / r ] ]

Where

PV = Present value of annuity   ; A = Amount of annuity    ; r = rate of interest ; n = no. of years

As per the information given in the question we have

r = 9 % = 0.09   ; n = 11 years   ; A = $ 32,000

Applying the above information in the formula we have

= $ 32,000 + [ $ 32000 * [ ( 1 - ( 1 / ( 1 + 0.09 ) ^{( 11 – 1 )} )) / 0.09 ] ]

= $ 32,000 + [ $ 32000 * [ ( 1 - ( 1 / ( 1.09 ) ^{( 10 )} )) / 0.09 ] ]

= $ 32,000 + [ $ 32000 * [ ( 1 - ( 1 / 2.367364 )) / 0.09 ] ]

= $ 32,000 + [ $ 32000 * [ ( 1 – 0.422411 ) / 0.09 ] ]

= $ 32,000 + [ $ 32000 * [ 0.577589 / 0.09 ] ]

= $ 32,000 + [ $ 32000 * 6.417658 ]

= $ 32,000 + $ 205,365.0464

= $ 237,365.05 ( When rounded off to two decimal places )

The present value of the annuity assuming that it is ​an annuity due = $ 237,365.05

Note: The value of ( 1.09 ) ¹⁰   is calculated using the Excel formula =POWER(Number,Power)

=POWER(1.09,10) = 2.367364

(2). In an annuity due the annuity is received at the beginning of the period whereas in case of an ordinary annuity, the annuity is received at the end of the period.

Thus the present value of annuity due is greater than present value of ordinary annuity.

The amount received under annuity due i.e., $ 237,365.05 is greater than the amount received under ordinary annuity i.e., $ 217,766.10   

Thus Annuity due is preferable over ordinary annuity.