## 2000 2000 2000 2000

Figure 4.3. Uniform series cash flow

Figure 4.3. Uniform series cash flow

Fortunately, uniform series occur frequently enough in practice to justify tabulating values to eliminate the need to repeatedly sum a series of (PI F,i,n) factors. To accommodate uniform series factors, we need to add a new symbol to our time value of money terminology in addition to the single sum symbols P and F. The symbol "A" is used to designate a uniform series of cash flows. When dealing with uniform series cash flows, the symbol A represents the amount of each annual cash flow and the n represents the number of cash flows in the series. The factor (PI A,i,n) is known as the uniform series, present worth factor and is read "to find P given A at i% for n years." Tables of (PIA,i,n) are provided in Appendix 4A. An algebraic expression can also be derived for the (PIA,i,n) factor which expresses P in terms of A, i, and n. The derivation of this formula is omitted here, but the resulting expression is shown in the summary table (Table 4.6) at the end of this section.

An important observation when using a (PIA,i,n) factor is that the "P" resulting from the calculation occurs one period prior to the first "A" cash flow. In our example the first withdrawal (the first "A") occurred one year after the deposit (the "P"). Restating the example problem above using a (PIA,i,n) factor, it becomes:

P = A * (PIA,i,n) P = 2000 * (PIA,9%,4) P = 2000 * (3.2397) = $6479.40

This result is identical (as expected) to the result using the (PI F,i,n) factors. In both cases the interpretation of the result is as follows: if we deposit $6479.40 in an account paying 9%/yr interest, we could make withdrawals of $2000 per year for four years starting one year after the initial deposit to deplete the account at the end of 4 years.

The reciprocal relationship between P and A is symbolized by the factor (AIP,i,n) and is called the uniform series, capital recovery factor. Tables of (A I P,i,n) are provided in Appendix 4A and the algebraic expression for (AIP,i,n) is shown in Table 4.6 at the end of this section. This factor enables us to determine the amount of the equal annual withdrawals "A" (starting one year after the deposit) that can be made from an initial deposit of "P."

### Example 9

Determine the equal annual withdrawals that can be made for 8 years from an initial deposit of $9000 in an account that pays 12%/yr. The first withdrawal is to be made one year after the initial deposit.

A = P * (AIP,12%,8) A = 9000 * (0.2013) A = $1811.70

Factors are also available for the relationships between a future worth (accumulated amount) and a uniform series. The factor (FIA,i,n) is known as the uniform series future worth factor and is read "to find F given A at i% for n years." The reciprocal factor, (AIF,i,n), is known as the uniform series sinking fund factor and is read "to find A given F at i% for n years." An important observation when using an (FIA,i,n) factor or an (AIF,i,n) factor is that the "F" resulting from the calculation occurs at the same point in time as to the last "A" cash flow. The algebraic expressions for (AIF,i,n) and (FIA,i,n) are shown in Table 4.6 at the end of this section.

### Example 10

If you deposit $2000 per year into an individual retirement account starting on your 24th birthday, how much will have accumulated in the account at the time of your deposit on your 65th birthday? The account pays 6%/yr.

n = 42 (birthdays between 24th and 65th, inclusive) F = A * (FIA,6%,42) F = 2000 * (175.9505) = $351,901

### Example 11

If you want to be a millionaire on your 65th birthday, what equal annual deposits must be made in an account starting on your 24th birthday? The account pays 10%/yr.

n = 42 (birthdays between 24th and 65th, inclusive) A = F * (AIF,10%,42)

4.6.8 Gradient Series

A gradient series of cash flows occurs when the value of a given cash flow is greater than the value of the previous period's cash flow by a constant amount. The symbol used to represent the constant increment is G. The factor (PIG,i,n) is known as the gradient series, present worth factor. Tables of (PIG,i,n) are provided in Appendix 4A. An algebraic expression can also be derived for the (PIG,i,n) factor which expresses P in terms of G, i, and n. The derivation of this formula is omitted here, but the resulting expression is shown in the summary table (Table 4.6) at the end of this section.

It is not uncommon to encounter a cash flow series that is the sum of a uniform series and a gradient series. Figure 4.4 illustrates such a series. The uniform component of this series has a value of 1000 and the gradient series has a value of 500. By convention the first element of a gradient series has a zero value. Therefore, in Figure 4.4, both the uniform series and the gradient series have length four (n=4). Like the uniform series factor, the "P" calculated by a (PI G,i,n) factor is located one period before the first element of the series (which is the zero element for a gradient series).

### Example 12

Assume you wish to make the series of withdrawals illustrated in Figure 4.4 from an account which pays 15%/yr. How much money would you have to deposit today such that the account is depleted at the time of the last withdrawal?

This problem is best solved by recognizing that the cash flows are a combination of a uniform series of value 1000 and length 4 (starting at time=1) plus a gradient series of size 500 and length 4 (starting at time=1).

P = A * (PIA,15%,4) + G * (PIG,15%,4) P = 1000 * (2.8550) + 500 * (3.7864)

Occasionally it is useful to convert a gradient series to an equivalent uniform series of the same length. Equivalence in this context means that the present value (P) calculated from the gradient series is numerically equal to the present value (P) calculated from the uniform series. One way to accomplish this task with the time value of money factors we have already considered is to convert the gradient series to a present value using a (PI G,i,n) factor and then convert this present value to a uniform series using an (A I P,i,n) factor. In other words:

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