Breakout Session Topic 9: Asset / liability management

Transcription

1 Coffee Break 1

2 Breakout Session Topic 9: Asset / liability management 11 September

3 The arbitrage-free equilibrium pricing of liabilities in an incomplete market: application to a South African retirement fund Rob Thomson 3

4 4 Agenda 1. Introduction 2. Liabilities specification & 3. Pricing method 4. Results and sensitivity 5. Conclusions

5 5 Introduction: aim Apply the pricing method of Thomson (2005) to: market-portfolio (Thomson unpublished); equilibrium asset-category (Thomson & Gott, 2009); a DB retirement-fund ; with a view to operationalising the pricing of such a fund and quantifying the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.

6 6 Introduction: method Marketportfolio Return on market portfolio Mortality Liabilities specification Risk-free return Inflation rate Salary Equilibrium asset-categories Returns on asset categories Pensionincreases Asset liability Price of liabilities

7 7 Introduction: method Marketportfolio Return on market portfolio Mortality Liabilities specification Risk-free return Inflation rate Salary Equilibrium asset-categories Returns on asset categories Pensionincreases Asset liability Price of liabilities

8 8 Introduction: method Marketportfolio Return on market portfolio Mortality Liabilities specification Risk-free return Inflation rate Salary Equilibrium asset-categories Returns on asset categories Pensionincreases Asset liability Price of liabilities

9 9 Liabilities specification & no exits before retirement mortality only after retirement projected unit method salaries and pensions expressed in real terms

10 10 Liabilities specification & : salaries S = S ξ + ζ mt m t t t exp ( ), 1 ξ = μ + b η + b η + σ ε t ξ ξ1 3t ξ2 7t ξ ξt ζ = μ + σ ε t ς x ς x ςt μςx = αμς + βμς exp( λμςx) σ 2 ζ x = σ 2 ζ x M x, t 1 σςx = ασς + βσς exp( λσςx ) μ ξ = 0,01 1 = 0, 005 b ξ 2 = 0, 005 σ = 0,03 b ξ ξ α μς = 0,016 β μς = 0,5 λ = 0,1 μς α σς = 0,042 β σς = 0,5 λ = 0,08 σς

11 11 Liabilities specification & : pension increases { ( )} P = P exp max 0, γ t t 1 t

12 12 Liabilities specification & : pensioner mortality ν ν PNL00 SAP98 { x} SAIL98 { x} = ν IL00 { x} ν{ x} ( ) ν ν μ SAP SAP98 { x} = { x} exp 10 ν where: ν ν χ ν SAP SAP exp { x} + t = { x} + t 1 ( t) χ = χ + μ + b η + σ ε ν t ν, t 1 ν ν 7t ν νt μ = 0,004 ν b ν = 0,001 σ ν = 0,005

13 13 Pricing method: primary & secondary simulations Time 0 Time 1 Time t -1 Time t Time T -1 Time T primary simulation primary simulation node secondary simulation

14 14 Pricing method: state-space vector x t ( s ) PIt 1 M PIt ( su ) PCt ( s1 ) M = PCt ( su ) θ t Pxt 1 M P xn t where: It ( ) = exp { ( )} P s Y s Ct ( ) = exp { ( )} P s Y s θt = exp( χ ν t) It Ct

15 15 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

16 16 Pricing method: secondary simulations Final year 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

17 17 Pricing method: secondary simulations year t 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

18 18 Pricing method: secondary simulations year 1 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

19 19 Pricing method σ σ Σˆ ˆ ˆ ˆ ˆ εt = Ft σfvt Vt σfvt ( ˆ f ) ˆ 1 zt = ΣVt μ Vt t1 ˆ μ = m μˆ Mt t Vt m t Mt 1 z z 1 = t t t ˆ σ 2 = m Σ ˆ m ˆ σ = m σˆ Vt t HMt t FVt ˆ β * Ft = ˆ σ + ˆ σ ˆ σ HMt εt Mt 2 ˆ σ Mt P = 1 ˆ ( ˆ f ) { * ˆ μ β μ } L, t 1 Ft Ft Mt t ft

20 20 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, , ,94 13,46 12,8 13,36-0, , ,00 13,49 12,4 13,39-0, , ,67 13,70 8,1 13,62-0, , ,36 13,78 3,1 13,78 0, ,1 75 9,53 9,69 1,7 9,69 0, ,7 85 5,96 6,01 0,9 6,01 0, ,9 total ,5

21 21 Results and sensitivity deterministic stochastic price valuation R million % incr Female ,5 Male ,6 Total ,1 Aggregate ,6 Adjusted to fund data ,6

22 22 Results and sensitivity Deterministic valuation of -point data difference due to risk-free stochastic pricing 11 Risk-free stochastic price hedge-portfolio risks 19 Stochastic price with hedge-portfolio risks residual risks 20 Stochastic price: hedge-portfolio & residual risks cost of guarantee 192 Stochastic price based on -point data adjustment to fund data 20 Stochastic price based on fund data 3 074

23 23 Results and sensitivity: major effects Parameter name description standard value b ξ 1 g σ M b γ test value Test result price (R million) change in price (%) standard values N/A general salary increase: sensitivity to inflation 0, ,14 return on market portfolio: sensitivity to risk-free rate 1,39 1, ,37 return on market portfolio: residual volatility 0,159 0, ,45 force of inflation: residual volatility 0, ,07

24 24 Conclusions Method computationally demanding, but not impossible: 41 hours.

25 25 Conclusions Method computationally demanding, but not impossible. Convergence complicated, but Sobol numbers expedite it.

26 26 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).

27 27 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.

28 28 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 intercohort 0,5%.

29 29 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 intercohort 0,5%. Major sensitivities: volatility of force of inflation in excess of conditional exante expected inflation; sensitivity of ex-ante expected returns on the market portfolio to positive risk-free returns; residual volatility of the return on the market portfolio.

30 30 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 intercohort 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%.

31 31 Contact details Rob Thomson School of Statistics & Actuarial Science University of the Witwatersrand, Johannesburg, South Africa +27 (0)