Time value of money Study Guide

📖 Core Concepts Time Value of Money (TVM): Money today is worth more than the same amount tomorrow because it can earn interest. Interest Rate (i): The periodic percentage earned (or paid) on a sum; the “price” of using money over one period. Present Value (PV): Current worth of a future cash flow, found by discounting at the appropriate rate. Future Value (FV): Value of a present sum after it grows at the interest rate for n periods. Annuity: A series of equal payments made at regular intervals (ordinary = end‑of‑period; due = beginning‑of‑period). Growing Annuity/Perpetuity: Payments increase each period by a constant growth rate g. Nominal vs Real: Nominal cash flows use the stated (inflated) rate; real cash flows are adjusted for inflation and use a real rate. --- 📌 Must Remember PV of a single sum: $PV = FV\,(1+i)^{-n}$ FV of a single sum: $FV = PV\,(1+i)^{n}$ PV of an ordinary annuity: $PV = A\;\dfrac{1-(1+i)^{-n}}{i}$ PV of an annuity due: $PV{\text{due}} = PV{\text{ordinary}}\,(1+i)$ FV of an ordinary annuity: $FV = A\;\dfrac{(1+i)^{n}-1}{i}$ PV of a perpetuity: $PV = \dfrac{A}{i}$ (only if payments continue forever). PV of a growing perpetuity (i > g): $PV = \dfrac{A}{i-g}$ (Gordon growth model). PV of a growing annuity (i ≠ g): $PV = A\;\dfrac{1-\left(\dfrac{1+g}{1+i}\right)^{n}}{i-g}$ Continuous discounting: $PV = FV\,e^{-rt}$ for constant continuous rate r. --- 🔄 Key Processes Identify the cash‑flow pattern (single sum, ordinary annuity, annuity due, growing, perpetuity). Choose the correct formula based on pattern and whether you need PV or FV. Plug in known values (payment A, rate i, growth g, periods n). Solve for the unknown (often i, n, or A). Check units (e.g., convert annual rate to monthly for mortgage problems). --- 🔍 Key Comparisons Ordinary Annuity vs. Annuity Due Ordinary: payments at period end → PV uses $1-(1+i)^{-n}$; FV uses $(1+i)^{n}-1$. Due: payments at period start → multiply ordinary PV/FV by $(1+i)$. Perpetuity vs. Growing Perpetuity Perpetuity: constant payment → $PV = A/i$. Growing: payment grows at g → $PV = A/(i-g)$ (requires $i>g$). Nominal vs. Real Discount Rate Nominal: includes inflation → discount nominal cash flows. Real: inflation‑adjusted → discount real cash flows. --- ⚠️ Common Misunderstandings Mixing nominal cash flows with real rates (or vice‑versa). Leads to incorrect PV/NPV. Using the perpetuity formula when i ≤ g. The series diverges; the formula is invalid. Treating an annuity due as ordinary without the extra (1+i) factor. Understates PV/FV. For continuous compounding, forgetting the exponent sign: PV = $FV\,e^{-rt}$, not $e^{+rt}$. --- 🧠 Mental Models / Intuition “Money snowballs” – each period’s interest adds to the principal, which then earns interest itself (compound growth). Discounting = “time‑travel backward” – imagine pulling a future payment back to today by shrinking it at the rate the market demands. Annuity = “stack of bricks” – each brick (payment) is smaller in present‑value terms the later it arrives; sum the shrunken bricks. --- 🚩 Exceptions & Edge Cases Zero interest rate (i = 0): PV = FV, annuity formulas reduce to $PV = A\cdot n$. Growth rate equals interest (g = i): Growing‑annuity formula divides by zero; use limit approach → PV = $A\,n/(1+i)$. Variable discount rates: Use $PV = FV\,\exp\!\left(-\int{0}^{T} r(t)dt\right)$ instead of simple $(1+i)^{-n}$. --- 📍 When to Use Which Single future cash flow → use single‑sum PV/FV formulas. Equal periodic payments → ordinary annuity (end) unless payments start immediately (due). Payments that grow → growing annuity or growing perpetuity formulas, check that i > g. Long‑term, “forever” streams → perpetuity or growing perpetuity (only if convergence condition holds). High‑frequency compounding (daily) → continuous compounding approximation $e^{rt}$ for simplicity. --- 👀 Patterns to Recognize Geometric‑series pattern: Any series of equal or proportionally growing payments will collapse to a compact formula. “(1+i)^{n} – 1” appears in FV of ordinary annuities. “1 – (1+i)^{-n}” appears in PV of ordinary annuities. Denominator “i – g” signals a growing cash‑flow formula (annuity or perpetuity). --- 🗂️ Exam Traps Choosing the wrong annuity type: Selecting ordinary when the problem says “first payment today” → miss the extra (1+i) factor. Applying perpetuity formula to a finite stream: Leads to an inflated PV. Ignoring inflation: Using nominal rate on real cash flows (or vice‑versa) gives a biased NPV. Sign error in continuous discounting: Using $e^{+rt}$ will give a value far larger than reality. Growth rate larger than discount rate: The “growing perpetuity” formula will produce a negative denominator → answer is invalid; the correct response is “no finite PV.” ---