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Uniform Annual Series and Present Value

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Uniform annual series and present value

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Suppose that there is a series of "n" uniform payments, uniform in amount and uniformly spaced, such as a payment every year. Let "A" be the amount of each uniform payment.

Let "P" be a single amount equivalent to the series, with "P" occurring one period before the first "A" payment. Note that although "P" is an abbreviation of "Present," the single amount "P" may actually occur in the future as long as it occurs exactly one period before the first "A" payment.

The relationship between P and A is

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

Example: Suppose that a recent college graduate has $3,000 available as a down payment on a new car. The graduate can afford a uniform car loan payment of no more than $500 per month for 48 months, beginning 1 month from now. Interest is 6%, compounded monthly. What is the most that the graduate can spend today on a new car?

Let X = most can spend (budget).

X = P + $3,000

A = $500 per month

i = 0.5% per month

n = 48 months

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

= $500 [ (1.005) 48 - 1 ] / [ (0.005) (1.005) 48 ] = $21,290

Or, using the 0.5% interest table, which is quicker:

P = A (P/A,0.5%,48) = $500 ( 42.580 ) = $21,290

X = $21,290 + $3,000 = $24,290

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Uniform annual series and present value

Question 1

Question 2

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Question 1.

Suppose that $30,000 is borrowed today at 12% interest. The loan is to be repaid by uniform annual payments for 5 years, beginning 1 year from now. Calculate the annual payment.

Choose an answer by clicking on one of the letters below, or click on "Review topic" if needed.

A A = P / n = $30,000 / 60 years = $500 per year

B A = P (A/P,12%,5) = $30,000 (0.2774) = $8,322 per year

C P = A (P/A,12%,5) = $30,000 ($3.6048)= $108,140 per year

D A = P (A/P,12%,60) = $30,000 (0.1201) = $3603.00 per year

Review topic

Question 2.

What single payment now is equivalent to a uniform series of $1000 per year for 20 years, beginning 6 years from now? Interest is 12%.

Choose an answer by clicking on one of the letters below, or click on "Review topic" if needed.

A P = A (P/A,12%,20) = $1000 (7.469) = $7,469

B P = A (P/A,12%,20) (P/F,12%,6) = $1000 (7.469) (0.5066) = $3,784

C P = A (P/A,12%,20) (P/F,12%,5) = $1000 (7.469) (0.5674) = $4,238

D P = A (P/A,12%,20) (F/P,12%,5) = $1000 (7.469) (1.762) = $13,160

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