Integrated Financial Risk Management: Capital Allocation Issues

A strategic optimization model provides the ideal setting for allocating the scarce capital of a financial intermediary such as an insurance company. The goal of management is to maximize shareholder value. Capital allocation serves three primary purposes: to compare managerial performance across business units, to provide a risk indicator for regulators and other stakeholders, and to develop a common basis for major decisions, including investment and underwriting strategies, and setting the corporate structure. Capital allocation puts diverse activities on an equal footing by adjusting profits and revenues for risks. We show that optimal allocation rules can be readily retrieved from the solution to a special dynamic financial analysis model of the firm. Motivation For any financial intermediary, a major concern involves growing the company’s wealth over time, consistent with taking on acceptable risks. In the same light, we are often told that senior management should render decisions so as to maximize shareholder value; it is the mantra of modern capitalism. Yet there are few practical mathematical modeling tools for assisting in the integrated optimization of a financial organization. We propose a dynamic financial analysis (DFA) for monitoring risks -by allocating capital, by setting underwriting limits, and by rewarding decision makers in concert with financial benchmarks. This system identifies the goals of a financial company as benchmark targets. A critical aspect of strategic planning is determining the underlying capital structure of a firm. Too little capital leads to excess risks for the shareholders and the public. Too much capital leads to inferior returns as compared with competing investments. There are several alternative approaches to capital allocation: Existing banking and insurance capital requirements, such as risk based capital required by regulators (e.g. 1992 NAIC capital adequacy requirements, and the 1988 Basle accord). Mostly, these rules are based on business activity such as premium to surplus ratios, and they emphasize credit risks. Measurements of risks based on probabilistic estimates and stochastic models: • Value at Risk – VaR, • Expected policyholder deficit – EPD and its generalization ETD, • Certainty equivalent returns – CER and its approximation, target planning. Each approach possesses limitations due to the conflicting purposes for capital allocation. Of course, a primary goal is to prevent an unacceptable loss of capital by the financial intermediary. In a large financial organization, there are multiple stakeholders who possess conflicting goals: shareholders, many of whom can diversify their financial stake, officers and employees, obligators, state and federal regulators, the ceding company, and others. Each group has an interest in seeing that the company survives economic downturns. However, the risk/reward tradeoff may be different for each group. Regulators, for example, may have less concern for higher short-term returns than do the stockholders. Many current capital allocation procedures are indirect indicators of risks since they fail to estimate the probability of losses to the surplus over a multi-period time horizon. Likewise, those methods based exclusively on historical returns may be inadequate when evaluating a new set of circumstances and future market dynamics. The remainder of this report is organized as follows. In the next section, we review key elements of the DFA model for the firm. Several alternative objectives are mentioned, depending upon the needs of the company and its stakeholders. In most cases, the model should display the values of the multiple objectives since there are so many ways to evaluate the health of a financial intermediary. In the third section, we propose a decentralized approach to the DFA system. In section 4, we discuss the capital allocation process. Section 5 presents a small example based on an insurance problem. Last, we make some concluding remarks. 1 Other domains have embraced optimization methods, for instance, see Integrated Logistics (Asad, et al. 1992). 1. Integrated Financial Risk Management This section presents the basic framework for integrating the major decisions of a financial intermediary. We propose a multi-period stochastic optimization model. The primary goal is to maximize shareholder value at the end of a long-term planning horizon, in which we propose several alternative ways to define “value”. Risk is taken into account by including a non-linear penalty for dropping below a benchmark capital level and by adding a set of risk-based constraints. The three primary elements of a strategic DFA system are: 1) a stochastic scenario generator; 2) a set of decision rules involving investments, insurance underwriting strategies, reinsurance, and the corporate structure; and 3) a dynamic optimization module. See Figure 1. The first two elements form the corporate simulation system; these are deployed before the optimization module seeks out the best compromise decisions for the company given the relevant business, policy, and regulatory constraints. In effect, the optimization runs the simulation by searching for decisions that best fits the proposed objective function over the multi-period planning period. Figure 1 The Primary Elements of an Integrated Financial Risk Management System In this section, we describe a tightly integrated model for the insurance firm. Major decisions are evaluated by running the system to determine the impact of the decisions on the company’s growth of capital and risks. For example, business mix decisions should be linked to the amount and type of reinsurance. All senior decision-makers should be comfortable with accessing and running the corporate planning system. Of course, the DFA model should be simple enough so that a substantial number of sensitivity analyses can be conducted (Mulvey and Madsen 1999). 2 Retrocession coverage for reinsurance company. Model Uncertainties Simulate Organization scenarios Risk aversion Calibrate and sample What ifs Basic Technology Optimize In a large organization, it is impractical to expect that the major decision-makers will be able to run a single planning system. There are information and organizational barriers to a centralized approach. Most managers favor operating in a rather independent fashion from other parts of the organization. It is generally more efficient for individuals to concentrate on a well-defined business. To accomplish the goals of an integrated risk management system and to operate in a semi-independent fashion can be difficult. Figure 2 shows an organizational structure wherein headquarters focus on decisions relating to joint resources and overall capacity constraints. For example, headquarters is responsible for maintaining and deploying the proper level of risk capital. Figure 2 Structure for De-centralized Organization 2.1 Model Framework In this section, we define the basic modeling framework in a relatively idealized setting. Many details are omitted in order to concentrate on issues relating to capital allocation. See Mulvey (1999) for a richer description of an optimization-based DFA system. The financial planning model consists of Τ time stages t = {1,2,3,...,Τ}. The spacing of the periods will depend upon the user’s needs, with small steps typically occurring early in the process. The end of the planning period Τ defines the horizon; it depicts a point in which the investor has some critical decision to make, such as the repayment date of a substantial liability, or a target point for planning purposes. At the beginning of each period, the model renders decisions regarding the asset mix, the liabilities, and the corporate mix of equity and debt. We assume that cashflows such as dividends and interest occur at the last instant of a time period. We employ a system of stochastic differential equations for modeling the stochastic parameters over time. These relate a set of key economic factors to remaining components, such as asset and liability returns. As far as the model is concerned, the future at any point is the scenario sub-tree emanating out of the relevant node (Figure 4). The set of possibilities is called the representative Linking Strategic and Tactical Tactical Asset Systems Tactical Liability Systems Strategic System Re-insurance contracts Prices of Risk (t,s) Target benchmarks Risk Adjusted Profit set of scenarios, s ∈ S. Cariño et al. (1994) and Mulvey and Thorlacius (1998) discuss scenario generators. ______|_________|_________|________|_________|________|________| T (Horizon) The Planning Period ( t = 1,2,3, ..., T) Figure 3 The primary variables involve asset categories, liability-related decisions, and the capital structure: xj,t s asset category j ∈ J, time t ∈ T, scenario s ∈ S, yi,t s activity category i ∈ I, time t ∈ T, scenario s ∈ S, zk,t s decisions regarding the firm’s capital structure (e.g. stock issues), k ∈ K, time t ∈ T, scenario s ∈ S. The asset and liability variables can be thought of as simple bundles of future cashflows. We are agnostic about the actual definitions themselves, except those categories should represent a set of well-defined continuing positions, as opposed to individual securities with a fixed termination date. At each time period t, the model maximizes its objective function by moving money between asset categories, adjusting liabilities, and accounting for the capital structure. There are several candidates for the objective function as discussed in the next section. We represent the future as paths in a scenario tree (Figure 4). At each node of this tree, the model must decide upon the best course of action based on information available to it at the time. To do this, we must calculate the company’s position in many dimensions, for instance, the market value of its assets. A single scenario is a complete path in this scenario tree. Each combination of a scenario and time point (s ∈ S and t ∈ T) references a single node, n ∈N in the tree. The index set Sn identifies all scenarios that pass through node n. For practical purposes, we limit the number of nodes in the scenario tree. Statistical sampling theory, such as variance reduction provides methods for doing this. 3 We would define assets (liabilities) as cashflows normally possessing largely negative (positive) cashflows at the beginning and projected positive (negative) cashflows in the latter stages. 4 Securities are excluded in order to keep the model’s size manageable, but they can be added with enhanced computer resources. 5 The level of a business should depend upon the economic environment embedded in the scenario. Thus, ongoing business assumptions will need to be made. At each node n, we can evaluate a given product. The Scenario Tree Figure 4 There are three basic equations to any financial planning system. These equations preserve the flow of funds so that money can be accounted for at each junction. Equation [1] for j asset category: xj,t+1 s = (xj,t s + rj,t ) qj,t s + pj,t s (1-tj) for asset j, time t, scenario s. where rj,t s = return for asset j, time t, scenario s, qj,t s = sales of asset j, time t, scenario s, pj,t s = purchase of asset j, time t, scenario s, tj = transaction costs for asset j. Equation [2] for the i th liability-related category: yi,t+1 s = (yi,t s + ri,s ) qi,t s + pi,t s (1-tj) for asset j, time t, scenario s. where ri,t s = return for liability i, time t, scenario s, qi,t s = reduction in liability category i, time t, scenario s, pi,t s = addition to liability i, time t, scenario s, ti = transaction costs for liability i. Equation [3] for the cash flow collector nodes: x1,t+1 s = (x1,t s + r1,t ) ∑ qj,t + ∑ pj,t (1 tj) ) ∑ qi,t + ∑ pi,t (1 tj) + zt where zt s = cash inflows or outflows at time t, scenario s, based on capital structure decisions. Note that cash is asset category 1. The return parameters are calculated in several ways depending upon the context. To find economic value, we tally the future cashflows for the asset or liability in question and discount at the appropriate risk adjusted rate. In other situations, we employ book or regulatory value for determining the company’s statutory or accounting surplus. For the core model, however, we fix the best estimates for economic value in equations 1 to 3 since these equations involve actual cashflows. The multi-stage model avoids looking into the future in an inappropriate fashion. The model must optimize over scenarios that represent a range of plausible outcome for the future. To prevent the model from taking advantage of future information, we add special equations, called nonanticipatory conditions. The general form of the constraints is: xj,t s1 = xj,t s2 and yi,t s1 = yi,t s2 for all scenarios s1 and s2 which inherent a common scenario path up to time t, i.e. they share a sequence of arcs up to node n (s,t). The financial planning system addresses these non-anticipatory conditions, either explicitly or implicitly, and special purpose algorithms are available for solving the resulting optimization model. Many issues complicate the decision making process, for instance, certain liabilities and assets cannot be simply bought or sold at any time or for any amount. To address these issues, we impose a set of linear constraints on the process. Some examples of the constraints are the following: limiting borrowing to certain ratios, addressing transactions costs whenever assets are bought or sold, constraining duration of the assets, or limiting the model’s ability to take advantage of investment opportunities. Insurance companies often address the duration of their assets and liabilities as a critical factor in protecting their surplus. This concept can be readily incorporated, but will have less importance as companies invest in assets and take on liabilities with diminishing relationships to interest rate fluctuations. An important topic involves the cost of changes to the corporate structure, for example, increased borrowing costs that occur when a company’s capital drops below a threshold level, say a point in which the rating agencies would decrease its bond ratings. These costs are added to the financial model for borrowing costs: Cost of zk = g (capital available at current conditions under scenario s). [4] The nonlinear constraints are included in the planning model along with the other general constraints. See Froot and Stein(1998) for an explanation of penalties for capital dropping below a threshold. To directly control risks throughout a large organization, we impose an additional set of constraints on the risk capital of the firm. The next subsection takes up this issue. 2.2 Risk Based Constraints The capital allocation issue involves in a critical manner the amount of risk that an insurance company should undertake. Greater capital requirements will reduce the chances of a negative event, but at the same time reduces the profitability of the enterprise. Several approaches are possible for addressing risks in the DFA model. For example, we could compute the amount of Value-at-Risk by setting a constraint on the quantiles of the company’s profit/loss distribution. Here is an example in which we limit economic capital. The same type of equations can be employed for other definitions of capital surplus. Define the capital of the firm at node n (time t, scenario s) based on economic value: Ct s ≡ market value of assets – market value of liabilities. Under this framework, capital is uncertain and depends upon the scenario at hand, scenario s. We can limit the probability that capital will drop below some threshold value at time t, say Rt with a constraint (called chance constraint) of the form: Probability (Ct s ≥ Rt ) ≥ L [5] Here the parameter L depicts the critical quantile on the cumulative distribution function. We are forcing the model to allocate adequate capital so that there is a high probability (e.g. L = 99%) that the firm’s capital will stay above the threshold Rt under the analyzed scenarios. This approach is called VaR capital allocation. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 P ro b ab ill it y Figure 5 Probability distribution for Capital and VaR point at 1/20 level 6 We have omitted these details in order to keep the presentation concise; see Mulvey (1999) for further details. An alternative approach to measuring and ultimately controlling risks is to compute the expected policyholder deficit (EPD), which determines the expected losses for the company, given that losses are great enough to outweigh investment and other income, and possibly the capital itself. Again, we can express this risk measurement as a constraint in the planning model: