Lesson 8 of 8Discrete Functions

Unit 3 · Lesson 8 of 8

Simple Annuities

An annuity is a series of equal, regular payments — think loan payments or automatic savings deposits. Learn the future value and present value formulas for ordinary simple annuities.

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Key things to remember

1Ordinary simple annuity: equal payments $R$ at the end of each period, with the payment interval matching the compounding interval.
2Future value: $F V = R \cdot \frac{( 1 + i ) ^{n} - 1}{i}$ .
3Present value: $P V = R \cdot \frac{1 - ( 1 + i ) ^{- n}}{i}$ .
4 $n$ is the TOTAL number of payments, and $i$ is the interest rate per payment period — same matching rule as compound interest.
5Both formulas come from a geometric series: the future value adds $R + R (1 + i) + \dots + R (1 + i)^{n - 1}$ .
6Total paid $= n R$ . Loan interest $= n R - P V$ ; savings interest $= F V - n R$ .

So far, compound interest has dealt with ONE lump sum growing over time. An annuity instead involves a SERIES of equal payments made at regular intervals — a car loan paid off monthly, rent, a mortgage, or 50 dollars automatically deposited into savings every month. Most real financial plans look like this: people rarely have a large lump sum, but they can commit to a steady payment. Each individual payment compounds for a different length of time, which is exactly why annuities connect to geometric SERIES, not just a single geometric term. The two questions you will answer over and over are "how much will my regular deposits grow to?" and "how large a loan do my regular payments pay off?"

This topic covers ORDINARY SIMPLE annuities: payments happen at the END of each period, and the payment interval matches the compounding interval exactly (e.g. monthly payments with interest compounded monthly). "Ordinary" rules out payments at the start of each period, and "simple" rules out mismatched intervals such as monthly payments with semi-annual compounding. Both of those variations need adjustments beyond this course. We use $R$ for the regular payment, $i$ for the interest rate per payment period, and $n$ for the total number of payments.

Fig 1Every payment sits on the timeline and compounds for a different number of periods — the earliest payment compounds the longest.

Future value: a geometric series in disguise

The future value, $F V$ , is the total accumulated value of all the payments PLUS their interest, right at the moment of the final payment. Follow each payment to the end of the term. The LAST payment is made at the end, so it earns nothing and is worth $R$ . The second-last payment has one period to grow, so it is worth $R (1 + i)$ . The FIRST payment is made at the end of period 1, so it grows for $n - 1$ periods, not $n$ , and is worth $R (1 + i)^{n - 1}$ . Adding these up, from the last payment back to the first, gives a geometric series with first term $R$ , common ratio $1 + i$ and $n$ terms. The geometric series formula finishes the job, and its denominator $(1 + i) - 1$ simplifies to just $i$ .

F V = R + R (1 + i) + R (1 + i)^{2} + \dots + R (1 + i)^{n - 1} = \frac{R [ ( 1 + i ) ^{n} - 1 ]}{( 1 + i ) - 1}

F V = R \cdot \frac{( 1 + i ) ^{n} - 1}{i}

R = regular payment amount, i = interest rate per payment period, n = total number of payments.

Payment (end of year)	Years of growth	Value at end of year 4
1	3	$1000 (1.05)^{3} = 1157.63$
2	2	$1000 (1.05)^{2} = 1102.50$
3	1	$1000 (1.05)^{1} = 1050.00$
4	0	$1000 (1.05)^{0} = 1000.00$
Total		$4310.13 = 1000 \cdot \frac{1.0 5 ^{4} - 1}{0.05}$

Four annual deposits of 1000 dollars at 5% compounded annually, added one by one. The formula gives the same total.

VideoWorked future value problems for regular end-of-period deposits. The video writes the payment as C and the rate per period as r, which play the roles of R and i in this lesson's formula.Watch on YouTube

Present value: what the payments are worth today

The present value, $P V$ , is the lump sum today that is exactly equivalent to making all those future payments — this is how loan amounts are calculated (a bank hands you $P V$ today in exchange for your $n$ future payments of $R$ ). Bring each payment back to today with the present-value idea from the last lesson: the first payment is worth $R (1 + i)^{- 1}$ now, the second $R (1 + i)^{- 2}$ , down to $R (1 + i)^{- n}$ for the last. That is another geometric series, and summing it gives the formula below. A quick consistency check links the two formulas: $P V = F V (1 + i)^{- n}$ , because the present value invested today as a lump sum would grow to exactly the annuity's future value.

P V = R \cdot \frac{1 - ( 1 + i ) ^{- n}}{i}

Tip

Savings goal ('how much will I have') → future value. Loan or 'lump sum today' problems ('how much can I borrow/afford') → present value. Reading which one a word problem wants is usually the hardest part. Ask yourself WHEN the single amount in the question sits on the timeline: at the end of the payments means future value; today, before the first payment, means present value.

VideoEdspira brings each future payment back to today and adds them, which is where the present value formula for a loan comes from. Match the letters the video uses for the payment and the rate to R and i.Watch on YouTube

Solving for the payment and the interest

Many questions give the target and ask for the payment. Rearrange the right formula for $R$ : for a savings goal, $R = \frac{F V \cdot i}{( 1 + i ) ^{n} - 1}$ ; for a loan, $R = \frac{P V \cdot i}{1 - ( 1 + i ) ^{- n}}$ . Always evaluate the power first, then the fraction, keeping full calculator precision. Once you know $R$ , the total amount paid is simply $n R$ . For a loan, the interest charged is $n R - P V$ ; for savings, the interest earned is $F V - n R$ . On a 15 000 dollar car loan at 8% compounded monthly over 3 years, the payment is about 470.05 dollars, so the borrower pays $36 (470.05) = 16 921.80$ dollars in total, including $1921.80$ dollars of interest. Use the sim to try different rates, payments and terms and watch how the interest portion grows. Solving for $i$ or $n$ is different: $i$ appears both inside a power and in the denominator, so it cannot be isolated with algebra in this course. Instead, test values (guess and check) or use a graph of the formula against the target and read off the intersection.

Loading interactive…

Switch to annuity mode, set a payment, rate and compounding frequency, and watch how deposits and interest build up the future value year by year.

Key idea

Exactly like compound interest,

i

and

n

must match the PAYMENT period. Monthly payments need a monthly

i = r /12

and

n

counted in months. Never use the lump-sum formula

A = P (1 + i)^{n}

for a stream of payments, and never use an annuity formula for a single deposit. A fast sanity check catches most errors: the future value must be MORE than the total deposited,

n R

(for two or more payments), because interest is added; the present value must be LESS than

n R

, because money paid later is worth less today. For 200 dollars a month for 5 years at 6% compounded monthly,

n R = 12000

dollars sits between

P V \approx 10 345.11

and

F V \approx 13 954.01

dollars. If your answer breaks that ordering, recheck

i

n

and the sign of the exponent.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

You deposit 200 dollars at the end of every month into an account earning 6%/year compounded monthly, for 5 years. Find the future value.

Example 2

Using the SAME payments (200 dollars/month, 6%/year compounded monthly, 5 years), find the present value — how much that annuity is worth as a single lump sum today.

Example 3

A car loan of 15 000 dollars is paid off with equal monthly payments over 3 years at 8%/year compounded monthly. Find the monthly payment.

Where marks get lost

Confusing future value (how much accumulates) with present value (equivalent lump sum today) — decide which one the question is actually asking for before picking a formula.
Using the annuity formula's $(1 + i)^{n} - 1$ but forgetting to divide by $i$ at the end.
Mismatching the payment period and the compounding period — an ordinary SIMPLE annuity requires them to be the same (monthly payments need monthly compounding).
Applying the compound interest formula $A = P (1 + i)^{n}$ to a series of payments — that formula is for a single lump sum, not repeated deposits.
Rounding $i$ too early in a long calculation, which can shift the final dollar amount noticeably over many periods.

Flashcards 10

Annuity

Ordinary simple annuity

Future value of an annuity

Present value of an annuity

+ 6 more

Practice

100 dollars is deposited at the end of every month for 1 year into an account earning 6%/year compounded monthly. Find the future value (round to the nearest cent).

dollars

Which situation calls for the PRESENT VALUE of an annuity formula?

Find the present value of an annuity paying 500 dollars every quarter for 5 years at 8%/year compounded quarterly (round to the nearest cent).

dollars

In an ORDINARY annuity, when do payments happen relative to each period?

For a car loan paid monthly over 4 years at a nominal 6%/year, what is

n

What must be true about the payment interval and the compounding interval in a SIMPLE annuity?

150 dollars is deposited at the end of every month for 2 years at 4%/year compounded monthly. Find the future value (round to the nearest cent).

dollars

A 20 000 dollar loan is repaid with equal payments at the end of every month for 5 years at 6%/year compounded monthly. Find the monthly payment (round to the nearest cent).

dollars

For the loan above (60 payments of 386.66 dollars on a 20 000 dollar loan), how much interest is paid in total?

How much must be deposited at the end of every quarter, at 5%/year compounded quarterly, to have 10 000 dollars after 4 years? Round to the nearest cent.

dollars

An annuity pays 300 dollars at the end of every month for 2 years at 12%/year compounded monthly. Order these amounts from smallest to largest.

1Total of the payments, $n R$
2Present value
3Future value

Match each question to the quantity you should calculate.

A. $R$ from the $F V$ formula
B. $R$ from the $P V$ formula
C.Future value $F V$
D.Present value $P V$

How much will my monthly deposits grow to?

How big a loan can these payments repay?

What deposit reaches my savings goal?

What payment pays off this loan?

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