What is Perpetuity and Deferred Perpetuity?

Perpetuity: 
It is a fixed series of payments received in infinite periods.
Example: Console bond has no maturity period and it pays fixed coupon.
Growing Perpetuity: It is a fixed series of payments receives at a constant growth rate for infinite periods.                            








Deferred Perpetuity: It is fixed series of cash flows received at a future date.                                

Perpetuity Vs Annuity:

In perpetuity payment received for infinite period and in annuity payment received for fixed period.The formula to calculate perpetuity and annuity is also different, in annuity the formula is  C[1-(1/(1+r)ⁿ/r) and the formula for perpetuity is C/r.

Formula:

	Simple  perpetuity	Growing  Perpetuity	Deferred  Perpetuity	Deferred Growing  Perpetuity
PV	C1/r	C1/r-g	PVp /(1+r) t	PV GP /(1+r) t

Where,

C₁= Initial cash flow

R= rate per period

G= growth rate

PV_P=Preset Value of perpetuity

PV_GP=Present Value of Growing Perpetuity

Perpetuity Problem:

Example 1: How much amount do you need to invest in perpetuity today and get Rs.5000 each year in future, starting from next year @ 8% per annum?

PV_P = C₁/r

=5000/0.08

=62500

Example 2: What is the present value of perpetuity if the future amount amount is Rs.6, 000 and you will start receiving payment after retirement at the rate of 10% quarterly.

Solution : PV_P = C₁/r

= 6, 000 / 0.025

= 2, 40, 000

Growing Perpetuity Problem:

Example 3: Dividend of a company A’s tock is expected to increase every quarter @ 0.5% and the rate of return of stock is 7 % annually and stock is paying dividend of Rs.20 each quarterly. Find out the present worth of stock?

Solution: 

PV_GP = C₁/r-g

=20/1.705^*-0.5

=3478

*EAR of 7% = (1+0.07) ^1/4-1 =0.01705 =1.705

Example 4: Find out the present value of perpetuity if the cash inflow increases from Rs. 2, 000 to Rs. 2, 050 in next year and Rs. 2, 124, Rs. 2, 209 in 3^rd and 4^th year respectively. The discount rate is 9% per annum.

Solution: First of all you have to find out the growth rate of 2^nd 3^rd and 4^th year that is-

(2, 124 – 2, 000) / 2, 000 = 2.5

(2, 124 – 2, 050) / 2050 = 3.6

(2, 209 – 2, 124) / 2, 124 = 4

Then calculate present value,

PV₀ = C₁ / (r-g) +  C₂ / (r-g) + C₃ / (r-g) + C₄ / (r-g)

= 2, 000/ (0.09 – 0) + 2, 050/ (0.09 – 0.025) + 2, 124/ (0.09 – 0.036) + 2, 209/ (0.09 – 0.04)

= 22, 222 + 31, 539 + 39, 333 + 44, 180

= 1, 37, 274

Deferred Perpetuity problem:

Example 5: Find out the present value of deferred perpetuity if you start receiving Rs.9000 cash annually from 3^rd year @ 6% interest?

Solution:

PV₂=C₁/r

= 9000/0.06

=150000

PV₀= PV₂/ (1+r) ^t

= 150000/ (1+0.06)²

=1, 33, 499.46

Example 6: Neha has turned 16 today and her father had taken a growing perpetuity. The first payment amounting Rs. 12000 will occur when Neha turn 19.Every year the amount she receives will increase @3% and also earns interest @7.8 % per annum. Find out present value of deferred growing annuity?

Solution:

PV₁₈= C₁/r-g

=12000/0.078-0.03

= 250000

PV ₁₆Or PV₀= PV₁₈/ (1+r) ²

= 250000/ (1+0.078)²

=215130.74