Encyclopedia of Financial Mathematics

Institutional Finance Formula Library & Glossary

Explore standard algebraic equations, CFA and ICAP accounting proofs, variable legends, and worked examples with instant calculator execution.

Showing 12 Financial Formulas

personal

Compound Interest Formula (Future Value)

A = P × (1 + r / n)^(n × t)

Calculates the total accumulated amount of an investment or loan including initial principal and compounded interest accrued over time.

Variable Definitions:
A:Final future amount including interest
P:Initial principal investment balance
r:Annual nominal interest rate (decimal)
n:Number of compounding periods per year
t:Number of years the money is invested
Worked Numerical Example:

Invest $10,000 at 8% annual return compounded monthly for 10 years: A = 10,000 × (1 + 0.08/12)^(120) = $22,196.40.

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realestate

Mortgage / Amortized Loan Monthly Payment (PMT)

M = P × [ i(1 + i)^n ] / [ (1 + i)^n – 1 ]

The standard annuity formula used by commercial and housing banks to determine fixed periodic monthly payments necessary to fully amortize a loan.

Variable Definitions:
M:Monthly mortgage payment amount
P:Net loan principal balance (Home Price – Down Payment)
i:Monthly interest rate (Annual Rate / 12 / 100)
n:Total number of monthly payments (Years × 12)
Worked Numerical Example:

Borrow $400,000 at 6.5% for 30 years (360 months): i = 0.065/12 = 0.005417. M = 400,000 × [0.005417(1.005417)^360] / [(1.005417)^360 – 1] = $2,528.27/month.

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islamic

Zakat Payable Formula (Islamic Shariah)

Zakat = 2.5% × (Total Zakatable Assets – Immediate Liabilities)

Shariah ruling for mandatory annual alms due from Muslims possessing wealth equal to or exceeding the Nisab threshold (87.48g gold or 612.36g silver) for one complete lunar year (Hawl).

Variable Definitions:
Zakatable Assets:Gold, silver, liquid bank cash, prize bonds, trade inventory, net rental yields
Immediate Liabilities:Short-term debts, unpaid bills, and immediate living obligations due
Nisab Benchmark:Cash equivalent of 7.5 Tolas Gold or 52.5 Tolas Silver
Worked Numerical Example:

Total cash savings & gold value = PKR 2,000,000; short-term debt = PKR 200,000. Net Zakatable = PKR 1,800,000. Zakat = 2.5% × 1,800,000 = PKR 45,000.

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corporate

EBITDA (Operating Cash Flow Benchmark)

EBITDA = Net Income + Interest + Taxes + Depreciation + Amortization

Core financial metric used by investment bankers, private equity analysts, and corporate controllers to measure raw operational profitability independent of capital structure and tax jurisdictions.

Variable Definitions:
Net Income:Bottom-line net earnings after all expenses
Interest:Finance costs and interest paid on corporate debt
Taxes:Corporate income tax obligations
D&A:Non-cash depreciation of fixed assets and amortization of intangibles
Worked Numerical Example:

Net Income $5M, Interest $1.2M, Tax $1.5M, D&A $2.3M. EBITDA = 5 + 1.2 + 1.5 + 2.3 = $10.0 Million.

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corporate

Break-Even Point (BEP)

BEP (Units) = Fixed Costs / (Price Per Unit – Variable Cost Per Unit)

The volume of product units a company must produce and sell to cover 100% of operating expenses with zero profit and zero loss.

Variable Definitions:
Fixed Costs:Overheads that do not fluctuate with production volume (rent, salaries)
Price Per Unit:Gross selling price per individual unit
Variable Cost:Direct production cost per unit (materials, labor, shipping)
Worked Numerical Example:

Fixed costs = $50,000/mo. Selling price = $100/unit, variable cost = $60/unit. Contribution margin = $40. BEP = $50,000 / $40 = 1,250 units.

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banking

Debt Service Coverage Ratio (DSCR)

DSCR = Net Operating Income (NOI) / Total Debt Service

The primary solvency metric commercial bankers and mortgage underwriters examine to determine whether cash flows from an asset can safely support annual loan principal and interest payments.

Variable Definitions:
NOI:Net Operating Income (Gross Revenue – Operating Expenses)
Total Debt Service:Annual principal, interest, and mandatory lease debt payments
Worked Numerical Example:

Commercial property produces $180,000 annual NOI; annual mortgage payments = $120,000. DSCR = 180,000 / 120,000 = 1.50x (Safe prime tier > 1.25x).

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corporate

Weighted Average Cost of Capital (WACC)

WACC = (E/V × Re) + [ (D/V × Rd) × (1 – Tc) ]

Calculates a firm's blended cost of capital across both equity and debt financing, adjusted for the corporate interest tax shield. Serves as the required hurdle discount rate in DCF models.

Variable Definitions:
E / V:Market value of equity divided by total firm value
Re:Cost of equity capital (derived via CAPM)
D / V:Market value of debt divided by total firm value
Rd:Cost of debt (pre-tax yield on company bonds/loans)
Tc:Marginal corporate tax rate
Worked Numerical Example:

Firm is 60% Equity (Cost 12%) and 40% Debt (Cost 6%, Tax 25%): WACC = (0.60 × 0.12) + [0.40 × 0.06 × (1 – 0.25)] = 7.2% + 1.8% = 9.0%.

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corporate

Days Sales Outstanding (DSO)

DSO = (Accounts Receivable / Total Credit Sales) × Days in Period

Core working capital and ICAP accounting ratio measuring the average number of days a company takes to collect cash receivables after a credit sale is finalized.

Variable Definitions:
Accounts Receivable:Total uncollected customer invoices at end of period
Total Credit Sales:Gross sales made on credit terms over the period
Days in Period:Standard 365 days (annual) or 90 days (quarterly)
Worked Numerical Example:

Accounts receivable = $120,000; Annual credit sales = $1,000,000. DSO = (120,000 / 1,000,000) × 365 = 43.8 Days.

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corporate

Inventory Turnover Ratio (ITR)

Inventory Turnover = Cost of Goods Sold (COGS) / Average Inventory

Measures operational supply chain velocity and liquidity: how many times a business turns over and replaces its inventory stock within a fiscal year.

Variable Definitions:
COGS:Cost of Goods Sold on the Income Statement
Average Inventory:(Beginning Inventory + Ending Inventory) / 2
Worked Numerical Example:

COGS = $800,000; Average inventory = $160,000. Turnover = 800,000 / 160,000 = 5.0x (Inventory converts to sales every 73 days).

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personal

Rule of 72 (Doubling Time Approximation)

Years to Double ≈ 72 / Annual Interest Rate (%)

A mental math shortcut that calculates how many years it will take for an investment to double in value at a given constant annual rate of compounded return.

Variable Definitions:
Years:Estimated time for initial principal to grow 2x
Rate (%):Annual percentage return without converting to decimal (e.g., 9% = 9)
Worked Numerical Example:

At 9% annual return: Years to double ≈ 72 / 9 = 8 Years. At 12% return: 72 / 12 = 6 Years.

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personal

Return on Investment (ROI)

ROI (%) = [ (Net Profit / Cost of Investment) ] × 100

Universal metric evaluating the efficiency or relative profitability of an investment outlay compared directly against initial capital invested.

Variable Definitions:
Net Profit:Final Value of Investment – Initial Cost of Investment
Cost of Investment:Total capital deployed to acquire and maintain asset
Worked Numerical Example:

Bought stock for $25,000, sold for $35,000. Net profit = $10,000. ROI = (10,000 / 25,000) × 100 = 40.0%.

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realestate

Capitalization Rate (Real Estate Cap Rate)

Cap Rate (%) = [ Net Operating Income (NOI) / Current Market Value ] × 100

Fundamental commercial real estate metric representing the unleveraged, cash-on-cash rate of return expected on an income-producing residential or commercial property.

Variable Definitions:
NOI:Annual gross rental revenue minus operating expenses (property taxes, insurance, repairs)
Market Value:Acquisition purchase price or current market property appraisal
Worked Numerical Example:

Apartment complex produces $90,000 annual NOI; purchase price = $1,200,000. Cap Rate = (90,000 / 1,200,000) × 100 = 7.50%.

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