Question

Suppose you inherit a perpetuity that will pay you $10,000 a year for the rest of your life.  You will receive the first $10,000 payment exactly three years from today. The interest rate is 5%.  

1. What is the perpetuity worth at the moment you start collecting payments (three years from now)
2. What is the perpetuity worth now?
3. A bank offers to exchange your perpetuity for another stream of yearly payments that also lasts forever but whose first payment will be made exactly one year from today instead of three.  What is the highest yearly payment that the bank is willing to offer you in exchange for your trust?

Answer 1

Part (a):

Present value at the time of commencement of perpetuity PV(Due)= PMT + (PMT/r)

Where PMT= periodical payment and r= Rate of interest.

Given, payments will commence 3 years from now.

PMT= $10,000.   Rate of interest r= 5%

Hence PV after 3 years from now= $10,000 + ($10,000/0.05)= $10,000 + $200,000 = $210,000

Part (b):

Worth of the perpetuity now is the present value of $210,000. The Present worth is $181,405.90 calculated as follows:

3 A B C D E 1 Present Value 2 3 Present value of an amount is calculated using the formula V=F/(1+r)^n 4 Where F= Amount after the specified period (Future Value), 5 n= Number of compounbding periods. 6 r= Rate of interest per period in decimals 7 All amounts in $ 8 Given that 9 Future value F= 210,000 No. of years (N)= 10 Discount Rate (% p.a)= R= 5 No. of times compounded a year (t) 11 12 Calculations: 13 Formula Cell reference Value 14 Discount rate per period in decimals (r )= R/(100*t) D10/(100*G10) 0.05 15 No. of periods (n)= N*t G9*G10 16 (1+r)= 1+G14 1.05 (1+r)^n= G16^G15 1.157625 18 Discounted value = F/(1+r)^n D9/617 181405.8957 Rounded to $181,405.90 20 17 19

Part (c):

The payment stream offered by the bank constitutes an ordinary perpetuity. Present value (as of today) of this perpetuity is equal to the one to be exchanged, ie., $181,405.90

PV of Ordinary perpetuity= PMT/r. Therefore, PMT= PV * r

Where PMT= periodical payment and r= Rate of interest.

Given, PV= $181,405.90 and r= 5% or 0.05

Therefore, maximum yearly payment to be offered (first payment at the end of year 1)

= $181,405.90 * 0.05 = $ 9,070.29