The effective allocation of capital is the best way to steer a portfolio towards higher profitability, according to Michel Dacorogna and Christoph Hummel

Recent crises in the financial markets and increased competition between insurers have made the industry realise that efficient capital management is a fundamental and necessary precondition for optimising insurance company performance. This may seem obvious and indeed true for all types of corporations, but here the word fundamental is emphasised. The primary function of capital in a financial institution (a bank or an insurer) is to absorb financial risk, rather than to provide funding for the assets of the business. Since this is its purpose, capital allocation cannot be ancillary to business processes, but should be a key step in managing risk. From this perspective, capital management and capital allocation to the various business units of a company are an essential instrument to optimise value creation in insurance.

Some banks already have different internal units trading their allocated capital among themselves. When one unit has no efficient use for its capital, it may want to sell it to another entity that identifies promising opportunities in the market. In other words, capital has become the "currency" for doing business and measuring it. In order to achieve this, the systems for measuring the capital at risk should be in place.

Reaching consensus

Moreover, the individual players must reach a consensus on how to calculate capital at risk. Despite the fact that risk-based capital is increasingly critical to commercial success, standards for defining it are unfortunately far from being common practice among insurance companies. For all stakeholders of the insurance industry capital has an essential but somehow contradictory function. On the one hand, regulators and policyholders want to ensure the company has sufficient capital to meet its obligations even if the claims are unusually high. On the other hand, shareholders want a decent return for the risk they assume in putting capital at the company's disposal. Therefore, they would want to limit this capital. Such conflicts make it even more important to arrive at a consensus on how to allocate capital to risk.

In the financial industry, capital is allocated to risks once a model for the probabilities of possible outcomes has been built. The amount of capital is then related to the tails of the distribution of a company's wealth at a future point in time. The company will hold sufficient capital to cover unexpected losses up to a certain probability and within a certain time horizon. For instance, the popular Value-at-Risk (VaR) measure refers usually to a confidence level of 99% or 99.6%. A confidence level of 99% means that in 1 in 100 cases the capital will not cover the liabilities at the end of the chosen time horizon. A confidence level of 99.6% corresponds to 1 in 250 cases. For banks, the Basel Committee has set these horizons at 1 and 10 business days. In insurance, the horizon will extend to some 1 to 5 years.

Insurance risks have usually much fatter tails than financial risks. This means that the very large losses' share in total losses is higher than in other financial markets. Some potentially large losses, however, might make next to no impact on VaR. Indeed, if an insurance company chooses a confidence level of 99% and writes an insurance contract for which the probability of turning a profit is very high and the probability of bankrupting the company is very low (eg 0.5%), it hardly increases the company's risk-based capital. Even worse, the higher the correlation between this contract and the rest of the portfolio, the lesser the increase of the capital required. These characteristics may be acceptable to shareholders, but from the policyholders' point of view, they make VaR an inappropriate risk measure for insurance companies. In addition, capital allocation within the company is problematic, because underwriters profit from adding exposure far out in the tail, thereby steering the portfolio in the "wrong" direction.

Tail-Value-at-Risk

That is why the International Actuarial Association recommends the use of Tail-Value-at-Risk (TVaR) to measure risk. TVaR is simply the average loss over a certain probability threshold. In practice, it corresponds to the average of the 1% worst scenarios within a stochastic simulation. Note that the insurance contract in the example above, which jeopardised the company, would require more capital in the TVaR setting. The higher its correlation with the remaining portfolio, the higher the capital required. Furthermore, the capital requirements computed according to TVaR have the advantage that they can be split among sub-units in a coherent fashion.

This mathematical property is very important because capital allocation should allow management to monitor the performance of the various sub-units of the company. The higher the risk of a sub-unit and the lower its diversification within the company's portfolio, the higher the capital allocated to it. Capital serves as tender for measuring business results. If the unit generates a profit which covers the cost of its capital, then the unit contributes to the bottom line of the company. The performance is not measured by premium volume generated within a unit, but rather by profit related to the contribution of its effectively assumed business to the company's risk.

Allocating capital

Once the risk-based capital has been computed for the entire company on the portfolio level, we still need to allocate it to the sub-units. This is done by averaging the sub-unit's contribution to the loss in the company's Net Asset Value (NAV) over each of the company's 1% worst scenarios. Under every single scenario, each sub-unit will contribute to the overall (bad) performance of the portfolio. Totalling the average contributions results in the total amount of capital computed for the entire portfolio. Not only does TVaR have the above-mentioned conceptual advantages over VaR, but the practical feasibility of allocating capital should count in its favour.

Reliable tools for calculating capital and allocating it to sub-units are a prerequisite for the implementation of such a concept within an entire company. The following three conditions must be met in order to effectively allocate capital: firstly, a satisfactory risk measure (as discussed earlier), secondly, proper individual modelling of each risk, and, thirdly, modelling of the interdependency of the various risks. Most of the time, insurance companies are able to fulfill the first two conditions. The third one is much harder to meet because it is based on the understanding that the dependency model can have a considerable impact on the amount of capital required.

In the example in table 1 shows the risk-based capital for two risks that are known to depend on each other with a linear correlation of approximately 0.4, as a function of the dependency model chosen.

On course for profitability

Why does one want to go through all this trouble in order to allocate capital to the various different units of a firm? Simply, because it is the best way to steer the portfolio towards higher profitability. If management is able to assign an amount of the company's equity to a business unit, it can also measure the performance of this unit and thus reduce or increase the exposure according to the potential results of that particular line of business. Moreover, explicitly assigning capital to a business unit and relating the amount to the risk assumed by that unit facilitates the development of a risk management culture within the company. The simple fact that an amount of money is assigned facilitates discussion among the various stakeholders as to the underlying assumptions for this amount. Such discussions tend to be the exception rather than the rule in the insurance industry, but with the coming Solvency II requirements and the need to improve business profitability they are becoming a key success factor for forward-looking companies.

Michel Dacorogna is head of financial analysis and risk modelling and Christoph Hummel is manager of pricing specialty lines at Converium.