Equilibrium Pricing of General Insurance Policies

Abstract

A model is developed for determining the price of general insurance policies in a competitive, noncooperative market. This model extends previous single-optimizer pricing models by supposing that each participant chooses an optimal pricing strategy. Specifically, prices are determined by finding a Nash equilibrium of an N-player differential game. In the game, a demand law describes the relationship between policy sales and premium, and each insurer aims to maximize its (expected) utility of wealth at the end of the planning horizon. Two features of the model are investigated in detail: the effect of limited total demand for policies, and the uncertainty in the calculation of the breakeven (or cost price) of an insurance policy.

It is found that if the demand for policies is unlimited, then the equilibrium pricing strategy is identical for all insurers, and it can be found analytically for particular model parameterizations. However, if the demand for policies is limited, then, for entrants to a new line of business, there are additional asymmetric Nash equilibria with insurers alternating between maximal and minimal selling. Consequently it is proposed that the actuarial cycle is a result of price competition, limited demand, and entry of new insurers into the market. If the breakeven premium is highly volatile, then the symmetric equilibrium premium loading tends to a constant, and it is suggested that this will dampen the oscillatory pricing of new entrants.     

Are tightened trading rules always bad? Evidence from the Chinese index futures market

This paper investigates the impact of tightened trading rules on the market efficiency and price discovery function of the Chinese stock index futures in 2015. The market efficiency and the price discovery of Chinese stock index futures do not deteriorate after these rule changes. Using variance ratio and spectral shape tests, we find that the Chinese index futures market becomes even more efficient after the tightened rules came into effect. Furthermore, by employing Schwarz and Szakmary [J. Futures Markets, 1994, 14(2), 147–167] and Hasbrouck [J. Finance, 1995, 50(4), 1175–1199] price discovery measures, we find that the price discovery function, to some extent, becomes better. This finding is consistent with Stein [J. Finance, 2009, 64(4), 1517–1548], who documents that regulations on leverage can be helpful in a bad market state, and Zhu [Rev. Financ. Stud., 2014, 27(3), 747–789.], who finds that price discovery can be improved with reduced liquidity. It also suggests that the new rules may effectively regulate the manipulation behaviour of the Chinese stock index futures market during a bad market state, and then positively affect its market efficiency and price discovery function.     

Bid ask spread in a competitive market with institutions and order size

Large orders, particularly from institutions, are quite common these days and hence there is interest to know if institutional trading has any bearing on the price effect associated with large trades. Recent empirical studies contradict earlier evidence of negative price effect on selling large blocks and find no price effect associated with large trades. Existing theoretical framework suggests a monotonic and increasing adverse price effect for large trades, where the motivation for a large trade is private information. We model a trading system where pure information, information-liquidity, and pure liquidity traders trade small and large sizes. The pure information traders strategically choose an order size. Institutions trade only large sizes because of their low execution costs for large trades; they are information-liquidity traders whose ability to use an information signal to determine their trades is subject to a binding liquidity constraint. We show that in such a market a separating equilibrium where trade size is informative does not exist and hence there is no price effect for large trades. Trade size may be revealing only if there is a buy sell asymmetry (large buy size is not equal to large sell size) or the corresponding price effect is asymmetric (price effect due to a large buy is not equal to that of a large sell). Further for a pooling equilibrium to exist, where trade size is not informative, the width of the market denoted by the ratio of order size (large size/small size) needs to be small, while the shallowness (inverse depth) of the market denoted by the ratio between pure information and institutional trades and the information signal needs to be stronger (higher). Our results on bid and ask prices and spread confirm recent empirical evidence on price effect of large and institutional trades found in the literature.

On derivatives with illiquid underlying and market manipulation

In illiquid markets, option traders may have an incentive to increase their portfolio value by using their impact on the dynamics of the underlying. We provide a mathematical framework to construct optimal trading strategies under market impact in a multi-player framework by introducing strategic interactions into the model of Almgren [Appl. Math. Finance, 2003, 10(1), 1–18]. Specifically, we consider a financial market model with several strategically interacting players who hold European contingent claims and whose trading decisions have an impact on the price evolution of the underlying. We establish the existence and uniqueness of equilibrium results for risk-neutral and CARA investors and show that the equilibrium dynamics can be characterized in terms of a coupled system of possibly nonlinear PDEs. For the linear cost function used by Almgren, we obtain a (semi) closed-form solution. Analysing this solution, we show how market manipulation can be reduced.     

Capital market governance: How do security laws affect market performance?

This paper examines the link between capital market governance (CMG) and several key measures of market performance. Using detailed data from individual stock exchanges, we develop a composite CMG index that captures three dimensions of security laws: the degree of earnings opacity, the enforcement of insider laws, and the effect of removing short-selling restrictions. We find that improvements in the CMG index are associated with decreases in the cost-of-equity capital (both implied and realized), increases in market liquidity (trading volume, market depth, and U.S. foreign investments), and increases in market pricing efficiency (reduced price synchronicity and IPO underpricing). The results are quite consistent across individual components of CMG and over alternative market performance measures.     

The Asymmetric Wealth Effect in the US Housing and Stock Markets: Evidence from the Threshold Cointegration Model

Previous studies commonly use a linear framework to investigate the long-run equilibrium relationship between the housing and stock markets. The linear approaches may not be appropriate if adjustments from disequilibrium are asymmetric in both markets. Nonlinear adjustments are likely to be observed since the two markets respond rather differently to negative shocks where the stock market is more volatile but price rigidity is found in the housing market. In this paper, we firstly propose two hypotheses on the long-run equilibrium relationship of the US housing and stock markets, and then employ the threshold cointegration model to investigate the potential asymmetric relationships between the two markets. Our empirical results reveal that cointegration exists among the markets, but adjustments toward its long-run equilibrium are asymmetric. Further evidence points out that a rapid mean reversion occurs in one regime where the stock price outperforms the housing price, and no significant reversion is found in the other regime, supporting the hypothesis of the existence of an asymmetric wealth effect among the two markets in the US. Furthermore, evidence from the asymmetric vector error correction model shows that significant error corrections toward the equilibrium exist in the short run only when the stock price exceeds the real estate price by the estimated threshold level, reassuring the finding of the asymmetric wealth effect.     

A financial CCAPM and economic inequalities

This paper considers a wealth heterogeneous multi-agent (MA) financial pricing CCAPM model. It is based on the following observations: (a) A distinction between what agents are willing to pay for consumption and what they actually pay. The former is a function of a number of factors including the agent’s wealth and risk preferences and the latter is a function of all other agents’ aggregate consumption or equivalently, their wealth committed to consumption. (b) Unlike traditional pricing models that define a representative agent underlying the pricing model, this paper assumes that each agent is in fact ‘Cournot-gaming’ a market defined by all other agents. This results in a decomposition of an n-agents game into n games of two agents, one a specific agent and the other a synthetic agent (a proxy for all other agents), on the basis of which an equilibrium consumption price solution is defined. The paper’s essential results are twofold. First, a Martingale pricing model is defined for each individual agent expressing the consumer willingness to pay (his utility price) and the market price—the price that all agents pay for consumption. In this sense, price is unique defined by each agent’s ‘Cournot game’ Agents’ consumption are then adjusted accordingly to meet the market price. Second, the pricing model defined is shown to account for agents wealth distribution pointing out that all agents valuations are a function of their and others’ wealth, the information they have about each other and other factors which are discussed in the text. When an agent has no wealth or cannot affect the market price of consumption, then this pricing model is reduced to the standard CCAPM model while any agent with an appreciable wealth compared to other agents, is shown to value returns (and thus future consumption) less than wealth-poor agents. As a result, this paper will argue that even in a financial market with an infinite number of agents, if there are some agents that are large enough to affect the market price by their decisions, such agents have an arbitrage advantage over the poorer agents. The financial CCAPM MA pricing model has a number of implications, some of which are considered in this paper. Finally, some simple examples are considered to highlight the applicability of this paper to specific financial issues.     

Adverse selection and the market for annuities

Adverse selection is often blamed for the malfunctioning of the annuities market. We simulate the impact of adverse selection on the consumption allocation of annuitants under alternative parameter values, and explore the resulting welfare implications. We show that, for most parameter values, the welfare losses associated with equilibriums that are subject to adverse selection correspond to a loss of wealth of around one percent in a first-best equilibrium. These losses are smaller than the corresponding losses associated with equilibriums with no access to an annuity market by an order of magnitude of ten. The existence of substitutes for annuities such as a bequest motive or a social security system intensifies the adverse selection but reduces its welfare impact.

现券市场做市报价点差的微观特征分析

交易机制与价格发现的关系一直是金融市场微观结构的重要研究方向,其目的是为了发现金融市场如何借由微观行为的不均衡实现市场层面的价格均衡过程。事实上,交易机制、交易习惯和市场均衡过程三者之间是相互影响的关系。分析和研究这三者的内在联系可以加深我们对市场的理解,并找到提高市场资源配置效率的途径。文章以做市报价收益率点差作为研究对象,详细分析了其时间分布、相关性、波动性及时间持续性等微观特征,为深入研究做市商报价和交易行为提供各种可能的切入点。     

Sources of the stock price fluctuations in Chinese equity market

This paper proposes a latent factor approach based on a state–space framework in order to identify which factor, if any, dominates price fluctuations in the Chinese stock markets. We also illustrate the connection of such stock price decomposition with several general equilibrium asset pricing models and show that the decomposition results can potentially offer useful insights with regard to the empirical relevance of asset pricing models. We use quarterly data of the Chinese A-Share equity market over the period 1995Q3–2011Q1 and find that the estimates of the state–space model suggest that the expected return is the primary driving force behind price fluctuations in the Chinese stock market. We show that the time-varying expected returns appear to be counter-cyclical and this result seems to be consistent with the habit formation model of Campbell and Cochrane [1999. By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107, no. 2: 205–51.]. However, we also note that there is a great deal of uncertainty with respect to this variance decomposition due to the resulting small signal-to-noise ratio in the estimated state–space model.     

Stock market reaction to policy interventions

ABSTRACT

We analyse stock price reactions to the announcements of monetary and fiscal policy actions in 12 stock exchanges worldwide between 1 June 2007 and 30 June 2012. While past papers have analysed the effect of policy interventions focusing on monetary policy actions (e.g. Ricci 2015), our paper focuses on stock indices either capturing the whole stock market or various industries. By estimating abnormal stock reactions around the announcement date, we show that (1) stock industry indices react to policy interventions in a different manner than the broad stock index does; (2) stock returns react negatively to restriction measures for general and non-banking sector indices; and (3) stock reaction to expansionary measures was stronger at the beginning of the financial crisis.     

An index of market shocks based on multiscale analysis*

Abstract

Financial markets are places of sudden and violent price movements. Nevertheless, financial crises lack a universally recognized way of assessing their gravity. This has motivated the measure recently proposed and applied to the exchange rates market by Zumbach et al (2000a Int. J. Theor. Appl. Finance 3 347–55). This measure relies on an analogy with geophysics: the scale of market shocks (SMS) is equivalent to the Richter scale used for earthquakes. More precisely, as a market is the place where economic agents—with different investment horizons—interact, the SMS definition is a weighted aggregation of volatility measures corresponding to these different horizons. In this paper, we implement and apply a similar measure to stock markets, and adapt it to take into account some extra features of these markets.

The volatilities are first described, and then used to assess the market instability perceived by a market participant. The evolution of our index of market shocks (IMS)—after rescaling for easy interpretation—is presented using different computational methods.

The IMS is then compared with another multiscale measure, the multifractal spectrum width, and we also investigate the links between the IMS and the daily close-to-close returns and volatility. Finally, we describe the recent turbulence on the French market using the IMS as an exploratory tool, concluding that the events of September 2001 proved to be a major shock compared to the Russian and Asian crises.     

Capital market equilibrium with externalities, production and heterogeneous agents

The paper studies general equilibrium in an economy with externalities, production and heterogeneous agents. The model developed builds on Brock [Brock, W.A., 1982. Asset prices in a production economy. In: McCall, J.J. (Ed.), The Economics of Information and Uncertainty. University of Chicago Press, Chicago, pp. 1–43] and Merton [Merton, R.C., 1987. A simple model of capital market equilibrium with incomplete information. Journal of Finance 42, 483–510]; it involves both a stock market and a market for loans, together with negative externalities produced by a subset of firms. Importantly, the technological production structure of the firms is reflected in the properties of the shares traded in the stock market. Agents are heterogeneous in their financial choices, potentially discriminating against the firms producing a negative externality. The model sheds light on the utility costs of the discriminating behavior and on the impact on the price of the stock issued by the firm which is responsible for the externality. The model is used to study the factors which may magnify or reduce the impact of discrimination. A set of discriminated firms may be seriously affected only if the discriminating investors command a large portion of overall wealth and/or they do not represent important diversification instruments. The model can be applied to understanding the effects of socially responsible investment, whereby investors discriminate against companies belonging to some sectors which are perceived as socially dangerous or unethical.     

On the theory of market information efficiency

This paper discusses the concept of market information efficiency and demonstrates the following: As the number of traders who participate in the market becomes large, the variations in the price of a security caused by the variations in traders' beliefs make the market price vary as if traders all knew the ‘true’ distribution of returns on the security. The empirical implications of the analysis are also explored.     

How efficiency shapes market impact

We develop a theory for the market impact of large trading orders, which we call metaorders because they are typically split into small pieces and executed incrementally. Market impact is empirically observed to be a concave function of metaorder size, i.e. the impact per share of large metaorders is smaller than that of small metaorders. We formulate a stylized model of an algorithmic execution service and derive a fair pricing condition, which says that the average transaction price of the metaorder is equal to the price after trading is completed. We show that at equilibrium the distribution of trading volume adjusts to reflect information, and dictates the shape of the impact function. The resulting theory makes empirically testable predictions for the functional form of both the temporary and permanent components of market impact. Based on the commonly observed asymptotic distribution for the volume of large trades, it says that market impact should increase asymptotically roughly as the square root of metaorder size, with average permanent impact relaxing to about two-thirds of peak impact.     

Ownership structure and open market stock repurchases in France

Abstract

This paper examines open market stock repurchases in France. We find a positive average market reaction to the repurchase announcement. However, the magnitude of the price reaction is found to depend on a number of corporate governance structure measures. The positive aspects of the announcement only appear for a company with a low likelihood of being taken over, and with a low risk of minority shareholder expropriation. Specifically, stock repurchase programmes are good news when the firm is supported by foreign institutional investors, and in the case of controlled firms, when the firm has a second large shareholder, which guarantees an effective balance of power for the controlling shareholders.     

Risk sharing in a financial market with endogenous option prices

This article investigates a financial market in which investors may trade in risk-free bonds, stock and put options written on the stock. In each period, stock and option prices are simultaneously determined by market clearing. While the introduction of put options will decrease the systematic risk in the financial market, it will increase the price of risk. Investors with mean-variance preferences will generally hold portfolios containing the primary asset and the put option and may use the option to increase the risk in their wealth position in exchange for higher returns. Aggregate wealth is unaffected by an option market when there are no spillover effects on stock prices, and it is shown that short selling of options will increase the volatility of individual wealth positions. Investors with erroneous beliefs may on average be better off not trading in put options.     

Asset market equilibrium with liquidity risk

This paper derives an equilibrium asset pricing model with endogenous liquidity risk. Liquidity risk is modeled as a stochastic quantity impact on the price from trading, where the size of the impact depends on trade size. Under a strong set of assumptions, we prove that a unique equilibrium liquidity cost process and a unique equilibrium price process exists for our economy. We characterize the market’s state price density, which enables the derivation of the risk-return relation for the stock’s expected return including liquidity risk. We derive a generalized intertemporal CAPM and consumption CAPM for these markets. In contrast to the traditional models without liquidity risk, there is an additional systematic liquidity risk factor which is related to the stock return’s covariation with the market’s stochastic liquidity cost. Traditional transaction costs are a special case of our formulation.     

Forecasting stock market returns over multiple time horizons

In this paper, we seek to demonstrate the predictability of stock market returns and explain the nature of this return predictability. To this end, we introduce investors with different investment horizons into the news-driven, analytic, agent-based market model developed in Gusev et al. [Algo. Finance, 2015, 4, 5–51]. This heterogeneous framework enables us to capture dynamics at multiple timescales, expanding the model’s applications and improving precision. We study the heterogeneous model theoretically and empirically to highlight essential mechanisms underlying certain market behaviours, such as transitions between bull and bear markets and the self-similar behaviour of price changes. Most importantly, we apply this model to show that the stock market is nearly efficient on intraday timescales, adjusting quickly to incoming news, but becomes inefficient on longer timescales, where news may have a long-lasting nonlinear impact on dynamics, attributable to a feedback mechanism acting over these horizons. Then, using the model, we design algorithmic strategies that utilize news flow, quantified and measured, as the only input to trade on market return forecasts over multiple horizons, from days to months. The backtested results suggest that the return is predictable to the extent that successful trading strategies can be constructed to harness this predictability.     

A spot market model for pricing derivatives in electricity markets

Abstract

In this paper, we analyse the evolution of prices in deregulated electricity markets. We present a general model that simultaneously takes into account the following features: seasonal patterns, price spikes, mean reversion, price dependent volatilities and long term non-stationarity. We estimate the parameters of the model using historical data from the European Energy Exchange. Finally, we demonstrate how it can be used for pricing derivatives via Monte Carlo simulation.