Question

Answer the following questions and draw the cash flow diagrams for each:

 (a) What are the equivalent annual payments over a 15-year period for a present value of $14,500 and a compound interest rate of 6% per year.

 (b) What are the annual equivalent payments for a present value of $445,000 in perpetuity and a compound interest rate of 6% per year. 

 (c) What is the present worth of $15,000/year for 30 years at a compound interest rate of 6% per year?

Answer 1

a)

Number of years (N) = 15

interest rate or Discount rate = 6% per annum

Present value(PV) = $14,500

Assuming Annual payment = $A

Calculating present value for each annual payment for next 15 years,

present value of all the $A payments made for the next N years from now is =

present value = $14,500 = + + - - - - - +

solving for A,

$14,500 = + + - - - - - +

by simplifying the equation using geometric series

$14,500 =

14,500*0.06 = A(1 - 0.417)

A =

A = $1492.29

so, equivalent annual payment that should be made for next 15 years is $1492.29.

b)

Perpetuity means that the annual payments continue to perpetuity (never ending)

using geometric series the standard formula derived for the value of perpetuities is :

value of perpetuity =

Assuming annual payment is $A

r is the interest rate per annum (Discount rate)

Given that the present value of perpetuity is $445,000

so,

445,000 = (According to above formula)

That is,

A = 445,000 * 0.06

= $26,700.

SO the equivalent annual payments = $26,700.

c)

Calculating present value for each annual payment for next 30 years,

Given annual payment A = $15,000 per annum

given interest rate = 6% per annum

present value of all the $A payments made for the next N years from now is =

present value = + + - - - - - +

similar to the first subpart equation reduces to

present value =

solving the equation,

present value = $206,472.46

so the present value of the payments made for next 30 years is $206,472.46