Pricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates

Abstract

We consider the pricing of variable annuities with the Guaranteed Minimum Withdrawal Benefit (GMWB) under the Vasicek stochastic interest rates framework. The holder of the variable annuity contract pays an initial purchase payment to the insurance company, which is then invested in a portfolio of risky assets. Under the GMWB, the holder can withdraw a specified amount periodically over the term of the contract such that the return of the entire initial investment is guaranteed, regardless of the market performance of the underlying asset portfolio. The investors have the equity participation in the reference investment portfolio with protection on the downside risk. The guarantee is financed by paying annual proportional fees. Under the assumption of deterministic withdrawal rates, we develop the pricing formulation of the value function of a variable annuity with the GMWB. In particular, we derive the analytic approximation solutions to the fair value of the GMWB under both equity and interest rate risks, obtaining both the lower and upper bound on the price functions. The pricing behavior of the embedded GMWB under various model parameter values is also examined.

Pricing guaranteed minimum/lifetime withdrawal benefits with various provisions under investment, interest rate and mortality risks

“…On the other hand, this neutralization effect becomes insignificant with a largerρ and introducing the interest rate risk will increase fair charges and option premiums. This finding is consistent to the finding ofPeng et al (2012). The above phenomena imply that ignoring the interest rate risk might significantly misprice…”

“…Lin and Tan (2003) and Kijima and Wong (2007) further consider the pricing of equity-indexed annuities and show that the interest rate risk becomes significant when valuing long-duration insurance policies. Peng et al (2012) first analyze the interest rate risk for pricing GMWBs by employing the Vasicek model (see Vasicek (1977)) that can not exactly fit the real market zero rate curves. Due to the mathematical complexity, they derive lower and upper bounds for the GMWB value without considering various provisions of GMWBs.…”

The valuation of GMWB variable annuities under alternative fund distributions and policyholder behaviours

“…Rider PH behaviour Fund process [12] GMWB static * GBM [32] GMWB static/dynamic GBM [8] GMWB static * /dynamic GBM [14] GMWB dynamic GBM [17] GMWB dynamic GBM [15] GMWB dynamic GBM/Merton [40] GLW static SIR+SV [34] GMWB static SIR [9] GMWB dynamic RS [35] GLW static GBM [3,4] In this paper, we show how an approach based on dynamic programming can accomodate any fund return distribution within the class of Lévy processes, allowing therefore a great variety of statistical features such as kurtosis and skewness.…”

Measuring Risk-Adjusted Performance and Product Attractiveness of a Life Annuity Portfolio

“…The attractiveness of the product is considered together with a prudential capital allocation and solvency management by means of appropriate thresholds inserted in the model. The choice of this specific ex-ample arise from the interest of the market in variable annuities, which is going to be steady and substantial, in particular for the contractual forms with minimum guaranteed benefits [7]. Moreover, the attractiveness of the product is interesting also according to the insured's point of view.…”