Financial Resources Formulary

by Clemens Werkmeister

Present values, perpetuities and annuities

Present value PV: value of a future payment C_t (in year t), discounted to year 0:

Future value FV_t: value of a present payment (in year 0), calculated by compounding to year t:

t year

r discount rate (interest rate)

discount factor with discount rate (interest rate) r for t years

C_t 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):

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

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

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:

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

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

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

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

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

Internal rate of return (IRR): discount rate that results in NPV = 0:

Profitability index: ratio of NPV to investment of a project:

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

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:

Effective annual rate with continuous compounding: effective annual rate for n → ∞ shorter periods: (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: → (being i the inflation rate)

Valuing bonds

Price of a bond:
with

C_t 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:

Modified duration: a measure of volatility (elasticity) of bond prices:

Valuing stocks

(Expected) Stock return r (equity cost of capital):

• Stock price P₀ in the single-period case:

• Dividend discount model for the stock price P0 in the multi-period case until time horizon H:

• Stock price P₀ with specific dividends until time horizon H and growing dividends after H:

• Stock price for a perpetual stream of dividends:

• Stock price for a perpetual stream of growing dividends:

• Stock price = Discounted earnings + growth opportunities:

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:

Return on Equity with market values:

Payout ratio: fraction of earnings paid out as dividends:

Plowback ratio: fraction of earnings retained by the firm:

Sustainable growth rate: rate at which a firm can steadily grow:

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:

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): (based on j observations)

Standard deviation: a measure of volatility of expected stock returns:

Expected portfolio return (with two assets):

Expected portfolio return (with j = 1, …, n assets):

x_j weight of asset j in the portfolio

r_j (expected) return of asset j

Variance of portfolio return (portfolio variance) in the case of two assets:

Covariance between asset i and j with

Correlation coefficient between asset i and j:

Variance of portfolio return (portfolio variance) in the case of n assets:

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

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

Expected return following the security market line equation (SML):

Expected return of a stock in event studies:

Abnormal return = actual return – expected return =

Capital Structure and Return

Rates of return:

Weighted average cost of capital (WACC):

r_D interest rate on debt resp. debt cost of capital

r_E return on equity resp. equity cost of capital

T_c corporate tax rate

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

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

Leverage-formula for equity beta: risk increases with debt/equity-ratio:

Exercises

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