# 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 - C
_{0}initial investment of a project (for normal investment projects: C_{0}< 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 C_{0}:

**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 C_{1} 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 C_{1} 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

- 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):**

**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

**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 the Financial Ratios or at our investment pages.