Asset Allocation, Longevity Risk, Annuitisation and Bequests Dr. David Schiess, B. Sc. Group for Mathematics & Statistics Bodanstrasse 6, 9000 St. Gallen, Switzerland david.schiess@unisg.ch www.mathstat.unisg.ch David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 1/24
Outline ■ Motivation Outline ● Outline ■ Model Assumptions Motivation ■ Optimisation Problem Assumptions ■ Results Optimisation Problem Results ■ Conclusions Conclusions Thanks Back-up David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 2/24
Relevance ■ Importance of the End of the Life-Cycle: Outline ◆ Rising Conditional Life Expectancies Motivation ● Relevance ◆ Growing Number of DC Plans ● Technical Problem ● Literature ◆ Continuing Wealth Concentration Among Pensioners ● Extensions ◆ Input for Labour Models Assumptions Optimisation Problem Results Conclusions Thanks Back-up David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 3/24
Technical Problem ■ Technical View of the Pensioner’s Problem: Outline ◆ Consumption/Portfolio Optimisation ( c, π ) Motivation ● Relevance → Financial Market Risk ● Technical Problem ● Literature ◆ Optimal Annuitisation Decision ( τ ) ● Extensions → Longevity Risk Assumptions ■ ⇒ Combined Optimal Stopping and Optimal Control Optimisation Problem Problem (COSOCP) Results Conclusions Thanks Back-up David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 4/24
Literature ■ Literature Overview: Outline ◆ Merton (1969) → Stochastic Control Motivation ● Relevance ◆ Vast Literature Imposing a Fixed or Infinite Planning ● Technical Problem ● Literature Horizon ● Extensions ◆ Yaari (1965) → Uncertain Lifetime Assumptions ◆ Richard (1975) → Reversible Annuities Optimisation Problem ■ Few Normative Models with Irreversible Annuities and Results Uncertain Lifetime, i.e. Conclusions ◆ Milevsky and Young (2007): Thanks Commitment to Predetermined Annuitisation Time Back-up ◆ Stabile (2006): Annuitisation Rule as a Controlled Stopping Time David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 5/24
Extensions ■ Our Extensions to the Model of Stabile (2006): Outline ◆ Inclusion of a Bequest Motive Motivation ● Relevance ◆ Prior Life Insurance and Subsistence Level of Bequest ● Technical Problem ● Literature ◆ Economically Relevant Risk Aversion ( γ > 1 ) ● Extensions ◆ New Solution Method with Duality Arguments Assumptions Optimisation Problem Results Conclusions Thanks Back-up David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 6/24
Model Assumptions ■ Utility Maximisation (Consumption, Annuity, Bequest; Outline Identical Relative Risk Aversion) Motivation ■ No Stochastic Income Assumptions ● Model Assumptions → No Labour Income Optimisation Problem ■ Prior Decision on Annuitisation and Life Insurance Taken as Results Given Conclusions ■ Annuitisation of Entire Wealth and Consumption of Entire Thanks Annuity Back-up ■ One Riskless Asset, One Risky Asset (Geometric Brownian Motion) ■ Exponential Mortality Law David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 7/24
Indirect Utility Total Expected Discounted Utility J c,π,τ ( w ) : Outline Motivation T x Assumptions � � W ( τ ) � � � e − δ S t E U 1 ( c ( t )) 1 { t ≤ τ } + U 2 1 { t>τ } dt Optimisation Problem a x + τ ¯ ● Indirect Utility ● COSOCP 0 ● Verification Theorem �� + ηe − δ S T x � ● Variational Inequality U 3 ( W ( T x ) + Z s ) 1 { T x ≤ τ } + U 3 ( Z s ) 1 { T x >τ } Results Conclusions Thanks Back-up David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 8/24
COSOCP General COSOCP with Exponential Mortality: Outline Motivation V ( w ) = sup J c,π,τ ( w ) for all w > 0 Assumptions ( c,π,τ ) ∈G ( w ) Optimisation Problem ● Indirect Utility τ ● COSOCP � ● Verification Theorem e − β S t f ( c ( t ) , W ( t )) dt + e − β S τ g ( W ( τ )) J c,π,τ ( w ) = E w ● Variational Inequality Results 0 Conclusions Thanks dW ( t ) = W ( t ) [ r + π ( t ) ( µ − r )] dt − c ( t ) dt + σπ ( t ) W ( t ) dB ( t ) Back-up David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 9/24
Verification Theorem ■ Optimal Strategies: Outline ◆ Annuitisation rule Motivation Assumptions τ ∗ = inf { t ≥ 0 | W ∗ ( t ) / ∈ D } Optimisation Problem ● Indirect Utility ● COSOCP with ● Verification Theorem ● Variational Inequality D = { W ( t ) ∈ G | v ( W ( t )) > g ( W ( t )) } Results ◆ Consumption rule Conclusions c ∗ = I ( v W ( W ∗ ( t ))) 1 { t ≤ τ ∗ } Thanks Back-up ◆ Investment rule v W ( W ∗ ( t )) π ∗ = − µ − r W ∗ ( t ) v W W ( W ∗ ( t ))1 { t ≤ τ ∗ } σ 2 David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 10/24
Variational Inequality ■ The Verification Theorem Reduces the COSOCP to the Outline Variational Inequality: Motivation Assumptions max { L com v ( W ( t )) , g ( W ( t )) − v ( W ( t )) } = 0 for W ( t ) > 0 Optimisation Problem ● Indirect Utility ● COSOCP with ● Verification Theorem ● Variational Inequality L com v ( W ( t )) = f ( c ( t ) , W ( t )) − β S v ( W ( t )) + Lv ( W ( t )) � � sup Results ( c,π ) ∈G τ ( W ( t )) Conclusions Thanks ■ subject to Back-up v ( W ( t )) = g ( W ( t )) for all W ( t ) ∈ ∂D ■ and v W ( W ( t )) = g W ( W ( t )) for all W ( t ) ∈ ∂D. David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 11/24
No-Bequest Case ■ Now-or-Never Annuitisation: M nb Outline Motivation ■ Natural Parameter Effects Assumptions ◆ Risk Aversion (A + ) Optimisation Problem ◆ Subjective Life Expectancy (A + ) Results ◆ Objective Life Expectancy (A − ) ● No-Bequest Case ● Bequest Case γ < 1 ◆ Identical Life Expectancy (A − ) ● Bequest Case γ > 1 ◆ Sharpe Ratio (A − ) Conclusions ■ Annuitisation in Most Parameter Settings Thanks → Important Inclusion of a Bequest Motive Back-up David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 12/24
Bequest Case γ < 1 ■ Bequest Case γ < 1 and Z s = 0 : Outline ◆ Now-or-Never Annuitisation: M b = M nb + λ S η Motivation ◆ Slight Tendency for the Financial Market Assumptions → Important Inclusion of Bequest Motive Optimisation Problem ◆ Natural Parameter Effects Results ● No-Bequest Case ◆ Natural Comparison to No-Bequest Case ● Bequest Case γ < 1 ● Bequest Case γ > 1 c b c nb W b < ■ W nb Conclusions ■ W b > W nb Thanks ■ π b = π nb Back-up c b W b decreases in η ■ David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 13/24
Bequest Case γ > 1 ■ Bequest Case γ > 1 and Z s > 0 : Outline ◆ Never Annuitisation or Wealth-Dependent Annuitisation Motivation with D = ( W, ∞ ) Assumptions ◆ Natural Comparison to No-Bequest Case Optimisation Problem ◆ Real COSOCP with D = ( W, ∞ ) : Results ● No-Bequest Case ■ Simplification via Duality Arguments ● Bequest Case γ < 1 ● Bequest Case γ > 1 ■ Free Boundary Value Problem Conclusions ■ Numerical Solution Algorithm Thanks → Boundaries Back-up → Value Function ■ Natural Parameter Effects: → Life Insurance ( A + ) → Bequest Motive ( A − ) ■ Heavy Consumption Smoothing ■ More Aggressive Investment Rule Compared to Merton → Additional Option of Annuitisation David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 14/24
Conclusions ■ Conclusions: Outline ◆ COSOCP: New Solution Method Motivation ◆ Economically Important Risk Aversion γ > 1 Assumptions ◆ Longevity Risk Is Absolutely Relevant Optimisation Problem → Modelling of Lifetime Results → Role of Pension Funds Conclusions ● Conclusions ◆ Essential Inclusion of a Bequest Motive Thanks → Consumption-Wealth Trade-off Back-up → Absurd Strong Tendency for the Annuity Market Vanishes David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 15/24
Thanks Thank you very much for your attention! Outline Motivation Assumptions Optimisation Problem Results Conclusions Thanks ● Thanks Back-up David Schiess, 27th of March 2008, MAF 2008 Asset Allocation, Longevity Risk, Annuitisation and Bequests - p. 16/24
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