Relative Guarantees

Many real-world financial contracts have some sort of minimum rate of return guarantee included. One class of these guarantees is so-called relative guarantees, i.e., guarantees where the minimum guaranteed rate of return is given as a function of the stochastic return on a reference portfolio. These guarantees are the topic of this paper. We analyse a wide range of different functional specifications for the minimum guaranteed rate of return, hereunder both so-called maturity and multi-period guarantees. Several closed form solutions are presented.     

Guaranteed Investment Contracts: Distributed and Undistributed Excess Return

Annual minimum rate of return guarantees are analyzed together with rules for distribution of positive excess return, i.e. investment returns in excess of the guaranteed minimum return. Together with the level of the annual minimum rate of return guarantee both the customer's and the insurer's fractions of the positive excess return are determined so that the market value of the insurer's capital inflow (determined by the fraction of the positive excess return) equals the market value of the insurer's capital outflow (determined by the minimum rate of return guarantee) at the inception of the contract. The analysis is undertaken both with and without a surplus distribution mechanism. The surplus distribution mechanism works through a bonus account that serves as a buffer in the following sense: in (‘bad’) years when the investment returns are lower than the minimum rate of return guarantee, funds are transferred from the bonus account to the customer's account. In (‘good’) years when the investment returns are above the minimum rate of return guarantee, a part of the positive excess return is credited to the bonus account. In addition to characterizations of fair combinations of the level of the annual minimum rate of return guarantee and the sharing rules of the positive excess return, our analysis indicates that the presence of a surplus distribution mechanism allows the insurer to offer a much wider menu of contracts to the customer than without a surplus distribution mechanism.     

Pricing Participating Inflation Retirement Funds Through Option Modeling and Copulas

Abstract

Pension plans and life insurances offering minimum performance guarantees are very common worldwide. In the Brazilian market, the customers of a common type of defined contribution plan have the right to receive, over their savings, the positive difference between the return of a specified investment fund, usually a fixed income fund, and the minimum guaranteed rate, commonly defined as the composition of a fixed interest rate and a floating inflation rate. This instrument can be characterized as an option to exchange one asset, the minimum guaranteed rate, for another, the return of the specified investment fund. In this paper we provide a closed formula to evaluate this liability that depends on two stochastic rates assuming bivariate normality. We also explore the use of copulas for the modeling of the dependence structure and price the options using Monte Carlo simulation to compare the effects of the copula specification in their values. An application with real data is provided. The model makes use of a one-factor Vasicek framework for the term structures of interest rate and inflation rate.     

A lattice-based model to evaluate variable annuities with guaranteed minimum withdrawal benefits under a regime-switching model

We consider the problem of evaluating variable annuities with a guaranteed minimum withdrawal benefit under a regime-switching model. We propose a trinomial lattice model to approximate the evolution of the investment fund value and the policy value at inception is computed through a backward induction scheme. Finally, the insurance fee is computed as the solution of the equation that makes the contract actuarially fair. Numerical results are reported to illustrate the consistency of the proposed model.     

Impacts of Jumps and Stochastic Interest Rates on the Fair Costs of Guaranteed Minimum Death Benefit Contracts

The authors offer a new perspective to the field of guaranteed minimum death benefit contracts, especially for simple return premium and rising floor guarantees. A particular feature of these contracts is a guaranteed capital upon the insured's death. A complete methodology based on the generalized Fourier transform is proposed to investigate the impacts of jumps and stochastic interest rates. This paper thus extends Milevsky and Posner (2001). If jumps alone are considered, similar results are obtained, but, when stochastic interest rates are introduced, the fair costs of the guarantee feature are found to be substantially higher in this more general economy.     

A Simulation-Based Investigation of Errors in Accounting-Based Surrogates for Internal Rate of Return

Accounting-based measures of a firm's ex post performance represent accessible, albeit imperfect, surrogates for its internal rate of return (IRR). Using a cross-sectional data set obtained via computer simulation, this study calculated the error with which the accounting rate of return (ARR) and conditional estimate of internal rate of return (CIRR) estimate IRR. The study compared the error with which both surrogates measure IRR, as well as the ability of growth in unit demand (gD), inventory cost flow assumption (INV) and depreciation method (DEP) to explain the measurement error in both surrogates.     

Paying for Minimum Interest Rate Guarantees: Who Should Compensate Who?

Defined contribution pension schemes often have a mandatory minimum interest rate guarantee as an integrated part of the contract. The guarantee is an embedded put option issued by the institution to the individual who is forced to invest in the option. As argued in this paper, the individual may in this way face a constraint on the feasible set of portfolio choices. We quantify the effect of the minimum interest rate guarantee constraint and demonstrate that guarantees may induce a significant utility loss. We also consider the effects of the interest rate guarantee in the case of heterogenous investors sharing a common portfolio on a pro rata basis.     

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

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.     

Exploiting the “Win But Does Not Cover” Phenomenon in College Basketball

Wolfers (2006) was the first to document that heavy favorites in college basketball win but fail to cover the pre‐game point spread at a statistically higher rate than expected. We generate a hedged strategy to exploit the “win but does not cover” phenomenon using two wagers: a bet on the underdog sides line and a bet on the favorite money line. While one bet is guaranteed to win regardless of the outcome, both bets win if the favorite wins but does not cover. We show that the minimum‐variance portfolio best exploits this anomaly, yielding an average return of 0.34% per game and a positive return in five of the seven seasons of college basketball analyzed.     

Risk based capital for guaranteed minimum withdrawal benefit

The guaranteed minimum withdrawal benefit (GMWB), which is sold as a rider to variable annuity contracts, guarantees the return of total purchase payment regardless of the performance of the underlying investment funds. The valuation of GMWB has been extensively covered in the previous literature, but a more challenging problem is the computation of the risk based capital for risk management and regulatory reasons. One needs to find the tail distribution of the profit–loss function, which differs from its expected payoff required for pricing the GMWB contract. GMWB has embedded two option-like features: Management fees are proportional to the current value of the policyholder’s account which results in an average price of the account. Thus the contract resembles an Asian option. However, the fees are charged only up to the time of the account hitting zero which resembles a barrier option payoff. Thus the GMWB is mathematically more complicated than Asian or barrier options traded on the financial markets. To the authors’ best knowledge, this is the first paper in the literature to formulate and analyse profit–loss distribution using PDE methods of such a product with intricate option-like features. Our approach is much more efficient than the current market practice of rather intensive and expensive Monte Carlo simulations due to the lack of samples for extreme cases.     

Good-Deal Option Price Bounds for a Non-Traded Event with Stochastic Return: A Note

Cochrane and Sa'a-Requejo (2000, Journal of Political Economy) proposed the good-deal price bounds for the European call option on an event that is not a traded asset, but is correlated with a traded asset that can be used as an approximate hedge. One remarkable feature of their model is that the return on an event process explicitly appears in the option price bounds formula, which offered a contrast with the standard option pricing model. We show that the good-deal option price bounds on a non-traded event are obtained as a closed-form formula, when the return on an event is governed by a mean reverting process.     

影响我国教育回报率因素研究

本文主要研究工资水平和教育投入、性别、家庭背景等因素的关联性,以及在明瑟方程的基础上加人性别和家庭背景的因素测算教育的回报率,以此探讨现如今我国教育回报率的影响因素。首先探讨教育回报率是否过低,通过对数据进行Mann—WhitneyU检验,发现存在过度教育现象。此外,通过教育回报率的影响因素分析,发现性别、教育程度、家庭背景对教育回报率存在显著的影响。     

A Study of Volatility and Externality Compensative Return of Internet Financial Products in the Case of Yuebao

We use the data of 10 thousand accrual of Zenglibao monetary fund of Celestica Fund and two indicators of the monetary fund market, WIND index of monetary fund and CSI money fund index, to analyze the volatility and compensative rate of return of Yuebao. Based on the time-variant capital asset pricing model (CAPM), we quantitatively show that the volatility of return of Yuebao is less than that of the market, and the correlation between the Yuebao and the market is relatively low. These two conditions make the beta coefficient lower than that in traditional financial products. In this article, we define the gap between return of Yuebao and the estimated return by CAPM as the externality compensative rate of return, which is the main explanation of the high-return property of Yuebao.     

Projections of pension fund solvency under alternative valuation regimes

This paper examines the impact of three alternative valuation regimes on perceived pension fund solvency. Deterministic valuation assumes smoothed valuation of assets and liabilities. National valuation is based on market valuation of assets and on smoothed valuation of liabilities. International valuation marks assets and liabilities to market values. Using closed-form methods based on the funding ratio return, we exemplify the dramatic effect that the choice of valuation approach has on long-horizon solvency projections.     

An analysis of private loan guarantees

Loans guaranteed by private parties other than government agencies experienced an explosive growth in the early 1980s. Althoughprivate financial guarantees are widely used, models for valuing them are not well developed. Using option-pricing theory, in a discrete-time framework we obtain closed-form formulas for evaluating private loan guarantees. We conduct numerical simulations and derive comparative statics for vulnerable loan guarantees under different conditions of debt subordination in conjunction with government default-free guarantees.     

Investment guarantees in unit-linked life insurance from the customer perspective

Interest rate guarantees are a typical contract feature in unit-linked-life insurance products. As the financial crisis of 2007/2008 has shown, these guarantees can be of substantial value for policyholders since they ensure that at least a minimum amount will be paid back even if the mutual fund value falls below a specific guaranteed level. However, from the insurance company’s view, these guarantees can be costly—especially in highly volatile markets—due to the required risk management measures which must be undertaken to secure the guarantees promised to the customers. Thus, the aim of this paper is to investigate whether customers really value these guarantees and if their willingness to pay (WTP) is sufficient to cover the guarantee costs. To elicit customer WTP, we use an online questionnaire and compare these results to the actual guarantee costs calculated with the Black and Scholes option pricing formula. One main finding is that even though most of the participants in the online questionnaire work in the financial industry, subjective prices are difficult to derive and are lower, on average, than the prices obtained using a financial pricing model. However, many participants are still willing to pay a substantially higher price.     

Bayesian multivariate regime-switching models and the impact of correlation structure misspecification in variable annuity pricing

ABSTRACT

We develop Bayesian multivariate regime-switching models for correlated assets, comparing three different ways to flexibly structure the correlation matrix. After developing the models, we examine their relative characteristics and performance, first in a straightforward asset simulation example, and later applied to a variable annuity product with guarantees. We find that the freedom allowed by the more flexible structures enables these models to more accurately reflect the actual asset dependence structure. We also show that the correlation structures inferred by the most commonly used (and simplest) model will result in significantly larger estimates of the cost of the annuity guarantees.     

Optimal pension insurance design

In this paper we analyze how the traditional life and pension contracts with a guaranteed rate of return can be optimized to increase customers’ welfare. Given that the contracts have to be priced correctly, we use individuals’ preferences to find the preferred design. Assuming CRRA utility, we cannot explain the existence of any form of guarantees. Through numerical solutions we quantify the difference (measured in certainty equivalents) to the preferred Merton solution of direct investments in a fixed proportion of risky and risk free assets. The largest welfare loss seems to come from the fact that guarantees are effective by the end of each year, not only by the expiry of the contract. However, the demand for products with guarantees may be explained through behavioral models. We use cumulative prospect theory as an example, showing that the optimal design is a simple contract with a life-time guarantee and no default option.     

The Expected Return and Exercise Time of Merton-style Real Options

We analyse the rate of return and expected exercise time of Merton-style options (1973) employed in many real option situations where the possibility of exercise is both perpetual and American in nature. Using risk-neutral and risk-adjusted pricing techniques, Merton-style options are shown to have an expected return that is a constant percentage of the option value and independent of the proximity to the critical exercise boundary. Merton options thus remain at the same point on the Security Market Line, unlike European options whose position and rate of return change dynamically. We also present formulae for the expected time and discounted times to exercise and analyse the dependency of these variables on volatility.     

多资产大宗商品期权定价研究

基于大宗商品收益率与便利收益服从均值回复过程的假设,建立带协整效应的多资产大宗商品期权定价模型,求解多资产大宗商品期权价格的解析解,将大宗商品期权定价推广到更一般情况.结果表明:标的资产收益率增加,期权价格上升,替代品期权价格下降;标的资产的便利收益增加,期权价格下降,相应替代品的期权价格上升.