CFA L2 Formulas

QUANTITATIVE METHODS QUANTITATIVE METHODS BASICSOF OFMULTIPLE MULTIPLEREGRESSION REGRESSIONAND AND BASICS UNDERLYING ASSUMPTIONS ASSUMPTIONS UNDERLYING Multiple Linear Regression Y: Dependent variable/explained variable Xi: Independent variable/explanatory variables Y! = b" + b# X#! + b$ X $! + ⋯ + b% X %! + ϵ, where b" is the intercept, b% ’s are the partial slope coefficients, and ϵ is the error term Assumptions of Multiple Linear Regression Model - Linear relationship between the dependent variable and the independent variables - Homoskedasticity (i.e., constant variance of residuals) - Independence of observations (independence of errors) - Normality of the residuals - Independence of independent variables EVALUATING REGRESSION REGRESSIONMODEL MODELFIT FITAND AND EVALUATING INTERPRETING MODEL MODELRESULTS RESULTS INTERPRETING Goodness of Fit Measures Coefficient of determination: SSR R$ = SST Adjusted-R2: n−1 +$ = 1 − . R 1 (1 − R$ ) n−k−1 SSE = 1 − 5n − k − 17 SST (n − 1) + $ will always be less than R$ because k is greater -R than 0. +$ to be negative. - It is possible for R + $ does not necessarily mean the - A high R regression is well specified. Analysis of variance (ANOVA) Sum of squares error (SSE): Unexplained variation in Y Sum of squares regression (SSR): Explained variation in Y Sum of squares total (SST): Total variation in Y SST = SSE + SSR Measures of Parsimony Akaike’s Information Criterion (AIC): SSE 1 + 2(k + 1) AIC = n × ln . n Schwarz’s Bayesian Information Criterion (SBC): SSE SBC = n × ln . 1 + ln(n) × (k + 1) n - Lower AIC/SBC is better (more parsimonious) - SBC is more conservative www.saltsolutions.com - AIC is preferred when the model is used for predictive purposes - SBC is preferred when the model is used for descriptive purposes Testing Joint Hypotheses for Coefficients t-test for Slope Coefficients t-statistic for a slope coefficient: b! − B! t= s&! - b' is the estimated value of slope coefficient - B' is the hypothesized value of slope coefficient - 𝑠𝑠&" is the standard error of the slope coefficient F-test for Joint Hypothesis F-statistic for a one-tailed test: (SSE( − SSE) )/q F= SSE) /(n − k − 1) - SSER is the sum of squared errors for the restricted model - SSEU is the sum of squared errors for the unrestricted model - q is the number of restrictions (omitted variables) - n is the number of observations - k is the number of independent variables General linear F-test: F-statistic: MSR F= MSE MODEL MISSPECIFICATION MISSPECIFICATION Principles of Model Specification - A regression model should be based on economic reasoning - A well-specified model should be parsimonious - A model should perform well when applied to out-of-sample data - The functional form for the variables should be appropriate - A model should uphold the multiple regression assumptions Misspecified Functional Form - Omitted variables - Inappropriate form of variables - Inappropriate scaling of variables - Inappropriate pooling of data Heteroskedasticity Variance of error term differs across observations - Heteroskedasticity is unconditional if error term is uncorrelated with independent variables and conditional if variance is correlated - Conditional heteroskedasticity is more problematic than the unconditional version Consequences of Heteroskedasticity - MSE is biased - Unreliable F-test and t-test - Incorrect standard error, test statistics - High Type I errors Testing Conditional Heteroskedasticity - Breusch-Pagan test (one-tailed) - Reject null hypothesis of no conditional heteroskedasticity if nR$ > critical χ$ value Correcting for Heteroskedasticity - Compute robust standard error - Generalized least squares method Serial Correlation Errors are correlated across observations Consequences of Serial Correlation Is not a Is a lagged lagged Independent value of value of variable… dependent dependent variable variable Invalid standard Yes Yes error estimates Invalid coefficient No Yes estimates - Positive (negative) serial correlation: An error in one direction increases (decreases) the chance of an error in the same direction in a subsequent observation - Inflated F-statistic due to underestimated MSE - Inflated t-statistic due to underestimated standard errors - High Type I errors Testing for Serial Correlation - Durbin-Watson (DW) test (for first-order serial correlation only) - Breusch-Godfrey (BG) test Correcting for Serial Correlation - Adjust coefficients’ standard errors Multicollinearity 2+ independent variables are highly correlated Consequences of Multicollinearity - Unreliable regression coefficient estimates - Inflated standard errors - Low t-statistics Detecting Multicollinearity - High R$ , significant F-statistic coupled with insignificant t-statistic for slope coefficients - High variation inflation factor (VIF) 1 VIF = 1 − R$* Correcting for Multicollinearity - Exclude one or more of the independent variables - Increase sample size - Use different proxies for an independent variable Copyright © 2024 Salt Solutions. 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Resale or distribution is prohibited. 1 EXTENSIONSOF OFMULTIPLE MULTIPLEREGRESSION REGRESSION EXTENSIONS TIME SERIES ANALYSIS TIME SERIES ANALYSIS Influence Analysis High-leverage points: Extreme values for independent variables Outliers: Extreme values for dependent variables Time Series Challenges - Linear regression assumptions violated - Correlated residual errors - Mean/variance of time series changes over time Leverage measure: - h!! quantifies the distance between the ith value of an independent variable and its mean - h!! > 3 K %+# , L indicates an influential observation Studentized residuals for the ith observation: t !∗ = $! e∗! n−k−1 = N s.∗ SSE(1 − h!! ) − e$! - e∗! : Residuals with the ith observation deleted - n: Number of observations - k: Number of independent variables - SSE: Sum of squared errors for the regression model - h!! : Leverage value for ith observation - s.∗ : Standard deviation of residuals Cook’s distance: e$! h!! P Q D! = k × MSE (1 − h!! )$ - Detect both potential high-leverage points and potential outliers - D! > 0.5: The observation should be investigated - D! > 1 or D! > 2Uk/n: The observation is likely influential Dummy Variables in a Multiple Linear Regression Dummy variable = 1, if true; 0, if false To distinguish among n categories, use n – 1 dummy variable. Intercept Dummies Y = b" + d" D + b# X + ϵ Slope Dummies Y = b" + b# X + d# (D × X) + ϵ Intercept and Slope Dummies Y = b" + d" D + b# X + d# (D × X) + ϵ Multiple Linear Regression with Qualitative Dependent Variables For models with qualitative dependent variables, it is often preferable to use logit model. - p is the probability that an event happens - The logistic transformation is ln K - / #0/ - ln K / #0/ L is known as the odds of an event happening / #0/ L is the natural logarithm of the odds of an event happening, which is known as log odds or logits - Coefficients are estimated using the maximum likelihood estimation (MLE) - Overall model fit is assessed using a likelihood ratio (LR) test (score closer to 0 indicates a better fit) www.saltsolutions.com Linear Trend Model y1 = b" + b# t + ε1 , t = 1, 2, … , T Log-Linear Trend Model y1 = e&%+&&1 , t = 1, 2, … , T t = 1, 2, … , T ln y1 = b" + b# t + ε1 , Autoregressive Time-Series Model A first-order autoregression, AR(1), predicts a variable (x1 ) based on its most recent value (x10# ): x1 = b" + b# x10# + ε1 A model using values for p periods, AR(p), is: x1 = b" + b# x10# + b$ x10$ + ⋯ + b/ x10/ + ε1 Covariance Stationary Assumption For inferences from AR models to be valid, it is assumed that the time series’ mean and variance are constant over time Detecting Serial Correlation of Error Residual autocorrelation t= 1⁄√T T = number of observations in time series Mean Reversion for an AR(1) Model Mean reverting level = - xd1+# = x1 when x1 = - xd1+# > x1 when x1 < - xd1+# < x1 when x1 > !! "#!" &% #0&& &% #0&& &% #0&& Root Mean Squared Error (RMSE) - In-sample forecast errors are the residuals from the time period used to estimate the parameters of the model. - Out-of-sample forecast errors are the residuals from a time period not used to fit the data. - The root mean squared error (RMSE) is the square root of the average squared error. A relatively low RMSE for out-of-sample data indicates a good fit. Random Walk and Unit Root x1 = x10# + ε1 E(ε1 ) = 0, E(ε$1 ) = σ$ , E(ε1 ε2 ) = 0, if t ≠ s Random Walk with Drift E(ε1 ) = 0 x1 = b" + x10# + ε1 , Take first difference before analyzing y1 = x1 − x10# b" ≠ 0 y1 = b " + ε1 , Unit Root Test of Nonstationarity For an AR(1) time series to be covariance stationary, the absolute value of b1 must be < 1. Dickey-Fuller Test x1 = b" + b# x10# + ε1 x1 − x10# = b" + (b# − 1)x10# + ε1 x1 − x10# = b" + g# x10# + ε1 H" : g# = 0 (has unit root) H3 : g# < 0 (does not have unit root) If the time series has a unit root, we can model the first-differenced series using an autoregressive time series. Moving Average Time-Series Model MA(q) model x1 = ε1 + θ# ε10# + θ$ ε10$ + ⋯ + θ/ x104 E(ε1 ) = 0, E(ε$1 ) = σ$ , E(ε1 ε2 ) = 0, if t ≠ s Seasonality in Time Series - Regular pattern within a year - Significant seasonal autocorrelation of error term Autoregressive Moving Average Model An ARMA(p, q) model includes p autoregressive parameters and q moving-average parameters - An AR(1) model has a “one-period memory” and all autocorrelations other than the first will be 0 Limitations of AR Models - Highly unstable parameters - Imperfect criteria for deciding p and q - Should not be used for <80 observations Autoregressive Conditional Heteroskedasticity Models ARCH(1) model ε1 ~ N(0, a" + a# ε$10# ) If a# = 0, variance of error in every period is a" . The variance is constant over time and does not depend upon past errors If a# > 0, variance of error in one period depends on how large the squared error was in the previous period. If a large error occurs in one period, variance of error in the next period will be larger: εd$1 = a" + a# εd$10# + u1 If a time-series model has ARCH(1) errors, the variance of error in period t+1 can be predicted in period t: o$1+# = ad" + ad# εd$1 σ Cointegration of Time Series Two time series are co-integrated when they have a financial or economic relationship that prevents them from diverging without bound in the long run. Cointegration Detection Engle-Granger or Dickey-Fuller test Other Issues in Time Series - Large forecast uncertainty - Need to consider uncertainty of error term and estimated parameters MACHINE LEARNING MACHINE LEARNING Used for client profiling, asset allocation, stock selection, portfolio construction, trading, etc. Supervised ML: Uses labeled data to infer patterns between inputs and outputs - Dependent variable (Y) is the target and independent variables (X) are features - Can be used for regression (linear and non-linear) and classification problems Copyright © 2024 Salt Solutions. All Rights Reserved. Personal copies permitted. Resale or distribution is prohibited. 2 Unsupervised ML: Finds patterns within unlabeled data; there is no dependent variable - Can be used for dimension reduction and clustering problems Deep Learning: Sophisticated algorithms for tasks such as image classification, face recognition, and natural language processing Reinforcement Learning (RL): An algorithm learns from the data that it generates Neural networks: Highly flexible ML algorithms used for classification, regression, deep learning, and reinforcement learning ML Methods for Different Types of Variables Supervised Unsupervised Variables ML ML Regression Dimensionality - Linear, Reduction LASSO - PCA Continuous - Logistic Clustering - CART - K-Means - Random - Hierarchical Forest Classification Dimensionality - Logit Reduction - SVM - PCA Categorical - KNN Clustering - CART - K-Means - Hierarchical Neural Neural Continuous networks, networks, Deep Deep Learning, or RL Categorical Learning, RL Training an ML Model: Sampling Training an ML model requires a dataset to be divided into three non-overlapping samples: 1. Training sample: In-sample data used to find relationships 2. Validation sample: In-sample data used to validate relationships found in training sample 3. Test sample: Out-of-sample data used to test the model’s predictive powers Overfit models explain the training data well but do not generalize to the out-of-sample data. K-fold cross-validation can be used to prevent overfitting. Training an ML Model: Errors Bias error: Does not explain training data well (underfit); more likely for linear functions Variance error: Model performs differently with out-of-sample data because it has incorporated noise from training data (overfit); more likely for non-linear functions Base error: Unavoidable errors due to randomness The trade-off between bias error and variance error can be shown on a fitting curve www.saltsolutions.com Model complexity Low High Bias error Variance error Higher Lower Lower Higher Supervised ML Methods Penalized regression/Regularization: Seeks to reduce the risk of overfitting by imposing a penalty on additional features; Least Absolute Shrinkage and Selection Operator (LASSO) uses a hyperparameter, λ, as a penalty Support Vector Machine (SVM): Linear classifier model used for binary classification, regression, and outlier detection; Soft margin classification is a non-linear alternative to SVM K-Nearest Neighbor (KNN): Non-parametric method typically used for classification (e.g., credit rating prediction), but also used for regression Classification and Regression Tree (CART): Produces a tree with a root node, decision nodes, and terminal nodes; Iterative structure is used to find relationships in non-linear data, but it is a black box method Ensemble learning: Use multiple models to reduce error rate relative to relying on one model. Examples include majority-vote classifier, bootstrap aggregating, and random forest. Unsupervised ML Methods Principal Components Analysis (PCA): Features are grouped together to reduce the number of independent variables in a model. It significantly reduces model complexity but is a black box method. Clustering: K-means clustering, hierarchical clustering, and dendrograms are used to organize observations into groups. DATA PROJECTS BIG DATA PROJECTS 4Vs of Big Data 1. Volume 2. Variety 3. Velocity 4. Veracity ML Model Building Steps 1. Conceptualization 2. Data Collection/Curation 3. Data Preparation and Wrangling - Data Cleansing - Data Preprocessing 4. Data Exploration - Exploratory Data Analysis - Feature Selection - Feature Engineering 5. Model Training - Method Selection - Performance Evaluation - Tuning Text ML Model Building 1. Text problem formulation 2. Text curation 3. Text preparation and wrangling 4. Text exploration Errors Addressed with Data Cleansing Incompleteness error: Data not present Invalidity error: Outside meaningful range Inaccuracy error: Not a measure of true value Inconsistency error: Conflicts between data points Non-uniformity error: Multiple formats used Duplication error: Duplicate observations present Data Wrangling Methods Feature extraction: Creating a new variable from an existing variable to improve analysis Aggregation: Combining similar variables Filtration: Removing irrelevant rows Selection: Removing irrelevant columns Conversion: Making adjustments to increase relevance Addressing Outlier Data Trimming: Removing the top/bottom X% Winsorization: Replacing extreme high/low observations with maximum/minimum values Normalization: X ! − X 7!, X ,56738!9.: = X 73; − X 7!, Standardization: X! − µ X 213,:36:!9.: = σ Text Wrangling Methods 1. Lowercasing 2. Stop words 3. Stemming 4. Lemmatization Model Performance Evaluation Actual Actual training training label - 1 label - 0 True False Predicted positive positive result - 1 (TP) (FP) False True Predicted negative negative result - 0 (FN) (Type (TN) II Error) Precision (P) = TP TP + FP Accuracy (A) = TP + TN TP + FP + TN + FN Recall (R) = F1 score = TP TP + FN 2 ×P×R P+R FP TN + FP TP True Positive Rate (TPR) = TP + FN False Positive Rate (FPR) = Copyright © 2024 Salt Solutions. All Rights Reserved. Personal copies permitted. Resale or distribution is prohibited. 3 Receiver Operator Characteristic (ROC) technique plots FPR on x-axis and TPR on y-axis. Root Mean Squared Error (RMSE) is calculated as the square root of the sum of mean squared errors. , RMSE = st !<# (Predicted! − Actual!)$ n A lower RMSE indicates potentially better model performance if historical relationships hold. ECONOMICS ECONOMICS CURRENCYEXCHANGE EXCHANGERATES RATES CURRENCY Factors Influencing Bid/Ask Spreads Currency pair: Wider for less liquid currencies Time of day: Wider when NY/London closed Market conditions: Wider when more volatile Contract term: Wider for longer-term forward contracts due to lower liquidity and increased exposure to credit risk and interest rate risk Low capital mobility Fiscal Policy Expansionary Restrictive Monetary Policy Expansionary DC ↓ Ambiguous Restrictive Ambiguous DC ↑ Currency Crisis Warning Signs - Recent liberalization of capital markets - Large foreign capital inflows, especially s/t funds - Banking crises, either just before or concurrent - Fixed or partially fixed exchange rates - Sudden, sharp decline in FX reserves - Recent spike in domestic currency value - Deteriorating terms of trade - Money supply growing faster than bank reserves - Recent high inflation ECONOMIC ECONOMICGROWTH GROWTH Covered Interest Rate Parity Days 1 + i= K L 360 w F=⁄: = S=⁄: u Days L 1 + i: K 360 Factors Affecting Growth: Developing Countries - Low rates of savings and investment - Poorly developed financial markets - Weak legal systems and failure to enforce laws - Lack of property rights and political stability - Poor public education and health services - Excessive taxes and regulations - Restrictions on international trade/capital flows Estimated Future Spot Rate F=⁄: − S=⁄: = %∆S=.⁄: = i= − i: S=⁄: Cobb-Douglas Production Function F (K, L) = K ? L#0? Uncovered Interest Rate Parity %∆S=.⁄: = i= − i: Potential Growth and Stock Market Returns 𝑃𝑃 𝐸𝐸 + %∆ %∆𝑃𝑃 = %∆𝐺𝐺𝐺𝐺𝐺𝐺 + %∆ 𝐸𝐸 𝐺𝐺𝐺𝐺𝐺𝐺 Absolute Version of PPP P= = S=⁄: × P: S=⁄: = P= ⁄ P: Capital Deepening vs. Technological Progress The Fisher Effect and Real Interest Rate Parity i = r + π. (i= − i: ) = (r= − r: ) + (π.= − π.: ) (r= − r: ) = (i= − i: ) − (π.= − π.: ) Real Interest Rate Parity (r= − r: ) = 0 International Fisher Effect (i= − i: ) = (π.= − π.: ) Expansionary Restrictive Monetary Policy Expansionary Restrictive DC ↓ Ambiguous Ambiguous www.saltsolutions.com ∆y ∆A ∆k = + α. 1 y A k ∆k Y = s. 1− δ − n k K ∆y θ Growth rate of output per capita = = y 1−α θ ∆Y = +n Growth rate of output = 1−α Y Implications: - Capital accumulation, capital deepening, and an increase in savings rate can only temporarily increase growth, but technological improvements can have a permanent impact - Per capita income growth will converge across countries Endogenous Growth Theory - Output per worker is proportional to stock of capital per worker (k . ) - c is constant marginal product of capital y. = f(k . ) = ck . ∆y.⁄y. = ∆k . ⁄k . = sc − δ − n - A higher savings rate can permanently increase the potential GDP growth rate Types of Regulation - Statutes enacted by legislatures - Administrative regulations for agencies - Judicial laws established by legal rulings Ex-ante Version of PPP %∆S=.⁄: ≅ π.= − π.: Fiscal Policy Neoclassical Model - Based on Cobb-Douglas production function - Both capital (K) and labor (L) are subject to diminishing marginal productivity - In the steady state, the output-to-capital ratio is constant because they grow at the same rate - In the long run, output per capita is driven by: 1. Savings/investment rate 2. Rate of technological change 3. Population growth ECONOMICS OF ECONOMICS OFREGULATION REGULATION Relative Version of PPP %∆S=⁄: ≅ π= − π: Mundell-Fleming Model High capital mobility Classical model failed because: 1. Population growth slowed as incomes rose 2. Diminishing marginal returns to labor input were more than offset by technological progress DC ↑ Growth Accounting ∆Y ∆A ∆K ∆L = + α . 1 + (1 − α) . 1 Y A K L Growth rate in potential GDP = L/T growth rate of labor force + L/T growth rate in labor productivity Labor Supply Inputs - Population growth - Labor force participation - Net migration - Average hours worked Classical Model - Labor productivity increases population growth - Population growth accelerates as per capita incomes increase - Diminishing marginal returns to labor input will lead to decline in per capita income Classification of Regulators - Independent regulators: Granted the ability to make regulations by government - Self-regulatory bodies: Private organizations that regulate members, typically industry peers - Self-regulatory organizations: Independent industry bodies that have been granted law enforcement powers - Standard-setting bodies: Establish rules but lack any enforcement powers (e.g., IFRS) Regulatory Interdependencies - Regulatory capture: Businesses use relationship with regulators to serve their interest - Regulatory competition: Regulators from different jurisdictions compete to attract certain entities - Regulatory arbitrage: Businesses exploit differences between economic substance and regulatory interpretation Copyright © 2024 Salt Solutions. 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Resale or distribution is prohibited. 4 Regulatory Tools - Price mechanisms (e.g., taxes, subsidies) - Mandates and restrictions - Provision of public goods - Public financing of private projects Cost-Benefit Analysis of Regulation Regulatory burden: Private costs of regulation, both direct and indirect Net burden: Private costs less private benefits FINANCIAL STATEMENT ANALYSIS FINANCIAL STATEMENT ANALYSIS INTERCORPORATEINVESTMENTS INVESTMENTS INTERCORPORATE Classification - Financial assets (<20%): Buyer has no significant control over the investee - Associates (20% - 50%): Buyer has significant influence but not control - Joint venture: Entity is operated by companies that share control - Business combinations (>50%): Buyer has control Investments in Financial Assets - Under IFRS 9, all financial assets are initially measured at fair value (cost basis at acquisition) - In subsequent periods, financial assets may be measured at either amortized cost or fair value - To be measured at amortized cost, financial assets must meet the following criteria: o Held to collect contractual cash flows o Cash flows may only be principal and interest - Financial assets that fail to meet both criteria must be measured at either fair value through profit or loss (FVPL) or fair value through other comprehensive income (FVOCI) - Securities are classified as follows: o Debt securities: Carried at amortized cost if held to maturity, but must be carried at fair value if it is possible that they may be sold o Equities: Can be held at FVPL or FVOCI o Derivatives: Must be carried at FVPL unless they are being used as hedging instruments Investments in Associates and Joint Ventures - Equity method (one-line consolidation): o Initial investment is recorded at cost on the balance sheet as a non-current asset o Carrying amount is adjusted upward to reflect a proportionate share of earnings o Dividends received are treated as a return of capital and reduce the carrying amount o If carrying value falls to 0, equity method is discontinued and no further losses recorded - Profits from transactions with associates cannot be realized until the products are sold/used Classifying Business Combination - Merger: A + B = A - Acquisition: A+ B = (A + B) - Consolidation: A + B = C www.saltsolutions.com Acquisition Method - The fair value of the consideration given by the acquiring company is used - Direct costs of the business combination are expensed as incurred - IFRS: Full or partial goodwill US GAAP: Full goodwill only Partial goodwill = Fair value of consideration given − Acquirer @ s shares of the fair value of A/L Full goodwill = Fair value of acquired@ s entity − Fair value of the entity @ s A/L - Non-controlling interest (NCI) is the portion of subsidiary’s equity held by third parties o Full goodwill: NCI is measured at fair value o Partial goodwill: NCI is based on the proportionate share of net identifiable assets EMPLOYEE COMPENSATION: COMPENSATION:POSTEMPLOYMENT AND SHARE-BASED POST-EMPLOYMENT AND SHARE-BASED Financial Reporting for Share-Based Compensation Restricted stock: - Total expense is allocated evenly over each year of vesting period - Share-based compensation reserve is recorded as owner’s equity in the balance sheet o Transfers to common equity on settlement date - No impact on statement of cash flows Stock options: - Total expense is allocated evenly over each year of vesting period - Share-based compensation reserve is recorded as owner’s equity in the balance sheet o Balance is reduced as the options are exercised - Financing cash inflow upon option exercise Share-Based Compensation Tax and Share Count Effects, Note Disclosures Share price change after grant date Tax impact IFRS US GAAP Increase Windfall Gain in equity Reduced income tax expense Decrease Shortfall Diluted shares outstanding Loss in equity Higher income tax expense = Basic shares outstanding + Shares from conversion or exercise − Share-Based Compensation and Financial Statement Modeling Basic shares outstanding (beginning of period) + RSUs vested during period + Options exercised during period + New shares issued + Share repurchases = Basic shares outstanding (end of period) Post-Employment Benefits: Disclosures and Modeling Defined contribution (DC): Employer’s obligation is limited to periodic contribution; Future benefits depend on investment performance Defined benefit (DB): Firm makes periodic payments to employee after retirement; Employer’s contributions may vary depending on assumptions and the performance of plan assets DB plan assumptions: - Assuming a higher discount rate reduces the pension liability and service costs - Assuming a higher rate of compensation growth increases liability and service costs - Assuming a higher expected return on assets reduces the pension expense under US GAAP but not IFRS and has no impact on the value of assets Financial Reporting for DB Plans Funded status = Fair value of plan assets − PV of defined benefit obligation Beginning fair value of plan assets + Actual return on plan assets + Employer contributions − Benefits paid = Ending fair value of plan assets Beginning pension obligation + Current service cost + Past service cost + Interest expense + Actuarial losses − Actuarial gains − Benefits paid = Ending pension obligation Net pension asset/liability: Overfunded - Plan assets exceed the pension obligation - Sponsor reports net pension asset Underfunded - Pension obligation exceeds plan assets - Sponsor reports net pension liability Periodic pension cost = (Ending funded status − Beginning funded status) − Employer contributions Assumed proceeds of conversion or exercise Average share price for the reporting period = (ITM options) × (Strike price) + Average unrecognized share-based compensation expense Copyright © 2024 Salt Solutions. 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Resale or distribution is prohibited. 5 Components of pension expense (IFRS): - Current service cost (P&L) - Past service cost (P&L) - Net interest expense (P&L) o Net liability × Interest rate - Net return on plan assets (OCI, not amortized) o Actual return − (Assets × Interest rate) - Actuarial losses (OCI, not amortized) Components of pension expense (US GAAP): - Current service cost (P&L) - Past service cost (OCI, then amortized) - Interest expense (P&L) o Liability × Interest rate - Expected return (P&L) o Assets × Expected return - Net return on plan assets (OCI, then amortized) o Actual return − (Assets × Expected return) - Actuarial losses (OCI, then amortized) MULTINATIONAL OPERATIONS OPERATIONS Presentation and Functional Currencies - Presentation currency: Currency in which the company presents its financial statements - Functional currency: Currency in which the company conducts its primary activity - Local currency: Used within the country in which the company operates - Often, functional currency of subsidiary ≠ functional and presentation currency of parent Remeasurement/Translation Methods Current rate method - Use when the subsidiary’s functional currency is different from the parent’s functional currency - FX gain/loss reported in shareholders’ equity as part of cumulative translation adjustment (CTA) - Exposure is net assets (assets minus liabilities) Temporal method - Use when the subsidiary’s functional currency is the same as the parent’s functional currency - Remeasurement gain/loss reported in income statement - Exposure is net monetary assets (monetary assets minus monetary liabilities) Exchange Rate for Each Line Item Monetary assets Monetary liabilities Nonmonetary assets Nonmonetary liabilities Common stock Revenues/ Expenses COGS Depreciation Current Rate Current Temporal Current Current Historical Current Current Historical Average Average Average www.saltsolutions.com Current Historical Historical Average Historical Historical Translation Adjustments Balance Foreign Currency Sheet Strengthens Weakens Exposure Positive Negative Net asset translation translation adjustment adjustment Negative Positive Net translation translation liability adjustment adjustment Effect of Translation Method on Ratios - Pure income statement and pure balance sheet ratios are unaffected by the current rate method - Under both the current rate and temporal methods, ratios using both income statement and balance sheets figures will be different from the same ratios calculated using local currency statements before translation Examples: 1. Receivables turnover (sales/receivables) is the same under both current and temporal methods - Sales are translated at the average exchange rate under both - Receivables are translated at the current exchange rate under both 2. Current ratio (current assets/current liabilities) will be different under the two methods - Inventory is translated at the current exchange rate under the current method, but at the historical exchange rate under the temporal method - If the subsidiary’s currency appreciates relative to the parent, the current ratio will be higher under the current rate method than the temporal method Subsidiaries in Hyperinflationary Economies Under IFRS - Restate subsidiary’s local currency financial statements for local inflation - Translate inflation-restated foreign currency financial statements into the parent’s presentation currency using the current FX rate Under U.S. GAAP - Use the temporal method to translate the subsidiary’s local currency financial statements - Include the resulting translation adjustment as a gain or loss in determining net income ANALYSIS OF OF FINANCIAL FINANCIALINSTITUTIONS INSTITUTIONS Key Components of Basel III: - Minimum capital requirements: Based on riskweighted assets - Minimum liquidity: Enough to handle a 30-day liquidity stress scenario - Stable funding: To cover needs over a one-year horizon; based on the length of the deposits and the type of depositor Minimum Capital Requirements under Basel III: - Common Equity Tier 1 capital ≥ 4.5% - Total Tier 1 capital ≥ 6.0% - Tier 1 plus Tier 2 capital ≥ 8.0% The CAMELS Approach 1. Capital adequacy 2. Asset quality 3. Management capabilities 4. Earnings sufficiency 5. Liquidity position 6. Sensitivity to market risk Relevant Factors Not Covered by CAMELS Banking-Specific Analytical Considerations - Government support - Government ownership - Mission of banking entity - Corporate culture Considerations relevant to any company - Competitive environment - Off-balance-sheet items - Segment information - Currency exposure - Risk factors - Basel III disclosures Analyzing Property and Casualty Insurers Loss and loss adjustment expense ratio: Loss expense + Loss adjustment expense Net premiums earned Underwriting expense ratio: Underwriting expense Net premiums written Combined ratio: Loss and loss adjustment expense ratio + Underwriting expense ratio Dividends to policyholders ratio: Dividends to policyholders Net premiums earned Combined ratio after dividends: Combined ratio + Dividends to policyholders ratio EVALUATING REPORTS EVALUATINGQUALITY QUALITYOF OFFINANCIAL FINANCIAL REPORTS Quality Spectrum of Financial Reports 1. GAAP-compliant; decision-useful, high-quality earnings (adequate return/sustainable) 2. GAAP-compliant; decision-useful, low-quality earnings (inadequate return/not sustainable) 3. GAAP-compliant; not decision-useful 4. GAAP-compliant; earnings management 5. Non-compliant accounting 6. Fictitious transactions Copyright © 2024 Salt Solutions. All Rights Reserved. Personal copies permitted. Resale or distribution is prohibited. 6 Beneish Model M-scores: Higher values indicate a greater likelihood of earnings manipulation - Threshold M-scores: o −1.78 (3.8% likelihood of manipulation) o −1.49 (6.8% likelihood of manipulation) - Days Sales in Receivables Index (DSRI): Large increase could mean revenue inflation - Gross Margin Index (GMI): >1 means gross margin has decreased, possible pressure to manipulate - Asset Quality Index (AQI): Increase could indicate excessive capitalization of expenses - Sales Growth Index (SGI): Rapid sales growth can create pressure to manipulate earnings - Depreciation Index (DEPI): >1 means depreciation rate has decreased, indicating manipulation - SG&A Expenses Index (SGAI): >1 means increasing SG&A expense, might encourage manipulation - Accruals: More accruals indicate manipulation - Leverage index (LEVI): >1 shows increasing debtto-asset ratio, greater pressure to manipulate In the Beneish model, SGAI and LEVI are negatively correlated with the M-score; other variables are positively correlated Measures of Earnings Persistence and Accruals - Earnings forecast should not include nonrecurring items - Cash components are more persistent than accrual components - Earnings with significant accruals will experience a faster mean reversion - A company that consistently performs slightly better than benchmark should be scrutinized - Regulatory enforcement actions and restatements of previous financial statements are red flags Altman model: A lower Z-score indicates a higher probability of bankruptcy Cash Flow Quality - A startup company might be expected to have negative operating and investing cash flows, funded by positive financing cash flows from equity issues or borrowing - For established companies, high-quality operating cash flow (OCF) should be: - Positive - Derived from sustainable sources - Enough for capex, dividends, and debt service - Stable relative to peers Balance Sheet Quality Indicators High financial reporting quality requires: - completeness - unbiased measurement - clear presentation A high-quality balance sheet requires: - optimal amount of leverage - adequate liquidity - economically successful asset allocation www.saltsolutions.com Sources of Information about Risk - Financial statements, including notes - Audit opinion or discretionary change in auditor - Management commentary - Regulatory disclosures - Financial press INTEGRATION INTEGRATION OF OF FINANCIAL FINANCIALSTATEMENT STATEMENT ANALYSIS TECHNIQUES ANALYSIS TECHNIQUES Framework for Analysis 1. Define purpose for analysis 2. Collect input data 3. Process data 4. Analyze/interpret processed data 5. Develop and communicate conclusions 6. Follow up Sources of Earnings and ROE Net Income Sales Assets ROE = × × Sales Assets Equity ROE = ROE = Net Profit Asset × × Leverage Margin Turnover NI EBT EBIT Sales Assets × × × × EBT EBIT Sales Assets Equity Accruals and Earnings Quality Balance sheet approach AccrualsAB = NOA.,: − NOA&.C Accruals RatioAB = NOA.,: − NOA&.C ìNOA.,: + NOA&.C î⁄2 Cash flow statement approach AccrualsDE = NI − CFO − CFI Accruals RatioDE = NI − CFO − CFI ìNOA.,: + NOA&.C î⁄2 FINANCIAL FINANCIAL STATEMENT STATEMENT MODELING MODELING Financial Modeling: Overview Income Statement Modeling: Revenue - Top-down approach - Bottom-up approach - Hybrid approach Income Statement Modeling: Operating Costs Cost analysis should match revenue analysis - Cost of goods sold (COGS) - Selling, general, and administrative expenses (SG&A) Income Statement Modeling: Non-Operating Costs Financing expenses are affected by debt level and interest rate Tax rates: - Statutory tax rate - Effective tax rate - Cash tax rate CORPORATE ISSUERS CORPORATE ISSUERS ANALYSIS DIVIDENDS AND AND SHARE ANALYSIS OF OF DIVIDENDS SHARE REPURCHASES REPURCHASES Stock Dividends - In lieu of paying a cash dividend, companies may distribute additional shares. - Such a distribution increases the number of shares outstanding without affecting the company's total market value. - The stock dividend does not affect the company’s balance sheet or income statement, so liquidity and financial leverage ratios are unchanged. Stock Splits - They have no economic effect on the company and should not impact shareholders’ wealth. - Reverse stock splits reduce the number of shares outstanding, but they still have no economic impact on the company or shareholder. Dividend Theories - Dividends are irrelevant: MM Propositions - Bird-in-the-hand argument: Investors prefer cash dividends over unrealized capital gains - Signalling: Managers want to increase dividends as a signal of strength to investors - Agency cost: Paying dividends to owners limits managers’ ability to fund negative NPV projects Factors Affecting Dividend Policy - Investment opportunities: Company with more (less) investment opportunities will pay out less (more) in dividends - Expected volatility of future earnings: Companies with greater earnings volatility are less likely to increase dividends; Increasing dividends increases chances of not maintaining the dividend - Financial flexibility: Companies seeking more flexibility are less likely to initiate or increase dividends - Tax considerations: Investors consider dividends on an after-tax basis - Flotation costs: Make using newly issued stock more expensive than internally generated funds; Smaller companies face higher flotation costs - Contractual and legal restrictions: Bond indentures, obligations to preferred shareholders, rules against capital impairment, etc. Tax System and Dividend Policy - Double Taxation: Corporate earnings are first taxed at the corporate level and then again at the shareholder level when distributed as dividends. ETR = Corp. tax rate +(1 − Corp. tax rate)(Ind. tax rate) Balance Sheet and Cash Flow Statement Modeling Return on invested capital (ROIC) Net operating profit less adjusted taxes (NOPLAT) = Invested capital Copyright © 2024 Salt Solutions. All Rights Reserved. Personal copies permitted. Resale or distribution is prohibited. 7 - Split-rate: Corporate earnings distributed as dividends are taxed at a lower rate than the earnings retained. Dividends are taxed as ordinary income for investors. The effective tax rate on dividends is calculated with the same formula used under a double taxation system, but the lower corporate tax rate is used for dividend distributions ETR = Corp. tax rateF +(1 − Corp. tax rateF )(Ind. tax rate) - Imputation: Corporate earnings distributed as dividends are taxed at the shareholder’s tax rate. Individual shareholders receive a tax credit if their marginal tax rate is less than the corporate tax rate. If their marginal tax rate is higher, they only pay tax based on the difference between their marginal tax rate and the corporate rate. Payout Policies Stable Dividend Policy D# = D" + (E# × PR G − D" ) × A where E# is expected earnings PR G is target payout ratio A is an adjustment factor (1/Years to target ratio) Constant Dividend Payout Dividend as a constant share of earnings Residual Dividend Payout Pay out earnings remaining after funding all positive NPV capital projects consistent with its target capital structure. D = max u0, Earnings − Capital Equity % w × Budget in capital structure Share Repurchase Methods - Buy in the open market - Buy back a fixed number of shares at a fixed price - Dutch auction - Repurchase by direct negotiation Financial Statement Effects of Repurchases Changes in Earnings per Share - If the net income stays the same, the EPS will increase after a share repurchase because there are fewer shares outstanding. - Share repurchases made with borrowed funds: If the earnings yield is greater (lower) than the after-tax cost of the funds, the EPS will increase (decrease). - An increase in EPS does not necessarily imply an increase in shareholders’ wealth. Changes in Book Value per Share - If the market price per share is greater (less) than its book value per share, a stock repurchase will decrease (increase) the book value per share. Valuation Equivalence - Ignoring the tax and information effects, cash dividends and share repurchases should have the same effect on shareholder wealth. www.saltsolutions.com - However, if shares are repurchased at a premium, wealth will be transferred from remaining shareholders to the seller. Advantages of Share Repurchases vs. Dividends - Potential tax advantages - Signaling - Managerial flexibility - Offsets dilution from employee stock options - Adjusting capital structure - Increasing EPS Analysis of Dividend Safety All else being equal, the risk of a dividend cut or omission increases when we have: - A higher dividend payout ratio (dividends/net income) - A lower dividend coverage ratio (net income/dividends) FCFE FCFE coverage = Dividends + Share repurchases ratio - If the ratio is equal to 1, the company is distributing all available cash to shareholders. - If the ratio is significantly greater than 1, the company is keeping some earnings to enhance liquidity. - If the ratio is significantly less than 1, the company is borrowing cash to pay dividends, thereby decreasing liquidity. This is unsustainable because the company is paying out more than it can afford. ESGCONSIDERATIONS CONSIDERATIONSINININVESTMENT ESG INVESTMENT ANALYSIS ANALYSIS Ownership Structure - Dispersed: Many small minority shareholders - Concentrated: One majority shareholder or a few shareholders with large minority positions - Horizontal: Cross-holding share arrangements - Vertical: Controlling interest in two or more holding companies with their own subsidiaries - Dual-class shares: Disproportionate voting rights for certain classes of shares Conflicts with Different Ownership Structures - Dispersed ownership, dispersed voting power: Leads to principal-agent problem where strong managers take advantage of relatively weak shareholders - Concentrated ownership, concentrated voting power: Leads to the principal-principal problem where a captive board makes decisions to the detriment of minority owners - Dispersed ownership, concentrated voting power: Also leads to the principal-principal problem, but for the benefit of large minority owners holding dual-class shares - Concentrated ownership, dispersed voting power: Can occur if voting caps are used Board Composition One-tier: Executive and non-executive directors Two-tier: Independent supervisory board oversees management board CEO duality: The same individual serves as both CEO and chair of a one-tier board, raising concerns about a lack of independent oversight Special voting arrangements: May require directors to secure dual majorities among both controlling and minority shareholders Stewardship codes: Encourage asset managers to increase engagement in corporate governance Board Policies and Practices - Board of Directors Structure - Board Independence - Board Committees - Board Skill and Experience - Board Composition - Other Considerations in Board Evaluation Executive Remuneration - Companies may have a say on pay provision that allows shareholders to vote or at least provide feedback on compensation. - A claw-back policy allows companies to regain previously paid compensation if mismanagement or misconduct is subsequently uncovered. Shareholder Voting Rights - Under a straight voting share structure, each share has the same voting rights. - In a dual-class structure, certain classes of shares have enhanced voting rights, which can create a conflict of interest between minority shareholders and founders/management. Approaches to Identify ESG Factors - Proprietary methods - Third-party vendors - Non-profit industry organizations and initiatives Evaluating ESG-related Risks and Opportunities - Equity analysts consider both the opportunities and risks related to ESG factors, while fixedincome analysts primarily consider the downside risks associated with ESG issues. - Companies may issue green bonds to raise capital for projects intended to benefit the environment. - Misrepresenting a project’s benefits when issuing green bonds is known as greenwashing. COST OF COST OFCAPITAL: CAPITAL: ADVANCED ADVANCEDTOPICS TOPICS Cost of Capital Factors Weighted average cost of capital: WACC = w: r:(1 − t) + w/ r/ + w. r. Top-down external factors: - Capital availability - Market conditions - Legal and regulatory considerations - Tax jurisdiction Bottom-up company specific factors: - Revenue, earnings and cash flow volatility - Asset nature and liquidity - Financial strength, profitability, and financial leverage - Security features Copyright © 2024 Salt Solutions. All Rights Reserved. Personal copies permitted. Resale or distribution is prohibited. 8 Estimating Cost of Debt - Traded debt - Non-traded debt - Bank debt - Leases - International considerations Estimating Equity Risk Premium Historical approach Factors: - Equity index selection - Time period - Mean type - Risk-free rate proxy Limitations: - Past does not represent the future - Index-based ERP can be inflated due to survivorship bias Forward-looking approach Survey-based estimates: - Limitations: Sampling bias, recency bias, and confirmation bias Dividend discount model estimates: - Gordon growth model V" = r. = D# r. − g D# + 𝑔𝑔 𝑉𝑉" ERP = r. − r= = D# + g − r= V" Macroeconomic modeling: - Grinold-Kroner model 𝐷𝐷# + Δ𝑃𝑃𝑃𝑃 − Δ𝑆𝑆 + 𝑖𝑖 + 𝑔𝑔 𝐸𝐸(𝑟𝑟H ) = 𝑃𝑃" Estimating Cost of Equity Dividend discount models D# r. = + 𝑔𝑔 𝑉𝑉" Bond yield plus risk premium (BYPRP) r. = r: + Risk premium = r: + (Mean equity index return − Mean corporate bond index return) Risk-based models - Capital asset pricing model (CAPM) r. = r= + β(ERP) = r= + β(r7 − r= ) - Fama-French model r! = R E + β# (ERM) + β$ (SMB) + βI (HML) - Expanded Fama-French model r! = R E + β# (ERM) + β$ (SMB) + βI (HML) + βJ(RMW) + βK (CMA) Estimating the Cost of Equity for Private Companies Expanded CAPM method r. = r= + β/..6 (ERP) + SP + IP + SCRP Build-up approach r. = r= + ERP + SP + IP + SCRP www.saltsolutions.com International Considerations Country spread model ERPLM = ERP for a developed market +(λ × Country risk premium) Country risk rating model Country risk premium = Sovereign yield spread × σL4N!1O σA5,: Global CAPM (GCAPM) Uses a global market index as a factor and assumes zero correlation between returns across countries International CAPM (ICAPM) r. = r= + βP ìrC7 − r= î + βD (rQ − r=) CORPORATE CORPORATE RESTRUCTURING RESTRUCTURING Corporate Structural Changes Type of Motivations Change - Realize synergies - Increase growth potential Investment - Improve capabilities actions - Access new resources - Acquire an undervalued firm - Improve operational focus - Improve valuation metrics Divestment - Meet liquidity needs actions - Address regulatory requirements - Improve return on capital Restructuring markets actions - Avoid/respond to financial challenges Investment Actions Equity investments: - The acquirer and the target operate as independent entities Joint ventures: - Separate entity created by two or more companies that contribute assets and other resources to pursue common objectives Acquisitions: - Target company becomes a subsidiary and its assets, liabilities, and revenues are consolidated as part of the acquirer's financial statements Divestment Actions Sales: - The seller transfers a segment or business line to an acquirer, which assumes control of the associated assets and liabilities Spin-offs: - The parent company's shareholders receive shares in the spun-off company, but the two companies issue their own securities and file separate financial reports Restructuring Actions Cost restructuring: - Improve operational efficiency and increase profit margins through outsourcing and offshoring Balance sheet restructuring: - Sale leasebacks: Address immediate liquidity needs and eliminate the risks associated with ownership - Dividend recapitalization: Debt-financed dividend payment that increases the leverage in a company's capital structure Reorganization: - Must be approved by a bankruptcy court Leveraged Buyouts (LBO) - PE firms raise debt to finance acquisition of private companies and aim to operate them more efficiently under private ownership - The private companies may return to the public market through IPO Evaluating Corporate Restructurings Step 1: Initial evaluation Step 2: Preliminary valuation - Comparable company analysis - Comparable transaction analysis - Premium paid analysis Takeover premium = Deal price − Stock price Stock price Step 3: Modeling and evaluation - EBIT/Sales Profitability - EBITDA/Sales - Standard deviation of revenues Volatility - Standard deviation of EBITDA Leverage - Debt/EBITDA Interest rates - Benchmark reference rates - Credit spreads Collateral - Asset liquidity EQUITY VALUATION EQUITY VALUATION VALUATION EQUITY VALUATION Equity Value vs. Market Price VL – P = (V – P) + (VL – V) VL : estimate of intrinsic value P: market price V: unobservable intrinsic value (V – P): “true” mispricing Definitions of Value - Going concern value: Assumes company will continue operating for the foreseeable future - Liquidation value: Assumes assets must be sold immediately, likely at a discount - Orderly liquidation value: Higher than if assets must be sold immediately - Fair market value: Price negotiated between a willing buyer and seller - Investment value: Subjective valuation for a specific investor Copyright © 2024 Salt Solutions. All Rights Reserved. Personal copies permitted. Resale or distribution is prohibited. 9 DISCOUNTEDDIVIDEND DIVIDENDVALUATION VALUATION DISCOUNTED Single Period D# + P# V" = (1 + r)# Multiple Period , D1 P, V" = t + (1 + r)1 (1 + r), 1<# Gordon Growth Model (GGM) D# D" (1 + g) V" = = r−g r−g Value of Fixed-rate Perpetual Preferred Stock D V" = r Present Value of Growth Opportunities E# V" = + PVGO r Required Rate of Return Using GGM D# +g r= P" Two-Stage DDM , D" (1 + g B )1 D" (1 + g B ), (1 + g R ) + V" = t (1 + r), (r − g R ) (1 + r)1 1<# H-Model V! = D! (1 + g " ) + D! H(g # − g " ) r − g" Required Return from H-Model D" r = . 1 [(1 + g R ) + H(g B − g R )] + g R P" Sustainable Growth Rate g = b × ROE b = retention rate = 1 − dividend payout ratio Dividend Growth Rate and ROE Analysis Net Income ROE = Shareholder′ equity Net income Sales Assets 1. 1. 1 =. Assets Equity Sales g=. NI − Dividends NI Sales Assets 1. 1. 1. 1 NI Sales Assets Equity FREE FREE CASH CASHFLOW FLOWVALUATION VALUATION FCFF Valuation Model V=!67 T FCFF1 t (1 + WACC)1 1<# Constant Growth Model FCFF# FCFF" (1 + g) V=!67 = = WACC − g WACC − g FCFE Valuation Model T V.4N!1O = t 1<# FCFE1 = V=!67 − MV:.&1 (1 + r)1 Constant Growth Model V.4N!1O = FCFE# FCFE" (1 + g) = r−g r−g www.saltsolutions.com Calculating FCFF from Net Income FCFF = Net income to common shareholders + Net non-cash charges + Interest expense (1 – tax rate) - Investment in fixed capital - Investment in working capital FCFF from NI: Adjustments for Noncash Items Adjustment to Noncash Item NI Depreciation/Amortization Added back Impairment of intangibles Added back Restructuring charges Added back Expense reversals Subtracted Losses Added back Gains Subtracted Amortization of long-term Added back bond discounts Amortization of long-term Subtracted bond premiums Added back Deferred taxes but special attention Calculating FCFF from Statement of Cash Flows FCFF = CFO + Int(1 − tax) − FCInv CFO = NI + NCC − WCInv FCInv = CapEx − Proceeds from sale of l/t assets Calculating FCFE from FCFF FCFE = FCFF − Int(I − tax) + Net borrowing Finding FCFF/FCFE from EBIT or EBITDA FCFF = EBIT(1 − tax) + Dep − FCInv − WCInv NI + Int(1 − tax) EBIT = 1 − tax FCFF = EBITDA(1 − tax) + Dep(tax) − FCInv − WCInv Uses of FCFF and FCFE Uses of FCFF = ± Δ Cash balance + Interest expense (1-t) + Debt repayment + Cash dividends + Share repurchases Uses of FCFE = ± Δ Cash balance + Cash dividends + Share repurchases Forecasting of FCFF and FCFE Assuming fixed debt ratio (DR) in capital structure Net borrowing = DR(FCInv – Dep) + DR(WCInv) FCFE = NI − (1 − DR)(FCInv − Dep) − (1 − DR)(WCInv) International Application of Single-Stage Model Required rate of return (real) = Country return(real) ± Industry adjustment ± Size adjustment ± Leverage adjustment Free Cash Flow Model Variations Sensitivity Analysis of FCFF and FCFE Valuations Sensitivity analysis can be performed on key variables such as beta, risk-free rate, equity risk premium, FCFE growth rates, and the initial FCFF or FCFE. Two-Stage Free Cash Flow Models , V=!67 = t 1<# FCFF1 (1 + WACC)1 + FCFF,+# 1 P Q WACC − g (1 + WACC), Three-Stage Growth Models Three-stage models usually assume one of the following: - Constant growth rate in all three stages - Constant growth rate in the first and third stages, with a declining growth rate in stage two Non-operating Assets and Firm Value Firm value = Value of operating assets + Value of non-operating assets MARKET-BASED VALUATION: PRICE AND ENTERPRISE VALUE MULTIPLES Trailing (current) P/E Current stock price Most recent four quarters’ EPS Forward (leading) P/E Current stock price Next year’s expected EPS Justified P/E P" D#⁄E# 1 − b Leading = = E# r−g r−g P" D" (1 + g)⁄E" (1 − b)(1 + g) Trailing = = E" r−g r−g P/E to Growth (PEG) Ratio - P/E divided by expected earnings growth - Stocks with lower PEG are more attractive than stocks with higher PEGs, all else equal Fed Model - Compares the S&P 500 earnings yield (inverse of P/E) to the 10-year Treasury yield - Market is overvalued if the earnings yield, based on next 12 months’ expected earnings, is less than the 10-year Treasury yield Yardeni Model CEY = CBY − b × LTEG + residual - CEY = current earnings yield on the market index - CBY = current yield on A-rated corporate bonds - LTEG = consensus 5-year market EPS growth rate - b = weight the market gives to earnings estimates 1 P = E (CBY − b × LTEG) Justified Price Multiples P" ROE − g Price/Book = B" r−g Price/Sales E" P" K £S" L (1 − b)(1 + g) = S" r−g Copyright © 2024 Salt Solutions. 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Resale or distribution is prohibited. 10 Price to Cash Flow Advantages vs. P/E - Cash flows are more stable than earnings - Cash flows are less subject to manipulation - Differences between CFs and earnings should be eliminated over time Disadvantages vs. P/E - Cash flows may still be manipulated - FCFE is theoretically better but more volatile than FCFF - Cash flows are different under IFRS vs. US GAAP Justified Dividend Yield D" r − g = P" 1 + g Enterprise Value to EBITDA Enterprise Value = Market Value of Common Equity + Market Value of Preferred Stock + Market Value of Debt + Minority Interest − Cash and Investment EBITDA = NI + Int + Taxes + Dep & Amort EV/EBITDA vs. P/E Advantages: - Controls for differences in leverage - Controls for differences in depreciation - EBITDA can be used even if earnings are negative Disadvantages: - EBITDA overestimates cash flows if working capital is growing - EBITDA ignores the impact of revenue recognition policies on CFO Momentum Indicators Earnings Surprise / Unexpected Earnings UE1 = EPS1 − E(EPS1 ) Percent earnings surprise = UE/Expected EPS Standardized Unexpected Earnings EPS1 − E(EPS1 ) SUE1 = σ[EPS1 − E(EPS1)] RESIDUAL RESIDUALINCOME INCOMEVALUATION VALUATION Residual Income (RI) RI = NI – Equity charge Economic Value Added (EVA) EVA = NOPAT − (C% × TC) NOPAT = EBIT(1 − tax) C% = Cost of capital TC = Total capital Market Value Added (MVA) = Market value of the company − Accounting book value of total capital Residual Income Model RI1 = E1 − rB10# = (ROE − r)B10# T RI1 V" = B" + t (1 + r)1 1<# www.saltsolutions.com Clean Surplus Relationship B1 = B10# + E1 − D1 Constant Growth ROE − r V" = B" + B r−g " Multistage RI Valuation Model G V" = B" + t 1<# E1 − rB10# PG − BG + (1 + r)1 (1 + r)G If RI fades over time: G0# V" = B" + t 1<# E1 − rB10# EG − rBG0# + (1 + r)1 (1 + r − ω)(1 + r)G0# - ω = persistence factor; 0 < ω < 1 - ω = 0: RI will fall to zero after forecast horizon - ω = 1: RI will persist at current level indefinitely Strengths - Terminal value does not make up a large portion of the total present value as in other models - RI models use readily available accounting data - Can be applied to companies that do not pay dividends or positive expected near-term FCF - Can be used when cash flows are unpredictable - Appealing focus on economic profitability Weaknesses - The models are based on accounting data that can be subject to manipulation by management - Accounting data used as inputs may require significant adjustments - Additional adjustments are needed when clean surplus accounting relationships do not hold - The RI model’s use of accounting income assumes that the cost of debt is reflected appropriately by interest expense PRIVATE COMPANY VALUATION VALUATION PRIVATE COMPANY Private Company Areas of Focus Enterprise value , Terminal Value FCFF! + =t (1 + WACC), (1 + WACC)! !<# FCFF= EBITDA(1 − t) + Depreciation(t) −ΔLT Assets − ΔWorking Capital Required Rate of Return Models Capital asset pricing model (CAPM) r. = r= + β(r7 − r= ) Expanded CAPM r. = r= + β(r7 − r= ) + Small-cap stock premium + Company-specific stock premium Build-up method r. = r= + (r7 − r=) + Small-cap stock premium + Company-specific stock premium ± Industry premium/discount Valuation Discounts and Premiums Discounts for lack of control 1 DLOC = 1 − 1 + Control premium Approaches to Valuation - Income approach - Market approach - Asset-based approach Capitalized Cash Flow Method FCFF%+# Enterprise value% = WACC − g EBIT%+# (1 − t)(1 − RIR) = WACC − g Equity value% = FCFE%+# r. − g Excess Earnings Method Step 1: Calculate required returns on working capital and fixed assets Required return on working capital = Working capital × rUD Required return on fixed assets = Fixed assets × rEV Step 2: Calculate residual income Normalized income − (Working capital × rUD ) − (Fixed assets × rEV ) = Residual income Step 3: Capitalize residual income 𝐕𝐕𝐕𝐕𝐕𝐕𝐕𝐕𝐕𝐕 𝐨𝐨𝐨𝐨 𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢 (𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑𝐑 𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢𝐢)(𝟏𝟏 + 𝐠𝐠) = 𝐫𝐫𝐑𝐑𝐑𝐑 − 𝐠𝐠 Step 4: Calculate enterprise value Value of intangibles + Fair value of working capital + Fair value of fixed assets = Total firm enterprise value Guideline Public Company Method βR.Y.6.: β),8.Y.6.: = [1 + (1 − t)(D/E)] 𝛃𝛃∗𝐋𝐋𝐋𝐋𝐋𝐋𝐋𝐋𝐋𝐋𝐋𝐋𝐋𝐋 = 𝛃𝛃𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔 [𝟏𝟏 + (𝟏𝟏 − 𝐭𝐭 ∗ )(𝐃𝐃∗ /𝐄𝐄 ∗ )] FIXED INCOME FIXED INCOME STRUCTURE AND TERM STRUCTURE AND INTEREST RATE DYNAMICSRATE DYNAMICS INTEREST Spot Rates DFb = 1 (1 + ZN )b Forward Pricing Model (1 + zB )B = (1 + zA )A ì1 + fA,B-A î 1 FV,A0V = A0V Ω1 + fV,A0V æ DFA FV,A0V = DFV B-A Relationship between Spot and Forward Rates For an upward-sloping (downward-sloping) spot curve, the forward curve will be above (below) the spot curve Copyright © 2024 Salt Solutions. 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Resale or distribution is prohibited. 11 Swap Rate Binomial Interest Rate Tree G sG 1 t + =1 (1 + z1 )1 (1 + zG )G 1<# G t sG DF1 + DFG = 1 Ho-Lee Model: 1<# Spreads - Swap spread = Swap rate − Treasury yield - I-spread = Bond yield − Swap rate - Z-spread: Spread added to each spot rate so that PV of bond’s cash flows equals bond’s market price - TED spread = LIBOR rate − T-bill rate - LIBOR-OIS spread = LIBOR rate − OIS rate Term Structure of Interest Rates: Theories Pure expectations: Forward rates are unbiased predictors of future spot rates Local expectations: All bonds are expected to generate the risk-free rate over short time periods Liquidity preference: Forward rates reflect expectations of future spot rates plus liquidity premiums for longer maturities Segmented markets: Yield curve shape is determined by preferences of borrowers and lenders Preferred habitat: Investors have preferred maturity ranges, but they will shift if yields are attractive Yield Curve Risk Based on Key Rate Durations Decomposes changes in yield curve into changes in level (L), steepness (S), and curvature (C): ∆P . 1 ≈ −DR ∆xR − DB ∆xB − DD ∆xD P Flattening and Steepening of Yield Curve Bullish steepening: Short-term rates falling faster than long-term rates; observed when central bank loosens monetary policy to stimulate economy Bearish flattening: Short-term rates increasing more than long-term rates; occurs when central bank raises rates by restricting money supply Bullish flattening: Long-term rates falling more than short-term rates; observed in the aftermath of market turmoil as investors flock to government bonds in a flight to quality ARBITRAGE-FREE VALUATION FRAMEWORK ARBITRAGE-FREE VALUATION FRAMEWORK Arbitrage Concepts Value additivity: Exists if the whole does not equal sum of parts Dominance: Exists if the present value of a risk-free payoff is non-zero Stripping: Turning coupon bonds into multiple zero-coupon bonds Reconstitution: Combining zero-coupon bonds to replicate a coupon-paying bond www.saltsolutions.com Arbitrage-Free Models - Based on the assumption that bond prices and the term structure implied by those prices are correct - Allow parameters to vary deterministically over time dr1 = θ1 dt + σdZ - Drift term: Time-dependent - Volatility: Constant - Negative rates: Possible i#,h = i#,R e$i Kalotay-Williams-Fabozzi (KWF) Model: d ln(r1) = θ1 dt + σdZ - Drift term: Time-dependent - Volatility: Constant - Negative rates: Not possible BONDS WITH EMBEDDED OPTIONS OPTIONS BONDS WITH Binomial Valuation Framework Bond value plus coupon payment if higher rate is realized. Bond value at any node Bond value plus coupon payment if lower rate is realized. VH + C VL + C + 1 1+i 1+i C + 0.5(VH + VL) = 1+i Bond value at node = 0.5 . Term Structure Models Equilibrium Models - Use fundamental economic variables to describe the term structure - Require parameters to be specified - Preferable for dynamic applications - Allow for possibility of multiple different future interest rate paths Valuation of Callable and Putable Bonds VQ3883&8. = V2163!Cj1 − VQ388 V/N13&8. = V2163!Cj1 + V/N1 Effects of Interest Rates and Yield Curve - Callable (putable) bond value is inversely (directly) related to interest rate volatility - Value of option-embedded bonds declines as upward-sloping yield curve flattens Option Adjusted Spread (OAS) Constant spread is added to each forward rate in a benchmark binomial interest rate tree to equate PV of credit risky bond’s cash flow to its market price - OASQ3883&8. is inversely related to assumed volatility of benchmark binomial tree rates - OAS/N13&8. is directly related to assumed volatility of benchmark binomial tree rates Effective Duration of Callable & Putable Bonds (PV0) − (𝑃𝑃𝑉𝑉+ ) Effective duration = 2 × (∆Curve)(PV" ) - EffDurQ3883&8. ≤ EffDur2163!Cj1 - EffDur/N13&8. ≤ EffDur2163!Cj1 Cox-Ingersoll-Ross (CIR) Model: dr1 = k(θ − r1 )dt + σUr1 dZ - Drift term: Mean reverting - Volatility: Proportional to the square root of the short-term rate - Negative rates: Not possible Vasicek Model: dr1 = k(θ − r1 )dt + σdZ - Drift term: Mean reverting - Volatility: Constant - Negative rates: Possible Copyright © 2024 Salt Solutions. All Rights Reserved. Personal copies permitted. Resale or distribution is prohibited. 12 Convexity Improves Duration-based Estimates (PV0 ) + (𝑃𝑃𝑉𝑉+ ) − [2 × (𝑃𝑃𝑉𝑉" )] Effective Convexity = (∆Curve)$ (PV") Convexity for Interest Rates and Bond Structure Interest Straight and Callable putable bonds bonds rates High Positive Low Positive Positive Negative Factors for the Term Structure of Credit Spreads - Credit quality - Financial conditions - Market supply and demand - Company-value model CREDIT ANALYSIS ANALYSIS MODELS MODELS CREDIT Measures of Credit Risk - Recovery rate = 100% − Loss severity - Loss given default = Exposure × (100% − RR) - The conditional probability of default in a given year is the probability of default assuming no prior defaults. If this probability of default is a constant rate, then the probability of survival to year t can be calculated as: (100% − Hazard rate)1 Measures of Credit Risk (continued) PV of Expected Loss = Discount factor × Expected loss = Discount factor × Loss given default × Prob. of default = Discount factor × (Exposure × Loss severity) × Prob. of default where Prob. of default = POSAl × Hazard rate1 Credit Valuation Adjustment (CVA) Capped or Floored Floating-rate Bonds Capped floater Bond with option that prevents coupon rate from rising above specified maximum VQ3//.: = V2163!Cj1 − V.7&.::.: =8531.6 =8531.6 Q3/ =8531.6 =8531.6 =8556 Floored floater Embedded option sets minimum coupon rate: V=8556.: = V2163!Cj1 + V.7&.::.: Convertible Bonds Bondholder has right to convert bond into common shares at conversion ratio: Market price Conversion Conversion × = value ratio of stock Market Market price Conversion conversion = of convertible¡ ratio price bond Market Market Market conversion = conversion − price of premium per share price stock Minimum value Straight Conversion , L of convertible = max K value value bond Callable and putable convertible bond value = V2163!Cj1 + VQ388 5, − VQ388 5, + V/N1 5, &5,: 215Q% &5,: www.saltsolutions.com &5,: , CVA = t PV of expected loss1 1<# Structural and Reduced Form Credit Models Structural credit models can interpret debt and equity values with option terminology. Debt and equity are also random values because they are functions of the asset random values. At time T, the balance sheet can be represented with: A(T) = D(T) + E(T) Assuming the maturity value at time T of the debt is K, a default will occur if the asset value at time T is less than K. The value of the equity and debt at time T are: - E(T) = max[A(T) − K, 0] - D(T) = A(T) − max[A(T) − K, 0] This implies that: - Equity is a call option on the assets with a strike price equal to the face value of the debt - Debtholders have written the call option to the equity holders The value of the equity and debt at time T can also be expressed as follows using put-call parity: - E(T) = A(T) − K + max[K − A(T),0] - D(T) = K − max[K − A(T),0] This implies that: - Equity can also be viewed as a long position in the assets, long put option, and short bond - Debt can be interpreted as a long bond and a short put option Factors for the Credit Analysis on Securitized Debt - Homogeneity - Granularity - Underlying collateral - Structure of the secured debt transaction CREDIT CREDIT DEFAULT DEFAULT SWAPS SWAPS Settlement Protocol - Loss given default (LGD) = 1 − Recovery rate (%) - Payout amount = LGD × Notional amount CDS Pricing Conventions PV of credit spread = Upfront premium + PV of fixed coupon Upfront Credit Fixed ≈. − 1 × CDS Duration premium spread coupon CDS price per 100 par = 100 − Upfront premium % Valuation Changes during CDS Term % Change in CDS price = Change in spread (bps) × Duration Applications of CDS - Naked credit default swap: Purchase of credit protection without holding the reference obligation - Long/short trade: Long position in one CDS and short position in another - Curve trade: Buy a CDS of one maturity and sell a CDS with a different maturity for the same reference entity - Basis trade: Exploit differences in credit spreads offered by the CDS market and the bond market DERIVATIVES DERIVATIVES PRICING AND VALUATION OF FORWARD COMMITMENTS Principles of Arbitrage-Free Pricing To implement the no-arbitrage argument, the following simplifying assumptions are made: - Replicating instruments are available. - Market frictions are absent. - Short selling is allowed. - Investors can borrow and lend at the risk-free rate. Pricing and Valuation of Forwards and Futures Forward Price F" (T) = FV",G (S" + CC" − CB" ) - Carry benefits could include dividends and bond coupon payments. They reduce the forward price. Copyright © 2024 Salt Solutions. All Rights Reserved. Personal copies permitted. Resale or distribution is prohibited. 13 - Carry costs could include waste, storage, and insurance. They increase the forward price. - If interest rates, carry benefits, and carry costs are continuously compounded, then: F" (T) = S" e(6'+DD0DA)G - If there is no carry benefit, then set CB = 0. - If there is no carry cost, then set CC = 0. Valuing Forward Contracts V1 (T) = PV1,G[F1 (T) − F" (t)] for long V1 (T) = PV1,G[F" (t) − F1 (T)] for short VG(T) = SG − F"(t) for long VG(T) = F" (t) − SG for short Futures Contract - Right before marking to market: v1 (T) = f1 (T) − f10(T) for long v1 (T) = f10 (T) − f1(T) for short - The value after the daily marking to market is 0. Interest Rate Forward and Futures Forward Rate Agreement - The two FRA counterparties are the fixed-rate receiver (short party) and the floating-rate receiver (long party). - FRAs are identified in an “X × Y” format. A 3 × 9 FRA will expire in three months with a payoff based on the six-month Libor when the FRA expires. - FRAs typically use the advanced set, advanced settled approach. This means interest rate is set and the FRA settlement is made at time h, when the FRA expires. - The settlement amount for the floating receiver (long position) is: NA[L7 − FRA" ]t 7 1 + D7 t 7 - The FRA fixed rate is the forward m-day rate in h days: 1 1 + LG t G − 1Q . 1 FRA" = P t7 1 + Lj t j - The value of an existing long FRA (floating receiver) can be calculated using an offsetting transaction at the new rate applicable when the FRA expires: ΩFRAC − FRA" æt 7 VC = 1 + DG0C t G0C Fixed-Income Forward and Futures Accrued interest Accrued interest = Accrual period × Periodic coupon amount C NAD 1×. 1 AI = . n NTD where: - NAD is the number of days since the last coupon payment - NTD is the total days during a coupon payment period - n is the number of coupon payments per year - C is the annual coupon amount If accrued interest is included in the bond price quote, then F" = FV",G (S" + CC" − CB" ). www.saltsolutions.com - S" is the full bond price (including accrued interest). - There are no carry costs, so CC" = 0. - The carry benefits are the coupon interest payments, so CB" = PVCI",G . If accrued interest is not included in the bond price quote, then: F" = FV",G ΩB" + AI" − PVCI",G æ = FV",G [B" + AI" ] − AIG − FVCI",G Often, the delivery of more than one bond is permitted. Then, the actual futures price is: F" = Q " × CF Valuing Fixed-Income Forwards and Futures - The value of a bond futures contract is the price change since the previous day’s settlement. - The value of a bond forward contract at a later date is the present value of the current forward price less the original forward price. Pricing and Valuing Swap Contracts Swap pricing equation 1.0 − PV",1( (1) rE'o = , ∑*<# PV",1! (1) VALUATION CONTINGENT CLAIMS VALUATION OF CONTINGENT CLAIMS Binomial Option Valuation Model Exercise values for calls and puts cG = Max(0, SG − X) pG = Max(0, X − SG ) Hedge ratio c+ − c0 h= + ≥ 0 for call S − S0 + 0 p −p h= + ≤ 0 for put S − S0 - A long call is equivalent to buying stocks with borrowed funds. - A long put is equivalent to selling stocks short and lending the proceeds. Expectations approach c = PV[πc + + (1 − π)c 0 ] p = PV[πp+ + (1 − π)p0 ] π= #+60: N0: American-Style Options Solve for the values from right to left. At each node, the value is the greater of the calculated value or the immediate exercise value. The Black-Scholes-Merton Formula Stock For nondividend-paying stock: c = SN(d# ) − e06G XN(d$ ) p = e06G XN(−d$ ) − SN(−d#) ln(S⁄X) + [r + (σ$ ⁄2)]T d# = σ√T d$ = d# − σ√T - N(d$ ) represents the probability the call option expires in the money. - 1 − N(d$ ) represents the probability the put option expires in the money. Futures c = F" (T)e06G N(d# ) − e06G XN(d$ ) p = e06G XN(−d$ ) − F" (T)e06G N(−d# ) d# = ln(F" (T)e06G ⁄X) + [r + (σ$ ⁄2)]T d$ = d# − σ√T σ√T Interest Rate Options - An interest rate cap is a portfolio of call options on interest rates (a.k.a. caplets), each with the same exercise rate and with sequential maturities. - An interest rate floor is a portfolio of put options on interest rates (a.k.a. floorlets), each with the same exercise rate and with sequential maturities. Swaptions A swaption is an option on a swap. It gives the holder the right to enter into a swap with specified terms, including the fixed rate. - A payer swaption is an option on a swap to pay a fixed rate and receive a floating rate. - A receiver swaption is an option on a swap to pay a floating rate and receive a fixed rate. Option Greeks Delta - Delta is the change in an instrument's value for a given change in the underlying value. - The delta of a long stock is 1. - Using the BSM model: DeltaQ = e0pG N(d# ); 0 ≤ DeltaQ ≤ e0pG Delta/ = −e0pG N(−d# ); −e0pG ≤ Delta/ ≤ 0 - As the stock price increases: DeltaQ → 1 Delta/ → 0 - Delta hedging is used to offset the exposure of the portfolio to changes in the underlying. Portfolio Delta Nh = − Deltah If Nh < 0, short the hedging instrument. If Nh > 0, long the hedging instrument. - Delta approximation: «prıce ≈ option price + DeltaìS» − Sî Optıon Gamma - Gamma measures the change in an instrument’s delta for a small change in the underlying stock. - The gamma for a stock position is 0. - GammaQ = Gamma/ - Gamma is always non-negative. - The largest value occurs when the option is near the money. - Delta-plus-gamma approximation: «prıce ≈ option price + DeltaìS» − Sî Optıon Gamma $ ìS» − Sî + 2 Theta - Theta is the change in an instrument's value for a small change in calendar time. - Stock theta is zero. Copyright © 2024 Salt Solutions. All Rights Reserved. Personal copies permitted. Resale or distribution is prohibited. 14 - Option theta is negative and more pronounced the closer the option is to expiration. Vega - Vega is the change in an instrument's value for a small change in volatility. - VegaQ = Vega/ - Positive for both call and put Rho - Rho is the change in an instrument's value for a small change in the risk-free interest rate. - Positive for calls, negative for put Implied Volatility - The implied volatility measure provides information about the market consensus on price uncertainty and demand for options. - The volatility smile is a plot of the implied volatility with respect to the exercise price. - The volatility surface is a plot of the implied volatility with respect to various combinations of time to expiration and exercise prices. ALTERNATIVE INVESTMENTS ALTERNATIVE INVESTMENTS INTRODUCTION INTRODUCTIONTO TOCOMMODITIES COMMODITIESAND AND COMMODITY DERIVATIVES COMMODITY DERIVATIES Commodities Overview - Energy (crude oil, natural gas, and refined products) - Grains - Industrial (base) metals - Livestock - Precious metals - Softs Spot and Futures Pricing Basis: Spot price − Futures price - Spot price > Futures price: Backwardation - Spot price < Futures price: Contango Calendar spread = Futures contract with an earlier expiration −Futures contract with a later expiration Components of Futures Returns - Price return - Roll return - Collateral return - Rebalance return Public Equity Debt www.saltsolutions.com Risk Factors of Real Estate Investment Property demand and supply: - Business conditions - Demographics - Excess supply Valuation: - Cost and availability of capital - Availability of information - Lack of liquidity - Rising interest rates Property operations: - Management - Lease provisions - Leverage - Environmental - Obsolescence - Recent and ongoing market disruption Benefits of Real Estate Investment - Current income - Price appreciation - Inflation hedge - Diversification - Tax benefits Types of Leases Gross lease: The owner pays all operating expenses Net lease: The tenant covers certain operating expenses Triple-net lease: The tenant pays their share of common area maintenance costs, property taxes, and building insurance Sale-leaseback: The owner sells its property and leases it back from the new owner Appraisal-Based Index Relevant appraisal data in the U.S. are provided by the NCREIF Property Index (NPI). Holding period return: HPR = OVERVIEW OF TYPES TYPESOF OFREAL REALESTATE INVESTMENT ESTATE INVESTMENTS REOC shares REIT shares ETFs/Index funds Mortgage REITs MBSs Unsecured REIT debt Real Estate Characteristics - Heterogeneity and fixed location - High unit value - Management intensive - High transaction costs - Depreciation - Need for debt capital - Illiquidity - Price determination Private Direct Indirect Mortgages Private debt Bank debt NOI − Capex + (End. MV − Beg. MV) Beginning MV Disadvantages: - Appraisal lag occurs - Volatility and correlations with other assets are understated - Index requires complicated unsmoothing techniques Transaction-Based Index Repeat sales index uses repeat sales of the same property. Hedonic index performs regression based on properties that are sold. Disadvantages: - Market trends contain random noises INVESTMENTS IN INVESTMENTS IN REAL REALESTATE ESTATE THROUGH THROUGH PUBLICLY TRADED TRADED SECURITIES SECURITIES PUBLICLY Publicly Traded Real Estate Securities Real estate investment trusts (REITs) - Tax-advantaged entities Real estate operating companies (REOCs): - Ordinary taxable entities Mortgage-backed securities (MBS): - Asset-backed securitized debt obligations Advantages and Disadvantages of REITs Advantages: - Liquidity - Transparency - Diversification - High-quality portfolios - Active professional management - High, stable income - Tax efficiency Disadvantage: - Lack of retained earnings - Regulatory costs - Reduced portfolio diversification benefits - Limitations on types of assets owned Net Asset Value Per Share (NAVPS) NAV NAVPS = Shares outstanding MV of assets − MV of liabilities = Shares outstanding Funds from Operations (FFO) = Accounting net earnings + Depreciation charges on real estate + Deferred tax charges + Loss from property sales & debt restructuring − Gain from property sales & debt restructuring Adjusted Funds from Operations (AFFO) = FFO − Non-cash rent − Maintenance-type capital expenditures and leasing costs Advantages of Using P/FFO and P/AFFO - Widely accepted in global stock markets - Valuations of REITs and REOCs are easily compared to other investment alternatives - FFO estimates are readily available - Multiples can be used with expected growth and leverage levels to deepen understanding Disadvantages of Using P/FFO and P/AFFO - FFO or AFFO may not capture all intrinsic value (like land parcels) - Does not adjust for the right recurring capex - Income statement rules have changed, so P/FFO and P/AFFO are difficult to calculate Copyright © 2024 Salt Solutions. All Rights Reserved. Personal copies permitted. Resale or distribution is prohibited. 15 HEDGE HEDGEFUND FUNDSTRATEGIES STRATEGIES Classification of Hedge Funds and Strategies Hedge Fund Characteristics - Legal/ Regulatory Overview - Flexible Mandates - Large Investment Universe - Aggressive Investment Styles - High Leverage - Liquidity Constraints - High Fees Equity Strategies - Long/Short Equity: Buy undervalued stocks and sell overvalued stocks short. - Dedicated Short-Selling and Short-Biased: Take short-only positions in equities but may vary the short positions by holding cash. - Equity Market-Neutral: Use offsetting long and short positions to establish an overall portfolio with near zero net exposure to equity market risk (beta). Event-Driven Strategies - Merger Arbitrage: Take positions based on their expectations about merger activity. - Distressed Securities: Focus on firms that are either already in the bankruptcy process or that are currently experiencing financial distress and are expected to file for bankruptcy in the near future. Relative Value Strategies - Fixed-Income Arbitrage: Seek to profit from relative mispricing. o Yield curve trades involve taking long and short positions based on expected changes in the shape of the yield curve in response to macroeconomic conditions. o Carry trades are executed by shorting low-yielding bonds and purchasing higher-yielding bonds. - Convertible Bond Arbitrage: Seek to take advantage of discrepancies between the prices of an issuer’s convertible bonds and their straight bonds. Opportunistic Strategies - Global Macro Strategies: Generally hold views on macro-level economic activity based on top-down analysis and express views by taking positions in a variety of asset classes and instruments. - Managed Futures: A highly technical and systematic strategy that is executed using primarily futures and options on futures. www.saltsolutions.com Specialist Strategies - Volatility Trading: Relative value volatility arbitrageurs can take advantage of differences in implied volatility for the same product across time zones (time-zone arbitrage) or markets (cross-asset volatility trading). - Reinsurance/Life Settlements: The sale of a life insurance policy to a party other than its originator is called a life settlement. Third-party brokers will then package the individual policies that they have acquired into pools to be resold to hedge funds. Multi-Manager Strategies - Fund-of-Funds: The FoF manager will make allocations to a diverse group of funds. - Multi-Strategy Hedge Funds: Multiple teams pursuing distinct strategies within the same organization. Multi-Manager Strategies - A conditional risk-factor model goes a step further to account for the possibility that factor exposures can change under different market conditions. - Using the example of a managed futures with exposure to equity and commodity risks, the conditional factor model is: 𝑟𝑟qr = 𝛼𝛼 + 𝛽𝛽s 𝑅𝑅s + 𝛽𝛽t 𝑅𝑅t + 𝐷𝐷𝛽𝛽s 𝑅𝑅s + 𝐷𝐷𝛽𝛽t 𝑅𝑅t where o 𝐷𝐷: Dummy variable with a value of 0 under normal market conditions and 1 during crises o 𝐷𝐷𝛽𝛽s 𝑅𝑅s : Incremental exposure to equity risk during periods of market stress PORTFOLIO MANAGEMENT PORTFOLIO MANAGEMENT EXCHANGE-TRADED FUNDS: FUNDS:MECHANICS AND EXCHANGE-TRADED MECHANICS AND APPLICATIONS APPLICATIONS The Creation/Redemption Process of ETFs - ETF shares are created or redeemed continuously to match supply and demand. - The primary market is over-the-counter (OTC) with trades between authorized participants (APs), which are large broker/dealers, and the ETF manager (a.k.a. ETF sponsor or ETF issuer). APs are the only investors who can create or redeem new shares of an ETF. Tracking Error - Tracking error is the standard deviation of return differentials between an ETF and its index. - Sources of ETF tracking error include: - Fees and expenses - Representative sampling/optimization - Depository receipts and ETFs - Index changes - Fund accounting practices - Regulatory and tax requirements - Asset manager operations ETF Bid-Ask Spreads ETF spreads are typically less than or equal to the sum of the factors listed below: - Direct trading costs, such as brokerage and exchange fees, as well as creation/redemption fees paid to the sponsor - Bid-ask spreads for the underlying securities - Compensation for the market maker’s risk of carrying positions - The market maker’s desired profit spread Premiums and Discounts Sources of ETF premiums and discounts to NAV include: - Timing differences - Stale pricing Total Costs of ETF Ownership The main components of ETF cost are: - Management fees - Trading costs (e.g., commissions, bid-ask spreads, premiums/discounts) - Taxes - Tracking error - Portfolio turnover - Security lending Trading costs are more significant for active shorter-term investors. Ongoing costs have an increasing impact on returns for longer-term investors. Risks - Exchange-traded notes (ETNs) carry counterparty risks of default. - Any fund that uses OTC derivatives will carry settlement risk because mark-to-market gains are subject to default. - ETFs often lend securities to short-sellers for additional income to investors, creating the risk of counterparty default. - The closing of an ETF fund can trigger unexpected tax liabilities and force investors to find another fund. Portfolio Uses of ETF - Efficient Portfolio Management - Asset Class Exposure Management - Active and Factor Investing USING MULTIFACTOR USING MULTIFACTOR MODELS MODELS Arbitrage Pricing Theory (APT) The APT is an equilibrium pricing model, but it makes these weaker assumptions: - A factor model describes the asset returns. - A well-diversified portfolio can be created to eliminate asset-specific risk. - No arbitrage opportunities exist in welldiversified portfolios. EìR / î = R E + λ# β/,# + ⋯ + λ% β/,% Carhart Model EΩR / æ = R E + β/,# RMRF + β/,$ SMB + β/,I HML + β/,J WML Copyright © 2024 Salt Solutions. All Rights Reserved. Personal copies permitted. Resale or distribution is prohibited. 16 Fundamental Factor Models R ! = a! + b!# F# + b!$ F$ + ⋯ + b!u Fu + ε! Factors are stated as returns rather than return surprises in relation to predicted values Macroeconomic Factor Models R ! = a! + b!# F# + b!$ F$ + ⋯ + b!u Fu + ε! Asset returns are correlated with the surprises in certain factors Surprise = Predicted value − Expected value Factor Models in Return Attribution Active return = R v − R A u Benchmark Portfolio Active = t P. 1 Q 1 −. sensitivity % sensitivity % return u<# Factor × K L return % + Security selection Factor Models in Risk Attribution Active risk = Tracking error = s(R v − R A ) Active risk square = s $ (R v − R A ) = Active factor risk + Active specific risk , Active specific risk = t(w!3)$ σ$w! Information Ratio +A +v − R R IR = s(R v − R A ) !<# MEASURING MEASURINGAND ANDMANAGING MANAGINGMARKET MARKETRISK RISK Understanding Value at Risk - Value at Risk (VaR) is the minimum loss expected a certain percentage of time over a certain period of time. - It can be measured in currency units or as a percentage of the portfolio value. Estimating VaR Parametric Method - This method (a.k.a. the analytical method or the variance-covariance method) typically estimates VaR by assuming normally-distributed returns - Advantage: Simplicity; Works best when the normal distribution assumption is reasonable and the parameter estimates are reliable - Disadvantage: Does not work well for portfolios that contain options because the returns on options are not normally distributed Historical Simulation Method - This method analyzes the return of the current portfolio composition over a historical period. The VaR is then calculated by sorting the returns from the largest loss to the largest gain and choosing the one based on the desired confidence interval. - Advantage: Can incorporate events that actually happened and it does not require specification of a distribution or the estimation of parameters; can also handle options - Disadvantage: Only useful if the historical period is representative of the future www.saltsolutions.com Monte Carlo Simulation Method - This method allows the analyst to develop assumptions about the statistical characteristics of the distribution, then use them to generate random outcomes. - Advantage: Very flexible; Works well with a portfolio that has many assets and not constrained by the normal distribution assumption - Disadvantage: Can be complex and time-consuming to use Advantages and Limitations of VaR Advantages of VaR include: - Simple concept - Easily communicated concept - Basis for risk comparison - Facilitates capital allocation decisions - Can be used for performance evaluation - Reliability can be verified - Widely accepted by regulators Limitations of VaR include: - Subjectivity - Underestimating the frequency of extreme events - Failure to consider liquidity - Sensitivity to correlation risk - Vulnerability to volatility regimes - Misunderstanding of its meaning - Oversimplification - Disregard of right-tail events Extensions of VaR - Conditional VaR (CVaR), a.k.a. expected shortfall or expected tail loss, measures the expected loss if VaR is exceeded. - Incremental VaR (IVaR) measures the impact of changing a position's size within a portfolio. - Marginal VaR (MVaR) is similar to incremental VaR in that it measures the change in VaR for a small change in a position, but it uses formulas derived from calculus. - Relative VaR, a.k.a. ex ante tracking error, indicates the amount that a portfolio may deviate from its benchmark. Sensitivity Risk Measures Equity Exposure Measures - Beta E(R ! ) = R E + β![E(R M ) − R E ] Fixed-Income Exposure Measures – Duration, Convexity ∆A A ≈ −D ∆O #+O # + C $ ∆O) (#+O)) Option Risk Measures – Delta, Gamma, Vega ∆(delta) ≈ Dj3,C. !, Y38N. 5= 5/1!5, Dj3,C. !, Y38N. 5= N,:.68O!,C Г(gamma) ≈ Dj3,C. !, :.813 Dj3,C. !, Y38N. 5= N,:.68O!,C # c + ∆c ≈ c + ∆Q ∆S + ГQ (∆S)$ Vega ≈ $ Dj3,C. !, Y38N. 5= 5/1!5, Dj3,C. !, Y5831!8!1O 5= N,:.68O!,C Scenario Risk Measures - Historical scenarios are scenarios that measure the portfolio return if historical markets repeat themselves. - Hypothetical scenarios model the impact of extreme movements and co-movements in different markets that have not previously happened. - Reverse stress testing is the process of targeting and stressing the portfolio’s material exposures. - Sensitivity and scenario risk measures complement VaR in the following ways: o Sensitivity measures address some of the shortcomings of position size measures. o They do not have to rely on historical volatility and correlation data, so their utility is not limited by the parameters of the lookback period. o Normal distributions do not have to be assumed. o Scenarios can target key exposures of the portfolio, such as concentrated positions. - One limitation of sensitivity risk measures is they often do not distinguish by volatility. - The limitations of scenario measures include: o Hypothetical scenarios are necessary to fill in the gaps of historical scenarios. o Hypothetical scenarios may incorrectly specify the correlation between assets or fail to adjust correctly for factors such as liquidity or concentrated positions. o Hypothetical scenarios are difficult to maintain. There is no certainty the range of scenarios tested is adequate. o It is difficult to establish limits based on scenario analysis. o Extreme hypothetical scenarios are not likely to be taken seriously by management. However, using only plausible scenarios could be too limiting. Using Constraints in Market Risk Management - Risk budgeting first sets limits for the entire firm, then allocates the firm's overall risk budget among sub-activities. The risk will generally be based on ex ante tracking error or VaR. - Position limits are effective controls against overconcentration. They may be specified in terms of the market value of securities or the notional principal of derivatives. - Scenario limits place limits on the loss in a given scenario. This can be used to address shortcomings in VaR. If results are not within the limits, corrective action should be taken. - Stop-loss limits require changes if losses over a given magnitude occur in a specified period. This can catch trending losses that are staying just below the VaR daily limits. - Capital allocation aligns risks and rewards by placing limits on capital assigned to each of the firm's activities. Copyright © 2024 Salt Solutions. All Rights Reserved. Personal copies permitted. Resale or distribution is prohibited. 17 BACKTESTING SIMULATION BACKTESTING AND SIMULATION b R V = t Δw! R ! ; Δw! = wv.! − wA,! Backtesting Process 1. Strategy design 2. Historical investment simulation 3. Analysis of backtesting output !<# b R V = t Δw! R V! ; R V! = R ! − R A !<# Decomposition of Value Added Backtesting Multifactor Models - Give equal weight to each parameter - Use a risk-based weighting scheme Common Problems in Backtesting - Survivorship bias: This can be combated using a point-in-time approach - Look-ahead bias: This includes reporting lags, data revisions, and addition of new companies to indexes/databases - Data snooping: To combat this practice, analysts may use a higher statistical t-value, perform cross-validation, or test theories on different data Simulation Analysis Historical simulation: A non-deterministic form of rolling window backtesting Monte Carlo simulation: A model that specifies a statistical distribution for the underlying data ECONOMICS AND ECONOMICS ANDINVESTMENT INVESTMENTMARKETS MARKETS Pricing a default-free nominal coupon paying bond b P1! = t B<# ! CF1+2 2 ì1 + l1,2 + θ1,2 + π1,2 î B<# ! Õ 1+2 æ E1 ΩCF ì1 + l1,2 + θ1,2 + π1,2 + γ!1,2î γ!1,2 : credit premium b 2 B<# ! Õ 1+2 æ E1 ΩCF ì1 + l1,2 + θ1,2 + π1,2 + γ!1,2 + κ!1,2 î 2 Commercial Real Estate b ! Õ 1+2 æ E1ΩCF 2 ì1 + l1,2 + θ1,2 + π1,2 + γ!1,2 + κ!1,2 + ϕ!1,2 î B<# ϕ!1,2 : liquidity risk premium ANALYSIS ANALYSIS OF OF ACTIVE PORTFOLIO MANAGEMENT MANAGEMENT Measuring Value Added RV = Rv − RA where: b R A = t wA,! R ! !<# b R v = t wv,! R ! Sharpe Ratio Rv − RE SR v = STD(R v ) Information Ratio Rv − RA IR = STD(R v − R A ) Constructing Optimal Portfolios SR$v = SR$A + IR$ IR STD(R A ) STD(R V ) = SR A Active Security Returns µ! = ICσ! S! Mean-variance active return weights µ! σV Δw!∗ = $ σ! IC√BR Ex Ante Measurement of Skill κ!1,2 : equity premium relative to credit risky bonds P1! = t *<# Ex Post Performance Measurement E(R V |IC( ) = (TC)(IC( )√BRσV R V = E(R V |IC( ) + Noise The portfolio’s active return variance can be divided into two parts: - Variation due to the realized information coefficient, TC $ - Variation due to constraint-induced noise, 1 − TC $ Equities and the Equity Risk Premium P1! = t *<# R V = t Δw* R A,* + t wv,* R V,* Full Fundamental Law E(R V) = (TC)(IC)√BRσV Credit Premiums and Business Cycle P1! = t M Basic Fundamental Law Expected active portfolio return E(R V)∗ = IC√BRσV l1,2 : yield to maturity on a real default-free investment today θ1,2 : expected inflation rate π1,2 : compensation for uncertainty in inflation b M σV = σ'D √Nσ(M IC E(R V) = σ σ'D V Practical Measure of Breadth N BR = 1 + (N − 1)ρ ETHICAL AND PROFESSIONAL STANDARDS ETHICAL AND PROFESSIONAL STANDARDS I(A) Knowledge of the Law Obey strictest applicable law. Disassociate immediately from any illegal or unethical activity. I(B) Independence and Objectivity Do not offer or accept gifts that might impair independence and objectivity. Gifts from clients may be permissible. I(C) Misrepresentation Cite sources. Do not plagiarize or omit important information. Act quickly to correct any errors. I(D) Misconduct Does not apply to personal behavior unless it reflects poorly on the investment profession. II(A) Material Nonpublic Information Do not act or cause others to act on material nonpublic information. Seek public dissemination. II(B) Market Manipulation Do not take any actions that distort prices or trading volume. Market making and legitimate trading strategies are allowed. III(A) Loyalty, Prudence, and Care Place clients’ interest above yours. Disclose policies on proxy voting and soft commissions. III(B) Fair Dealing Treat all clients fairly. Treat non-immediate family like other clients. Communicate investment recommendations and changes simultaneously. III(C) Suitability Use a regularly updated IPS during investment decisions. Evaluate decisions in a portfolio context. III(D) Performance Presentation Performance data should be fair, accurate, and complete. Do not promise returns for risky assets. III(E) Preservation of Confidentiality Keep all client information confidential unless: client is involved in illegal activity, you are legally required, or you have the client’s permission. IV(A) Loyalty Get permission before taking outside work (even unpaid) that competes with employer. Abide by non-compete agreement (if applicable) and do not take employer’s property. IV(B) Additional Compensation Arrangements Obtain written permission from all parties before receiving any compensation for outside work. IV(C) Responsibilities of Supervisors Supervisors must adequately train and monitor subordinates. Responsibilities may be delegated. !<# www.saltsolutions.com Copyright © 2024 Salt Solutions. All Rights Reserved. Personal copies permitted. Resale or distribution is prohibited. 18 V(A) Diligence and Reasonable Basis Exercise diligence and thoroughness. Support actions with research and investigation. V(B) Communication with Clients and Prospective Clients Make appropriate disclosures. Distinguish between fact and opinion in analysis and recommendations. V(C) Record Retention Maintain records to support recommendations and decisions. 7-year retention period recommended. VI(A) Disclosure of Conflicts Disclose any matters that may impair independence and objectivity, prominently and in plain language VI(B) Priority of Transactions Execute clients’ transactions before accounts in which you have a beneficial interest. VI(C) Referral Fees Disclose referral fees to clients and employer, including non-monetary arrangements. VII(A) Conduct as Participants in CFA Institute Program Do not share confidential exam details. Expressing opinions about CFAI policies is permissible. VII(B) Reference to CFA Institute, the CFA Designation, and the CFA Program Do not misrepresent the meaning of CFA Institute membership, designation, or candidacy. BA II PLUS TIPS BACALCULATOR II PLUS CALCULATOR TIPS Basic Operations 2ND : Access secondary functions (in yellow) ENTER : Send value to a variable 2ND + ENTER : Toggle between options ↑ ↓ : Navigate between variables/options STO + 0 - 9 : Store current value into memory RCL + 0 - 9 : Recall value from memory Time Value of Money (TVM) For annuity, loan, and bond calculations N : Number of periods I/Y : Effective interest rate per period (in %) PV : Present value PMT : Payment/coupon amount FV : Future value/redemption value CPT + one of the above : Solve for unknown 2ND + BGN : Toggle between ordinary annuity and annuity due 2ND + CLR TVM : Clear TVM worksheet Note: - Always clear the TVM worksheet before starting a new calculation. - For bonds, PMT and FV should have the same sign, and opposite signs for PV. Cash Flow Worksheet ( CF , NPV , IRR ) For non-level payments Input ( CF ) CF0: Initial cash flow C01: 1st distinct cash flow after initial cash flow F01: Frequency of CO1 C0n: nth distinct cash flow F0n: Frequency of C0n Note: - Always clear the CF worksheet before starting a new calculation. - The use of F0n is optional. You can leave them as 1 and input repeating cash flows multiple times. If you do so, C01 will be the cash flow at time 1, C02 will be the cash flow at time 2, and so on. Output ( NPV , IRR ) I: Effective interest rate per period (in %) NPV + CPT : Solve for net present value IRR + CPT : Solve for internal rate of return www.saltsolutions.com Copyright © 2024 Salt Solutions. All Rights Reserved. Personal copies permitted. Resale or distribution is prohibited. 19