1 the arbitrage-free equilibrium pricing of liabilities in an incomplete market: application to a...

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The arbitrage-free equilibrium pricing of liabilities in an incomplete market:application to a South African retirement fund

Hacettepe University 27/6/2011

Rob Thomson

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Agenda

1. Introduction

2. Liabilities specification & model

3. Pricing method

4. Results and sensitivity

5. Conclusions

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Introduction: aim

Apply the pricing method of Thomson (2005) to:• market-portfolio model (Thomson, unpublished);• equilibrium asset-category model (Thomson & Gott, 2009);• a DB retirement-fund model;with a view to operationalising the pricing of such a fund and

quantifying the effects of:• non-additivity due to incompleteness;• guarantees implicit in reasonable expectations of pension

increases; and• the sensitivity of the price of illustrative liabilities to

sources of risk and parameters of the model.

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Introduction: method

Market-portfolio model

Return on market

portfolio

Mortality model

Liabilities specification

Risk-free return

Inflation rate

Salary model

Equilibrium asset-categories

model

Returns on asset

categories

Pension-increases

model

Asset-liability model

Price of liabilities

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Return on market

portfolio

Mortality model


Risk-free return

Inflation rate

Salary model


model

Returns on asset

categories

Pension-increases

model



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Return on market

portfolio

Mortality model


Risk-free return

Inflation rate

Salary model


model

Returns on asset

categories

Pension-increases

model



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Liabilities specification & model

• no exits before retirement

• mortality only after retirement

• projected unit method

• salaries and pensions expressed in real terms

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Liabilities specification & model: salaries model

, 1 expmt m t t tS S -

1 3 2 7t t t tb b

t x x t

expx x -

expx x -

22

, 1

x

xx tM

-

0,01

1 0,005b -

2 0,005b 0,03

0,016 0,5 0,1

0,042 0,5

0,08

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Liabilities specification & model:pension increases

1 exp max 0,t t tP P - -

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Liabilities specification & model: pensioner mortality

00{ }98 98

{ } { }00{ }

PNLxSAP SAIL

x xILx

{ } { } 1 expSAP SAPx t x t t -

98{ } { } exp 10SAP SAP

x x

where:

, 1 7t t t tb - 0,004 -

0,001b -

0,005

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Pricing method: primary & secondary simulations

primary simulation secondary simulation

Time T -1 Time T

primary simulation node

Time 0 Time 1 Time t -1 Time t

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Pricing method: state-space vector

1

1

1

N

It

It u

Ct

tCt u

t

x t

x t

P s

P s

P s

P s

P

P

x

where:

exp ( )It ItP s Y s -

exp ( )Ct CtP s Y s -

expt t

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Pricing method: primary simulations of the state-space vector

time 0 time 1 time 1 time T - 1

primary simulation 1 x 11 x 21 x T - 1,1

primary simulation 2 x 12 x 22 x T - 1,2

x 0

primary simulation I x 1I x 2I x T - 1,I

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Pricing method: secondary simulations

time T - 1 time T time T - 1 time T

x* T 11, p* T 11 x* T 11, p* T 11

x T - 1,1 x* T 12, p* T 12 x T - 1,1, p T - 1,1 x* T 12, p* T 12

x* T 1J, p* T 1J x* T 1J, p* T 1J

x* T 21, p* T 21 x* T 21, p* T 21

x T - 1,2 x* T 22, p* T 22 x T - 1,2, p T - 1,2 x* T 22, p* T 22

x* T 2J , p* T 2J x* T 2J , p* T 2J

x* TI 1, p* TI 1 x* TI 1, p* TI 1

x T - 1,I x* TI 2, p* TI 2 x T - 1,I , p T - 1,I x* TI 2, p* TI 2

x* TIJ , p* TIJ x* TIJ , p* TIJ

Final year

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time t - 1 time t time t time t time t - 1 time t

x* t 11 x* t 11, p* t 11 x t 1, p t 1 x* t 11, p* t 11

x t - 1,1 x* t 12 x* t 12, p* t 12 x t 2, p t 2 x t - 1,1, p t - 1,1 x* t 12, p* t 12

x* t 1J x* t 1J, p* t 1J x* t 1J, p* t 1J

x* t 21 x* t 21, p* t 21 x* t 21, p* t 21

x t - 1,2 x* t 22 x* t 22, p* t 22 x t - 1,2, p t - 1,2 x* t 22, p* t 22

x* t 2J x* t 2J , p* t 2J x* t 2J , p* t 2J

x* tI 1 x* tI 1, p* tI 1 x* tI 1, p* tI 1

x t - 1,I x* tI 2 x* tI 2, p* tI 2 x t - 1,I , p t - 1,I x* tI 2, p* tI 2

x* tIJ x* tIJ , p* tIJ x tI , p tI x* tIJ , p* tIJ

year t

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time 0 time 1 time 1 time 1 time 0 time 1

x* 11 x* 11, p* 11 x 11, p 11 x* 11, p* 11

x 0 x* 12 x* 12, p* 12 x 12, p t 12 x 0, p 0 x* 12, p* 12

x* 1J x* 1J , p* 1J x 1I , p 1I x* 1J , p* 1J

year 1

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Pricing method

2 2 1ˆˆ ˆ ˆ ˆt Ft FVt Vt FVt Σσ σ- -

1ˆ ˆt Vt tfΣ 1Vtz μ- -1

t tt

m zz 1

VttMt μm ˆˆ tVttMt mΣm ˆˆ 2 ˆ ˆHMt t FVt m σ

*2

ˆ ˆ ˆˆˆ

HMt t MtFt

Mt

*, 1

1 ˆˆ ˆL t Ft Ft Mt tt

P ff

- - -

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Results and sensitivity

Sex Age Value per unit accrued pension Aggregate value deterministic

valuation stochastic

price deterministic

valuation stochastic price

1 member entire cohort %

incr %

incr R’million %

incr (1) (2) (3) (4) (5) (6) (7) (8) Female

25 14,11 16,00 13,4 15,84 -1,0 17 19 12,2 35 11,94 13,46 12,8 13,36 -0,8 172 193 11,8 45 12,00 13,49 12,4 13,39 -0,7 311 347 11,6 55 12,67 13,70 8,1 13,62 -0,6 347 372 7,5 65 13,36 13,78 3,1 13,78 0,0 434 447 3,1 75 9,53 9,69 1,7 9,69 0,0 236 240 1,7 85 5,96 6,01 0,9 6,01 0,0 62 63 0,9 total 1 579 1 681 6,5

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deterministic valuation

stochastic price

R’million % incr Female 1 579 1 681 6,5 Male 1 351 1 427 5,6 Total 2 930 3 109 6,1 Aggregate 2 930 3 094 5,6 Adjusted to fund data 2 912 3 074 5,6

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Deterministic valuation of model-point data 2 930difference due to risk-free stochastic pricing 11Risk-free stochastic price 2 941hedge-portfolio risks –19Stochastic price with hedge-portfolio risks 2 922residual risks –20Stochastic price: hedge-portfolio & residual risks 2 902cost of guarantee 192Stochastic price based on model-point data 3 094adjustment to fund data –20Stochastic price based on fund data 3 074

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Results and sensitivity: major effects

Parameter Test result name description standard

value test

value price

(R’million) change in price

(%) standard values 3 094 N/A

1b general salary increase: sensitivity to inflation

–0,005 0 3 098 0,14

g return on market portfolio: sensitivity to risk-free rate

1,39 1,2 3 082 -0,37

M return on market portfolio: residual volatility

0,159 0,1 3 108 0,45

b force of inflation: residual volatility

–0,01379 0 3 061 –1,07

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Conclusions• Method computationally demanding, but not impossible: 41 hours.

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Conclusions• Method computationally demanding, but not impossible.• Convergence complicated, but Sobol numbers expedite it.

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Conclusions• Method computationally demanding, but not impossible.• Convergence complicated, but Sobol numbers expedite it.• Stochastic price 5,6% higher than deterministic: because of pension

guarantee (may be overstated).

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Conclusions• Method computationally demanding, but not impossible.• Convergence complicated, but Sobol numbers expedite it.• Stochastic price 5,6% higher than deterministic.• Without guarantee, stochastic price only 1% less than deterministic:

If the valuation of the liabilities should allow for a risk premium only to the extent that the trustees are unable to avoid risk, then the valuation basis must be much closer to a risk-free basis than that produced by the risk premiums typically used.

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Conclusions

• Method computationally demanding, but not impossible.• Convergence complicated, but Sobol numbers expedite it.• Stochastic price 5,6% higher than deterministic: because of

pension guarantee.• Without guarantee, stochastic price only 1% less than

deterministic.• Effects of non-additivity: intra-cohort 0-1% plus inter-cohort

0,5%.

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Conclusions




0,5%.• Major sensitivities:

volatility of force of inflation in excess of conditional ex-ante expected inflation;sensitivity of ex-ante expected returns on the market portfolio to risk-free returns;residual volatility of the return on the market portfolio.

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Conclusions




0,5%.• Major sensitivities.• Overall effect: Excluding uncertainties common to deterministic

and stochastic valuations, an error of about 5,6% is reduced to uncertainty of about 1%.

1 the arbitrage-free equilibrium pricing of liabilities in an incomplete market: application to a...

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sensitivity slide

salaries model slide

conclusions method

pricing method of thomson

rob thomson slide

major effects slide

riskfree stochastic

statespace vector slide