Compound interest · Key concepts

It includes interest earned on prior interest

Compounding adds interest to the balance so later interest is earned on the enlarged amount.

The accumulated amount at the end of the investment period

Future value, or compound amount, includes principal and accumulated interest.

The amount needed now for a specified future amount

Present value discounts a future target back to today's equivalent amount.

n = mt

Multiply years by the number of periods in each year.

i = r/m

Divide the annual nominal rate among the m compounding periods.

2

Semiannual means twice per year, at six-month intervals.

4

A quarter is three months, so there are four quarters in a year.

12

There are twelve monthly compounding periods in a year.

FV = PV(1 + i)^n

Each period multiplies the balance by 1 + i; repeating n times gives the exponent.

PV = FV / (1 + i)^n

Discounting reverses the compounding multiplication.

FV - PV

Interest is the growth beyond the original deposit.

The stated annual rate before the effect of within-year compounding

The nominal rate is divided by m to obtain the periodic rate.

The effective return over one year including compounding

APY standardizes the effect of annual rate and compounding frequency over one year.

APY = (1 + r/m)^m - 1

One year's growth factor minus 1 is the annual effective yield.

They are equal

There is only one period in the year, so (1 + r)^1 - 1 = r.

APY is higher

Interest earned within the year itself earns additional interest.

Monthly rather than quarterly compounding

More frequent compounding earns interest on accumulated interest sooner.

Greater than 1

Compounding multiplies the deposit by a growth factor greater than 1.

Between zero and 1

A future dollar is discounted to less than one present dollar when the rate is positive.

They are reciprocals

One is (1 + i)^n and the other is its inverse.

The rate per compounding period

The table must use periodic rate i together with the total number of periods n.

The total number of compounding periods

The row counts periods, so quarterly compounding requires four rows' worth of periods per year.

Table factors are rounded

Multiplying by a rounded factor can produce a small difference from full-precision exponentiation.

Compound it forward using the same rate and periods

PV(1 + i)^n should reproduce the target FV, subject to cent rounding.

PV decreases

More time permits more growth, so less must be invested now.

PV decreases

A larger growth factor appears in the PV denominator.

Compound interest is greater

Interest on earlier interest produces additional growth after the first period.

They are equal

Both are principal multiplied by 1 + i after a single period.

0

No recurring payments occur, so the payment register must be zero.

As a negative PV

The deposit is money paid out; the later withdrawal is money received.

Old settings or values can contaminate a new calculation

A leftover payment or other variable can make the calculator solve a different scenario.

Three years after time 0

Years 1, 2, and 3 have elapsed when Year 4 begins.

Compound each for its own remaining time, then add

Each lump sum has a different number of periods until the common valuation date.

The growth is overstated

The annual nominal rate should first be divided by four.

Too few compounding periods are used

n must count quarters, not years, when i is quarterly.

Money available now can earn interest and avoids waiting risks

Earning potential and risks such as nonpayment and inflation explain why timing matters.

It equals PV

The growth factor is (1 + 0)^n = 1.

APY under the stated assumptions

APY incorporates compounding, allowing a consistent one-year comparison.

That quotient is a five-year return, not a one-year yield

APY is based on a one-year horizon; a multi-year cumulative return is different.

An amount that must be paid or invested now

The sign describes cash-flow direction; the deposit's magnitude is the funding required.