EME 460
Geo-Resources Evaluation and Investment Analysis

Compound Interest Formulas III

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5. Uniform Series Present-Worth Factor

The fifth group in Table 1-5 covers a set of problems that uniform series of equal investments, A, occurred at the end of each time period for n number of periods at the compound interest rate of i. In this case, the cumulated present value of all investments, P, needs to be calculated. In summary, P is unknown and A, i, and n are given parameters. And the problem can be noted as P/Ai,n and displayed as:

P=? A A A A 0

0 1 2 ... n-1 n

Figure 1-7:Uniform Series Present-Worth Factor, P/Ai,n

If we replace substitute F in Equation 1-3 from Equation 1-2, we will have the present value as:

Equation 1-3: F = A[(1+i)n-1]/i
Equation 1-2: F = P(1+i)n
P(1+i)n =A[(1+i)n-1]/i

Then,

P =A[(1+i)n-1]/[i(1+i)n]

Equation 1-5

Equation 1-5 gives the cumulated present value, P, of all uniform series of equal investments, A, as P = A[(1+i)n-1]/[i(1+i)n]. And also can be noted as: P = A * P/Ai,n.The factor [(1+i)n−1]/[i(1+i)n] is called the “uniform series present-worth factor” and is designated by P/Ai,n. This factor is used to calculate the present sum, P that is equivalent to a uniform of equal end of period payments, A. Then P/Ai,n= A[(1+i)n−1]/[i(1+i)n].

Note that n is the number of time periods that equal series of payments occur.

Please review the following video, Uniform Series Present Worth Factor (Time 3:35).

Uniform Series Present Worth Factor
Click for the transcript of "Uniform Series Present Worth Factor" video.

PRESENTER: The fifth group covers the set of problems that P is a known parameter, A, i and n are given variables. In these problems, we have uniform series of equal investments, A, in the end of each time period, for n number of periods, at the compound interest rate of I.

And the problem asks you to calculate the accumulated present value of all investments, P. We can summarize these questions using the factor notation. P is the unknown variable, and should be on the left side. And A is the given, and should be written on the right side.

As explained before, equation 1-3 returns the future value, F, from A, i and n. And equation 1-2 calculates the future value, F, from present value, P, interest rates, i and n number of periods. So if we substitute F in equation 1-3 from equation 1-2, we will have this new equation-- 1-5. This equation gives us the accumulated present value of equal series payments, A, paid for n period, at interest rate of i.

Equation 1-5 can also be written according to factor notation. P equals A times the factor P over A. This factor is called Uniform Series Present-Worth Factor, which is used to calculate the presence on P that is equivalent to a uniform series of equal payments, and of the period payments, A.

For example, what would be the present value of 10 uniform investments of \$2,000, invested at the end of each year, for interest rate of 12%, compounded annually? First, we draw the time line. Left hand side is a present time, time zero payment, which needs to be calculated. N equals 10, because there are 10 uniform investments.

So we have 10 years. And above each year, we have \$2,000, starting from year one to year 10. So A equals \$2,000, n is 10, and interest rate is 12%. Using the factorization, P equals A, multiply the factor-- i is 12%, and n is 10. And the result.

So if you save \$2,000 per year, at the end of each year for 10 years, starting from year one to year 10, the accumulated money is equal to \$11,300 at present time. It has the same value as \$11,300 at the present time.

Credit: Farid Tayari

Example 1-5:

Calculate the present value of 10 uniform investments of 2000 dollars to be invested at the end of each year for interest rate 12% per year compound annually.

P=? A=$2000 A=$2000 A=$2000 A=$2000 0

0 1 2 ... 9 10

So,
A =$2000
n =10
i =12%
P=?

Using Equation 1-5, we will have:
P = A * P/Ai,n =A[(1+i)n−1]/[i(1+i)n]
P = A * P/A12%,10=2000*[(1+0.12)10−1]/[0.12(1+0.12)10]
P = 2000*5.650223 = $11,300.45

Note that we use the factor P/Ai,n  when we have equal series of payments. i is the interest rate and n is the number of equal payments. There is an important assumption here, the first payment has to start from year 1. In that case P/Ai,n  will return the equivalent present value of the equal payments.

Now let's consider the case that we have equal series of payments and the first payment doesn't start from year 1. In that case the factor P/Ai,n  will give us the equivalent single value of equal series of payments in the year before the first payment. However, we want the present value of them (at year 0). So, we need to multiply that with the factor P/Fi,n and discount it to the present time (year 0).
 

Example: 

P=? A=$2000 A=$2000 A=$2000 0

0 1 2 ... 10 11

Note that there are 10 equal series of $2,000 payments. But the first payment is not in year 1. The factor P/A12%,10 returns the equivalent value of these 10 payments to the year before the first payment, which is year 1.

P=? $2000(P/A12%,10) 0

0 1 2 ... 10 11

However, we want the present value. So, we need to discount the value by one year to have the present value of 10 equal payments.

P=? $2000(P/A12%,10)(P/F12%,1) 0

0 1 2 ... 10 11

Present value = 2,000(P/A12%,10)(P/F12%,1)

Example: Now consider the the following case that the first payment starts at year 3:

P=? A=$2000 A=$2000 A=$2000 0

0 1 2 3 ... 10 12

Present value = 2,000(P/A12%,10)(P/F12%,2)

Table 1-10: Uniform Series Present-Worth Factor, P/Ai,n
Factor Name Formula Requested variable Given variables
P/Ai,n Uniform Series Present-Worth Factor [(1+i)n−1]/[i(1+i)n] P: Present value of uniform series ofequalinvestments A: uniform series of equal investments
n: number of time periods
i: interest rate

6.Capital-Recovery Factor

The sixth group in Table 1-5 belongs to set of problems that A is unknown and P, i, and n are given parameters. In this category, uniform series of an equal sum, A, is invested at the end of each time period for n periods at the compound interest rate of i. In this case, the cumulated present value of all investments, P, is given and A needs to be calculated. It can be noted as A/Pi,n.

P A=? A=? A=? A=? 0

0 1 2 ... n-1 n

Figure 1-8: Capital-Recovery Factor, A/Pi,n

Equation 1-5 can be rewritten for A (as unknown) to solve these problems:

A =P[i(1+i)n]/[(1+i)n-1]

Equation 1-6

Equation 1-6 determines the uniform series of equal investments, A, from cumulated present value, P, as A = P[i(1+i)n]/[(1+i)n-1].The factor [i(1+i)n]/[(1+i)n−1] is called the “capital-recovery factor” and is designated by A/Pi,n. This factor is used to calculate a uniform series of end of period payment, A that are equivalent to present single sum of money P.

Note that n is the number of time periods that equal series of payments occur.

Please watch the following video, Capital Recovery Factor (Time 3:37).

Capital Recovery Factor
Click for the transcript of "Capital Recovery Factor " video.

PRESENTER: The sixth group belongs to the set of problems that A is unknown and P, i, and n are given parameters. This category is similar to the fifth group, but P is given and A needs to be calculated. In this category of problems, we know the present value P, or accumulated present value of all payments. And we want to calculate the uniform series of equal sum A that are invested in the end of each time period for n periods at the compound interest rate of i.

So we have present value P, and we want to calculate equivalent A, given interest rate of i and number of periods n. The proper factor to summarize these questions is A over P, or A/P. A is the unknown variable, is on the left side, and P, given variable, on the right side.

Equation to calculate A is straightforward. We just need to rewrite the equation in 1-5 for A as unknown, and we will have equation 1-6 that calculates A from P, i, and n. If we write the equation 1-6 according to the factor notation, we will have factor A over P. The factor is called capital recovery factor and is used to calculate uniform sales of end of period payments A that are equivalent to present single sum of money P.

Let's work on this example. We want to know the uniform series of equal investment for five years at interest rate of 4% which are equivalent to \$25,000 today. Let's say you want to buy a car today for \$25,000, and you can finance the car for five years and 4% of interest rate per year, compounded annually. And you want to know how much you have to pay each year.

First, we draw the timeline. Left side is the present time, which we have \$25,000. n equals 5, and above each year, starting from year one to year five, we have A that has to be calculated. For the factor, we have i equal 4% and n is five and the result, which tells us \$25,000 at present time is equivalent to five uniform payments of \$5,616 starting from year one to year five with 4% annual interest rate. Or \$25,000 at present time has the same value of five uniform payments of \$5,616 starting from year one to year five with 4% annual interest rate.

Credit: Farid Tayari

Example 1-6:

Calculate uniform series of equal investment for 5 years from present at an interest rate of 4% per year compound annually which are equivalent to 25,000 dollars today. (Assume you want to buy a car today for 25000 dollars and you can finance the car for 5 years with 4% of interest rate per year compound annually, how much you have to pay each year?)

P=$25,000 A=? A=? A=? A=? A=? 0

0 1 2 3 4 5

Using Equation 1-6, we will have:
A = P * A/Pi,n = P[i(1+i)n]/[(1+i)n−1]
A = P * A/P4%,5 = 25,000 * [0.04(1+0.04)5/[(1+0.04)5-1]
A = 25,000 *0.224627=5615.68

So, having $25,000 at the present time is equivalent to investing $5,615.68 each year (at the end of the year) for 5 years at annual compound interest rate of 4%.

Table 1-11: Capital-Recovery Factor, A/Pi,n
Factor Name Formula Requested variable Given variables
A/Pi,n Capital-Recovery Factor [i(1+i)n]/[(1+i)n-1] A:uniform series of equal investments P: Present value of uniform series of equal investments
n: number of time periods
i: interest rate

Note that
A/Pi,n=A/Fi,n*F/Pi,n=P[i(1+i)n]/[(1+i)n−1]

Using these six techniques, we can solve more complicated questions.

Example 1-7:

Assume a person invests 1000 dollars in the first year, 1500 dollars in the second year, 1800 dollars in the third year, 1200 dollars in the fourth year and 2000 dollars in the fifth year. At an interest rate of 8%:
1) Calculate time zero lump sum settlement “P
2) Calculate end of year five lump sum settlement “F”, that is equivalent to receiving the end of the period payments
3) Calculate five uniform series of equal payments "A", starting at year one, that is equivalent to above values.

P=? 1000 1500 1800 1200 2000 F=?

0 1 2 3 4 5

1) Time zero lump sum settlement “P” equals the summation of present values:
P =1000 *(P/F8%,1)+ 1500 *(P/F8%,2)+ 1800 *(P/F8%,3)+ 1200 *(P/F8%,4)+ 2000 *(P/F8%,5)
P =1000 * 0.92593 + 1500 * 0.85734 + 1800 * 0.79383 + 1200 * 0.73503 + 2000 * 0.68058
P =5884.03

2) End of year five lump sum settlement “F”, that is equivalent to receiving the end of the period payments equals the summation of future values:
F= 1000 *(F/P8%,(5-1))+ 1500 *(F/P8%,(5-2))+ 1800 *(F/P8%,(5-3))+ 1200 *(F/P8%,(5-4))+ 2000
F= 1000 *(F/P8%,4)+ 1500 *(F/P8%,3)+ 1800 *(F/P8%,2)+ 1200 *(F/P8%,1)+ 2000
F= 1000 * 1.36049 + 1500 * 1.25971 + 1800 * 1.1664 + 1200 * 1.08 + 2000
F= 8645.58

Please note thatin the factor subscript,nis the number of time period difference betweenF(the time that future value has to be calculated) andP(the time that the payment occurred). For example, 1800 payment occurs in year 3 but we need its future value in year 5 (2 year after) and time difference is 2 years. So, the proper factor would be:(F/P8%,(5-3)) or(F/P8%,2).

3) Uniform series of equal payments "A" can be calculated from either P or F :
A = 5884.03 *A/P8%,5 = 5884.03 * 0.25046 = 1473.7
or
A = 8645.58 *A/F8%,5 = 8800.71 * 0.17046 = 1473.7

Example 1-8: repeat your calculations for the following payments:

P=? 800 1000 1000 1600 1400 F=?

0 1 2 3 4 5

1) Time zero lump sum settlement “P” equals the summation of present values:
P= 800 + 1000 *(P/F8%,1)+ 1000 *(P/F8%,2)+ 1600 *(P/F8%,3)+ 1400 *(P/F8%,4)
P= 800 + 1000 * 0.92593 + 1000 * 0.85734 + 1600 * 0.79383 + 1400 * 0.73503
P= 4882.44

2) End of year five lump sum settlement “F”, that is equivalent to receiving the end of the period payments equals the summation of future values:
F= 800 *(F/P8%,5) +1000 *(F/P8%,4)+ 1000 *(F/P8%,3)+ 1600 *(F/P8%,2)+ 1400 *(F/P8%,1)
F= 800 * 1.46933 + 1000 * 1.36049 + 1000 * 1.25971 + 1600 * 1.1664 + 1400 * 1.08
F= 7173.9

3) Uniform series of equal payments "A" can be calculated from eitherPorF:
A= 4882.44 *A/P8%,5= 4882.44 * 0.25046 = 1222.84
or
A= 7173.9 *A/F8%,5= 7173.9 * 0.17046 = 1222.84