Sunflower Intelligence
Finance & Accounting

F04 · Introductory

Mathematics of Finance

9 lessons

F04 · Module 3, lesson 1 · Lesson 5 of 9

Annuities

Loan instalments, rent, leases, pensions and savings plans are all streams of equal payments. Learn to value such a stream in one step, whether the payments come at the end or the start of each period, start now or later, stay level or grow, and to solve for the payment, the number of payments or the rate.

About 80 minutes

Before you start

You will be able to

  • Calculate the present and future value of an ordinary annuity and of an annuity due, and explain why they differ by a factor of (1 + r).
  • Calculate the value of a deferred annuity and of a growing annuity by moving the stream to a common date.
  • Calculate the payment or the number of payments of an annuity, using logarithms for the number of payments.
  • Find the interest rate of an annuity by trial and error or with a spreadsheet’s RATE function.
  • Evaluate an instalment plan sold as “0% interest” by finding its present value and its implied interest rate.

Review first

  • Simple and compound interest — Solving for the number of payments uses natural logarithms, introduced there.
  • Present and future value — Every annuity formula is a shortcut for discounting or compounding single sums, and you need to move an amount from one date to another.
  • Discounting cash flows — An annuity is a cash-flow stream whose payments happen to be equal: discounting each payment and adding is the definition behind every formula here.
  • Effective annual rates — Payments are often monthly while rates are quoted a year at a time; you need the rate per period that matches the payment period.
  • Powers, negative powers and the natural logarithm (ln) on a calculator.

Fictional case A fictional coffee business founded by Ana Lim. It starts as a catering and barista-training service, opens its own cafés and an online shop for beans, and plans a canned cold brew.

The idea

Fictional case

March 2027. Lantern Coffee Co. wants a cold-brew brewing system that costs $24,000. The supplier offers its own instalment plan: 24 monthly payments of $1,050, the first one month after delivery, advertised as “0% interest”. Coastline Bank, Lantern’s bank, would instead lend the $24,000 at 7.2% a year compounded monthly, repaid over four years. Ana adds up the instalments: 24×1,050=25,20024 \times 1{,}050 = 25{,}200 dollars, which is $1,200 more than the cash price. So is the plan free, as the advertisement says, or does it cost $1,200? Neither answer is right, because the 24 payments are spread over two years, and a dollar paid in two years is worth less than a dollar paid today. To compare the plan with the cash price, Ana needs the value today of 24 equal payments: the value of an annuity.

Streams of equal payments

An annuity is a series of equal payments made at equal intervals for a fixed number of periods. Loan instalments, rent, equipment leases, insurance premiums, pension payments and a savings plan that puts the same amount aside every month are all annuities. So are the 24 payments of $1,050 in the supplier’s plan.

In everyday speech an annuity is also an insurance product: a contract with an insurance company that pays an income, usually in retirement. Those products are built from the same mathematics, but in this course the word means any level stream of payments, whoever pays it.

When the payments fall within each period matters, so there are two basic kinds:

Ordinary annuity
Each payment is made at the end of its period, so the first payment comes one period from now. Loan repayments and the supplier’s plan work this way. This is the default: “annuity” on its own means an ordinary annuity.
Annuity due
Each payment is made at the start of its period, so the first payment is made today. Rent, leases and insurance premiums are usually paid in advance and work this way.

Four numbers describe an annuity: the payment per period CC, the number of payments nn, the interest rate per period rr, and the timing (end or start). The rate and the payments must refer to the same period. For monthly payments with a rate quoted as an APR compounded monthly, the monthly rate is the APR divided by 12; Coastline’s 7.2% a year is 0.6% a month.

Valuing an annuity one payment at a time

You already know how to value a stream of payments: discount each payment to today and add up the results. An annuity is no different. Take $2,000 at the end of each year for five years, valued at 6% a year. Each payment is multiplied by its discount factor, 1÷1.06t1 \div 1.06^t, where tt is the year in which it is paid.

$2,000 at the end of each year for five years, valued at 6% a year
YearPaymentDiscount factorPresent value
1$2,0000.943396$1,886.79
2$2,0000.889996$1,779.99
3$2,0000.839619$1,679.24
4$2,0000.792094$1,584.19
5$2,0000.747258$1,494.52
Total$10,0004.212364$8,424.73

Discount factors are 1 ÷ 1.06 raised to the power of the year, to six decimal places; present values are rounded to the cent, and the totals are computed unrounded.

The five payments add up to $10,000, but they are worth $8,424.73 today: the later the payment, the less it is worth now. Look at the total of the discount factors, 4.212364. Because every payment is the same, the present value is the payment times that total. The total is called the annuity factor, and there is a formula for it, so you never need the table:

PV=C×1−(1+r)−nr=2,000×1−1.06−50.06=2,000×4.212364=8,424.73\text{PV} = C \times \frac{1 - (1 + r)^{-n}}{r} = 2{,}000 \times \frac{1 - 1.06^{-5}}{0.06} = 2{,}000 \times 4.212364 = 8{,}424.73

Here 1.06−5=1÷1.065≈0.7472581.06^{-5} = 1 \div 1.06^5 \approx 0.747258, the discount factor for the last payment. The formula gives the same answer as the table in one line, whether there are five payments or 360.

The same n payments of C on two timelines. In an ordinary annuity the payments fall at times 1 to n; in an annuity due every payment is one period earlier, at times 0 to n − 1. The present-value formula values an ordinary annuity at time 0, one period before its first payment.

Payments at the start: the annuity due

If the same five payments of $2,000 were made at the start of each year instead of the end, each would arrive one year sooner and so be worth 6% more in today’s money. The whole annuity due is therefore worth 8,424.73×1.06≈8,930.218{,}424.73 \times 1.06 \approx 8{,}930.21 dollars. The rule is general: an annuity due is worth an ordinary annuity times (1+r)(1 + r), for present values and for future values alike. The difference is one period’s interest on the whole stream, not a different formula.

Looking forward: the future value of an annuity

A savings plan asks the opposite question: if you deposit the same amount every period, how much will you have at the end? Suppose Lantern sets aside $400 at the end of every month for three years, in a business savings account paying 3% a year compounded monthly (0.25% a month), to replace its espresso machine. Each deposit grows until the end of the 36 months; the first grows for 35 months and the last not at all. Adding them up gives the future value of the annuity:

FV=C×(1+r)n−1r=400×1.002536−10.0025=400×37.620560≈15,048.22\text{FV} = C \times \frac{(1 + r)^{n} - 1}{r} = 400 \times \frac{1.0025^{36} - 1}{0.0025} = 400 \times 37.620560 \approx 15{,}048.22

Lantern deposits 36×400=14,40036 \times 400 = 14{,}400 dollars and the account adds $648.22 of interest. (The account, the rate and the plan are fictional.) The future value is the present value moved forward nn periods: FV=PV×(1+r)n\text{FV} = \text{PV} \times (1 + r)^n, so once you know one, you know the other.

Is the supplier’s “0% interest” plan free?

Now Ana can value the plan. If Lantern pays cash, it must find $24,000 now, and the obvious place to find it is Coastline Bank at 7.2% a year compounded monthly. So she values the 24 payments at 0.6% a month, the rate at which Lantern can move money through time:

PV=1,050×1−1.006−240.006=1,050×22.289933≈23,404.43\text{PV} = 1{,}050 \times \frac{1 - 1.006^{-24}}{0.006} = 1{,}050 \times 22.289933 \approx 23{,}404.43

At Lantern’s 7.2% borrowing rate, the plan costs the same as paying $23,404.43 today: about $595.57 less than the $24,000 cash price. The benefit of paying later outweighs the extra $1,200 that the instalments add up to. But the plan is not free either. A plan that really charged 0% would ask for 24,000÷24=1,00024{,}000 \div 24 = 1{,}000 dollars a month. The extra $50 a month is interest by another name. The rate it implies is the rate at which the 24 payments are worth exactly $24,000; the In depth section finds it, about 0.394% a month, an APR of about 4.73%. That is cheaper than Coastline’s 7.2%, which is why the plan wins at that rate.

In depth

Why the formula works

The present value of an ordinary annuity is a sum of nn discounted payments, each one (1+r)(1 + r) times smaller than the one before: a geometric series. Multiply the sum by (1+r)(1 + r) and every term moves one period closer; subtract the original and all but two terms cancel:

PV=C1+r+C(1+r)2+⋯+C(1+r)n⟹(1+r) PV−PV=C−C(1+r)n\text{PV} = \frac{C}{1 + r} + \frac{C}{(1 + r)^2} + \cdots + \frac{C}{(1 + r)^n} \quad\Longrightarrow\quad (1 + r)\,\text{PV} - \text{PV} = C - \frac{C}{(1 + r)^{n}}

The left side is r×PVr \times \text{PV}, so PV=C [1−(1+r)−n]÷r\text{PV} = C\,[1 - (1 + r)^{-n}] \div r. A second way to see it uses the next lesson, perpetuities-and-growth: a payment of CC every period forever is worth C÷rC \div r. An annuity is a perpetuity that stops after nn payments, so it is worth a perpetuity starting now minus a perpetuity whose payments start after period nn, which is worth (C÷r)(C \div r) discounted by (1+r)n(1 + r)^n. For the five payments of $2,000: 33,333.33−24,908.61≈8,424.7333{,}333.33 - 24{,}908.61 \approx 8{,}424.73 dollars (subtracting the unrounded values).

The future-value formula is the present value moved forward: multiply by (1+r)n(1 + r)^n and you get C [(1+r)n−1]÷rC\,[(1 + r)^n - 1] \div r. If the rate is zero both formulas break down (they divide by zero), but then there is nothing to discount: the value is just n×Cn \times C.

Ordinary annuity or annuity due?

Moving every payment one period earlier multiplies each present value, and each future value, by (1+r)(1 + r), so the conversion is a single multiplication. You can also think of an annuity due as one payment now plus an ordinary annuity of n−1n - 1 payments: C+C [1−(1+r)−(n−1)]÷rC + C\,[1 - (1 + r)^{-(n-1)}] \div r, which gives the same number. Read the contract to decide which you have. A loan’s first payment is normally due a period after the money is lent: an ordinary annuity. A lease or a rent is normally paid in advance: an annuity due. Spreadsheets and financial calculators handle both with a timing switch (described below).

Deferred annuities

A deferred annuity starts later than one period from now. University fees that begin in ten years, or a pension that starts at 65, are examples. Value it in two moves. First apply the annuity formula: it gives the value one period before the first payment. Then discount that single amount back to today. If the first payment comes at the end of period k+1k + 1, the annuity formula values the stream at time kk, so you discount by kk periods, not k+1k + 1:

PV0=1(1+r)k×C×1−(1+r)−nr\text{PV}_0 = \frac{1}{(1 + r)^{k}} \times C \times \frac{1 - (1 + r)^{-n}}{r}

A check that never fails: the deferred annuity is worth an annuity of k+nk + n payments minus an annuity of the first kk payments, which never happen. Worked example 2 uses both routes.

Growing annuities

Some streams rise at a steady rate: a supply contract with a price increase each year, or a salary that rises with inflation. A growing annuity pays C1C_1 at the end of the first period, then grows at gg per period, so the payment in period tt is C1(1+g)t−1C_1(1 + g)^{t-1}. The same geometric-series argument gives:

PV=C1r−g[1−(1+g1+r)n]\text{PV} = \frac{C_1}{r - g}\left[1 - \left(\frac{1 + g}{1 + r}\right)^{n}\right]

Note that C1C_1 is the first payment, not today’s amount. The formula works whether gg is below or above rr; it fails only when they are equal, and then each payment is worth exactly C1÷(1+r)C_1 \div (1 + r) today, so the present value is n×C1÷(1+r)n \times C_1 \div (1 + r). Worked example 3 values a supply contract whose price rises 3% a year.

Solving for the payment

Rearranging the present-value formula gives the payment that a present amount will support: C=PV×r÷[1−(1+r)−n]C = \text{PV} \times r \div [1 - (1 + r)^{-n}]. This is the loan payment: a lender who hands over PV\text{PV} today and wants its money back with interest at rr needs payments of CC. The loan-amortisation lesson builds on it. Rearranging the future-value formula gives the deposit needed to reach a target, C=FV×r÷[(1+r)n−1]C = \text{FV} \times r \div [(1 + r)^n - 1]; a fund built this way is called a sinking fund. For example, to have $20,000 in four years at 3% a year compounded monthly, deposit 20,000×0.0025÷(1.002548−1)≈392.6920{,}000 \times 0.0025 \div (1.0025^{48} - 1) \approx 392.69 dollars at the end of each month (a fictional target).

Solving for the number of payments

How many payments of CC will repay an amount PV\text{PV}? Rearrange the formula until the power stands alone, (1+r)−n=1−PV×r÷C(1 + r)^{-n} = 1 - \text{PV} \times r \div C, then take natural logarithms of both sides. A logarithm turns a power into a multiplication, ln⁡(xn)=nln⁡x\ln(x^n) = n \ln x, which is what lets you solve for nn:

n=−ln⁡(1−PV×rC)ln⁡(1+r)n = \frac{-\ln\left(1 - \dfrac{\text{PV} \times r}{C}\right)}{\ln(1 + r)}

Take a fictional $10,000 loan at 9% a year compounded monthly (0.75% a month) repaid at $500 a month: n=−ln⁡(1−10,000×0.0075÷500)÷ln⁡1.0075=−ln⁡0.85÷ln⁡1.0075≈21.75n = -\ln(1 - 10{,}000 \times 0.0075 \div 500) \div \ln 1.0075 = -\ln 0.85 \div \ln 1.0075 \approx 21.75. The answer is rarely a whole number. In practice the borrower makes 21 full payments of $500 and a smaller 22nd payment that clears what is left, about $375.52 here (the balance after 21 payments, $372.73, plus a month’s interest). The formula has a condition: the payment must be larger than the interest on the amount, C>PV×rC > \text{PV} \times r. At $75 a month the payments would only cover the interest and the debt would never be repaid; below $75 it would grow. For a savings target the matching formula is n=ln⁡(1+FV×r÷C)÷ln⁡(1+r)n = \ln(1 + \text{FV} \times r \div C) \div \ln(1 + r).

Solving for the rate

There is no practical formula that isolates rr in the annuity equation once there are more than two payments. The rate is found by trial and error: guess a rate, value the payments, and adjust the guess until the value matches the price. Higher rates give lower values, so each trial tells you which way to move. Here is the search for the rate in the supplier’s plan, where 24 payments of $1,050 must be worth $24,000:

Searching for the rate at which 24 payments of $1,050 are worth $24,000
APR triedMonthly rateValue of the 24 paymentsCompared with $24,000
0%0%$25,200.00too high: raise the rate
4.0%0.3333%$24,179.66too high: raise the rate
5.0%0.4167%$23,933.59too low: lower the rate
4.7%0.3917%$24,007.05just too high
4.8%0.4000%$23,982.53too low
4.73%0.3942%$23,999.6931 cents low: close enough

Each value is 1,050 × (1 − (1 + r)^−24) ÷ r at the monthly rate r = APR ÷ 12, rounded to the cent.

Halving the gap between a guess that is too high and one that is too low (bisection) closes in on the answer in a few dozen steps; a computer does it instantly. The plan’s implied monthly rate is about 0.394%, an APR of about 4.73%, and an effective annual rate of about 4.83% (1.0039412−11.00394^{12} - 1). This is the plan’s implied interest rate: the rate the “0%” plan really charges, given that a buyer paying cash would pay $24,000.

Spreadsheets do the search for you. In Excel and most other spreadsheets, =RATE(24, -1050, 24000) returns the rate per period, about 0.394%; multiply by 12 for the APR. Microsoft’s documentation says RATE is calculated by iteration and can have zero or more solutions, and that it returns an error if it fails to settle on an answer after 20 tries, in which case you can supply a guess. The companion functions PV, FV, PMT and NPER solve for the other unknowns.

Assumptions and limits

  • The payments are equal (or, for a growing annuity, grow at a constant rate) and fall at regular intervals.
  • One interest rate applies to every period. If rates are expected to change, discount each payment at its own rate, as in the present-and-future-value lesson.
  • The rate period matches the payment period. Monthly payments with quarterly compounding, for example, need the rate converted first.
  • The value depends on the rate chosen. A present value at 7.2% answers “what is this worth to someone who can borrow or lend at 7.2%?”; a different rate answers a different question.
  • The payments are certain. Risky payments are discounted at a higher rate, a topic for F05 Corporate Finance.

Formulas

Present value of an ordinary annuity

PV=C×1−(1+r)−nr\text{PV} = C \times \frac{1 - (1 + r)^{-n}}{r}
PV\text{PV}
Present value: the value of the whole stream one period before the first payment$
CC
Payment per period$
rr
Interest rate per period, as a decimal (an APR compounded monthly ÷ 12 for monthly payments)per period
nn
Number of paymentsperiods

Assumes. Equal payments at the end of each period; one constant rate; the rate and the payments refer to the same period.

The fraction is the annuity factor. With r = 0 the value is n × C. Spreadsheet: =PV(r, n, −C).

Future value of an ordinary annuity

FV=C×(1+r)n−1r\text{FV} = C \times \frac{(1 + r)^{n} - 1}{r}
FV\text{FV}
Future value: the value of the stream at the date of the last payment$
CC
Payment (deposit) per period$
rr
Interest rate per period, as a decimalper period
nn
Number of paymentsperiods

Assumes. Deposits at the end of each period, each earning the same rate until the last deposit is made.

FV = PV × (1 + r)^n. Spreadsheet: =FV(r, n, −C).

Annuity due (payments at the start of each period)

PVdue=PVord×(1+r),FVdue=FVord×(1+r)\text{PV}_{\text{due}} = \text{PV}_{\text{ord}} \times (1 + r), \qquad \text{FV}_{\text{due}} = \text{FV}_{\text{ord}} \times (1 + r)
PVdue,FVdue\text{PV}_{\text{due}}, \text{FV}_{\text{due}}
Values when each payment is made at the start of its period: the present value today (time 0, the date of the first payment) and the future value at the end of period n, one period after the last payment$
PVord,FVord\text{PV}_{\text{ord}}, \text{FV}_{\text{ord}}
The same values for an ordinary annuity with the same payment, rate and number of payments$
rr
Interest rate per period, as a decimalper period

Assumes. Same payment, rate and number of payments; only the timing differs.

Spreadsheet: set the type argument to 1.

Deferred annuity (first payment at the end of period k + 1)

PV0=1(1+r)k×C×1−(1+r)−nr\text{PV}_0 = \frac{1}{(1 + r)^{k}} \times C \times \frac{1 - (1 + r)^{-n}}{r}
PV0\text{PV}_0
Value today$
CC
Payment per period$
rr
Interest rate per period, as a decimalper period
nn
Number of paymentsperiods
kk
Number of periods before the annuity formula’s valuation date: one less than the period of the first paymentperiods

Assumes. An ordinary annuity whose first payment is at the end of period k + 1.

Check: an annuity of k + n payments minus an annuity of k payments gives the same value.

Present value of a growing annuity

PV=C1r−g[1−(1+g1+r)n]\text{PV} = \frac{C_1}{r - g}\left[1 - \left(\frac{1 + g}{1 + r}\right)^{n}\right]
PV\text{PV}
Present value, one period before the first payment$
C1C_1
The first payment, made at the end of period 1$
rr
Interest rate per period, as a decimalper period
gg
Growth rate of the payment per period, as a decimalper period
nn
Number of paymentsperiods

Assumes. Each payment is (1 + g) times the one before; r ≠ g.

If r = g, PV = n × C₁ ÷ (1 + r).

The payment of an annuity

C=PV×r1−(1+r)−norC=FV×r(1+r)n−1C = \frac{\text{PV} \times r}{1 - (1 + r)^{-n}} \qquad\text{or}\qquad C = \frac{\text{FV} \times r}{(1 + r)^{n} - 1}
CC
Payment per period$
PV\text{PV}
Amount to be repaid or drawn down (a loan, a fund)$
FV\text{FV}
Target to be reached by saving (a sinking fund)$
rr
Interest rate per period, as a decimalper period
nn
Number of paymentsperiods

Assumes. Ordinary annuity; for payments at the start of each period divide the result by (1 + r).

Spreadsheet: =PMT(r, n, PV) or =PMT(r, n, 0, FV); the sign of the answer follows the sign convention.

The number of payments

n=−ln⁡(1−PV×rC)ln⁡(1+r)n = \frac{-\ln\left(1 - \dfrac{\text{PV} \times r}{C}\right)}{\ln(1 + r)}
nn
Number of payments (usually not a whole number: round up, with a smaller last payment)periods
PV\text{PV}
Amount to be repaid$
CC
Payment per period; must exceed PV × r$
rr
Interest rate per period, as a decimalper period

Assumes. Ordinary annuity with C > PV × r.

For a savings target: n = ln(1 + FV × r ÷ C) ÷ ln(1 + r). Spreadsheet: =NPER(r, −C, PV).

Worked examples

A first example: A lease paid in advance

Fictional figures

A physiotherapy clinic leases a treatment machine for three years at $500 a month. The lease asks for each payment at the start of the month. The clinic can borrow at 6% a year compounded monthly. What are the 36 lease payments worth today, and how much more is that than if the payments were made at the end of each month?

  1. Match the rate to the payments: monthly rate r=6%÷12=0.5%=0.005r = 6\% \div 12 = 0.5\% = 0.005; number of payments n=36n = 36.
  2. Value the stream as if it were an ordinary annuity: PVord=500×(1−1.005−36)÷0.005=500×32.871016≈16,435.51\text{PV}_{\text{ord}} = 500 \times (1 - 1.005^{-36}) \div 0.005 = 500 \times 32.871016 \approx 16{,}435.51.
  3. The payments are made at the start of each month, so this is an annuity due: multiply by 1.0051.005. PVdue=16,435.51×1.005≈16,517.69\text{PV}_{\text{due}} = 16{,}435.51 \times 1.005 \approx 16{,}517.69.
  4. Difference: 16,517.69−16,435.51=82.1816{,}517.69 - 16{,}435.51 = 82.18, which is one month’s interest on the whole stream (16,435.51×0.00516{,}435.51 \times 0.005).

Answer. The lease payments are worth about $16,517.69 today, $82.18 more than the same payments made at the end of each month ($16,435.51). Both are well below the $18,000 the payments add up to.

What it means. Paying in advance costs the clinic one month’s interest on the whole stream. The present value, not the $18,000 total, is the figure to set against the price of buying the machine outright, together with what the machine would still be worth after three years.

A harder example: University fees that start in ten years

Fictional figures

Parents want to set money aside today for their daughter’s university fees: $12,000 a year for four years, with the first payment exactly ten years from today. The money will earn 5% a year. How much must they deposit today?

  1. Draw the timeline: payments at the end of years 10, 11, 12 and 13. The first payment is in year 10, so the annuity formula values the four payments at year 9, one year before the first.
  2. Value at year 9: 12,000×(1−1.05−4)÷0.05=12,000×3.545951≈42,551.4112{,}000 \times (1 - 1.05^{-4}) \div 0.05 = 12{,}000 \times 3.545951 \approx 42{,}551.41.
  3. Discount that single amount nine years to today: 1.059≈1.5513281.05^9 \approx 1.551328, so 42,551.41÷1.551328≈27,429.0242{,}551.41 \div 1.551328 \approx 27{,}429.02.
  4. Check by the second route: an annuity of 13 payments minus an annuity of the 9 payments that do not happen, 12,000×(9.3935730−7.1078217)=12,000×2.2857513≈27,429.0212{,}000 \times (9.3935730 - 7.1078217) = 12{,}000 \times 2.2857513 \approx 27{,}429.02. The two routes agree.

Answer. They must deposit about $27,429.02 today. Discounting by ten years instead of nine would give $26,122.87, about $1,306 too little.

What it means. A deposit of $27,429.02 grows to exactly what is needed to pay $12,000 in each of years 10 to 13. The most common error with deferred annuities is the extra year of discounting: the formula already values the stream one period before the first payment.

A harder example: A supply contract with a yearly price rise

Fictional figures

A bakery chain signs a five-year contract to buy flour from a mill. It pays at the end of each year: $30,000 for the first year, then 3% more each year to cover rising costs. The bakery values future payments at 7% a year. What is the contract worth today, and how does it compare with a flat $30,000 a year?

  1. This is a growing annuity: C1=30,000C_1 = 30{,}000, g=0.03g = 0.03, r=0.07r = 0.07, n=5n = 5. The payments are $30,000, $30,900, $31,827, $32,781.81 and $33,765.26, a total of $159,274.07.
  2. Growth factor: (1.03÷1.07)5≈0.82654639(1.03 \div 1.07)^5 \approx 0.82654639, so the bracket is 1−0.82654639=0.173453611 - 0.82654639 = 0.17345361.
  3. Present value: 30,000÷(0.07−0.03)×0.17345361=750,000×0.17345361≈130,090.2130{,}000 \div (0.07 - 0.03) \times 0.17345361 = 750{,}000 \times 0.17345361 \approx 130{,}090.21.
  4. A flat $30,000 a year would be worth 30,000×(1−1.07−5)÷0.07=30,000×4.1001974≈123,005.9230{,}000 \times (1 - 1.07^{-5}) \div 0.07 = 30{,}000 \times 4.1001974 \approx 123{,}005.92.

Answer. The contract is worth about $130,090.21 today, about $7,084.28 more than a flat $30,000 a year ($123,005.92).

What it means. The 3% rises add $9,274.07 to the payments but only about $7,084 to their present value, because the rises come later. When negotiating, the bakery should compare offers by present value: a flat price slightly above $30,000 might cost less than a lower starting price that rises each year.

Applied: a business decision: Lantern and the supplier’s “0% interest” plan

Fictional figures

The brewer costs $24,000 in cash. The supplier’s plan asks for 24 monthly payments of $1,050, the first one month after delivery. Lantern can borrow from Coastline Bank at 7.2% a year compounded monthly. Should Lantern take the plan rather than pay cash with money borrowed at 7.2%, and what rate does the plan really charge?

  1. The payments are at the end of each month: an ordinary annuity with C=1,050C = 1{,}050, n=24n = 24 and, at Coastline’s rate, r=0.072÷12=0.006r = 0.072 \div 12 = 0.006.
  2. Present value at 7.2%: 1,050×(1−1.006−24)÷0.006=1,050×22.289933≈23,404.431{,}050 \times (1 - 1.006^{-24}) \div 0.006 = 1{,}050 \times 22.289933 \approx 23{,}404.43. That is 24,000−23,404.43=595.5724{,}000 - 23{,}404.43 = 595.57 dollars less than the cash price.
  3. Implied rate: search for the monthly rate at which the 24 payments are worth exactly $24,000 (the table in “In depth”), or use =RATE(24, -1050, 24000). Either gives about 0.394% a month.
  4. Convert: APR ≈12×0.394%≈4.73%\approx 12 \times 0.394\% \approx 4.73\%; effective annual rate ≈1.0039412−1≈4.83%\approx 1.00394^{12} - 1 \approx 4.83\%.

Answer. Take the plan, on cost. At Lantern’s 7.2% borrowing rate it is worth $23,404.43 today, $595.57 less than $24,000 in cash. It is not interest-free: its implied rate is about 4.73% a year (APR), or 4.83% effective, which is still below Coastline’s 7.2%.

What it means. Two methods, one conclusion: the plan costs less than borrowing at 7.2% because its implied rate is lower than 7.2%. “0% interest” describes the sticker, not the economics: the extra $50 a month above $1,000 is the interest. The plan does demand $1,050 a month for two years, much more than a 48-month bank loan would. The loan-amortisation lesson weighs that cash-flow cost.

Explore

Start from the supplier’s plan: 24 monthly payments of $1,050 at 7.2%. Switch the timing to the start of each period and the value rises by one month’s interest. Then stretch the payments over 30 years and watch the value of each payment, in today’s money, shrink as it moves further away.

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Find the rate hidden in an instalment plan. The curve shows what the instalments are worth at each interest rate; the plan’s implied rate is where the curve crosses the cash price. Raise the instalment and watch the rate climb; set it to cash price ÷ number of payments and the rate falls to zero.

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Common mistakes

  • Mixing an annual rate with monthly payments

    With monthly payments you need the monthly rate and the number of months: 0.6% and 24 for the supplier’s plan, not 7.2% and 24, or 0.6% and 2. Using the annual rate with 24 payments values the plan at about $11,834; using 0.6% with two payments values it at about $2,081. If a result looks absurd, check the periods first.

  • Using the ordinary formula for payments made in advance

    Rent, leases and premiums are usually paid at the start of each period. Valuing them as an ordinary annuity understates their present value by one period’s interest. Read the timing from the contract and multiply by (1 + r) for an annuity due.

  • Discounting a deferred annuity one period too far

    The annuity formula already values the stream one period before the first payment. If the first payment is in year 10, the formula gives a value at year 9, so discount by nine years, not ten. Check with the difference of two annuities.

  • Adding up the payments

    “The plan costs $25,200” treats a dollar in two years as worth a dollar today. Payments at different dates can be compared only after moving them to the same date. The sum of the payments is useful for one thing only: the gap between it and the price (or the amount borrowed) is the total interest, in undiscounted dollars.

  • Believing “0%” when there is a cash price

    If a buyer who pays cash pays less than the instalments add up to, the plan charges interest whatever the advertisement says. Find the implied rate from the cash price, the payment and the number of payments, and compare it with what borrowing would cost.

  • Entering everything as positive in a spreadsheet

    RATE, NPER and PV need the sign convention: money received positive, money paid negative. =RATE(24, 1050, 24000) has no answer, because it describes receiving both the $24,000 and the payments. Write =RATE(24, -1050, 24000).

Guided practice

Work through these with hints when you need them. Every hint and every worked solution is there to be used.

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On your own

No hints this time. Answer, check, and read the feedback; the solution appears once you have answered or if you ask for it. Your first answers count toward mastery.

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In practice

Annuity arithmetic sits behind every decision that swaps a lump sum for a stream of payments, or a stream for a lump sum.

Leasing or buying equipment. The finance manager of a fictional battery-storage installer must choose between buying a $90,000 crane truck and leasing it for $2,100 a month, paid in advance, for four years. She values the lease payments as an annuity due at the rate the company would pay to borrow the purchase price, then adds what the company would lose by not owning the truck at the end of the lease (its expected resale value, discounted to today). The cheaper option in present value wins, unless the lease brings servicing or flexibility worth paying for. Adding up the 48 lease payments ($100,800) and comparing them with $90,000 would answer the wrong question.

Lump sum or instalments. A fictional court settlement offers an injured cyclist either $400,000 now or $30,000 a year for 20 years, the first payment in a year. At 6% the instalments are worth about $344,097.64, so the lump sum is worth more to anyone who can invest at 6%; at 4% they are worth about $407,709.79 and the instalments win. The decision turns on the rate, so a careful adviser shows the value at several rates and asks how safe the payer is. Lottery prizes and pension options pose the same question.

Saving for a known cost. A fictional hotel knows it must refurbish its rooms in five years at a cost of about $250,000. Rather than borrow at the last minute, its owners set up a sinking fund: equal monthly deposits that, with interest, reach the target on time. The payment formula tells them the deposit, and the future-value formula lets them check each year whether the fund is on track.

Fictional case

Ana now knows that the supplier’s “0% interest” plan is worth $23,404.43 at Lantern’s 7.2% borrowing rate, and that it really charges about 4.73% a year. On cost alone it beats paying cash with money borrowed from Coastline. Cost is not the only question, though. The plan takes $1,050 a month for two years; a four-year bank loan would take much less each month but last twice as long. The loan-amortisation lesson builds the bank loan’s repayment schedule and puts the two side by side.

Knowledge check

Six questions, one attempt each, no hints. Score 80% or more to complete the lesson; you can take it again at any time.

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Summary

  • An annuity is a series of equal payments at equal intervals. In an ordinary annuity each payment is at the end of its period; in an annuity due, at the start.
  • PV = C × [1 − (1 + r)^−n] ÷ r and FV = C × [(1 + r)^n − 1] ÷ r, with r and n in the same period as the payments. FV = PV × (1 + r)^n.
  • An annuity due is worth an ordinary annuity × (1 + r). A deferred annuity is valued with the annuity formula one period before its first payment, then discounted to today.
  • A growing annuity is worth C₁ ÷ (r − g) × [1 − ((1 + g) ÷ (1 + r))^n], where C₁ is the first payment.
  • Rearranging gives the payment; logarithms give the number of payments; the rate needs trial and error or a spreadsheet’s RATE function.
  • The supplier’s 24 × $1,050 plan is worth $23,404.43 at 7.2%, $595.57 less than $24,000 in cash, but it is not free: its implied rate is about 4.73% a year (APR).

Key formulas

Terms introduced

Annuity
A series of equal payments made at equal intervals for a fixed number of periods, such as loan instalments, rent or a regular savings deposit. Also the name of an insurance product that pays an income.
Ordinary annuity
An annuity whose payments are made at the end of each period, so the first payment comes one period from now. The default meaning of “annuity”.
Annuity due
An annuity whose payments are made at the start of each period, so the first payment is made now. Worth an ordinary annuity × (1 + r).
Annuity factor
The present value of $1 a period for n periods at rate r: [1 − (1 + r)^−n] ÷ r. Multiplied by the payment, it gives the present value of an ordinary annuity.
Deferred annuity
An annuity whose first payment comes later than one period from now. Valued with the annuity formula one period before the first payment, then discounted to today.
Growing annuity
A stream of payments for a fixed number of periods in which each payment is a constant percentage larger than the one before.
Implied interest rate
The interest rate at which the present value of a stream of payments equals the amount borrowed or the cash price: the rate a credit or instalment arrangement really charges.

Sources

  1. Dahlquist, J. & Knight, R. (2022). Principles of Finance, section 8.2, “Annuities”. OpenStax. openstax.org/books/principles-finance/pages/8-2-annuities
  2. Dahlquist, J. & Knight, R. (2022). Principles of Finance, section 8.5, “Equal Payments with a Financial Calculator and Excel”. OpenStax. openstax.org/books/principles-finance/pages/8-5-equal-paymen…
  3. Microsoft Support. RATE function (Excel). support.microsoft.com/en-us/office/rate-function-9f665657-4a…
  4. US Securities and Exchange Commission, Investor.gov. Annuities. investor.gov/introduction-investing/investing-basics/investm…