Fast Portfolio Computations

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Fast Portfolio Computations

• Thomas F. Coleman
• Dean, Faculty of Mathematics
• Professor, Combinatorics and Optimization
• University of Waterloo

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Alternative title
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• Portfolio Speed with Brains or Brawn?

or
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Portfolio Trends

• Larger and more complex portfolios
• Less restrictive assumptions ? simulations
  required
• Movement to real-time/near-time computations ?
  (semi-) automated trading systems
• Complex hedging strategies (long-dated)
• Increased competitiveness
• Increased regulatory requirements (e.g.,
  VaR-related)
• Increased appreciation of robustness properties
• (more) speed is needed

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• Should (increased) speed be achieved by
• Brains
• Or
• Brawn.

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Speed by Brains

• Many portfolio computations are structured ?
  structure can be used to extract speed.
• Speed gains can be stunning
• Textbook codes will not do
• Packages will (likely) have to be tailored
• Hire an expert in computational mathematics,
  scientific computing, practical optimization

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A (very) Simple Example

• Background
• Solving systems of linear equations fundamental
  to everything (computational)
• 
  Ax b
• If A is n-by-n,
• If A is sparse (i.e., mostly zero),
• (eg work n, or work n?logn).

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A (very) Simple Example

• Background
• Solving systems of linear equations fundamental
  to everything (computational)
• 
  Ax b
• If A is n-by-n,
• If A is sparse (i.e., mostly zero),
• (eg work n, or work n?logn).

!
!
!
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Comparing Costs
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Example continued

• Suppose

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Example continued

• Suppose

sparse

Note A is dense!
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Solve Axb
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(No Transcript)
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Solve Axb
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Comparing Costs
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Matlab experiment T tridiagonal
Speedup
n
Subtle
Straight
1000
1.4
.06
23
5000
1200
.1
120
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Application to Portfolio Optimization (simplified)

• Portfolio optimization often contains the form

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Application to Portfolio Optimization (simplified)

• Portfolio optimization often contains the form

n-by-n
t
n
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Application to Portfolio Optimization (simplified)

• Portfolio optimization often contains the form

n-by-n
t
n
t ltlt n
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Application to Portfolio Optimization (simplified)

• Portfolio optimization often contains the form

Covariance
(OPT)
n-by-n
t
Additional linear relationships
n
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Solution
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Factor Assumption

• Many portfolio modelers make a factor assumption
• Assume q risk factors,
• Q

Residual risk mtx. Diagonal
Covariance mtx of risk factors. q-by-q
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Factor Assumption

• Many portfolio modelers make a factor assumption
• Assume q risk factors,
• Q

Residual risk mtx. Diagonal
Covariance mtx of risk factors. q-by-q
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Factor Assumption

• Many portfolio modelers make a factor assumption
• Assume q risk factors,
• Q

Structure!

Covariance mtx of risk factors. q-by-q
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Solve OPT

• Method 1 (Straight)

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Solve OPT

• Method 2 (Subtle)
• Form Matrix C, vector d.
• Solve Cy d (a standard dense solve)
• Form b
• Solve (a diagonal system!)

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(qt)-by-(qt)
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Solve OPT

• Method Subtle
• Form Matrix C, vector d.
• Solve Cy d (a standard dense solve)
• Form b
• Solve (a diagonal system!)

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Solve OPT

• Method Subtle
• Form Matrix C, vector d.
• Solve Cy d (a standard dense solve)
• Form b
• Solve (a diagonal system!)
• Note!

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Comparing costs
n
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Speedup
n
Subtle
Straight
1000
1.4
.06
23
5000
1200
.1
120
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Speed by Brawn

• Just evaluation of a portfolio (sensitivities)
  can be very expensive e.g., portfolios of
  derivatives may require extensive simultations
• VaR..
• Optimization including VaR

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• How to calculate future portfolio prices/hedging
  solutions/sensitivities from an Excel spreadsheet
  with existing modest resources FAST
• An expensive supercomputer is not required.
• Moving to a specialized environment is not
  necessary!

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Example questions you may be afraid to ask now
(because you wont get an answer in useful
time/reasonable cost)

• How will my portfolio value change under a shift
  and a twist in the yield curve? How are VaR/CVaR
  affected?
• In a range of possible yield curve scenarios,
  with a range of possible stock values, what are
  the worst-case scenarios for my portfolio of
  convertibles 3 months from now, 6 months from
  now?
• How sensitive is my portfolio to default of the
  bond holders?

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Our Solution Framework

• Use parallelism, under .Net/web services.

The Environment
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Why .NET/webservices?

• (Increasingly) commonly available.
• A commodity environment not specialized and
  expensive.
• Rich master environment all the familiar
  tools are available.
• The parallel back-end can be many things a
  shared resource existing resource, a collection
  of workstations, a dedicated rack of
  processors,the only requirement is that they be
  identified as web servers

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Some of the problems amenable to this (brawny)
approach

• Computing the fair price of portfolios of
  financial instruments
• Future values under different scenarios.
• Computing sensitivities to various risk factors
• Hedging, hedging, hedging.
• Value at risk, Conditional value-at-risk!!

Dont cheat!
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Example Pricing Portfolios of Complex Bonds (on
a Windows cluster, under .Net, using Web Services)

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Pricing a callable bond

• Very complex to price American style feature
  exercise decision and current value depend on
  stochastic interest rates
• Least Squares Monte Carlo (LSM)
• Longstaff and Schwartz (2001)
• Equips Monte Carlo method with regression
  analysis to compute fair prices of any future
  cash flow.
• Callable bond price Non-callable bond price
  call option price

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demo
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Concluding Remarks

• Brains or Brawn?

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A Hierarchy
Increasing computational cost

• Price a complex bond
• Price a portfolio of complex bonds
• Compute VaR of the portfolio
• Optimize

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A Hierarchy
Increasing computational cost

• Price a complex bond
• Price a portfolio of complex bonds
• Compute VaR of the portfolio
• Optimize

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A Hierarchy
Increasing computational cost

• Price a complex bond
• Price a portfolio of complex bonds
• Compute VaR of the portfolio
• Optimize

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A Hierarchy
Increasing computational cost

• Price a complex bond
• Price a portfolio of complex bonds
• Compute VaR of the portfolio
• Optimize

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Thank you for listening!

• Feel free to email me with any questions, etc
• tfcoleman_at_uwaterloo.ca