1. STOCHASTIC MODELS FOR ACTUARIAL USE: THE EQUILIBRIUM MODELLING OF LOCAL MARKETS Rob Thomson, Dmitri Gott Hacettepe University 24 th June 2011

2. 2 Agenda Introduction Assumptions Prices and returns Notional risky assets Development of the model Summary of the model Parameter estimation Illustrative results

3. 3 Introduction Predictive model of returns on the market portfolio Predictive equilibrium model of:  (real) returns on major asset categories  (real) risk-free rates  inflation rates  (interdependent) factors  (independent) notional risky assets Descriptive estimation of the equilibrium model

4. 4 Agenda Introduction Assumptions Prices and returns Notional risky assets Development of the model Summary of the model Parameter estimation Illustrative results

5. 5 Assumptions The local market: default-free index-linked zero-coupon bonds; default-free conventional zero-coupon bonds; and ‘equity’.

6. 6 Assumptions ctd. Market participants:  have homogeneous expectations  are able to borrow or lend unlimited amounts at the same risk-free return. The market is frictionless. At the end of a year means and variances of factors affecting the return on each asset during the forthcoming year are known. ‘return’: the aggregate instantaneous real rate of return. At the beginning of the year, portfolios are selected by optimisation in mean–variance space so that the market is in equilibrium. Conditional CAPM

7. 7 Agenda Introduction Assumptions Prices and returns Notional risky assets Development of the model Summary of the model Parameter estimation Illustrative results

8. 8 Prices & returns Index-linked (zero-coupon) bonds (Real) risk-free rate Conventional (zero-coupon) bonds Inflation Equity

9. 9 Index-linked bonds: price The price at time t = 0,…, T of an index-linked bond maturing at time t + s is: where: T is the time horizon to which projections will be required; is the expected value of, the aggregate force of return to maturity;

10. 10 (Real) Risk-free Rate The risk-free rate during year t is:

11. 11 Inflation The average instantaneous rate of inflation during year t is: where:

12. 12 Conventional bonds: price The price at time t of a conventional bond maturing at time n is: where:

13. 13 Constant inflation risk premium The inflation risk premium is constant, so that, for all t:

14. 14 Equity: price The price of equity at time t is: where:

15. 15 Agenda Introduction Assumptions Prices and returns Notional risky assets Development of the model Summary of the model Parameter estimation Illustrative results

16. 16 Notional risky assets: Total return If there are N risky assets in a market and an investor maintains constant exposure w i (at market prices) to asset i during a year then, if all income is reinvested when received, the total return is: where  i is the aggregate return on asset i during that year.

17. 17 Notional risky assets: No arbitrage The returns on all asset categories are linear functions of the factors  j,t. The factors are linear functions of  i,t. The returns on the notional risky assets are linear functions of  i,t. The returns on all asset categories are therefore linear functions of the returns on the notional risky assets Thus: portfolios of bonds and equities can be replicated out of the notional risky assets and vice versa no arbitrage

18. 18 The Factors & the Market Let: Then it may be shown that:

19. 19 Agenda Introduction Assumptions Prices and returns Notional risky assets Development of the model Summary of the model Parameter estimation Illustrative results

20. 20 Development of the model The market price of covariance Index-linked bonds Conventional bonds Equity

21. 21 Development: The Market Price of Covariance In order for an asset to satisfy the CAPM during year t, we require that: where:

22. 22 Development: Index-linked bonds For each index-linked bond: where:

23. 23 Development: Conventional bonds For each conventional bond: where:

24. 24 Development: Equity For equity: where:

25. 25 Agenda Introduction Assumptions Prices and returns Notional risky assets Development of the model Summary of the model Parameter estimation Illustrative results

26. 26 Summary of the model Parameters Variables

27. 27 Equilibrium Model: Summary: Parameters The parameters required are: for all required values of s :  ; and for i = 1,…, N and j = 1,…, 6:

28. 28 Equilibrium Model: Summary: Variables

29. 29 Equilibrium Model: Summary: Variables

30. 30 Equilibrium Model: Summary: Variables

31. 31 Summary of the model The equilibrium model: allows for any type of model of the market portfolio models bonds, ‘equity’ and inflation maintains equilibrium each year is arbitrage-free is linear uses discrete time but allows for intra-year variability assumes mean–variance decision-making uses a conditional CAPM focuses on a local market

32. 32 Agenda Introduction Assumptions Prices and returns Notional risky assets Development of the model Summary of the model Parameter estimation Illustrative results

33. 33 Parameter estimation:Variables used for various asset classes Equity: FTSE All-Share TRI Conventional and Index-Linked Bonds: UK DMO zero-coupon curves Inflation: UK retail prices index

34. 34 Parameter estimation: Methodology Market-portfolio model:

35. 35 Parameter estimation: Methodology Market-portfolio return parameters estimated using:  historical returns on equities and zero-coupon bonds of different maturities  historical market capitalisation of equity and bond markets future payments on bonds decomposed into zero-coupon bonds, split by year of payment

36. 36 Parameter estimation: Methodology (ctd.) Market price of risk (price of covariance)  Expected return on market portfolio is a multiple of risk-free return (regression)  Standard deviation is assumed constant Expected returns on assets are derived from historical covariance with market portfolio returns and MPR

37. 37 Parameter estimation: Methodology (ctd.) Parameters of interest rate models are derived from:  yield curve at the estimation date  PCA of deviations of zcb returns from expected Inflation risk premium  arbitrary at present  area for further research

38. 38 Agenda Introduction Assumptions Prices and returns Notional risky assets Development of the model Summary of the model Parameter estimation Illustrative results

39. 39 Illustrative results 10 000 simulations for each economic variable 20-year projection mean and 95% CI

40. 40 Illustrative results: inflation

41. 41 Illustrative results: equity

42. 42 Illustrative results: long-term index-linked bond yield

43. 43 Illustrative results: short-term index-linked bonds

44. 44 Illustrative results: long-term conventional bond yield

45. 45 Illustrative results: short-term conventional bond yield

46. 46 Conclusion The equilibrium model: allows for any type of model of the market portfolio models bonds, ‘equity’ and inflation maintains equilibrium each year is arbitrage-free is linear uses discrete time but allows for intra-year variability assumes mean–variance decision-making uses a conditional CAPM focuses on a local market