Exam 2 Study Guide — Investments (Bodie, Kane & Mrcus)

Portfolio Expected Return Formula

E(rp) = Sum of wi x E(ri) | Weighted average of each asset's expected return, where wi = weight of asset i and weights sum to 1.

Portfolio Variance Formula (2 assets)

σp² = w1²·σ1² + w2²·σ2² + 2·w1·w2·Cov(r1,r2) | σ = standard deviation. Risk falls below weighted average of individual risks whenever the correlation between assets is less than 1. The lower the correlation, the greater the risk reduction.

Covariance Formula

Cov(r1,r2) = Sum of P(s)(r1(s) - mean1)(r2(s) - mean2) | For each scenario: probability times deviation of r1 from its mean times deviation of r2 from its mean. Sum all scenarios.

Correlation Coefficient Formula

ρ = Cov(r1,r2) / (σ1 x σ2) | Always between -1 and +1. ρ = -1: perfect negative correlation, maximum diversification (can eliminate all risk). ρ = +1: perfect positive, no diversification benefit. σ = standard deviation of each asset.

Beta Formula

Beta_i = Cov(ri, rM) / Var(rM) | Beta is the SLOPE of the Security Characteristic Line (SCL). Measures systematic risk sensitivity to the market.

CAPM / Security Market Line Formula

E(ri) = rf + Beta_i[E(rM) - rf] | rf = risk-free rate; [E(rM) - rf] = market risk premium; Beta = systematic risk. All assets in equilibrium plot on the SML.

Security Characteristic Line (SCL) Regression

Ri = alpha + beta x RM + ei | Ri = stock excess return; RM = market excess return; alpha = intercept; beta = slope; ei = firm-specific residual.

Current Yield Formula

Current Yield = Annual Coupon / Current Bond Price | Example: $80 coupon on a $950 bond = 8.42%. Does NOT account for price appreciation or depreciation to maturity.

Accrued Interest Formula

AI = (Coupon/2) x (Days since last coupon / Days in coupon period) | Example: 6% coupon, $1,000 par, 45 days since last payment: AI = $30 x (45/180) = $7.50

R-squared (Index Model)

R² = (β²·Var(rM)) / Var(ri) | Ratio of systematic variance to total variance. Tells you what fraction of a stock's return variation is explained by the market. β = beta, Var(rM) = market variance, Var(ri) = total stock variance.

Adjusted Beta Formula

Adjusted Beta = (2/3)(historical beta) + (1/3)(1.0) | Betas mean-revert toward 1.0 over time, so historical estimates are adjusted before forecasting.

Invoice Price Formula

Invoice Price = Flat (Quoted) Price + Accrued Interest | Quoted prices in financial publications exclude accrued interest. You pay the invoice (dirty) price.

APT Multi-Factor Formula

Single-factor: E(rp) = rf + βp x (Factor Risk Premium). Multi-factor: E(ri) = rf + β1(RP1) + β2(RP2) + ... Each beta measures sensitivity to one systematic risk factor. Each RP is that factor's risk premium. Derived from no-arbitrage, not equilibrium.

Bond Indenture

The legal contract between bond issuer and bondholder defining all terms, rights, and obligations of the bond. Enforced by a trustee.

Sinking Fund (Bond)

Indenture provision requiring the issuer to periodically repurchase a portion of outstanding bonds before maturity. Reduces default risk over time.

Subordination Clause

Indenture restriction stipulating that senior bondholders must be paid first in bankruptcy before subordinated (junior) debtholders receive anything.

Collateral vs Debenture

Collateral: specific asset pledged against default. Debenture: bond NOT backed by specific collateral -- bondholders are general creditors in bankruptcy.

Negative Pledge Clause

If the issuer secures any other debt with assets, existing noteholders will also be secured equally. Protects bondholders from being subordinated by new secured debt.

Callable Bond

Issuer may repurchase the bond at a specified call price during a defined call period. Premium bonds are more likely to be called when rates fall.

Convertible Bond

Bondholder may exchange the bond for a specified number of common stock shares at the holder's option.

Puttable Bond

Holder may exchange the bond for par value OR extend for a given number of years -- gives the power to the bondholder, not the issuer.

Bond Rating Scale (S&P vs Moody's)

Very High: AAA/Aaa, AA/Aa | High: A/A, BBB/Baa | Speculative: BB/Ba, B/B | Very Poor: CCC/Caa, D/C. S&P uses +/-; Moody's uses 1/2/3 modifiers.

Investment Grade vs Junk Bond

Investment Grade: BBB/Baa and ABOVE. Junk (Speculative): BB/Ba and BELOW. Junk bonds pay higher yields to compensate investors for higher default risk.

Coverage Ratios (Bond Safety)

Company earnings relative to fixed costs. Key: EBITA-to-interest coverage. Aaa firms ~12x coverage. C-rated firms below 1x. Higher = safer.

Leverage Ratios (Bond Safety)

Debt/Equity and Debt/(Debt+Equity). Aaa firms ~42.8% debt ratio. C-rated firms ~79.8%. Higher leverage = lower safety = lower rating.

Cash Flow-to-Debt (Bond Safety)

Funds from Operations / Total Debt. Aaa: 40.4%. C-rated: 5.7%. Measures ability to service debt from operating cash flow.

Bond Pricing Rule

Bond Price = PV of coupons + PV of par value. Price = Coupon x Annuity Factor(r,T) + Par x PV Factor(r,T). r = market yield, T = maturity.

Price-Yield Inverse Relationship

When market interest rates RISE, bond prices FALL -- and vice versa. This inverse relationship is the primary source of bond market interest rate risk.

Maturity and Price Sensitivity

Longer maturity bonds are MORE sensitive to interest rate changes. A 30-year bond price swings far more than a 1-year bond for the same rate move.

Premium Bond vs Discount Bond

Premium: Coupon Rate > Market Yield, Price > Par. Discount: Coupon Rate < Market Yield, Price < Par. Par: Coupon Rate = Market Yield, Price = $1,000.

Zero-Coupon Bond

Pays NO periodic coupons. Sells at a deep discount from par. All return comes from price appreciation at maturity. Example: STRIPS.

STRIPS

Separate Trading of Registered Interest and Principal of Securities. Treasury creates zero-coupon bonds by stripping interest from coupon-bearing notes.

Zero-Coupon Bond Taxation (OID)

IRS taxes built-in price appreciation as PHANTOM INCOME each year even though no cash is received. Called Original Issue Discount (OID) treatment.

Why Zeros Suit Tax-Deferred Accounts

OID creates a cash-flow mismatch: taxes owed on phantom income without actual cash received. IRAs and 401(k)s defer these taxes until withdrawal.

Covariance -- Step by Step

1) Compute mean return for each asset. 2) For each scenario: find deviations from mean. 3) Multiply the two deviations. 4) Weight by probability. 5) Sum all scenarios.

Random Walk Theory

Stock price changes are random and unpredictable. Past prices contain no information about future prices. Successive changes are independent.

Efficient Market Hypothesis (EMH)

Security prices fully reflect ALL available information. Beating the market consistently requires superior information, not just better analysis of public data.

Weak-Form EMH

Prices already reflect all PAST TRADING DATA (prices and volume). Implication: technical analysis cannot consistently earn excess returns.

Semistrong-Form EMH

Prices reflect all PUBLICLY AVAILABLE INFORMATION. Implication: fundamental analysis also cannot consistently beat the market after costs.

Strong-Form EMH

Prices reflect ALL information including private/inside information. Even insiders cannot consistently earn excess returns.

Momentum Effect (EMH Anomaly)

Well- or poorly-performing stocks tend to CONTINUE their abnormal performance over short horizons. Contradicts weak-form EMH.

Reversal Effect (EMH Anomaly)

Over LONG horizons, past winners tend to underperform and past losers tend to outperform. Suggests mean-reversion in returns.

Small-Firm Effect (EMH Anomaly)

Stocks of small firms earn abnormal risk-adjusted returns, primarily in January. One of the most documented anomalies; may reflect liquidity or risk premia.

EMH -- Selection Bias Issue

Successful strategies are rarely shared publicly. Observable results are biased toward failures, making anomalies appear more common than they are.

Beta Interpretation

Beta = 1.0: moves with market. Beta > 1.0: more volatile (aggressive). Beta < 1.0: less volatile (defensive). Beta = 0: no systematic risk.

Beta -- Slope of the SCL

Beta is the SLOPE of the Security Characteristic Line -- the regression of a stock's excess return on the market excess return. Steeper line = higher beta.

Alpha -- Intercept of the SCL

Alpha is the Y-INTERCEPT of the SCL -- the stock's expected excess return when market excess return = 0. Positive alpha: plots above SML (underpriced). Negative alpha: overpriced.

Portfolio Weighted Return -- Example

Asset A: 40% weight, 12% return. Asset B: 60% weight, 8% return. E(rp) = (0.40)(12%) + (0.60)(8%) = 4.8% + 4.8% = 9.6%.

Why Diversification Reduces Risk But Not Return

Expected return = always the simple weighted average (no reduction). Risk is reduced because the covariance term in the variance formula can be small or negative.

CAPM Assumptions

1) Rational mean-variance optimizers. 2) Single-period horizon. 3) Homogeneous expectations. 4) All assets publicly traded. 5) No taxes or transaction costs. 6) Can borrow/lend at rf.

Security Market Line (SML)

Graph of E(ri) = rf + Beta[E(rM) - rf]. All assets in CAPM equilibrium plot ON the SML. Positive-alpha stocks plot ABOVE the SML (underpriced).

CAPM vs APT

CAPM: single market factor, equilibrium-based, strict assumptions. APT: multiple factors, no-arbitrage based, less restrictive. Both: risk premium = beta x factor premium.

Market Portfolio (CAPM)

Portfolio where each security is held in proportion to its market value. In CAPM equilibrium all investors hold the market portfolio as their optimal risky portfolio.

APT -- Well-Diversified Portfolio

Nonsystematic risk is negligible in a well-diversified portfolio. Only factor (systematic) risk remains, so expected return depends only on factor exposures.

APT -- Factor Portfolio

A well-diversified portfolio with Beta = 1.0 on ONE specific factor and Beta = 0 on all others. Used to identify the risk premium for each factor.

Fama-French Three-Factor Model

Adds SMB (small-minus-big, size premium) and HML (high-minus-low book-to-market, value premium) to CAPM. Higher R-squared and lower unexplained alpha.

Arbitrage -- Definition

Exploiting relative mispricing to earn a RISKLESS profit with ZERO NET INVESTMENT. Requires: zero net investment + positive return + no risk.

Arbitrage in APT Context

If two well-diversified portfolios have identical factor exposures but different returns, buy the underpriced and short the overpriced until prices converge.

Limits to Arbitrage -- Magnitude Issue

Small mispricings may not be exploitable after transaction costs and execution risk. Market efficiency is relative, not binary.

Limits to Arbitrage -- Short-Selling Constraints

Costly or impossible to borrow shares to short-sell, preventing arbitrage from correcting overpriced securities efficiently.

Limits to Arbitrage -- Bubble Persistence

Keynes: Markets can stay irrational longer than you can stay solvent. Correct positions can lose money before mispricing corrects.

SCL -- Slope vs Intercept Summary

SLOPE = Beta (systematic risk, market sensitivity). INTERCEPT = Alpha (abnormal return when market excess return = 0). SCL equation: Ri = alpha + beta*RM + ei

Residual (ei) in Regression

Actual return minus predicted return. Represents FIRM-SPECIFIC (nonsystematic) risk -- the portion of returns not explained by the market factor.

R-squared in the SCL

R² = (β²·Var(rM)) / Var(ri) | β = beta, Var(rM) = market variance, Var(ri) = total stock variance. High R² means the stock moves closely with the market (low firm-specific risk). Low R² means firm-specific risk dominates.

Technical Analysis

Studies recurrent price patterns using historical trading data. Uses resistance/support levels, moving averages, volume. Contradicted by weak-form EMH.

Resistance Level

A price CEILING based on historical patterns -- the stock is unlikely to rise above this level. Used in technical analysis to identify sell signals.

Support Level

A price FLOOR based on historical patterns -- the stock is unlikely to fall below this level. Used in technical analysis to identify buy signals.

Fundamental Analysis

Research on determinants of stock value: earnings, dividends, future interest rates, firm risk. Assumes price = PV of expected future cash flows. Contradicted by semistrong EMH.

Active vs Passive Management

Active: assumes market inefficiency, seeks mispriced securities. Passive: consistent with semistrong EMH, holds index fund in proportion to market weights.

Exchange Rate

The rate at which domestic currency can be converted into foreign currency. Critically affects the USD return on international investments.

Exchange Rate Impact on Returns

Total USD return = local market return + currency effect. Strengthening dollar HURTS foreign returns. Weakening dollar HELPS. Currency risk is additive to market risk.

Exchange Rate -- 2022 Examples

σp² = w1²σ1² + w2²σ2² + 2·w1·w2·ρ·σ1·σ2 | The last term (2·w1·w2·ρ·σ1·σ2) is what diversification affects. Lower ρ (correlation) → smaller last term → lower portfolio variance. When ρ = -1, risk can be fully eliminated.

Correlation -- Range and Meaning

Always between -1 and +1. rho = +1: perfect positive (no diversification). rho = 0: no relationship. rho = -1: perfect negative (maximum diversification possible).

Why Diversification Works

Portfolio variance approaches the AVERAGE COVARIANCE among stocks as you add more securities. Individual (firm-specific) risks cancel out; market risk remains.

Correlation and Portfolio Variance

Lower (or negative) correlation = lower portfolio variance. The 2w1w2rhoo1*o2 term in the variance formula shrinks as correlation falls, reducing total risk.

Federal Funds Rate

The rate banks charge each other for OVERNIGHT loans of excess reserves. Market-determined but targeted by the Fed through open market operations.

Discount Rate (Fed Lending)

Rate the Fed charges commercial banks borrowing directly from the discount window. Set by the Fed. Used when banks cannot find other funding sources.

Fed Funds Rate vs Discount Rate

Federal Funds: bank-to-bank, market rate. Discount Rate: Fed-to-bank, set by Fed. Fed funds rate is the key benchmark short-term rate in the U.S.

Yield Curve Spread as Recession Signal

Spread = 10-year T-bond yield minus federal funds rate. Positive = healthy economy expected. INVERTED (negative spread) = historically reliable recession predictor. Also an LEI component.

Open Market Operations

The Fed's PRIMARY monetary policy tool. Buying Treasuries injects money (rates fall). Selling Treasuries removes money (rates rise).

LEI -- Index of Leading Economic Indicators

Published by The Conference Board monthly. 10 components predicting economic activity 6-12 months ahead. Grouped into: Labor, Orders, Real Estate, Financial Markets, Sentiment.

LEI -- Labor Components (2)

1) Average weekly hours of production workers (manufacturing). 2) Initial claims for unemployment insurance (inverted -- fewer claims is positive).

LEI -- Orders Components (3)

3) Manufacturers new orders for consumer goods. 4) ISM Index of New Orders. 5) New orders for nondefense capital goods (excluding aircraft).

LEI -- Real Estate and Financial Components (3)

6) New private housing units authorized by building permits. 7) Yield curve spread (10-yr T-bond minus fed funds rate). 8) S&P 500 stock prices.

LEI -- Money and Sentiment Components (2)

9) M2 money supply growth rate (inflation-adjusted). 10) Index of consumer expectations for business conditions.

Coincident Indicators

Economic series that move WITH the economy simultaneously: nonfarm payrolls, personal income less transfer payments, industrial production, manufacturing and trade sales.

Lagging Indicators

Economic series that trail the economy AFTER turning points: average unemployment duration, inventory-to-sales ratio, prime rate, commercial loans, CPI for services.

GDP and Stock Market

Rising GDP signals corporate earnings growth and stock prices tend to rise. GDP = market value of all goods and services produced over a time period.

Inflation and Asset Prices

Rising inflation leads the Fed to raise rates. Higher rates lower bond prices (directly) and lower stock prices (by raising the discount rate applied to future cash flows).

Demand Shock vs Supply Shock

Demand Shock: affects demand for goods/services (e.g., stimulus, confidence collapse). Supply Shock: affects production costs/capacity (e.g., oil embargo, pandemic).

Business Cycle Phases

Peak: end of expansion, start of contraction. Trough: end of recession, start of recovery. Cyclical industries most sensitive; defensive industries least sensitive.

Sector Rotation Strategy

Post-peak: shift to defensive industries. At trough: capital goods outperform. In expansion: cyclical industries outperform. Based on macroeconomic cycle forecasting.

Fiscal vs Monetary vs Supply-Side Policy

Fiscal: government spending and taxes (Congress). Monetary: Fed actions on money supply and interest rates. Supply-Side: incentives to increase productive capacity.

Current Yield -- Worked Example

Bond: $1,000 par, 8% coupon, priced at $950. Annual Coupon = $80. CY = $80 / $950 = 8.42%. Current yield only measures income return, not total return.

Current Yield vs YTM

Discount bond: YTM > CY (price appreciation adds to total return). Premium bond: YTM < CY (price depreciation reduces total return). YTM is the complete return measure.

Accrued Interest -- Worked Example

$1,000 par, 6% coupon (semi-annual payments), 45 days since last coupon. AI = $30 x (45/180) = $7.50. Invoice Price = Quoted Price + $7.50.

Clean Price vs Dirty Price

Clean (Quoted): price listed in financial media, excludes accrued interest. Dirty (Invoice): what you actually pay = Clean Price + Accrued Interest.

Bond Price Without TVM

Coupon Rate > Market Yield = Premium (Price > $1,000). Coupon Rate = Market Yield = Par ($1,000). Coupon Rate < Market Yield = Discount (Price < $1,000).