Perpetuity

Most investments pay for a fixed time. A 5-year deposit pays interest for 5 years, a 10-year annuity pays for 10 years, and then the payments stop.

A perpetuity is a cash flow that goes on forever. The workbook defines it as "a cash flow from an investment that goes on forever", with no finite period for which the cash flow is received.

Its two examples are:

• Perpetual bonds. The workbook calls these "the best example of real life perpetuity". The interest keeps coming year after year.
• Pensions payable to a pensioner for life. The workbook gives this as the everyday example.

You might expect a payment that never ends to be worth an infinite amount. It is not, because of the time value of money. A rupee received far in the future is worth very little today. Each later payment adds less than the one before it, and the total settles at a fixed figure.

The workbook values a perpetuity the same way it values any cash flow: it finds the present value. Each payment is discounted back to today, and the payments are added up:

PV = C/(1+r)^1 + C/(1+r)^2 + C/(1+r)^3 + .....

Because the cash flow is the same every period and never stops, this endless sum works out to a very short formula. In the workbook's words, the calculation ends up as "the cash flow divided by the discount rate".

The workbook's example (Chapter 2). A perpetual bond pays Rs 10,000 of interest every year. At a discount rate of 8%:

PV = 10,000 / 0.08 = Rs 1,25,000

The same idea in bond pricing (Chapter 9). A perpetual bond pays its agreed coupons but never repays the face value and has no maturity date. Its value is Coupon / Yield. The workbook's example: an 8% coupon on a face value of 100, when the investor wants an annual yield of 6%, is worth (100 × 8%) / 6% = 133.33. The coupon rate is above the yield, so the bond is worth more than its face value.

Two things drive the answer, and they pull in opposite directions:

The rate matters a great deal. Every payment is discounted at r, forever, so a small change in r makes a big change in value.

Scope. The X-A workbook teaches only the level perpetuity, where the cash flow stays constant. It does not give a formula for a perpetuity whose payments grow, so do not bring one into an X-A answer.

PV = C / r

PV = Present value
C  = Cash flow received each period
r  = Discount rate for the same period

Perpetual bond:  Value = Coupon / Yield

Keep C and r on the same period. Use an annual cash flow with an annual rate, or a monthly cash flow with a monthly rate.

1. The workbook's corpus question. One of the Module 6 case studies has X, in his early thirties, wanting a guaranteed Rs 7,500 a month for his parents' medicines. The yield on offer is 6%, and the hint says the money is needed for perpetuity.

Annual income needed   7,500 x 12     = Rs 90,000
Corpus                 90,000 / 0.06  = Rs 15,00,000

Same answer in monthly terms:
Monthly rate           6% / 12        = 0.5%
Corpus                 7,500 / 0.005  = Rs 15,00,000

The answer is Rs 15,00,000. The trap is to divide Rs 7,500 by 0.06. That gives Rs 1,25,000, because it mixes a monthly cash flow with an annual rate.

2. A family scholarship (illustrative numbers). Mrs Rao of Visakhapatnam wants to fund a scholarship of Rs 60,000 a year, paid at the end of each year, for as long as the school exists. How much must she set aside?

At 8%    60,000 / 0.08  = Rs 7,50,000
At 6%    60,000 / 0.06  = Rs 10,00,000
At 10%   60,000 / 0.10  = Rs 6,00,000

If the assumed rate falls by 2 percentage points, from 8% to 6%, she needs Rs 2,50,000 more. That is a third more money for the same scholarship.

Why an endless stream has a limit. Treat only the first 30 years as an ordinary annuity at 8%:

First 10 years   60,000 x 6.7101   = Rs 4,02,605   (about 54% of Rs 7,50,000)
First 30 years   60,000 x 11.2578  = Rs 6,75,467   (about 90% of Rs 7,50,000)
All years after year 30            = Rs 74,533

Every payment from year 31 onwards, added together, is worth less than Rs 75,000 today. That is why the value of a stream with no end still comes to a fixed number.

Chapter 2 (Time Value of Money) names the "Concept of Perpetuity" as a learning objective. Section 2.2.7 defines it, gives the formula PV = C / r and works the Rs 10,000 bond at 8%. The section comes straight after annuities (2.2.6), and questions often ask you to tell the two apart. Chapter 9, section 9.4.5 (Perpetual bonds and pricing of perpetual bonds), values an 8% coupon at a 6% yield using Coupon / Yield. The Module 6 case study on X uses the same division to find the corpus behind a monthly income for his parents. Expect a direct C / r calculation, sometimes with the cash flow given monthly.

• Divide by the rate as a decimal. Rs 10,000 / 0.08 = Rs 1,25,000, not Rs 10,000 / 8.
• Match the periods. A monthly income against an annual yield must be turned into annual terms first. Rs 7,500 a month at 6% needs Rs 15,00,000, not Rs 1,25,000.
• A perpetuity is not an annuity. An annuity pays for a set number of periods, and you need an annuity factor or the PV function. A perpetuity never ends, so C / r is enough. The workbook's case-study hint calls C / r the "annuity formula" and then says the money is needed "for perpetuity". Use C / r whenever the payments never stop.
• A perpetual bond never repays its face value. It has no maturity date, so there is no principal repayment to discount. Only the coupons count.
• A higher discount rate means a lower value. If the required yield is above the coupon rate, the perpetual bond is worth less than face value. If it is below, as in the workbook's 8% coupon at a 6% yield, it is worth more.
• Level cash flows only. The X-A formula assumes C stays the same forever. The workbook gives no growing-perpetuity version.
• Different usage in Series II-B (cross-series note). The II-B workbook says equity capital is "for perpetuity", meaning the company never has to repay it. That describes how long the capital stays in the company. It is not a valuation, and it is not what "perpetuity" means in X-A's time value of money chapter.