Question

Given a 6 percent discount rate compounded quarterly, what is the present value of a perpetuity of $100 per month if the first payment does not begin until the end of year five? Please give explanation and formula! Thanks!

Answer 1

Annual Rate = 6%

Compounding = 4

SO, Interest rate per period = 6% / 4

= 1.5%

Effective annual rate = ( 1 + Interest rate per period)^ Number of compounding - 1

= ( 1 + 1.5%)^4 - 1

= 1.015^4 - 1

= 0.0613635 or 6.13635%

So, Interest rate per month = ( 1 + EAR) ^ ( 1/ Number of period) - 1

= ( 1 + 6.13635%) ^ ( 1/12) - 1

= (1.0613635)^(1/12) - 1

= 0.0049752 or 0.4975202%

Now, Value of perpetuity = Cash Flow per period / Interest rate per period

= 100 / 0.4975202%

= 100 /  0.0049752

= 20099.6944

Value of Perpetuity = 20099.6944

But, we know that the value of perpetuity is after the end of year 5, because the payment was not made at that period. So, we will discount the future value to arrive at present value

Present Value = Future Value / ( 1 + interest rate) ^ Number of Periods

= 20099.6944 / ( 1 +  6.13635%)^5

=  20099.6944 / (1.0613635)^5

=  20099.6944 / 1.34685468534

=14923.432

Present Value of perpetuity = 14923.432 [ Approximately]