Investment Analysis Skill

Comprehensive investment analysis frameworks covering portfolio theory, asset pricing, fixed income, equity valuation, derivatives, and portfolio management. Based on the canonical graduate-level textbook by Bodie, Kane, and Marcus.

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Part I: Investment Environment Foundations

Asset Classification

• Real assets: Land, buildings, machines, knowledge — generate net income to the economy
• Financial assets: Stocks, bonds, derivatives — claims on income generated by real assets
• Three broad types of financial assets:
  1. Fixed-income/debt: Promised fixed or formula-determined stream of income (money market, capital market)
  2. Equity: Ownership share; residual claim on assets after debt paid
  3. Derivatives: Value derived from prices of other assets (options, futures, swaps)

Money Market Instruments

• Treasury bills (T-bills): Short-term government debt, sold at discount
• Certificates of deposit (CDs): Bank time deposits
• Commercial paper: Short-term unsecured corporate debt
• Bankers’ acceptances, Eurodollars, repos, federal funds

Capital Market Instruments

• Treasury bonds and notes: Government debt, maturities 1-30 years
• Municipal bonds: Tax-exempt state/local government debt
• Corporate bonds: Long-term corporate debt
• Common and preferred stock
• Mortgage-backed securities

Market Structure

• Primary market: New securities issued (IPOs, seasoned offerings)
• Secondary market: Existing securities traded
• Order types: Market orders, limit orders, stop orders
• Trading mechanisms: Dealer markets, electronic communication networks (ECNs), specialist/market maker systems
• Margin trading: Buying on margin (initial margin ~50%, maintenance margin ~25%)
• Short selling: Borrowing and selling securities expecting price decline

Investment Companies

• Open-end mutual funds: Shares redeemed at NAV; NAV = (Market value of assets - Liabilities) / Shares outstanding
• Closed-end funds: Fixed number of shares trading at premium/discount to NAV
• ETFs: Trade continuously like stocks; lower expense ratios; tax-efficient
• Fees: Front-end loads, back-end loads (CDSC), 12b-1 fees, management fees, expense ratios
• Key finding: Average actively managed equity fund underperforms passive index by ~1% annually

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Part II: Portfolio Theory and Practice

Ch 5: Risk, Return, and the Historical Record

Interest Rate Fundamentals

• Fisher equation: r_nominal = r_real + E(inflation)
  • r_n = r_r + E(i)
  • After-tax real rate: r_r(1-t) - i*t (inflation penalty = tax rate x inflation rate)

Return Measures

• Holding period return (HPR): HPR = (Ending price - Beginning price + Dividends) / Beginning price
• Effective annual rate (EAR): 1 + EAR = [1 + r_f(T)]^(1/T)
• Annual percentage rate (APR): APR = n x r_f(T) where n = compounding periods per year
• Continuous compounding: 1 + EAR = e^(r_cc); r_cc = ln(1 + EAR)
• Relationship: 1 + EAR = (1 + T x APR)^(1/T)

Expected Return and Risk

• Expected return: E(r) = Sum[p(s) x r(s)] for all scenarios s
• Variance: sigma^2 = Sum[p(s) x (r(s) - E(r))^2]
• Standard deviation: sigma = sqrt(variance)
• Risk premium: E(r) - r_f (expected excess return over risk-free rate)
• Sharpe ratio: S = [E(r_P) - r_f] / sigma_P (reward-to-volatility ratio)

From Historical Data

• Arithmetic average: Best estimate of expected future return for single period
• Geometric average: g = [(1+r_1)(1+r_2)…(1+r_n)]^(1/n) - 1; better measure of actual historical growth
• Relationship: E(g) approx= arithmetic_avg - (1/2)*sigma^2

Risk Measures

• Value at Risk (VaR): Loss corresponding to a specified percentile (e.g., 5th percentile)
• Expected Shortfall (ES/CVaR): Expected loss given that loss exceeds VaR; accounts for severity of tail losses
• Skewness: Measure of asymmetry; negative skew means more extreme losses than gains
• Kurtosis: Measure of tail thickness; excess kurtosis > 0 means fatter tails than normal

Historical U.S. Equity Performance (1926-2012 approximate)

• Large stocks: ~12% arithmetic mean, ~20% SD, Sharpe ~0.37
• Small stocks: ~18% arithmetic mean, ~33% SD
• Long-term T-bonds: ~6% arithmetic mean, ~8% SD
• T-bills: ~3.5% arithmetic mean, ~3% SD

Ch 6: Capital Allocation to Risky Assets

Utility Function

• Mean-variance utility: U = E(r) - (1/2) * A * sigma^2
  • A = coefficient of risk aversion (typical values: 2-4)
  • A > 0: risk averse; A = 0: risk neutral; A < 0: risk lover
• Certainty equivalent rate: Risk-free rate that makes investor indifferent to the risky portfolio

Capital Allocation Line (CAL)

• Complete portfolio: C = y*P + (1-y)*F, where y = fraction in risky portfolio P, F = risk-free asset
• Expected return: E(r_C) = r_f + y * [E(r_P) - r_f]
• Standard deviation: sigma_C = y * sigma_P
• CAL equation: E(r_C) = r_f + [(E(r_P) - r_f) / sigma_P] * sigma_C
• CAL slope = Sharpe ratio of risky portfolio = [E(r_P) - r_f] / sigma_P

Optimal Capital Allocation

• Optimal risky allocation: y* = [E(r_P) - r_f] / (A * sigma_P^2)
  • Higher risk premium -> more in risky portfolio
  • Higher risk aversion -> less in risky portfolio
  • Higher variance -> less in risky portfolio

Passive vs Active Strategies

• Passive strategy: Invest in broad market index + T-bills (Capital Market Line)
• Active strategy: Security analysis to identify mispriced securities
• CML is the CAL when the risky portfolio is the market portfolio

Ch 7: Optimal Risky Portfolios

Two-Asset Portfolio

• Expected return: E(r_P) = w_1E(r_1) + w_2E(r_2)
• Variance: sigma_P^2 = w_1^2sigma_1^2 + w_2^2sigma_2^2 + 2w_1w_2sigma_1sigma_2*rho_12
• Covariance: Cov(r_1, r_2) = rho_12 * sigma_1 * sigma_2
• Correlation: rho_12 = Cov(r_1, r_2) / (sigma_1 * sigma_2)

Diversification Effects

• Perfect positive correlation (rho=+1): No diversification benefit; portfolio SD is weighted average
• Perfect negative correlation (rho=-1): Complete hedging possible; portfolio SD can be zero
• Uncorrelated (rho=0): Significant risk reduction
• As n assets increase: sigma_P^2 approaches average covariance (systematic risk remains)

Minimum Variance Portfolio (2 assets)

• w_1* = [sigma_2^2 - Cov(1,2)] / [sigma_1^2 + sigma_2^2 - 2*Cov(1,2)]

Efficient Frontier

• Markowitz portfolio selection: Set of portfolios offering maximum expected return for each level of risk
• Portfolios below the minimum-variance portfolio are inefficient
• Optimal risky portfolio: Tangency point of CAL with efficient frontier (maximizes Sharpe ratio)

Separation Property

• All investors hold the same optimal risky portfolio (tangency portfolio)
• Differ only in capital allocation between risky and risk-free assets based on risk aversion
• Two independent tasks: (1) determine optimal risky portfolio; (2) allocate between risky and risk-free

Ch 8: Index Models

Single-Index Model

• Regression equation: R_i = alpha_i + beta_i * R_M + e_i
  • R_i = excess return on security i
  • R_M = excess return on market index
  • alpha_i = security’s abnormal return (intercept)
  • beta_i = sensitivity to market (slope)
  • e_i = firm-specific/residual return (zero mean, uncorrelated with R_M)

Risk Decomposition

• Total risk: sigma_i^2 = beta_i^2 * sigma_M^2 + sigma^2(e_i)
• Systematic risk: beta_i^2 * sigma_M^2 (market/nondiversifiable risk)
• Firm-specific risk: sigma^2(e_i) (diversifiable risk)
• R-squared: Fraction of total variance explained by market = beta_i^2 * sigma_M^2 / sigma_i^2

Covariance in Index Model

• Cov(r_i, r_j) = beta_i * beta_j * sigma_M^2
• Dramatically reduces estimation burden: n betas + n residual variances + 1 market variance vs n(n+1)/2 covariances

Diversification in Index Model

• Portfolio beta: beta_P = (1/n) * Sum(beta_i) for equally weighted
• Portfolio variance: sigma_P^2 = beta_P^2 * sigma_M^2 + sigma^2(e_P)
• Diversifiable component: sigma^2(e_P) = (1/n) * avg(sigma^2(e_i)) -> 0 as n -> infinity
• Key insight: As diversification increases, total variance approaches systematic variance

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Part III: Equilibrium in Capital Markets

Ch 9: Capital Asset Pricing Model (CAPM)

CAPM Assumptions

1. Investors are price-taking, mean-variance optimizers
2. Single-period investment horizon
3. Investments limited to publicly traded financial assets and risk-free borrowing/lending
4. No taxes or transaction costs
5. Homogeneous expectations (all investors analyze securities the same way)

CAPM Results

• Market portfolio: Tangency portfolio on the efficient frontier; all investors hold it
• Capital Market Line (CML): E(r_P) = r_f + [(E(r_M) - r_f) / sigma_M] * sigma_P
  • Applies only to efficient portfolios (combinations of market portfolio and risk-free asset)

Security Market Line (SML)

• CAPM equation: E(r_i) = r_f + beta_i * [E(r_M) - r_f]
  • Applies to ALL assets and portfolios
  • beta_i = Cov(r_i, r_M) / Var(r_M) = Cov(r_i, r_M) / sigma_M^2
  • beta_M = 1 (market beta)
  • Risk premium of asset i proportional to its beta

Alpha

• Jensen’s alpha: alpha_i = E(r_i) - {r_f + beta_i * [E(r_M) - r_f]}
  • alpha > 0: Security plots above SML; underpriced
  • alpha < 0: Security plots below SML; overpriced
  • alpha = 0: Fairly priced per CAPM

Extensions

• Zero-beta CAPM: Replaces risk-free rate with return on zero-beta portfolio (when borrowing restricted)
• CAPM with labor income and nontraded assets: Effective market portfolio includes human capital
• Liquidity-adjusted CAPM: Expected return increases with illiquidity cost and illiquidity beta
  • E(r_i) = r_f + beta_i*[E(r_M) - r_f] + f(liquidity_i)

Ch 10: Arbitrage Pricing Theory (APT) and Multifactor Models

APT Framework

• Returns driven by multiple systematic factors: R_i = E(R_i) + beta_i1F_1 + beta_i2F_2 + … + e_i
• No-arbitrage condition: Well-diversified portfolios must satisfy SML relationship
• APT pricing: E(r_i) = r_f + beta_i1RP_1 + beta_i2RP_2 + … + beta_iK*RP_K
  • RP_k = risk premium for factor k
  • Does NOT require identification of market portfolio (advantage over CAPM)

Multifactor Models

• Chen, Roll, Ross factors: Industrial production growth, expected inflation, unexpected inflation, credit spread, term spread
• Fama-French Three-Factor Model: E(r_i) - r_f = beta_i*[E(r_M) - r_f] + s_iE[SMB] + h_iE[HML]
  • SMB = Small Minus Big (size factor): Return on small-cap minus large-cap portfolio
  • HML = High Minus Low (value factor): Return on high B/M minus low B/M portfolio
• Carhart Four-Factor Model: Adds WML (Winners Minus Losers) momentum factor
• Fama-French Five-Factor Model (later addition): Adds RMW (profitability) and CMA (investment) factors

Ch 11: Efficient Market Hypothesis (EMH)

Three Forms of Market Efficiency

1. Weak form: Prices reflect all past trading data (price, volume). Technical analysis cannot generate alpha.
2. Semi-strong form: Prices reflect all publicly available information. Fundamental analysis cannot generate alpha.
3. Strong form: Prices reflect ALL information including private/insider info. Even insider trading cannot generate alpha.

Implications

• Stock prices follow a random walk (unpredictable changes)
• Active management cannot systematically outperform passive indexing after costs
• New information incorporated rapidly into prices
• Portfolio managers add value through: diversification, tax management, risk-level selection

Evidence Supporting EMH

• Most mutual funds underperform passive index by ~1% annually
• Event studies show rapid price adjustment to new information
• Weak serial correlation in short-term stock returns
• Professional analysts’ stock picks show little persistent skill

Anomalies Challenging EMH

• Size effect: Small-cap stocks earn higher risk-adjusted returns
• Value effect: High book-to-market stocks outperform growth stocks
• Momentum: Past winners continue to outperform past losers (3-12 month horizons)
• Post-earnings announcement drift: Prices drift in direction of earnings surprise
• January effect: Higher returns in January, especially for small stocks
• Low volatility anomaly: Low-beta stocks earn higher risk-adjusted returns

Ch 12: Behavioral Finance

Key Behavioral Biases

• Overconfidence: Investors overestimate precision of their beliefs
• Conservatism/anchoring: Slow to update beliefs in face of new evidence
• Representativeness: Overweight recent experience; extrapolate recent trends
• Disposition effect: Tendency to sell winners too early and hold losers too long
• Framing: Decisions affected by how choices are presented
• Mental accounting: Segregate decisions that should be combined
• Regret avoidance: Avoidance of actions that could produce regret
• Prospect theory (Kahneman & Tversky): Losses hurt roughly 2x more than equivalent gains feel good; risk-seeking in domain of losses

Limits to Arbitrage

• Fundamental risk: Even if mispriced, prices may diverge further before correcting
• Implementation costs: Short-selling constraints, transaction costs, model risk
• Model risk: Apparent mispricing may reflect model error, not true opportunity

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Part IV: Fixed-Income Securities

Ch 14: Bond Prices and Yields

Bond Pricing

• Price of a coupon bond: P = Sum[C/(1+y)^t, t=1..T] + FV/(1+y)^T
  • C = coupon payment, y = yield to maturity per period, FV = face value, T = maturity
• Price of a zero-coupon bond: P = FV / (1+y)^T
• Price-yield relationship: Inverse; as yields rise, prices fall (and vice versa)
  • Convex relationship: Price increase from yield decrease > price decrease from equal yield increase

Yield Measures

• Yield to maturity (YTM): Internal rate of return assuming held to maturity and coupons reinvested at YTM
• Current yield: Annual coupon / Current price
• Yield to call: YTM computed to first call date using call price
• Realized compound return: Actual return accounting for reinvestment rates
• Bond equivalent yield (BEY): Semiannual yield x 2

Bond Pricing Relationships

1. Prices and yields move inversely
2. For a given change in yield, price increase from yield decrease exceeds price decrease from yield increase (convexity)
3. Long-term bonds are more price-sensitive to yield changes than short-term bonds
4. Price sensitivity increases at a decreasing rate as maturity increases
5. Price sensitivity is inversely related to coupon rate
6. Price sensitivity is inversely related to yield level at which bond is currently selling

Credit Risk and Bond Ratings

• Investment grade: Moody’s Baa/S&P BBB and above
• Speculative/junk: Below Baa/BBB
• Default premium: Yield spread over comparable Treasury
• Altman Z-score: Z = 3.3*(EBIT/Assets) + 1.0*(Sales/Assets) + 0.6*(MV equity/BV debt) + 1.4*(Retained earnings/Assets) + 1.2*(Working capital/Assets)
  • Z > 3.0: Safe zone; Z < 1.81: Distress zone

Ch 15: Term Structure of Interest Rates

Spot and Forward Rates

• Spot rate (y_n): Yield on zero-coupon bond maturing in n years
• Forward rate (f_n): Implied rate for a future period
  • (1+f_n) = (1+y_n)^n / (1+y_{n-1})^{n-1}
• Yield from forward rates: (1+y_n) = [(1+r_1)(1+f_2)(1+f_3)…(1+f_n)]^(1/n)
• Bootstrapping: Extracting spot rates from coupon bond prices

Term Structure Theories

1. Expectations hypothesis: Forward rates = expected future spot rates; yield curve reflects expectations of future interest rates
   • Upward-sloping curve => market expects rates to rise
2. Liquidity preference theory: Forward rates = expected spot rates + liquidity premium
   • Investors demand premium for longer-term bonds (greater price risk)
   • Liquidity premium > 0, so forward rates are upward-biased predictors of future spot rates
   • Yield curve can be upward-sloping even if rates expected to stay constant
3. Market segmentation theory: Supply/demand within each maturity segment determines rates independently

Ch 16: Managing Bond Portfolios

Duration

• Macaulay duration: D = Sum[t * w_t] where w_t = [CF_t/(1+y)^t] / Price
  • Weighted average time to receive bond’s cash flows
• Modified duration: D* = D / (1+y)
  • Measures price sensitivity: dP/P approx= -D* * dy
  • Key formula: Delta_P/P = -D* * Delta_y
• Duration rules:
  1. Duration of a zero-coupon bond equals its maturity
  2. Duration decreases with coupon rate (for given maturity)
  3. Duration generally increases with maturity
  4. Duration decreases with yield to maturity
  5. Duration of a perpetuity = (1+y)/y

Convexity

• Convexity: C = (1/P) * Sum[t*(t+1)*CF_t / (1+y)^(t+2)]
• Price change with convexity: Delta_P/P = -D* * Delta_y + (1/2) * Convexity * (Delta_y)^2
• Convexity is desirable: Greater convexity means bond gains more from rate decreases and loses less from rate increases
• Callable bonds have negative convexity at low yields (price ceiling near call price)

Immunization Strategies

• Single liability immunization: Match duration of asset portfolio to duration of liability
  • Protects against parallel yield curve shifts
  • Must rebalance as time passes (duration changes)
• Cash flow matching/dedication: Match timing of asset cash flows exactly to liabilities
• Contingent immunization: Active management until cushion spread is exhausted, then switch to immunization
• Duration of portfolio: D_P = Sum[w_i * D_i] (weighted average of component durations)

Active Bond Management

• Substitution swap: Exchange one bond for a similar but mispriced bond
• Intermarket spread swap: Exploit abnormal yield spread between sectors
• Rate anticipation swap: Shift duration based on interest rate forecast
• Pure yield pickup swap: Move to higher-yielding bonds (accepting any risk changes)
• Tax swap: Realize tax losses while maintaining similar portfolio exposure

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Part V: Security Analysis

Ch 17: Macroeconomic and Industry Analysis

Top-Down Analysis Framework

1. Global economy: GDP growth, trade, currency movements
2. Domestic macroeconomy: GDP, employment, inflation, interest rates, fiscal/monetary policy
3. Industry analysis: Sensitivity to business cycle, industry life cycle, competitive structure
4. Company analysis: Financial statements, competitive position, management quality

Key Economic Indicators

• Leading indicators: Stock prices, money supply, building permits, consumer expectations, new orders, average workweek
• Coincident indicators: Industrial production, personal income, manufacturing/trade sales
• Lagging indicators: Average duration of unemployment, CPI services, bank prime rate, ratio of inventories to sales

Business Cycle Sensitivity Factors

1. Sales sensitivity: Necessities (food, drugs) low sensitivity vs. discretionary (luxury, durables) high sensitivity
2. Operating leverage: DOL = 1 + Fixed costs / Profits; higher fixed costs = greater earnings sensitivity
3. Financial leverage: Interest payments are fixed costs that amplify business cycle sensitivity

Sector Rotation Strategy

• Peak: Energy, natural resources
• Contraction: Consumer staples, health care, utilities (defensive)
• Trough: Financials (lower rates), capital goods (anticipating recovery)
• Expansion: Technology, consumer discretionary, industrials (cyclicals)

Industry Life Cycle

1. Start-up: Rapid growth, high risk, low/no dividends
2. Consolidation: Above-average growth, leaders emerging, increasing predictability
3. Maturity: Growth matches economy, price competition, cash cows, high dividends
4. Relative decline: Below-economy growth or shrinkage, obsolescence risk

Porter’s Five Forces (Industry Structure)

1. Threat of new entrants
2. Rivalry among existing competitors
3. Threat of substitute products
4. Bargaining power of buyers
5. Bargaining power of suppliers

Ch 18: Equity Valuation Models

Dividend Discount Models (DDM)

• General DDM: V_0 = Sum[D_t / (1+k)^t, t=1..infinity]
  • k = required rate of return (from CAPM or other model)
• Constant growth (Gordon model): V_0 = D_1 / (k - g)
  • D_1 = D_0 * (1+g); g = sustainable growth rate; requires k > g
  • Implied return: k = D_1/P_0 + g (dividend yield + capital gains rate)
• Sustainable growth rate: g = ROE * b (where b = plowback/retention ratio = 1 - payout ratio)
  • g = ROE * (1 - Dividends/Earnings)
• Two-stage DDM: High growth for T years, then constant growth: V_0 = Sum[D_0*(1+g_1)^t / (1+k)^t, t=1..T] + [D_{T+1}/(k-g_2)] / (1+k)^T
• H-model: Growth rate declines linearly from g_s to g_l over 2H years: V_0 = D_0 * [(1+g_l) + H*(g_s - g_l)] / (k - g_l)

Present Value of Growth Opportunities (PVGO)

• Stock price decomposition: P_0 = E_1/k + PVGO
  • E_1/k = value as no-growth perpetuity (all earnings paid as dividends)
  • PVGO = value of future growth opportunities
  • Growth is valuable only if ROE > k (firm invests in positive-NPV projects)

Price-Earnings (P/E) Ratios

• Trailing P/E: Price / Past 12 months EPS
• Forward P/E: Price / Expected next 12 months EPS
• From Gordon model: P_0/E_1 = (1-b) / (k-g) = payout ratio / (k - ROE*b)
  • Higher P/E justified by: lower k (less risk), higher g (more growth), higher ROE
  • Pitfalls: Earnings can be negative; earnings are volatile; accounting differences

Other Valuation Multiples

• Price/Book (P/B): Market value / Book value of equity
  • P/B > 1 when market believes firm earns more than required return on equity
• Price/Sales (P/S): Useful when earnings are negative
• Price/Cash Flow (P/CF): Less subject to accounting manipulation
• EV/EBITDA: Enterprise value / EBITDA; useful for comparing firms with different capital structures

Free Cash Flow (FCF) Valuation

• FCFF (to firm) = EBIT*(1-tax) + Depreciation - Capital expenditures - Increase in NWC
• FCFE (to equity) = FCFF - Interest*(1-tax) + Net borrowing
• Firm value: V = Sum[FCFF_t / (1+WACC)^t] + Terminal value / (1+WACC)^T
• Equity value: Firm value - Market value of debt
• Terminal value: FCFF_{T+1} / (WACC - g) using Gordon growth model

Ch 19: Financial Statement Analysis

Key Financial Ratios

Profitability:

• ROE = Net income / Equity
• ROA = Net income / Total assets
• Profit margin = Net income / Sales
• Operating margin = Operating income / Sales

DuPont Decomposition:

• ROE = Profit margin x Asset turnover x Equity multiplier
• ROE = (Net income/Sales) x (Sales/Assets) x (Assets/Equity)
• Identifies source of changes in ROE: margins, efficiency, or leverage

Leverage:

• Debt-to-equity = Total debt / Equity
• Interest coverage (TIE) = EBIT / Interest expense
• Equity multiplier = Assets / Equity

Liquidity:

• Current ratio = Current assets / Current liabilities
• Quick ratio = (Cash + Receivables + Marketable securities) / Current liabilities

Efficiency/Turnover:

• Asset turnover = Sales / Total assets
• Inventory turnover = COGS / Average inventory
• Receivables turnover = Sales / Average receivables
• Days sales outstanding = 365 / Receivables turnover

Market Value:

• P/E, P/B, P/S, dividend yield, market cap

Quality of Earnings Considerations

• Revenue recognition timing
• Depreciation and amortization methods
• Inventory accounting (LIFO vs FIFO in inflationary environments)
• Off-balance-sheet items
• Non-recurring items and restructuring charges
• Stock option expense treatment
• Pension and OPEB assumptions

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Part VI: Options, Futures, and Other Derivatives

Ch 20: Options Markets

Option Fundamentals

• Call option: Right (not obligation) to BUY at strike price X before/at expiration
• Put option: Right (not obligation) to SELL at strike price X before/at expiration
• European: Exercisable only at expiration
• American: Exercisable any time up to and including expiration
• In the money: Call: S > X; Put: S < X
• At the money: S = X
• Out of the money: Call: S < X; Put: S > X

Option Payoffs at Expiration

• Long call: max(S_T - X, 0); Profit = max(S_T - X, 0) - Premium
• Long put: max(X - S_T, 0); Profit = max(X - S_T, 0) - Premium
• Short call: -max(S_T - X, 0); Profit = Premium - max(S_T - X, 0)
• Short put: -max(X - S_T, 0); Profit = Premium - max(X - S_T, 0)

Put-Call Parity (European, no dividends)

• C + PV(X) = P + S
  • C = call price, P = put price, S = stock price, PV(X) = X/(1+r_f)^T
  • Equivalently: C - P = S - PV(X)
• With dividends: C + PV(X) = P + S - PV(Dividends)

Option Strategies

• Protective put: Stock + Long put (portfolio insurance; floor on losses)
• Covered call: Stock + Short call (income generation; cap on gains)
• Straddle: Long call + Long put (same X, same T); profits from large price moves either direction
• Strangle: Long call (high X) + Long put (low X); cheaper than straddle, needs larger move
• Bull spread: Long call(X_1) + Short call(X_2), X_1 < X_2; profits from moderate price increases
• Bear spread: Long put(X_2) + Short put(X_1), X_1 < X_2; profits from moderate price decreases
• Butterfly spread: Long call(X_1) + 2 Short calls(X_2) + Long call(X_3); profits from low volatility
• Collar: Stock + Long put(X_1) + Short call(X_2), X_1 < S < X_2; limits both upside and downside

Ch 21: Option Valuation

Factors Affecting Option Value

Factor	Call Value	Put Value
Stock price (S)	+	-
Strike price (X)	-	+
Time to expiration (T)	+	+
Volatility (sigma)	+	+
Risk-free rate (r_f)	+	-
Dividends	-	+

Binomial Option Pricing

• Stock can move up (u) or down (d) in each period
• Risk-neutral probability: p = (1+r_f - d) / (u - d)
• Call value: C = [p*C_u + (1-p)*C_d] / (1+r_f)
• Hedge ratio (delta): H = (C_u - C_d) / (S_u - S_d)
  • Number of shares to hold per option written for riskless hedge

Black-Scholes Option Pricing Model

• Call price: C = S_0 * N(d_1) - X * e^(-r_f*T) * N(d_2)
• Put price: P = X * e^(-r_f*T) * N(-d_2) - S_0 * N(-d_1)
• Where:
  • d_1 = [ln(S_0/X) + (r_f + sigma^2/2)T] / (sigmasqrt(T))
  • d_2 = d_1 - sigma*sqrt(T)
  • N(.) = cumulative standard normal distribution function
  • sigma = annualized standard deviation of continuously compounded stock returns

Black-Scholes Assumptions

1. Stock price follows geometric Brownian motion with constant drift and volatility
2. No dividends during option life (can be adjusted)
3. Continuous trading possible
4. No transaction costs or taxes
5. Risk-free rate constant and known
6. No arbitrage opportunities

• With dividends: Replace S_0 with S_0 - PV(dividends) or use S_0e^(-deltaT) for continuous dividend yield delta

The Greeks

• Delta: dC/dS = N(d_1) for calls; N(d_1) - 1 for puts; measures sensitivity to stock price
• Gamma: d(Delta)/dS; measures rate of change of delta; highest for ATM near expiration
• Theta: dC/dT (negative for most options); time decay
• Vega (kappa): dC/d(sigma); sensitivity to volatility; always positive; highest ATM
• Rho: dC/d(r_f); sensitivity to interest rate

Implied Volatility

• Volatility that equates Black-Scholes price to observed market price
• Volatility smile/smirk: Implied volatility varies with strike price
  • Typically higher for OTM puts (crash protection) = volatility skew
• VIX: CBOE Volatility Index; “fear gauge”; implied vol of S&P 500 index options

Ch 22: Futures Markets

Futures vs. Forwards

• Futures: Standardized, exchange-traded, daily marking to market, clearinghouse guarantees
• Forwards: Customized, OTC, settled at maturity, counterparty risk

Futures Pricing

• Spot-futures parity (cost of carry): F_0 = S_0 * (1 + r_f - d)^T
  • Or: F_0 = S_0 * e^((r_f - d)*T) for continuous compounding
  • d = dividend yield or convenience yield
  • r_f = risk-free rate
• For stock index futures: F_0 = S_0 * (1 + r_f - d)^T where d = dividend yield
• For commodities: F_0 = S_0 * (1 + r_f + storage costs - convenience yield)^T
• Basis: Spot price - Futures price; converges to zero at expiration

Hedging with Futures

• Short hedge: Sell futures to protect against price decline (e.g., farmer selling wheat futures)
• Long hedge: Buy futures to lock in purchase price (e.g., baker buying wheat futures)
• Hedge ratio: h* = beta_portfolio * (Portfolio value / Futures contract value)
  • Number of contracts = h* (using beta to adjust for imperfect correlation)
• Cross-hedging: Using futures on a related but different asset

Ch 23: Futures, Swaps, and Risk Management

Stock Index Futures Applications

• Synthetic stock position: T-bills + Long index futures = Stock index exposure
  • Total payoff at maturity: S_T (equivalent to holding index)
• Hedging market risk: Number of contracts = -(Portfolio beta * Portfolio value) / (Futures price * Contract multiplier)
  • To reduce beta to zero: sell h* contracts; to adjust beta: sell proportional amount
• Index arbitrage: Exploit deviations from spot-futures parity via program trading

Interest Rate Futures

• Treasury bond futures: Delivery of T-bonds; cheapest-to-deliver determines pricing
• Duration-based hedging: N = -(D_P * P) / (D_F * F) where D=duration, P=portfolio value, F=futures price

Swaps

• Interest rate swap: Exchange fixed-rate for floating-rate payments on notional principal
  • Valued as: Long fixed-rate bond + Short floating-rate bond (or vice versa)
• Currency swap: Exchange cash flows in different currencies
• Credit default swap (CDS): Insurance against default; buyer pays premium, seller pays if credit event occurs
• Total return swap: Exchange total return on an asset for a fixed/floating rate

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Part VII: Applied Portfolio Management

Ch 24: Portfolio Performance Evaluation

Risk-Adjusted Performance Measures

1. Sharpe ratio: S = (r_P - r_f) / sigma_P

   • Excess return per unit of TOTAL risk
   • Appropriate when portfolio represents investor’s entire risky investment
   • Higher is better

2. Treynor ratio: T = (r_P - r_f) / beta_P

   • Excess return per unit of SYSTEMATIC risk
   • Appropriate for evaluating a sub-portfolio within a larger diversified portfolio
   • Higher is better

3. Jensen’s alpha: alpha_P = r_P - [r_f + beta_P*(r_M - r_f)]

   • Abnormal return relative to CAPM prediction
   • Positive alpha indicates outperformance
   • Regression-based: regress excess returns on market excess returns; intercept = alpha

4. Information ratio: IR = alpha_P / sigma(e_P)

   • Alpha per unit of nonsystematic (diversifiable) risk
   • Measures consistency of active management skill
   • sigma(e_P) = tracking error (standard deviation of residuals from benchmark regression)

5. M-squared (M^2): M^2 = r_P* - r_M

   • Where r_P* is return on a leveraged/de-leveraged version of P with same total risk as market
   • r_P* = r_f + (sigma_M / sigma_P) * (r_P - r_f)
   • Easy to interpret: percentage points of outperformance at same risk as market

Attribution Analysis

• Asset allocation: Contribution from over/underweighting asset classes vs. benchmark
• Security selection: Contribution from individual security picks within each asset class
• Market timing: Contribution from varying portfolio beta over time
  • Regress: r_P - r_f = a + b*(r_M - r_f) + c*(r_M - r_f)^2 + e
  • c > 0 suggests successful market timing (higher beta when market rises)
  • Equivalently: Henriksson-Merton model uses binary variable for up/down markets

Style Analysis (Sharpe)

• Regress fund returns on set of style indexes (e.g., large growth, small value, bonds)
• Coefficients reveal fund’s effective asset allocation
• R-squared indicates fraction of return variance explained by style
• Residual return = selection return (alpha from security selection)

Ch 25: International Diversification

Benefits

• Low correlation among international markets increases diversification potential
• Access to broader investment opportunity set
• Emerging markets may offer higher expected returns

Currency Risk

• Dollar return on foreign investment: r_$ = r_foreign + r_currency + r_foreign * r_currency
  • Approximately: r_$ ≈ r_foreign + r_currency (for small returns)
• Interest rate parity: F/S = (1 + r_domestic) / (1 + r_foreign)
  • F = forward exchange rate, S = spot rate
• Purchasing power parity: Expected change in exchange rate equals inflation differential
• Currency hedging: Use forward contracts or currency futures to eliminate exchange rate risk

International CAPM

• World market portfolio is the relevant benchmark
• Country-specific and currency risk factors may carry risk premiums
• Home bias: Investors overweight domestic assets relative to world market capitalization weights

Ch 26: Hedge Funds

Common Strategies

• Long-short equity: Long undervalued, short overvalued; reduces market exposure
• Merger/event-driven arbitrage: Long target, short acquirer in announced mergers
• Convertible bond arbitrage: Long convertible bond, short underlying stock
• Fixed-income arbitrage: Exploit relative mispricings in bond markets
• Global macro: Directional bets on macroeconomic trends across markets
• Market neutral: Zero beta; profit from security selection only
• Managed futures (CTA): Systematic trend-following in futures markets
• Distressed securities: Invest in firms near or in bankruptcy

Performance Measurement Issues

• Survivorship bias: Only surviving funds in databases; overstates average returns by 2-4%/year
• Backfill bias: New funds add historical returns selectively
• Stale pricing: Illiquid holdings may be marked at stale prices, understating volatility
• Non-normal returns: Fat tails, negative skew common; Sharpe ratio misleading
• Tail risk: Many strategies resemble short put positions (steady small gains, occasional large losses)
• Alpha decay: Strategies may lose effectiveness as capital floods in

Fee Structure

• “2 and 20”: 2% management fee + 20% of profits (performance fee)
• High-water mark: Performance fee only on new profits above previous peak NAV
• Incentive fee asymmetry: Manager gains 20% of profits but does not share in losses

Ch 27: Theory of Active Portfolio Management

Treynor-Black Model

• Combines active security selection with passive index fund
• Active portfolio: Constructed from securities with nonzero alpha forecasts
  • Weight of each security in active portfolio proportional to: alpha_i / sigma^2(e_i)
• Optimal risky portfolio: Combination of active portfolio (A) and passive market index (M)
  • Weight of active portfolio: w_A* = [alpha_A / sigma^2(e_A)] / [E(r_M) - r_f] / sigma_M^2]
  • Adjusted for beta: w_A = w_A* / [1 + (1 - beta_A) * w_A*]

Information Ratio and Optimal Active Management

• Squared Sharpe ratio: S_P^2 = S_M^2 + IR_A^2
  • IR_A = alpha_A / sigma(e_A) = information ratio of active portfolio
  • Active management adds value proportional to squared information ratio
• Fundamental law of active management (Grinold):
  • IR ≈ IC * sqrt(BR)
  • IC = information coefficient (correlation between forecasts and realizations)
  • BR = breadth (number of independent forecasts per year)
  • Even small IC can generate useful IR with high breadth

Practical Implications

• Concentrate active bets in securities with highest alpha/residual-risk ratio
• Diversify active positions to reduce tracking error
• Optimal portfolio is a tilt from passive index toward active picks
• Forecast quality matters more than quantity: better to have few good forecasts than many poor ones

Ch 28: Investment Policy (CFA Framework)

Investment Policy Statement (IPS) Components

1. Return objectives: Required return vs. desired return; nominal vs. real
2. Risk tolerance: Ability to take risk (time horizon, wealth, income stability) + willingness to take risk (psychological)
3. Time horizon: Short, intermediate, long; may have multiple stages
4. Liquidity needs: Expected and unexpected cash needs
5. Tax considerations: Tax-deferred vs. taxable accounts; capital gains vs. income
6. Legal and regulatory constraints: ERISA, prudent investor rule, etc.
7. Unique circumstances: ESG preferences, concentrated holdings, restricted stock

Asset Allocation by Investor Type

Investor Type	Return Req.	Risk Tolerance	Time Horizon	Liquidity
Individual	Varies	Varies	Varies	Varies
Pension (DB)	Actuarial rate	Depends on funded status	Long	Low (predictable payouts)
Endowment	Spending rate + inflation	Above average	Very long	Predictable (spending policy)
Insurance (Life)	Spread over cost of funds	Below average	Long	Low to moderate
Bank	Spread income	Low	Short	High

Tax-Efficient Investing

• Place high-tax assets (bonds, REITs) in tax-deferred accounts
• Place low-tax assets (growth stocks, index funds) in taxable accounts
• Tax-loss harvesting: Realize losses to offset gains
• Hold appreciated assets to defer capital gains
• Municipal bonds for high-tax-bracket investors in taxable accounts

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Key Formulas Quick Reference

Portfolio Theory

E(r_P) = Sum[w_i * E(r_i)]
sigma_P^2 = Sum_i Sum_j [w_i * w_j * Cov(r_i, r_j)]
Sharpe = [E(r_P) - r_f] / sigma_P
y* = [E(r_P) - r_f] / (A * sigma_P^2)

CAPM

E(r_i) = r_f + beta_i * [E(r_M) - r_f]
beta_i = Cov(r_i, r_M) / Var(r_M)
alpha_i = r_i - {r_f + beta_i * [r_M - r_f]}

Index Model

R_i = alpha_i + beta_i * R_M + e_i
sigma_i^2 = beta_i^2 * sigma_M^2 + sigma^2(e_i)
Cov(r_i, r_j) = beta_i * beta_j * sigma_M^2

Fama-French Three-Factor

E(r_i) - r_f = b_i*[E(r_M)-r_f] + s_i*E[SMB] + h_i*E[HML]

Bond Pricing & Duration

P = Sum[C/(1+y)^t] + FV/(1+y)^T
D* = D / (1+y)
Delta_P/P = -D* * Delta_y + (1/2) * Convexity * (Delta_y)^2
Forward rate: (1+f_n) = (1+y_n)^n / (1+y_{n-1})^{n-1}

Equity Valuation

V_0 = D_1 / (k - g)          [Gordon Growth Model]
g = ROE * (1 - payout ratio)  [Sustainable growth]
P/E = (1 - b) / (k - g)      [Justified P/E]
FCFF = EBIT(1-t) + Dep - CapEx - Delta_NWC

Black-Scholes

C = S_0 * N(d_1) - X * e^(-rT) * N(d_2)
P = X * e^(-rT) * N(-d_2) - S_0 * N(-d_1)
d_1 = [ln(S_0/X) + (r + sigma^2/2)*T] / (sigma*sqrt(T))
d_2 = d_1 - sigma*sqrt(T)

Put-Call Parity

C + PV(X) = P + S

Futures Pricing

F_0 = S_0 * (1 + r_f - d)^T

Performance Evaluation

Sharpe = (r_P - r_f) / sigma_P
Treynor = (r_P - r_f) / beta_P
Jensen's alpha = r_P - [r_f + beta_P * (r_M - r_f)]
Information Ratio = alpha_P / sigma(e_P)
M^2 = r_f + (sigma_M/sigma_P)*(r_P - r_f) - r_M

Fundamental Law of Active Management

IR = IC * sqrt(BR)
S_P^2 = S_M^2 + IR^2

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Decision Frameworks

Which Valuation Method to Use?

Situation	Preferred Method
Stable dividend-paying firm	Gordon Growth DDM
High-growth firm transitioning	Two-stage or H-model DDM
Firm with negative earnings	P/S, EV/EBITDA, or FCF
Comparing across capital structures	EV/EBITDA
Real estate, asset-heavy firms	P/B, NAV
Relative valuation	P/E, P/B, P/S comparables
Intrinsic valuation	DCF (FCFF or FCFE)

Which Risk Measure?

Context	Use
Entire wealth at risk	Standard deviation (total risk)
Well-diversified portfolio addition	Beta (systematic risk)
Fixed-income portfolio	Duration and convexity
Tail risk assessment	VaR, Expected Shortfall
Performance of sub-portfolio	Information ratio, Treynor
Performance of total portfolio	Sharpe ratio, M-squared

When to Hedge?

Scenario	Strategy
Own stock, fear downside	Protective put
Own stock, willing to cap upside	Covered call
Own stock, zero-cost protection	Collar (buy put, sell call)
Portfolio manager, fear market crash	Sell index futures
Bond portfolio, fear rising rates	Sell Treasury futures, reduce duration
Foreign investment, fear currency loss	Sell currency forward
Expect large move, unknown direction	Long straddle or strangle

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Vocabulary

• Alpha: Abnormal return relative to a benchmark risk-adjusted model (e.g., CAPM)
• Arbitrage: Riskless profit from mispricing; simultaneous buying and selling of equivalent assets
• Basis risk: Risk that the hedge instrument does not move perfectly with the hedged position
• Beta: Measure of systematic risk; sensitivity of a security’s return to the market portfolio return
• Convexity: Second-order measure of bond price sensitivity to yield changes; curvature of price-yield relationship
• Correlation: Standardized measure of co-movement between two variables, ranging from -1 to +1
• Covariance: Measure of how two variables move together; Cov(X,Y) = E[(X-mu_X)(Y-mu_Y)]
• Diversification: Risk reduction through combining imperfectly correlated assets
• Duration: Weighted average time to receive a bond’s cash flows; measures first-order price sensitivity to yield
• Efficient frontier: Set of portfolios offering maximum return for each level of risk
• Efficient market: Market where security prices fully reflect available information
• Forward rate: Interest rate for a future period implied by current spot rates
• Immunization: Strategy to protect portfolio value against interest rate changes by matching asset and liability durations
• Information ratio: Alpha divided by residual standard deviation; measures risk-adjusted active return
• Market portfolio: Value-weighted portfolio of all investable assets; the tangency portfolio in CAPM
• Moral hazard: Risk that one party changes behavior when insured against loss
• Nonsystematic risk: Firm-specific, diversifiable risk that can be eliminated through portfolio construction
• Payout ratio: Fraction of earnings paid as dividends; 1 - plowback ratio
• Risk premium: Expected return in excess of the risk-free rate; compensation for bearing risk
• Security Market Line (SML): Graphical representation of CAPM; plots expected return vs. beta
• Sharpe ratio: Excess return per unit of total risk (standard deviation)
• Short selling: Selling a security you do not own by borrowing it; profit from price decline
• Spot rate: Yield to maturity on a zero-coupon bond
• Systematic risk: Market-wide, nondiversifiable risk affecting all securities
• Tracking error: Standard deviation of the difference between portfolio and benchmark returns
• Yield curve: Plot of yields to maturity vs. maturity for bonds of similar credit quality
• Yield spread: Difference in yields between two bonds, typically corporate vs. Treasury
• Zero-coupon bond: Bond paying no coupons; sold at discount, returns par value at maturity