Prudence Revisited The use of expected-utility theory for decision-making by the trustees of a retirement fund

Transcription

1 Prudence Revisited The use of expected-utility theory for decision-making by the trustees of a retirement fund Rob Thomson Actuarial Society of South Africa Retirement Matters seminar 6 th June 2011

2 Outline Introduction Argument of the utility function Functional form of the utility function Weighted average relative risk aversion Summary and conclusion 2

3 Introduction Normative validity of EU theory: for an individual: Raiffa (1961), Pratt (1964), Thomson (2003) for trustees: Arrow (1951), Harsanyi (1975), Sen (1973) Trusteeship requires prudence 3

4 Outline Introduction Argument of the utility function Functional form of the utility function Weighted average relative risk aversion Summary and conclusion 4

5 Argument of the utility function Individual /DC member: current and future consumption (& bequests) (Von Neumann & Morgenstern, 1947; Savage, 1954) wealth at a time horizon (Tobin, 1958; Samuelson, 1969; Owen & Rabinovitch, 1983) wealth at retirement (Nielsen, unpublished); net replacement ratio at retirement (Thomson & Levitan, 2009) one-period-ahead funding ratio relative to reasonable expectations 5

6 Argument of the utility function DB fund: one-period-ahead surplus (Cardinale et al, 2006) single-period surplus (Sherris, 1993) one-period-ahead funding ratio (Leibowitz et al, 1994: Hoevenaars et al, 2008) 6

7 Outline Introduction Argument of the utility function Functional form of the utility function Weighted average relative risk aversion Summary and conclusion 7

8 Functional form of the utility function HARA class has become standard rx ( ) u( x) 1 u( x) a bx 1 x for b 0, 1; 1 ux ( ) ln( x ) for b 0, 1; exp x for b0. 8

9 Functional form of the utility function Problem with the HARA class: It does not allow decreasing relative risk aversion Evidence of decreasing relative risk aversion (Projector & Weiss, 1966; Friend, 1973; Bossons, 1973; Cohn et al, 1975) Prudence suggests non-increasing relative risk aversion 9

10 Functional form of the utility function uz ( ) Booth (1995): ln( z) for z 1; ln( z) h for z 1 utility funding ratio 10

11 Functional form of the utility function Khorasanee & Smith (1997): uz () ln( z) for z 1; * * p ln( z) for z<1; p 1. utility relative risk aversion funding ratio 11 utility RRA

12 Functional form of the utility function Advantages of non-increasing relative risk aversion in the one-period-ahead funding ratio: consistent with empirical results not speculative, so consistence with prudence equivalent to non-increasing absolute risk aversion in the one-period-ahead force of return period may be defined as the interval between decisions 12

13 Functional form of the utility function Elicitation of trustees utility functions: consensus compromise Harsanyi s (1975) individualism postulate: M u( x) cmum( x) m1 13

14 Functional form of the utility function Prudence (Kimball, 1990): coefficient of absolute prudence: u( x) px ( ) u ( x) coefficient of relative prudence: u( x) ( x) x u ( x ) Problems with Kimball s prudence: Positive prudence does not necessarily imply positive risk aversion Positive prudence does not necessarily even imply non-satiation 14

15 Functional form of the utility function In order to satisfy the requirements of prudence, the trustees utility function u() should conform to the following criteria: 1. It should map the open interval (0, ) into the open interval (, ) or into a subset of that interval. 2. It should be at least twice differentiable and, if it is not thrice differentiable, then u() should have a finite number of finite jump-discontinuities. 3. The trustees should be unsatiated, so that, for all z, u( z) For all z, 5. For any * ( z) 1. * z and for all z * * z, z z. 15

16 Outline Introduction Argument of the utility function Functional form of the utility function Weighted average relative risk aversion Summary and conclusion 16

17 Weighted average relative risk aversion: the basic WARRA class where: u 0 ( z) uz ( ) 1 0 z 1 ; 1 1 z 1 u ( z) ; 1 c 0 ; and u ( ) ( ) 0 z cu z 1 c ; 17

18 Weighted average relative risk aversion: the basic WARRA class Relative risk aversion: where: 0 ( z) c z 0 1 cz whence: and: z0 z 0 lim ( ) lim ( z) z and d ( z) 0 dz 18

19 WARRA: parameterisation Cumulative distribution Average relative risk aversion 19

20 WARRA: parameterisation Relative risk aversion a 1b 1c 2a 2b Funding ratio 20

21 21 WARRA: generalisation ( ) exp 1 y z c x u z dx dy x cx 0 ( ) 1 c z z cz

22 WARRA: generalisation Relative risk aversion Funding ratio 22 lambda = 1 lambda = 3 lambda = 10 lambda = inf

23 WARRA: extension to group utility aggregation where: u m ( z) 1 m z 1 1 m M 1 u( z) um( z) M m 1 and hence: ( z) z1 z0 M m1 M z m1 m z m z 1 lim ( ) m ; lim ( z) M ; z d ( z) 0 dz ; and lim ( z) m1 ; M 1 m. M 23

24 Outline Introduction Argument of the utility function Functional form of the utility function Weighted average relative risk aversion Summary and conclusion 24

25 Summary and conclusion Basic WARRA class: u ( ) ( ) 0 z cu z uz ( ) 1 c where: 1 0 z 1 u0( z) ; 1 1 z 1 u ( z) ; 1 c 0 ; and ; Generalisation of the basic class to faster transition: z y 1 0 c x u( z) exp dx dy x 1 cx 0 1 Extension of the basic class to group utility aggregation: M 1 u( z) um( z) M m 1 where: 1 m z 1 um( z) 1 m 25

26 Summary and conclusion Operationalising prudent utility-based retirementfund risk management: qualitative information to assist trustees in implementation of the process data to inform trustees about appropriate levels of RRA and transition rates risk-adjustment of returns earned application to allocation of assets, pricing of liabilities and reinsurance of risks 26

27 Prudence Revisited The use of expected-utility theory for decision-making by the trustees of a retirement fund Contact details: Rob Thomson rthomson@icon.co.za

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29 Argument of the utility function CobbDouglas utility function: L, t A t u Lt At u u Stax Lt 1 29