Annuities and perpetuities
Many financial products involve regular payments: loans, mortgages, pensions, and savings plans. Annuity formulas value them.
What an annuity is
An annuity is a series of equal payments at regular intervals for a fixed number of periods, such as monthly loan payments or annual pension payments. An ordinary annuity pays at the end of each period; an annuity due pays at the beginning, as with rent.
Future value of an annuity
The future value of an ordinary annuity is \(FV = PMT \times \frac{(1+r)^n - 1}{r}\). Saving 200 dollars a month for 30 years at 6% annual (0.5% monthly) grows to about \(200 \times \frac{1.005^{360} - 1}{0.005} \approx 200{,}903\) dollars, from total contributions of 72,000 dollars.
Present value of an annuity
The present value of an ordinary annuity is \(PV = PMT \times \frac{1 - (1+r)^{-n}}{r}\). It answers questions like how much a stream of future payments is worth today, or how large a loan given payments can support.
PV = PMT\frac{1-(1+r)^{-n}}{r}Loan payments
Rearranging gives the payment on a loan: \(PMT = \frac{PV \times r}{1 - (1+r)^{-n}}\). A 300,000-dollar, 30-year mortgage at 6% annual (0.5% monthly) has a monthly payment of about 1,799 dollars. Over 30 years, total payments are about 647,000 dollars, of which about 347,000 is interest.
Amortisation
An amortisation schedule shows how each loan payment splits between interest and principal. Early payments are mostly interest because the balance is large; later payments are mostly principal. Making extra principal payments early saves a great deal of interest and shortens the loan.
Perpetuities
A perpetuity pays a constant amount forever. Its present value is \(PV = \frac{PMT}{r}\). A perpetuity paying 1,000 dollars a year at a 5% discount rate is worth 20,000 dollars. A growing perpetuity, with payments rising at rate g, is worth \(\frac{PMT_1}{r - g}\), provided r > g.
Annuities due
For an annuity due, each payment occurs one period earlier, so its present and future values are the ordinary annuity values multiplied by (1 + r). Leases and insurance premiums often follow this pattern.
Worked example
A car loan of 25,000 dollars over 5 years at 7.2% annual interest (0.6% monthly, 60 payments) has a monthly payment of \(\frac{25{,}000 \times 0.006}{1 - 1.006^{-60}} \approx 497\) dollars. Total payments are about 29,840 dollars, so the interest cost is about 4,840 dollars.
Key takeaways
Annuities are equal periodic payments; ordinary annuities pay at period end, annuities due at the start. FV of an annuity = PMT x ((1 + r)^n - 1) / r; PV = PMT x (1 - (1 + r)^-n) / r. Loan payments come from rearranging the PV formula. Early payments are mostly interest; extra principal payments save interest. Perpetuities are worth PMT / r; growing perpetuities PMT / (r - g). Annuities due multiply by (1 + r).
Flashcard
Equal payments at regular intervals
Check your understanding
A perpetuity pays 500 dollars a year forever. At an 8% discount rate, what is its present value?