🔹 Holding Period Return

Return for holding a stock with dividend D and price change:
Return = (Dividend + Ending Price − Beginning Price) / Beginning Price

🔹 Dividend Yield

Dividend Yield = Dividend / Price

🔹 Gordon Growth Model (Constant Growth DDM)

• Dividends grow at a constant rate (g).
• Value of stock = Next Year’s Dividend / (Required Return − Growth Rate)
• Next Year’s Dividend = Last Dividend × (1 + g)
• Required Return = Risk-Free Rate + Beta × (Market Return − Risk-Free Rate)

Supporting Ratios:

• Growth Rate = Plowback Ratio × Return on Equity
• Plowback Ratio = 1 − Payout Ratio
• Payout Ratio = Dividends per Share / Earnings per Share
• Return on Equity = Net Income / Book Value of Equity

🔹 No-Growth Value

• Assumes 100% earnings payout
  NGV = EPS / Required Return

🔹 PVGO (Present Value of Growth Opportunities)

PVGO = Price − NGV
or
PVGO = (D0 × (1 + g) / (r − g)) − (E1 / r)

🔹 PE Ratio with Growth

Price = (Earnings × (1 − Plowback Ratio)) / (r − (Plowback Ratio × ROE))

🔹 Residual Dividend Policy

Let Project Return = rp, and Cost of Equity = re:

• If rp > re → Create value
• If rp = re → Indifferent
• If rp < re → Destroy value

🔹 Gordon Two-Stage Model

• First 3 years: Dividends grow at variable rates g1, g2, g3
• From year 4 onward: Growth stabilizes at g4

Dividends:

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

Present Value (PV):

• Stage 1 (Years 1–3): PV of D1, D2, D3
• Stage 2: Terminal Value at year 3 = D4 / (r − g4), discounted back
• Total Value = Stage 1 + Stage 2

🔹 EBITDA Multiples

Fair Value per Share = Equity Value / Shares Outstanding

EBITDA Multiple = EV / EBITDA = 1 / (r − g)

Higher risk or lower growth → Lower multiple

Enterprise Value = EBITDA × Multiple

Equity Value = EV + Excess Cash − Long-Term Debt

🔹 Effective Annual Rate (EAR)

When compounding is more frequent than annually:
EAR = (1 + Periodic Rate) ^ T − 1
Also:
EAR = (1 + APR × (T/1)) ^ (1/T) − 1

🔹 Annual Percentage Rate (APR)

APR = Periodic Rate × Number of Periods

🔹 Continuous Compounding

Continuously Compounded Rate = ln(1 + EAR)

🔹 Real vs. Nominal Interest Rates

• Real Rate ≈ Nominal Rate − Inflation
• Nominal Rate = Real Rate + Inflation + (Real × Inflation)
• Precise:
  r_real = (1 + r_nom) / (1 + i) − 1

🔹 Forward Rates

• Expectation of future interest rates based on current yields.
• n-Year Forward Rate = ((1 + r_t)^n / (1 + r_t)^t)^(1/(n−t)) − 1

🔹 Liquidity Preference Theory

Forward rates reflect:
Forward Rate = Expected Future Rate + Liquidity Premium

🔹 Deferred Loan Rates

• Implied rate for loans starting in future years:
  f = [(1 + r_Y-1) ^ (X+Y−1) / (1 + r_Y−1)^X] − 1

🔹 Bond Equivalent Yield

• Converts semi-annual yield to annual by doubling it.
• BEY = 2 × Semi-Annual Yield

🔹 Realized Yield to Maturity

• Accounts for reinvestment of coupons
  Real YTM = [(Total Proceeds / Price) ^ (1/T)] − 1

🔹 Equivalent Annual Cost (EAC)

Used for comparing projects with different lifespans

• Annuity Factor = [1 − 1 / (1 + r)^n] / r
• EAC = NPV / Annuity Factor

🔹 Net Present Value (NPV)

NPV = −Initial Outlay + Σ (Cash Flow / (1 + r)^t)

🔹 Internal Rate of Return (IRR)

• Discount rate where NPV = 0
• IRR > discount rate → Accept
• IRR < discount rate → Reject

🔹 Bonds

• YTM = IRR of bond cash flows
• If Coupon = YTM → Par
• Coupon < YTM → Discount
• Coupon > YTM → Premium
• YTM ≈ RFR + Default Risk + Interest Rate Risk + Option Premiums

🔹 Bond Valuation

With coupons:
PV = PV(Coupons) + PV(Face Value)
Without coupons (zero):
PV = Face Value / (1 + r)^T

🔹 Duration (Interest Rate Risk)

• Duration = Weighted average time of cash flows
• Higher duration = more interest rate sensitivity
• Portfolio Duration = Weighted average of individual durations

🔹 Modified Duration

Estimates % price change for a 1% change in yield:
Mod Duration = Duration / (1 + Yield)
Approx. %ΔPrice ≈ −ModDur × ΔYield

🔹 Trade Discounts

For early payment:
r_d = WACC × (1 / (1 − Discount Rate)) ^ (365 / Days Early) − 1

🔹 Cost of Debt

Cost of debt = Risk-free rate + Credit spread
Credit spread depends on interest coverage:
Interest Coverage = EBIT / Interest Expense

🔹 Cost of Equity (From CAPM)

Cost of Equity = Risk-free rate + Beta × Market Risk Premium

🔹 Weighted Average Cost of Capital (WACC)

WACC = (1 − Tax Rate) × (Debt / (Debt + Equity)) × Cost of Debt + (Equity / (Debt + Equity)) × Cost of Equity

🔹 Levered vs. Unlevered Return

• Unlevered return = Return on assets
• Levered return = Amplifies return with debt
  rE = rA + (D/E) × (rA − rD) × (1 − Tax Rate)

🔹 Levered vs. Unlevered Beta

• Beta with debt = Higher risk
• Unlevered Beta:
  Beta_U = Beta_L / (1 + (1 − Tax Rate) × D/E)
• Levered Beta:
  Beta_L = Beta_U × (1 + (1 − Tax Rate) × D/E)

🔹 Tax Shields

• Permanent debt adds value:
  Value of Tax Shield = Tax Rate × Debt
• Total Firm Value = All-Equity Firm Value + Tax Shield
  Leverage adds value but also financial distress risk — optimize accordingly.

🔹 Value of Government Claim (if firm pays taxes)

PV(Tax Shield) = (E + T) × Tax Rate / (1 − Tax Rate)

🔹 Sustainable Growth Rate (SGR)

Rate at which firm grows with internally generated equity.
SGR = ROE × Plowback Ratio
Expanded:
SGR = (Retained Earnings / Beginning Equity)
or
SGR = Net Income / Sales × Sales / Net Assets × Net Assets / Equity

🔹 Buybacks and Value

• Increase in firm value if excess cash/debt used for repurchase
• Value to Shareholders = Premium × Volume
• Tax shield:
  PV(Tax Shield) = ΔDebt × Tax Rate

🔹 Total Enterprise Value (TEV)

TEV = Debt + Equity − Non-operating Cash Flows
Also:
TEV = EBIT × (1 − Tax Rate) / WACC

🔹 EVA & Economic Profit

• EVA (Economic Value Added) = Income − Cost of Capital × Investment
• EP (Economic Profit) = (ROI − r) × Capital Invested

🔹 Key Metrics

• EPS = Net Income / Shares Outstanding
• ROE = Net Income / Equity
• ROA = Net Operating Profit After Tax / Total Assets
• P/E = Share Price / EPS

🔹 Dividend Tax Strategy

Compare:

1. Immediate payout: Pay personal taxes
2. Delay payout: Accumulate at corporate rate then pay personal tax

• If corporate tax > personal tax → Immediate payout better
• If corporate tax < personal tax → Delay payout better

🔹 Franking Credit (Australia-specific)

If corporate tax has been paid, franking credits offset shareholder tax
Franking Credit = Dividend × (Tax Rate / (1 − Tax Rate))

🔹 Dividend Policy Guidance

• If Project Return < Cost of Equity → Return cash (bad projects)
• If Project Return > Cost of Equity → Keep cash (invest)

🔹 Working Capital Management

ΔWorking Capital = ΔCurrent Assets − ΔCurrent Liabilities

NPV = −ΔWorking Capital − Present Value of FCF

FCF = (1 − Tax Rate) × EBIT / r

🔹 Law of One Price / Synthetic Bonds

• If you can replicate a cash flow stream with different assets, those assets must have the same price.
• Example:
  If C3 = A × C1 + (1 − A) × C2, then
  A = (C3 − C2) / (C1 − C2)

🔹 Two-Asset Portfolio

Expected Return (μ_p):
μ_p = ω_A × μ_A + ω_B × μ_B

Portfolio Variance (σ²_p):
Includes individual variances and covariance:
σ²_p = ω²_Aσ²_A + ω²_Bσ²_B + 2ω_Aω_B × Cov(A, B)

Covariance:
Cov(A, B) = ρ_AB × σ_A × σ_B

🔹 Minimum Variance Portfolio

Optimal weights to minimize risk:
Weight of A = (σ²_B − Cov(A, B)) / (σ²_A + σ²_B − 2 × Cov(A, B))

🔹 Optimal Portfolio (Tangency Portfolio)

Maximizes Sharpe Ratio:
Weight = [(μ_A − r_f)σ²_B − (μ_B − r_f)Cov(A, B)] / Denominator
(denominator is a long formula involving variances and covariances)

🔹 Sharpe Ratio (Risk-adjusted return)

Sharpe = (Portfolio Return − Risk-Free Rate) / Portfolio Standard Deviation

Maximized on the Capital Market Line (CML).

🔹 Capital Market Line (CML)

Return = r_f + (Sharpe × Portfolio σ)
Used when adding risk-free assets to portfolios.

🔹 Capital Asset Pricing Model (CAPM)

Expected Return = r_f + β × (Market Return − r_f)

🔹 Beta (β)

Measures sensitivity to market movements:
β = Cov(Stock, Market) / Var(Market)

• β > 1: Aggressive stock
• β < 1: Defensive stock

🔹 Realized CAPM (With error term)

R_i = α_i + β_i × R_m + e_i

🔹 Variance of Stock Returns

σ² = β² × σ²_Market + σ²(Error)

🔹 R² (Proportion of Systematic Risk)

R² = (β² × σ²_Market) / σ²_Total

🔹 Portfolio Beta

β_P = Σ (Weight × β_i)

🔹 Fama-French 3-Factor Model

Improves on CAPM by adding:

1. Market Risk
2. Size (Small − Large)
3. Value (High Book-to-Market − Low)

α = r_f + β_Market × (r_M − r_f) + β_Size × (r_Small − r_Large) + β_Value × (r_HighBM − r_LowBM)

🔹 Utility Theory

Utility = E(R) − 0.5 × A × σ²

Max utility when:
ω = (μ_P − r_f) / (A × σ²_P)

A = Risk aversion

🔹 Option Payoffs

Call Option (Right to Buy):

• Payoff = Max(0, S − K)
  Put Option (Right to Sell):
• Payoff = Max(0, K − S)

Where:

• S = Stock price
• K = Strike price

🔹 Protective Put

• Own stock + buy a put
• Downside protected, unlimited upside
• Payoff = Max(K, S) − Put Premium

🔹 Covered Call

• Own stock + sell a call
• Upside capped, earns premium
• Payoff = Min(S, K) + Call Premium

🔹 Straddle

• Buy a call and a put at the same strike
• Profits from volatility, losses if price is stable
• Payoff = Max(S − K, K − S) − Total Premiums

🔹 Collar

• Long stock, buy a put (downside protection), sell a call (cap upside)
• Lower cost strategy for hedging

🔹 Put-Call Parity

Call − Put = S − PV(K)
Where PV(K) = Present value of strike
Holds for European options (no early exercise)

🔹 Binomial Model

• Model option pricing over discrete time periods
• Stock goes up by u or down by d
• Risk-neutral probability (q):
  q = (1 + r − d) / (u − d)

🔹 Black-Scholes Formula (for European Calls)

• Assumes lognormal returns, no dividends
• Inputs: S, K, T, r, σ
• Core idea: Value = Expected payoff discounted at risk-free rate, under risk-neutral probabilities

🔹 Greeks

• Delta (Δ): Sensitivity to stock price
• Gamma (Γ): Sensitivity of delta
• Theta (Θ): Time decay
• Vega (ν): Sensitivity to volatility
• Rho (ρ): Sensitivity to interest rates

🔹 Hedging

Delta Hedging:

• Hold opposite position to offset delta risk
• Rebalance as delta changes (not static)

🔹 Futures & Forwards

• Forward Contract: Private, customizable, settled at maturity
• Futures Contract: Standardized, traded on exchange, marked to market

Futures Price = Spot × e^(r × T) (no arbitrage condition)

🔹 Hedging with Futures

Hedge Ratio = Δ Value of Asset / Δ Value of Futures

🔹 Value at Risk (VaR)

Estimates max loss at a confidence level:
VaR = z × σ × √T × Portfolio Value

Where:

• z = z-score (e.g., 1.65 for 95%)
• σ = volatility
• T = time in years

🔹 Credit Risk Tools

• Credit Default Swap (CDS): Insurance against default
• Z-spread: Yield spread over risk-free curve to match bond price
• Distance to Default: Based on equity volatility and firm leverage

🔹 Interest Rate Swaps

Net cash flow = Difference in fixed vs. floating rates × Notional

Exchange fixed vs. floating interest

Common for managing interest exposure

🔹 Capital Budgeting Rules

Net Present Value (NPV):

• Accept if NPV > 0
• Measures value created
  NPV = Σ (Cash Flow / (1 + r)^t) − Initial Outlay

Internal Rate of Return (IRR):

• Discount rate where NPV = 0
• Accept if IRR > required return
• Caution: Multiple IRRs or no IRR possible with non-conventional cash flows

Payback Period:

• Time until initial investment is recovered
• Ignores time value of money and cash flows beyond recovery

🔹 Profitability Index (PI)

PI = Present Value of Cash Inflows / Initial Investment

• Accept if PI > 1
• Used when capital is constrained

🔹 Free Cash Flow (FCF)

FCF = EBIT × (1 − Tax Rate) + Depreciation − CapEx − ΔWorking Capital

🔹 Economic Value Added (EVA)

EVA = Net Operating Profit After Tax − (WACC × Capital Invested)

• Tells you if the project earns more than its cost of capital

🔹 Real Options

• Flexibility adds value
  Types:
• Abandonment Option: Stop project if it’s not working
• Expansion Option: Scale up if project does well
• Timing Option: Wait for better conditions
• Switching Option: Change inputs or outputs as needed

🔹 Scenario & Sensitivity Analysis

• Scenario: Change multiple inputs at once
• Sensitivity: Change one input at a time to see impact

🔹 Monte Carlo Simulation

• Run thousands of randomized scenarios to estimate risk profile
• Output: Distribution of NPV or IRR

🔹 Hurdle Rate

• Minimum acceptable return on a project
• Usually equal to WACC unless risk-adjusted

🔹 Project Risk Adjustment

• Riskier projects → Higher discount rate
• Use divisional WACC or adjust for project-specific beta

🔹 Replacement Chain Analysis

• Equalize project lengths by replicating shorter projects
• Compare total NPV over least common multiple of years

🔹 Inflation in Valuation

• Nominal Cash Flows → Nominal Discount Rate
• Real Cash Flows → Real Discount Rate
  Mixing = valuation errors

🔹 Working Capital in Projects

• Increase in working capital = cash outflow
• Released at end of project = inflow

🔹 Terminal Value

Discount back to present with appropriate rate

Estimate value beyond forecast period
TV = FCF_n × (1 + g) / (r − g)