Compound interest

A concise reviewer for Slater Chapter 19.

Use i = r/m and n = mt. Keep full formula precision; round final money to cents and APY to 0.01 percentage point. Periodic-rate questions specify four decimal places. Use supplied table factors exactly. Daily compounding uses 365 periods per year; weekly uses 52. Only stated cash flows occur.

Compound growth

Each period adds interest to the balance, so the next period starts from a larger base.

Future value (FV), also called the compound amount, contains the original present value (PV) plus accumulated interest. Simple interest uses a constant principal; compound interest builds on previous interest.

For the same positive periodic rate and more than one whole period, compounding produces more than simple interest. Over exactly one period, the results agree.

FV = PV(1 + i)ⁿ
Compound interest = FV − PV

Worked example: Grow a single deposit

Invest $5,000 for 3 years at 6% compounded quarterly. Find the balance and total interest.

Periodic rate: i = 0.06/4 = 0.015. Total periods: n = 3 × 4 = 12.
FV = 5,000(1.015)¹² = $5,978.09.
Compound interest = 5,978.09 − 5,000 = $978.09.

Future value: $5,978.09 · Interest: $978.09

Watch out: Future value is not the same as interest. Subtract the initial deposit when the question asks for earnings alone.

Slater Ch. 19, pp. 499–505

Periods, rates & tables

Use a periodic rate and a total period count—not the annual rate and years by default.

Let m be periods per year: annual 1, semiannual 2, quarterly 4, monthly 12, weekly 52, and daily 365 in this study set. Divide the nominal annual rate by m and multiply years by m.

For table lookup, choose the periodic-rate column and total-periods row. Multiply PV by the FV factor, or FV by the PV factor. Rounded table factors can differ slightly from full-precision formulas.

i = r / m
n = m × t
FV = PV × future-value table factor

Worked example: Quarterly compounding for five years

Find i and n for 8% nominal annual interest compounded quarterly over 5 years.

There are m = 4 periods per year.
i = 8% / 4 = 2% per quarter (0.02 in formulas).
n = 5 × 4 = 20 periods. Use the 20-period row and 2% column in a table.

i = 2% per quarter · n = 20 quarters

Watch out: Do not use 8% in every quarter or only 5 periods. Both inputs must describe the same period length.

Slater Ch. 19, pp. 500–504, 509

Nominal rate vs. APY

APY includes the effect of compounding over one year.

The nominal annual rate is the stated rate before the effect of within-year compounding. APY is the one-year effective return. At a positive nominal rate, more frequent compounding gives a higher APY.

Annual compounding gives APY equal to the nominal rate. With more than one compounding period per year, APY exceeds a positive nominal rate. A multiyear total return is not APY.

APY = (1 + r/m)ᵐ − 1
APY = interest for one year / initial principal

Worked example: A 6% account with quarterly interest

Find APY for a nominal annual rate of 6% compounded quarterly.

Periodic rate = 0.06 / 4 = 0.015.
One-year growth factor = (1.015)⁴ = 1.06136355…
APY = 1.06136355… − 1 = 0.06136355… = 6.14%.

APY = 6.14%

Watch out: Use one year for APY. Dividing a five-year gain by the original deposit produces a five-year return.

Slater Ch. 19, pp. 506–507

Present value

Present value tells you how much to invest now to reach a target later.

Discounting to PV reverses compound growth. Use the same periodic rate and number of periods, but divide the future amount by the growth factor.

For a fixed future target and positive rate, a longer term or higher rate reduces required PV. A PV table factor is between 0 and 1. Check the answer by compounding it forward.

PV = FV / (1 + i)ⁿ
Check: PV(1 + i)ⁿ = FV

Worked example: Fund a purchase five years away

Need $20,000 in 5 years at 8% compounded semiannually. How much is needed now?

i = 0.08/2 = 0.04; n = 5 × 2 = 10.
PV = 20,000 / (1.04)¹⁰ = $13,511.28.
Compounding the unrounded PV by (1.04)¹⁰ returns $20,000. A deposit rounded to cents may give a small difference.

Required deposit: $13,511.28

Watch out: Multiplying by the growth factor moves forward in time. To find present value, divide by it.

Slater Ch. 19, pp. 508–511

Deposits at different dates

Move each cash flow to a common date before adding or comparing.

The beginning of Year 4 is three years after time 0. A deposit made then has fewer periods to grow than the initial deposit. Compound each amount for its own remaining term and add the results.

For successive rates, grow the balance through each stage in order. To handle a withdrawal, grow to its date, subtract it, then grow the remainder. For additional funding today, find the target’s PV and subtract funds already available today.

Total FV = sum of each deposit’s own future value
Future shortfall = future goal − future savings
Additional deposit now = required PV − available funds

Worked example: A deposit now and one later

Deposit $4,000 now and $2,500 at the beginning of Year 4. Find the balance at the end of Year 6 at 6% compounded semiannually.

Initial deposit grows for 6 years: 4,000(1.03)¹² = $5,703.04.
Later deposit is made 3 years after time 0, so it grows for 3 years: 2,500(1.03)⁶ = $2,985.13.
Add the unrounded results: total FV = $8,688.17.

Total future value: $8,688.17

Watch out: Do not combine all deposits and give them the longest time horizon. Only the first deposit was invested at time 0.

Slater Ch. 19, pp. 517–522