Mitigating Interest Rate Risk in Variable Annuities: An Analysis of Hedging Effectiveness under Model Risk

ABSTRACT

Variable annuities are investment vehicles offered by insurance companies that combine a life insurance policy with long-term financial guarantees. These guarantees expose the insurer to market risks, such as volatility and interest rate risks, which can be managed only with a hedging strategy. The objective of this article is to study the effectiveness of dynamic delta-rho hedging strategies for mitigating interest rate risk in variable annuities with either a guaranteed minimum death benefit or guaranteed minimum withdrawal benefit rider. Our analysis centers on three important practical issues: (1) the robustness of delta-rho hedging strategies to model uncertainty, (2) the impact of guarantee features (maturity versus withdrawal benefits) on the performance of the hedging strategy, and (3) the importance of hedging interest rate risk in either a low and stable or rising interest rate environment. Overall, we find that the impact of interest rate risk is equally felt for the two types of products considered, and that interest rate hedges do lead to a significant risk reduction for the insurer, even when the ongoing low interest rate environment is factored in.

Notes

1 The “Report of the Task Force on Segregated Fund Liability and Capital Methodologies” produced by the Canadian Institute of Actuaries (2010) states that “typically delta and rho are hedged while vega and gamma are only monitored.” The survey conducted by Towers Watson (2013) also shows that U.S. insurers mainly focus on hedging delta and rho risks in the context of VAs.

2 Note that the parameter μ in Kaeck and Alexander (2013, table 6) is defined as the drift of the log-return process, whereas the parameter μ in Equation (1(1) ) is defined as the drift of the stock price. Therefore, to obtain a consistent value of μ in Equation (1(1) ), we added θ/2 to the drift estimated by Kaeck and Alexander (2013, table 6).

3 The “Report of the Task Force on Segregated Fund Liability and Capital Methodologies” produced by the Canadian Institute of Actuaries (2010) states that “some companies use implied volatilities in the short term and grade toward a long-term assumption, while others are using only a long-term assumption.” The Committee on Life Insurance Financial Reporting (CLIFR) of the Canadian Institute of Actuaries prepared an educational note titled “Guidance for the 2016 Valuation of Insurance Contract Liabilities of Life insurers” (Canadian Institute of Actuaries 2016), which stated that the CLIFR formed a subcommittee in 2016 to provide a recommendation regarding the volatility of equity returns in the context of hedging. This subcommittee expects to release a paper in 2017 on that matter, but at the time of writing (December 2016) this report is not yet available.

4 For example, Equation (4(4) ) shows that the GMDB/GMAB net liability at time t is a function of the value of put options with maturities ranging from 0 to T − t. Therefore, to compute the net liability's delta we must have an estimate of the whole implied volatility surface up to a (T − t)–year maturity.

5 Not all authors employ the term net liability to represent an expected present value. For example, Feng (2014) and Feng and Huang (2016) use it to represent the present value of the insurer's loss (), which is a random variable. Therefore, we note that there are variations in terminology in the actuarial literature with respect to the net liability.

6 The Standard Ultimate Survival Model assumes that the force of mortality is modeled as u_x = A + Bc^x, with A = 0.00022, B = 2.7 × 10^{− 6}, and c = 1.124 (Makeham's law).

7 We performed exact maximum likelihood estimation of the one-factor CIR model using the daily three-month CMT rate over the time period January 4, 1982, to June 30, 2014 (see Iacus 2008, for more details on the estimation approach). We obtain a₁ = 0.328, b₁ = 0.0329, and σ_{r, 1} = 0.0790 (under ). The interest rate risk premium parameter was determined by calibrating the model to the 10-year Treasury yield curve rate observed on June 30, 2014 (2.53%), which gives λ₁ = −0.0591.

8 Chen et al. (2008) and Donnelly et al. (2014) show that interest rate and volatility assumptions significantly impact the fair fee. For example, by assuming a higher average interest rate and a lower average volatility, Bauer et al. (2008) obtain much lower fair fees than those presented in Table 3. However, Bacinello et al. (2011) consider assumptions that are closer to ours and report fair fees of a similar order of magnitude.

9 The statistical definitions of the RMSE and CTE (1 − p)% are where denotes the ith largest hedged loss among the N = 100, 000 simulated market scenarios.

10 Assume that the BS model is calibrated at time t to observed zero-coupon bond prices, P_{t, v}, v ⩾ t. Then, for v ⩾ t, the deterministic time-varying risk-free rate parameter, r_v, must satisfy P_{t, v} = exp ( − ∫^v_tr_s ds), which implies r_v = −∂log P_{t, v}/∂v. Therefore, the time-varying risk-free rate parameter in the BS model is directly obtained from the term structure of interest rates.

11 Even though the CBOE does not publish the VIX at a one-year timeframe, this volatility measure could potentially be calculated from market data. Moreover, we experimented calibrating σ_S to a 30-day VIX and found this choice to be significantly inferior to both the one-year VIX and our historical estimate. This is simply due to the fact that the 30-day VIX is subject to abrupt changes over time and is not representative of future long-term volatility.