Editorial

3 pages, 256 KiB

Open AccessEditorial

Special Issue “Interplay between Financial and Actuarial Mathematics”

by Corina Constantinescu and Julia Eisenberg

Risks 2021, 9(8), 139; https://0-doi-org.brum.beds.ac.uk/10.3390/risks9080139 - 26 Jul 2021

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The Special Issue aims to highlight the interaction between actuarial and financial mathematics, which, due to the recent low interest rates and implications of COVID-19, requires an interlace between actuarial and financial methods, along with control theory, machine learning, mortality models, option pricing, [...] Read more.

The Special Issue aims to highlight the interaction between actuarial and financial mathematics, which, due to the recent low interest rates and implications of COVID-19, requires an interlace between actuarial and financial methods, along with control theory, machine learning, mortality models, option pricing, hedging, unit-linked contracts and drawdown analysis, among others [...] Full article

(This article belongs to the Special Issue Interplay between Financial and Actuarial Mathematics)

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25 pages, 730 KiB

Open AccessArticle

Optimal Surplus-Dependent Reinsurance under Regime-Switching in a Brownian Risk Model

by Julia Eisenberg, Lukas Fabrykowski and Maren Diane Schmeck

Risks 2021, 9(4), 73; https://0-doi-org.brum.beds.ac.uk/10.3390/risks9040073 - 13 Apr 2021

Cited by 2 | Viewed by 2235

Abstract

In this paper, we consider a company that wishes to determine the optimal reinsurance strategy minimising the total expected discounted amount of capital injections needed to prevent the ruin. The company’s surplus process is assumed to follow a Brownian motion with drift, and [...] Read more.

In this paper, we consider a company that wishes to determine the optimal reinsurance strategy minimising the total expected discounted amount of capital injections needed to prevent the ruin. The company’s surplus process is assumed to follow a Brownian motion with drift, and the reinsurance price is modelled by a continuous-time Markov chain with two states. The presence of regime-switching substantially complicates the optimal reinsurance problem, as the surplus-independent strategies turn out to be suboptimal. We develop a recursive approach that allows to represent a solution to the corresponding Hamilton–Jacobi–Bellman (HJB) equation and the corresponding reinsurance strategy as the unique limits of the sequence of solutions to ordinary differential equations and their first- and second-order derivatives. Via Ito’s formula, we prove the constructed function to be the value function. Two examples illustrate the recursive procedure along with a numerical approach yielding the direct solution to the HJB equation. Full article

(This article belongs to the Special Issue Interplay between Financial and Actuarial Mathematics)

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20 pages, 642 KiB

Open AccessEditor’s ChoiceArticle

A Machine Learning Approach for Micro-Credit Scoring

by Apostolos Ampountolas, Titus Nyarko Nde, Paresh Date and Corina Constantinescu

Risks 2021, 9(3), 50; https://0-doi-org.brum.beds.ac.uk/10.3390/risks9030050 - 09 Mar 2021

Cited by 23 | Viewed by 14403

Abstract

In micro-lending markets, lack of recorded credit history is a significant impediment to assessing individual borrowers’ creditworthiness and therefore deciding fair interest rates. This research compares various machine learning algorithms on real micro-lending data to test their efficacy at classifying borrowers into various [...] Read more.

In micro-lending markets, lack of recorded credit history is a significant impediment to assessing individual borrowers’ creditworthiness and therefore deciding fair interest rates. This research compares various machine learning algorithms on real micro-lending data to test their efficacy at classifying borrowers into various credit categories. We demonstrate that off-the-shelf multi-class classifiers such as random forest algorithms can perform this task very well, using readily available data about customers (such as age, occupation, and location). This presents inexpensive and reliable means to micro-lending institutions around the developing world with which to assess creditworthiness in the absence of credit history or central credit databases. Full article

(This article belongs to the Special Issue Interplay between Financial and Actuarial Mathematics)

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19 pages, 997 KiB

Open AccessArticle

A Two-Population Mortality Model to Assess Longevity Basis Risk

by Selin Özen and Şule Şahin

Risks 2021, 9(2), 44; https://0-doi-org.brum.beds.ac.uk/10.3390/risks9020044 - 20 Feb 2021

Cited by 2 | Viewed by 2809

Abstract

Index-based hedging solutions are used to transfer the longevity risk to the capital markets. However, mismatches between the liability of the hedger and the hedging instrument cause longevity basis risk. Therefore, an appropriate two-population model to measure and assess longevity basis risk is [...] Read more.

Index-based hedging solutions are used to transfer the longevity risk to the capital markets. However, mismatches between the liability of the hedger and the hedging instrument cause longevity basis risk. Therefore, an appropriate two-population model to measure and assess longevity basis risk is required. In this paper, we aim to construct a two-population mortality model to provide an effective hedge against the basis risk. The reference population is modelled by using the Lee–Carter model with the renewal process and exponential jumps, and the dynamics of the book population are specified. The analysis based on the U.K. mortality data indicate that the proposed model for the reference population and the common age effect model for the book population provide a better fit compared to the other models considered in the paper. Different two-population models are used to investigate the impact of sampling risk on the index-based hedge, as well as to analyse the risk reduction regarding hedge effectiveness. The results show that the proposed model provides a significant risk reduction when mortality jumps and sampling risk are taken into account. Full article

(This article belongs to the Special Issue Interplay between Financial and Actuarial Mathematics)

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18 pages, 908 KiB

Open AccessArticle

Minimal Expected Time in Drawdown through Investment for an Insurance Diffusion Model

by Leonie Violetta Brinker

Risks 2021, 9(1), 17; https://0-doi-org.brum.beds.ac.uk/10.3390/risks9010017 - 06 Jan 2021

Cited by 2 | Viewed by 2222

Abstract

Consider an insurance company whose surplus is modelled by an arithmetic Brownian motion of not necessarily positive drift. Additionally, the insurer has the possibility to invest in a stock modelled by a geometric Brownian motion independent of the surplus. Our key variable is [...] Read more.

Consider an insurance company whose surplus is modelled by an arithmetic Brownian motion of not necessarily positive drift. Additionally, the insurer has the possibility to invest in a stock modelled by a geometric Brownian motion independent of the surplus. Our key variable is the (absolute) drawdown

Δ

of the surplus X, defined as the distance to its running maximum

\bar{X}

. Large, long-lasting drawdowns are unfavourable for the insurance company. We consider the stochastic optimisation problem of minimising the expected time that the drawdown is larger than a positive critical value (weighted by a discounting factor) under investment. A fixed-point argument is used to show that the value function is the unique solution to the Hamilton–Jacobi–Bellman equation related to the problem. It turns out that the optimal investment strategy is given by a piecewise monotone and continuously differentiable function of the current drawdown. Several numerical examples illustrate our findings. Full article

(This article belongs to the Special Issue Interplay between Financial and Actuarial Mathematics)

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28 pages, 658 KiB

Open AccessArticle

Retrospective Reserves and Bonus with Policyholder Behavior

by Debbie Kusch Falden and Anna Kamille Nyegaard

Risks 2021, 9(1), 15; https://0-doi-org.brum.beds.ac.uk/10.3390/risks9010015 - 05 Jan 2021

Cited by 8 | Viewed by 2488

Abstract

Legislation imposes insurance companies to project their assets and liabilities in various financial scenarios. Within the setup of with-profit life insurance, we consider retrospective reserves and bonus, and we study projection of balances with and without policyholder behavior. The projection resides in a [...] Read more.

Legislation imposes insurance companies to project their assets and liabilities in various financial scenarios. Within the setup of with-profit life insurance, we consider retrospective reserves and bonus, and we study projection of balances with and without policyholder behavior. The projection resides in a system of differential equations of the savings account and the surplus, and the policyholder behavior options surrender and conversion to free-policy are included. The inclusion results in a structure where the system of differential equations of the savings account and the surplus is non-trivial. We consider a case where we are able to find accurate differential equations and suggest an approximation method to project the savings account and the surplus, including policyholder behavior, in general. To highlight the practical applications of the results in this paper, we study a numerical example. Full article

(This article belongs to the Special Issue Interplay between Financial and Actuarial Mathematics)

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28 pages, 563 KiB

Open AccessArticle

Comparing Two Different Option Pricing Methods

by Alessandro Bondi, Dragana Radojičić and Thorsten Rheinländer

Risks 2020, 8(4), 108; https://0-doi-org.brum.beds.ac.uk/10.3390/risks8040108 - 19 Oct 2020

Cited by 1 | Viewed by 2484

Abstract

Motivated by new financial markets where there is no canonical choice of a risk-neutral measure, we compared two different methods for pricing options: calibration with an entropic penalty term and valuation by the Esscher measure. The main aim of this paper is to [...] Read more.

Motivated by new financial markets where there is no canonical choice of a risk-neutral measure, we compared two different methods for pricing options: calibration with an entropic penalty term and valuation by the Esscher measure. The main aim of this paper is to contrast the outcomes of those two methods with real-traded call option prices in a liquid market like NASDAQ stock exchange, using data referring to the period 2019–2020. Although the Esscher measure method slightly underperforms the calibration method in terms of absolute values of the percentage difference between real and model prices, it could be the only feasible choice if there are not many liquidly traded derivatives in the market. Full article

(This article belongs to the Special Issue Interplay between Financial and Actuarial Mathematics)

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23 pages, 865 KiB

Open AccessFeature PaperArticle

Variance and Interest Rate Risk in Unit-Linked Insurance Policies

by David Baños, Marc Lagunas-Merino and Salvador Ortiz-Latorre

Risks 2020, 8(3), 84; https://0-doi-org.brum.beds.ac.uk/10.3390/risks8030084 - 06 Aug 2020

Cited by 1 | Viewed by 3822

Abstract

One of the risks derived from selling long-term policies that any insurance company has arises from interest rates. In this paper, we consider a general class of stochastic volatility models written in forward variance form. We also deal with stochastic interest rates to [...] Read more.

One of the risks derived from selling long-term policies that any insurance company has arises from interest rates. In this paper, we consider a general class of stochastic volatility models written in forward variance form. We also deal with stochastic interest rates to obtain the risk-free price for unit-linked life insurance contracts, as well as providing a perfect hedging strategy by completing the market. We conclude with a simulation experiment, where we price unit-linked policies using Norwegian mortality rates. In addition, we compare prices for the classical Black-Scholes model against the Heston stochastic volatility model with a Vasicek interest rate model. Full article

(This article belongs to the Special Issue Interplay between Financial and Actuarial Mathematics)

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31 pages, 1260 KiB

Open AccessFeature PaperArticle

Multivariate Collective Risk Model: Dependent Claim Numbers and Panjer’s Recursion

by Cordelia Rudolph and Uwe Schmock

Risks 2020, 8(2), 43; https://0-doi-org.brum.beds.ac.uk/10.3390/risks8020043 - 02 May 2020

Cited by 1 | Viewed by 3490

Abstract

In this paper, we discuss a generalization of the collective risk model and of Panjer’s recursion. The model we consider consists of several business lines with dependent claim numbers. The distributions of the claim numbers are assumed to be Poisson mixture distributions. We [...] Read more.

In this paper, we discuss a generalization of the collective risk model and of Panjer’s recursion. The model we consider consists of several business lines with dependent claim numbers. The distributions of the claim numbers are assumed to be Poisson mixture distributions. We let the claim causes have certain dependence structures and prove that Panjer’s recursion is also applicable by finding an appropriate equivalent representation of the claim numbers. These dependence structures are of a stochastic non-negative linear nature and may also produce negative correlations between the claim causes. The consideration of risk groups also includes dependence between claim sizes. Compounding the claim causes by common distributions also keeps Panjer’s recursion applicable. Full article

(This article belongs to the Special Issue Interplay between Financial and Actuarial Mathematics)

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28 pages, 621 KiB

Open AccessArticle

Stochastic Mortality Modelling for Dependent Coupled Lives

by Kira Henshaw, Corina Constantinescu and Olivier Menoukeu Pamen

Risks 2020, 8(1), 17; https://0-doi-org.brum.beds.ac.uk/10.3390/risks8010017 - 11 Feb 2020

Cited by 5 | Viewed by 5918

Abstract

Broken-heart syndrome is the most common form of short-term dependence, inducing a temporary increase in an individual’s force of mortality upon the occurrence of extreme events, such as the loss of a spouse. Socioeconomic influences on bereavement processes allow for suggestion of variability [...] Read more.

Broken-heart syndrome is the most common form of short-term dependence, inducing a temporary increase in an individual’s force of mortality upon the occurrence of extreme events, such as the loss of a spouse. Socioeconomic influences on bereavement processes allow for suggestion of variability in the significance of short-term dependence between couples in countries of differing levels of economic development. Motivated by analysis of a Ghanaian data set, we propose a stochastic mortality model of the joint mortality of paired lives and the causal relation between their death times, in a less economically developed country than those considered in existing studies. The paired mortality intensities are assumed to be non-mean-reverting Cox–Ingersoll–Ross processes, reflecting the reduced concentration of the initial loss impact apparent in the data set. The effect of the death on the mortality intensity of the surviving spouse is given by a mean-reverting Ornstein–Uhlenbeck process which captures the subsiding nature of the mortality increase characteristic of broken-heart syndrome. Inclusion of a population wide volatility parameter in the Ornstein–Uhlenbeck bereavement process gives rise to a significant non-diversifiable risk, heightening the importance of the dependence assumption in this case. Applying the model proposed to an insurance pricing problem, we obtain the appropriate premium under consideration of dependence between coupled lives through application of the indifference pricing principle. Full article

(This article belongs to the Special Issue Interplay between Financial and Actuarial Mathematics)

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