The Mathematics of Finance

Symbols and Meanings

• $Per\:\:annum$ OR $Per\:\:year$ OR $Annually$ OR $Yearly$ means for a year (per $1$ year)
• $SI$ OR $I$ = Simple Interest or Dividend or Yield or Return ($)
• $P$ = Principal or Present Value or Investment ($)
• $r$ = Rate or Annual Interest Rate or Annual Percentage Rate (%)
• $APR$ = Rate or Annual Percentage Rate (%)
• $t$ = Time $(years)$
• $A$ = Amount or Future Value ($)
• $CI$ OR $I$ = Compound Interest or Yield or Dividend or Return($)
• $m$ = Number of Compounding Periods Per Year
• $n$ = Total Number of Compounding Periods $(years)$
• $i$ = Interest Rate Per Period $(\%) \:\:per\:\: period$
• $CCI$ = Continuous Compound Interest or Yield or Dividend or Return($)
• $APY$ = Annual Percentage Yield or Effective Interest Rate or True Interest Rate (%)
• $FV$ = Future Value ($)
• $PMT$ = periodic payment ($)
• $PMTs$ = total periodic payments ($)
• $PV$ = Present Value of all payments ($)
• $a_{n\i}$ = $a$ angle $n$ at $i$
• $s_{n\i}$ = $s$ angle $n$ at $i$
• $PV$ = Present Value of all payments ($)
• $payoff$ = payoff amount for a mortgage
• $k$ = number of remaining payments
• $CFV$ = Combined Future Value ($)
• Rule of $78$
• $UI$ = Unearned Interest or Interest Refund ($)
• $PMT$ = Monthly Payment ($)
• $k$ = Number of remaining monthly payments OR remaining number of monthly payments
• $TI$ = Total Interest ($)
• $n$ = original number of payments
• $TP$ = Total Payments ($)
• $LA$ = Loan Amount or Finance Charge ($)
• $RF$ = Refund Fraction
• $LAR$ = Loan Amount Refund or Finance Charge Refund ($)
• Bonds
• $CP$ = Coupon Payment ($)
• $CR$ = Coupon Rate (%)
• $FV$ = Face Value or Par Value of the Bond ($)
• $m$ = Number of Coupon Payments per year
  This is similar to the number of compounding periods per year.
• $BP$ = Bond Price ($)
• $YTM$ = Yield to Maturity ($\%$)
• $t$ = time to reach maturity (years)
• $Annualized\:\:YTM$ = Annualized Yield to Maturity ($\%$)

Mathematics of Finance Formulas

Financial Mathematics Literacy

(1.) Monthly interest payment = Monthly interest rate * Average balance

(2.) Net monthly cash flow = Monthly income − Monthly expenses

Simple Interest

$ (1.)\:\: SI = Prt \\[3ex] (2.)\:\: SI = A - P \\[3ex] (3.)\:\: P = \dfrac{SI}{rt} \\[5ex] (4.)\:\: t = \dfrac{SI}{Pr} \\[5ex] (5.)\:\: r = \dfrac{SI}{Pt} \\[5ex] (6.)\:\: A = P + SI \\[3ex] (7.)\:\: A = P(1 + rt) \\[3ex] (8.)\:\: P = \dfrac{A}{1 + rt} \\[5ex] (9.)\:\: t = \dfrac{A - P}{Pr} \\[5ex] (10.)\:\: r = \dfrac{A - P}{Pt} \\[5ex] (11.)\:\: SI = \dfrac{Art}{1 + rt} $

Compound Interest

$ (1.)\:\: A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[7ex] (2.)\:\: P = \dfrac{A}{\left(1 + \dfrac{r}{m}\right)^{mt}} \\[10ex] (3.)\:\: r = m\left[\left(\dfrac{A}{P}\right)^{\dfrac{1}{mt}} - 1\right] \\[10ex] (4.)\:\: r = m\left(10^{\dfrac{\log\left(\dfrac{A}{P}\right)}{mt}} - 1\right) \\[10ex] (5.)\:\: t = \dfrac{\log\left(\dfrac{A}{P}\right)}{m\log\left(1 + \dfrac{r}{m}\right)} \\[7ex] (6.)\:\: A = P + CI \\[3ex] (7.)\:\: CI = A - P \\[3ex] (8.)\:\: A = P(1 + i)^n \\[4ex] (9.)\:\: P = \dfrac{A}{(1 + i)^n} \\[7ex] (10.)\:\: i = \dfrac{r}{m} \\[5ex] (11.)\:\: n = mt \\[3ex] (12.)\;\; Total\;\;Return = \dfrac{A - P}{P} * 100\% \\[7ex] (13.)\;\; Annual\;\;Return = \left[\left(\dfrac{A}{P}\right)^{\dfrac{1}{t}} - 1\right] * 100\% \\[7ex] $ Future Value (Amount) of Cash Flows (Principal) for Several Years

$ (13.)\:\:At\:\:the\:\:end\:\:of\:\:each\:\:year:\:\: FV = PV\left(1 + \dfrac{r}{m}\right)^{m(last\:\:year - that\:\:year)} \\[7ex] (14.)\:\: Total\:FV = \Sigma FV $

Values of $m$

Continuous Compound Interest

$ (1.)\:\: A = Pe^{rt} \\[4ex] (2.)\:\: P = \dfrac{A}{e^{rt}} \\[7ex] (3.)\:\: t = \dfrac{\ln \left(\dfrac{A}{P}\right)}{r} \\[7ex] (4.)\:\: r = \dfrac{\ln \left(\dfrac{A}{P}\right)}{t} \\[7ex] (5.)\;\; Total\;\;Return = \dfrac{A - P}{P} * 100\% \\[7ex] (6.)\;\; Annual\;\;Return = \left[\left(\dfrac{A}{P}\right)^{\dfrac{1}{t}} - 1\right] * 100\% $

APY for Compound Interest

$ (1.)\:\: APY = \left(1 + \dfrac{r}{m}\right)^m - 1 \\[7ex] (2.)\:\: r = m\left[(APY + 1)^{\dfrac{1}{m}} - 1\right] \\[7ex] (3.)\:\: r = m\left(\sqrt[m]{APY + 1} - 1\right) $

APY for Continuous Compound Interest

$ (1.)\:\: APY = e^r - 1 \\[4ex] (2.)\:\: r = \ln(APY + 1) $

Investments

(1.) Market Capitalization (Market Cap) = Total Number of Outstanding Shares * Current Share Price

Income Taxes

(1.) Gross income (GI) = sum of all Income a person receives during the​ year, including​ wages, tips, profits from a​ business, interest or dividends from investments.

(2.) Adjusted Gross Income (AGI) = Gross Income − Contributions for individual retirement accounts or any other​ tax-deferred savings plans

(3.) Taxable Income (TI) = Adjusted Gross Income − Exemptions and Deductions
Between Standard deduction and Itemized deductions, use whichever is higher.

(4.) If Standard Deduction > Itemized Deductions:
Use Standard Deduction
Savings in Taxable Income = Standard Deduction − Itemized Deductions

(5.) If Itemized Deductions > Standard Deductions:
Use Itemized Deductions
Savings in Taxable Income = Itemized Deductions − Standard Deduction

Future Value of Ordinary Annuity

$ (1.)\:\: FV = m * PMT * \left[\dfrac{\left(1 + \dfrac{r}{m}\right)^{mt} - 1}{r}\right] \\[10ex] (2.)\;\; FV = PMT * \dfrac{\left[\left(1 + \dfrac{r}{m}\right)^{mt} - 1\right]}{\dfrac{r}{m}} \\[10ex] (3.)\:\: t = \dfrac{\log\left[\dfrac{r * FV}{m * PMT} + 1\right]}{m * \log\left(1 + \dfrac{r}{m}\right)} \\[10ex] (4.)\:\: Total\:\:PMTs = PMT * m * t \\[3ex] (5.)\:\: CI = FV - Total\:\:PMTs \\[5ex] (6.)\:\: FV = PMT * \left[\dfrac{(1 + i)^n - 1}{i}\right] \\[7ex] (7.)\:\: n = \dfrac{\log\left[\dfrac{i * FV}{PMT} + 1\right]}{\log(1 + i)} \\[10ex] (8.)\:\: s_{n\i} = \dfrac{m}{r} * \left[\left(1 + \dfrac{r}{m}\right)^{mt} - 1\right] \\[7ex] (9.)\:\: s_{n\i} = \dfrac{(1 + i)^n - 1}{i} \\[5ex] (10.)\:\: FV = PMT * s_{n\i} \\[3ex] (11.)\:\: i = \dfrac{r}{m} \\[5ex] (12.)\:\: n = mt \\[3ex] (13.)\:\: Annual\:\:Fuel\:\:Expense = \dfrac{Annual\:\:Miles\:\:Driven}{Miles\:\:per\:\:Gallon} * Price\:\:per\:\:Gallon $

Sinking Fund

$ (1.)\:\: PMT = \dfrac{r * FV}{m * \left[\left(1 + \dfrac{r}{m}\right)^{mt} - 1\right]} \\[10ex] (2.)\:\: t = \dfrac{\log\left[\dfrac{r * FV}{m * PMT} + 1\right]}{m * \log\left(1 + \dfrac{r}{m}\right)} \\[10ex] (3.)\:\: Total\:\:PMTs = PMT * m * t \\[3ex] (4.)\:\: CI = FV - Total\:\:PMTs \\[3ex] (5.)\:\: PMT = \dfrac{i * FV}{(1 + i)^n - 1} \\[7ex] (6.)\:\: n = \dfrac{\log\left[\dfrac{i * FV + PMT}{PMT}\right]}{\log(1 + i)} \\[10ex] (7.)\:\: s_{n\i} = \dfrac{m}{r} * \left[\left(1 + \dfrac{r}{m}\right)^{mt} - 1\right] \\[7ex] (8.)\:\: s_{n\i} = \dfrac{(1 + i)^n - 1}{i} \\[5ex] (9.)\:\: i = \dfrac{r}{m} \\[5ex] (10.)\:\: n = mt $

Present Value of Ordinary Annuity

$ (1.)\:\: PV = m * PMT * \left[\dfrac{1 - \left(1 + \dfrac{r}{m}\right)^{-mt}}{r}\right] \\[10ex] (2.)\:\: t = -\dfrac{\log\left[1 - \dfrac{r * PV}{m * PMT}\right]}{m * \log\left(1 + \dfrac{r}{m}\right)} \\[10ex] (3.)\:\: PV = PMT * \left[\dfrac{1 - (1 + i)^{-n}}{i}\right] \\[7ex] (4.)\:\: n = \dfrac{\log \left[\dfrac{PMT}{PMT - i * PV}\right]}{\log(1 + i)} \\[10ex] (5.)\:\: a_{n\i} = \dfrac{m}{r} * \left[1 - \left(1 + \dfrac{r}{m}\right)^{-mt}\right] \\[7ex] (6.)\:\: a_{n\i} = \dfrac{1 - (1 + i)^{-n}}{i} \\[5ex] (7.)\:\: PV = PMT * a_{n\i} \\[3ex] (8.)\:\: i = \dfrac{r}{m} \\[5ex] (9.)\:\: n = mt \\[3ex] (10.)\:\: Total\:\:PMTs = PMT * m * t \\[3ex] (11.)\:\: CI = Total\:\:PMTs - PV $

Amortization

$ (1.)\:\: PMT = \dfrac{PV}{m} * \left[\dfrac{r}{1 - \left(1 + \dfrac{r}{m}\right)^{-mt}}\right] \\[10ex] (2.)\:\: t = -\dfrac{\log\left[1 - \dfrac{r * PV}{m * PMT}\right]}{m * \log\left(1 + \dfrac{r}{m}\right)} \\[10ex] (3.)\:\: PMT = \dfrac{i * PV}{1 - (1 + i)^{-n}} \\[7ex] (4.)\:\: n = \dfrac{\log \left[\dfrac{PMT}{PMT - i * PV}\right]}{\log(1 + i)} \\[10ex] (5.)\:\: i = \dfrac{r}{m} \\[5ex] (6.)\:\: n = mt \\[3ex] (7.)\:\: Payoff = PMT * n * \left[\dfrac{1 - \left(1 + \dfrac{r}{n}\right)^{-k}}{r}\right] \\[10ex] (8.)\:\: Total\:\:PMTs = PMT * m * t \\[3ex] (9.)\:\: CI = Total\:\:PMTs - PV \\[3ex] (10.)\:\: CI = PMT * m * t - PV \\[3ex] (11.)\:\: Number\:\:of\:\:payments = m * t \\[3ex] (12.)\:\: Down\:\:Payment = Given\:\:Rate * Purchase\:\:Price \\[3ex] (13.)\:\: Amount\:\:of\:\:Mortgage = Purchase\:\:Price - Down\:\:Payment \\[3ex] (14.)\:\: Payment\:\:for\:\:x\:\:points\:\:at\:closing = x\:\:as\:\:\% * Amount\:\:of\:\:Mortgage $

Future Value of an Annuity Due

$ (1.)\:\: FV = m * PMT * \left[\dfrac{\left(1 + \dfrac{r}{m}\right)^{mt} - 1}{r}\right] * \left(1 + \dfrac{r}{m}\right) \\[10ex] (2.)\:\: PMT = \dfrac{r * FV}{(m + r) * \left[\left(1 + \dfrac{r}{m}\right)^{mt} - 1\right]} \\[10ex] (3.)\:\: t = \dfrac{\log\left[\dfrac{r * FV}{PMT(m + r)} + 1\right]}{m * \log\left(1 + \dfrac{r}{m}\right)} \\[10ex] (4.)\:\: Total\:\:PMTs = PMT * m * t \\[3ex] (5.)\:\: CI = FV - Total\:\:PMTs \\[3ex] (6.)\:\: FV = PMT * \left[\dfrac{(1 + i)^n - 1}{i}\right] * (1 + i) \\[7ex] (7.)\:\: PMT = \dfrac{i * FV}{(1 + i)\left[(1 + i)^n - 1\right]} \\[7ex] (8.)\:\: n = \dfrac{\log\left[\dfrac{i * FV}{PMT(1 + i)} + 1\right]}{\log(1 + i)} \\[10ex] (9.)\:\: i = \dfrac{r}{m} \\[5ex] (10.)\:\: n = mt \\[3ex] (11.)\:\: CFV = P\left(1 + \dfrac{r}{m}\right)^{mt} + m * PMT * \left[\dfrac{\left(1 + \dfrac{r}{m}\right)^{mt} - 1}{r}\right] * \left(1 + \dfrac{r}{m}\right) \\[10ex] (12.)\:\: t = \dfrac{\log\left[\dfrac{rCFV + PMT(m + r)}{rP + PMT(m + r)}\right]}{m\log\left(1 + \dfrac{r}{m}\right)} $

Rule of 78

$ \underline{Monthly} \\[3ex] (1.)\:\: UI = \dfrac{TI * k * (k + 1)}{n(n + 1)} \\[5ex] (2.)\:\: TP = n * PMT \\[3ex] (3.)\:\: TI = TP - LA \\[3ex] (4.)\:\: RF = \dfrac{UI}{TI} \\[5ex] (5.)\:\: RF = \dfrac{sum\:\:of\:\:digits\:\:for\:\:up\:\:to\:\:k}{sum\:\:of\:\:digits\:\:for\:\:up\:\:to\:\:n} \\[5ex] (6.)\:\: LAR = LA * RF \\[3ex] (7.)\:\: UI = TI * RF $

Zero-Coupon Bonds

$ (1.)\:\: CP = \dfrac{CR * FV}{m} \\[7ex] (2.)\:\: YTM = \left(\dfrac{FV}{BP}\right)^{\dfrac{1}{t}} - 1 \\[7ex] (3.)\:\: BP = \dfrac{FV}{(YTM + 1)^t} \\[7ex] (4.)\:\: FV = BP * (YTM + 1)^t \\[5ex] (5.)\:\: t = \dfrac{\log\left(\dfrac{FV}{BP}\right)}{\log(YTM + 1)} $

Coupon Bonds

$ (1.)\:\: CP = \dfrac{CR * FV}{m} \\[5ex] (2.)\:\: BP = \dfrac{FV * CR}{YTM} * \left[1 - \dfrac{1}{\left(1 + \dfrac{YTM}{m}\right)^{mt}}\right] + \dfrac{FV}{\left(1 + \dfrac{YTM}{m}\right)^{mt}} \\[10ex] (3.)\:\: YTM \approx \dfrac{m * t * CP + FV - BP}{t(FV + BP)} \\[7ex] (4.)\:\: Annualized\:\:YTM \approx \dfrac{2(m * t * CP + FV - BP)}{t(FV + BP)} \\[7ex] (5.)\:\: YTM \approx \dfrac{t * CR * FV + FV - BP}{t(FV + BP)} \\[7ex] (6.)\:\: Annualized\:\:YTM \approx \dfrac{2(t * CR * FV + FV - BP)}{t(FV + BP)} \\[7ex] $