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Every finance formula on this site

57 formulas, each with what its symbols stand for and a working calculator underneath it. Every one is applied in a worked example whose numbers are recomputed by an independent program before the site can build.

Class and first paycheck

  • Simple interest calculator and formula

    I=PrtA=P(1+rt)I = Prt \qquad A = P(1 + rt)

    PP is the original principal, rr the annual rate as a decimal, and tt time in years. II is the interest. AA is principal plus interest. Eighteen months is t=1.5t = 1.5.

  • Hourly wage to annual salary

    A=h×w×nA = h \times w \times n

    hh is the hourly wage, ww paid hours in a week, and nn paid weeks in a year. Weekly pay is h×wh \times w. Monthly pay on this page is A/12A / 12, not four weeks of pay.

  • Paycheck calculator and FICA split

    Net=G−0.062min⁡(G,B/n)−0.0145G−fG−sG\text{Net} = G - 0.062\min(G, B/n) - 0.0145 G - fG - sG

    GG is gross pay this check, nn the number of checks a year, BB the Social Security wage base, ff federal withholding as a decimal, and ss state withholding as a decimal. The min term spreads the annual Social Security cap evenly across the year's checks.

  • 50/30/20 budget split calculator

    N=0.50HW=0.30HS=0.20HN = 0.50H \quad W = 0.30H \quad S = 0.20H

    HH is take-home pay. The three percents are yours to type. 50, 30 and 20 are the usual starting split, not a law. Leftover is HH minus the three buckets.

  • Net worth calculator and formula

    Net worth=A−D\text{Net worth} = A - D

    AA is everything owned at today's prices. DD is everything owed at today's payoff balances. The difference can be negative.

  • Credit utilization calculator

    U=BLU = \frac{B}{L}

    BB is the reported revolving balance and LL the total revolving limit. The page prints UU in percentage points: 30, not 0.30. Instalment loans sit outside this ratio.

  • Emergency fund calculator: size and time

    D=(T−S(1+r12)N)r12(1+r12)N−1D = \frac{\left(T - S\left(1 + \frac{r}{12}\right)^{N}\right)\frac{r}{12}}{\left(1 + \frac{r}{12}\right)^{N} - 1}

    TT is the target, which is your essential monthly costs times the months of cover you want. SS is what is already set aside, rr the nominal annual rate as a decimal, before compounding is folded into it, NN the number of monthly deposits, and DD the deposit at the end of each month.

Saving and growth

  • Compound interest calculator and formula

    A=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt}

    AA is the ending balance, PP the starting amount, rr the annual rate as a decimal, nn how many times a year interest is added, and tt the number of years.

  • Savings goal calculator: monthly deposit

    PMT=(FV−P(1+rn)nt)rn(1+rn)nt−1PMT = \frac{\left(FV - P\left(1 + \frac{r}{n}\right)^{nt}\right)\frac{r}{n}}{\left(1 + \frac{r}{n}\right)^{nt} - 1}

    FVFV is the target, PP what you have already saved, rr the annual rate as a decimal, nn how many times a year interest is added and money goes in, and tt the number of years. PMTPMT is the deposit at the end of each period.

  • Rule of 72 calculator and doubling time

    t≈72rt=ln⁡2ln⁡(1+r/100)t \approx \frac{72}{r} \qquad t = \frac{\ln 2}{\ln(1 + r/100)}

    rr is the annual growth rate in percentage points, so 7 rather than 0.07, and tt is the number of years the money takes to double. The expression on the left is the shortcut. The one on the right is the exact answer it is standing in for.

  • Inflation calculator and buying power

    Pt=P0(1+i)tBt=P0(1+i)tP_t = P_0 (1 + i)^t \qquad B_t = \frac{P_0}{(1 + i)^t}

    P0P_0 is the sum today, ii the inflation rate for one year as a decimal, and tt the number of years. PtP_t is what the same basket of goods costs then. BtB_t is what P0P_0 still buys then, measured in today's money.

  • Real return after inflation calculator

    rreal=1+rnominal1+i−1r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + i} - 1

    rnominalr_{\text{nominal}} is the return you are quoted, ii is inflation over the same period, and rrealr_{\text{real}} is what the money actually buys. Both rates go in as decimals, so 7 percent is 0.07.

  • Future value of an annuity calculator

    FV=PMT×(1+i)n−1iFV = PMT \times \frac{(1 + i)^n - 1}{i}

    FVFV is the value at the end, PMTPMT the payment made every period, ii the rate for one period, and nn the number of payments.

  • Present value calculator

    PV=F(1+i)n+PMT×1−(1+i)−niPV = \frac{F}{(1+i)^n} + PMT \times \frac{1 - (1+i)^{-n}}{i}

    FF is a lump due after nn periods, PMTPMT a level end-of-period payment, and ii the rate per period. At a zero rate the present value is just the undiscounted sum, because nothing is being given up by waiting.

  • CAGR calculator and formula

    CAGR=(VtV0)1t−1\text{CAGR} = \left(\frac{V_t}{V_0}\right)^{\frac{1}{t}} - 1

    V0V_0 is the value at the start, VtV_t the value at the end, and tt the number of years between them. The answer comes out as a decimal, so 0.1029 means 10.29 percent a year.

  • APR vs APY calculator and formula

    APY=(1+APRn)n−1APY = \left(1 + \frac{APR}{n}\right)^{n} - 1

    APRAPR is the nominal annual rate as a decimal and nn is how many times a year interest is added. The result is the effective annual rate.

Borrowing

  • Loan payment calculator and formula

    M=P×i1−(1+i)−nM = P \times \frac{i}{1 - (1 + i)^{-n}}

    MM is the payment, PP the amount borrowed, ii the rate for one period (the annual rate divided by 12 for a monthly loan), and nn the total number of payments.

  • Mortgage affordability calculator and formula

    max loan=(L×I−D−T)(1−(1+i)−n)i\text{max loan} = \frac{\left(L \times I - D - T\right)\left(1 - (1 + i)^{-n}\right)}{i}

    LL is the lender's debt-to-income limit as a decimal, II gross monthly income, DD the other monthly debt payments and TT monthly tax and insurance. What survives those subtractions is the budget for principal and interest, and the fraction turns that budget into a loan size at period rate ii over nn payments.

  • Mortgage extra payment calculator

    n=ln⁡(1−iBp)−ln⁡(1+i)n = \frac{\ln\left(1 - \frac{iB}{p}\right)}{-\ln(1+i)}

    BB is the balance, ii the monthly rate, and pp the total monthly payment including extra principal. nn is the number of months until the balance hits zero. If pp does not cover the first month's interest, there is no finite nn.

  • Credit card payoff calculator and formula

    n=−ln⁡(1−iBM)ln⁡(1+i)n = -\frac{\ln\left(1 - \frac{iB}{M}\right)}{\ln(1 + i)}

    nn is the number of monthly payments, BB the balance you owe now, MM the fixed payment you make each month, and ii the monthly rate. A card quotes a nominal annual rate, so ii is that rate divided by 12, not the smaller monthly rate that would compound up to it over a year.

  • Car loan calculator: payment and total cost

    M=P×i1−(1+i)−nM = P \times \frac{i}{1 - (1 + i)^{-n}}

    MM is the monthly payment, PP the amount financed, ii the monthly rate (the nominal annual rate divided by 12, not an effective annual rate), and nn the number of payments, which is the term in years times 12.

  • Student loan payoff calculator

    n=−ln⁡(1−iBM)ln⁡(1+i)n = \frac{-\ln\left(1 - \frac{i B}{M}\right)}{\ln(1 + i)}

    BB is the balance, ii the monthly rate (the nominal annual rate divided by 12, not an effective annual rate), MM the payment you actually make each month, and nn the number of months until the balance reaches zero.

  • Debt-to-income ratio calculator and formula

    DTI=total monthly debt paymentsgross monthly income×100%\text{DTI} = \frac{\text{total monthly debt payments}}{\text{gross monthly income}} \times 100\%

    Above the line goes every payment you are required to make each month. Below it goes gross pay, before tax and deductions. The division gives a decimal, so multiply by 100100 for the percentage a lender quotes.

  • Refinance break-even calculator and formula

    months=CMold−Mnew\text{months} = \frac{C}{M_{\text{old}} - M_{\text{new}}}

    CC is the closing costs, MoldM_{\text{old}} is the payment on the loan you have and MnewM_{\text{new}} the payment on the loan replacing it. Both payments come from the same level-payment formula, so the only things that set them are the balance, the rate and the term on each side.

  • PMI calculator and formula

    monthly PMI=L×p12until Bt≤0.8×H\text{monthly PMI} = \frac{L \times p}{12} \quad \text{until } B_t \le 0.8 \times H

    LL is the loan, pp the annual PMI rate as a decimal, HH the original purchase price, BtB_t the amortised balance. The 80 percent threshold is the borrower-request cancellation point under US rules; automatic cancellation sits a little later.

  • Mortgage points calculator

    cost=B×pnBE=costM0−M1\text{cost} = B \times p \qquad n_{BE} = \frac{\text{cost}}{M_0 - M_1}

    BB is the loan, pp points as a decimal (1 percent is 0.01), M0M_0 the payment at the higher rate and M1M_1 at the lower. Break-even nBEn_{BE} is months of payment saving against the cash cost, a cash-flow test, not a present-value test.

Work and retirement

  • Employer match calculator

    M=S×min⁡(d,c)×mM = S \times \min(d, c) \times m

    SS is salary, dd the share of pay you defer, cc the cap the match applies to, and mm the match rate, all as decimals. Deferring past the cap still raises your own contribution and does not raise the match.

  • Retirement withdrawal calculator and formula

    Bn=P(1+i)n−W×(1+i)n−1iB_n = P(1 + i)^n - W \times \frac{(1 + i)^n - 1}{i}

    BnB_n is what is left after nn withdrawals, PP the pot you start with, WW the amount you take each period, and ii the return for one period, which is the nominal annual return divided by 12 when you withdraw monthly, not an effective annual rate.

  • FIRE number calculator

    F=Sw(1+r)t=F+C/rB+C/rF = \frac{S}{w} \qquad (1+r)^t = \frac{F + C/r}{B + C/r}

    SS is annual spending, ww the withdrawal rate as a decimal (4 percent is 0.04), BB what is already saved, CC saved each year, rr the annual return. tt is years to reach FF. The 4 percent rule is w=0.04w = 0.04, which is also F=25SF = 25S.

Investing and valuation

  • Price to earnings calculator

    P/E=PEPS=Market capEarnings\text{P/E} = \frac{P}{\text{EPS}} = \frac{\text{Market cap}}{\text{Earnings}}

    PP is the share price and EPS is earnings per share. Multiplying both by the share count gives market cap over total earnings, which is the same ratio.

  • DCF calculator and formula

    V=∑t=1nFt(1+r)t+Fn(1+g)(r−g)(1+r)nV = \sum_{t=1}^{n} \frac{F_t}{(1+r)^t} + \frac{F_n(1+g)}{(r-g)(1+r)^n}

    FtF_t is free cash flow in year tt, rr the discount rate (WACC as a decimal), gg perpetual growth after year nn. The second term is Gordon growth on the year-after-forecast flow, then discounted back nn years. gg must stay below rr.

  • Dividend discount calculator

    P=D0(1+g)k−g=D1k−gP = \frac{D_0(1+g)}{k-g} = \frac{D_1}{k-g}

    D0D_0 is the dividend just paid, gg perpetual growth, kk the required return, all as decimals. D1=D0(1+g)D_1 = D_0(1+g) is next year's dividend. kk must stay above gg or the growing perpetuity has no finite price. The implied yield D1/PD_1/P equals k−gk-g.

  • Bond price calculator and current yield

    P=∑t=1nC(1+i)t+F(1+i)nP = \sum_{t=1}^{n} \frac{C}{(1+i)^t} + \frac{F}{(1+i)^n}

    PP is the price, CC the coupon paid each period, FF the face value repaid at maturity, ii the market rate for one period, and nn the number of periods left.

  • Bond duration calculator

    DMac=∑tt PV(CFt)PDMod=DMac1+y/kD_{Mac} = \frac{\sum_t t \, PV(CF_t)}{P} \qquad D_{Mod} = \frac{D_{Mac}}{1 + y/k}

    PV(CFt)PV(CF_t) is the present value of the cash flow at time tt in years, PP the price, yy the annual yield and kk the number of coupon periods a year. DV01 is modified duration times price over 10,000: the dollar change for a one basis point fall in yield.

  • Cap rate calculator

    cap=NOIpriceV=NOIcap\text{cap} = \frac{\text{NOI}}{\text{price}} \qquad V = \frac{\text{NOI}}{\text{cap}}

    NOI is rent minus operating costs, before debt service and tax. The two forms are the same identity run forwards and backwards: a higher cap is a lower price for the same income.

  • Tax-equivalent yield calculator

    yte=yex1−ty_{te} = \frac{y_{ex}}{1 - t}

    yexy_{ex} is the tax-exempt yield and tt is the marginal tax rate, both as decimals. The identity is just 'undo the tax': a taxable yield, after tax, should equal the exempt yield.

  • Expense ratio calculator and fee impact

    A=P(1+g−fn)ntA = P\left(1 + \frac{g - f}{n}\right)^{nt}

    AA is the ending balance, PP what you start with, gg the return before costs as a decimal, ff the expense ratio as a decimal, nn how many times a year the money compounds, which is twelve on this page, and tt the number of years.

Business maths

  • Enterprise value and EV/EBITDA

    EV=E+D−CEV = E + D - C

    EE is the value of the equity, DD interest-bearing debt, and CC surplus cash. Net debt is D−CD - C, so EV=E+net debtEV = E + \text{net debt}. The multiple is EVEV divided by EBITDA when EBITDA is positive.

  • Unlevered free cash flow calculator

    FCF=EBIT(1−t)+DA−Capex−ΔNWCFCF = EBIT(1-t) + DA - Capex - \Delta NWC

    EBIT(1−t)EBIT(1-t) is NOPAT. DADA is depreciation and amortisation, a non-cash charge added back. CapexCapex is capital expenditure. ΔNWC\Delta NWC is the increase in net working capital. A fall in working capital is a source of cash, so a negative delta raises FCF.

  • ROIC calculator and NOPAT split

    ROIC=EBIT(1−t)ICROIC = \frac{EBIT(1-t)}{IC}

    EBIT(1−t)EBIT(1-t) is NOPAT. ICIC is invested capital, the operating capital tied up in the firm. On a teaching sheet that is equity plus interest-bearing debt minus surplus cash. The ratio is in percentage points: 15, not 0.15.

  • ROE calculator and formula

    ROE=Net incomeEquityROE = \frac{\text{Net income}}{\text{Equity}}

    Net income is profit after interest and tax. Equity is book equity, not market cap. The ratio is in percentage points: 15, not 0.15.

  • Cash conversion cycle calculator

    CCC=DSO+DIO−DPOCCC = DSO + DIO - DPO

    DSO is days sales outstanding, DIO days inventory outstanding, and DPO days payable outstanding. The operating cycle is DSO plus DIO. CCC subtracts the payable days from that.

  • Interest coverage calculator

    Coverage=EBITInterest\text{Coverage} = \frac{\text{EBIT}}{\text{Interest}}

    EBIT is operating profit before interest and tax. Interest is the period's interest expense. The ratio is a multiple: 8, not 8 percent. A zero interest line is not a coverage ratio.

  • Unlevered beta calculator

    βU=βE1+(1−t)(D/E)\beta_U = \frac{\beta_E}{1 + (1 - t)(D/E)}

    βE\beta_E is the equity beta, tt the tax rate as a decimal, and D/ED/E the debt-to-equity ratio. βU\beta_U is the asset beta, the beta the operations would have if they were all-equity financed. Debt beta is assumed to be zero.

  • Offer premium calculator

    Premium=Poffer−PunaffectedPunaffected\text{Premium} = \frac{P_{\text{offer}} - P_{\text{unaffected}}}{P_{\text{unaffected}}}

    The unaffected price is the close before the offer leaked, not the last trade. A negative figure is a discount to that close. The page prints percentage points: 30, not 0.30.

  • NPV calculator and net present value formula

    NPV=∑t=0nCFt(1+r)tNPV = \sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}

    CFtCF_t is the cash flow in year tt, rr is the discount rate as a decimal, and nn is the last year. At t=0t = 0 the divisor is 1, so money paid or received today is not discounted.

  • IRR calculator: internal rate of return

    0=∑t=0nCt(1+IRR)t0 = \sum_{t=0}^{n} \frac{C_t}{(1 + IRR)^t}

    CtC_t is the cash flow at time tt, negative when money goes out and positive when it comes back. IRRIRR is the one rate that makes the whole series sum to zero.

  • WACC formula and calculator

    WACC=EV×Re+DV×Rd×(1−t)\text{WACC} = \frac{E}{V} \times R_e + \frac{D}{V} \times R_d \times (1 - t)

    EE is the market value of equity, DD the market value of debt, and V=E+DV = E + D the two added together. ReR_e is the cost of equity, RdR_d the cost of debt before tax, and tt the marginal tax rate that applies to the company.

  • Leverage ratio formula and calculator

    D/E=DED/A=DAEM=AE\text{D/E} = \frac{D}{E} \qquad \text{D/A} = \frac{D}{A} \qquad \text{EM} = \frac{A}{E}

    DD is total debt, EE is equity and AA is total assets. Equity is assets minus everything the company owes, so one balance sheet fixes all three at once. When the debt line is total liabilities, debt-to-equity plus 1 is the equity multiplier.

  • Break-even point calculator and formula

    Q=FP−VQ = \frac{F}{P - V}

    QQ is the number of units you have to sell, FF the fixed costs, PP the price per unit and VV the variable cost per unit. P−VP - V is the contribution margin, the part of each sale left over for the fixed costs.

  • Payback period calculator and formula

    t=n+C−∑k=1nCFkCFn+1t = n + \frac{C - \sum_{k=1}^{n} CF_k}{CF_{n+1}}

    CC is the cash paid up front, CFkCF_k is the cash the project returns in year kk, and nn is the last full year at which the running total is still short of CC. The fraction finishes the job: what is still owed, over the cash arriving in the next year. When every year pays the same, this collapses to t=C/CFt = C / CF.

  • Current ratio and gross margin calculator

    CR=CACLQR=CA−ICLGM=R−CR\text{CR} = \frac{CA}{CL} \qquad \text{QR} = \frac{CA - I}{CL} \qquad \text{GM} = \frac{R - C}{R}

    CACA is current assets, CLCL current liabilities and II inventory, all three read off the balance sheet. RR is revenue and CC the cost of goods sold, both off the income statement. Gross margin comes out as a decimal, so multiply by 100100 for the percentage.

Factor formulas

The five factors behind the printed tables. Each one turns a single multiplication into an answer, which is why they survived long after calculators replaced the books they came from.

  • Present value of 1

    PV factor=1(1+r)nPV\text{ factor} = \frac{1}{(1 + r)^{n}}

    Multiply a single future amount by the factor to bring it back to today.

  • Future value of 1

    FV factor=(1+r)nFV\text{ factor} = (1 + r)^{n}

    Multiply a single amount today by the factor to carry it forward.

  • Present value of an annuity of 1

    PVIFA=1−(1+r)−nrPVIFA = \frac{1 - (1 + r)^{-n}}{r}

    Multiply a level payment by the factor to value the whole stream today.

  • Future value of an annuity of 1

    FVIFA=(1+r)n−1rFVIFA = \frac{(1 + r)^{n} - 1}{r}

    Multiply a level payment by the factor to get what the stream is worth at the end.

  • Loan payment factor

    CRF=r1−(1+r)−nCRF = \frac{r}{1 - (1 + r)^{-n}}

    Multiply the amount borrowed by the factor to get the payment per period.

Free to use and free to cite. Educational material, not financial advice.