Ratedrill / Cheat sheet

Time value of money formulas cheat sheet

Every time value of money formula from a finance course on one sheet, three ways: the algebra, the Excel function with the sign convention that trips everyone, and the BA II Plus keys. Most entries have a problem with fresh numbers every time. Print the page or get the PDF by email.

The ones worth learning first

What	How
Future value of a lump sum	`FV = PV (1 + r)ⁿ =FV(r, n, 0, -PV)`
Present value of a lump sum	`PV = FV / (1 + r)ⁿ =PV(r, n, 0, -FV)`
Payment on a loan	`PMT = PV · r / [1 - (1 + r)⁻ⁿ] =PMT(r, n, -PV)`
Present value of an annuity	`PV = PMT · [1 - (1 + r)⁻ⁿ] / r =PV(r, n, -PMT)`
Future value of an annuity	`FV = PMT · [(1 + r)ⁿ - 1] / r =FV(r, n, -PMT)`
Periodic rate	`r = APR / m, n = years x m`
Effective annual rate	`EAR = (1 + APR/m)ᵐ - 1 =EFFECT(APR, m)`
Years to a goal	`n = ln(FV/PV) / ln(1 + r) =NPER(r, 0, -PV, FV)`
Perpetuity	`PV = PMT / r`
Sign convention	`Excel and the calculator: money out negative, money in positive`

Interest and rates

Simple interest	`I = P · r · t total = P (1 + r t)` drill
Compound growth	`FV = PV (1 + r)ⁿ` drill
Simple vs compound	`Difference = PV (1 + r)ⁿ - PV (1 + r n)` drill
Monthly compounding	`FV = PV (1 + APR/12)^(12 t)` drill
Periodic rate and periods	`r = APR / m, n = years x m (m = 12 monthly, 4 quarterly, 365 daily)`
Effective annual rate	`EAR = (1 + APR/m)ᵐ - 1 =EFFECT(APR, m)` drill
Daily compounding EAR	`EAR = (1 + APR/365)³⁶⁵ - 1` drill
Continuous compounding	`FV = PV e^(r t), EAR = e^r - 1 =EXP(r)-1` drill
APR from an EAR	`APR = m [ (1 + EAR)^(1/m) - 1 ] =NOMINAL(EAR, m)`
Real rate (Fisher)	`1 + nominal = (1 + real)(1 + inflation)`
Rule of 72	`years to double ≈ 72 / rate in percent`
Exact doubling time	`n = ln 2 / ln(1 + r) =NPER(r, 0, -1, 2)` drill

Lump sums

Future value	`FV = PV (1 + r)ⁿ =FV(r, n, 0, -PV) N I/Y PV CPT FV` drill
Present value	`PV = FV / (1 + r)ⁿ =PV(r, n, 0, -FV)` drill
Rate needed	`r = (FV/PV)^(1/n) - 1 =RATE(n, 0, -PV, FV)` drill
Years needed	`n = ln(FV/PV) / ln(1 + r) =NPER(r, 0, -PV, FV)` drill
Discount factor	`DF = 1 / (1 + r)ⁿ`
CAGR	`(end/start)^(1/years) - 1 =RRI(years, start, end)`

Annuities

PV of an ordinary annuity	`PV = PMT [1 - (1 + r)⁻ⁿ] / r =PV(r, n, -PMT)` drill
FV of an ordinary annuity	`FV = PMT [(1 + r)ⁿ - 1] / r =FV(r, n, -PMT)` drill
Annuity due (payments at the start)	`multiply the ordinary result by (1 + r) =FV(r, n, -PMT, 0, 1) BGN mode` drill
PV of an annuity due (lease)	`PV = PMT [1 - (1 + r)⁻ⁿ] / r · (1 + r) =PV(r, n, -PMT, 0, 1)` drill
Payment to reach a goal	`PMT = FV · r / [(1 + r)ⁿ - 1] =PMT(r, n, 0, -FV)` drill
Perpetuity	`PV = PMT / r` drill
Growing perpetuity	`PV = PMT₁ / (r - g) (first payment one period out, r > g)` drill
Growing annuity	`PV = PMT₁/(r - g) · [1 - ((1+g)/(1+r))ⁿ]` drill
Deferred annuity	`PV today = PV of the annuity at its start / (1 + r)^(periods of delay)` drill
How long a balance lasts	`n = -ln(1 - PV r / PMT) / ln(1 + r) =NPER(r, PMT, -PV)` drill
Income from savings	`PMT = PV · r / [1 - (1 + r)⁻ⁿ] =PMT(r, n, -PV)` drill
Present value of uneven cash flows	`PV = Σ CFₜ / (1 + r)ᵗ =NPV(r, CF1:CFn) + CF0`

Loans

Mortgage payment	`PMT = P · r / [1 - (1 + r)⁻ⁿ], r = APR/12, n = years x 12 =PMT(APR/12, years*12, -P)` drill
Car loan payment	`same formula, n = months =PMT(APR/12, months, -P)` drill
Balance after k payments	`B_k = PMT [1 - (1 + r)⁻⁽ⁿ⁻ᵏ⁾] / r (PV of the remaining payments) =PV(r, n-k, -PMT)` drill
Interest in payment k	`I_k = B_(k-1) · r =IPMT(r, k, n, -P)` drill
Principal in payment k	`P_k = PMT - I_k =PPMT(r, k, n, -P)` drill
Total interest	`n · PMT - P =-CUMIPMT(r, n, P, 1, n, 0)` drill
Payoff time with extra payments	`=NPER(r, -(PMT + extra), P)` drill
Balloon payment	`balloon = FV of P minus FV of the payments made =FV(r, n, PMT, -P)` drill
APR implied by a payment	`=RATE(n, -PMT, P) x 12` drill
How much you can borrow	`P = PMT [1 - (1 + r)⁻ⁿ] / r =PV(r, n, -PMT)` drill
Amortization row	`interest = opening x r; principal = PMT - interest; closing = opening - principal`
Points and fees in the real rate	`=RATE(n, -PMT, P - fees) x 12`

Excel and calculator

Excel sign convention	`Cash paid out is negative, cash received is positive. =PMT(r, n, -PV) returns a positive payment.`
Excel rate and periods	`Always the periodic rate with the matching number of periods: 6%/12 and 30*12, never 6% and 30.`
Type argument	`0 (or omitted) = end of period; 1 = beginning (annuity due)`
BA II Plus basics	`Set P/Y = 1 and enter the periodic rate as I/Y; clear with 2ND CLR TVM before each problem`
BA II Plus annuity due	`2ND BGN, 2ND SET, 2ND QUIT; set back to END after`
BA II Plus amortization	`2ND AMORT, P1 and P2, then arrows for BAL, PRN, INT`
HP 12C	`n i PV PMT FV keys; g BEG / g END for timing; negative with CHS`
Check	`Longer term: lower payment, more total interest. Higher rate: higher payment. PV of an annuity is always less than n x PMT.`

ratedrill.com: practice every one of these with instant checking. 50 entries.

Questions

Why does Excel give me a negative number?

Excel follows the cash flow sign convention: what you pay out is negative and what you receive is positive. Enter the present value as a negative (money borrowed is received, so for a loan payment enter -PV to get a positive PMT) or wrap the result in ABS.

Annual rate or monthly rate?

Always match the rate to the period of the payments. A 6% APR with monthly payments is 0.5% per period for years x 12 periods. Using 6% with 12 x years periods is the single most common exam mistake.

Ordinary annuity or annuity due?

Ordinary: payments at the end of each period (loans, most bonds). Due: payments at the start (rent, leases, savings deposited on the first of the month). The due value is the ordinary value multiplied by (1 + r).

APR or EAR?

APR is the quoted nominal rate; EAR is what you actually earn or pay after compounding within the year. Compare offers on EAR; use APR divided by the periods per year for calculations.

A cheat sheet does not make them stick

Doing them with instant checking does. The free exercises need no account.

Start practicing