Present Value: Some Additional Cash Flow Patterns - Financial Management

The discussion of present value thus far has focused on two cash flow patterns: single payments and annuities. The present value of three additional types of cash flow streams are examined in this section: namely, perpetuities, uneven cash flows, and deferred annuities. Examples of these types of cash flows are encountered in many different areas of financial decision making.

Timeline of a Present Value of an Annuity Due (PMT = $1,000; i = 6%; n = 5)


A perpetuity is a financial instrument that promises to pay an equal cash flow per period forever; that is, an infinite series of payments. Therefore, a perpetuity may be thought of as an infinite annuity. Some bonds (and some preferred stocks) take the form of a perpetuity because these special securities never mature; that is, there is no obligation on the part of the issuer to redeem these bonds at their face value at any time in the future. A financial instrument such as this provides the holder with a series of equal, periodic payments into the indefinite future.

Consider, for example, a financial instrument that promises to pay an infinite stream of equal, annual payments (cash flows) of PMTt = PMT for t = 1, 2, 3, . . . years; that is, PMT1 = PMT2 = PMT3 = . . . = PMT. If we wish to find the present value (PVPER0) of this financial instrument, it can be represented as follows:

PVPER0=PMT/(1+i) + PMT/(1+ i)2 + PMT/(1+i)3


where i equals the rate of return required by an investor in this financial instrument. It should be apparent that Equation represents a special type of annuity where the number of periods for the annuity equals infinity. This type of problem cannot be solved using Table. For example, assume that Kansas City Power & Light series E preferred stock promises payments of $4.50 per year forever and that an investor requires a 10 percent rate of return on this type of investment. How much would the investor be willing to pay for this security? An examination of the PVIFA interest factors for 10 percent indicates that the value in the 10 percent column increases as the number of years increases, but at a decreasing rate. For example, the PVIFA factor for 10 percent and 10 years is 6.145, whereas the factor for 10 percent and 20 years is only 8.514 (much less than twice the 10-year factor).

The limiting value in any column of Table is 1 divided by the interest rate of that column, i. In the case of a 10 percent perpetuity, the appropriate interest factor is 1/0.10, or 10. Thus Equation can be rewritten as follows:

PV = PMT /i

In this example, the value of a $4.50 perpetuity at a 10 percent required rate of return is given as

$4.50 PVPER0 = ——— 0.10 = $45

Present Value of an Uneven Payment Stream

Many problems in finance —particularly in the area of capital budgeting —cannot be solved according to the simplified format of the present value of an annuity because the periodic cash flows are not equal. Consider an investment that is expected to produce a series of unequal payments (cash flows),PMT1,PMT2,PMT3, . . . ,PMTn, over the next n periods. The present value of this uneven payment stream is equal to the sum of the present values of the individual payments (cash flows). Algebraically, the present value can be represented as

PV0 =PMT1/(1+i) + PMT2/(1+i)2 + PMT3(1+I)3 + .....+ PMTn/(1+i)n

Present Value of an Uneven Payment Stream

where i is the interest rate (that is, required rate of return) on this investment and PVIFi, t is the appropriate interest factor from Table. It should be noted that the payments can be either positive (cash inflows) or negative (outflows). Consider the following example. Suppose The Gillette Company is evaluating an investment in new equipment that will be used to manufacture a new product it has developed. The equipment is expected to have a useful life of five years and yield the following stream of cash flows (payments) over the 5-year period:


Note that in year 3, the cash flow is negative. (This is due to a new law that requires the company to purchase and install pollution abatement equipment.) The present value of these cash flows, assuming an interest rate (required rate of return) of 10 percent, is calculated using Equation as follows:

PV0 = $100,000 (PVIF0.10, 1) + $150,000(PVIF0.10, 2)
– $50,000(PVIF0.10, 3) + $200,000(PVIF0.10, 4)
+ $100,000(PVIF0.10, 5)
= $100,000(0.909) + $150,000(0.826)
*$50,000(0.751) + $200,000(0.683) + $100,000(0.621)
= $375,950

illustrates a timeline for this investment. The present value of the cash flows ($375,950) would be compared with the initial cash outlay (that is, net investment in year 0) in deciding whether to purchase the equipment and manufacture the product. As will be seen later in the text, during the discussion of capital budgeting, calculations of this type are extremely important when making decisions to accept or reject investment projects.

Timeline of a Present Value of Unequal Payments (i = 10%, n = 5)

Timeline of a Present Value of Unequal Payments (i = 10%, n = 5)


Present Value of Deferred Annuities

Frequently, in finance, one encounters problems where an annuity begins more than one year in the future. For example, suppose that you wish to provide for the college education of your daughter. She will begin college five years from now, and you wish to have $15,000 available for her at the beginning of each year in college.How much must be invested today at a 12 percent annual rate of return in order to provide the 4-year, $15,000 annuity for your daughter?

This problem can be illustrated in the timeline given in Figure. Four payments of $15,000 each are required at the end of years 5, 6, 7, and 8. Of course, this problem could be solved by finding the sum of the present values of each of the payments as follows:


Present Value of Deferred Annuity = $28,950

Present Value of Deferred Annuity = $28,950

It should be apparent that this would be an extremely tedious method of calculation in the case of a 10 -year -deferred annuity, for example. Figure illustrates one alternative means of solving this problem. First, you can calculate the present value of the 4-year annuity, evaluated at the end of year 4 (remember that this is the same as the beginning of year 5). This calculation is made by multiplying the annuity amount ($15,000) by the PVIFA for a 4-year, 12 percent annuity. This factor is 3.037 and can be obtained from Table IV. Next the present value of the annuity ($45,555), evaluated at the end of year 4 (PVAN4),must be discounted back to the present time (PV0). Hence, we multiply $45,555 by a PVIF for 12 percent and four years.

This factor, obtained from Table, is equal to 0.636. The present value of the deferred annuity is $28,973. (This differs from the amount calculated earlier due to rounding in the tables. No difference will exist if this problem is solved with a calculator or tables that are carried out to more decimal places.) If you have $28,973 today and invest it in an account earning 12 percent per year, there will be exactly enough money in the account to permit your daughter to withdraw $15,000 at the beginning of each year in college. After the last withdrawal, the account balance will be zero.

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