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Key Formulas Reference

Quick reference for all important formulas in Principles of Finance.

Time Value of Money​

Present Value (PV)​

PV=FV(1+r)t\text{PV} = \frac{\text{FV}}{(1 + r)^t}

Where:

  • FV = Future Value
  • r = Discount rate (interest rate)
  • t = Time period

Use: Calculate today's value of future cash flows.

Future Value (FV)​

FV=PV×(1+r)t\text{FV} = \text{PV} \times (1 + r)^t

Where:

  • PV = Present Value
  • r = Interest rate
  • t = Time period

Use: Calculate future value of money invested today.

Net Present Value (NPV)​

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

Or simplified:

NPV=Present Value of Future Cash Flows−Initial Investment\text{NPV} = \text{Present Value of Future Cash Flows} - \text{Initial Investment}

Where:

  • CF_t = Cash flow in period t
  • r = Discount rate
  • t = Time period

Decision Rule:

  • NPV > 0 → Accept project
  • NPV < 0 → Reject project
  • NPV = 0 → Indifferent

Interest Calculations​

Simple Interest​

Interest=Principal×Rate×Time\text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time}

Use: Interest calculated only on principal amount.

Compound Interest​

Future Value=Principal×(1+Rate)Time\text{Future Value} = \text{Principal} \times (1 + \text{Rate})^{\text{Time}}

Use: Interest calculated on principal plus accumulated interest.

Annuities​

Present Value of Ordinary Annuity​

PVannuity=Payment×1−(1+r)−nr\text{PV}_{\text{annuity}} = \text{Payment} \times \frac{1 - (1 + r)^{-n}}{r}

Where:

  • Payment = Regular payment amount
  • r = Interest rate per period
  • n = Number of periods

Use: Value today of series of equal future payments.

Future Value of Annuity​

FVannuity=Payment×(1+r)n−1r\text{FV}_{\text{annuity}} = \text{Payment} \times \frac{(1 + r)^n - 1}{r}

Use: Value at end of period of series of equal payments.

Present Value of Growing Annuity​

PVgrowing annuity=Payment×1−(1+g1+r)nr−g\text{PV}_{\text{growing annuity}} = \text{Payment} \times \frac{1 - \left(\frac{1 + g}{1 + r}\right)^n}{r - g}

Where:

  • Payment = First payment amount
  • g = Growth rate
  • r = Discount rate (must be > g)
  • n = Number of periods

Use: Value of payments growing at constant rate over fixed period.

Perpetuities​

Present Value of Perpetuity​

PVperpetuity=Paymentr\text{PV}_{\text{perpetuity}} = \frac{\text{Payment}}{r}

Where:

  • Payment = Constant payment amount
  • r = Required rate of return

Use: Value of constant payments continuing forever.

Present Value of Growing Perpetuity​

PVgrowing perpetuity=Paymentr−g\text{PV}_{\text{growing perpetuity}} = \frac{\text{Payment}}{r - g}

Where:

  • Payment = First payment amount
  • r = Required rate of return (must be > g)
  • g = Growth rate

Use: Value of payments growing forever at constant rate.

Stock Valuation​

Gordon Growth Model (Constant Growth Model)​

Stock Price=D1r−g\text{Stock Price} = \frac{D_1}{r - g}

Where:

  • D₁ = Next year's expected dividend
  • r = Required rate of return (must be > g)
  • g = Constant dividend growth rate

Use: Value stocks with constant dividend growth.

Important: Always use D₁ (next year's dividend), not D₀ (current dividend).

Holding Period Return (HPR)​

HPR=P1−P0P0\text{HPR} = \frac{P_1 - P_0}{P_0}

Where:

  • P₁ = Ending price
  • P₀ = Beginning price

Use: Calculate return over a holding period.

Required Return Components​

Required Return Decomposition​

Required Return=Real Rate+Expected Inflation+Risk Premium\text{Required Return} = \text{Real Rate} + \text{Expected Inflation} + \text{Risk Premium}

Components:

  • Real Rate: Compensation for delaying consumption (2-3%)
  • Expected Inflation: Purchasing power protection
  • Risk Premium: Compensation for uncertainty

Merton's Model - Payoff Functions​

Senior Debt Payoff​

Senior Debt Payoff=min⁡(VT,F1)\text{Senior Debt Payoff} = \min(V_T, F_1)

Alternative form:

Senior Debt Payoff=VT−max⁡(VT−F1,0)\text{Senior Debt Payoff} = V_T - \max(V_T - F_1, 0)

Where:

  • V_T = Asset value at maturity
  • F₁ = Promised payment to senior debt

Interpretation: Risky Bond = Risk-Free Bond - Put Option

Junior Debt Payoff​

Junior Debt=min⁡(F2,max⁡(VT−F1,0))\text{Junior Debt} = \min(F_2, \max(V_T - F_1, 0))

Alternative form:

Junior Debt=VT−Senior Debt−Equity\text{Junior Debt} = V_T - \text{Senior Debt} - \text{Equity}

Where:

  • F₂ = Promised payment to junior debt
  • F₁ = Promised payment to senior debt
  • V_T = Asset value at maturity

Equity Payoff​

Equity Payoff=max⁡(VT−(F1+F2),0)\text{Equity Payoff} = \max(V_T - (F_1 + F_2), 0)

Where:

  • V_T = Asset value at maturity
  • F₁ + F₂ = Total debt obligation

Interpretation: Equity = Call option on firm's assets with strike price = total debt

Asset Pricing Models​

Geometric Brownian Motion​

dV=μV dt+σV dWdV = \mu V \, dt + \sigma V \, dW

Where:

  • V = Asset value
  • μ = Expected return (drift)
  • σ = Volatility
  • dW = Random shock (Brownian motion)

Use: Model random asset price movements.

Put-Call Parity for Corporate Debt​

min⁡(VT,F1)=F1−max⁡(F1−VT,0)\min(V_T, F_1) = F_1 - \max(F_1 - V_T, 0)

Interpretation: Risky corporate bond = Risk-free bond - Put option

Quick Reference Table​

FormulaEquationUse Case
Present ValuePV=FV(1+r)t\text{PV} = \frac{FV}{(1+r)^t}Discount future cash flows
Future ValueFV=PV×(1+r)t\text{FV} = PV \times (1+r)^tCompound present value
NPVNPV=∑CFt(1+r)t\text{NPV} = \sum \frac{CF_t}{(1+r)^t}Evaluate projects
Annuity PVPV=PMT×1−(1+r)−nr\text{PV} = \text{PMT} \times \frac{1-(1+r)^{-n}}{r}Value equal payments
Perpetuity PVPV=PMTr\text{PV} = \frac{\text{PMT}}{r}Value infinite payments
Growing PerpetuityPV=PMTr−g\text{PV} = \frac{\text{PMT}}{r-g}Value growing infinite payments
Gordon ModelP0=D1r−gP_0 = \frac{D_1}{r-g}Value stocks
HPRHPR=P1−P0P0\text{HPR} = \frac{P_1 - P_0}{P_0}Calculate returns

Important Rules to Remember​

  1. NPV Rule: Accept projects with NPV > 0
  2. Mutually Exclusive: Choose highest NPV
  3. Growing Formulas: r must be > g
  4. Stock Valuation: Use D₁ (next dividend), not D₀
  5. Compound Interest: Interest on interest accelerates growth
  6. Discount Rate: Higher rate → lower present value
  7. Time Horizon: Longer period → greater compounding effect

Common Mistakes to Avoid​

❌ Don't​

  • Use D₀ instead of D₁ in Gordon model
  • Assume r ≤ g in growing perpetuity formulas
  • Ignore time value of money in multi-period analysis
  • Add cash flows from different periods without discounting
  • Confuse simple and compound interest

✅ Do​

  • Always discount future cash flows to present value
  • Use appropriate discount rate for risk level
  • Check that r > g for growing formulas
  • Include opportunity costs in NPV calculations
  • Consider all relevant cash flows