Financial Mathematics Reference

Definitions

Simple interest

I = P r t

Interest calculated only on the original principal.

Example:

$1, 000

5%

for

2

years earns

$100

Compound interest

A = P (1 + i)^{n}

Interest calculated on principal plus previously accumulated interest.

Example:

$1, 000

compounded annually at

5%

grows to

$1, 102.50

2

years.

Annuity

A sequence of equal payments made at regular intervals.

Example: Monthly retirement contributions form an annuity.

Annuity due

An annuity with payments at the beginning of each period.

Example: Monthly rent paid at the start of the month.

Perpetuity

P V = R / i

An equal-payment stream that continues indefinitely.

Example:

$100

yearly forever at

5%

has

P V = $2, 000

Effective rate

i_{e f f} = (1 + r / m)^{m} - 1

The actual annual growth rate after accounting for compounding within the year.

Example:

6%

nominal compounded monthly is slightly more than

6%

effective.

Continuous compounding

A = P e^{r t}

The limiting growth model as compounding becomes continuous.

Example:

$1, 000

4%

continuously for

3

years is

1000 e^{0.12}

Formula sheet

9 formulas

Simple interest

I = P r t, A = P (1 + r t)

Variables: $P$ : principal; $r$ : annual decimal rate; $t$ : years
Remember: Interest is calculated on original principal only.

Compound future value

A = P (1 + \frac{r}{m})^{m t}

Variables: $m$ : compounding periods per year
Remember: Rate and number of periods must use matching units.

Compound present value

P = \frac{A}{( 1 + i ) ^{n}}

Variables: $i$ : rate per period; $n$ : periods
Remember: Discount a future amount back to today.

Ordinary annuity future value

F V = R \frac{( 1 + i ) ^{n} - 1}{i}

Variables: $R$ : end-of-period payment; $i$ : period rate
Remember: Payments occur at the end of each period.

Ordinary annuity present value

P V = R \frac{1 - ( 1 + i ) ^{- n}}{i}

Variables: $R$ : payment
Remember: Present value of end-of-period payments.

Annuity due

F V_{d u e} = F V_{or d ina r y} (1 + i)

Variables: $i$ : period rate
Remember: Beginning-of-period payments earn one extra period.

Perpetuity

P V = \frac{R}{i}

Variables: $R$ : regular payment; $i$ : period rate
Remember: Assumes the payment continues forever and $i > 0$ .

Effective annual rate

i_{e f f} = (1 + \frac{r}{m})^{m} - 1

Variables: $r$ : nominal annual rate; $m$ : compounds per year
Remember: Reports actual annual growth.

Continuous compounding

A = P e^{r t}

Variables: $e$ : Euler’s number
Remember: The limiting compound-growth model.

Worked examples

Compound an investment

Find the value of $$2, 000$ at $6%$ compounded monthly for $3$ years.

Show solution

$P = 2000, r = 0.06, m = 12, t = 3$ .
$A = 2000 (1 + \frac{0.06}{12})^{12 \cdot 3}$ .
$A = 2000 (1.005)^{36}$ .
$A \approx $2, 393.36$ .

Comparison table

Financial models
Model	Payment timing	Core idea
Simple interest	No periodic payment	Interest on original principal
Compound interest	No periodic payment	Interest on accumulated balance
Ordinary annuity	End of each period	Equal repeating payments
Annuity due	Beginning of each period	Each payment earns one extra period
Perpetuity	Regular forever	Infinite equal-payment stream
Continuous compounding	No periodic payment	Limiting exponential growth model

Variable glossary

$P, P V$: principal or present value, in dollars
$A, F V$: accumulated or future value, in dollars
$R$: regular payment, in dollars per period
$r$: nominal annual rate as a decimal
$i$: interest rate per payment/compounding period
$n$: number of periods

Student tools

Open graph studio Guided calculators