Corporate Finance, Financial Management and Investments Formula Sheet

Tacettin İKİZ · March 03, 2025, 08:46:03 AM

Corporate Finance, Financial Management and Investments Formula Sheet

Tacettin İKİZ · March 03, 2025, 08:48:23 AM

Corporate Finance, Financial Management and Investments Formula Sheet

Equity

Holding Period Return
For a stock of price S₀, with dividend D:

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E(r₁) = (D₁ + S₁ - S₀) / S₀
Dividend Yield

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r_d = D₁ / S₀
Gordon Model
Assume constant growth of dividends g, required rate of return r.

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D₁ = D₀ (1 + g)
P = D₁ / (r - g)
r = r_f + β × (r_m - r_f)
g = Plowback ratio × Return on equity

Payout ratio:

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Dividend per share / Earnings per share
Plowback ratio:

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1 - Payout ratio
Return on equity:

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ROE = Earnings per share / Book value per share

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ROE = Net income / Book value of equity
Earnings per share:

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Net income / Number of outstanding shares
Book value:

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Common stock + Retained earnings
Common stock:

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Face value of shares

No-Growth Value

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NGV = EPS₁ / r
Present Value of Growth Opportunities (PVGO)

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PVGO = P - NGV

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PVGO = (D₀(1 + g) / (r - g)) - (E₁ / r)
PE Ratios with Growth

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P₀ = D₁ / (r - g) = E₁(1 - b) / (r - (b × ROE))

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P₀ / E₁ = (1 - b) / (r - (b × ROE))
Residual Dividend Approach
Project returns: rₚ
Shareholder cost of equity: rₑ

If:
-

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rₚ > rₑ → company creates value for shareholders
-

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rₚ = rₑ → shareholders indifferent
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rₚ < rₑ → company destroys value for shareholders

Gordon Two-Stage Model

Last year's dividend D₀

Example: 3 years' variable growth (g₁, g₂, g₃), after which growth becomes constant (g₄).

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D₁ = D₀ (1 + g₁)
D₂ = D₁ (1 + g₂)
D₃ = D₂ (1 + g₃)
D₄ = D₃ (1 + g₄)

Present value of future constant growth at t = 3:

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P₃ = D₄ / (r - g₄)
Stage 1: Sum of discounted D₁, D₂, and D₀ to t = 0

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S₁ = Σ (Dₜ / (1 + r)ₜ) from t=1 to 3
Stage 2: Discount P₃ to t = 0

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S₂ = P₃ / (1 + r)³ = (D₃(1 + g₄)) / ((r - g₄)(1 + r)³)
Final Value:

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PV = S₁ + S₂

EBITDA Multiples

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EBITDA Multiple = EV / EBITDA ≈ 1 / (r - g)
- ↓ Multiple → ↑ Risk
- ↑ Multiple → ↓ Growth

Enterprise Value Calculation

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Enterprise Value = EBITDA × Multiple
Equity Valuation

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Fair value of Equity = EV + Excess cash - Long term debt

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Fair value per share = Equity / # of shares outstanding

Residual Income Model

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PV = BV₀ + ((ROE₁ - rₑ)BV₀) / (1 + rₑ)  
   + ((ROE₂ - rₑ)BV₁) / (1 + rₑ)²  
   + ((ROE₃ - rₑ)BV₂) / (1 + rₑ)³ + ...

Dupont Analysis

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ROE = Net Income / Shareholder Equity
Key Components:

Asset Turnover:

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Sales / Avg. Assets
"How much revenue from assets?"

Operating Efficiency:

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NOPAT / Sales
"What portion of sales is left to operations & shareholders after covering expenses?"

Interest Efficiency:

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Net Income / NOPAT
"How well does the firm use debt financing?"

Leverage:

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Avg. Assets / Avg. Equity
"What assets beyond equity does the company hold?"

Tacettin İKİZ · March 03, 2025, 08:50:58 AM

Corporate Finance, Financial Management and Investments Formula Sheet

Debt

Effective Annual Rate
With a compounding period T (measured in years, e.g., 1 month T = 1/12), return over that period r₀(T):

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EAR = [1 + r₀(T)]^(1/T) - 1
    = (1 + APR × T)^(1/T) - 1
    = exp(r_cc) - 1

Annual percentage rate:

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APR = r₀(T) × (1/T)
Continuously compounding rate:

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r_cc = ln(1 + EAR)

Real and Nominal Interest Rates
i = inflation

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r_real = (r_nom - i) / (1 + i)
i = r_nom - r_real
r_nom = r_real + i
r_real ≈ r_nom - i

Forward Rates

n-year forward rate:

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fₙ = ( (1 + rₜ)ⁿ / (1 + rₜ₋₁)ⁿ⁻¹ ) - 1
Forward rates give an expectation of interest rates, as short rates are based on yields to maturity of different-duration assets. Discount future cash flows by the product of relevant forward rates (taken from t=0).

Liquidity Preference Theory

This theory asserts that forward rates reflect expectations about future interest rates plus a liquidity premium that increases with maturity:

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Interest Rate = Forward Rate - Liquidity Premium

Deferred Loans

Implied annually compounded forward rate for a deferred loan of length X beginning in year Y, based on the yields r is:

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f = ( (1 + rₓ₊ᵧ₋₁)^(X+Y-1) / (1 + rᵧ₋₁) )^(1/X) - 1

Bond Equivalent YTM

Convention: Work out semi-annual yield to maturity (i.e., 2x number of periods), then double it.

Realised Compounding Yield to Maturity

Yield with reinvestment of coupons at a given interest rate. Add up total proceeds, including T-compounded value of all coupons.

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Real. ytm = (Total Proceeds / Current Price)^(1/T) - 1

Equivalent Annual Costs (Loan Repayments)

Take out a loan of NPV=$L, paid over a period of n with interest rate r. n-year annuity factor is:

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AFₙ = r / (1 - (1 + r)⁻ⁿ)
Equivalent per-period costs (repayments) is:

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C = NPV / AFₙ

NPV (Net Present Value)

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NPV = -C₀ + C₁/(1 + r) + C₂/(1 + r)² + ... + Cₜ/(1 + r)ᵀ

IRR (Internal Rate of Return)

Expected rate of return. Use IRR(Cashflows) in Excel:

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IRR = r ↔ NPV = 0
-

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IRR > Discount → Accept
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IRR < Discount → Reject

Bonds: YTM = IRR

(BBB+ investor, BBB- junk)

-

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YTM = coupon → "Par"
-

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YTM < coupon → "Premium"
-

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YTM > coupon → "Discount"

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YTM ≈ RFR + Default risk + Interest rate risk + Premium for embedded options
Interest rates ↑ → Bond price ↓, Interest rates ↓ → Bond price ↑

With coupons C, YTM r, num periods T, and face value F:

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PV(coupon + FV) = PV(r, T, C, FV)
PV(coupon only) = PV(r, T, C)
PV(FV only) = PV(r, T, F)

YTM formula:

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YTM = RATE(#periods, payments, PV, FV) or IRR(Cashflows)
Check semi-annual, for rates ≠ yields. PV must be negative.

Bid = sell $ to broker, Ask = buy $ from broker

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Liquidity ∝ 1 / (Ask - Bid)

Duration

Weighted average of the times when the bond's cash payments are received.

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Duration = (1 × PV(C₁) + 2 × PV(C₂) + ... + T × PV(Cₜ)) / PV

Portfolio Duration

Total equity in portfolio = Σ Assets - Σ Liabilities

Portfolio duration = Multiply component durations by their values (liabilities negative), divide by total equity.

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Portfolio duration = (Σ Asset × Durat. - Σ Liab. × Durat.) / Total equity in portfolio

Modified Duration

Percentage change in bond price for a 1 percentage-point change in yield (adjusts for accuracy).

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Mod dur = volatility (%) = Duration / (1 + yield)
Approximates % change in bond price for 1% change in yield. Units in years.

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%Δ(price) = [yr] × [%Δrate/year] = %

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Mod dur = (ΔP / P) / (Δy / (1 + y))

Trade Discounts

Discount rₐ for payment at t₁ days rather than usual t₂ day payment terms.

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WACC = (1 - rₐ)^(365 / (t₂ - t₁)) - 1

Tacettin İKİZ · March 03, 2025, 08:53:27 AM

Corporate Finance, Financial Management and Investments Formula Sheet

Markets

Synthetic Bonds / Law of One Price

Consider three cash flows C₁, C₂, and C₃. If:

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C₃ = A × C₁ + (1 - A) × C₂
Then:

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A = (C₃ - C₂) / (C₁ - C₂)
With:

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PV₃ = A × PV₁ + (1 - A) × PV₂
Example: C₁ = 100, C₂ = 0, and C₃ = 50

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A = (50 - 0) / (100 - 0) = 50%

Two-Asset Portfolio

Mean returns μₐ, μᵦ, risk (StDev) σₐ, σᵦ, correlation:

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ρ_AB = COV(μ_A, μ_B) / (σ_A × σ_B)
Risk-free rate γᵣ. Weights ωₐ and ωᵦ = 1 - ωₐ.

Mean portfolio return:

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μ_P = ω_A × μ_A + ω_B × μ_B
Variance of portfolio:

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σ_P² = ω_A² × σ_A² + ω_B² × σ_B² + 2 × ω_A × ω_B × COV(γ_A, γ_B)

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σ_P² = ω_A² × σ_A² + ω_B² × σ_B² + 2 × ω_A × ω_B × ρ_AB × σ_A × σ_B

Minimum-Variance Portfolio

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ω_A = (σ_B² - COV(γ_A, γ_B)) / (σ_A² + σ_B² - 2 × COV(γ_A, γ_B))
Optimal (tangent) portfolio is given by A-stock weighting:

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ω_A = ((μ_A - γ_r) × σ_B² - (μ_B - γ_r) × ρ_AB × σ_A × σ_B) / ((μ_A - γ_r) × σ_B² + (μ_B - γ_r) × σ_A² - (μ_A + μ_B - 2 × γ_r) × ρ_AB × σ_A × σ_B)

Sharpe ratio / 'reward to risk' is:

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Sharpe ratio = (μ_P - γ_r) / σ_P
Sharpe is max along Tangent line/Capital Market Line ('CML'):

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μ_P = Sharpe × σ_P + γ_r
For CML portfolio with risky asset allocation:

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ω_P = 1 - ω_f

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μ_C = (1 - ω_f) × γ_r + ω_f × μ_P

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σ_C = ω_f × σ_P
To achieve a target return μ_T:

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ω_P = (μ_T - γ_r) / (μ_P - γ_r)

Utility

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U = E(γ) - 0.5 × A × σ²
- A > 0: Risk aversion, preference for lower risk
- A = 0: Risk neutral, highest return sought
- A < 0: Pathological risk seeking, lower returns ok with ↑ risk

With risk-free rate γᵣ and risky asset γᵖ, σₚ², maximum level of risk aversion for which risky asset is preferred:

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A = (2 × (γ_P - γ_r)) / σ_P²

Capital Asset Pricing Model (CAPM)

Market Risk Premium:

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r_m - r_f
Required return on risky asset (systematic risk only):

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r = r_f + β × (r_m - r_f)
Asset Beta:

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β = COV(stock, market) / VAR(market)
-

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β > 1 → Aggressive stock
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