📌 Simple Interest

Simple Interest:

\( I = P \cdot r \cdot t \)

Future Value (Simple Interest):

\( A = P (1 + r t) \)

Present Value (Simple Interest):

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

Interest Rate (Simple):

\( r = \frac{I}{P \cdot t} \)

Time (Simple Interest):

\( t = \frac{I}{P \cdot r} \)

Principal (Simple Interest):

\( P = \frac{I}{r \cdot t} \)

Annual Interest Earned:

\( I_{\text{annual}} = P \cdot r \)

Total Amount with Multiple Periods:

\( A = P + P r t \)

📌 Compound Interest

Compound Interest Formula:

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

Compound Interest Earned:

\( I = P \left(1 + \frac{r}{n}\right)^{n t} - P \)

Present Value (Compound):

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

Effective Annual Rate (EAR):

\( EAR = \left(1 + \frac{r}{n}\right)^n - 1 \)

Continuous Compound Interest:

\( A = P e^{r t} \)

Continuous Interest Earned:

\( I = P e^{r t} - P \)

Present Value (Continuous):

\( P = A e^{-r t} \)

Time to Double (Rule of 72):

\( t \approx \frac{72}{r} \) (where \( r \) is in percentage)

Time to Double (Exact):

\( t = \frac{\ln 2}{r} \) (continuous), \( t = \frac{\ln 2}{n \ln \left(1 + \frac{r}{n}\right)} \) (compound)

Growth Rate:

\( r = n \left[ \left( \frac{A}{P} \right)^{\frac{1}{n t}} - 1 \right] \)

📌 Annuities

Future Value of Ordinary Annuity:

\( FV = PMT \cdot \frac{\left(1 + r\right)^n - 1}{r} \)

Present Value of Ordinary Annuity:

\( PV = PMT \cdot \frac{1 - \left(1 + r\right)^{-n}}{r} \)

Future Value of Annuity Due:

\( FV = PMT \cdot \frac{\left(1 + r\right)^n - 1}{r} \cdot (1 + r) \)

Present Value of Annuity Due:

\( PV = PMT \cdot \frac{1 - \left(1 + r\right)^{-n}}{r} \cdot (1 + r) \)

Payment (Ordinary Annuity FV):

\( PMT = \frac{FV \cdot r}{\left(1 + r\right)^n - 1} \)

Payment (Ordinary Annuity PV):

\( PMT = \frac{PV \cdot r}{1 - \left(1 + r\right)^{-n}} \)

Perpetuity Present Value:

\( PV = \frac{PMT}{r} \)

Growing Perpetuity PV:

\( PV = \frac{PMT}{r - g} \), where \( r > g \)

Growing Annuity FV:

\( FV = PMT \cdot \frac{\left(1 + r\right)^n - (1 + g)^n}{r - g} \), \( r \neq g \)

Growing Annuity PV:

\( PV = PMT \cdot \frac{1 - \left(\frac{1 + g}{1 + r}\right)^n}{r - g} \), \( r \neq g \)

📌 Loans and Amortization

Loan Payment (Fixed):

\( PMT = \frac{P \cdot r (1 + r)^n}{(1 + r)^n - 1} \) (where \( r \) is periodic rate)

Loan Balance After \( k \) Payments:

\( B_k = P (1 + r)^k - PMT \frac{(1 + r)^k - 1}{r} \)

Total Interest Paid:

\( I = n \cdot PMT - P \)

Remaining Principal:

\( P_{\text{rem}} = \frac{PMT}{r} \left[ 1 - \left(1 + r\right)^{-(n-k)} \right] \)

Interest Portion of Payment:

\( I_k = B_{k-1} \cdot r \)

Principal Portion of Payment:

\( P_k = PMT - I_k \)

Loan Amount from Payment:

\( P = PMT \cdot \frac{(1 + r)^n - 1}{r (1 + r)^n} \)

Number of Payments:

\( n = \frac{\ln \left( \frac{PMT}{PMT - P \cdot r} \right)}{\ln (1 + r)} \)

Mortgage Interest Rate:

\( r = \left( \frac{PMT (1 - (1 + r)^{-n})}{P} \right) \) (requires iteration)

Balloon Payment:

\( B = P (1 + r)^n - PMT \frac{(1 + r)^n - 1}{r} \)

📌 Investments

Net Present Value (NPV):

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

Internal Rate of Return (IRR):

\( 0 = \sum_{t=0}^{n} \frac{CF_t}{(1 + IRR)^t} \) (solve for IRR)

Payback Period:

\( PP = \text{Time until } \sum CF_t \geq \text{Initial Investment} \)

Annualized Return:

\( r_{\text{ann}} = \left( \frac{A}{P} \right)^{\frac{1}{n}} - 1 \)

Dividend Discount Model (DDM):

\( P_0 = \frac{D_1}{r - g} \) (constant growth)

Gordon Growth Model:

\( P_0 = \frac{D_0 (1 + g)}{r - g} \)

Capital Asset Pricing Model (CAPM):

\( E(R_i) = R_f + \beta_i (E(R_m) - R_f) \)

Sharpe Ratio:

\( S = \frac{E(R_p) - R_f}{\sigma_p} \)

Portfolio Expected Return:

\( E(R_p) = w_1 R_1 + w_2 R_2 \)

Portfolio Variance (2 Assets):

\( \sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \text{Cov}(R_1, R_2) \)

📌 Bonds

Bond Price (Fixed Coupon):

\( P = \frac{C}{r} \left( 1 - (1 + r)^{-n} \right) + \frac{F}{(1 + r)^n} \)

Yield to Maturity (YTM):

\( P = \sum_{t=1}^{n} \frac{C}{(1 + YTM)^t} + \frac{F}{(1 + YTM)^n} \) (solve for YTM)

Current Yield:

\( CY = \frac{C}{P} \)

Coupon Payment:

\( C = F \cdot \text{Coupon Rate} \)

Zero-Coupon Bond Price:

\( P = \frac{F}{(1 + r)^n} \)

Duration (Macaulay):

\( D = \frac{\sum_{t=1}^{n} t \cdot \frac{C_t}{(1 + r)^t}}{P} \)

Modified Duration:

\( D_{\text{mod}} = \frac{D}{1 + r} \)

Price Change (Duration Approximation):

\( \Delta P \approx -D_{\text{mod}} \cdot \Delta r \cdot P \)

Convexity:

\( \text{Conv} = \frac{\sum_{t=1}^{n} t (t + 1) \frac{C_t}{(1 + r)^{t+2}}}{P} \)

Bond Price with Convexity:

\( \Delta P \approx P \left( -D_{\text{mod}} \Delta r + \frac{1}{2} \text{Conv} (\Delta r)^2 \right) \)

📌 Depreciation

Straight-Line Depreciation:

\( D = \frac{C - S}{n} \)

Book Value (Straight-Line):

\( BV_t = C - t \cdot D \)

Double-Declining Balance:

\( D_t = \frac{2}{n} \cdot BV_{t-1} \)

Book Value (Double-Declining):

\( BV_t = BV_{t-1} - D_t \)

Sum-of-the-Years’-Digits (SYD):

\( D_t = (C - S) \cdot \frac{n - t + 1}{\sum_{k=1}^{n} k} \)

SYD Total:

\( \sum_{k=1}^{n} k = \frac{n (n + 1)}{2} \)

Units of Production Depreciation:

\( D_t = (C - S) \cdot \frac{U_t}{\sum U} \)

Salvage Value Remaining:

\( BV_t = C - \sum_{k=1}^{t} D_k \)

Annual Depreciation Rate:

\( r = \frac{D}{C} \)

Accelerated Depreciation Factor:

\( D_t = C \cdot r \cdot (1 - r)^{t-1} \)

📌 Financial Ratios

Current Ratio:

\( CR = \frac{\text{Current Assets}}{\text{Current Liabilities}} \)

Debt-to-Equity Ratio:

\( D/E = \frac{\text{Total Debt}}{\text{Total Equity}} \)

Return on Investment (ROI):

\( ROI = \frac{\text{Net Profit}}{\text{Investment Cost}} \)

Return on Equity (ROE):

\( ROE = \frac{\text{Net Income}}{\text{Shareholders’ Equity}} \)

Return on Assets (ROA):

\( ROA = \frac{\text{Net Income}}{\text{Total Assets}} \)

Profit Margin:

\( PM = \frac{\text{Net Income}}{\text{Revenue}} \)

Price-to-Earnings Ratio (P/E):

\( P/E = \frac{\text{Market Price per Share}}{\text{Earnings per Share}} \)

Earnings per Share (EPS):

\( EPS = \frac{\text{Net Income}}{\text{Number of Shares}} \)

Quick Ratio:

\( QR = \frac{\text{Current Assets} - \text{Inventory}}{\text{Current Liabilities}} \)

Asset Turnover:

\( AT = \frac{\text{Revenue}}{\text{Total Assets}} \)

📌 Options and Derivatives

Black-Scholes Option Price (Call):

\( C = S_0 N(d_1) - K e^{-r t} N(d_2) \)

Black-Scholes \( d_1 \):

\( d_1 = \frac{\ln \left( \frac{S_0}{K} \right) + \left( r + \frac{\sigma^2}{2} \right) t}{\sigma \sqrt{t}} \)

Black-Scholes \( d_2 \):

\( d_2 = d_1 - \sigma \sqrt{t} \)

Put Option Price (Black-Scholes):

\( P = K e^{-r t} N(-d_2) - S_0 N(-d_1) \)

Put-Call Parity:

\( C - P = S_0 - K e^{-r t} \)

Delta (Call):

\( \Delta_C = N(d_1) \)

Delta (Put):

\( \Delta_P = N(d_1) - 1 \)

Gamma:

\( \Gamma = \frac{N'(d_1)}{\sigma S_0 \sqrt{t}} \), where \( N'(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \)

Theta (Call):

\( \Theta_C = -\frac{S_0 \sigma N'(d_1)}{2 \sqrt{t}} - r K e^{-r t} N(d_2) \)

Vega:

\( \nu = S_0 \sqrt{t} N'(d_1) \)