Computational Finance Applications in Investment Management Prepared by John Lafare Vice President & Director, Information Technology Group 2007 The Capital Group Companies, Inc. The Capital Group Companies 1 Sacred Symbols of Wall Street? 2007 The Capital Group Companies, Inc. 2 About the Capital Group Who we are: One of the worlds largest and most respected investment management organizations American Funds and Capital Group International 2007 The Capital Group Companies, Inc. Investment manager of mutual funds, separately managed accounts and pooled investment funds 9,500 associates in 19 offices worldwide Privately held organization founded more than 75 years ago

Experienced, long-tenured investment professionals Known for exceptional customer service 3 What is Computational Finance? Tools Used Types of Decisions Mathematical finance Numerical methods Computer simulations Trading Hedging Investment Computational Computational Finance Finance (or (or Financial Financial Engineering) Engineering) Objective: Risk Management

Determine the financial risk that certain financial instruments create 2007 The Capital Group Companies, Inc. 4 What Are The Tools of The Trade? The Quants Toolkit Linear Algebra Multivariate Calculus Stochastic Calculus Differential Equations Statistical Inference Probability Theory 2007 The Capital Group Companies, Inc. 5 Toolkit Example: MapleSoft 2007 The Capital Group Companies, Inc. 6 The Role Of The Quantitative Analyst

Application of mathematical and software technology to the realization of financial models Models are abstractions written in the language of mathematics and used to simplify decision-making The job of the quantitative analyst is to make sure they amount to credible representations of complex financial or economic realities Modeling is about using the logic of science to explore information from financial data without prejudice 2007 The Capital Group Companies, Inc. 7 Risk-Seeking Behavior Then Men Men wanted wanted for for hazardous hazardous journey, journey, small small wages, wages, bitter bitter cold, cold, long

long months months of of complete complete darkness, darkness, constant constant dangers, dangers, safe safe return return doubtful. doubtful. Honour Honour and and recognition recognition in in case case of of success. success. Advertisement Advertisement placed placed by by Earnest Earnest Shackleton Shackleton in in 1914. 1914. He He received received 5000 5000 replies. replies. 10 years ago Long-Term Long-Term Capital

Capital Management Management calculated calculated that that its its daily daily value value at at risk risk was was $35 $35 million million and and reduced reduced its its equity equity to to 3%. 3%. Later Later that that month, month, it it dropped dropped $550 $550 million million in in aa day. day. They They eventually eventually loose loose $4.5 $4.5 billion billion

and now Long-Term Long-Term Capital Capital Managements Managements former former broker, broker, Bear Bear Stearns Stearns also also gambles gambles that that it it can can survive survive with with aa similarly similarly weak weak balance balance sheet sheet (an (an identical identical 3% 3% equity) equity) and and is is pushed pushed to to the the brink brink of of bankruptcy bankruptcy before before aa government-orchestrated

government-orchestrated rescue rescue 2007 The Capital Group Companies, Inc. 8 What Happened? Before 1929, a computer model would have calculated very slim odds of a great depression; after it considerably greater odds. Before August 1998, Russia had not defaulted on its debt since 1917; when it did, the credit markets behaved in way LTCM didnt predict and wasnt prepared for. Before 2007, the United States had never suffered a nationwide contraction in housing prices; risk models assumed the pattern would hold. The lesson is that risk is commonly expressed as a function of potential market movement based on historical data, but history is at best an imprecise guide 2007 The Capital Group Companies, Inc. 9 Sobering Thoughts

Truth is Power Banner hanging in the BYU Computer Science Department Mathematics will draw the soul towards truth Plato Algorithms will draw the soul towards truth BYU Computer Science Faculty? But then there is data and there are assumptions 2007 The Capital Group Companies, Inc. 10 Common Flaws of Modern Finance It tends to eschew data that is messy and potentially misleading but nonetheless very real Instead, it favors abstraction which is perfect but theoretical Rather than evaluate financial assets case by case, financial models rely on the notion of randomness This has huge implications for diversification: two investments are safer than one, three safer than two, etc The theory of option pricing, devised by two LTCM partners, is based on the idea that each new price is random, like a coin flip

LTCM was therefore shocked when trades spanning multiple asset classes crashed in unison In other words, markets arent so random and in times of stress correlations rise 2007 The Capital Group Companies, Inc. 11 Exact and Inexact Models The logic of the scientific method requires that we let the complete set of financial data speak for itself, without prejudice This doesnt mean that we cannot transform the original data but we have to be very careful about how we do it The original data may be exact or inexact Realization refers to the construction of a model from exact data and implies that all the available data could have been produced by experimenting with the model Building a model from inexact data is called identification and means that not all the available data can be produced by experimentation

Garbage in, garbage out: the inexactness of the data is expressed in terms of uncertainty about the model and the model can only be as exact as the data used by its identification 2007 The Capital Group Companies, Inc. 12 Inexact Data: Investment Bank Interview Q Please explain to me how it was possible that the '100-year flood' stress tests that your Risk Management group supposedly runs did not do a better job of catching the impact of an increase in sub-prime mortgage cumulative losses to levels that we are seeing today, in an environment where people have been talking about poor underwriting standards, a potentially overleveraged consumer, and a housing bubble for at least the past two years? A Our stress tests on our CDO super senior holdings assumed they were AAA-rated assets and would behave as such in our stress tests. In other words, we did not stress test the collateral. We are now making the appropriate adjustments to the structure of the risk management committees. Q Are you kidding me??? A But, it is not as if other Wall Street firms didn't make the same mistakes .. 2007 The Capital Group Companies, Inc. 13 Computational Finance in Practice Equity Portfolio Optimization Portfolio Value-at-

Performance Risk Attribution Mortgage PrePayment Risk Analysis Some Models Portfolio Stress Used At Testing The Capital Group Portfolio Exposure Credit Risk Modeling Analysis Fixed Income Analytics 2007 The Capital Group Companies, Inc. 14 Portfolio Optimization: Modern Portfolio Theory

2007 The Capital Group Companies, Inc. For a given standard deviation, a rational investor will choose the portfolio with the greatest return. For a given level of return, a rational investor will choose the portfolio with the lowest standard deviation. A portfolio is said to be efficient if there is no portfolio having the same standard deviation with a greater expected return and there is no portfolio having the same return with a lesser standard deviation. The efficient frontier represents the collection of efficient portfolios 15 The Portfolio Construction Problem

Assume returns are normally distributed with mean r and covariance matrix Q The portfolio has an expected return of z = rTw and variance of 2 = wTQw Risk preference is captured by the constant k > 0 The optimal portfolio is determined by solving for the weighting parameter, w. Minimize variance subject to given return k minimize wT Qw subject to in1 riwi r * (minimum acceptable return) 2 n i1 wi 1 and w 0 Maximize return subject to given variance n k maximize i 1 riwi subject to wT Qw 2 2 Combining the models: balancing risk and return n i 1 wi 1 and w 0 a commonly used utility function describing the 1 2 2

1 - exp (-kz krelationship ) between risk and return 2 2007 The Capital Group Companies, Inc. 16 Portfolio Construction Example Optimal Portfolio Statistics : . Avg Return (Mthly%): 1.46 . Standard Deviation : 2.46 (Avg) . Return (annualized, pct) 18.9 Partial view of correlation matrix AT&T ALCOA ALLIED SIGNAL BETHLEEM STEEL 2007 The Capital Group Companies, Inc. AT&T 1.00 0.31 -0.12 0.19 ALCOA 0.31 1.00 0.02 0.64

ALLIED SIGNAL -0.12 0.02 1.00 -0.03 BETHLEEM STEEL 0.19 0.64 -0.03 1.00 17 Value at Risk (VaR): Concept Most commonly used to measure the market risk of portfolios Maximum loss that will occur within a given period of time, T, and a given probability prob (St+T - St -VaR) =

2007 The Capital Group Companies, Inc. My portfolio is worth $100,000 today and chances that by the end of the week it will be worth only $90,000 or less, are 5%. VaR does not give any information about the severity of loss by which it is exceeded Both its strength and weakness is that it relies on historical data of relatively short horizon 18 Calculating Value at Risk 2007 The Capital Group Companies, Inc. Historical Method. Simply re-organizes actual historical returns, ordering them from worst to best; assumes that history will repeat itself, from a risk perspective. Variance-Covariance Method. Assumes returns are normally distributed; requires estimation of only two factors, expected return and standard deviation, to plot a normal distribution curve.

Monte Carlo Simulation. Refers to any method that randomly generates trials; involves developing a model for future returns and running multiple hypothetical trials through the model. 19 Stress Testing Determine the size though not the frequency of potential losses related to specific scenarios A scenario may consist of large changes in the value of risk factors: interest rates, credit spreads, exchange rates, equity prices, commodity prices, etc A scenario may also correspond to extreme historical events such as the stock market crash of October 1987 (when stocks fell 22 times their daily standard deviation) The calculation that tells us how much the portfolio might loose under such scenarios is called as stress test During market crises, correlations change as volatilities increase The major benefit of stress testing is the identification of a portfolios potential vulnerability

2007 The Capital Group Companies, Inc. 20 Stress Test Example 1 Visualization of the portfolios behavior in terms of profits and losses as a function of changes in two risk factors: yield curve and volatility In this chart, which focuses on the two basic risk factors affecting these securities, the outcomes look quite good. There is considerable upside and very little potential for loss. 2007 The Capital Group Companies, Inc. 21 Stress Test Example 2 Catastrophic Stress Tests (Horizon: One Month; Unit: Basis Points) Historical Events Hypothetical Events 1 Monthly

Risk Factor Duration Period 1 Period 2 10-yr Treas. Spot Rate 6.0 79 -11 37 ATM MBS OAS 15.0 12 45 Mtg/Treas. Basis -2.0 12 10-yr AAA Corp. Spread

7.0 Implied Volatility 3.0 Loss (% of Market Value) SD Mortgage 3 SD Event Spreads Simultaneous 110 0 100 11 32 50 75 45 11

32 50 75 7 27 9 28 0 75 75 80 69 208 0 300 -8.98% -9.54%

-6.31% -18.94% -6.50% -30.00% Example above presents a variety of historical and hypothetical stress tests for a fixed income portfolio It expresses sensitivities to a selected number of systematic risk factors: interest rates, mortgage spreads, corporate spreads, implied volatilities 2007 The Capital Group Companies, Inc. 22 Portfolio Credit Risk Modeling

Portfolios of bonds or loans generally show asymmetric returns, with limited upside and substantial, though rare, downside risk associated with multiple corporate defaults. Credit risk is the risk of losing contractually obligated cash flows promised by a corporation, financial institution, government, etc. (the counterparty) due to default on the debt obligation. The key modeling questions cover a range of issues from default probability to correlation of defaults to recovery. Because corporate behavior is so complex and the reasons for issuing debt and for default are so varied, models tend to rely on empirical data and economic intuition over mathematical processes. In some cases mathematical models are chosen for their analytic tractability and are calibrated and adjusted to match empirical evidence. 2007 The Capital Group Companies, Inc. 23 Credit Risk Modeling Example: Moodys KMV Estimate asset value and asset volatility: Market Value Defaulted of Assets

Default Point November 2001 Value Distribution of asset value at horizon Solve for implied asset value and volatility Calculate Distance to Default: (Liabilities Due) Equity is a call option on asset value Contractual obligations determine Default Point Number of standard deviations from default Scale Distance to Default to Expected Default Frequency: Assign EDF using actual historical default rates Asset Volatility (1 Std Dev)

Asset Value Distance-to-Default = Default Point 3 Standard deviations EDF Today 2007 The Capital Group Companies, Inc. 1 Yr Time 24 Performance Attribution Attribution is an analytical technique used to evaluate the performance of a portfolio relative to a benchmark It tells us where value was added or subtracted as a result of the managers decision The primary focus is to find the source of the portfolios excess return Contribution refers to the impact of various sub-portfolio

elements impact the portfolios overall return Multi-Currency Fixed Income Portfolio Return Decomposition Capital Portfolio Managers Decisions 1. 2. 3. 4. Income Currency allocation Total Return Yield curve position (duration) Sector allocation Duration Bond selection Slope 2007 The Capital Group Companies, Inc. Risk Premium Shift Spread Selection

Currency Allocation Timing 25 Performance Attribution Example 6.8% 6.1% 3.5% 3.3% 0.2% 2.8% 0.5% 3.5% 2007 The Capital Group Companies, Inc. 26 Portfolio Exposure Analysis Interest Rate Risk Assess exposure to shifts in the Spread Risk

reference yield curve Risk Assess risk exposure for global Assess exposure to general shifts in credit spreads Global Currency/ Curve/Country Concentration Risk portfolios Assess exposure to specific issuers 2007 The Capital Group Companies, Inc. 27 Portfolio Exposure Analysis Examples 2007 The Capital Group Companies, Inc. 28 Prepayment Risk Analysis

Prepayments are usually the primary distinguishing feature of mortgage-backed securities (MBS) Understanding and forecasting prepayments is essential to the successful evaluation of MBS as investments Prepayments arise from moving, refinancing, and default Technically, defaults are not prepayments but to security holders the effect is the same The key to understanding prepayments is the data which has become increasingly available at the loan-level, and for a variety of loan types and interest rates environments Prepayment forecasting faces the challenge of seeking to estimate future events in a changing world But the primary guide remains past prepayments and models seek to explain past prepayments using econometric or statistical approaches The hope is that this information provides valuable insights into future prepayments 2007 The Capital Group Companies, Inc. 29 Prepayment Analysis Using the Ad Co Model In addition to using historical data to fit model parameters, the model is adjusted using stress tests and scenario analysis The impact of changes in the prepayment model parameters is

also evaluated using a variety of valuation and risk measures Model Factors Turnover & seasonality Refinance incentive Yield curve spread effect Home price (cash-out) effect Credit cure effect Active vs. passive borrowers 2007 The Capital Group Companies, Inc. 30 Financial Analysis of IT Investments Questions? 2007 The Capital Group Companies, Inc. 31