Additional Notes & Examples on Time Value of Money

Growing Annuity
A growing annuity, is a stream of cash flows for a fixed period of time, t, where the initial cash flow, C, is growing (or declining, i.e., a negative growth rate) at a constant rate g. If the interest rate is denoted with r, we have the following formula for the present value (=price) of a growing annuity:

PV = C [1/(r-g) - (1/(r-g))*((1+g)/(1+r))t ],
where:

PV = Present Value of the growing annuity
C = Initial cash flow
r = Interest rate
g = Growth rate
t = # of time periods

Example I:

Suppose you have just won the first prize in a lottery. The lottery offers you two possibilities for receiving your prize. The first possibility is to receive a payment of \$10,000 at the end of the year, and then, for the next 15 years this payment will be repeated, but it will grow at a rate of 5%.  The interest rate is 12% during the entire period. The second possibility is to receive \$100,000 right now. Which of the two possibilities would you take?

You want to compare the PV of the growing annuity to the PV of receiving \$100,000 right now (which is, obviously just \$100,000). So, here are the numbers:

C = \$10,000
r = 0.12
g = 0.05
t = 16

PV = 10,000 [(1/0.07) - (1/0.07)*(1.05/1.12)16] = \$91,989.41 < \$100,000, therefore, you would prefer to be paid out right now.

Example II:

Assume the same situation as in Example I, but with the difference that you can now make a choice between receiving a payment of 10,000 at the end of year 1, which will then grow at 5% per year, and be paid out to you for the next 15 years. Or, you can receive \$85,000 right now. What would you do?

We know from Example I that the present value of the growing annuity is equal to \$91,989.41. However, the annuity starts only at the end of year 1, and hence, we need to bring this value back one additional period before we can compare it to the \$85,000 to received right now. Thus:

PV = \$91,989.41 / (1.12) = \$82,133.40 < 85,000, so we still prefer to be paid out immediately.

Growing Perpetuity

A growing perpetuity is the same as a regular perpetuity (C/r), but just like we saw above, the cash flow is growing (or declining) each year. A perpetuity has no limit to the number of cash flows, it will go indefinitely.  The growing perpetuity is in that way just the same as a growing annuity with an extremely large t.

PV = C / (r-g),
where:

PV = Present Value of the growing perpetuity
C = Initial cash flow
r = Interest rate
g = Growth rate

Example I:

What would you be willing to pay (given that you could live forever, and hence could receive all the cash flows) for a preferred share of stock in the University of Pittsburgh, that promises you to pay a cash dividend to you at the end of the year of \$25, which will increase every year by 1%, forever. The interest rate is fixed at 4.75%.