Alan Brown MSc, FIAA, AIA

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Risk Management for DC Pension Plans

• Alan Brown MSc, FIAA, AIA
• Swinburne Univerisity
• Larry Weldon PhD
• Simon Fraser University

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Talk Outline

• Investment Environment - DC Plan
• Traditional Approach - Risk Mgt.
• Special Features of Long Term Horizon
• Practicality of Short Term Management
• Exponential Utility and Risk Adjustment
• Optimal Asset Class Mix
• Concluding Remarks

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Investment Environment

• SFUs Defined Contribution Pension Plan
• No member choice until 2001
• 87 still in default plan.
• Heterogeneous mix in default plan.
• Usually long term in plan (25 yrs )

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Asset Class Mix?

• Balance of Risk and Return
• Balance of Variability and Return ?
• Is Risk Variability?

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Investment Risk

• Downside variability, over a particular time
  period
• How to quantize this?
• Depends on the time period
• What if 25 years or more?

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(No Transcript)
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Cumulative Market ReturnsCanada 1956-2004
Log Scale -gt
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Experience 1956-2004

• Equities outperform bonds in every 25 year span.
  What is risky?
• Can it be counted on for the future?
• Simulation approach Mimic stochastic features of
  the past without constraining the result

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Calibration of Simulation

• Equities 10 pa
• Bonds 7 pa
• Daily changes
• Equities Exponential with mean .5,
• Up with prob .547
• Bonds Exponential with mean .3 for down
• Up with prob.544
• Bonds Gamma with mean .3 and shape 2 for up

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Simulation 100, 25-year-periods
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Big Picture

• Longer term investment -gt longer term returns are
  realized
• Pension investing is very long term.
• Equities will produce the greatest long term
  returns (from risk premium).

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100 equities for long term ?

• Legal liability in a bad year
• Election as trustee in jeopardy
• Fund Managers dont like short term variability
• Therefore, need to reduce variability
• (i.e. include bonds)

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Short-term risk aversion

• Down-side variability to be avoided
• Down-side variability of large asset classes most
  important to control
• Consider the utility function for asset x u(x, R)
  1 exp(- x/R) for R gt 0R is the Total
  Amount at Risk.
• Note this is Relative Utility, max1.

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Risk Adjusted Value

• The Risk Adjusted Value of a random amount Y is
  defined as the unique solution of
• 1-exp(-z/R) 1-Eexp(-Y/R)
• This implies
• z ? rav(Y, R) - R log( Eexp(-Y/R) )
• Exponential Premium Principle -Buhlmann (1980) )

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Connection with Cumulants

• KY(t) log( Eexp(Yt) )
• rav(Y, R) - R log( Eexp(-Y/R) )
• rav(Y, R) - R KY(-1/R)

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Portfolio mixtures

• Z pX where p is a constant multiple then
• KZ(t) KX(pt)
• Also,
• Z pX qY where X and Y are independent
• Then
• KZ(t) KX(pt) KY(qt)

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Dependent Variables

• KZ(t) KX(pt) KY(qt) ?XY cX cY KX(pt) KY(qt)
• (X,Y are assets in two asset classes)

?XY - correlation (X,Y) cX- coefficient of
variation of X A useful approximation - data
available. Generalizable to pgt2 components
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Markowitz Problem

• Maximize Portfolio Return for given Var

? ?pi ?i subject to v ? 2 ? ? pi pj ?ij
?i ?j
?pi 1 and pigt0
?i is force of return for asset class i pi is
proportion of assets in asset class i
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Markowitz Efficient Frontier
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Utility (RAV) Maximization Problem

• maximise rav(?) rav( ?pi ?i )
• (mean return, adjusted for risk)

?pi 1 and pigt0
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RAV Maximization
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Mechanics of Optimization

• We used Solver in Excel, Cumulants for each asset
  class from past data (1984-2004), and past
  correlations used as well.
• Setting R1, we obtained optimum of

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Adjusting for Future

• old correlation matrix replaced by forecast
• exogenous prediction of future asset class
  correlations to assess sensitivity to correlation
  specification -gt result

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Sensitivity to Correlation shift?
A higher correlation between US Equities and
Foreign Equities has resulted in Foreign Equities
being dropped.
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E-mail Contacts

• Alan Brown MSc, FIAA, AIA
• Swinburne Univerisity
• abrown_at_labyrinth.net.au
• Larry Weldon PhD
• Simon Fraser University
• weldon_at_sfu.ca