Variance Premium Research Articles

Recovering the Variance Premium

This paper generalizes Ross (2015) recovery theory to accommodate growth, including the Black-Scholes and stochastic volatility option models. The new theory recovers information about equity risk premia and variance risk premia from options prices. In the Heston (1993) stochastic volatility model, the theory predicts an exact (negative) value for the variance risk premium as a function of the equity premium. Recovery theory also predicts that the stochastic discount factor is the reciprocal return on a model-free portfolio of index options. This paper tests the theory on returns from one-month VIX and three-month VIX3M option portfolios from 2007-2018. Recovery theory links the equity pre- mium to the values of both conditional and unconditional variance premia. It also predicts how VIX3M option variance is a biased predictor of future one-month VIX variance. Empirically, recovery theory simultaneously matches the average S&P 500 equity premium, the average variance premium, and observed biases in the variance expectations hypothesis. Autocorrelation properties of VIX indices imply a 12% annual equity premium.

Beyond the Yield Curve: Understanding the Effect of FOMC Announcements on the Stock Market

A large literature uses high-frequency changes in interest rates around FOMC announcements to study monetary policy. These yield changes have puzzlingly low explanatory power for the stock market - even in a narrow 30-minute window. We propose a new approach to test whether the unexplained variation represents monetary policy news or just noise. In particular, we allow for a latent Fed non-yield curve shock'', which we estimate via a heteroskedasticity-based procedure. Using a test for weak identification, we show that our shock is well identified, that is, the unexplained variation is not just noise. We then go on to show that the shock, signed to increase stock prices, leads to sizable declines in the equity and variance premium, an increase in the 10-year term premium, an increase in short-run inflation expectations, as well as a dollar depreciation against multiple non-safe-haven currencies. Hence, the evidence supports the interpretation that the shock affects risk-appetite and leads to a reverse flight-to-safety'' effect. Lastly, using a method from the computational linguistics literature, we show that our shock can be linked to specific topics discussed in FOMC statements, suggesting that it reflects written communication by the Federal Reserve.

Jump, Diffusion, and Long-Term Volatility Risks with Incremental Information in VIX Assets

A reduced-form model is proposed to disentangle the dynamics of positive-jump, diffusion, and long-term risks in VIX assets. A dissection of the three component risks in volatility is undertaken to determine their distinctive dynamics, risk prices, and market information. The dynamics of three component volatility risks play important and distinctive roles in the VIX derivatives market. The risk dissection substantially improves VIX option valuation. The component risks considerably explain VIX asset returns with a distinct exposure pattern and require separate risk premia. The risks are priced distinctively in the volatility market and are sensitive to different macroeconomic conditions. A comparison of the VIX derivatives market with the S&P 500 market reveals incremental information from the component risks beyond that provided by the traditional S&P 500 and its option market. TOPICS:Derivatives, options, factors, risk premia Key Findings ▪ A reduced-form model is used to disentangle the jump risk dynamic from the diffusion and long-term risks in the volatility market. The dynamic volatility jump risk is shown to be critical in the VIX derivatives market. ▪ The authors find unique properties in the different volatility risks. The three component risks are separately priced in the VIX market, the risk premia compose the total variance premium by different patterns, and they have unique responding for the macroeconomic conditions. ▪ They compare the pricing information in the S&P 500 and volatility markets and show that component risk dynamics in the VIX derivatives market contain incremental and critical pricing information beyond what is embedded in the S&P 500 and its option market.

Stochastic differential investment and reinsurance games with nonlinear risk processes and VaR constraints

This paper investigates a class of non-zero-sum stochastic differential investment and reinsurance games between two insurance companies. We allow both insurers to purchase a proportional reinsurance contract and invest in risky and risk-free assets. When applying the generalized mean–variance premium principle in determining reinsurance premium, the surplus process becomes quadratic in the retained proportion of the claims. The optimization criterion of each insurer is to maximize the expected utility of the insurer’s terminal performance relative to that of his competitor. In addition, we incorporate dynamic Value-at-Risk (VaR) constraints in the optimization problems of both insurers to satisfy the capital requirements from regulators. The results show that this game problem can be converted to solving a system of nonlinear equations by means of dynamic programming principle and Karush–Kuhn–Tucker (KKT) conditions. Specifically, when both insurers are constant absolute risk aversion (CARA) institutions and the reinsurance premium principle reduces to the expected value principle, we derive the simplified expressions for the Nash equilibrium strategies. Finally, we use some numerical examples to illustrate the effects of several model parameters on the Nash equilibrium strategies under three different scenarios.

Optimal dividends and reinsurance with capital injection under thinning dependence

In this paper, we adopt the variance premium principle to investigate the problem of optimal dividends and reinsurance in a diffusion approximation risk model with thinning-dependence structure. We first study the optimal problem without capital injection. We then consider the incorporation of forced capital injection into the model whenever the reserve level drops below zero. We finally turn to the general problem in which capital injection is allowed but not compulsory. For the three optimal problems, we apply the technique of stochastic control theory to obtain closed-form expressions for the optimal strategies and the corresponding value functions for two classes of insurance business with thinning dependence. Under the assumption of non cheap reinsurance, we obtain results that are quite different from those in the case of cheap reinsurance for both bounded and unbounded dividend rates. Furthermore some numerical examples are presented to show the effect of parameter values on the optimal policies.

Optimal investment and reinsurance policies for an insurer with ambiguity aversion

In this paper, we study the optimal investment and reinsurance problem for an insurer based on the variance premium principle, in which three cases are considered. First, we assume that the financial market does not exist. The insurer only holds an insurance business, and the optimal reinsurance problem is studied. Subsequently, we assume that there exists a financial market with an accurately modeled risky asset. The optimal investment and reinsurance problem is investigated under these conditions. Finally, we consider the general case in which the insurer is concerned about the model ambiguity of both the insurance market and the financial market. In all three cases, the value function is set to maximize the expected utility of terminal wealth. By employing the dynamic programming principle, we derive the Hamilton–Jacobi–Bellman (HJB) equations, which are satisfied by the value functions and obtain closed-form solutions for optimal reinsurance and investment policies and the value functions in all three cases. Most interestingly, we elucidate how investment improves the insurer’s utility and find that the existence of ambiguity can significantly affect the optimal policies and value functions. We also compare the ambiguities in the two markets and find that ambiguity in the insurance market has much more significant impact on the value function than the ambiguity in the financial market. It implies that it is more valuable for insurer to precisely evaluate the insurance risk. We also provide some numerical examples and economic explanations to illustrate our results.

A continuous-time theory of reinsurance chains

A chain of reinsurance is a hierarchical system formed by the subsequent interactions among multiple (re)insurance agents, which is quite often encountered in practice. This paper proposes a novel continuous-time framework for studying the optimal reinsurance strategies within a chain of reinsurance. The transactions between reinsurance buyers and sellers are formulated by means of Stackelberg games, in order to reflect the conflicting interests and unequal negotiation powers in the bargaining process. Assuming the variance premium principle and the mean–variance criterion on the surplus processes, we solve the time-consistent optimal reinsurance demands and pricing strategies in explicit forms, which are surprisingly plain.Based on the proposed reinsurance chain models, our in-depth theoretical analysis shows that: (a.) it is optimal to situate more (resp. less) risk averse reinsurers to the latter (resp. former) positions in a chain of reinsurance; (b.) adding new reinsurers will lower the reinsurance prices at all levels in a chain of reinsurance, promoting the existing agents to rationally control their respective risk exposures; and essentially (c.) alleviate the systemic risk in the chain structure.

A Desirable Aspect in the Variance Premium in a Collective Risk Model

This paper focuses on the study of the Collective and Bayes Premiums, under the Variance Premium Principle, in the classic Collective Risk Poisson-Exponential Model. A bivariate prior distribution is considered for both the parameter of the distribution of the number of claims and that of the distribution of the claim amount, assuming independence between these parameters. Furthermore, we analyze the consequences on these premiums of small levels of contamination in the structure functions, and find that the premiums are not sensitive to small levels of uncertainty. These results extend the conclusions obtained in Gómez-Déniz et al. (2000), where only variations in the parameter for the number of claims and its effects on premiums were studied.

Minimizing the Probability of Absolute Ruin Under Ambiguity Aversion

In this paper, we consider an optimal robust reinsurance problem in a diffusion model for an ambiguity-averse insurer, who worries about ambiguity and aims to minimize the robust value involving the probability of absolute ruin and a penalization of model ambiguity. It is assumed that the insurer is allowed to purchase per-claim reinsurance to transfer its risk exposure, and that the reinsurance premium is computed according to the mean-variance premium principle which is a combination of the expected-value and variance premium principles. The optimal reinsurance strategy and the associated value function are derived explicitly by applying stochastic dynamic programming and by solving the corresponding boundary-value problem. We prove that there exists a unique point of inflection which relies on the penalty parameter greatly such that the robust value function is strictly concave up to the unique point of inflection and is strictly convex afterwards. It is also interesting to observe that the expression of the optimal robust reinsurance strategy is independent of the penalty parameter and coincides with the one in the benchmark case without ambiguity. Finally, some numerical examples are presented to illustrate the effect of ambiguity aversion on our optimal results.

The Heavy-Tailed Exponential Distribution: Risk Measures, Estimation, and Application to Actuarial Data

Modeling insurance data using heavy-tailed distributions is of great interest for actuaries. Probability distributions present a description of risk exposure, where the level of exposure to the risk can be determined by “key risk indicators” that usually are functions of the model. Actuaries and risk managers often use such key risk indicators to determine the degree to which their companies are subject to particular aspects of risk, which arise from changes in underlying variables such as prices of equity, interest rates, or exchange rates. The present study proposes a new heavy-tailed exponential distribution that accommodates bathtub, upside-down bathtub, decreasing, decreasing-constant, and increasing hazard rates. Actuarial measures including value at risk, tail value at risk, tail variance, and tail variance premium are derived. A computational study for these actuarial measures is conducted, proving that the proposed distribution has a heavier tail as compared with the alpha power exponential, exponentiated exponential, and exponential distributions. We adopt six estimation approaches for estimating its parameters, and assess the performance of these estimators via Monte Carlo simulations. Finally, an actuarial real data set is analyzed, proving that the proposed model can be used effectively to model insurance data as compared with fifteen competing distributions.

Downside Variance Premium, Firm Fundamentals, and Expected Corporate Bond Returns

We find a positive relationship between individual downside variance premia, the difference between risk-neutral and physical expected downside variances, and future corporate bond returns. The hedge portfolio earns the economically substantial and statistically significant excess return of 0.35% (0.39%) per month in value (equal)-weighted returns. The predictive power of downside variance premium is stronger in non-investment-grade (long-maturity) corporate bonds than in investment-grade (short-maturity) ones. We show that downside variance premium positively relates to the likelihood of future default and cash flow uncertainty and negatively relates to future cash flow. When rational investors anticipate a high likelihood of future default, high cash flow uncertainty, or low future cash flow, the current bond price has to decline, resulting in higher future bond returns.

Optimal reinsurance to minimize the probability of drawdown under the mean-variance premium principle

In this paper, we determine the optimal reinsurance strategy to minimize the probability of drawdown, namely, the probability that the insurer's surplus process reaches some fixed fraction of its maximum value to date. We assume that the reinsurance premium is computed according to the mean-variance premium principle, a combination of the expected-value and variance premium principles. We derive closed-form expressions of the optimal reinsurance strategy and the corresponding minimum probability of drawdown. Then, under the variance premium principle, we show that the safe level can never be reached before drawdown under the optimally controlled surplus process. Finally, we present some numerical examples to show the impact of model parameters on the optimal results.

Asset Variance Risk Premium and Capital Structure

AbstractThis article investigates how the asset-return variance risk premium changes leverage. I find that the premium reduces leverage by increasing risk-neutral bankruptcy probability and costs in a model where asset returns have stochastic variance with the risk premium. Empirically, the model calibrations verify a significant reduction in optimal leverage, closer to observed leverage than the model without the premium. In model-free regressions, I document that leverage correlates negatively with the variance premium. The highest negative correlation is among investment-grade firms with low asset beta and historical variance but high variance premiums because their assets have high exposure to the market’s variance premium.

Optimal Reinsurance Strategy for an Insurer and a Reinsurer with Generalized Variance Premium Principle

This paper focuses on the optimal reinsurance problem with consideration of joint interests of an insurer and a reinsurer. In our model, the risk process is assumed to follow a Brownian motion with drift. The insurer can transfer the risk to the reinsurer via proportional reinsurance, and the reinsurance premium is calculated according to the variance and standard deviation premium principles. The objective is to maximize the expected exponential utility of the weighted sum of the insurer’s and the reinsurer’s terminal wealth, where the weight can be viewed as a regularization parameter to measure the importance of each party. By applying stochastic control theory, we establish the Hamilton–Jacobi–Bellman equation and obtain explicit expressions of optimal reinsurance strategies and optimal value functions. Furthermore, we provide some numerical simulations to illustrate the effects of model parameters on the optimal reinsurance strategies.

Modelling insurance losses using a new beta power transformed family of distributions

Actuaries are often in search of new distributions suitable for modeling financial and insurance losses. In this work, we propose a new family of distributions, called a new beta power transformed family of distributions. A special sub-model of the proposed class, called a new beta power transformed Weibull, suitable for modeling heavy tailed data in the scenario of actuarial statistics and finance, is considered in detail. The proposed distribution possesses desirable properties relevant to actuarial sciences. Expressions for the actuarial quantities such as value at risk, tail value at risk, tailed variance and tailed variance premium are derived. A simulation study is conducted to evaluate the behavior of the proposed distribution in actuarial sciences. Some distributional properties with estimation of parameters using maximum likelihood method are also discussed. Finally, a practical application of the proposed model to insurance data is presented.

Optimal reinsurance under the mean–variance premium principle to minimize the probability of ruin

We consider the problem of minimizing the probability of ruin by purchasing reinsurance whose premium is computed according to the mean–variance premium principle, a combination of the expected-value and variance premium principles. We derive closed-form expressions of the optimal reinsurance strategy and the corresponding minimum probability of ruin under the diffusion approximation of the classical Cramér–Lundberg risk process perturbed by a diffusion. We find an explicit expression for the reinsurance strategy that maximizes the adjustment coefficient for the classical risk process perturbed by a diffusion. Also, for this risk process, we use stochastic Perron’s method to prove that the minimum probability of ruin is the unique viscosity solution of its Hamilton–Jacobi–Bellman equation with appropriate boundary conditions. Finally, we prove that, under an appropriate scaling of the classical risk process, the minimum probability of ruin converges to the minimum probability of ruin under the diffusion approximation.

Return predictability of variance differences: A fractionally cointegrated approach

AbstractThis paper examines the fractional cointegration between downside (upside) components of realized and implied variances. A positive association is found between the strength of their cofractional relation and the return predictability of their differences. That association is established via the common long‐memory component of the variances that are fractionally cointegrated, which represents the volatility‐of‐volatility factor that determines the variance premium. Our results indicate that market fears play a critical role not only in driving the long‐run equilibrium relationship between implied‐realized variances but also in understanding the return predictability. A simulation study further verifies these claims.

Liquidity, implied volatility and tail risk: A comparison of liquidity measures

Liquidity is easily perceived but not easily measured in financial markets. Researchers and practitioners develop and test new measures of liquidity which may be good candidates for measuring this elusive concept. In this study, we present a comparison of variables within two empirical exercises using up to eight traditional liquidity proxies and two proposed proxies based on semi-deviations in a sample of NYSE-listed stocks. The first empirical exercise analyzes the relationship between liquidity and implied volatility, showing that increases in implied volatility impacts increases illiquidity. Using a decomposition of the squared VIX, we show that both conditional variance and variance premium components affects liquidity. Our second empirical exercise investigates the relationship between common factors in liquidity and tail risk. Common factors increase individual stocks' Value-at-Risk and Expected Shortfall, although the effect is not significant for market risk. In both applications, most of the studied proxies present results aligned with the body of literature.

Robust optimal investment–reinsurance strategies for an insurer with multiple dependent risks

This paper considers a robust optimal investment and reinsurance problem with multiple dependent risks for an Ambiguity-Averse Insurer (AAI), who is uncertain about the model parameters. We assume that the surplus of the insurance company can be allocated to the financial market consisting of one risk-free asset and one risky asset whose price process satisfies square root factor process. Under the objective of maximizing the expected utility of the terminal surplus, by adopting the technique of stochastic control, closed-form expressions of the robust optimal strategy and the corresponding value function are derived. The verification theorem is also provided. Finally, by presenting some numerical examples, the impact of some parameters on the optimal strategy is illustrated and some economic explanations are also given. We find that the robust optimal reinsurance strategies under the generalized mean–variance premium are very different from that under the variance premium principle. In addition, ignoring model uncertainty risk will lead to significant utility loss for the AAI.

Investors’ risk aversion integration and quantitative easing

PurposeThe purpose of this paper is to examine the spillover effects in international financial markets related to investors’ risk aversion as proxied by the variance premium, and how these relationships were affected by the quantitative easing (QE) announcements by the Federal Reserve.Design/methodology/approachThe empirical analysis employs a multivariate exponential generalized autoregressive conditionally heteroskedastic (VAR-EGARCH) specification, which includes the USA, the UK, Germany, France and Switzerland.FindingsTwo main findings are raised from the empirical analysis. First, the VAR-EGARCH model identifies statistically significant spillover effects identifying the USA as the leading source driving investors’ risk aversion. Second, unconventional monetary easing announcement by the Fed has had significant effects on investors’ risk perspectives.Practical implicationsAccounting for the dynamic volatility of variance premium inter-dependencies, the authors show that the correlations among variance premia increase during the QE announcements by the Federal Reserve, suggesting a herding behavior that may potentially lead to stock price bubbles and undermine financial stability.Originality/valueThis is an empirical attempt that investigates the unexplored effects of unconventional monetary policy decisions in relation with investors’ attitudes toward risk.