Pricing catastrophe options with stochastic claim arrival intensity in claim time

We model claim arrival and loss uncertainties jointly in a doubly-binomial framework to price an Asian-style catastrophe (CAT) option with a non-traded underlying loss index using the no-arbitrage martingale pricing methodology. We span these uncertainties by benchmarking to the shadow price of a one-claim bond and the premium of a reinsurance contract. We implement a stochastic time change from calendar time to claim time to more efficiently price the CAT option as a random sum – a binomial sum of claim time binomial Asian option prices. This choice of the operational time dimension allows us to incorporate different patterns of catastrophe arrivals by adjusting the claim arrival probability. We demonstrate this versatility by incorporating a mean-reverting Ornstein-Uhlenbeck intensity arrival process. Simulation results verify our model predictions and demonstrate how the claim arrival probability varies with the expected claim arrival intensity.     

Pricing mortgage insurance contracts under housing price cycles with jump risk: evidence from the U.K. housing market

Previous studies have investigated the determinants of housing price cycles in the housing market; however, we observed the phenomenon of housing price jumps in the 2007 subprime crisis. This paper presents a discussion on the housing price cycle and abnormal price jumps to describe the behavior of housing prices in the United Kingdom. The empirical results show that the impact factors of housing cycles are market risk and the switching factor. Furthermore, the impact factors of jump risks include the bursting of the housing bubble and financial crises. Therefore, in this paper, we employ the Markov switching model with jump risks to value the MI contracts and analyze the influences of housing price cycles, jump risks, risks of market interest rate, and the prepayment risks on MI premiums. The results of sensitivity analysis show that more volatile housing price index returns, as well as longer periods of higher volatility in housing prices, raise MI premiums. Moreover, the MI premium is positively related to the absolute value of the average jump amplitude and the shock frequency of abnormal events. There is the tradeoff between the market interest rate and the prepayment risk. The influences of market interest rate are different on MI premium with/without prepayment risks.     

The Effect of Regulation on Insurance Pricing: The Case of Germany

This article analyzes the impact of policy form regulation on the unit price of insurance and determinants of premium changes using the 1994 deregulation of the German property–liability market as a natural experiment. Our result show that policy form regulation did not increase prices above competitive levels. Factors influencing premium changes are significantly different for the two time periods, pre‐ and post‐deregulation, indicating that regulation affects insurance pricing. Focusing on highly competitive lines after deregulation, we find a significant price decrease, and this decrease is offset by higher prices in the remaining other lines.     

我国保险机构无赔款优待机制的评价研究

我国汽车保险费率自2003年实行市场化改革以来,各家保险公司分别设计了独具特色的奖惩系统。为了比较分析我国汽车保险市场上的五个主要奖惩系统,需要从平均保费水平、风险区分度和弹性三个方面探讨无赔款优待机制的评价问题、奖惩系统最具竞争力保险公司以及我国保险机构优化车险奖惩的激励机制。     

A rational expectations equilibrium model of inventories of finished goods and employment

A critical roadblock to modeling inventories of finished goods has been the claim that the production and inventory decisions of a perfectly competitive firm are determined independently of each other. A basic goal of this study is to specify fundamental preferences of economic agents, technologies, constraints and market structures that are, in a rough way, capable of generating patterns of serial correlations and cross correlations between inventories and employment of factors of production that are consistent with those observed in the data. The claim is made that the time series for inventories, output and employment can, in principle, be interpreted as emerging from a well-specified dynamic, stochastic competitive equilibrium in which economic agents are assumed to form rational expectations about variables not included in their information sets. Inventories and employment will not be related in a direct way if and only if the price elasticity of demand for output is equal to infinity.     

中国产险业价格竞争有效性的实证研究

中国产险市场的迅速发展,吸引了大量机构进入,同时也加剧了价格竞争。本文用赔付率作为保费价格的代理变量,采用横截面数据及最小二乘法分别对中美两国产险公司保费收入增长与保费价格、保险公司费用率之间的关系进行了研究。结果表明,在产险市场中,虽然价格竞争对增加保费收入有效,但其却极可能导致利润减少;而降低保险公司费用率可以同时增加保费收入和保险公司利润。以上结论对中国产险业尤其有效,最后本文建议中国产险业应以降低保险公司费用率作为首要选择;同时要限制价格过度竞争。     

Credibility pseudo-estimators

We treat a model with independent claim numbers and claim amounts, conditional on stochastic parameters. Groups are categorized into a smaller number of classes, which likely differ in risk premium. The collective claim frequency and mean claim for a group are modeled as those of the class the group belongs to. For each group we find the Best Linear Predictor, also known as Credibility Estimator, in a generic model covering claim frequency and mean claim, as a weighted mean of the group’s individual estimate and the collective estimate. Assuming Poisson distributed claim numbers and some distributional properties of claim amounts, we find estimators of variance components, estimation errors of the collective claim frequency and mean claim, and covariances between observations, estimators, and stochastic parameters. Pseudo-estimators, i.e. estimators which are defined by expressions that contain them and which must be solved numerically, are given for between-groups variance components. Simulation results, where some of the assumptions are violated, indicate when they are preferable over non-pseudo-estimators.     

Dynamic equilibrium and the structure of premiums in a reinsurance market

In this paper we present an economic equilibrium analysis of a reinsurance market. The continuous-time model contains the principal components of uncertainty; about the time instants at which accidents take place, and about claim sizes given that accidents have occurred. We give sufficient conditions on preferences for a general equilibrium to exist, with a Pareto optimal allocation, and derive the premium functional via a representative agent pricing theory. The marginal utility process of the reinsurance market is represented by the density process for random measures, which opens up for numerous applications to premium calculations, some of which are presented in the last section. The Hamilton-Jacobi-Bellman equations of individual dynamic optimization are established for proportional treaties, and the term structure of interest rates is found in this reinsurance syndicate. The paper attempts to reach a synthesis between the classical actuarial risk theory of insurance, in which virtually no economic reasoning takes place but where the net reserve is represented by a stochastic process, and the theory of equilibrium price formation at the heart of the economics of uncertainty.     

The effect of large audit firm mergers on audit pricing in the UK

This paper examines the effects on UK audit market concentration and pricing of mergers between the large audit firms and the demise of Andersen. Based on data over the period 1985–2002, it appears that mergers contributed to a rise in concentration ratios to levels that suggest concern about the potential for monopoly pricing. The high concentration ratios have not improved the level of price competition in the UK audit market. Our pooled models suggest that concentration ratios are associated with higher audit fees. The evidence suggests that the effects of mergers between big firms on brand name fee premium and on price competition vary depending on the particular circumstances. The brand name premium is strongest for the largest quartile of companies prior to the mergers. After the Big Six mergers, the premium increases for average‐sized companies but falls for the smallest and largest companies. Following the PricewaterhouseCoopers merger, the premium increases for below median‐sized clients but decreases for above‐median sized clients. For the Deloitte‐Andersen transaction, the premium falls for the smallest and largest companies but increases for those in the second quartile. Our results provide evidence that auditees are likely to pay higher fees if their auditor merges with a larger counterpart. We attribute merger‐related fee hikes to product differentiation, rather than anti‐competitive pricing.     

Probability of ruin with variable premium rate

Abstract

The probability of ruin is investigated under the influence of a premium rate which varies with the level of free reserves. Section 4 develops a number of inequalities for the ruin probability, establishing upper and lower bounds for it in Theorem 4. Theorem 5 gives an expression for the ruin probability, and it is seen in Section 5 that this amounts to a generalization of the ruin probability given by Gerber for the special case of a negative exponential claim size distribution. In that same section it is shown the Lundberg's inequality is not derivable from the generalized theory of Section 4, and this is seen as a drawback of the methods used there. Sections 6 and 7 deal with some special cases, including claim size distributions with monotone failure rates. Section 8 shows that, in contrast with the result for a constant premium that the probability of ruin for zero initial reserve is independent of the claim size distribution, the same result does not hold when the premium rate is allowed to vary. Section 9 gives some comments on the possible effect of “dangerousness” of a claim size distribution on ruin probability.     

Alternative Methods of Increasing the Precision of Weighted Repeat Sales House Prices Indices

Weighted repeat sales house price indices have become one of the primary indicators used to identify housing market conditions and to estimate the amount of equity homeowners have gained through house price appreciation. The primary reason for the acceptance of this methodology is that it derives a location specific (typically, census division, state or metropolitan area) average change in house prices from repeated observations of individual house prices. It is this repeat attribute that allows repeat sales price indices to claim that it is a preferable index which does a better job of holding quality constant. The amount of time between the two observed prices for a single property is determined by when the home transacts. Some homes transact twice in a period of months and others do not transact for decades. It is likely that individual house price appreciation rates vary from the mean appreciation rate, as estimated by the index, in a systematic fashion. In general, the longer the time between transactions the more variance there is in individual house price appreciation. This paper extends this concept to include new dimensions. For instance, houses that appreciate faster than the mean, as estimated by the index for that location, may experience a different variation structure than homes that appreciate slower. This process can be viewed as an asymmetric treatment of the variance of house price appreciation around the estimated index. In addition, the variance of expensive and affordable homes may also be different and time varying. This paper finds evidence that adding the dimensions of price tiers and asymmetry to the variance estimate has merit and does affect the estimated index as well as homeowner equity estimates. Homeowner equity estimates are especially sensitive to these added dimensions because they depend on both the revised index and the estimated variances, which are specific to each dimension considered—time between transaction, asymmetry, and price tier.     

Performance of utility-based strategies for hedging basis risk

The performance of optimal strategies for hedging a claim on a non-traded asset is analysed. The claim is valued and hedged in a utility maximization framework, using exponential utility. A traded asset, correlated with that underlying the claim, is used for hedging, with the correlation ρ typically close to 1. Using a distortion method (Zariphopoulou 2001 Finance Stochastics 5 61–82) we derive a nonlinear expectation representation for the claim’s ask price and a formula for the optimal hedging strategy. We generate a perturbation expansion for the price and hedging strategy in powers of ε²?=1?ρ². The terms in the price expansion are proportional to the central moments of the claim payoff under the minimal martingale measure. The resulting fast computation capability is used to carry out a simulation-based test of the optimal hedging program, computing the terminal hedging error over many asset price paths. These errors are compared with those from a naive strategy which uses the traded asset as a proxy for the non-traded one. The distribution of the hedging error acts as a suitable metric to analyse hedging performance. We find that the optimal policy improves hedging performance, in that the hedging error distribution is more sharply peaked around a non-negative profit. The frequency of profits over losses is increased, and this is measured by the median of the distribution, which is always increased by the optimal strategies. An empirical example illustrates the application of the method to the hedging of a stock basket using index futures.     

Upper bounds on stop-loss premiums under constraints on claim size distribution

Abstract

The problem of maximal stop-loss premium under prescribed constraints on claim size distribution is taken up again. The methods of linear programming are used to show that the recent results of others are intuitively obvious. These results are then extended by the linear programming technique to cases of more general constraints, e.g. prescribed claim size variance, or prescribed minimum frequency of excess claims. In particular it is shown that, typically, the upper bound on stop-loss premiums is generated by a claim size distribution which has all its mass concentrated at very few points. In contrast with the results obtained by others recently, it is seen that the claim size distribution which produces the maximal stop-loss premium is not generally independent of the excess. Some numerical examples are given showing that the methods used here can sometimes improve considerably the recent results of others. The case of a compound Poisson distribution is treated briefly.     

Price Regulation in Secondary Insurance Markets

Secondary life insurance markets are growing rapidly. From nearly no transactions in 1980, a wide variety of similar products in this market has developed, including viatical settlements, accelerated death benefits, and life settlements and as the population ages, these markets will become increasingly popular. Eight state governments, in a bid to guarantee sellers a “fair” price, have passed regulations setting a price floor on secondary life insurance market transactions, and more are considering doing the same. Using data from a unique random sample of HIV+ patients, we estimate welfare losses from transactions prevented by binding price floors in the viatical settlements market (an important segment of the secondary life insurance market). We find that price floors bind on HIV patients with greater than 4 years of life expectancy. Furthermore, HIV patients from states with price floors are significantly less likely to viaticate than similarly healthy HIV patients from other states. If price floors were adopted nationwide, they would rule out transactions worth $119 million per year. We find that the magnitude of welfare loss from these blocked transactions would be highest for consumers who are relatively poor, have weak bequest motives, and have a high rate of time preference.     

The Impact of Introducing Insurance Guaranty Schemes on Pricing and Capital Structure

The introduction of an insurance guaranty scheme can have significant influence on the pricing and capital structures in a competitive market. The aim of this article is to study this effect on competitive equity–premium combinations while considering a framework with policyholders and equity holders where guaranty fund charges are volume‐based, as levied in existing schemes. Several settings with regard to the origin of the fund contributions are assessed and the immediate effects on the incentives of the policyholders and equity holders are analyzed through a one‐period contingent claim approach. One result is that introducing a guaranty scheme in a market with competitive conditions entails a shift of equity capital towards minimum solvency requirements. Hence, adverse incentives may arise with regard to the overall security level of the industry.     

The Forward Premium in the Nord Pool Power Market

This article investigates the forward premium of futures contracts in the Nordic power market for the time period from January 2004 to December 2013. We find that futures prices are biased predictors of the subsequent spot prices and that there is a significant forward premium in the Nord Pool market, particularly during the winter and autumn. We analyze the impact from several factors on the forward premium. The spot price, and the deviation of water inflow from its usual level, positively affect the forward premium. The variance of the spot price also has a positive effect on the forward premium, but only for the contract closest to delivery.     

The early exercise premium in American put option prices

This study estimates the value of the early exercise premium in American put option prices using Swedish equity options data. The value of the premium is found as the deviation of the American put price from European put-call parity, and in addition a theoretical estimate of the premium is computed. The empirically found premium is also used in a modified version of the control variate approach to value American puts. The results indicate a substantial value of the early exercise premium, where the premium derived from put-call parity is higher than the theoretical premium. The premium also increases with moneyness and time left to expiration, while the effect of interest rate and volatility depends on the moneyness of the option. The modified control variate technique works reasonably well relative to the theoretical models. In particular, for deep in-the-money options, this technique is superior.     

The index premium and its hidden cost for index funds

This paper empirically investigates the index premium and its implications from 1990 to 2005. For additions to the S&P 500 and Russell 2000, we find that the price impact from announcement to effective day has averaged + 8.8% and + 4.7%, respectively, and −15.1% and −4.6% for deletions. The premia have been growing over time, peaking in 2000, and declining since then. The implied price elasticity of demand increases with firm size and decreases with idiosyncratic risk, supporting theoretical predictions. We also introduce a new concept that we label the index turnover cost, which represents a hidden cost borne by index funds (and the indexes themselves) due to the index premium. We illustrate this cost and estimate its lower bound as 21-28 bp annually for the S&P 500 and 38-77 bp annually for the Russell 2000.     

Optimal dynamic reinsurance with dependent risks: variance premium principle

In this paper, we consider the optimal proportional reinsurance strategy in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. Under the criterion of maximizing the expected exponential utility with the variance premium principle, we adopt a nonstandard approach to examining the existence and uniqueness of the optimal reinsurance strategy. Using the technique of stochastic control theory, closed-form expressions for the optimal strategy and the value function are derived for the compound Poisson risk model as well as for the Brownian motion risk model. From the numerical examples, we see that the optimal results for the compound Poisson risk model are very different from those for the diffusion model. The former depends not only on the safety loading, time, and the interest rate, but also on the claim size distributions and the claim number processes, while the latter depends only on the safety loading, time, and the interest rate.     

Credibility for Severity Revisited

Abstract

It is basic actuarial knowledge that the pure premium of an insurance contract can be written as the product of the expected claim number and the expected claim amount. Actuaries use credibility theory to incorporate the contract’s individual experience into this calculation in a statistically optimal way. For many years, however, the use of credibility was limited to the frequency component. Starting with the paper by Hewitt (1971), there have been various suggestions as to how credibility theory also can be applied to the severity component of the pure premium. The latest such suggestion, Frees (2003), revived the interest in the problem.

In this paper, we review four different formulas incorporating frequency and severity into credibility calculations. We then compare by simulation which one is most accurate at predicting a contract’s next-year outcome. It is found that the classical formula of Bühlmann (1967) is as good as the other ones in many cases. Alternatives, however, may offer easier analysis of the separate effects of frequency and severity on the premium.

We also show that all the formulas reviewed in this paper stem from the same minimization problem, and we present a general, integrated, solution. At the same time, we complete Gerber (1972) by providing a proof to the main result of this paper and by stating required additional assumptions.