The Intermediate Model for the Solvency of a Financial Institution
Zaks Y., Dhaene J., 2015
Available on SSRN.
Granted by SOA
Keywords: Solvency II, Basel II, IRB, Risk Management
The regulator that is required to approve the statistical model and its parameters relies on the data that the insurance company provides. In case the regulator approves the model, he has some responsibility for it. The inability to validate the model, due to lack of resources or lack of understanding, causes the regulator to reject the proposed internal model and to ask the company to work with the standard model. This issue raises mainly in a small financial institutions and in a small line of business at big financial institutions. To overcome this conflict, we suggest the financial institution to follow an intermediate model, which is a weighted average of the internal model and the standard one. - Read more
An alternative proof to Markowitz’s model
Zaks Y., 2013
Journal of Finance and Economics, Vol.1(3), pp.33-35.
Keywords: portfolio selection, Markowitz model, quadratic programing, euclidean projection
In the fundamental paper on portfolio selection, Markowitz (1952) described via geometric reasoning his innovative theory and provided the explicit optimal selection for the cases of 3 and 4 assets. Merton (1972) obtained for the general case the efficient portfolio frontiers explicitly by using Lagrange multipliers. In this paper, we suggest a geometric approach to achieve the explicit optimal selection for the general case thus generalizing Markowitz’s original approach to achieve the explicit presentation of the desired selection.
Solving an Asset-Liability Problem as an Investment Portfolio Problem
Zaks Y. and Landsman Z.,Available on SSRN.
Keywords: portfolio selection, asset-liability management, capital allocation, Markowitz model, quadratic optimization
One of the fundamental questions in finance is how to select an investment portfolio? The most popular model is the Mean-Variance (MV) model that was presented by Markowitz in 1952. In the MV model, the optimization problem is a constrained quadratic functional. An optimal portfolio selection for asset-liability management problem (ALM) can be obtained by transferring the the ALM problem into the classical MV optimization problem, see Panjer et al. (2001). In this paper we show a technique to transfer the ALM problem into a standard investment portfolio problem in some other models. - Read more
Optimal Capital Allocation in a Hierarchical Corporate Structure
Zaks Y. and Tsanakas A., 2014
Insurance: Mathematics and Economics,Vol.56, pp. 48-55.
Published version on IME, Draft on SSRN
Granted by the Institute and Faculty of Actuaries
Keywords: Capital allocation, Solvency II, Basel II, Weighted capital allocation
Hierarchical firms
We consider capital allocation in a hierarchical corporate structure where stakeholders at two organizational levels (e.g., board members vs line managers) may have conflicting objectives, preferences, and beliefs about risk. Capital allocation is considered as the solution to an optimization problem whereby a quadratic deviation measure between individual losses (at both levels) and allocated capital amounts is minimized. Thus, this paper generalizes the framework of Dhaene et al. (2012), by allowing potentially diverging risk preferences in a hierarchical structure. An explicit unique solution to this optimization problem is given. In several examples, it is shown how the optimal capital allocation achieves a compromise between conflicting views of risk within the organization. - Read more
Optimal capital allocation - A generalization of the optimization problem
Zaks Y., 2013
Insurance Markets and Companies: Analyses and Actuarial Computations, Vol.4(2), pp. 29-34.
Granted by Katholieke Universiteit Leuven, Fortis Chair in Financial and Actuarial Risk Management, Belgium
: optimization in insurance, premium calculation, convex optimization
Several papers consider the capital allocation and the premium pricing problems as optimization problems. In these papers, the author considers some constraints on the convex function and finds an explicit solution in each case. In this paper we prove the existence of a unique solution where each class has a different convex function. Moreover, it gives an explicit solution in case that the convex functions have first order derivatives. - Read more
Zaks Y., 2013
European Actuarial Journal, Vol.3(1), pp 69-95. Draft on SSRN.
Granted by the Institute and Faculty of Actuaries, UK
Keywords: Portfolio selection, Asset-Liability Management, Capital allocation, Optimization, Short sale, Solvency
In this paper we present an optimization framework to deal with the asset-liability portfolio selection problem. We consider a financial institution that has multiple lines of business. The capital allocation is obtained by minimizing the sum of the expected squared differences between the liability in each line of business and the value of the corresponding investment portfolio. We show that in certain circumstances the bottom-up approach is consistent with the top–down approach, where the optimal capital is determined for the whole portfolio rather than its individual components. Such a case happens for example if the same weight function is used for all lines of business in the two approaches. Finally, we obtain investment portfolios under some limitations on short sales. - Read more
The Optimal claiming strategies in a Bonus-Malus Systems and the monotony property
Zaks Y., 2008
The Scandinavian Actuarial Journal, 2008(1), pp. 34-40
Keywords: Optimal strategy, Policyholder's behavior
In the classical Bonus-Malus System (BMS) there are several premium levels, e.g., M. A premium level consists of a premium and a deductible. In this paper, we consider the expected cost for a horizon of n periods of a policyholder in level i. When damage of size x occurs, the policyholder should decide whether or not to claim. We analyze the policyholder decision as an optimization problem related to the expected premiums and damages in the following years. - Read more
Pricing a heterogeneous portfolio based on a demand function
Zaks Y., Frostig E. and Levikson B., 2008
The North American Actuarial Journal, 12(1), pp. 65-73
Consider a portfolio containing number of risk classes. Each class has its own demand function, which determines the number of insureds in this class as a function of the premium. The insurer determines the premiums based on the number of insureds in each class. The “market” reacts by updating the number of the policyholders, then the insurer updates the premium, and so on. We show that this process has an equilibrium point, and then we characterize this point. - Read more
Optimal pricing for a heterogeneous portfolio for a given risk factor and convex distance measure
Frostig E., Zaks Y. and Levikson B., 2007
Insurance: Mathematics and Economics, 40(3), pp. 459-467
Keywords: Heterogeneous portfolio, Majorization, Schur convex functions
Consider a portfolio containing heterogeneous risks. The premiums of the policyholders might not cover the amount of the payments which an insurance company pays the policyholders. When setting the premium, this risk has to be taken into consideration. On the other hand the premium that the insured pays has to be fair. This fairness is measured by a function of the difference between the risk and the premium paid—we call this function a distance function. For a given small probability of insolvency, we find the premium for each class, such that the distance function is minimized. Next we formulate and solve the dual problem, which is minimizing the insolvency probability under the constraint that the distance function does not exceed a given level. This paper generalizes a previous paper [Zaks, Y., Frostig, E., Levikson, B., 2006. Optimal pricing of a heterogeneous portfolio for a given risk level. Astin Bull. 36 (1), 161–185] where only a square distance function was considered. - Read more
Optimal pricing of a heterogeneous portfolio for a given risk level
Zaks Y., Frostig E. and Levikson B., 2006
Astin Bulletin, 36(1), pp. 161-185
Keywords: Heterogeneous portfolio, Positive Definite Matrix, non-linear programming, dual problem
Consider a portfolio containing heterogeneous risks, where the policyholders’ premiums to the insurance company might not cover the claim payments. This risk has to be taken into consideration in the premium pricing. On the other hand, the premium that the insureds pay has to be fair. This fairness is measured by the distance between the risk and the premium paid. We apply a non-linear programming formulation to find the optimal premium for each class so that the risk is below a given level and the weighted distance between the risk and the premium is minimized. We consider also the dual problem: minimizing the risk level for a given weighted distance between risks and premium. - Read more