Time Value of Money

foundation

Future Value of a Single Sum

FV = PV(1 + r)^n

$FV$: future value — what the money has become by the end
$PV$: present value — the amount you start with today
$r$: the interest rate per period, written as a decimal (8% is 0.08)
$(1+r)$: the growth factor: the 1 keeps your original money, the r adds one period's interest on top
$(1+r)^n$: the little raised n means multiply by that growth factor n times — once for every period

Play with it

The gap between the straight line and the curve is interest-on-interest — the entire difference between simple and compound growth.

Rate8.0%Horizon20y

2.6×

Simple @ 20y

4.7×

Compound @ 20y

2.1×

Interest on interest

The gap between the straight line and the curves is interest earning interest. Continuous compounding is the ceiling — barely above annual at low rates, decisive at high ones.

Why it must be true

Money grows because each period you earn a return on everything you have so far — including past interest. That is all compounding is: multiply by $(1+r)$ once per period, so after $n$ periods you have multiplied by $(1+r)^n$ .

If interest were paid only on the original principal (simple interest), you would add $r \cdot PV$ each period and get $PV(1+rn)$ — always less than the compound answer for $n>1$ . The gap between the two is interest-on-interest.

The derivation

Start with one period: you invest $PV$ and earn the rate $r$ on it.

V_1 = PV + r \cdot PV = PV(1+r)

The second period pays $r$ on the whole balance $V_1$ , not just on $PV$ :

V_2 = V_1(1+r) = PV(1+r)^2

Each period multiplies the balance by the same factor $(1+r)$ , so after $n$ periods:

FV = PV(1+r)^n

When to reach for it

One lump sum, one constant compounded rate, no cash flows in between. If money is deposited once and left alone, this is the formula.

Listen for

“compounded annually / quarterly / monthly”“grows to … after n years”“a single deposit today”“what will it be worth in …”“CAGR / compound annual growth”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

≈
Rule of 72: money doubles in about 72 ÷ rate% periods. At 8%, ~9 periods per doubling — so 18 periods ≈ 4× your money. Count doublings to bracket any answer.
≈
The answer must beat simple interest: quickly compute $PV(1 + rn)$ in your head — the correct choice is the one just above it (the gap grows with $n$ ).
≈
For short horizons, $(1+r)^n \approx 1 + rn + \frac{n(n-1)}{2}r^2$ — the compound answer exceeds simple by roughly half of $(rn)^2$ .

Traps in applying it

✗Mismatching rate and period — an annual rate with monthly periods needs r/12 and n in months, together.
✗Applying it to a stream of payments — multiple deposits need the annuity formula, not repeated lump sums by shortcut.
✗Reading 'interest of 5% over 3 years' as per-period when it is total, or vice versa.

Limits & criticisms

It assumes one constant rate for the whole horizon. Real reinvestment rates float, so a 20-year projection at a fixed r is a scenario, not a forecast. It is also silent on taxes, fees and inflation — the growth is nominal and gross, and a "real" answer needs a real (inflation-adjusted) rate.

Where it came from

Compound interest is one of the oldest pieces of applied mathematics: Babylonian clay tablets from around 2000 BCE pose problems of interest accumulating on interest. Fibonacci's Liber Abaci (1202) brought systematic interest arithmetic to Europe, and Simon Stevin published practical compound-interest tables in 1582 — bankers' tools centuries before calculators.

In practice today it is everywhere growth is reported: savings products, CAGR figures in fund marketing, and every retirement projection. It is also the seed of everything downstream — discounting, annuities and bond pricing are all this one identity, rearranged.

One identity, 4 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

Grow money forward

FV = PV(1+r)^n

The base case: one growth factor $(1+r)$ per period, compounded. Every other face is this identity read in another direction.

Drill this face →

Discount money back

PV = \frac{FV}{(1+r)^n}

The same equation solved for today. Dividing by the growth factor is discounting — the price today of a future dollar.

Drill this face →

Implied compound return

r = \left(\frac{FV}{PV}\right)^{1/n} - 1

Given start and end values, the $n$ -th root spreads total growth evenly across periods — this is how track-record CAGRs are computed.

Drill this face →

Time to grow

n = \frac{\ln(FV/PV)}{\ln(1+r)}

Solving for the exponent needs logarithms. "How long until my money doubles?" is this face with $FV/PV = 2$ — the rule of 72 approximates it.

Drill this face →

On the BA II Plus

Worked example: An investment will be worth $20,481.35 after 10 periods of compounding at 2.5% per period. What is it worth today?

1.[2ND] [CLR TVM]always clear the worksheet first
2.10 [N]
3.2.5 [I/Y]the rate per period, as a percent
4.0 [PMT]no recurring payment — zero it explicitly
5.20,481.35 [FV]
6.[CPT] [PV]compute the unknown

→ $16,000.00

Where it leads

Master this and the following come almost for free:

Present Value of a Single Sum Effective Annual Rate (EAR)Future Value of an Ordinary Annuity Implied Forward Rate Forward Price

Drill this formula Back to the library