Financial Modeling PDF

lecture 1 Needed for pricing, hedging, forecasting, Hedge fund etc. - Lots of restrictions Models are always a simplification of the Hedge exchange rate risk with derivativ...

lecture 1 Needed for pricing, hedging, forecasting, Hedge fund etc. - Lots of restrictions Models are always a simplification of the Hedge exchange rate risk with derivatives real world Only need to capture main features CAPM does not price value, momentum relevant for pricing and size premium Models can always be rejected by the What Is a Quantitative Model? data “All models are wrong, but some are Model is abstract representation of useful” real-world financial situation Relevant question: “How inaccurate is Typically involves mathematical the model?” equations and random variables Need to understand purposes (scope) Simplification of more complex structure and limitations of models Relies on set of assumptions for simplification “A model is just a toy, though occasionally Often implemented using spreadsheet or a very good one, in which case people call programming language it a theory” “Models can be worse than nothing, the In corporate finance/accounting, financial equivalent of a dangerous operation on a modeling refers to financial statement patient who would stand a better chance if forecasting, capital budgeting, untreated” valuation, etc. Often deterministic models in discrete 2008 collapse cause of models (too much) time (e.g., discounted cash flow model) could not assess the real situation In investments/quant finance, financial modeling refers to asset pricing, return prediction, portfolio optimization, risk measurement Often probabilistic models in continuous time (require advanced math/stats) Examples: CAPM, Fama-French X factor model (discrete models), Black-Scholes (options), Vasicek (term structure), GARCH (discrete volatility), Heston (continuous volatility), etc. Model Classifications (I) Financial modeling is as much an art as it 1. Empirical (data driven) vs. theoretical is a science model Why Do We Need Models? Empirical model tries to describe underlying data generating process as Models are essential for financial good as possible: from data to model engineering and trading e.g., machine-learning model (how test validity? no data left) 1  - great way to see what works Panel: track multiple units over multiple time periods Theoretical model makes testable predictions based on abstract reasoning Classifications are not mutually exclusive, and set of assumptions: from theory to e.g., Black-Scholes model is example of model e.g., Black-Scholes model theoretical, stochastic, - hypothesis testable predictions continuous-variables, continuous-time model 2. Deterministic vs. probabilistic Model Optimization (stochastic) model Models often involve some type of Deterministic model does not have optimization (max/min output var) random/uncertain term e.g., CAPM OLS regression model: minimize sum of squared errors Probabilistic model does have Profit model: find optimal price level that random/uncertain term e.g., regression maximizes firm’s profit (line datapoints, residual = random Mean-variance portfolio choice: quantity that might be added) maximize return for given level of risk 3. Discrete vs. continuous variables model Two main approaches for optimization Analytical: use calculus to derive Discrete variable can take only discrete first-order condition and set to zero (specific) values binomial tree Numerical: use brute force approach (try many different values of input variables) Continuous variable can take any value Analytical approach is fast and allows within certain range BS model calculating sensitivity to inputs, but not always feasible Model Classifications (II) Numerical approach is slower but more flexible (e.g., easily handle constraints) 4. Discrete vs. continuous time model Numerical approach does not guarantee unique solution (sensitive to starting Discrete time: value of variable can values) change at fixed points in time (binomial tree) Simple optimization can be performed using Excel’s Solver tool Continuous time: value of variable can change at any point in time (BS model) Model Simulation 5. Cross-sectional vs. time-series vs. Monte Carlo simulation is powerful panel data model computer-based technique used to analyze and account for uncertainty in Cross-sectional: comparing multiple units financial decision making (companies, stocks) at point in time MC simulation algorithm involves repeated sampling of model input Time series: track one unit (company, variables from probability distribution to stock, country) over time obtain distribution of model output variables 2  Provides numerical solution to math Examples: incorrect price for product, problem when no analytical solution exists incorrect risk exposure/hedging Example 1: simulate random cash flows Financial institutions have dedicated to generate probability distribution of departments for model validation project NPV makes risk of investment explicit in capital budgeting Pros and Cons of Models decisions Main drawbacks: Example 2: simulate from stochastic Model can give false of precision and process by drawing random numbers to lead to generate many possible paths that overconfidence stock price can take in future widely used Model can be hard to understand for for risk management and pricing of broader audience complex derivatives products Model can miss important aspects (“soft” information) Main benefits: Model formalizes way of thinking and highlights main mechanisms Model makes key assumptions explicit Model preserves institutional knowledge Model testing can provide useful new insights Model-based decision making less prone Model Risk to behavioral biases Definition: “Potential loss institution may incur as consequence of decisions that In practice, complex model not always are principally based on the output superior to simple model! of internal models, as a result of errors in the development, implementation, or use Module 1.1: Introduction to Hedge Funds of such models” (KPMG, 2016) What are Hedge Funds? A private investment pool, open to Sources of model risk: institutional or wealthy Specification (design): conceptual investors, that is largely exempt from SEC mistakes, misspecification of statistical regulation and can model pursue more speculative investment Implementation: policies than mutual funds spreadsheet/programming flaws, wrong Idea: Sophisticated/wealthier investors inputs, estimation error need less protection Application (scope): function creep (less Broad term that encompasses funds that aware of assumptions) , misinterpretation follow very different of model outputs strategies and have different risk and Environment: time-varying parameters, return profile relations break down (regime shifts) Hedge fund strategies focused on absolute returns 3 Hedge Funds Private investment vehicle May use extensive leverage May engage in short selling May use derivatives Large minimum investment Liquidity often low - It takes a while for the investor to get out of the hedge fund - Redemption period 1 or 2 months - In crisis periods, investor get locked in for several months Flexible strategies Manager pay based on performance Manager invests own capital Aim for absolute return Mutuals funds SEC registered (in US) Limited use of leverage Usually no short selling Usually no derivatives Small minimum investment Daily redemption Limited strategy flexibility Manager paid salary and bonus Typically no own capital invested Aim to outperform benchmark Commodity: In finance, a commodity refers to a standardized, tangible asset or raw material that is bought and sold, often Domiciles offer HF attractive tax and via futures contracts, in commodity regulatory climates markets. Sharpe ratio:The Sharpe Ratio is a Module 1.2: Hedge Fund Fees and financial metric used to measure the Performance risk-adjusted return of an investment or portfolio. average return in excess of rf / std excess returns. return compensation per a unit of risk 4 diversification hedge fund limited Global macro: Involves long and short diversification benefits of hedge funds positions in capital or derivative markets increased across the world. Portfolio positions reflect Module 1.3: Hedge Fund Strategies views on broad market conditions and major economic trends. Long/short equity hedge: Equity-oriented positions on either side of the market (i.e. long or short), depending on out-look. Not meant to be market neutral. May establish a concentrated focus regionally (e.g. U.S. or Europe) or in a specific sector (e.g. tech or health care stocks). Derivatives may be used to hedge positions. Hedge Fund Strategies Managed futures: Uses financial, currency, Convertible arbitrage: Hedged investing in or commodity futures. May make use of convertible securities, typically long technical trading rules or a less structured convertible bonds and short stock. judgmental approach. Dedicated short bias: Net short position, Multistrategy: Opportunistic choice of usually in equities, as opposed to pure strategy depending on outlook. short exposure. Fund of funds: Fund allocates its cash to Emerging markets: Goal is to exploit several other hedges funds to be market inefficiencies in emerging markets. managed. Typically long-only because short-selling is not feasible in many of these markets. - Hedge fund follow dynamic strategies Equity market neutral: Commonly uses long/short hedges. Typically controls for -CAPM does not evaluate hedge fund industry, sector, size, and other performance well exposures, and establishes market-neutral positions designed to exploit some market -systematic beta= market beta inefficiency. Commonly involves leverage. Event driven: Attempts to profit from situations such as mergers, acquisitions, restructuring, bankruptcy, or reorganization. Fixed-income arbitrage: Attempts to profit from price anomalies in related interest rate securities, includes interest rate swap arbitrage. U.S. versus non-U.S. government bond arbitrage, yield-curve arbitrage, and mortgage-backed arbitrage. Alpha Transfer - Re equal to CAPM Rf + Alpha 5  - Long position stocks and short in index - See where you have skill and do long Investment Flexibility - Pension funds are only allowed to have investment rate bonds - -Alpha Transfer: Actual Return can still deviate based on the following factors: 1. Whether the CAMP holds 2. Was the Alpha forecast actually correct? 3. Beta forecast was correct? 4. Idiosyncratic risk that you cannot hedge away : positive or negative? Module 1.4: Why Do/Did Hedge Funds Do So Well? - Performance hedge funds hard to measure - HF used to outperform MF 6 Methodological issues - CAPM slope is equal to the market beta Methodological issues (VII) - Non-linear relation between trend followers and equity market - Trend followers have high returns when market does very well or very poor - Similar payoff profile as option straddle, even though trend-following funds do not take option positions 7  - Trend followers invest more as market rises (synthetical call) and go short as market falls (synthetic put) - Implication: alpha should be measured relative to passive straddle strategy, not equity index Exercises - Straddle: long position in call, long position put with same strike price - Correlation zero because non - Residuals are the components linear of the returns that cannot be - Up beta positive and down explained by markets negative How many contracts? Methodological Issues (VIII) dollar value market exposure needed (= portfolio x beta) / dollar value per - Deal is going to fail (Key risk) contract (=contract multiplier x where - Arbitrage related to put the contract is at) = number of - Loss during market downturns contracts Buy or sell? Do opposite action of what have done before (long position = sell, short position = buy) “Assuming that monthly returns are normally distributed, what is the probability that this market-neutral strategy will lose money over the next month?” Rf + alpha Calculate by hand: Z = ( x - average ) / std Looking up probability In Excel: =norm.dist 8 “Suppose you hold an equally ○ Investors buy weighted portfolio of x stocks with redeemable trust the same alpha, beta and residual certificates. standard deviation. Assume that ○ Typically focused on residual returns (e) on each of these specific asset types (e.g., stocks are independent of each municipal bonds). other. What is the residual standard 2. Managed Investment deviation of the portfolio?” Companies: ○ Active management of Independent so factor (x) less. the portfolio. Residual variance= root of ( 1/x). from ○ Includes closed-end variance to std by deleting the root. funds (traded on Doing the answer times the residual exchanges) and std deviation. open-end funds (redeem shares on r of a portfolio = rf + B(Rm -Rf + e + a demand at NAV). (CAPM) NAV stands for Net Asset Value. It Fund Of Funds Return and Stand- represents the per-share value of a Alone Fund Return fund's assets minus its liabilities. For each individual hedge fund Key Points: calculate: start of year value, end of 1. Closed-End Funds: year value (= start of year value + ○ NAV is calculated, but gross portfolio rate of return), end of these funds trade on year value after fee, rate of return exchanges. (=difference in start and end year ○ Market prices of the percentage) shares can deviate from NAV due to supply and Incentive fee cannot be lower than 0 demand. 2. Open-End Funds: FF larger fee than SA, individual fee ○ Shares are bought or paid for SA sometimes 0 redeemed directly with the fund at NAV. Chapter 4: Mutual Funds and ○ NAV is typically Other Investment Companies calculated at the end of Types of Investment Companies each trading day. 1. Unit Investment Trusts: NAV is crucial because it provides a ○ Fixed portfolio; no active benchmark to determine whether the management. market price of a closed-end fund is trading at a premium (above NAV) or a discount (below NAV). 9 On demand means that investors can 2. Equity Funds: Subcategories buy or sell (redeem) their shares include growth, value, or directly with the fund whenever they sector-specific funds. choose, typically at the fund's Net 3. Bond Funds: May specialize in Asset Value (NAV). government, municipal, or high-yield bonds. For open-end funds, this generally 4. Balanced and Asset works as follows: Allocation Funds: Combine stocks and bonds for risk 1. Buying shares: Investors can balancing. purchase shares from the fund 5. Index Funds: Aim to replicate a at the current NAV. market index at a low cost. 2. Redeeming shares: Investors can sell their shares back to the Costs of Mutual Funds: fund, again at the current NAV. 1. Operating Expenses: This process happens at the end of Management fees and other each trading day when the NAV is administrative costs. calculated, ensuring transactions 2. Load Fees: reflect the most accurate valuation of ○ Front-end load: Fee the fund's underlying assets. when purchasing shares. ○ Back-end load: Fee 3. Exchange-Traded Funds upon selling shares. (ETFs): ○ 12b-1 Fees: Marketing ○ Traded like stocks, and distribution costs. typically tracking market indexes. Performance Considerations: ○ Offer liquidity and low-cost diversification. Mutual fund managers often 4. Real Estate Investment Trusts underperform broad market (REITs): indexes due to fees and ○ Focused on real estate turnover. or mortgages. Use of benchmarks like the ○ Two main types: Equity Wilshire 5000 Index to compare trusts (invest in active vs. passive strategies. properties) and Mortgage trusts (invest in loans). Chapter 26: Alternative Assets Mutual Fund Categories: The Alternative Asset Universe: 1. Money Market Funds: Invest in 1. Hedge Funds: short-term, low-risk instruments. ○ Often structured as private partnerships. 10  ○ Strategies include ○ Include management derivatives, leverage, fees (1-2%) and and short selling. incentive fees (e.g., 20% ○ Focus on "absolute of returns above returns" rather than benchmarks). benchmarking. 2. Private Equity: Role in Portfolios: ○ Includes venture capital Low correlation with traditional (early-stage investments) asset classes provides and leveraged buyouts. diversification. ○ Active management to Can enhance portfolio Sharpe improve firm value. ratios by improving ○ Typically illiquid with long risk-adjusted returns. investment horizons (e.g., 10 years). The Sharpe Ratio is a measure of 3. Real Assets: risk-adjusted return that helps ○ Examples: Real estate, investors understand how much natural resources, and excess return they are receiving for the commodities. extra risk they are taking with an ○ Investment can be direct investment. (e.g., owning property) or indirect (e.g., REITs). 4. Structured Products: ○ Combine securities and derivatives for Performance Challenges: custom-designed outcomes. Alternative assets often face ○ Examples include survivorship and backfill biases mortgage-backed in reported performance data. securities. Long-term horizons and illiquidity complicate evaluation. Key Characteristics of Alternative Assets: Excel Workshop 1 1. Transparency: - Aggregate if missing numbers ○ Limited compared to or 3rd largest value etc. mutual funds; often minimal public disclosure. 2. Liquidity: ○ Typically low; long lock-up periods. 3. Fee Structures: 11  Matrix transposition = shifting lows with columns mmult for matrix times data table for changing variables solver for max etc. - left etc formulas to split text xlookup - - trim - text formula for formatting - date for creating dates part 2: and/ or for conditions and use of or 12  Minverse Row Vector: Matrix with one row. Column Vector: Matrix with one column. Square Matrix: Equal rows and columns. ○ Symmetric Matrices: Important in finance (e.g., variance-covariance matrices). 2. Matrix Operations: ○ Scalar Multiplication: Multiply every element by a scalar (k for example). ○ Addition: Add corresponding elements of matrices with the same dimensions. 3. Matrix Multiplication: ○ Non-commutative property: AB≠BA. Matrix Inverses: Identity Matrix: Square matrix with ones on the diagonal and zeroes elsewhere. ○ Chapter 30: Excel Functions 1. Financial Functions: Chapter 29: Matrices ○ NPV: Calculates net 1. Matrix Basics: present value of cash ○ Definitions: flows.(voer getal rate in en values in in excel) 13  ○ IRR: Finds the rate ○ Multi-line cells: Use where NPV equals [Alt] + [Enter] or zero.(voer values in in Wrap Text. excel) 3. Advanced Techniques: ○ PMT: Computes periodic ○ Naming cells for better payments for formula clarity. loans.(nper= #payments) ○ Using Personal Macro ○ XIRR: Modified IRR for Workbook for uneven cash flows. automating repetitive 2. Date Functions: tasks. ○ DATE, NOW, TODAY: Handle and manipulate Module 2.1: Introduction to dates. Performance Evaluation ○ YEARFRAC: Fraction of - Choice for performance metrics the year between two depends on the portfolio dates. - passive strategy = investing in ○ Conditional and etf Reference Functions: - goal of performance evaluation ○ IF, COUNTIF, INDEX ways is to separate risk from RETURNS VALUE, skill MATCH: Logical and lookup functions. 1. INDEX Function Purpose: Returns the value of a cell based on its row and column numbers in a specified range. 2. MATCH Function Purpose: Finds the position of a specific value in a range of cells. Chapter 32: Excel Hints 1. Efficiency Tips: ○ Create series using Fill | Series. 2. Dynamic Features: 14 - use leverage to move along the line Jensen’s Alpha - Measures excess return relative to set benchmark factors - Commodity Trading Advisors (CTAs) - TS= time series - Small Minus Big (SMB): A factor representing the return difference between small-cap and large-cap stocks (size effect). - High Minus Low (HML): A factor representing the return difference between high book-to-market (value) stocks and low book-to-market (growth) stocks. 15 - Error Term: Represents the part of the portfolio return unexplained by the model (idiosyncratic or unsystematic risk). Module 2.2: Performance Measurement: Empirical Issues Skill, Luck, or Risk? - Being successful is probability, no consensus how to measure - Mom= momentum - AUM stands for Assets Under Management Survivorship bias occurs when only successful investments, funds, or companies are analyzed, while those that failed are ignored. The survivor effect focuses on the traits or strategies of investments or fund managers that succeeded, often attributing their success to specific actions or skills while ignoring luck or external factors. 16  - Market beta by Treynor: b+2xcxmarket risk premium positive c= steeper slope characteristic line c positive->fund higher exposure market index when Module 2.3: Performance the market premium high Measurement: Style Analysis such a fund good timing ability (?) - little evidence timing ability MF Impact or Risk Switching on MF Performance 17  - MF observe holdings every quarter - HF don’t observe holdings at all - MF return depends more on asset allocation than actual which within the category Suggested Exercises 18 Chapter 5: Risk, Return, and the Historical Record 1. Measuring Returns Chapter 6: Capital Allocation to Risky Assets 1. Capital Allocation Line (CAL) Represents combinations of the risk-free asset and a risky portfolio. Slope = Sharpe Ratio 2. Optimal Portfolio Choice Utility Function: Balances risk and return for an investor A: Risk aversion coefficient. 19  Optimal Risky Portfolio: The estimate the mean return point on the CAL where utility is for each asset. maximized. ○ Variances (σi2​): Quantify the total risk of individual asset returns. ○ Covariances Chapter 7: Efficient (Cov(Ri,Rj)): Measure Diversification how the returns of assets co-move. Correlation 2. Covariance and Correlation coefficients can Covariance Measures how standardize these returns on two assets move relationships. together. 2. Construct the Portfolio: Correlation Coefficient (ρ): ○ Define portfolio weights Standardized covariance, (wiwi​) to allocate capital ranges from -1 to +1. to each asset. 3. Efficient Frontier Represents the set of portfolios with the highest return for a given level of risk. Portfolios on the efficient frontier are optimal compared to 3. Find optimal weights (Excel those below it. Solver) Markowitz Portfolio Optimization 4. Identify the Efficient Frontier Model 1. Purpose of the Model Minimize portfolio risk (variance) for a given level of return. Optimize asset allocation to Monte Carlo Simulations: achieve the most efficient ○ Generate thousands of risk-return trade-off. portfolio combinations to visualize and 2. Key Steps in the Model validate the efficient frontier. 1. Estimate Inputs: ○ Expected Returns (E[Ri]): Use historical data or forecasts to Chapter 24: Portfolio Performance Evaluation 20  ○ Passive managers focus on matching a Attribution Analysis benchmark. ○ Active managers aim to 1. Purpose: Decomposes portfolio outperform, justified by performance to identify sources positive alpha. of returns. 2. Key Components: Lab session 2 ○ Allocation Effect: Impact of asset allocation choices across sectors or asset classes. ○ Selection Effect: Impact of specific security selection within asset classes. ○ Interaction Effect: Combined impact of allocation and selection decisions. Challenges in Performance Evaluation 1. Survivorship Bias: ○ Performance looks better when failed or underperforming funds are excluded from analysis. 2. Backfill Bias: - log better accumulate over time ○ Occurs when fund returns are only included after successful performance, skewing historical results. Applications in Real-Life Portfolio Evaluation 1. Passive vs. Active Management: 21 - matrix multiplication depends on the amount of columns/rows - Formulas work for small ammount of stocks 22 Σ: Covariance matrix of the asset returns. - c= think of a line The "s.t." in the mathematical optimization problem stands for "subject to". It introduces the constraints of the optimization problem, which are conditions that the solution must satisfy. - u= weights - variance, covariance matrix (S)= - c= constant , z= vector 23 - check whether right with 0,1 - maximum drawdown= which period loses most of p 24  replicating portfolio Ordinary Least Squares (OLS) regression is a statistical method used - for solver put some input values to model the relationship between a - solver, things with 0 not dependent variable (also called the diversified response variable) and one or more - regression, 0 short long independent variables (predictors or - more = better explanatory variables). It is one of the - if one can replicate the portfolio, most widely used techniques for linear the manager doesnt have regression. superior skills plotting a chart showing the efficient frontier and the capital market line 1. scatter plot 2. 3. select data 4. 5. dubbel click graph->data labels 25 6. 7. (plot envelope) 13. 8. select data new series 14. Plotting the efficient market portfolio and risk-free asset 9. ( Plotting combinations of the envelope portfolios to obtain the efficient frontier) 15. select data add new series 16. Plotting combinations of the market portfolio and risk-free asset to obtain the CML 10. select data add new series 17. select data dd new series 11. 12. 18. 19. Book 26  - The efficient frontier represents the best risk-return trade-off. - The tangent portfolio maximizes the Sharpe ratio. Module 3.1: Modern Portfolio - fix the input or the output Theory in Practice: Problems (constraints) 1. basics p optimization 2. builds on mean-variance theory,excess returns based on - In-sample=In context of portfolio beta and RP choice refers to returns you 4. Efficient Market Hypothesis: p would’ve realized if you reflect all info would’ve implemented the portfolio strategy over the same period that is used to estimate the input parameters. Same period to evaluate input parameters and to evaluate performance of the strategy. You cannot realize this performance in practice. You cannot implement the portfolio before security returns that you need to estimate (the input parameters) have been realized. Timing of estimating 27  parameters and inputting the portfolio. Otherwise look ahead bias. - Oos= estimation and evaluation period differ. - if you evaluate the performance of a strategy over the same period that you use to estimate the input parameters of that strategy, it is not very surprising that the strategy would do well in that in sample period. however, the real test of the strategy is whether it does well in a future period that has not been used for estimating this strategy inputs. - Inter= international - EW = equal weights portfolio - Low risk stocks higher risk adjusted performance - Overweights low risk stocks - Bonds use to perform well so risk parity may depend on doing well then 28  (conclude) the markets opinion, about future returns. These market implied expect returns are then used as a starting point and can be combined when the investor's own view about expected returns. Module 3.2: Modern Portfolio Theory in Practice: Solutions Robustness refers to the ability of a system, model, or process to remain effective and reliable under a variety of conditions, including stress, uncertainty, or changing environments. - factors models= which factors to use - and another method to obtain better estimates of the covariance matrix is to use a shrinkage estimator that combines the standard sample estimate of the covariance matrix with a factor based estimate to mitigate the impact of extreme covariance estimates, caused by - however, a drawback of this measurement error. method is that the covariance - b if Er constant estimates that we obtain are - A black Letterman model, which biased if the factor structure that uses a reverse mean variant we assume does not optimization procedure to infer adequately describe the source of comovement between asset 29  return..in that case, the assumption that the idiosyncratic returns are uncorrelated will be violated, covariance estimates are wrong. a potential solution to this problem is to replace a single factor model by another factor model. But, this implies that additional factor betas must be estimated. - for asset iii at time ttt. - noisy - reduce oos performance - Black littner fices this and makes it more practical 30  Article Plenary Lecture 2 Book - the weird thing = precision - weighted average of the believe and ratings 31 32