Question

What is the difference between the payment (PMT) and present value (PV) in a perpetuity? What are the proper definitions of each?

For example, this problem that I'm solving now:

"What should you be willing to pay in order to receive $5,000 every six months forever, if you require 8% per year on the investment"?

Using the formula:

Perpetuity Present Value = Payment/ Interest Rate

I initially started solving for Payment (PMT) (because it directly asks you to), but my teacher solved it looking for Present Value (PV) instead.

Why is this and how can I learn do differentiate what they're actually asking for?

Answer 1

What is the difference between the payment (PMT) and present value (PV) in a perpetuity? What are the proper definitions of each?

There is no dearth of definition for PMT and PV on internet and textbooks. But let me try to explain to you quite simplistically and to some extent intuitively.

Payment in a case of perpetuity stands for the regular, periodic and perpetual cash flow associated with the perpetuity. Let's say you have to make a payment of $ 100 every month in perpetuity. So, there is a regular (every month), period (monthly) and perpetual (forever) cash outflow of $ 100 under this perpetuity. This $ 100 is payment.

However, this payment of $ 100 is going to occur at different point in time, perpetually. Totay, if i want to caclulate the worth of this perpetuity or total value of this perpetuity, I will calculate the present value of each of the regular, periodic payment of $ 100 that i will make under this perpetuity. So, Present value of perpetuity = PV = $ 100 / (1 + r) + 100 / (1 + r)² + 100 / (1 + r)³ + .....till infinity = $ 100 / r = Payment / interest rate or discount rate.

Thus payment is the regular payment that you make under perpetuity. And PV is the present value of all such payments made under this perpetuity.

While using the formula:

Perpetuity Present Value = Payment / Interest Rate; please note that payment is already known to you. The payment that you are receiving under perpetuity is $ 5,000 every six month. What you should be willing to pay today for this perpetuity is nothing but PV of all the future payments I am going to receive in perpetuity. Hence, the solution should be like this:

Interest rate = 8%

Payment received per annum = 2 x 5,000 = $ 10,000

PV of all the payment received = Payment received per annum / interest rate per annum = 10,000 / 8% = $ 125,000

Hence, you should be willing to pay $ 125,000 today, in order to receive $5,000 every six months forever, if you require 8% per year on the investment.

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Though you have not stated explicitly, I strongly believe that you got confused between the excel or financial calculator function of PMT and PV.

So, let me try and explain to you this untold problem.

Please note that the usual PMT and PV functions that help you on excel of financial calculator are all designed to given you results for annuity i,e. for cash flows occurring over finite time period. In each of the two function PMT and PV, you need to provide an input for period or nperiod or n to get the output. The value of period, nperiod or n has to be finite to get an output using PMT or PV function. In our case, it's a perpetuity. Hence, you won't be able to get answers from the usual PMT and PV functions. So even if you started solving for PMT, you would have got stuck and, subsequently would have been forced to change your direction.

You have to necessarily apply your understanding of the situation of the perpetuity to answer a question related to perpetuity. The solutions to such problems csn't be found using PMT or PV functions.