The Effect of Risk Aversion and Loss Aversion on Equity-Linked Life Insurance With Surrender Guarantees

We price equity-linked life insurance with surrender guarantees and account for risk preferences in the form of risk-averse and loss-averse policyholders in continuous time. Risk-averse policyholders surrender their policy for higher equity index values. Compared to optimally surrendered policies, this behavior creates substantial policy value losses. In contrast, loss-averse policyholders surrender once the surrender benefit realizes a gain but keep under-performing policies. This disposition effect reduces the policy value relative to both optimally surrendered policies and policies surrendered by risk-averse policyholders. Insurers in competitive markets need to estimate their policyholders’ risk preferences accurately.     

Fair Valuation of Equity-Linked Policies under Insurer Default Risk

Abstract

We consider the problem of computing the fair value of equity-linked policies with an interestrate guarantee when the insurer is subject to credit risk. The framework is developed based on modern financial theory using the no-arbitrage principle. In this context, an equity-linked policy is considered as a vulnerable contingent claim that expires before maturity if the firm asset value reaches a prespecified default threshold depending on the firm’s liabilities. We derive a closedform formula in a continuous-time environment to compute the fair value of the contract. We also develop a discrete-time model that allows us to address fair evaluation when the policy embeds a surrender option.     

Voluntary Termination of Life Insurance Policies

Abstract

In this paper, one error-correction model (ECM) that is able to avoid the problem of producing noise within traditional multiple cointegration vectors has been employed to explore the dynamics of surrender behavior. The evidence shows that both the emergency fund hypothesis and interest rate hypothesis are sustained in the short run as well as in the long run. A unique cointegration relationship within the surrender dynamics has been validated. In addition, a new hypothesis test that stresses the competition for the withdrawal of life insurance policy cash values has also been conducted. Such a crowding-out effect between policy loans and policy surrenders might be attributed to the motivation that keeps a life policy in force, the existence of surrender charges, and the automatic premium loan provision.     

终身寿险退保的利率敏感性分析——基于保单保障水平的测算

市场利率变化对于不分红终身寿险产品退保的影响很大,终身寿险保单不能够通过选择较长的缴费期来缓解市场利率变化引起的退保压力,调整预定利率能缓解退保压力;适当的分红政策能够大大缓解市场利率变化对分红型终身寿险退保的影响,但是经营效益差,分红水平低于市场预期的寿险公司产品可能会遭遇到更大的退保压力。     

The Impacts of Surrender Options on Reserve Durations

Estimating the interest rate risk of life insurance reserves is essential for insurers, and surrender options are critical to the estimation. This article advances our understanding of how surrender options affect the durations of reserves. We identify a pattern of the reserve duration with respect to the interest rate that is important in explaining how surrender rate levels and the interest-rate sensitivity of surrenders affect reserve durations. We further found that the surrender behavior that is more positively related to the interest rate produces larger/smaller effective dollar durations when the interest rate is low/high.     

Forecasting Surrender Rates Using Elliptical Copulas and Financial Variables

A multistage stochastic model to forecast surrender rates for life insurance and pension plans is proposed. Surrender rates are forecasted by means of Monte Carlo simulation after a sequence of GLM, ARMA-GARCH, and copula fitting is executed. The model is illustrated by applying it to age-specific time series of surrender rates derived from pension plans with annuity payments of a Brazilian insurer. In the GLM process, the only macroeconomic variable used as an explanatory variable is the Brazilian real short-term interest rate. The advantage of such a variable is that we can take future market expectation through the current term structure of interest rates. The GLM residuals of each age/gender group are then modeled by ARMA-GARCH processes to generate i.i.d. residuals. The dependence among these residuals is then modeled by multivariate Gaussian and Student's t copulas. To produce a conditional forecast on a stock market index, in our application we used the residuals of an ARMA-GARCH model fitted to the Brazilian stock market index (Ibovespa) returns, which generates one of the marginal distributions used in the dependence modeling through copulas. This strategy is adopted to explain the high and uncommon surrender rates observed during the recent economic crisis. After applying known simulation methods for elliptical copulas, we proceeded backwards to obtain the forecasted distributions of surrender rates by application, in the sequel, of ARMA-GARCH and GLM models. Additionally, our approach produced an algorithm able to simulate multivariate elliptical copulas conditioned on a marginal distribution. Using this algorithm, surrender rates can be simulated conditioned on stock index residuals (in our case, the residuals of the Ibovespa returns), which allows insurers and pension funds to simulate future surrender rates assuming a financial stress scenario with no need to predict the stock market index.     

Pricing Ratchet Equity-Indexed Annuities with Early Surrender Risk in a CIR++ Model

In this article we propose a lattice algorithm for pricing simple Ratchet equity-indexed annuities (EIAs) with early surrender risk and global minimum contract value when the asset value depends on the CIR++ stochastic interest rates. In addition we present an asymptotic expansion technique that permits us to obtain a first-order approximation formula for the price of simple Ratchet EIAs without early surrender risk and without a global minimum contract value. Numerical comparisons show the reliability of the proposed methods.     

Valuation of quanto options in a Markovian regime-switching market: A Markov-modulated Gaussian HJM model

We consider the valuation of European quanto call options in an incomplete market where the domestic and foreign forward interest rates are allowed to exhibit regime shifts under the Heath–Jarrow–Morton (HJM) framework, and the foreign price dynamics is exogenously driven by a regime switching jump-diffusion model with Markov-modulated Poisson processes. We derive closed-form solutions for four different types of quanto call options, which include: options struck in a foreign currency, a foreign equity call struck in domestic currency, a foreign equity call option with a guaranteed exchange rate, and an equity-linked foreign exchange-rate call.     

Pricing securities with multiple risks: A case of exchangeable debt

Building on the work of Das and Sundaram (2007), we develop a widely applicable model to price securities subject to interest rate, equity, and default risks and use it to price exchangeable bonds. The extension features a trivariate recombining lattice instead of the original model’s bivariate recombining lattice. We also show how to estimate some critical non-observable inputs to implement the model by using current market data so that the model’s prices reflect current market information. We test the model on a sample of exchangeable bonds to determine the model’s empirical performance. Besides exchangeable bonds, we can also use the model to price securities such as reverse exchangeable bonds, bonds exchangeable to indexes, and bonds exchangeable to commodities.     

基于格兰杰因果关系检验的我国银行保险退保因素分析

我国银行保险内存不足。通过格兰杰因果关系检验对“财务危机”和“利率替代”两种假说进行实证检验后得出：短期利率波动对银行保险退保影响不显著,长期利率波动对银行保险退保影响较显著,失业对银行保险退保短期影响较显著,证券投资波动对银行保险退保影响最为显著。为了降低我国银行保险的退保率,要深化银保合作机制,为了降低市场因素带来的影响,可适当提高银保产品的收益率,银保产品不要过于理财化,可适当增加一些保障功能。     

The Distributions of Policy Reserves Considering the Policy-Year Structures of Surrender Rates and Expense Ratios

In this article, we examine how the policy-year structures of expense ratios and surrender rates affect the distributions of policy reserves. Our results show that a convex expense ratio curve, though reduces the mean and the uncertainty of reserves, could make the beneficial impact of surrenders on insurers become detrimental. Our results also show that the convexity of the surrender rate curve is favorable to insurers while the volatilities of surrender rates are unfavorable. We further find that neglecting the policy-year structures of surrender rates and expense ratios may result in overestimation of the mean and the uncertainty of reserves.     

Variable annuities in a Lévy-based hybrid model with surrender risk

This paper proposes a market consistent valuation framework for variable annuities (VAs) with guaranteed minimum accumulation benefit, death benefit and surrender benefit features. The setup is based on a hybrid model for the financial market and uses time-inhomogeneous Lévy processes as risk drivers. Further, we allow for dependence between financial and surrender risks. Our model leads to explicit analytical formulas for the quantities of interest, and practical and efficient numerical procedures for the evaluation of these formulas. We illustrate the tractability of this approach by means of a detailed sensitivity analysis of the fair value of the VA and its components with respect to the model parameters. The results highlight the role played by the surrender behaviour and the importance of its appropriate modelling.     

Fair valuation of cliquet-style return guarantees in (homogeneous and) heterogeneous life insurance portfolios

Participating life insurance contracts allow the policyholder to participate in the annual return of a reference portfolio. Additionally, they are often equipped with an annual (cliquet-style) return guarantee. The current low interest rate environment has again refreshed the discussion on risk management and fair valuation of such embedded options. While this problem is typically discussed from the viewpoint of a single contract or a homogeneous* insurance portfolio, contracts are, in practice, managed within a heterogeneous insurance portfolio. Their valuation must then – unlike the case of asset portfolios – take account of portfolio effects: Their premiums are invested in the same reference portfolio; the contracts interact by a joint reserve, individual surrender options and joint default risk of the policy sponsor. Here, we discuss the impact of portfolio effects on the fair valuation of insurance contracts jointly managed in (homogeneous and) heterogeneous life insurance portfolios. First, in a rather general setting, including stochastic interest rates, we consider the case that otherwise homogeneous contracts interact due to the default risk of the policy sponsor. Second, and more importantly, we then also consider the case when policies are allowed to differ in further aspects like the guaranteed rate or time to maturity. We also provide an extensive numerical example for further analysis.     

Lapse rate modeling: a rational expectation approach

The surrender option embedded in many life insurance products is a clause that allows policyholders to terminate the contract early. Pricing techniques based on the American Contingent Claim (ACC) theory are often used, though the actual policyholders' behavior is far from optimal. Inspired by many prepayment models for mortgage backed securities, this paper builds a Rational Expectation (RE) model describing the policyholders' behavior in lapsing the contract. A market model with stochastic interest rates is considered, and the pricing is carried out through numerical approximation of the corresponding two-space-dimensional parabolic partial differential equation. Extensive numerical experiments show the differences in terms of pricing and interest rate elasticity between the ACC and RE approaches as well as the sensitivity of the contract price with respect to changes in the policyholders' behavior.     

基于广义线性模型的我国投资型寿险产品退保率实证研究

基于某一款万能险产品38个月的退保率样本数据,结合我国寿险业务的实际情况,利用广义线性模型对影响退保率的主要因素进行了分析。研究结果显示,源于消费者在购买保险的经济环境方面的差异,国外的退保率模型在我国并不适用。从我国实际情况出发,通过选择合适的解释变量,构建了新退保率模型,并分析了各解释变量对退保率的影响。     

International linkages and macroeconomic news effects on interest rate volatility — Australia and the US

We examine international linkages between daily time series of US and Australian 3-month treasury bills and 10-year government bonds from 1987–1995, paying particular attention to the effects of macroeconomic announcements in both countries. The two countries' interest rate data are modeled by a bivariate exponential generalized autoregressive conditional heteroscedasticity (EGARCH) formulation. The results suggest that market participants believed the Reserve Bank of Australia targeted the consumer price index (CPI), while the Federal Reserve targeted economic activity. Monetary policy announcements had significant effects on interest rates, as well as on their volatility in the short term. US macroeconomic activity announcements significantly moved Australian interest rates, particularly at the short end. Australian interest rates moved significantly in response to the previous day's US interest rate shocks. The conditional volatility of the Australian interest rate changes was also significantly influenced by lagged US interest rate shocks, as well as by surprises in US macroeconomic announcements. Some macroeconomic news announcements raised conditional volatilities, while others reduced them. Overall, there was a remarkable and complex array of linkages between the two countries.     

利率政策环境与央行货币政策选择

央行的货币政策实施受货币政策环境制约,特别是实施价格型货币政策时,在不同的利率环境下央行可能选择不同的利率调控模式。为此,通过构建门限回归模型对货币政策调控模式与利率区制的相依性进行实证检验,结果发现：在不同的利率区制内中央银行的政策偏好和利率调控模式存在显著差异。在高利率区制,中央银行存在针对产出缺口调整名义利率的政策偏好;在低利率区制,中央银行则不存在这种政策偏好,低利率环境下货币政策效应弱化甚至失效是导致此区制中央银行不针对产出缺口调整利率的主要原因。     

Fair Valuation of a Guaranteed Life Insurance Participating Contract Embedding a Surrender Option

In this article we deal with the problem of pricing a guaranteed life insurance participating policy, sold in the Italian market, which embeds a surrender option. This feature is an American‐style put option that enables the policyholder to sell back the contract to the insurer at the cash surrender value. Employing a recursive binomial formula patterned after the Cox, Ross, and Rubinstein (1979) discrete option pricing model we compute, first of all, the total price of the contract, which also includes a compensation for the participation feature (“participation option,” henceforth). Then this price is split into the value of three components: the basic contract, the participation option, and the surrender option. The numerical implementation of the model allows us to catch some comparative statics properties and to tackle the problem of suitably fixing the contractual parameters in order to obtain the premium computed by insurance companies according to standard actuarial practice.     

Determinacy under Inflation Targeting Interest Rate Policy in a Sticky Price Model with Investment (and Labor Bargaining)

In a sticky price model with investment spending, recent research shows that inflation-forecast targeting interest rate policy makes determinacy of equilibrium essentially impossible. We examine a necessary and sufficient condition for determinacy under interest rate policy that responds to a weighted average of an inflation forecast and current inflation. This condition demonstrates that the average-inflation targeting policy ensures determinacy as long as both the response to average inflation and the relative weight of current inflation are large enough. We also find that interest rate policy that responds solely to past inflation guarantees determinacy when its response satisfies the Taylor principle and is not large. These results still hold even when wages and hours worked are determined by Nash bargaining.     

利率市场化改革与货币政策银行利率传导

现代货币政策框架的内涵之一是畅通的货币政策传导机制。在货币政策对银行利率的传导中,政策利率尤其是再融资工具利率能够通过市场利率向贷款利率和存款利率顺畅传导。本文构造了一个多部门局部均衡模型,刻画利率市场化程度的时变特征,从理论上讨论利率市场化程度与货币政策银行利率传导之间的内生关系,两者共同决定了利率市场化改革的成效。基于时变因子扩展向量自回归模型,本文进一步验证了理论分析的结论。研究发现:(1)利率市场化改革具有阶段性波动特征,利率市场化程度不是一直上升的,改革可能会曲折迂回。(2)利率市场化改革具有动态政策效应,多种改革政策共同推动渐进的利率市场化,但从货币政策银行利率传导的有效性进行考察,利率市场化改革仍然存在进一步推进的空间。因此,有必要完善贷款市场报价利率的形成、传导与调控机制,理顺不同利率之间的联动关系,疏通货币政策对银行利率的传导,充分发挥利率市场化改革的潜力。