#14154 - Corporate Finance Formulas Sheet - Corporate Finance

FORMULAS – CORPORATE FINANCE

Future Value = FV = P • (1 + r)^t . . . (P = principle; r = risk free market rate; t = # of periods)

Present Value Formulas

PV = C₁/(1+r)^t = DF • C₁ . . . (C₁ = future cash flow; DF = discount factor)

Discount Factor = DF = 1/(1+r)^t . . . (r = risk free market rate; t = # of periods)

Net PV = NPV = PV – required investment = C₀ + [i=1 to T] C_i/(1+r)ⁱ

Present value of what an investment gives you over the market (+ = good investment)

C₀ = initial investment (normally negative!); required investment = same thing

Perpetuity: set cash payment in every year (perpetuity that starts in year zero, begins payments in year 1)

PV of Perpetuity = Cash Flow / Market Rate = C / r . . . (see proof on p. 27) (first cash flow at t₁)

PV of Delayed Perpetuity = C/(r • (1 + r)^t) . . . (i.e. PV of CF discounted for delay) (perpetuity starts in year t, but the first payment comes in year t+1: delayed t years after normal perpetuity)

PV of constant growth perpetuity = PV₀ = C₁ / (r – g) . . . (g = growth rate of the cash flow)

PV of constant growth perpetuity at any time = PV_t = C_t+1 / (r – g)

Annuity: annuity received in year zero, starts payments of C at year 1, and ceases at year t (i.e. t payments)

PV of Annuity year 1 to t = (C/r) – (C/r)•(1/(1+r)^t) = (C/r)• (1 – (1/(1+r)^t)) = perpetuity – delayed perpetuity

FV of Annuity year 1 to t = (C/r) • [(1+r)^t-1]

Annuity Factor: (1/r) – (1/r)•(1/(1+r)^t)

Equivalent annual annuity = present value of cash flows / annuity factor

Use this to find the cash flow per period (annuity) that has the same present value as the actual cash flow of the project

Bonds: purchase in year zero, first payment either at 6 months or 1 year

PV of Bond = annuity + deferred maturity value = C/(1+r)¹ ₊ C/(1+r)² + . . . + (maturity value + C)/(1+r)^N

Normally all Coupons (C) are equal, at C = coupon rate • maturity value

If paid semi-annually, half the market rate r for similar bonds, half of coupons C (assuming stated annually), and take periods as 6 months

Yield to Maturity = YTM = the market rate for similar bonds (note: this is essentially the return you will get after discount or premium)

Duration = [t=1 to T] t •PV(C_t)/PV = see back of book for easier formula using yield

(t = period; T is maturity time; PV(C_t) = present value of the payment in year t, PV is the current PV)

Duration measures how long before the bond price is paid via cash flows

Modified Duration = volatility(%) = duration / (1 + YTM)

Sensitivity of the bond to the market: percentage change in bond price for a 1 percentage-point change in the yield

Stock:

With Fixed Rate of Growth:

P₀ = Div₁ / (r – g) (for constant growth of dividends, value at t=0 with div1 paying out at t=1 NOT t=0)

In sum, you accumulate in year zero, use dividend in year 1 to calculate

Get “r” by CAPM; get “Div₁” by ROE•payout ratio (“POR”); get “g” by ROE•PBR

Dividend growth rate = g = ROE • plowback ratio (conservative estimate)

Payout Ratio: fraction of earnings paid as dividends

Plowback Ratio: fraction of earnings retained by firm (POR + PBR = 1)

Without Growth (Fixed Dividends)

PV(stock) = P₀ = PV(expected future dividends) = [t = 1 to inf.] Div₁ / (1 + r)^t . . . (r = expected return)

If Only Next Year’s Dividends are Known:

P₀ = (Div₁ + P₁)/(1+r)

P₁ = P₀ • (1 + r) – Div₁

In General

Expected Return= (Dividend + appreciation)/price = r = (Div₁ + P₁ - P₀)/P₀ = g + (Div₁/P₀) = g + dividend yield

(P₀=current price; P₁=price in one year; Div₁=dividend to be paid in one year); (g=firm growth rate)

AKA Cost of Equity Capital, Market Capitalization Rate, Opportunity Cost of Capital

Dividend Yield = Div₁/P₀

Note: the dividend yield = r (expected return) when there is no growth

ROE = return on equity = Earnings per share / book value(equity) per share = earnings / book value(equity)

PVGO = present value of growth opportunities = P₀ with new plowback – P₀ without new plowback

P₀ = (EPS/r) + PVGO

Valuing Business or Project

PV = PV_H / (1+r)^H + [t = 1 to H] FCF_t / (1+r)^t = FCF₁ / (1+r)¹ + FCF₂ / (1+r)² + . . . + PV_H / (1+r)^H

(H = valuation horizon, FCF = future cash flow, r = opportunity cost of capital)

First term is PV of free cash flows, whereas the second is PV of horizon value

Risk: a range of possible outcomes

r = r_f + r_p . . . (r_f = risk free rate; r_p = risk premium)

Standard Deviation and Variance

Variance= σ² = (ř – r)² . . . (ř = actual return, r = expected return)

= sum of all probabilities, each multiplied by the squared deviation (difference) from expected return associated with that probability (see Table 7.2, page 165)

Standard Deviation of ř_m = σ = (variance(ř_m))

Portfolio Analysis & Theory:

Portfolio Rate of Return = [i=0 to n] (fraction of port. in asset i) x (rate of return on asset i) . . . (n=# assets)

Portfolio Variance (of two stocks) = x₁²σ₁² + x₂²σ₂² + 2(x₁x₂ρ₁₂σ₁σ₂) . . . (x_n = weight, or amount of stock; ρ₁₂ = correlation coefficient between two stocks)

Covariance = σ₁₂ = ρ₁₂ • σ₁ • σ₂ (i.e. covariance between the two stocks)

Sharpe Ratio = r_p – r_f / σ_p . . . (subscript “p” denotes portfolio) (r_f is a point on the vertical axis)

SR is the slope of the line intersecting the risk free premium, and the portfolio point (σ_p, r_p) you chose in standard deviation-expected return space (e.g., Fig. 8.5) — it is y=mx+b

You want to MAXIMIZE the Sharpe ratio: you can get highest returns for given standard deviation

If the portfolio is the entire market, then ^ is the market return (the market is perfectly efficient

Capital Asset Pricing Model (CAPM): r – r_f = β(r_m – r_f) . . . (r = expected rate of return for a particular stock, (r – r_f) = expected risk premium for this stock, β = beta for this stock, r_m = market rate, r_f = risk-free rate, (r_m – r_f) = market risk premium).

If you know a company’s β, as well as r_m and r_f you can get the required return r for the company - this is r_E, aka your expected stock return. See WACC below

Beta: how susceptible a stock is to market variation

β = σ_im / σ²_m = ρ_im • σ_i / σ_m . . . (σ_im = covariance between stock and market, σ²_m variance of market returns)

β > 1 returns correlated with market but greater changes

β = 1 returns correlated exactly with market

0 < β < 1 returns correlated with market but lesser changes

β = 0 returns perfectly uncorrelated with market (i.e. no correlation ρ_im = 0, or no stock move σ_i =0)

-1 < β < 0 returns inversely correlated with market but lesser changes

β = -1 returns perfectly inversely correlated

β < -1 returns perfectly inversely correlated but greater change

Net Present Value Rule

Real discount rate = [(1 + nominal discount rate) / (1 + inflation rate)] – 1

nominal discount rate...