Financial Risk Management: The Temporal Nature of Risk

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Financial Risk ManagementThe Temporal Nature of
Risk

• Part 2 of a series on
• A Multidimensional Approach to Risk Management.

Presented by Prof Chris Visser Sanlam Chair in
Investment Management School of Business and
Finance University of the Western Cape To the
Southern African Finance Association 18th Annual
Conference 14-16 January 2009
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Agenda

• The Global Financial Crisis
• Elements of Risk
• The Dimensions of Risk
• Revisiting the Square-Root-of-Time Rule
• Brownian and Fractural Brownian Motion
• The Hurst Coefficient and Memory Models
• Incorporating Memory into Financial Models
• Some Empirical Evidence

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What does the following people have in common?

• Botanist Robert Brown
• Theorist Albert Einstein
• Academics Myron Scholes and Fischer Black
• Hydrologist Harold Edwin Hurst
• Answer
• They basically faced the same problem.
• All studied various processes that takes place
  over time.

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The Global Financial Crisis Quick perspective

• Financial risk management has taken centre stage.
• Quants have both been blamed for the crisis and
  hailed as the saviour of the financial system.
• Herds of cloned quant analysts and financial
  engineers was let loose on Wall Street.
• Some must take blame for the bull-run that ended
  in tears.
• How did the wiz-kids get it so wrong?
• They neglect something critical from their market
  models. Why?
• Financial engineers built models based on
  physical systems, not behavioural systems.
• The dynamic nature of risk was not taken into
  account.

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The changing risk profile of US Mortgage Debt
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We need new ways to measure risk!

• To manage you have to measure.
• To measure you have to understand.
• That may require a rethink!
• Textbook definitions and models such as CAPM must
  be revisited.
• Equating risk to volatility is assuming too much.

Business Report - December 23, 2008
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Which is more risky?
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Defining Financial Risk

• Text book Financial (Market) Risk is
• Risk of loss from changes in financial markets or
  conditions.
• No reference to time or uncertainty.
• A More Concise Definition
• A probability of a
• loss (or below threshold returns)
• during a certain time period.
• 3 Elements to Risk
• Uncertainty measured by probability.
• A magnitude of loss (relative to something).
• Time horizon.
• Exposure is NOT Risk.

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Have a Good Look at Risk!
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The Multidimensional Nature of Risk

• Dimensions of Risk
• Relevance
• Direction
• Distribution
• Depth (Time horizon)
• Memory
• Attitude
• Perspective
• Association

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The Square-Root-of-Time Rule

• Risk increases with the root of the time horizon.
• This rule is prevalent in finance and risk models
• Value-at-risk
• VaR -W?(1-?)?t 0.5
• B-S Option Pricing
• d1ln(S/X)(r-q0.5s2)t/(st0.5)
• d2d1-st0.5
• This rule is being applied blindly in finance
  based on the assumption that price movements
  follow the gBm process.

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Brownian Motion
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Fractural Brownian Motion (fBm)

• Geometric Brownian Motion (gBm)
• Relative change in price is a static mean times a
  time period plus a uncertain component with a
  Gaussian distribution time the square-root of the
  time period.
• Alternative Model
• Lift the assumption that sequential price
  movements are independent (No memory).
• Log Returns follow a self-similar process called
  the Fractural (as in Mandelbrot) Brownian Motion.
• Call for the introduction of additional model
  parameter call the Hurst Coefficient (H)

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Implications of Autocorrelation in Returns

• If we assume that the 1st and higher order
  auto-correlation of consecutive returns are not
  zero.
• The square-root-of-time rule becomes
• Were ?n is the nth order autocorrelation
  coefficient of the stochastic process.
• Problem with this model
• Higher order correlations are needed as
  parameters.
• This is a discrete model. Fractions cannot be
  used.

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The Hurst Coefficient for Self-Similar processes

• Autocorrelation function time horizon (t) and the
  self similarity parameter Hurst Coefficient (H)
  or "index of dependence".
• Limits for H 0 ? H ? 1.
• H 0 strong negative autocorrelation or ?-0.5
  
• Risk therefore not related to time
• 0 lt H lt 0.5 partial negative autocorrelation or
  -0.5 lt? lt 0
• Short-term memory model or reverting process.
• H 0.5 zero autocorrelation or ?0
• Zero memory model. Random process.
• 0.5 lt H lt 1 partial positive correlation or 0
  lt? lt 1
• long-term memory model. Trending process
• H 1 indicates perfect positive correlation or ?
  lt 1.
• Perfect memory. Linear process. No Risk

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Fractural Brownian Motion Simulated
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Advantages of Hurst in Self-Similar Process

• The term horizon (t) is not limited to whole
  numbers. Any positive fraction or real number can
  be used
• A requirement since the fBm is a continuous
  process.
• The coefficient can easily be estimate using more
  that one method.
• It fits neatly into currently used risk models.
• Higher-order correlation coefficients are not
  required since this process is self-similar.

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Estimating Hurst Coefficient (H)
According to Beran(1994) the autocorrelation
function of a such a self similar process is
given as 1
If t1 then
If ??(1), then from previous equation we get

It can be shown that the time risk relationship
is
1 Beran, J.
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Estimating Hurst for SA Asset Classes
Hurst Coefficient based on monthly returns
From this table it is clear that Equities do
not have significant memory Bonds have short
memory and Exchange rate and gold have long
memory. This means that the risk projection over
the medium- to long-term for these assets will be
different and not follow the simple
square-root-of-time rule.
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Modification of Risk Models
To compensate for autocorrelation of returns
every
should be replaced with
Where
the self similarity parameter of the self
similar stochastic process.
Using this modified square root of time rule, the
basic VaR formula becomes VaR -W?(1-?)?t H
The d1 and d2 parameters in B-S option pricing
model becomes d1ln(S/X)(r-q0.5s2)t/(stH)
d2d1-stH
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Modified Value at Risk (MVar)

• The complete MVaR formula then becomes
• where
• W the market value of the asset/portfolio
• µ the mean of the natural log returns
• zc the adjusted Gaussian critical value for
  probability (a)
• s The standard deviation of the natural log of
  returns
• H Hurst coefficient of autocorrelation of log
  returns
• t Fraction of multiple of time horizon
  relative to frequency of returns.

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Applying MVaR - Time Horizon is Important.
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Conclusions

• Markets (People) have memory.
• Brownian motion process is assuming too much.
• Self-similar process of a fractural Brownian is a
  more realistic choice.
• Hurst coefficient (H) required to be added to
  models.
• Option pricing and value-at-risk models can
  easily be modified with H.
• Requires the estimation of an additional
  parameter.
• H is easy to estimate.
• Effort is justified for MT LT market models.

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Thank You

• Questions?

Contact Details Prof CF Visser Cell 083 675
6939 Email cvisser_at_uwc.ac.za chris.visser_at_telko
msa.net