Financial Resources Formulary

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by Clemens Werkmeister

Present values, perpetuities and annuities

Present value PV: value of a future payment Ct (in year t), discounted to year 0: Form PV.png

Future value FVt: value of a present payment (in year 0), calculated by compounding to year t: Form FV.png

t year
r discount rate (interest rate)
Form Disc Fact.png discount factor with discount rate (interest rate) r for t years
Ct cash flow in year t
C0 initial investment of a project (for normal investment projects: C0 < 0)
T number of years of the project

The sum of several present values is a PV, too (additivity of present values):Form PV Sum.png

Net present value NPV: PV of future payments (of a project or a company) plus the - usually negative - initial investment C0: Form NPV.png

Perpetuity (console): a periodic (annual) payment C that is received or paid forever (beginning with the first payment at the end of year 1):Form PV Perp.png

Annuity: a payment of a level cash flow C during a specified number of years (from year 1 to n). Its present value can be calculated as difference between two perpetuities: Form PV Ann.png

Annuity (recovery) factor: average payment at the end of n periods, corresponding to a present value PV and considering for interest rate r:Form Recov factor.png

Annuity present value factor: factor for the PV of n equal payments at the end of years 1 to n: Form Ann PV factor.png

The annuity C for years 1 to n corresponding to a present value PV and discount rate r is: Form Ann PV.png

Growing perpetuity: a perpetuity starting with cash flow C1 in year 1 and increasing by the annual growth rate g forever:Form C growth.png (for t = 1, 2, …, ∞ ;g < r) and Form PV perp growth.png

Growing annuity: an annuity starting with cash flow C1 in year 1 and increasing by the annual growth rate g for n years: Form PV ann growth.png (with t = 1, 2, …, n)

Internal rate of return (IRR): discount rate that results in NPV = 0: Form NPV zero.png

Profitability index: ratio of NPV to investment of a project: Form PI.png

Equivalent annual cash flow (EAC): cash flow per year with the same present value as the actual cash flow of the project: Cw EAC.png

Interest and discount rates

Effective annual rate (EAR): annualized rate of shorter period interest rates (monthly, daily rates) using compound interest:

- EAR for a monthly rate m: EAR = (1+m)12 – 1
- EAR for a daily rate d: EAR = (1+d)360 – 1 (for 360 days per year)

Given an annual percentage rate (APR) of r, the corresponding EAR with respect to n shorter periods of equal length is: Form EAR.png

Effective annual rate with continuous compounding: effective annual rate for n → ∞ shorter periods: Form EAR cc.png (being r the simple annual rate)

Annual percentage rate (APR) or simple rate: annualized rate of shorter period interest rates (monthly, daily rates) using simple interest.

- APR for a monthly rate m: APR = 12 * m

Real rate of return: rate of return adjusted for inflation: Form Real Int 1.pngForm Real Int 2.png (being i the inflation rate)

Valuing bonds

Price of a bond: Form PV Bond.png

Ct annual coupon interest payment
F face value (or principal)
r discount rate (yield to maturity)
N maturity

Duration of a bond with maturity N: weighted average period of bond payments: Form Duration.png

Modified duration: a measure of volatility (elasticity) of bond prices: Form Mod Duration.png

Valuing stocks

(Expected) Stock return r (equity cost of capital): Form Stock Return.png

  • Stock price P0 in the single-period case: Form P0 einperiodig.png
  • Dividend discount model for the stock price P0 in the multi-period case until time horizon H: Cw DDM v2.png
  • Stock price P0 with specific dividends until time horizon H and growing dividends after H: Form P0 growth.png
  • Stock price for a perpetual stream of dividends: Form P0 perpetual.png
  • Stock price for a perpetual stream of growing dividends: Form P0 perp growth.png
  • Stock price = Discounted earnings + growth opportunities: Cw PVGO.png

Present value of growth opportunities (PVGO): net present value of a firm's future investments.

Return (Equity cost of capital) of a perpetual stream of dividends with growth: Form RoE growth.png

Return on Equity with market values: Form RoE.png

Payout ratio: fraction of earnings paid out as dividends: Form Payout.png

Plowback ratio: fraction of earnings retained by the firm: Form Plowback.png

Sustainable growth rate: rate at which a firm can steadily grow: Form Sustainable.png

Discounted cash flow (DCF): value of the free cash flows that are available to investors plus company value at the planning horizon, all discounted to present: Form DCF.png

Risk and return

Risk premium of an asset: asset return – return of risk-free asset.

Variance: expected value of squared deviations of observations from their expected value (mean): Form Var.png (based on j observations)

Standard deviation: a measure of volatility of expected stock returns: Form Standard Dev.png

Expected portfolio return (with two assets): Form EV zwei.png

Expected portfolio return (with j = 1, …, n assets): Form EV n.png

xj weight of asset j in the portfolio
rj (expected) return of asset j

Variance of portfolio return (portfolio variance) in the case of two assets: Form Port Var zwei.png

Covariance between asset i and j with Form Covariance Formel.png

Correlation coefficient between asset i and j: Form Corr Coeff Formel.png

Variance of portfolio return (portfolio variance) in the case of n assets: Form Port Var n.png

Sharpe-Ratio: ratio of risk premium to risk (standard deviation): Cw Sharpe ratio.png

Beta of the return of asset j to the market return (return of market portfolio m): Form Beta.png

Expected return following the security market line equation (SML): Form Expected Return.png

Expected return of a stock in event studies: Form Normal Return.png

Abnormal return = actual return – expected return = Form Abnormal.png

Capital Structure and Return

Rates of return:
Form RoI.png
Form RoA.png
Form RoC.png

Weighted average cost of capital (WACC): Form WACC after Tax.png

rD interest rate on debt resp. debt cost of capital
rE return on equity resp. equity cost of capital
Tc corporate tax rate

Weighted average cost of capital (WACC) with a zero-tax rate: Form WACC.png

Leverage-formula for return on equity: return on equity increases with debt/equity-ratio: Form Leverage.png

Leverage-formula for equity beta: risk increases with debt/equity-ratio: Form Leverage Beta.png


Please try our Financial Exercises or have a look at the Financial Ratios or at our investment pages.