Annuities

An annuity is a series of equal payments made at regular intervals. Monthly rent, car loan payments, mortgage payments, regular contributions to a retirement plan, lease payments, pension income and bond interest are all annuities. Instead of moving each payment separately with the compound interest formula, annuity formulas move the whole series at once. This lesson introduces the main types of annuity, the formulas for their future and present values, and how to solve for the payment, the term and the rate, by hand and with spreadsheet functions.

Types of annuity

The term of an annuity runs from the start of the first payment interval to the end of the last. For an ordinary annuity the first payment comes one period after the start; for an annuity due it comes immediately.

Future value of an ordinary annuity

The future value of an ordinary annuity is the sum of the future values of all the payments at the end of the term. The last payment earns no interest, the second-last earns one period of interest, and so on. Adding them up gives a geometric series with a neat closed form.

Annuities due

In an annuity due, every payment is made one period earlier than in the matching ordinary annuity, so every payment earns one extra period of interest. Both the future value and the present value of an annuity due equal the ordinary annuity value multiplied by (1 + i).

Present value of an ordinary annuity

The present value of an annuity is the single amount today that is equivalent to the whole series of payments. It is what a lender will advance in exchange for a series of loan payments, or what an investment that pays a fixed income is worth today.

Finding the payment

Often the future or present value is known and the payment must be found. Rearrange either formula.

General annuities

When payments and compounding happen at different frequencies, first convert the rate to the equivalent periodic rate per payment interval, using the method from lesson 7. Then use the simple annuity formulas with that rate.

Canadian mortgages are the most important general annuity in everyday life: payments are usually monthly or biweekly while the rate compounds semi-annually. Lesson 9 works through them in detail.

Finding the number of payments

When the payment, rate and present or future value are known, the number of payments is found with logarithms. The answer is rarely a whole number; the final payment is then smaller than the others.

Finding the interest rate of an annuity has no algebraic solution; it is done by trial and error or, in practice, with a financial calculator or the spreadsheet RATE function.

Perpetuities and deferred annuities

A perpetuity pays forever. Its present value is finite because distant payments are worth very little today. If the payment interval matches the compounding period, the present value is simply the payment divided by the periodic rate.

A deferred annuity begins after a delay. Value it in two steps: find the present value of the annuity at the start of its term, then discount that amount back to today over the deferral period.

Comparing payment plans

Annuity values turn a choice between payment plans into a straightforward comparison: convert each plan to its present value at the rate the business can earn or must pay, and choose the plan with the lowest present cost. Totals of payments are misleading, because they ignore when the money is paid.

Spreadsheet functions

Spreadsheets have built-in functions for annuities: FV(rate, nper, pmt, [pv], [type]), PV(rate, nper, pmt, [fv], [type]), PMT(rate, nper, pv, [fv], [type]), NPER(rate, pmt, pv, [fv], [type]) and RATE(nper, pmt, pv, [fv], [type], [guess]). The rate is the periodic rate, nper is the number of payments, and type is 0 for an ordinary annuity and 1 for an annuity due. Money paid out is entered as a negative number and money received as positive, so =PMT(0.05/12, 120, 0, 100000) returns −643.99: a deposit of $643.99 each month.

# Annuity functions matching the spreadsheet conventions (end-of-period payments)
import math
def fv_annuity(pmt, i, n, due=False): return pmt * ((1 + i) ** n - 1) / i * ((1 + i) if due else 1)
def pv_annuity(pmt, i, n, due=False): return pmt * (1 - (1 + i) ** -n) / i * ((1 + i) if due else 1)
def pmt_for_fv(fv, i, n): return fv * i / ((1 + i) ** n - 1)
def nper(pv, pmt, i): return -math.log(1 - pv * i / pmt) / math.log(1 + i)

print(f"RRSP/401(k) at 65: {fv_annuity(300, .005, 300):,.2f}; due: {fv_annuity(300, .005, 300, True):,.2f}")
print(f"Lease value: {pv_annuity(2500, .006, 60):,.2f}")
print(f"Sinking fund deposit: {pmt_for_fv(100000, .05 / 12, 120):.2f}")
p = 1.005 ** 3 - 1
print(f"Quarterly deposits FV: {fv_annuity(1500, p, 32):,.2f}")
print(f"Months the retirement fund lasts: {nper(500000, 3500, .045 / 12):.1f}")

RRSP/401(k) at 65: 207,898.19; due: 208,937.68
Lease value: 125,655.33
Sinking fund deposit: 643.99
Quarterly deposits FV: 61,108.22
Months the retirement fund lasts: 205.0