LEVEL II SCHWESER’S Quick Sheet

LEVEL II SCHWESER’S QuickSheet Critical Concepts for the 2025 CFA® Exam QUANTITATIVE METHODS MULTIPLE REGRESSION Coefficient of Determination, R2 total variation - unexplained variation R2 =     ​​      ​​ total variation explained variation _RSS ​​  SST - SSE    ​​     SST SST total variation _ _ _ _ _ _ _ _ _ ​​; _ _ _ _ _ R2 = _​​  RSS _ _ _ _ _ _ _ _ _ _ _ _ _ _ ​​ MSE = ​​  SSE  ​​; MSR = _​​  RSS SST n-k-1 k Adjusted R 2 ​​R​  ​ ​ = 1-​[​(_ ​  n - 1  ​)​×(​ 1-​R​​  2)​ ​]​​ n-k-1 2 a Covariance stationary: Mean and variance stable over time. To conclude a time series is covariance stationary: (1) plot data, (2) regress an AR model and test correlations, or (3) do Dickey-Fuller test. Unit root: Coeff on lagged dependent variable = 1. Series with unit root is not covariance stationary. First differencing will often eliminate the unit root. Autoregressive (AR) model: Specified correctly if autocorrelation of residuals not significant. ​b​  0​​ Mean-reverting level for AR(1) = _ ​​   ​​ (1 − ​b​  1​​) RMSE: Square root of average squared error. Random Walk Time Series: ​​x​  t​​ ​ = x​  t-1​​ + ε​ t​  ​​​ Akaike’s information criterion (AIC) and Schwarz’s Bayesian information criteria (BIC): Seasonality: Indicated by statistically significant lagged error term. Correct by adding lagged term. AIC is used if the goal is to have a better forecast, while BIC is used if the goal is a better goodness of fit. Lower values of each are better. SSE _ AIC = n×ln​​(​  n ​ )​ + 2​(k + 1)​​ SSE _ BIC = n×ln​​(​  n ​ )​ + ln(n)×​(k + 1)​​ ARCH: Detected by estimating ARCH(1) model: ​​​ε ​​ ̂ 2t​  ​ = ​a​  0​​  + ​a​  1​​ ​​ε ​​ ̂ 2t−1  ​ ​+ ​μ​  t​​​ F-statistic to evaluate nested models: (​ ​SSE​  R​​  - ​SSE​  U​​)​ / q  ​​ F = ______________ ​​       (​ ​SSE​  U)​​ ​ / ​(n - k - 1)​ with q and (n-k-1) degrees of freedom. F-test statistic to evaluate overall model fit: ​(​RSS​  U​​)​/ k F = _________ ​​        ​​ ​(​SSE​  U​​)​ / ​(n - k - 1)​ Model Misspecification • • • • Omitting a variable (that should be included). Variable should be transformed (for linearity). Inappropriate scaling of the variable. Incorrectly pooling data (e.g., different regimes). Regression Analysis—Problems • Heteroskedasticity: Non-constant error variance. Detect with scatter plots or Breusch–Pagan test. Correct with White-corrected standard errors. • Autocorrelation: Correlation among error terms. Detect with Durbin–Watson (DW) test or Breusch–Godfrey (BG) test. Correct using robust (Newey–West corrected) standard errors. • Multicollinearity: High correlation among Xs. (F-test significant, t-tests insig.). Detect using VIF. Correct by dropping correlated X variables. Variance inflation factor (VIF) to quantify multicollinearity: VIFj = 1 / (1 - Rj2) Logistic regression (logit) models: p ln​​ _ ​   ​ ​ = b0 + b1X1 + b2X2 + .. + ε​ ( 1-p ) odds = ​ ​​​e​​  ​y ​​​̂ P= ​ ​odds / (1 + ​ ​odds) = 1 / (1+ ​​e-​​  ​y ​​​)̂ Likelihood ratio (LR) test for logistic regressions: LR = −2 (log likelihood restricted model − log likelihood unrestricted model) TIME-SERIES ANALYSIS Linear trend model: ​​y​  t​​ = ​b​  0​​  + ​b​  1​​  t + ​ε​  t​​​ Log-linear trend model: ​ln ( ​y​  t​​  ) = ​b​  0​​  + ​b​  1​​  t + ​ε​  t​​​ Variance of ARCH series: ​​​σ̂ ​​  2t+1  ​​= ​​a ​​ ̂0​​  + ​​a ​​ ̂1​​ ​​ε ​​ ̂ 2t​  ​​ MACHINE LEARNING Supervised learning: Algorithm uses labeled training data to model relationships. Unsupervised learning: Algorithm uses unlabeled data to determine the structure of the data. Deep learning algorithms: E.g., neural networks and reinforced learning. Learn from their own prediction errors. Used in image recognition, etc. BIG DATA PROJECTS Preparing Data ECONOMICS CURRENCY EXCHANGE RATES Bid-ask spread = ask quote - bid quote Cross rates with bid-ask spreads: ​​​(_ ​  A  ​)​​  ​​  = ​​(_ ​  A ​)​​  ​​  ×  ​​(_ ​  B  ​)​​  ​​  B bid C bid C bid ​​(_ ​  A  ​)​​  ​​  = ​​(_ ​  A ​)​​  ​​  ×  ​​(_ ​  B  ​)​​  ​​​ B offer C offer C offer Currency arbitrage: “Up the bid and down the ask.” Forward premium = (forward price) - (spot price) Value of fwd currency contract prior to expiration: (F​P​  t​​  − FP)(contract size) _____________________ ​​V​  t​​ =    ​      ​​ days ​ 1 + ​R​  A( ​​​ _ ​   ​)] ​​ [ 360 Covered interest rate parity: days ​ 1 + ​R​  A( ​​​ _ ​   ​ ​ ​ ​S​  ​​ [ 360 )] 0 ________________ ​F =    ​      ​​ days ​[1 + ​R​  B​​​(_ ​   ​)​]​ 360 Uncovered interest rate parity: E(%DS)(A/B) = RA - RB Fisher relation: Rnominal = Rreal + E(inflation) International Fisher relation: Rnominal A - Rnominal B = E(inflationA) - E(inflationB) Relative purchasing power parity: High inflation rates lead to currency depreciation. %DS(A/B) = inflation(A) - inflation(B) where: %DS(A/B) = change in spot price (A/B) Normalization: Scales values between 0 and 1. ​X​  i​​  − ​X​  min​​ ​​normalized X​  i​​ = ​ _  ​​ ​X​  max​​  − ​ X​  min​​ Profit on FX carry trade = interest differential – change in the spot rate of investment currency Standardization: Centered at 0; scaled as std devs. ​X​  i​​  −μ ​​standardized X​  i​​ = ​ _ σ ​​ Dornbusch overshooting model: Restrictive monetary policy → short-term appreciation of currency, then slow depreciation to PPP value. Fit of a Machine Learning Algorithm true positives ​precision (P) = _________________________    ​       ​ (true positives +​false positives) true positives _________________________ ​recall (R) = ​        ​ (true positives +​false negatives) (true positives + true negatives) _________________________ ​accuracy = ​          ​​ (all positives and negatives) F1 score = (2 × P × R) / (P + R) true positive rate (TPR) = TP / (TP + FN) false positive rate (FPR) = FP / (FP + TN) Receiver operating characteristic (ROC): Shows tradeoff between false positives and true positives. Root mean square error (RMSE): Used when the target variable is _________________ continuous. √ n ​ ∑​​(​   ​predicted​  i​​​ – actual​  i​​  )​​2​ _________________ ​ MSE = ​    R ​  i=1    ​ ​​ n Mundell-Fleming model: Impact of monetary & fiscal policies on interest rates & exchange rates. ECONOMIC GROWTH Cobb-Douglas production function: Y = TKαL(1–α) Labor productivity: output per worker Y/L = T(K/L)α Growth accounting: growth rate in potential GDP = long-term growth rate of technology + α (long-term growth rate of capital) + (1 - α) (long-term growth rate of labor) growth rate in potential GDP = long-term growth rate of labor force + long-term growth in labor productivity Classical Growth Theory • Real GDP/person reverts to subsistence level. Neoclassical Growth Theory • Sustainable growth rate is a function of population growth, labor’s share of income, and the rate of technological advancement. • Growth rate in labor productivity driven only by improvement in technology. • Assumes diminishing returns to capital. ​g* = _ ​  θ  ​​  ​G* = _ ​  θ  ​  + ΔL​ (1 − α) (1 − α) Pension Accounting Endogenous Growth Theory • Ending PBO = beginning PBO + interest cost + current service cost + past service cost +/- actuarial losses/gains - benefits paid • Investment in capital can have constant returns. • ↑ in savings rate → permanent ↑ in growth rate. • R&D expenditures ↑ technological progress. Balance Sheet • Funded status = fair value of plan assets - PBO = balance sheet asset under GAAP and IFRS. Dividends to policyholders ratio dividends to policyholders _____________________ ​= ​        ​ ​ net premiums earned Combined ratio after dividends = combined ratio + divs to policyholders ratio Total investment return ratio = total investment income / invested assets Income Statement QUALITY OF FINANCIAL REPORTS FINANCIAL STATEMENT ANALYSIS INTERCORPORATE INVESTMENTS Accounting for Intercorporate Investments Investment in financial assets: <20% owned, no significant influence. • Amortized cost on balance sheet; interest and realized gain/loss on income statement. • FVOCI at FMV with gains/losses in equity on B/S; dividends, interest on I/S. • FVPL at FMV; dividends, interest, realized and unrealized gains/losses on I/S. Investments in associates: 20%–50% owned, significant influence. With equity method, pro rata share of the investee’s earnings increases B/S investment account, also in I/S. Dividends received decrease investment account (div. not in I/S). Business combinations: >50% owned, control. Acquisition method required under U.S. GAAP and IFRS. Goodwill not amortized, subject to annual impairment test. All assets, liabilities, revenue, and expenses of subsidiary are combined with parent, excluding intercomp. trans. If <100%, minority interest account for share not owned. Joint venture: 50% shared control. Equity method. Financial Effect of Choice of Method Equity, acquisition, & proportionate consolidation: • All three methods report same net income. • Assets, liabilities, equity, revenues, and expenses higher under acquisition vs. equity method. IFRS AND U.S. GAAP DIFFERENCES Fair value accounting, investment in associates: IFRS – Only for venture capital, mutual funds, etc. U.S. GAAP – Fair value accounting allowed for all. Goodwill: • IFRS permits either “partial goodwill” or “full goodwill.” U.S. GAAP requires full goodwill. Goodwill impairment processes: IFRS – 1 step (recoverable amount vs. carrying value). U.S. GAAP – 2 steps (identify; measure amount). Acquisition method contingent asset recognition: IFRS – Contingent assets are not recognized. U.S. GAAP – Recognized; recorded at fair value. Prior service cost: IFRS – Recognized as an expense in P&L. U.S. GAAP – Reported in OCI; amortized to P&L. Actuarial gains/losses: IFRS – Remeasurements in OCI and not amortized. U.S. GAAP – OCI, amortized with corridor approach. Dividend/interest income and interest expense: IFRS – Either operating or financing cash flows. U.S. GAAP – Classify as operating cash flow. EMPLOYEE COMPENSATION Stock Grant number of treasury shares = assumed proceeds ¸ average share price during the reporting period assumed proceeds = cash proceeds + average unrecognized share-based compensation expense • IFRS & GAAP differ on periodic pension cost. • Under GAAP, periodic pension cost in P&L = service cost + interest cost ± amortization of actuarial (gains) and losses + amortization of past service cost - expected return on plan assets. • Under IFRS, reported pension expense = service cost + past service cost + net interest expense. • Under IFRS, discount rate = expected rate of return on plan assets. GAAP interest cost = discount rate × begin. PBO Beneish model: Detects earnings manipulation using eight variables. M-score > −1.78 (i.e., less negative) → potential earnings manipulation. High-quality earnings are: 1. Sustainable: Expected to recur in future periods. 2. Adequate: To cover the company’s cost of capital. Earnings mean reversion: Faster for accruals-based earnings, especially if accruals are discretionary. IFRS net interest income (expense) = discount rate × beginning funded status Indicators of Balance Sheet Quality: Unbiased measurement; completeness; clarity of presentation. MULTINATIONAL OPERATIONS INTEGRATION OF FSA TECHNIQUES ROE decomposed (extended DuPont equation) Multinational Operations: Choice of Method For self-contained subsidiary, functional ≠ presentation currency; use current rate method: • Assets/liabilities at current rate. • Common stock at historical rate. • Income statement at average rate. • Exposure = shareholders’ equity. • Dividends at rate when paid. For integrated subsidiary, functional = presentation currency, use temporal method: • Monetary assets/liabilities at current rate. • Nonmonetary assets/liabilities at historical rate. • Sales, SGA at average rate. • COGS, depreciation at historical rate. • Exposure = monetary assets - monetary liabilities. Net asset position & depr. foreign currency = loss. Net liab. position & depr. foreign currency = gain. Original Financial Statements vs. All-Current • Pure B/S and I/S ratios unchanged. • If LC depreciating (appreciating), translated mixed ratios will be larger (smaller). Hyperinflation: GAAP vs. IFRS Hyperinfl. = cumulative inflation > 100% over 3 yrs. GAAP: use temporal method. IFRS: 1st, restate foreign curr. st. for infl. 2nd, translate with current rates. Net purch. power gain/loss reported in income. ANALYSIS OF FINANCIAL INSTITUTIONS Financial institutions differ from other companies due to systemic importance and regulated status. Tax Burden Interest Burden EBIT Margin EBT ​  × ​ _ EBIT ​  × ​ROE = _ ​  NI ​  × ​ _ EBT EBIT revenue Total Asset Turnover Financial Leverage average assets revenue ________________________  ​ _______________________    ​​ average assets ​  × ​    average equity Accruals Ratio (balance sheet approach) (NO ​A​  END​​  − NO ​A​  BEG​​  ) _____________________ ​​accruals ratio​​  BS​ =    ​      ​​ (NO ​A​  END​​  + NO ​A​  BEG​​  )  / 2 Accruals Ratio (cash flow statement approach) (NI − CFO − CFI) _____________________ ​​accruals ratio​​  CF​ =    ​     ​​ (NO ​A​  END​​  + NO ​A​  BEG​​  )  / 2 cash generated from operations (CGO) = operating cash flow + cash interest + cash taxes = EBIT + noncash charges - ↑ working capital RATIOS USED IN FINANCIAL ANALYSIS current assets  ​​ ​current ratio = ______________    ​     current liabilities cash + marktbl. securities + receivables ​quick ratio = _______________________     ​       ​​ current liabilities cash + marketable securities ​cash ratio = _______________________     ​      ​​ current liabilities cash + mkt. sec. + receivables ​defensive interval = _________________________     ​      ​​ daily cash expenditures Basel III: Minimum levels of capital and liquidity. annual sales  ​​ _______________ ​receivables turnover =    ​     average receivables CAMELS: Capital adequacy, Asset quality, Management, Earnings, Liquidity, and Sensitivity. cost of goods sold ______________ ​inventory turnover =    ​   ​​ average inventory highly liquid assets ​liquidity coverage ratio = ​  ​ __________________       ​​ expected cash outflows available stable funding ​net stable funding ratio = ​  ​ ___________________    ​​ required stable funding Insurance Company Key Ratios Underwriting loss ratio claims paid + Δ loss reserves ​= ​ _______________________         ​ ​ net premium earned Expense ratio underwriting expenses incl. commissions ________________________________ ​= ​           ​ ​ net premium written Loss and loss adjustment expense ratio loss expense + loss adjustment expense _______________________________ ​= ​          ​​ net premiums earned 365 ​days of sales outstanding = ________________ ​      ​​ receivables turnover 365 ​number of days payables = ​ __________________     ​​ payables turnover ratio revenue ​total asset turnover = _______________ ​      ​​ average total assets revenue ​fixed asset turnover = _______________ ​      ​​ average fixed assets gross profit ​gross profit margin = _ ​  revenue ​​ operating profit _ EBIT _____________ ​operating profit margin =    ​  revenue ​ = ​  net sales ​​ income ​ ​net profit margin = _ ​  net revenue ​ EBIT ​interest coverage ratio = ___________ ​      ​​ interest expense net income  ​​ _______________ ​return on assets =  ​       average total assets total assets  ​​ ​financial leverage ratio =  ​ _ total equity dividends paid ​payout ratio =  ​ ____________     ​​ net income r​ etention ratio = 1-payout ratio​ e​ arnings per share net income - preferred dividends = ​ _____________________________         ​​ average common shares outstanding ​book value per share common stockholders’ equity =  ​ ___________________________________          ​​ total number of common shares outstanding CORPORATE RESTRUCTURING Actions can include investment (to increase the size and scope), divestment (to decrease size or scope), or restructuring (to improve performance). Investment actions include equity investments, joint ventures, and acquisitions to pursue growth, synergies, or undervalued targets. Divestment actions, including sales and spin-offs, are made to increase growth or profitability or reduce risk. Restructuring actions: Cost cutting, balance sheet restructurings, reorganizations. To improve ROIC. Materiality is defined by both size and fit. Large actions are greater than 10% of EV. Fit refers to the alignment between the action and expectations. Valuation methods for corporate restructurings include comparable company, comparable transaction, and premium paid analysis. CORPORATE ISSUERS EQUITY DIVIDENDS AND SHARE REPURCHASES EQUITY VALUATION Effective tax rate = corporate tax rate + (1 − corporate tax rate) ´ (individual tax rate) Porter’s Five Forces of Industry Structure Target Payout Adjustment Model expected increase in dividends = expected target previous adjustment ​​ ​ ​  earnings​​​ × ​payout​ ​​ ​ – ​dividend ​​​ ​ × ​  factor​​ ​​ ) [( ] ratio adjustment factor = 1 / years of adjustment Dividend Coverage Ratios dividend coverage = net income / dividends FCFE coverage ratio = FCFE / (dividends + share repurchases) Share Repurchases • Share repurchase is equivalent to cash dividend, assuming equal tax treatment. • Unexpected share repurchase is good news. Rationale: (1) tax advantages, (2) share price support, (3) increase flexibility, (4) offsetting dilution by stock options, and (5) ↑ leverage. ESG CONSIDERATIONS Board of directors can be structured either as a 1-tier board of internal (executive) and external (non-executive) directors, or a 2-tier board (management board overseen by supervisory board). CEO duality: CEO is also chairperson of the board. ESG-related risk exposures: In fixed-income analysis, ESG considerations are primarily concerned with downside risk. In equity analysis, both upside opportunities and downside risk are considered. COST OF CAPITAL: ADVANCED TOPICS Grinold-Kroner model: ERP = [DY + ΔP/E + i + G - ΔS] - rf Cost of equity based on DDM: cost of equity (re) = dividend yield (DY) + capital gains yield (CGY) Fama-French model: required return of stock = rf + β1 ERP + β2 SMB + β3 HML Five-factor Fama-French extended model: required return of stock = rf + β1 ERP + β2 SMB + β3 HML + β4 RMW + β5 CMA Expanded CAPM: required return = rf + βpeer × ERP + SP + SCRP Build-up approach: required return = rf + ERP + SP + IP + SCRP • • • • • Rivalry (intra-industry) Threat of new entrants Threat of substitutes Bargaining power of suppliers Bargaining power of buyers Discounted Cash Flow (DCF) Methods Use dividend discount models (DDM) when: • Firm has dividend history. • Dividend policy is related to earnings. • Minority shareholder perspective. Use free cash flow (FCF) models when: • Firm lacks stable dividend policy. • Dividend policy not related to earnings. • FCF is related to profitability. • Controlling shareholder perspective. Use residual income (RI) when: • Firm lacks dividend history. • Expected FCF is negative. Gordon Growth Model (GGM) Assumes perpetual dividend growth rate: ​D​  0​​​(1 + g)​ _ ​D​  1​​ ​​V​  0​​ = ​ _ r − g ​ = ​  r − g ​​ Most appropriate for mature, stable firms. Limitations are: • Very sensitive to estimates of r and g. • Difficult with non-dividend stocks. • Difficult with unpredictable growth patterns (use multi-stage model). ​D​  p​​ ​Value of perpetual shares: ​V​ p​​  = ​ _ ​r​  p ​​​​ Present Value of Growth Opportunities ​E​  1​​ ​​V​  0​​ = ​ _ r ​  + PVGO​ H-Model ​[​D​  0​​  × ​(1 + ​g​  L)​​ ​]​ [​ ​D​  0​​  × H × ​(​g​  S​​  − ​g​  L​​)]​ ​ __________________ ​​V​  0​​ = ​ _____________       r − ​g​  ​​ ​  + ​    r − ​g​  ​​ ​​ L where: H = half-life (in years) of high-growth period Sustainable growth rate: b × ROE. Required Return From Gordon Growth Model r = (D1 / P0) + g FREE CASH FLOW VALUATION Free Cash Flow to Firm (FCFF) Assuming depreciation is the only NCC: • FCFF = NI + Dep + [Int × (1 - tax rate)] - FCInv - WCInv. • FCFF = [EBIT × (1 - tax rate)] + Dep - FCInv - WCInv. • FCFF = [EBITDA × (1 - tax rate)] + (Dep × tax rate) - FCInv - WCInv. • FCFF = CFO + [Int × (1 - tax rate)] - FCInv. Free Cash Flow to Equity (FCFE) • FCFE = FCFF - [Int × (1 - tax rate)] + Net borrowing. • FCFE = NI + Dep - FCInv - WCInv + Net borrowing. • FCFE = CFO - FCInv + net borrowing • FCFE = NI - [(1 - DR) × (FCInv - Dep)] - [(1 - DR) × WCInv]. (Used to forecast.) Weighted average cost of capital: WACC = (we × r) + [wd × rd × (1 − tax rate)] Single-Stage FCFF & FCFE Models DISCOUNTED DIVIDEND VALUATION L Sustainable growth rate (PRAT Model) net income ​ − dividends ​g = ​​(_____________________ ​​​        ​​)​​​ ​ net income × ​​(_ ​​​  net income  ​​ ​​​​× ​​ _ ​​​  sales  ​​ ​​​ ​ sales ) ( total assets ) total assets _________________ × ​​ ​​ ​        ​​ ​​​​ ( stockholders’ equity ) FCF ​F​  1​​ •​ ​​Value of the firm =  ​V​  0​​  = ​ _  ​​ WACC − g FCF ​E​  1​​ •​ ​​Value of equity =  ​V​  0​​  = ​ _ r − g ​​ PRICE AND EV MULTIPLES Price-to-Earnings (P/E) Ratio Problems with P/E: • If earnings < 0, P/E meaningless. • Volatile, transitory portion of earnings makes interpretation difficult. • Management discretion over accounting choices affects reported earnings. Price multiples: ​trailing P/E ​ market price per share =  ​ _________________________        ​​ EPS over previous 12 months leading P/E ​ market price per share _______________________________ =  ​         ​​ forecasted EPS over next 12 months market value of equity ​P/B ratio ​= ​ ___________________       ​ ​ book value of equity market value of equity ​P/S ratio ​= ​ ___________________        ​ ​​ total sales market value of equity ___________________ ​P/CF ratio ​= ​        ​​  ​​​ cash flow enterprise value EV/EBITDA ratio ​= ​ ______________     ​​ EBITDA 4 × most recent quarterly dividend ______________________________ ​trailing D/P ​= ​          ​ market price per share next 4 quarter forecast dividends ​leading D/P ​= ​ _____________________________________          ​​ market price per share Justified P/E ​P​  ​​ 1 − b ​justified leading P/E = _ ​  0 ​ = _ ​   ​ ​E​  1​​ r − g ​P​  ​​ ​(1 − b)​​(1 + g)​ justified trailing P/E = _ ​  0 ​ = ________ ​    r − g ​​ ​E​  0​​ Justified Dividend Yield ​D​  ​​ r − g ​​ _0 ​ = ​ _ ​​ ​P​  0​​ 1 + g Normalization Methods • Historical average EPS. • Average ROE. PEG ratio: P/E multiple to earnings growth: ​PEG ratio = _ ​  P/E g ​​ Price-to-Book (P/B) Ratio Advantages: • BV almost always > 0. • BV more stable than EPS. • Measures NAV of financial institutions. Disadvantages: • Size differences cause misleading comparisons. • Influenced by accounting choices. • BV ≠ MV due to inflation/technology. ROE − g ​justified P / B = _ ​  r − g ​​ Price-to-Sales (P/S) Ratio Advantages: • Meaningful even for distressed firms. • Sales revenue not easily manipulated. • Not as volatile as P/E ratios. • Useful for mature, cyclical, and start-up firms. Disadvantages: • High sales ≠ imply high profits and cash flows. • Does not capture cost structure differences. • Revenue recognition practices still distort sales. (​E​  0​​/S0) × (1 − b)(1 + g) ​justified P / S =  ​ ___________________    r   − g ​​ DuPont Model ​ROE = ​[_ ​  net income  ​ ​  × ​ _ ​  sales  ​ ​  × ​ _ ​  total assets  ​​​ sales ] [ total assets ] [ equity ] Price-to-Cash Flow Ratios Advantages: • Cash flow harder to manipulate than EPS. • More stable than P/E. • Mitigates earnings quality concerns. Disadvantages: • Difficult to estimate true CFO. • FCFE better but more volatile. (1 + g) Justified P/CF = ______________ ​ ​ r − g ​​   1 ​weighted harmonic mean =  ​ _ n ​w​  ​​  ​​ ​∑​ _ ​ i ​​ i=1 ​X​  i​​ Method of Comparables • Firm multiple > benchmark implies overvalued. • Firm multiple < benchmark implies undervalued. • Fundamentals that affect multiple should be similar between firm and benchmark. RESIDUAL INCOME VALUATION Residual Income Models • RI = Et - (r × Bt-1) = (ROE - r) × Bt-1 • Single-stage RI model: (​ ROE − r)​  × ​B​  0​​ ​​V​  0​​  = ​B​  0​​  + ​[______________    ​  ]​​ r − g ​ • Multistage RI valuation: V0 = B0 + (PV of interim high-growth RI) + (PV of continuing RI) Economic Value Added® • EVA = NOPAT - $WACC Market Value Added MVA = market value - total capital NOPAT = EBIT × (1 − t) = (sales − COGS − SGA − dep) × (1 − t) PRIVATE COMPANY VALUATION Private company beta estimate: ​β​  public​​ ​​β​  unlevered​​  = ​ ______________    ​​ ​[1 + (1 – t)​(_ ​  D ​)​]​ E * ​​β​  private company​​  = ​β​  unlevered​​​[1 + (1 - t*) ​​(_ ​  D ​)​​​ ​]​​ E Private Equity Valuation 1 DLOC = 1 - ​​ _________________ ​      ​ ​​ [ 1 + control premium ] Total discount = 1 - [(1 - DLOC)(1 – DLOM)]. Capitalized cash flow method: ​FCFF​  1​​ ​value of the firm = ​ _  ​​ WACC − g EBIT1(1 − T)(1 − b) ​value of the firm = ​ ______________        ​​ WACC − g ​ CFE​  1​​ F ​value of equity = ​ _ r − g ​​ FIXED INCOME TERM STRUCTURE OF INTEREST RATES Price of a T-period zero-coupon bond: 1 ​​P​  T​​ = _ ​   ​​ ​​(1 + ​S​  T​​)​​​  T​ Forward price of zero-coupon bond: 1 ​​F​  ​(j,k)​​​ = ___________ ​   ​​ ​​[1 + f (​ j,k)​]​​​  k​ Forward pricing model: ​​P​  ​(j+k)​​​ = ​Pj​​F​  ​(j,k)​​​ ​ ​P​  ​ j+k ​​​ ​​F​  ​(j,k)​​​ = _ ​  (  ​​) ​P​  j​​ Forward rate model: [1 + f (j,k)]k = [1 + S(j+k)](j+k) / (1 + Sj)j “Rolling down the yield curve”: Buying bonds with maturity > than investment horizon. Outperforms when yield curve is upward-sloping. Swap spreadt = swap ratet - treasury yieldt • Kalotay-Williams-Fabozzi (KWF) model: d ln(rt) = θtdt + σdzt BONDS WITH EMBEDDED OPTIONS Value of option embedded in a bond: Vcall = Vstraight bond - Vcallable bond Vput = Vputable bond - Vstraight bond When interest rate volatility increases: Vcall option↑, Vput option↑, Vcallable bond↓, Vputable bond↑ Upward-sloping yield curve: Results in lower call value and higher put value. When binomial tree assumed volatility increases: • Computed OAS of a callable bond decreases. • Computed OAS of a putable bond increases. B​V​ -Δy​​  − B​V​ +Δy​​ _____________ ​effective duration =    ​   ​​ 2 × B​V​ 0​​  × Δy B​V​ -Δy​​  + B​V​ +Δy​​  − (2 × B​V​  0​​  ) ________________________ ​effective convexity =     ​      ​​ B​V​ 0​​  × Δ​y​​  2​ Effective duration: • ED (callable bond) ≤ ED (straight bond). • ED (putable bond) ≤ ED (straight bond). • ED (zero-coupon) ≈ maturity of the bond. • ED fixed-rate bond < maturity of the bond. • ED of floater ≈ time (years) to next reset. One-sided durations: Callables have lower downduration; putables have lower up-duration. Value of a capped floater = straight floater value - embedded cap value Value of a floored floater = straight floater value + embedded floor value Minimum value of convertible bond = greater of conversion value or straight value Conversion value of convertible bond = market price of stock × conversion ratio Market conversion price market price of convertible bond = __________________________     ​ ​     ​ ​ conversion ratio Market conversion premium per share = market conversion price - stock’s market price Z-spread: When added to yield curve, makes PV of a bond’s cash flows = bond’s market price. Market conversion premium ratio market conversion premium per share ______________________________ =     ​ ​      ​ ​ market price of common stock TED spread: = (3-month MRR rate) - (3-month T-bill rate) Premium over straight value MRR-OIS spread: = MRR rate - “overnight indexed swap” rate Term Structure of Interest Rates Traditional theories: • Unbiased (pure) expectations theory. • Local expectations theory. • Liquidity preference theory. • Segmented markets theory. • Preferred habitat theory. Portfolio value vs. changes in the yield curve: ΔP/P ≈ −DLΔxL − DSΔxS − DCΔxC (L = level, S = steepness, C = curvature) Yield volatility: Long-term ← uncertainty regarding the real economy and inflation. Short term ← uncertainty re: monetary policy. Long-term yield volatility is generally lower than the volatility in short-term yields. ARBITRAGE-FREE VALUATION Modern term structure models: _ • Cox-Ingersoll-Ross: dr = a(b−r)dt +​σ​√r ​  dz​ • Vasicek model: dr = a(b - r)dt + σdz • Ho-Lee model: drt = θtdt + σdzt market price of convertible bond = ​ __________________________     ​      ​ ​  − 1​ ( ) straight value Callable and putable convertible bond value = straight value of bond + value of call option on stock - value of call option on bond + value of put option on bond CREDIT ANALYSIS MODELS Expected exposure: Amount a risky bond investor stands to lose before any recovery is factored in. Loss given default = loss severity × exposure Probability of default: Likelihood in a given year. recovery rate = % of $ received upon issuer default loss given default (%) = 100 – recovery rate expected loss = prob. default × loss given default present value of expected loss (PVEL) = (value of risk-free bond) – (value of credit-risky bond) Credit valuation adjustment (CVA): Sum of the present values of expected losses for each period. Credit score/rating: Ordinal rank; higher = better. Return from bond credit rating migration: D%P = -(modified duration) × (D spread) Structural models of corporate credit risk: • value of risky debt = value of risk-free debt - value of put option on the company’s assets • equity ≈ European call on company assets Reduced-form models: Do not explain why default occurs, but statistically model when default occurs. Credit spread on a risky bond = YTM of risky bond - YTM of benchmark CREDIT DEFAULT SWAPS Credit default swap (CDS): Upon credit event, protection buyer compensated by protection seller. Index CDS: Multiple borrowers, equally weighted. Default: Occurrence of a credit event. Value of interest rate swap to fixed payer days ​ ΣZ × (SF​R​  New​​  − SF​R​  Old​​  )   × ​ _ ​  × notional​ = 360 VALUING CONTINGENT CLAIMS Binomial stock tree probabilities: 1 + ​R​  f​​  − D pU = probability of “up” move = _ ​​   ​​ U−D pD = probability of “down” move = (1 - pU) Put-call parity: S0 + P0 = C0 + PV(X) synthetic call = put + stock − riskless bond synthetic put = call − stock + riskless bond Put-call parity when the stock pays dividends: P0 + S0e-δT = C0 + e-rTX Dynamic delta hedging: Common credit events in CDS agreements: Bankruptcy, failure to pay, restructuring. CDS spread: Higher for a higher probability of default and for a higher loss given default. Hazard rate = conditional probability of default Expected loss = prob. default × loss given default Upfront CDS payment (by protection buyer) = PV(protection leg) - PV(premium leg) ≈ (CDS spread - CDS coupon) × duration × NP Change in CDS value = protection buyer’s profit ≈ Δ in spread × duration × notional principal # shares hedged _______________ ​# of short call options = ​       ​​ delta of call option # shares hedged ​# of long put options = - ​ _______________       ​​ delta of put option Change in option value: ΔC ≈ (call delta × ΔS) + (½ gamma × ΔS2) ΔP ≈ (put delta × ΔS) + (½ gamma × ΔS2) Hedge ratio: + Calls: h = _______ ​​  ​C+​​  ​  - ​C​​ -  ​​​ ​S​​  ​  - ​S​​  ​ ​P​​  +​  - ​P​​  -​ Puts: h = ​​  _______  ​​ ​S​​  +​  - ​S​​  -​ Black–Scholes–Merton option valuation model: C0 = S0e-δTN(d1) - e-rTXN(d2) DERIVATIVES P0 = e-rTXN(-d2) - S0e-δTN(-d1) FORWARD COMMITMENTS Forward contract price (cost-of-carry model): ​FP = ​S​  0​​  × ​​(1 + ​R​  f​​)​​​  T​   ​S​ 0​​ = _ ​  FP T ​​ (​​ 1 + ​R​  f)​​ ​​​  ​ Price of equity forward with discrete dividends: FP(on an equity security) = (S0 - PVD)×(1+R f)T Value of forward on dividend-paying stock: ​​V​  t​​(long position) Forward on equity index w/ continuous dividend: ​ P =  ​S​  0​​  ×  ​e​​  ​(​R​  ​  ​−​δ​​  ​)​×T ​ = (​ ​S​  0​​  × ​e​​  –​δ​​  ​×T​)​  × ​e​​  ​R​  ​  ​×T​​ F where: ​𝛿​ ​​  c​​ = continuously compounded dividend yield c c δ​ = continuously compounded dividend yiewld ​S​  0​​ ​e​​  −δT​ = stock price, less PV of dividends​ ALTERNATIVE INVESTMENTS COMMODITIES AND DERIVATIVES Contango: futures prices > spot prices Backwardation: futures prices < spot prices FP = ​[​S​  t​​  − PV​D​  t​​]​  − ​ _ ​   ​ ​ [ ​(1 + ​R​  f​​  )​​  (T−t)​] c f where: c f Forward price on a coupon-paying bond: ​FP = ​(​S​  0​​  − PVC)​  ×  ​​(1 + ​R​  f​​)​​​  T​ = ​S​  0​​  × ​(1 + ​R​  f​​  )​​  T​  − FVC​ Value of a forward on a coupon-paying bond: FP ​​V​  t​​(long) = ​[​S​  t​​  − PV​C​  t​​]​  − ​ _ ​   ​ ​​ [ ​(1 + ​R​  f​​)​​  (T−t)​] Price of a bond futures contract: FP = [(full price)(1 + R f)T - AIT - FVC] full price = quoted spot price + AI0 Quoted bond futures price: ​QFP = forward price / conversion factor = ​[​(full price)(1+R​  f​​ ​)​​  T​  − ​AI​  T​​  − FVC]​​(_ ​  1  ​)​​ CF Forward rate agreement: “Price” of a 2 × 3 FRA is the implied 30-day forward rate in 60 days. Swap fixed rate: SFR(periodic) 1- ​Z​  3​​ 1   - final discount factor ​​ ​= ​ ________________     ​​ = ​ ​_____________________________ ​​    ​Z​  1​​  + ​Z​  2​​  + ​Z​  3​​ sum of discount factors where: Zn = 1/(1+Rn    ) = price of zero-coupon $1 bond Term Structure of Commodity Futures 1. Insurance theory: Contract buyers compensated for providing protection to commodity producers. Implies backwardation is normal. 2. Hedging pressure hypothesis: Like insurance theory, but includes both long hedgers ( → contango) and short hedgers (→ backwardation). 3. Theory of storage: Spot and futures prices related via storage costs and convenience yield. Futures price = spot price + storage cost - conv yield Total return on fully collateralized long futures = collateral return + price return + roll return Roll return: Positive in backwardation: long-dated contracts are cheaper than expiring contracts. price of expiring price of new ​ ​   ​​ ​  - ​ ​ ​ ​​ ( ) ( ) futures contract futures contract __________________________________ ​​roll​ ​​ =      ​​       ​​ return price of expiring futures contract The Real Estate Cycle: Four phases that reflect the health and direction of the property market: • Recovery: A trough, with little construction due to an uncertain outlook and tight credit. • Expansion: Economic growth and easing credit condition ® Higher prices and new construction. • Oversupply: Economic growth slows, leading to a property glut, and lower prices and rents. • Recession: Economic slowdown and tight credit lead to lows in construction starts and prices. Portfolio benefits of real estate: Current income, appreciation, inflation hedging, and diversification. Valuation approaches for real estate investments: • Cost approach: Value = land value + building replacement cost - depreciation - obsolescence. Useful for unusual properties lacking comparables. • Sales comparison approach: Comparable sales are adjusted for differences with subject property. • Income approach (direct capitalization): A “goingin” cap rate is applied to estimated NOI. • Income approach (DCF): NOI is estimated for the first “n” periods, then terminal value added. NCREIF appraisal index holding period return NOI − Capex + ​(Ending MV − Beginning MV)​ ​= ​ _______________________________________           ​​ Beginning Market value Transaction-based indexes: Based on: (1) repeat sales, (2) a hedonic model, or (3) public securities. Appraisal-based indexes: Lag transaction-based. Appear to have lower volatility & correlations. PUBLICLY TRADED REAL ESTATE NAV approach to REIT share valuation: estimated cash NOI ¸ assumed cap rate = estimated value of operating real estate + cash & accounts receivable - debt and other liabilities = net asset value ¸ shares outstanding = NAV/share Price-to-FFO approach to REIT share valuation: funds from operations (FFO) ¸ shares outstanding = FFO/share × sector average P/FFO multiple = NAV/share Price-to-AFFO approach to REIT share valuation: funds from operations (FFO) - non-cash rents - recurring maintenance-type capital expenditures = AFFO ¸ shares outstanding = AFFO/share × property subsector average P/AFFO multiple = NAV/share Discounted cash flow REIT share valuation: value of a REIT share = PV(dividends for years 1 through n) + PV(terminal value at the end of year n) HEDGE FUND STRATEGIES TYPES OF REAL ESTATE INVESTMENT Long/short equity: Stock picking; seeks alpha like a long-only approach, with lower std. dev. Metrics to assess real estate viability and risk: Dedicated short-selling: 60%–120% short at all times. debt service coverage (DSCR) = NOI1 ¸ debt service Short-biased strategies: 30%–60% net short, vs. long. loan-to-value (LTV) = loan amount ¸ appraised value Equity market-neutral (EMN): Profits from shortterm stock mispricing (w/leverage). Beta risk is low. equity dividend rate = (NOI – debt service) ¸ equity Merger arbitrage: Bets on a corporate takeover succeeding. High Sharpe ratio, with left-tail risk. ACTIVE PORTFOLIO MANAGEMENT Distressed securities: Seeks mispriced assets. Long biased, high illiquidity, low leverage, high returns. Portfolio return = ​​R​  P​​ = ​∑ ​​  ​w​  P,i​​​R​  i​​ Fixed-income arbitrage: E.g., yield curve and carry trades. Seeks mispriced bonds. Uses high leverage. Benchmark return = ​​R​  B​​ = ​∑ ​​  ​w​  B,i​​​R​  i​​ Convertible arbitrage: Extracts “underpriced” implied volatility. E.g., 300% long, 200% short. Opportunistic hedge fund strategies: e.g. global macro, managed futures. High liquidity & leverage. Active return = portfolio return − benchmark return n i=1 n i=1 Information ratio ​R​  P​​  − ​R​  B​​ _ ​R​  A​​ _ active return ​= ​ _ ​𝜎​  (​R​  ​​ − ​R​  ​​) ​​​ = ​  ​𝜎​  A ​​​ = ​  active risk ​​ P B Global macro: Exploits trends in global markets. ​R​  ​​  − ​R​  F​​ Portfolio Sharpe ratio = ​S​R​  P​​ = _ ​  P  ​​ STD(​R​  P​​) Managed futures: Contracts actively managed for diversification. Right skew in market turmoil. Optimal level of active risk: _________ Sharpe ratio = √ ​​ S​R​  2B ​ ​  + I​R​  2P ​ ​​​  Specialist strategies: Generate uncorrelated returns in market niches. Requires particular knowledge. Volatility traders: Exploit Δs in vol term structure. Life settlements: Buy policies w/ low surrender value, low premiums, insured likely to die soon. Multi-manager: Use strategy diversification to produce low-volatility, steady returns. Funds-of-funds: Can lack transparency, have slow tactical execution, and expose investor to netting risk. Multi-strategy funds: Diverse strategies under one roof. Better fee structure & tactical allocation vs. FoF. Conditional linear factor models: Four-factor model: Equity, currency, volatility, and credit risk. Impact of a portfolio allocation to hedge funds: ↓ std dev, ↑ Sharpe and Sortino, and ↓ max drawdown. PORTFOLIO MANAGEMENT ECONOMICS & INVESTMENT MARKETS ​ ​  t​​ u Inter-temporal rate of substitution = ​​m​  t​​ = ​ _ ​u​  0 ​​​​ marginal utility of consuming 1 unit in the future ​= ​ _______________________________________           ​​ marginal utility of current consumption of 1 unit Real risk-free rate of return 1 − ​P​  0​​ = ​R = _ ​   ​ = ​ _ ​  1  ​ ​  − 1​ ​P​  0​​ [ E( ​m​  1​​) ] Default-free, inflation-indexed, zero coupon: E(​P​  1​​) bond price = ​​P​  0​​ = _ ​   ​  + cov (​P​  1​​, ​m​  1​​)​ (1 + R) Nominal short-term interest rate (r) = real risk-free rate (R) + expected inflation (π) Nominal long-term interest rate = R + π + θ where θ = risk premium for inflation uncertainty Taylor rule: r = Rn + π + 0.5(π - π*)+ 0.5(y – y*) where: π* = central bank’s target inflation rate y* = log of central bank’s target (sustainable) output Break-even inflation rate (BEI) = yieldnon-inflation indexed bond - yieldinflation indexed bond BEI for longer maturity bonds = expected inflation (π) + infl. risk premium (θ) Credit risky bonds required return = R + π + θ + γ where γ = risk premium (spread) for credit risk Discount rate for equity = R + π + θ + γ + κ λ = equity risk premium = γ + κ κ = risk premium for equity vs. risky debt Discount rate for commercial real estate =R+π+θ+γ+κ+φ κ = terminal value risk, φ = illiquidity premium total portfolio risk: σP2 = σB2 + σA2 BACKTESTING & SIMULATION Problems in a backtest of an investment strategy: • Survivorship bias: Using data that only includes entities that have persisted until today. • Look-ahead bias: Using information that was unavailable at the time of the investment decision. • Data snooping: Model is chosen by backtesting performance (i.e. large t-stat or small p-value). Cross-validation: Model is first fitted using training data, then assessed using separate testing data. Scenario analysis: Investigates a strategy’s performance and risk under different structural regimes (e.g., high volatility vs. low volatility). Stress testing: Examines performance under the worst combinations of events and scenarios. _ Information ratio: I​ R = TC × IC × ​√BR ​​ Expected active return: E(RA) = IR × σA “Full” fundamental law of active management: _ ​E(​R​  A​​) = (TC)(IC)​√BR ​ ​σ​  A​​​ Sharpe-ratio-maximizing aggressiveness level: ​σA*= _ ​  IR  ​​ σ​  B​​​ S​R​  B​​ EXCHANGE-TRADED FUNDS ETF spreads: Positively related to cost of creation/ redemption, spread on underlying securities, risk premium for carrying trades until close of trade, and AP’s normal profit margin. Negatively related to probability of completing an offsetting trade. ETF premium (discount) % = (ETF price - NAV per share) / NAV per share USING MULTIFACTOR MODELS Arbitrage Pricing Theory E(RP) = RF + βP,1(λ1) + βP,2(λ2) + … + βP,k(λk) Expected return = risk-free rate + Σ(factor sensitivity) × (factor risk premium) Multifactor model return attribution: k ​factor return = ​∑ ​​  (β​ ​  pi​​  − ​β​  bi​​) × (​λ​  k​​)​​ i=1 Active return = factor return + security selection return Active risk squared = active factor risk + active specific risk n ​Active specific risk = ​∑ ​​  ​(​w​  pi​​  − ​w​  bi​​)​​  2​ ​​𝜎​  εi​​​​  2​​​ i=1 MEASURING & MANAGING MARKET RISK Value at risk (VaR): Minimum ($ or %) loss with a given probability over a specified period. 5% annual $VaR = [mean annual return - (1.65 × annual standard deviation)] × portfolio value Conditional VaR (CVaR): The expected loss given that the loss exceeds the VaR. Incremental VaR (IVaR): The change in VaR from a specific change in the size of a portfolio position. Marginal VaR (MVaR): ∆ in porfolio VaR for small ∆ in position. Estimates contribution to overall VaR. Historical simulation: Observations are randomly chosen from the historical dataset. Monte Carlo simulation: Statistical distribution is specified and calibrated using historical return data. Bootstrapping: Samples drawn with replacement. Useful when # of simulations >> the size of dataset. Sensitivity analysis: Overcomes the shortcomings of a traditional Monte Carlo simulation by taking into account fat tails and negative skewness. ETHICAL AND PROFESSIONAL STANDARDS I Professionalism I (A) Knowledge of the Law I (B) Independence and Objectivity I (C) Misrepresentation I (D) Misconduct I (E) Competence II Integrity of Capital Markets II (A) Material Nonpublic Information II (B) Market Manipulation III Duties to Clients III (A) Loyalty, Prudence, and Care III (B) Fair Dealing III (C) Suitability III (D) Performance Presentation III (E) Preservation of Confidentiality IV Duties to Employers IV (A) Loyalty IV (B) Additional Compensation Arrangements IV (C) Responsibilities of Supervisors V Investment Analysis, Recommendations, and Action V (A) Diligence and Reasonable Basis V (B) Communication with Clients and Prospective Clients V (C) Record Retention VI Conflicts of Interest VI (A) Avoid or Disclose Conflicts VI (B) Priority of Transactions VI (C) Referral Fees VII Responsibilities as a CFA Institute Member or CFA Candidate VII (A) Conduct as Participants in CFA Institute Programs VII (B) Reference to CFA Institute, the CFA Designation, and the CFA Program Variance for WA% fund A + WB% fund B ​​σ​ 2Portfolio  ​​= ​W​  2A ​ ​​  σ​  2A ​ ​  + ​W​  2B ​ ​​  σ​  2B ​ ​  + 2​W​  A​​ ​W​  B​​Co​v​  AB​​​ Annualized standard deviation _ =√ ​​ 250 ​​ × (daily standard deviation) % change in value vs. change in YTM = -duration (ΔY) + ½ convexity (ΔY)2 for Macaulay duration, replace ΔY by ΔY / (1+Y) ISBN: 978-1-0788-4644- 8 9 781078 846448 © 2024 Kaplan, Inc. All Rights Reserved.