Distortion Risk Measures and Economic Capital

Abstract

To provide incentive for active risk management, it is argued that a sound coherent distortion risk measure should preserve some higher degree stop-loss orders, at least the degree-three convex order. Such risk measures are called tail-preserving risk measures. It is shown that, under some common axioms and other plausible conditions, a tail-preserving coherent distortion risk measure identifies necessarily with the Wang right-tail measure or the expected value measure. This main result is applied to derive an optimal economic capital formula.     

On Hattendorff's theorem in the theory of risk

Abstract

1. Two of the most important measures of dispersion are the {istandard }deviation and the {iaverage deviation}¹ which, if we are concerned with the financial effects of deviations from an assumed mortality, are called the {imean risk} and the {iaverage risk} and are denoted by {iM} and {iR} respectively.     

Measuring Basis Risk in Longevity Hedges

Abstract

In examining basis risk in index longevity hedges, it is important not to ignore the dependence between the population underlying the hedging instrument and the population being hedged. We consider four extensions to the Lee-Carter model that incorporate such dependence: Both populations are jointly driven by the same single time-varying index, the two populations are cointegrated, the populations depend on a common age factor, and there is an augmented common factor model in which a population-specific time-varying index is added to the common factor model with the property that it will tend toward a certain constant level over time. Using data from the female populations of Canada and the United States, we show the augmented common factor model is preferred in terms of both goodness-of-fit and ex post forecasting performance. This model is then used to quantify the basis risk in a longevity hedge of 65-year old Canadian females structured using a portfolio of q-forward contracts predicated on U.S. female population mortality. The hedge effectiveness is estimated at 56% on the basis of longevity value-at-risk and 81.61% on the basis of longevity risk reduction.     

A Regime-Switching Model of Long-Term Stock Returns

Abstract

In this paper I first define the regime-switching lognormal model. Monthly data from the Standard and Poor’s 500 and the Toronto Stock Exchange 300 indices are used to fit the model parameters, using maximum likelihood estimation. The fit of the regime-switching model to the data is compared with other common econometric models, including the generalized autoregressive conditionally heteroskedastic model. The distribution function of the regime-switching model is derived. Prices of European options using the regime-switching model are derived and implied volatilities explored. Finally, an example of the application of the model to maturity guarantees under equity-linked insurance is presented. Equations for quantile and conditional tail expectation (Tail-VaR) risk measures are derived, and a numerical example compares the regime-switching lognormal model results with those using the more traditional lognormal stock return model.     

Efficient Post-Retirement Asset Allocation

Abstract

To examine post-retirement asset allocation, an extension to the classic Markowitz risk-return framework is suggested. Assuming that retirees make constant (real dollar) annual withdrawals from their portfolios, reward and risk measures are defined to be the mean and standard deviation of wealth remaining at end of life. Asset returns and time of death are both treated as random variables. Assuming constant lifetime asset allocation, the risk and reward measures can be evaluated analytically, and an efficient frontier can be determined. Life annuities can be used to extend the left-hand (low-risk) side of the efficient frontier. The desired level of wealth at end of life can be used to choose a desirable portfolio on the efficient frontier. The desirable portfolio strongly depends on the withdrawal rate. It is suggested (although not proven) that asset allocations strategies that vary with age do not add efficiency in this model, and asset allocation strategies that vary with wealth can add efficiency.     

Relative Importance of Risk Sources in Insurance Systems

Abstract

Actuaries, and other managers of uncertainty, identify factors in modeling insurance risks because they believe (1) that these factors affect the outcome of a risk or (2) that the factors can be managed, thus allowing analysts a degree of control over the insurance system. This paper shows how to use a statistical measure, the coefficient of determination, for quantifying the relative importance of a source of uncertainty. With a quantitative measure of relative importance, risk managers can sharpen their intuition about the relative importance of risk factors and become better custodians of financial security systems.

This paper shows that the coefficient of determination is intuitively appealing in assessing the effectiveness of basic risk management techniques including risk exchange, pooling, and financial risk management. A single source common to all risks reduces the effectiveness of a pool; the risk measure quantifies the relative importance of this common source. The coefficient of determination is shown to have roots in the economics as well as the statistics literature. This connection provides further motivation for using the coefficient of determination and also suggests alternative measures for quantifying relative importance. The risk measure is useful in multivariate situations in which several factors affect a risk simultaneously. The paper illustrates this usefulness by considering a pool of policies that is subject to mortality, a common disaster, and a common investment environment.     

CAPM and APT-like models with risk measures

The paper deals with optimal portfolio choice problems when risk levels are given by coherent risk measures, expectation bounded risk measures or general deviations. Both static and dynamic pricing models may be involved. Unbounded problems are characterized by new notions such as (strong) compatibility between prices and risks. Surprisingly, the lack of bounded optimal risk and/or return levels arises for important pricing models (Black and Scholes) and risk measures (VaR, CVaR, absolute deviation, etc.). Bounded problems present a Market Price of Risk and generate a pair of benchmarks. From these benchmarks we introduce APT and CAPM-like analyses, in the sense that the level of correlation between every available security and some economic factors explains the security expected return. The risk level non correlated with these factors has no influence on any return, despite the fact that we are dealing with risk functions beyond the standard deviation.     

Volatility as an Asset Class: European Evidence

Abstract

Volatility movements are known to be negatively correlated with stock index returns. Hence, investing in volatility appears to be attractive for investors seeking risk diversification. The most common instruments for investing in pure volatility are variance swaps, which now enjoy an active over-the-counter (OTC) market. This paper investigates the risk-return tradeoff of variance swaps on the Deutscher Aktienindex and Euro STOXX 50 index over the time period from 1995 to 2004. We synthetically derive variance swap rates from the smile in option prices. Using quotes from two large investment banks over two months, we validate that the synthetic values are close to OTC market prices. We find that variance swap returns exhibit an option-like profile compared to returns of the underlying index. Given this pattern, it is crucial to account for the non-normality of returns in measuring the performance of variance swap investments. As in the US, the average returns of selling variance swaps are found to be strongly positive and too large to be compatible with standard equilibrium models. The magnitude of the estimated risk premium is related to variance uncertainty and past index returns. This indicates that the variance swap rate does not seem to incorporate all past information relevant for forecasting future realized variance.     

Time Period and Risk Measures in the General Risk Equation

In an earlier paper, a general risk equation, applicable to all non growth systems, and inclusive of financial systems, was derived. It related expected throughput capacity of any system to both system resources and positive risk of loss of throughput capacity. Two risk measures were required, a new MEL‐risk measure, and the conventional standard‐deviation risk measure.

In this paper we show that the two apparently distinct risk measures are intimately related, and that which one is appropriate depends merely on the time period over which the risk is calculated. We show, ultimately by application of the Central Limit Theorem, that if we merely sufficiently alter the time period, at some point the need for one measure will transition into the need for the other, without any change in the underlying physical system.

This leads to a comprehensive risk measure that defaults to either the MEL‐risk measure, or standard‐deviation measure, depending not on the physical system, but merely on the time period over which the risk is calculated.     

Noise sensitivity of portfolio selection under various risk measures

We study the sensitivity to estimation error of portfolios optimized under various risk measures, including variance, absolute deviation, expected shortfall and maximal loss. We introduce a measure of portfolio sensitivity and test the various risk measures by considering simulated portfolios of varying sizes N and for different lengths T of the time series. We find that the effect of noise is very strong in all the investigated cases, asymptotically it only depends on the ratio N/T, and diverges (goes to infinity) at a critical value of N/T, that depends on the risk measure in question. This divergence is the manifestation of a phase transition, analogous to the algorithmic phase transitions recently discovered in a number of hard computational problems. The transition is accompanied by a number of critical phenomena, including the divergent sample to sample fluctuations of portfolio weights. While the optimization under variance and mean absolute deviation is always feasible below the critical value of N/T, expected shortfall and maximal loss display a probabilistic feasibility problem, in that they can become unbounded from below already for small values of the ratio N/T, and then no solution exists to the optimization problem under these risk measures. Although powerful filtering techniques exist for the mitigation of the above instability in the case of variance, our findings point to the necessity of developing similar filtering procedures adapted to the other risk measures where they are much less developed or non-existent. Another important message of this study is that the requirement of robustness (noise-tolerance) should be given special attention when considering the theoretical and practical criteria to be imposed on a risk measure.     

Portfolio Optimization under Lower Partial Risk Measures

Portfolio management using lower partial risk (downside risk) measures is attracting more attention of practitioners in recent years. The purpose of this paper is to review important characteristics of these riskmeasures and conduct simulation using four alternative measures, lower semi-variance, lower semi-absolute deviation, first order below targetrisk and conditional value-at-risk.We will show that these risk measures are useful to control downside risk whenthe distribution of assets is non-symmetric. Further, we will propose a computational scheme to resolve the difficultyassociated with solving a large dense linear programming problems resulting from these models. We will demonstrate that this method can in fact solve problems consisting of10⁴ assets and 10⁵ scenarios within a practical amount of CPU time.     

Utilitarianism and risk

Abstract

In day-to-day life, we are continuously exposed to different kinds of risk. Unfortunately, avoiding risk can often come at societal or individual costs. Hence, an important task within risk management is deciding how much it can be justified to expose members of society to risk x in order to avoid societal and individual costs y – and vice versa. We can refer to this as the task of setting an acceptable risk threshold. Judging whether a risk threshold is justified requires normative reasoning about what levels of risk exposure that are permissible. One such prominent normative theory is utilitarianism. According to utilitarians, the preferred risk threshold is the one that yields more utility for the most people compared to alternative risk thresholds. In this paper, I investigate whether and the extent to which utilitarian theory can be used to normatively ground a particular risk threshold in this way. In particular, I argue that there are (at least) seven different utilitarian approaches to setting an acceptable risk threshold. I discuss each of these approaches in turn and argue that neither can satisfactorily ground an acceptable risk threshold.     

Hedging Equity-Linked Life Insurance Contracts

Abstract

This paper examines a portfolio of equity-linked life insurance contracts and determines risk-minimizing hedging strategies within a discrete-time setup. As a principal example, I consider the Cox-Ross-Rubinstein model and an equity-linked pure endowment contract under which the policyholder receives max(S_T , K) at time T if he or she is then alive, where S_T is the value of a stock index at the term T of the contract and K is a guarantee stipulated by the contract. In contrast to most of the existing literature, I view the contracts as contingent claims in an incomplete model and discuss the problem of choosing an optimality criterion for hedging strategies. The subsequent analysis leads to a comparison of the risk (measured by the variance of the insurer’s loss) inherent in equity-linked contracts in the two situations where the insurer applies the risk-minimizing strategy and the insurer does not hedge. The paper includes numerical results that can be used to quantify the effect of hedging and describe how this effect varies with the size of the insurance portfolio and assumptions concerning the mortality.     

Comparative and qualitative robustness for law-invariant risk measures

When estimating the risk of a P&L from historical data or Monte Carlo simulation, the robustness of the estimate is important. We argue here that Hampel’s classical notion of qualitative robustness is not suitable for risk measurement, and we propose and analyze a refined notion of robustness that applies to tail-dependent law-invariant convex risk measures on Orlicz spaces. This concept captures the tradeoff between robustness and sensitivity and can be quantified by an index of qualitative robustness. By means of this index, we can compare various risk measures, such as distortion risk measures, in regard to their degree of robustness. Our analysis also yields results of independent interest such as continuity properties and consistency of estimators for risk measures, or a Skorohod representation theorem for ψ-weak convergence.     

Combining the Effects of OLS and Spread on Futures Hedging: Evidence from the Taiwan Stock Index

Abstract:

This study proposes a dynamic hedge ratio, the combined ordinary least squares spread (COLSS), which combines the hedge ratio of ordinary least squares and the value of spread. Using this dynamic ratio for hedging with futures contracts, one can replace spot risk with spread risk. The COLSS captures not only the long-run equilibrium between spot and futures returns, but also the short-run deviation from equilibrium. The spread is forecast by one-period lagged stock market factors and high-order moments that are estimated by an options model. In the in-sample and out-of-sample tests, the COLSS strategy achieves significant risk reduction and outperforms the alternative models by a large utility improvement.     

Economic Capital Allocation Derived from Risk Measures

Abstract

We examine properties of risk measures that can be considered to be in line with some “best practice” rules in insurance, based on solvency margins. We give ample motivation that all economic aspects related to an insurance portfolio should be considered in the definition of a risk measure. As a consequence, conditions arise for comparison as well as for addition of risk measures. We demonstrate that imposing properties that are generally valid for risk measures, in all possible dependency structures, based on the difference of the risk and the solvency margin, though providing opportunities to derive nice mathematical results, violates best practice rules. We show that so-called coherent risk measures lead to problems. In particular we consider an exponential risk measure related to a discrete ruin model, depending on the initial surplus, the desired ruin probability, and the risk distribution.     

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.     

A firefly algorithm modified support vector machine for the credit risk assessment of supply chain finance

PurposeNowadays, Supply Chain Finance (SCF) has been developing rapidly since the emergence of credit risk. Therefore, this paper used SVM optimized by the firefly algorithm, which is called firefly algorithm support vector machine (FA-SVM), and applied it to SCF evaluation with a different indicator selection.Design/methodology/approachIn this paper, we used FA-SVM to assess the credit risk of supply chain finance with extracted index through correlation and appraisal analysis, and finally determined 3 first-level indicators and 15 third-level indicators. Through the application analysis, 39 SMEs (117 sample data) were selected from the Computer and Electronic Communications Manufacturing Industry as the characteristics for the input variables, to verify the improvement effect of the method relative to the LIBSVM and the classification pretest effect in the credit risk assessment of the SCF.FindingsThe results showed that FA-SVM could improve the accuracy of classification prediction compared with LIBSVM, and decrease the error rate of falseness recognize credible enterprise to untrusted enterprise.Originality/valueThis paper appliedthe firefly support vector machine in the supply chain financial evaluation for the first time. The output variable was described in a more detailed manner during the index define, and the random selection set in the process of FA-SVM data training.     

Risk Measures for Hedge Funds: a Cross‐sectional Approach

This paper analyses the risk‐return trade‐off in the hedge fund industry. We compare semi‐deviation, value‐at‐risk (VaR), Expected Shortfall (ES) and Tail Risk (TR) with standard deviation at the individual fund level as well as the portfolio level. Using the Fama and French (1992) methodology and the combined live and defunct hedge fund data from TASS, we find that the left‐tail risk captured by Expected Shortfall (ES) and Tail Risk (TR) explains the cross‐sectional variation in hedge fund returns very well, while the other risk measures provide statistically insignificant or marginally significant results. During the period between January 1995 and December 2004, hedge funds with high ES outperform those with low ES by an annual return difference of 7%. We provide empirical evidence on the theoretical argument by Artzner et al. (1999) that ES is superior to VaR as a downside risk measure. We also find the Cornish‐Fisher (1937) expansion is superior to the nonparametric method in estimating ES and TR.