Training on Financial Risk ManagementInstructor : Abhijit Biswas Indian Institute of Quantitative Finance
Abhijit Biswas
Director and Head of Product Development at Risk Infotech Solutions, one of Indias first company for Portfolio Risk Management Software Products. Founder Director of IIQF. With over twelve years worth of experience in research and development in the field of Financial Engineering, is one of the pioneers of Risk Modelling Technologies in India. He is also an expert in Monte-Carlo Simulation theories and systems and advanced simulation technologies applied to finance and general business risks. As a Quantitative Finance professional, created numerous breakthroughs in Risk Modelling Technology in India. Developed Indias first commercial grade large scale Monte Carlo Simulation system for business analytics using Excel spreadsheet models. Consultant to major global financial institutions in risk management domain. Conducts trainings on Risk Management, Statistics, Econometrics, Simulation and related disciplines at Stock Exchanges and Financial Institutions.
OutlineIntroduction to Financial Risk Management
Financial disaster case study - LTCM
Financial Economics
Log Normal property of Returns, Skewness & Kurtosis: Implications for Risk Management Factor models and Arbitrage Pricing Theory (APT) Implementing an APT Based Risk Model in Excel Employing CAPM for performance evaluation of Portfolios/Funds
OutlineMeasures of Risk
Coherent measures of risk Metrics of Market Risk Fixed Income Duration, DV01/PV01/PVBP, Standard Deviation and VaR Equities Beta, Standard Deviation / Volatility and VaR
Value-at-Risk Single-Index Model and Beta Systematic/Market/Non-Diversifiable Risk Non-Systematic/Residual/Diversifiable Risk
Conditional VaR / Tail VaR / Expected Shortfall Downside Semi-Variance
Introduction to Financial Risk Management
Risk Basics
Notion of Risk and Financial Risk Types of Financial Risk Market : the risk that declining prices or volatility of prices in the financial markets will result in a loss. Credit : the possibility of default by a counter-party in a financial transaction, and the monetary exposure to credit risk is a function of the Probability of Default and the Loss Given Default. Operational : the risk from various Operational Failures, Frauds etc. Liquidity : the possibility of sustaining significant losses due to the inability to sufficiently liquidate a position at a fair price. Legal, Business, Catastrophe, etc.
LTCM Case StudyBackdrop
Started in 1994 by John Meriwether Former Head of Bond Trading at Salomon Brothers. Included Nobel laureates Myron Scholes and Robert C. Merton (they got the prize in 1997). High net worth investors minimum ticket size of $10mn Started trading in early 1994 with equity base of about $1 billion. Awarded returns in excess of 40% to its investors in its initial years of operations.
LTCM Case StudyStrategies : LTCMs main strategies were:
Fixed-Income Arbitrage Convergence Trades : Buying the cheaper off-the-run bonds and short selling the more expensive, but more liquid, on-the-run bond, it would be possible to make a profit as the difference in the value of the bonds narrowed. Merger Arbitrage Pairs Trading Statistical Arbitrage
LTCM Case StudyProblem :
As differences in Convergence Trades were low so to make significant profit the fund took highly leveraged positions as high as 25:1 debt-to-equity. At the beginning of 1998, the firm had equity of $4.72bn and had borrowed over $124.5bn with assets of around $129 billion. As the capital base grew there were not enough good bondarbitrage opportunities, so they adopted aggressive strategies like merger arbitrage, statistical arbitrage (specially they were short S&P 500 vega),etc.
LTCM Case StudyProblem :
East Asian financial crisis 1997 significant losses in May/June 1998. Losses aggravated by Salomon Brothers exiting arbitrage business in July 1998. Aug/Sept 1998 Russian Govt. defaulted on their bonds.
LTCM Case StudyProblem :
Flight to quality panicked investors sold European and Japanese bonds to buy US treasury bonds resulting in increasing prices of on-the-run bonds and hence divergence instead of convergence Result huge losses to the tune of $1.85bn. Margin pressure and flight-to-safety leading to more liquidations at highly unfavorable prices and suffers further losses in pairs trades.
LTCM Case StudyLessons :
Model risk Mathematical models used by LTCM assumed historical relationships would predict future in a reliable manner Correlation between low frequency high severity events understated. Therefore tail risk not captured.
Inadequacy of VaR to capture tail risks. LTCM management believed in risk management and tracked 1-month VaR seriously. However, they failed to look beyond that and did not look at scenario analysis and stress tests. Need of better risk metrics to capture tail risks. Also failed to consider liquidity risk into consideration
Financial Economics
Financial Economics
Log Normal Property of financial returns: ln(P1) ln(P2) ~ N(Q,W)
Stock returns are supposed to be log normal. Implications for risk estimation.
Financial Economics
Skewness and Kurtosis of return distributions Implications for Risk Management Black Swans : Assuming normality in the presence of skewness and/or kurtosis leads to under-estimation of risk.
Factor Models and Arbitrage Pricing Theory (APT)
CAPM deficiencyOne of the limitations of CAPM is the determination of market portfolio as a stock market index. Its use is largely restricted to the equity markets and index taken as market portfolio does not contain any other asset classes CAPM also assumes complete knowledge of risk premium
Arbitrage Pricing TheoryIt is a multiple factor model of excess returns It assumes that excess return of a portfolio can be expressed as an impact of k factors and is given by E(Rp) = Xp,k * mk + up where Xp,k = exposure/loading of stock p to factor k mk = factor return for factor k un = stocks return that cannot be explained by other factors APT says excess return of a stock/portfolio can be determined by its factor exposure and factor returns, although it does not specify a clear method to find it APT calls for regressing with certain identified factors
APT and CAPM
APT with k=1, factor loading as beta of the stock and factor exposure as market risk premium, reduces to CAPM model Alternatively, APT betas can be treated as components to the overall Beta used in CAPM CAPM requires knowledge of expected market returns. APT requires no such data. It simply uses risky assets expected return in relation to certain economic and company factors CAPM is generally suited for a more stable market, where beta remains relatively constant. APT is more suited for out-of-sample forecasting
APT ModelAny APT model that is capable of explaining the risk of a well-diversified portfolio is a qualified model Thus, if our APT model can explain the variations in returns of a broad index, say S&P 500, they qualify to be an effective APT model Once we have a qualified APT model, the next step is concerned with identifying the factor betas or factor loadings Once we have a model with factor exposures, we need to consider factor returns. This is generally done by a cross-sectional regression.
Arbitrage Pricing Theory (APT)Choosing factors for APT models: Factors that are quantifiable Factors that are common to several stocks, but have differential effects on each of them or we can say that the companies have different exposures to each of the factors Factors whose effect persists over time Factors that have intuitive or theoretical significance
APT Model
In general, we should only consider factors for which a reasonable forecast can be made APT models can be structural or statistical models Under the structural model, we consider factors as specific variables which have an underlying relationship with stocks and thus carry a definite explanation. For example, a stocks industry group, earnings yield, size and so on With structural model, factor forecasts are easier and the relationship turns intuitive There are a host of statistical techniques available to estimate the relationship between a stock and various indices. However, explanation is not intuitive.
Assumptions of APT Model
APT model is less restrictive compared to CAPM. It requires only three of its assumptions: Investors are risk-averse and utility maximisers Investors can borrow and lend at risk-free rate There are no taxes or transaction costs
Further, there are two additional assumptions for APT There is no arbitrage opportunity Investor agree on number and identity of factors that are necessary for pricing of assets
Multifactor Model
Specify a factor model Measure a-priori factor exposures for each return period Carry out cross-sectional regression for each period to get the factor returns Estimate the variance-covariance matrix of factor returns
Multifactor Model
Stock variance-covariance matrix is given by V = E * F * ET Portfolio variance is given by: S = HT * V * H
Where V = Stock variance-covariance matrix E = Stock factor exposure matrix F = Factor variance-covariance matrix H = Portfolio holdings vector S = Portfolio Variance
Multifactor Model
Excel Workshop
CAPM and Performance Evaluation of Funds/Portfolios
Performance AnalysisIt is the process of assessing the amount of risk being taken while investing in a particular asset or portfolio vis-a-vis the amount of return being generated by it. It further involves analyzing the returns generated and attributing it to various sources. The typical structure of a risk-adjusted performance measure is: risk-adjusted performance = Performance / Risk
Goal of Performance Analysis?
The goal of performance analysis is to distinguish skilled and unskilled investment managers.
It helps us to know whether the manager is earning alpha returns above the risk adjusted benchmark returns. It helps us to know whether the returns are because of skill or luck.
Performance Analysis
Measuring Returns Arithmetic Return R = (P1 P0) / P0 Logarithmic Return R = ln(P1 / P0)
Performance Analysis
Average Returns Arithmetic Average Return
Geometric Average Return 1 + Rg = [(1+R1)*(1+R2)*.*(1+Rt)]^ (1/T)
Performance Analysis
Measures of Realized Returns Excess Return = Portfolio Return Risk-free Rate Active Return = Portfolio Return Benchmark Return
Performance Analysis
Measures of Realized Risk Portfolio Returns SD Portfolio Excess Returns SD Portfolio Active Returns SD Beta
Performance Analysis
Benchmark Selection Benchmarks for index funds Benchmarks for stylized funds Benchmarks for sector funds
Treynor Ratio
Measures the systematic risk-adjusted return performance of the portfolio. Risk is measured as the Beta of the Portfolio. Treynor Ratio = Mean Excess Return Beta
Treynor Ratio
A high positive value indicates that the manager has achieved superior beta-adjusted returns. It compares the returns with the systematic risk of the portfolio. It is more suitable for well diversified portfolios.
Sharpe Ratio
Measures the risk-adjusted return performance of the portfolio. Risk is measured as the total Standard Deviation of Realized Portfolio Excess Returns. Sharpe Ratio = Mean Excess Return SD of Excess Return
Sharpe Ratio
A value greater than 1 indicates that the manager has achieved good risk-adjusted returns. A value higher than the benchmarks SR indicates superior performance. It compares the returns with the total risk of the portfolio. It is more suitable for non diversified portfolios.
Sharpe Ratio
The statistical significance of the Sharpe Ratio is measured with the t-statistic. A t-stat of 2 or more is indicative of skill at 95% confidence level.
t = (SRp SRb) / Sqrt(2/N) Where : SRp = Sharpe Ratio of portfolio SRb = Sharpe Ratio of benchmark N = Number of time periods
Sortino Ratio
Measures the portfolio return performance against the realized downside risk i.e. the standard deviation of the negative returns. Sortino Ratio = Mean Excess Return SD of Negative Excess Returns
Sortino Ratio
It compares the portfolio return performance against the downside risk assumed by it. It is more appropriate for hedge fund performance analysis.
Information RatioMeasures the return performance of the portfolio vis--vis its benchmark. Measures the Active Return generated per unit of Active Risk taken by the manager.
Information Ratio = Mean Active Return SD of Active Return
Information RatioIR is a measure of the value-added by the manager. The bigger the value of IR the higher is the value added and better is the performance. The statistical significance of the IR is measured by the t-test:
t = IR / Standard Error of IR Standard Error of IR is approximated as : SE(IR) = 1/ Square-root(number of observation)
Jensens Alpha
It uses the CAPM predicted return to assess the performance of the portfolio. Rp Rf = E + F * (Rm Rf)
where Rp = Portfolio Return Rf = Risk-free Rate Rm = Market Return E = Jensens alpha F = CAPM Beta
Jensens Alpha
Statistically significant positive Alpha indicates that the manager has earned better risk adjusted returns. Statistically significant positive Alphas over a period of time would indicate that the manager has skills rather than being lucky. t = (E 0) / Standard Error of E
More on Performance Attribution
Additive Attribution (BHB)The portfolio return in excess of the benchmark return is broken into three components Allocation, Selection and Interaction. Rp Rb = A + S + I A S I = 7 [np [nb) Rbn = 7 [nb (Rpn Rbn) = 7 [np [nb) (Rpn Rbn)
Additive AttributionRp Rb A S I [np [nb Rpn Rbn = Portfolio return = Benchmark return = Allocation Effect = Selection Effect = Interaction Effect = Weight of sector n in portfolio = Weight of sector n in benchmark = Return of sector n in portfolio = Return of sector n in benchmark
Excel Workshop
Sharpe Ratio Treynor Ratio Jensens Alpha Sortino Ratio
Measures of Risk
Coherent Measures of RiskAny measure of risk, say R, is called a Coherent Measure of risk, if it satisfies the following properties:
Monotonicity Positive Homogeneity Translation Invariance Sub-additivity
Coherent Measures of RiskMonotonicity: If X = R(Y)
That is, if we have two portfolios X and Y such that portfolio Y always has better values than portfolio X under all scenarios then the risk of Y should be less than the risk of X
Coherent Measures of RiskPositive Homogeneity : For b >= 0, R(bX) = b R(X)
Means that if a portfolio is multiplied by a constant b then the risk is also multiplied by b.
Coherent Measures of RiskTranslation Invariance For constant c, R(X + c) = R(X) c
The value c is just like adding cash to your portfolio X, the risk of X + c is less than the risk of X, and the difference is exactly the added cash c.
Coherent Measures of RiskSub-additivity : R(X + Y)
