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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] Ci/(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 t 1) 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 = Ct+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(Ct)/PV = see back of book for easier formula using yield (t = period; T is maturity time; PV(Ct) = 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
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