Variable Annuity and Interest Rate Risk

Transcription

1 Variable Annuity and Interest Rate Risk Ling-Ni Boon I,II and Bas J.M. Werker I October 13 th, 2017 Netspar Pension Day, Utrecht. I Tilburg University and Netspar II Université Paris-Dauphine

2 Financial Decision of a Retiree How much to consume? How to invest? C t C t+1 C t+2 C t+h Consumption Horizon, h Variable Annuity and Interest Rate Risk 2

3 Intertemporal Consumption and Investment Problem Single account Wealth Single investment portfolio Invested in the financial market r "# :" Wealth + Financial Market Returns Consumption Horizon, h C t C t+1 C t+2 C t+h Variable Annuity and Interest Rate Risk 3

4 Stock Market and Interest Rate Risk Merton s (1973) solution: θ "&' 1. Investment Rule Function of the financial assets expected returns, standard deviation of returns, and the individual s level of risk aversion. C "&' 2. Consumption Rule Function of realized financial market outcome. Single account Single investment portfolio C t Wealth r "# :" C t+1 C t+2 C t+h Consumption Horizon, h Variable Annuity and Interest Rate Risk 4

5 Two-Stage Formulation Wealth Multiple accounts W t W t+1 W t+2 W t+h Investment Rule Investment Rule Investment Rule Investment Rule θ " t θ " t + 1 θ " t + 2 θ " t + h Multiple investment portfolios r "# :" t r "# :"&. t + 1 r "# :"&0 t + 2 r "# :"&' t + h Consumption Horizon, h C t C t+1 C t+2 C t+h = W " 1 + r "# :" t = W "&. 1 + r "# :"&. t + 1 = W "&' 1 + r "# :"&' t + h Variable Annuity and Interest Rate Risk 5

6 Variable Annuity Wealth Multiple accounts W t W t+1 W t+2 W t+h Indexed to a universal reference portfolio Single investment portfolio r "# :" r "# :"&. r "# :"&0 r "# :"&' C t C t+1 C t+2 = W " 1 + r "# :" = W "&. 1 + r "# :"&. = W "&' 1 + r "# :"&' C t+h Consumption Horizon, h Variable Annuity and Interest Rate Risk 6

7 Variable Annuity Welfare Losses The variable annuity fails to optimally hedge interest rate risk at every consumption period. + 7 to 19% Equivalent Utility Levels Wealth Wealth Variable Annuity Optimal Consumption and Investment Rule A straightforward refinement of unit-linked contracts (e.g., variable annuity) to allow dependence of the unit on the consumption period improves individual welfare. Variable Annuity and Interest Rate Risk 7

8 Setup Financial Market Interest rate (Vasicek, 1977): dr " = κ μ 5 r " dt + σ 5 dz 9," Stock market index: ds " = S " r " + λ = σ = dt + S " σ = dz =," Constant τ maturity bond fund; and, Stochastic discount factor: dm " = M " r " dt + M " ϕ 5 dz 9," +M " ϕ = dz =," Individual Preference Certain lifetime between ages 65 in year t A to 110 in year T. Initial endowment, W "# = 1 Annual retirement benefits, C "# &', h = 0,, T t A. CRRA Utility: QG" e GH' I J#KL MNO # A.GP dh, γ = 2,, 10. Variable Annuity and Interest Rate Risk 8

9 Optimal Consumption Initial endowment Division of wealth over the consumption horizon, h Stage 1: Choose X "# Indexation Portfolio Stage 2: Choose Y "# &' The two decision variables X "# and Y "# &' allude to the two-stage formulation. Variable Annuity and Interest Rate Risk 9

10 Optimal Investment Time-invariant speculative demands Time-varying (h & u) hedge demand The bond hedge demand is a weighted average over the horizon: Variable Annuity and Interest Rate Risk 10

11 Two-Stage (2S) Formulation Stage 1: Choose X 0= "# given any investment policy that defines Y 0= "# &'. Stage 2: Regard every date t A + h as a standalone terminal wealth utility maximization problem to determine the investment policy that defines Y 0= "# &'. Both formulations are equivalent. Variable Annuity and Interest Rate Risk 11

12 Variable Annuity Initial endowment Division of wealth over the consumption horizon, h Similar three-term representation. Indexation Portfolio a "# h is the Assumed Interest Rate. Variable Annuity and Interest Rate Risk 12

13 Variable Annuity Indexation Portfolio Variable Annuity Two-Stage depends only on the planning horizon. depends on the planning and consumption horizons. Dependence on both horizons arise from the bond hedge demand: Consequence VA fails to optimally hedge for interest rate risk for all consumption periods. Variable Annuity and Interest Rate Risk 13

14 Welfare Losses (1/2) Financial market parameters calibrated to U.S. data. Quantify welfare losses with the Certainty Equivalent Wealth Loading (CEWL). Proportional loading on the initial wealth that the individual requires to purchase a VA* that yields the same expected utility as with the 2S. Non-monotonicity with respect to risk aversion level, γ. Variable Annuity and Interest Rate Risk 14

15 Welfare Losses (2/2) Welfare losses are more severe when there is: Lower subjective discount factor; Higher interest rate risk; Lower interest rate mean reversion rate, or mean reversion rate level, and; Higher correlation between interest rate and stock market risks. Variable Annuity and Interest Rate Risk 15

16 Conclusion The optimal consumption and investment problem is equivalent to a twostage problem that: 1. Selects the optimal division of initial wealth over the consumption horizon. 2. Sets the optimal investment policy for each portion of divided wealth. Constraining the investment policy to vary only along the planning horizon, the variable annuity fails to optimally hedge interest rate risk at every consumption period. The utility loss arising from the VA s deficient interest rate risk hedge is substantial: 7-19%. Applicable to all unit-linked contracts without guarantees: Pension contract design discussions in Australia (CIPR) and Europe (PEPP), especially the Netherlands (PPR) and Denmark. CIPRs: Comprehensive Income Products for Retirement. PEPP: Pan-European Personal Pension Product. PPR: Personal Pension with Risk-Sharing (Bovenberg and Nijman, 2016; Bovenberg and Nijman, 2017). Variable Annuity and Interest Rate Risk 16

17 Appendices Variable Annuity and Interest Rate Risk 17

18 Financial Market Parameters Maximum likelihood parameter estimates of the interest rate and the stock return, obtained by implementing the Kalman filter on monthly US government bond yields of 3-month, 1, 5, and 10-year maturities, and the return on the CRSP value- weighted stock index, from August 1971 till December Standard errors are by the outer product of gradients. Variable Annuity and Interest Rate Risk 18

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