Annuity (engineering economics)

An annuity in engineering economics is a series of equal cash flows, $A$, occurring at equal time intervals for a fixed number of periods. Mortgages, car-loan payments, bond coupons, lease payments, and equal-instalment savings plans are all annuities.

The standard assumptions: payments are equal in size, equally spaced in time, and each payment occurs at the end of its period (the ordinary annuity convention). Less commonly, an annuity due has payments at the beginning of each period — same idea, shifted by one period.

Four standard equivalences relate an annuity to a present value $P$ at time zero or a future value $F$ at time $N$:

• $F=A\cdot (F/A,i,N)$ — accumulate the annuity to a future lump sum.
• $A=F\cdot (A/F,i,N)$ — find the equal deposit needed to accumulate to $F$ (sinking fund).
• $P=A\cdot (P/A,i,N)$ — present value of the annuity stream.
• $A=P\cdot (A/P,i,N)$ — equal payment that amortises $P$ over $N$ periods (capital recovery).

The factors are computed from compounding: see Compound interest factor for the explicit formulas.

A worked instance: you want to save up $50,000 for a down-payment in 5 years, and the savings account pays 4% effective annual interest. How much do you need to deposit at the end of each year?

$A=F\cdot (A/F,4\%,5)=50,000\cdot \frac{0.04}{(1.04{)}^5-1}=50,000\cdot \frac{0.04}{0.2167}\approx \$9,231.$

So roughly $9,231/year for 5 years builds up to $50,000 at 4%.

A second instance: a $300,000 mortgage at 5% over 25 years. What’s the annual payment?

$A=P\cdot (A/P,5\%,25)=300,000\cdot \frac{0.05\cdot (1.05{)}^{25}}{(1.05{)}^{25}-1}\approx 300,000\cdot 0.0710\approx \$21,300/\text{year}.$

(Real Canadian mortgages compound semi-annually but are paid monthly, so the actual conversion is a bit more involved. See Nominal vs effective interest rate for handling period mismatches.)

Capitalized value is the present-worth of an annuity that runs forever — a perpetual annuity. Taking the limit as $N\to \infty$ in $(P/A,i,N)$ gives the clean result

$P_{\infty }=\frac{A}{i}.$

So $1,000/year forever at 5% is worth $20,000 today. Capitalised value applies to long-lived assets where the difference between “very long” and “forever” is negligible — civic infrastructure, endowments, perpetual bonds (rare in practice but conceptually clean).

For each individual factor in detail, see Sinking fund factor, Capital recovery factor, Series present worth factor, Uniform series compound amount factor. For non-uniform series, see Arithmetic gradient series and Geometric gradient series.