Compound interest: future and present value

Under simple interest, only the original principal earns interest. Under compound interest, interest is added to the balance at the end of each compounding period, and from then on the interest itself earns interest. Over a few months the difference is small, but over years it becomes enormous. Compound interest drives savings and investment growth, the cost of long-term loans and mortgages, the valuation of businesses and bonds, and the long-run effect of inflation. This lesson develops the core formulas for moving a single amount forward and backward in time, and for comparing rates that compound at different frequencies.

Nominal rates and periodic rates

Compound rates are quoted as a nominal annual rate together with a compounding frequency, such as 6 % compounded monthly. The nominal rate (written j) is divided by the number of compounding periods per year (m) to give the periodic rate (i) actually applied each period. The total number of periods is the number of years times m.

Future value

If a principal PV is invested at a periodic rate i, after one period it grows to PV(1 + i), after two periods to PV(1 + i)(1 + i), and after n periods to the future value FV.

The factor \( (1 + i)^n \) is the compounding factor. Because n is an exponent, the growth accelerates: each period's interest is larger than the last.

The more often interest compounds, the faster the balance grows, but the effect shrinks as the frequency rises. Daily compounding beats monthly by only a few dollars on $10,000 over 10 years. The limit, continuous compounding, gives \( FV = PV e^{jt} \).

Present value

Rearranging the future value formula gives the present value: the amount that must be invested today to reach a given future amount. Finding a present value is called discounting, and the factor \( (1 + i)^{-n} \) is the discount factor.

Present value is one of the most important ideas in finance. It lets amounts at different dates be compared on equal terms, and it underlies the valuation of loans, bonds, leases, pensions and whole businesses. Lesson 10 uses it to evaluate investment decisions.

Effective and equivalent rates

Because compounding frequencies differ, nominal rates cannot be compared directly. A rate of 6 % compounded monthly earns more than 6 % compounded annually. The effective annual rate (f) is the annually compounded rate that gives the same growth in one year. It is the fairest single number for comparing loans and investments.

Sometimes a rate stated with one frequency must be converted to an equivalent rate with another, for example when payments are monthly but the rate compounds semi-annually. Two rates are equivalent when they give the same effective annual rate. To convert, find the periodic rate for the new frequency that produces the same growth over a year.

Solving for the rate and the time

When the present value, future value and number of periods are known, the periodic rate is found by taking the n-th root. When the rate is known, the number of periods is found with logarithms.

Changing rates

When the rate changes during the term, compound the amount in stages: grow it at the first rate for the first stretch, then use the result as the starting value for the next stretch at the new rate.

Inflation and real returns

Inflation erodes what money can buy. If prices rise 2.4 % a year, $1,000 a year from now buys what $976.56 buys today. The nominal return on an investment is the stated rate; the real return is the growth in purchasing power after inflation. Over long periods inflation compounds exactly like interest: at 2.5 % a year, prices roughly double every 28 years.

Taxes reduce returns further. Interest earned in an ordinary account is fully taxable each year, but interest earned inside a tax-free savings account (TFSA) is never taxed, and growth inside a registered retirement savings plan (RRSP) is taxed only when withdrawn. The compounding formulas are the same; what changes is the rate that is left after tax.

# Future value, effective rates, equivalent rates and solving for n
import math
PV, j = 10000, 0.06
for m, name in [(1, "annual"), (2, "semi-annual"), (4, "quarterly"), (12, "monthly"), (365, "daily")]:
    fv = PV * (1 + j / m) ** (m * 10)
    eff = (1 + j / m) ** m - 1
    print(f"{name:12s} FV {fv:10,.2f}  effective {eff:.4%}")
print(f"PV of 50,000 in 7 years at 4.5% quarterly: {50000 / 1.01125 ** 28:,.2f}")
print(f"Monthly rate equivalent to 5.5% semi-annual: {1.0275 ** (1 / 6) - 1:.5%}")
print(f"Years to double at 7%: {math.log(2) / math.log(1.07):.2f}")

annual       FV  17,908.48  effective 6.0000%
semi-annual  FV  18,061.11  effective 6.0900%
quarterly    FV  18,140.18  effective 6.1364%
monthly      FV  18,193.97  effective 6.1678%
daily        FV  18,220.29  effective 6.1831%
PV of 50,000 in 7 years at 4.5% quarterly: 36,553.67
Monthly rate equivalent to 5.5% semi-annual: 0.45317%
Years to double at 7%: 10.24