A stream of equal payments
An annuity is a series of equal payments made at equal intervals of time: $1000 paid at the end of each year for five years, for example. Regular savings, a pension and the repayments on a loan are all annuities.
Two questions can be asked about one. What will the payments have grown to by the end, if each is saved as it is paid? And what are they worth today? Both answers come from adding the payments after moving each one to the same date.
The years run from 0, today, to 5. A payment of $1000 is made at the end of each of years 1 to 5, and none today.
Growing each payment to the end
Save each payment in an account paying 5% a year. The payment made at the end of year 1 earns interest for the 4 years left, so by the end of year 5 it has grown to dollars. The payment at the end of year 2 earns interest for 3 years and grows to dollars. And so on down to the last payment, made at the end of year 5, which earns nothing. Each payment compounds for a different number of years.
The value of the annuity at the end, its future value, is the sum of the five grown payments. Written from the last payment to the first, it is FV . That is a geometric series, with first term 1000, common ratio 1.05 and 5 terms.
Each row is the $1000 paid at the end of that year: the years it earns 5% interest, and the dollars it grows to by the end of year 5. The five add up to 5525.63125, which is $5525.63 to the nearest cent.
Summing the series
A geometric series with first term a, common ratio r and n terms sums to . Here a = 1000, r = 1.05 and n = 5, and , so FV dollars, to the nearest cent.
Check by adding the terms: 1215.50625 + 1157.625 + 1102.5 + 1050 + 1000 = 5525.63125, the same total. The five payments of $1000 put in $5000, and the other $525.63 is interest.
In general, a payment PMT at the end of each period for n periods, at a rate i per period written as a decimal, grows to FV = PMT . The r − 1 of the series is (1 + i) − 1, which is i.
Bringing each payment back to today
The same payments can be valued today instead. Each one is discounted by the years it waits: dividing by 1.05 for each year. The payment one year off is worth 1000 ÷ 1.05 = 952.38 dollars today, the one two years off dollars, and the last, five years off, only dollars, each to the nearest cent. The further off a payment, the less it is worth today.
Each $1000 payment valued today at 5% a year: $952.38, $907.03, $863.84, $822.70 and $783.53. Each bar is the one before it divided by 1.05.
Another geometric series
The value of the annuity today, its present value, is the sum PV . This is a geometric series too, with first term and common ratio , which is 1 ÷ 1.05.
The ratio is less than 1, so use the sum in the form . The denominator is 1 − 1 ÷ 1.05 = 0.05 ÷ 1.05, and dividing by it multiplies by 1.05 ÷ 0.05. That 1.05 cancels the ÷ 1.05 in the first term, which leaves PV . Since , PV = 1000 × 0.2164738335 ÷ 0.05 = 4329.48 dollars, to the nearest cent.
Check by adding the terms: 952.38 + 907.03 + 863.84 + 822.70 + 783.53 = 4329.48. In general, PV = PMT .
One stream, two dates
The present value and the future value describe the same payments on two dates, five years apart, so moving one forward five years gives the other. The unrounded present value is 4329.4767, and , to the nearest cent, the future value.
Both differ from the $5000 paid in. At the end the stream is worth more than $5000, because every payment but the last has earned interest. Today it is worth less than $5000, because every payment has to wait.
The usual mistakes
Adding the payments as they stand. 5 × $1000 = $5000 is neither value: each payment has to be moved to the same date first.
Answering on the wrong date. The value today is found by dividing by the growth each payment waits through, and the value at the end by multiplying by the growth each payment earns. $5525.63 is not what the stream is worth today, and $4329.48 is not what it grows to.
Giving the last payment interest. The payment at the end of year 5 is made on the last day and earns nothing by then; only the earlier ones grow.
Worked example: Saving the Same Amount Every Year Toward a Deposit
Question Ana wants $20000 for a deposit on a flat. She can save $3000 at the end of each year into an account paying 6% a year. At that rate $1 saved at the end of each year grows to $5.637093 after 5 years and to $6.975319 after 6 years. (a) Find what she has after 5 years, and how far short of $20000 that leaves her. (b) Find what she has after 6 years, and say whether that reaches the target and by how much.
1.The deposits are made at the end of each year, so the first $3000 earns interest for 4 years, the next for 3 years, and the fifth earns nothing. The factor 5.637093 is those five deposits added up for one dollar a year.
Each column is what she holds after that year’s deposit. The first $3000 earns for four years and the fifth earns nothing. 2.After 5 years Ana has 3000 × 5.637093 = $16911.28, to the nearest cent.
The factor adds the five deposits up for one dollar a year, so she has 3000 × 5.637093 = $16911.28. 3.(a) That is 20000 − 16911.28 = $3088.72 short of the target.
(a) The dashed line is the target of $20000, and the fifth column is $3088.72 below it. 4.After 6 years she has 3000 × 6.975319 = $20925.96.
A sixth year gives 3000 × 6.975319 = $20925.96, the first column above the line. 5.(b) That passes the target, by 20925.96 − 20000 = $925.96. Check line by line: the $16911.28 earns 6% and becomes $17925.96, and the sixth deposit of $3000 brings it to $20925.96.
(b) She passes the target by $925.96, and the line-by-line check agrees: 16911.28 × 1.06 + 3000 = 20925.96.
Answer: (a) She has $16911.28, which is $3088.72 short of the target; (b) she has $20925.96, which passes $20000 by $925.96
Common mistakes
- Adding the deposits up as 5 × 3000 = $15000 and then adding 6% once, giving $15900. Each deposit earns for a different number of years, so they cannot be added first and grown afterwards; the factor 5.637093 is what does that work correctly.
- Giving the fifth deposit a year of interest as well, so multiplying $16911.28 by 1.06. The fifth deposit is paid in at the end of the fifth year, so it has earned nothing by then, and 5 × 1.06 years of interest is not what the factor counts.
Worked example: A Prize Paid as Twenty Yearly Payments Against a Single Cash Sum
Question A lottery pays its prize as $50000 at the end of each year for 20 years, or as a single payment of $650000 today. Money is worth 4% a year. At that rate $1 paid at the end of each year for 20 years is worth $13.590326 today, and 1.0420 = 2.191123. (a) Find the value today of the twenty payments, and say which offer is worth more and by how much. (b) The prize is advertised as $1000000. Find how much of that is lost to the waiting.
1.The first payment arrives in a year and is worth 50000 ÷ 1.04 = $48076.92 today; the twentieth arrives in 20 years and is worth 50000 ÷ 2.191123 = $22819.35 today. Every payment in between lies between those two.
Each column is one payment: the gold part is what it is worth today and the faint part is what the waiting takes. The dashed line is the $50000 written on every payment. 2.The annuity factor adds all twenty of those values together for one dollar a year, so the twenty payments are worth 50000 × 13.590326 = $679516.30 today.
The annuity factor adds all twenty gold parts together for one dollar a year, so the run is worth 50000 × 13.590326 = $679516.30. 3.(a) The single payment is $650000, so the yearly payments are worth 679516.30 − 650000 = $29516.30 more today.
(a) The payments are worth $679516.30 today against $650000 offered, so they are $29516.30 better. 4.The cash the yearly payments actually hand over is 20 × 50000 = $1000000, which is the advertised prize.
The cash the payments hand over is 20 × 50000 = $1000000, which is the advertised prize. 5.(b) Of that advertised million, 1000000 − 679516.30 = $320483.70 is lost to the waiting. Check: the last payment alone keeps 22819.3550000 of its face value, which is under half, and the twenty payments together keep about 68% of theirs.
(b) Of the advertised million, $320483.70 is lost to the waiting and $679516.30 is what it is worth today.
Answer: (a) The twenty payments are worth $679516.30 today, so they beat the single payment of $650000 by $29516.30; (b) $320483.70 of the advertised million is lost to the waiting
Common mistakes
- Comparing the advertised $1000000 with the $650000 and taking the yearly payments by a wide margin. The million is handed over across twenty years, and the last of it waits twenty years to arrive, so it is worth $679516.30 today, not a million.
- Multiplying $50000 by twenty and then dividing once by 2.191123, giving $456386.98. That would be right only if all twenty payments arrived together in year 20. They arrive one a year, and the early ones are discounted far less, so the true value is higher.