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PART IX — MATHEMATICAL FINANCE🔵 Intermediate

How Do Regular Payments Add Up Over Time?

Time Value of Money — Advanced Compounding

Continuous compounding, effective annual rates, annuities, and growing perpetuities — the mathematical bedrock beneath every DCF, bond price, and option model.

math financeTVMcompoundingEARannuitycontinuousperpetuity

Live Result

$13,488.5

ƒ: pv * (1 + r/n)^(n*t)raw: 13488.5

ƒ(x)

Use variable names from the panel above (e.g. FV, r, n) — or type numbers directly: 10000 / (1 + 0.08)^10

pv$

Present value — initial principal ($)

pv = 10000

r% as decimal

Annual interest rate (decimal, e.g. 0.06 = 6%)

r = 0.06

Compounding periods per year (12 = monthly)

n = 12

Time horizon in years

t = 5

pmt

Periodic payment for annuity calculations ($)

pmt = 1000

g% as decimal

Perpetuity growth rate (decimal)

g = 0.03

cf$

Next period cash flow for Gordon Growth ($)

cf = 5

💡 You can also enter values directly in the formula: 10000 / (1 + 0.08)^10

⬇ Export Calculation

Exports a plain .txt file with your expression, formula, all variable values, result, and educational notes — ready to paste into any report, Word doc, Notion, or Google Docs.

The exported file includes the formula in standard mathematical notation — you can paste it directly into Excel, Google Sheets, or back into FinanceSheep.

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Learn: Time Value of Money — Advanced Compounding

PART IX — MATHEMATICAL FINANCE · Educational Guide

The Core Idea

Jake invests $10,000 in a savings account yielding 6% annually, compounded monthly. How much does he actually have in 5 years? The answer requires understanding that how often interest compounds dramatically changes the outcome — and that at the mathematical limit, compounding becomes continuous, which is the framework used in Black-Scholes and bond pricing.

How It Works

Discrete FV = PV × (1 + r/n)^(nt). As n→∞, this becomes Continuous FV = PV × e^(rt). The Effective Annual Rate (EAR) is the true annual cost/yield when compounding is sub-annual: EAR = (1 + r/n)^n − 1. A perpetuity with constant growth rate g discounted at rate r has PV = CF / (r − g) — the Gordon Growth Model, foundation of the DDM and terminal value.

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Real-World Example: $10,000 at 6% compounded monthly for 5 years: FV = $10,000 × (1 + 0.06/12)^60 = $13,488.50. Continuous: $10,000 × e^(0.06×5) = $13,498.59. Difference is tiny but continuous math is preferred in calculus-based pricing. EAR at 6% monthly = (1.005)^12 − 1 = 6.168% — the true annual rate. A mortgage at '6% compounded monthly' actually costs 6.168% per year. Gordon Growth: stock paying $5 dividend next year, growing 3% forever, discounted at 9% → PV = $5 / (0.09 − 0.03) = $83.33.

How Do Regular Payments Add Up Over Time?

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