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Computational Finance
Applications in Investment
Management
Prepared by John Lafare
Vice President & Director, Information Technology Group
The Capital Group Companies
© 2007 The Capital Group Companies, Inc.
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Sacred Symbols of Wall Street?
© 2007 The Capital Group Companies, Inc.
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About the Capital Group
Who we are:
One of the world’s largest and most respected
investment management organizations
American Funds and Capital Group International
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Investment manager of mutual funds, separately managed
accounts and pooled investment funds
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9,500 associates in 19 offices worldwide
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Privately held organization founded more than 75 years ago
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Experienced, long-tenured investment professionals
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Known for exceptional customer service
© 2007 The Capital Group Companies, Inc.
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What is Computational Finance?
Tools Used
Types of Decisions
• Mathematical finance
• Numerical methods
• Computer simulations
• Trading
• Hedging
• Investment
Computational Finance (or
Financial Engineering)
Objective: Risk Management
Determine the financial risk that
certain financial instruments create
© 2007 The Capital Group Companies, Inc.
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What Are The Tools of The Trade?
The Quant’s Toolkit
Linear
Algebra
Multivariate
Calculus
Stochastic
Calculus
Probability
Theory
© 2007 The Capital Group Companies, Inc.
Differential
Equations
Statistical
Inference
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Toolkit Example: MapleSoft
© 2007 The Capital Group Companies, Inc.
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The Role Of The Quantitative Analyst
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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.
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Risk-Seeking Behavior
Then …
“Men wanted for hazardous journey, small wages, bitter cold, long months of
complete darkness, constant dangers, safe return doubtful. Honour and
recognition in case of success.”
Advertisement placed by Earnest Shackleton in 1914. He received 5000 replies.
… 10 years ago
Long-Term Capital Management calculated that it’s daily “value at risk” was $35
million and reduced its equity to 3%. Later that month, it dropped $550 million in a
day. They eventually loose $4.5 billion
… and now
Long-Term Capital Management’s former broker, Bear Stearns also gambles that
it can survive with a similarly weak balance sheet (an identical 3% equity) and is
pushed to the brink of bankruptcy before a government-orchestrated rescue
© 2007 The Capital Group Companies, Inc.
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What Happened?
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Before 1929, a computer model would have calculated very slim
odds of a great depression; after it considerably greater odds.
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Before August 1998, Russia had not defaulted on its debt since
1917; when it did, the credit markets behaved in way LTCM
didn’t predict and wasn’t prepared for.
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Before 2007, the United States had never suffered a nationwide
contraction in housing prices; risk models assumed the pattern
would hold.
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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.
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Sobering Thoughts
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Truth is Power
Banner hanging in the BYU Computer Science Department
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Mathematics will draw the soul towards truth
Plato
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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.
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Common Flaws of Modern Finance
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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 aren’t so random and in times of stress
correlations rise
© 2007 The Capital Group Companies, Inc.
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Exact and Inexact Models
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The logic of the scientific method requires that we let the
complete set of financial data speak for itself, without prejudice
This doesn’t 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.
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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.
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Computational Finance in Practice
Equity Portfolio
Optimization
Portfolio Valueat-Risk
Mortgage PrePayment Risk
Analysis
Performance
Attribution
Some Models
Used At
The Capital Group
Portfolio Stress
Testing
Portfolio Exposure
Analysis
Credit Risk
Modeling
Fixed Income
Analytics
© 2007 The Capital Group Companies, Inc.
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Portfolio Optimization: Modern Portfolio Theory
© 2007 The Capital Group Companies, Inc.
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For a given standard deviation, a
rational investor will choose the
portfolio with the greatest return.
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For a given level of return, a
rational investor will choose the
portfolio with the lowest standard
deviation.
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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.
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The efficient frontier represents
the collection of efficient portfolios
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The Portfolio Construction Problem
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Assume returns are normally distributed with mean r and covariance matrix Q
The portfolio has an expected return of z = rTw and variance of σ2 = 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 T
minimize
w Qw subject to  in1 riwi  r * (minimum acceptable return)
2

n
i 1
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wi  1 and w  0
Maximize return subject to given variance
k
n
maximize i 1 riwi subject to wT Qw   2
2

n
i 1
wi  1 and w  0
Combining the models: balancing risk and return
1 - exp (-kz 
1 2 2
k  ) a commonly used utility function describing the
2
relationship between risk and return
© 2007 The Capital Group Companies, Inc.
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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
AT&T
1.00
0.31
-0.12
0.19
ALCOA
0.31
1.00
0.02
0.64
© 2007 The Capital Group Companies, Inc.
ALLIED
SIGNAL
-0.12
0.02
1.00
-0.03
BETHLEEM
STEEL
0.19
0.64
-0.03
1.00
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Value at Risk (VaR): Concept
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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 α
prob (St+T - St ≤ -VaR) = α
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© 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
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Calculating Value at Risk
© 2007 The Capital Group Companies, Inc.
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Historical Method. Simply re-organizes
actual historical returns, ordering them
from worst to best; assumes that history
will repeat itself, from a risk perspective.
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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.
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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.
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Stress Testing
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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
portfolio’s potential vulnerability
© 2007 The Capital Group Companies, Inc.
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Stress Test Example 1
Visualization of the
portfolio’s 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.
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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%
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Example above presents a variety of historical and
hypothetical stress tests for a fixed income portfolio
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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.
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Portfolio Credit Risk Modeling
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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.
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Credit Risk Modeling Example: Moody’s KMV
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Estimate asset value and asset
volatility:
–
Defaulted
November 2001
Market Value
of Assets
–
Default Point
(Liabilities Due)
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Calculate Distance to Default:
–
Value
Distribution of
asset value at
horizon
Asset
Value
Asset Volatility
(1 Std Dev)
Distance-to-Default =
3 Standard deviations
Default Point
EDF
Today
© 2007 The Capital Group Companies, Inc.
1 Yr
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Equity is a call option on
asset value
Solve for implied asset value
and volatility
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
Time
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Performance Attribution
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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
manager’s decision
The primary focus is to find the source of the portfolio’s excess
return
Contribution refers to the impact of various sub-portfolio
elements impact the portfolio’s overall return
Multi-Currency Fixed Income Portfolio Return Decomposition
Portfolio Manager’s Decisions
1. Currency allocation
2. Yield curve position (duration)
3. Sector allocation
4. Bond selection
Capital
Total Return
Duration
Slope
© 2007 The Capital Group Companies, Inc.
Income
Shift
Risk Premium
Spread
Selection
Currency
Allocation
Timing
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Performance Attribution Example
6.8%
3.5%
6.1%
3.3%
0.2%
2.8%
0.5%
3.5%
© 2007 The Capital Group Companies, Inc.
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Portfolio Exposure Analysis
Interest Rate Risk
Spread Risk
Assess exposure to
general shifts in
credit spreads
Assess exposure to
shifts in the
reference yield curve
Global Currency/
Curve/Country Risk
Concentration Risk
Assess risk exposure
for global portfolios
Assess exposure to
specific issuers
© 2007 The Capital Group Companies, Inc.
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Portfolio Exposure Analysis Examples
© 2007 The Capital Group Companies, Inc.
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Prepayment Risk Analysis
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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.
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Prepayment Analysis Using the Ad Co Model
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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.
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Financial Analysis of IT Investments
Questions?
© 2007 The Capital Group Companies, Inc.
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