NPV and the Time Value of Money

In Chapter 3, we discussed the time value of money and its related cash flows over a single period. We now take this concept a step further and study the concept of the time value of money with cash flows lasting over several periods.

The Timeline

A series of cash flows lasting several periods is referred to as a stream of cash flows. These streams of cash flows can be represented by the use of a timeline, which is a linear representation of the timing of the expected cash flows. With a timeline, inflows of cash (cash flows received) are represented by a positive number whereas outflows of cash (cash flows paid out) are represented by a negative number.

Valuing Cash Flows at Different Points in Time

It is only possible to compare or combine values at the same point in time.

In order to compare or combine cash flows that happen to occur at different points in time, these cash flows must first be converted into the same units by moving them to the same point in time.

The process of moving forward along the timeline to determine a cash flow's value in the future (future value) is known as compounding.
To calculate a cash flow's future value, it must be compounded.
We do this by multiplying the cash flow by (1+r) with r being the interest rate. When money is compounded, you earn interest on interest... that is you are earning interest on your original principle in addition to the earned interest. This phenomenon is referred to as compound interest. The follow formula represents the FV of a Cash Flow:

n = C * (1+r)n

The process of moving backwards on the timeline to find the equivalent value of a future cash flow today
(present value) is known as discounting. To calculate the value of a future cash flow at an earlier point in time, it must be discounted. Instead of multiplying the cash flow by (1+r), we now divide. The following formula represents the PV of a Cash Flow:

PVn = C / (1+r)n

Valuing a Stream of Cash Flows

The PV of a cash flow stream is the sum of the present values of each cash flow.

PV = C0 + C1/(1+r) + C2/(1+r)2 + Cn/(1+r)n

The Net Present Value of a Stream of Cash Flows

According to the valuation principle, the value of any decision is the value of its benefits minus its costs.

NPV = PV(benefits) - PV(costs) = PV(benefits - costs)

Perpetuities, Annuities, and Other Special Cases

A perpetuity is a stream of equal cash flows that occur at regular intervals and last forever. An example of perpetuity would be a consol, or perpetual bond, which is a bond issued by the British government. With a perpetuity, it is important to remember that the first cash flow does not start the end of the first period. This is often referred to as payment in arrears. The PV of a perpetuity is calculated as follows:

PV(C in perpetuity) = C/r or C = r * P where P is the principle, r is the interest rate, and C is the amount paid out

A growing perpetuity, on the other hand, is a stream of cash flows that occur at regular intervals and grow at a constant rate forever. The PV of a growing perpetuity is as follows:

PV(growing perpetuity) = C/(r-g) where g is the rate of growth

(Note: if the rate of growth is larger than the discount rate, the PV is infinite)

An annuity is a stream of N equal cash flows paid at regular intervals. An annuity eventually ends after so many periods, unlike a perpetuity. Just as with a perpetuity, annuity payments begin at the end of the first period.

PV(annuity) = C * 1/r (1 - 1 / (1+r)n)

FV(annuity) = C * 1/r ((1+r)n - 1)

Solving for Variables Other Than Present or Future Value

When dealing with loans, remember that the amount borrowed represents the present value of the loan. If you know how long the loan is for and the interest rate, you can easily solve for the payment, C. Notice that this formula for a loan payment is the inverse of the annuity formula.

Loan Payment: C = P / (1/r) (1 - 1/(1+r)n)

The internal rate of return is the interest rate that sets the NPV of the cash flows equal to zero.