Question

What is the future value of an annuity due that pays $1000 for 5 years if the appropriate discount rate is five percent? What’s the present value of this annuity due?

please show all work and steps not with excel

Answer 1

Solution:

Calculation of Future Value of an annuity due :

The formula for calculating the Future value of annuity due is

FV = A * [ ( ( 1 + r )ⁿ – 1 ) / r ] * ( 1 + r )

Where

FV = Future value of annuity ; A = Annuity amount   ; r = discount rate   ; n = No. of years

As per the information given in the question we have

A = $ 1,000   ;   r = 5 % = 0.05 ;   n = 5 years

Applying the above information in the formula we have

= $ 1,000 * [ ( ( 1 + 0.05 )⁵ – 1 ) / 0.05 ] * ( 1 + 0.05 )

= $ 1,000 * [ ( ( 1.05 )⁵ – 1 ) / 0.05 ] * 1.05                  

= $ 1,000 * [ ( 1.276282– 1 ) / 0.05 ] * 1.05

= $ 1,000 * [ 0.276282 / 0.05 ] * 1.05                          

= $ 1,000 * 5.525631 * 1.05

= $ 5,801.9128

Thus the Future value of annuity due that pays $1000 for 5 years if the appropriate discount rate is 5 % is = $ 5,801.9128

= $ 5,801.91 ( when rounded off to two decimal places )              

Note: The value of ( 1.05 ) ⁵   is calculated using the Excel formula =POWER(Number,Power)

=POWER(1.05,5) = 1.276282       

Calculation of Present Value of an annuity due :

The formula for calculating the Present value of annuity due is

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

Where

PV = Present value of annuity ; A = Annuity amount   ; r = discount rate   ; n = No. of years

As per the information given in the question we have

A = $ 1,000   ;   r = 5 % = 0.05 ; n = 5 years

Applying the above information in the formula we have

= $ 1,000 * [ ( 1 – ( 1 / ( 1 + 0.05 )⁵ ) )/ 0.05 ] * ( 1 + 0.05 )

= $ 1,000 * [ ( 1 – ( 1 / ( 1.05 )⁵ ) )/ 0.05 ] * 1.05

= $ 1,000 * [ ( 1 – ( 1 / 1.276282) )/ 0.05 ] * 1.05

= $ 1,000 * [ ( 1 – 0.783526 ) / 0.05 ] * 1.05

= $ 1,000 * [ 0.216474 / 0.05 ] * 1.05

= $ 1,000 * 4.329477 * 1.05

= $ 4,545.9505

Thus the present value of annuity due that pays $1000 for 5 years if the appropriate discount rate is 5 % is = $ 4,545.9505

= $ 4,545.95 ( when rounded off to two decimal places )              

Note: The value of ( 1.05 ) ⁵   is calculated using the Excel formula =POWER(Number,Power)

=POWER(1.05,5) = 1.276282