Finance Formulae

NPV = PV_inflows – PV_outflows
IRR = i that makes NPV = 0
UAS – annuity that = PV of irregular cash flow pattern

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R_i= realized asset = expected return

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ß=1 ⇒ average risk
ß<1 ⇒ below average risk

ß>1 ⇒ above average risk

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CAPM relates a security’s relevant risk (ß) to expected return.

CFAT (indirect) = market value of asset sold – tax liability = MV – (MV – BV)(𝜏)
Increase in working capital, ΔWC = CA – CL
ΔCFAT = (ΔS – ΔC – ΔD)(1 – 𝜏) + ΔD {S = sales, C = costs, D = depreciation}

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NPV (loan) = Amount borrowed – PV (all after-tax payments)

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APV (Adjusted PV) = All equity value – PV(FC) + NPV(loan)

1. Calculate NPV for all-equity financing
2. Adjust NPV for flotation costs
3. Adjust NPV for tax shield + subsidization

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WACC (weighted average cost of capital)

{D + E = V}

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Pure Play Method

1. Find publicly traded company with business similar

to proposed project.

2. Determine equity beta (ß_e) for pre-play firm’s stock.
3. Calculate pure-play assets ß_A

Loan $ = Gross Proceeds (1 – FC%)

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When to Lease?
If there is a tax differential or if the lease lowers transaction costs and/or reduces uncertainty

Lessee (user): 0 = I_o – Max[1-𝜏_c^u] × PVFA[T,r_D(1-𝜏_c^u)] – D × 𝜏_c^u × PVFA[T,r_D(1-𝜏_c^u)] Lessor (owner): 0 = -I_o – Min[1-𝜏_c^o] × PVFA[T,r_D(1-𝜏_c^o)] + D × 𝜏_c^o × PVFA[T,r_D(1-𝜏_c^o)]

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RRR_D = R_f + ß_D[R_m – R_f] RRR_A = R_f + ß_A[R_m – R_f] ROE = ROA + D/E[ROA – R_f] ROA = RRR_Eu = R_f + ß_A[R_m – R_f] – expected return on an un-levered firm’s assets
RRR_El = RRR_Eu + D/E [RRR_Eu – RRR_D] ROA = [E/(D+E)](ROE)  + [D/(D+E)](R_f)
ß_A = [D/(D+E)]( b_D)  + [E/(D+E)]( ß_E)
ß_E = ß_A + D/E(ß_A – ß_D)
E(R_E) = R_f + ß_E  [E(R_m) – R_f] – M&M CAPM (Equity)

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Payout Ratio = % Earnings paid as dividends
Retention Ratio = 1 – Payout Ratio
g = (Retention Ratio)(ROE)
ROE = NI/Earnings = Earnings/Book Equity

PVGO = PV[PV(cash inflows at time t) – PV(cash outflows at time t)] – present value growth option

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Value_firm = Cost of Assets + PV(OCF)
E = V_F – D {D = amount borrowed/r_D}

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1 share offered for $70 + 4 rights {150,000 new shares offered ⇒ $10,500,000 additional equity}
⇒ 4 rights = $94 – $70 = $24
⇒ 1 right = $6

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M&M (Modigliani and Miller)
Proposition 1: method of structuring D/E has no effect on firm value
Proposition 2: cost of equity capital increases as amount of debt increases

Assumes:

1. No taxes
2. No transaction costs
3. No changes in investment policy