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Introduction
Capital Allocation for Insurance Companies:
Issues and Methods
Paul Mumford
Jens Perch Nielsen
Rolf Poulsen
Capital Allocation for Insurance Companies:
Issues and Methods
Abstract
We address some key issues related to risk and capital allocation in insurance companies. We argue that the Froot-Stein approach to risk is relevant to a number of important problems in the daily management of an insurance portfolio and that – taken to its consequence – this approach will lead to a close co-operation between actuarial and financial departments with a particular view to the actuaries ability to separate financial risk from insurance risk and the ability of the finance department to hedge the right amount of their complicated financial risk at a good price. Further, we introduce a new capital allocation principle based on the solvency of the entire company and we claim that this principle is more robust than traditional methods. We also introduce a new way of combining capital allocation and pricing of insurance policies. Through a performance measure of return on capital, we suggest a way of evaluating the price of capital for a single policy, that this money is held as a reserve and that this reserve should run off in such a way that old underwriting does not harm the performance of any given business line.
Introduction
Insurance companies expect to pay claims throughout a working year. However they cannot be sure in advance how many claims they will have to pay, they cannot be sure when the claims will be paid and they cannot be sure how much the claims will be.
Using statistics companies can predict on average how much money they should hold to pay all the claims in a year. This amount is called the reserve. However what if there are a higher than average number of claims? Then they will have to payout more money than the reserve that is being held. Hence insurance companies hold more money than they actually expect to pay out in claims so that they honour unexpected claims. This money is called capital.
So how much capital should an insurance company hold? Since capital is held to cover risks this is actually a question of risk. Does a company want to hold enough money to cover all possible claims, which would tie up a huge sum of capital? Capital that they must somehow raise. Or would a company rather free up capital for other uses but become vulnerable to not being able to pay a large unexpected claim? In other words how much risk should the company take?
The tautological answer is “as little as possible”. A quantitative or practical answer is closely related to the classic actuarial discipline of calculating/estimating probabilities of rare events; i.e. ruin probabilities. And this should be combined with the insights modern financial theory for an explicit modelling of investment risks and their trade-off with expected return.
But that is not all. Insurance companies have had to deal with an increasing level of regulation and explicit legal requirements (such as Basel and Solvency directives.) There has also been more focus on corporate risk management from investors/rating agencies (with rules such as “multiply capital by 3”; which isn’t actually as bad as it sounds, see the discussion in Leippold and Vanini (2003)) and academics (with an axiomatic approach pioneered by Artzner et al. (1999)). Actuarial literature has therefore devoted increasing attention to the necessity of understanding the stochastics of the entire insurance company, see Cummins (2000), Kaufmann, Gadmer and Klett (2001) and Laeven and Goovaerts (2004) for some early contributions.
However, this choice of overall/company wide capital level, say C, it not the focus of this paper. We take it as exogenously given. Or as a very concrete yardstick: The FSA require that enough capital is held to ensure there is only a 1 in 200 chance that the insurance company is unable to honour its claims in the next year. In reality an insurance company may wish to hold even more capital -for example, it may hold sufficient capital to maintain an S&P A rating.
In this paper we look into some practical problems related to capital allocation. Firstly, as we discuss in Section 2, capital must be allocated to individual lines. We introduce a robust allocation principle that allocates capital to each line according to its loss given that the whole company can cover its losses. This approach is robust because it is less dependent on the extreme tail of the considered distribution. Furthermore, in case the whole company cannot cover its losses, a major restructuring takes place, and an “ex-ante” (beforehand) line-by-line allocation plays little role. In other words, this approach puts weight on cases where line-by-line allocation matters. Secondly, over the years there have been great confusion on the relationship between performance, reserving and price for capital. This has lead to under-pricing of business lines that lock up capital for a long period of time, for example liability lines. In Section 3, we introduce a simple but powerful way of linking together price of capital, reserving and performance. One simply has to calculate the discounted price of capital for each policy – based on the asked for return on capital – take this amount as part of the premium for each policy, hold it as a return-on-capital-reserve for each policy and release it over the time as the discounted future need of capital decreases. Thirdly, there has been an important development in the theory of financial mathematics that has great practical and theoretical consequences for insurance, namely the award-winning paper of Froot and Stein (1998) with the powerful conclusion that all companies owned by stock-holders should use financial markets to hedge, but should not speculate; the stock-holders can do that themselves. In Section 4, we go through some of the practical consequences of this paper for the handling of insurance risk. Finally, we conclude in Section 5.
2 Allocating Capital to Groups/Lines: A robust principle
We have to allocate capital to individual lines. The law requires this to some extent, since any insurance company – and its subsidiaries – will be required to have its own capital. However, a good allocation of capital to individual business lines can be useful for many reasons. It helps evaluate performance, it will allow better pricing of new business, It forces an individual and quantitative assessment of the lines, thus telling central management how each line of business is performing. Also, allocating capital “ex ante” (beforehand) can help avoid moral hazard/agency problems among managers.
Once an insurance company has chosen its overall capital allocation C, it must then decide how it wants to split this capital between its different groups or lines of business. A multinational insurance company consists of many lines of business each of which require capital. As an Example Figure 1.1 is a simplified diagram of the Royal&SunAlliance (R&SA) group that operates in more than 130 countries, has over 20 million customers and the R&SA UK company itself has over 30 subdivisions.
Figure 1: The tree structure of groups/lines in a multinational insurance company.
A simple and intuitive way of allocating capital is using the celebrated Tail Value at Risk (TVaR), see for example Klueppelberg, Urban, Dittrich and Stoelting (2004), Venter (2004) and Kreps (2005). When allocating capital to different lines of business in an insurance company, one has to remember to centre claims, or at the very least be consistent. If this is forgotten the result will be distorted by factors not related to the risks involved. One could consider centring around premiums rather the expected losses. But by doing the latter, we are able to separate the effect of each item. Capital depends on “risk” (this section); premiums depend on expected losses and (cost of) capital (next section). Otherwise we would get a “feed-back effect”: Capital depends on premiums, premiums depend on capital. So, we suggest that one centre the considered losses with their expected values. In the following we introduce a robust allocation principle starting from these centred variables.
2.1. A robust capital allocation principle
In this section we consider the traditional TVaR allocation principle and show how it can be modified into a more robust allocation principle.
2.1.1 The traditional TVaR allocation principle
The company and the individual lines have some (random) claims/losses and we let X = Present value (claims – E(claims)) denote the discounted centred versions of these, ie. the unexpected losses. By
å
=
i
i
X
X
we denote the total (present value of unexpected) losses we let and
)
(
X
q
a
is the
a
-quantile, which is the minimum loss incurred in the
)
1
(
a
-
% worst cases.
Once the overall capital
C
held by the whole group has been decided it can then be allocated.
In the TVaR method the capital allocated to the the ith of business is given by:
)]
(
|
[
)]
(
|
[
X
q
X
X
E
X
q
X
X
E
C
C
i
i
a
a
³
³
´
=
.
This principle is an example of proportional allocation with respect to a risk co-measure. Note that
))
(
|
(
)
(
X
q
X
X
E
X
i
i
a
r
³
=
is not a risk measure in the sense of Artzner et al. (1999) because i) we condition on the whole company’s losses, and ii) we work only with centred variables.
This allocation method some advantages. First, proportionality is easily explained. Second, it is based on TailVaR (aka. expected shortfall) which is a coherent risk measure (for continuous distributions) meaning in short that it does not do “obviously silly things”. Conditioning on (the full) X ensures additivity and that (some) attention is taken to the joint distribution of losses; correlation matters in other words, and in a multinational company, there could be subtle cross-country-correlations.
Consider a company with two lines that both have expected loss of 50 and standard deviation 10. However, one line is lognormally distributed, the other pareto distributed, and thus has a much heavier (power law type) tail. The following table shows how the TailVaR method allocates capital to the two lines depending on which quantile level,
)
(
X
q
a
, is used:
Quantile
% of capital
to lognormal
% of capital
to pareto
0.5
51
49
0.75
51
49
0.95
46
54
0.995
34
66
0.99975
21
79
The TVaR allocation method is is the method currently used by the RSA group where the
%
50
-quantile is used. Typically when stated in literature a higher quantile such as the
%
95
-quantile or even
%
99
-quantile is associated with this method. (And remember that the overall capital level C is often chosen as the 99.975%-quantile.)
The effect of using a lower quantile is that more weight is assigned to the more likely small losses which cumulatively become important, rather than basing business decisions on highly unlikely circumstances. Interestingly, the same results and be obtained with a different motivational story: Allocate capital to each line according to its loss given that the whole company can cover its losses. This is the simple modification to the TVAR suggested in the next section.
2.1.2 A modified and more robust TVAR allocation principle
Consider the simple modification to the TVAR principle, where we allocate capital to each line according to its loss given that the whole company can cover its losses:
]
|
[
]
|
[
C
X
X
E
C
X
X
E
C
C
i
i
£
£
´
=
Line-by-line allocation is a moot point in the cases where the company defaults. The whole company is going to be restructured anyway.
In the lognormal/pareto case from before this leads the these allocations:
Overall capital quantile
% of capital
To lognormal
% of capital
to pareto
0.5
48.7
51.3
0.75
49.8
50.2
0.95
50.3
49.7
0.995
50.2
49.8
0.99975
50.0
50.0
The allocations are much more stable, and the numbers comparable to what is being done. Note also, that the obtained numbers are quite close to the result of the 50% quantile in the previous section. The method currently used by Royal&SunAlliance. The reason that this method is more robust is that it depends less on the tail behaviour of the involved losses. For practical purposes this point is even more important than the above example indicates, because the tail behaviour of insurance losses can only be estimated with a high degree of uncertainty.
3 Performance and Return on Capital
In this section, we introduce a new approach to pricing and reserving of capital based on a return on capital measure, and we consider some subtleties on the relationship between capital and the credit rating of an insurance company.
3.1 A consistent link between capital allocation, pricing and performance
Another important point when it comes to capital allocation is how well is the capital used after it has been allocated? As long as it remains accessible capital can be invested and used to make money. Currently insurance managers are assessed using the ratio of the return made over the capital held, this is called the return on capital (ROC). Again this is a very simple way of measuring performance. So we must ask, is this the fairest way to evaluate an investment manager and does it provide the best business incentives.
In other words, how do we link capital allocation and performance and how do we incorporate this in our pricing models?
To answer this question, we must be more precise by exactly what we mean by capital. Some capital is held for the new underwriting of policies, whilst the rest is held for old underwriting. The capital held for old underwriting is used for claims which must be paid in the future for policies which were written in previous years. The amount held for the expected value of old underwriting is often termed the reserve, but to avoid confusion with the expected claims we will just call it old underwriting. A well run business would ensure that adequate levels of old underwriting capital were held at all times. This keeps the risk of default to an acceptable level. Any excess capital can then be used for new underwriting.
The return on capital ratio merely provides insurance managers the incentive to make as high return on their allotted capital as possible. The intention is to find and reward managers who are running their portfolios well and efficiently and making the highest returns. However the ROC doesn’t necessarily encourage managers to run their portfolios well or efficiently, it only encourages them to make a high return.
One way they can do this is to hold very little capital for old underwriting, which frees up more capital for new underwriting and means they write lots of new policies which will bring in lots of premiums. For a year or two, the manager may get lucky and produce fantastic ROC figures and maybe earn himself a lucrative promotion. However the fund is being poorly run and is susceptible to an unacceptably high risk of default. This may be a risk worth taking for the insurance manager but it is not in the groups best interests.
The natural way to prevent managers from doing this is to consider the return on new underwriting. This removes the incentive for insurance managers to use too much capital for new underwriting. However even this performance measure still ignores the actual split between old and new underwriting and doesn’t necessarily reward a well run business.
A crude method to set premiums to achieve a particular ROC is:
Premiums
=
Present value (
(
E
Claims
)
EMBED Equation.3
+
Expenses)
+
ROC% Capital
So, how much does it cost to hold capital?
For any given policy, capital will need to be held for a certain number of years. We’d expect the amount of capital which must be held to decrease each year, so that the level of capital needed each year resembles figure 1.5, where C 3 represents the capital held in the third year of this portfolio.
C1
C2
C3
C4
C5
C6
Figure 2
The amount of capital held in
j
years time is
j
C
. This will earn
%
i
interest in a risk-free bond. Hence the present value of the interest we make by investing all our capital in risk free bonds is:
å
¥
=
+
1
)
1
(
j
j
j
i
iC
.
However we hope to make a ROC of
%
r
. So the present value of the return we hope to make is:
å
¥
=
+
1
)
1
(
j
j
j
i
rC
.
Hence the extra we must charge in the premiums is make the required return is:
(
)
å
¥
=
+
-
1
)
1
(
j
j
j
i
i
r
C
.
This then gives us the pricing formula:
Premiums
=
PV (
(
E
Claims
)
EMBED Equation.3
+
Expenses)
+
EMBED Equation.3
(
)
å
¥
=
+
-
1
)
1
(
j
j
j
i
i
r
C
By using this formula a well maintained capital structure is encouraged and business performance can be measured. In fact, this formula gives us a 1-1 relationship between premiums and performance. This allows us to set premiums to meet performance targets or set performance targets from market values. Just after a policy has been sold the capital reserve linked to this particular policy is
(
)
å
¥
=
+
-
1
)
1
(
j
j
j
i
i
r
C
. After one year the first years capital has been freed and the expected capital to hold is
(
)
å
¥
=
+
+
-
1
1
)
1
(
j
j
j
i
i
r
C
. The entire capital absorbed by writing the policy will be freed during the lifetime of the policy. Moreover, the reserve will free capital in such a way that the business line will get subsidised with an amount corresponding to an expected return on capital locked equal to the ROC of
%
r
. Therefore, this method ensures that old underwriting does not harm performance while properly managed – whilst if it is mismanaged, it will do harm. This makes perfect sense and encourages any business line manager to have a close eye on old underwriting.
One result for the policyholder of using this formula is that long tailed insurance such as liability will become more expensive. The reason for this being that they must now pay upfront for capital to be held for a long time. On the other hand short tail insurance such as property insurance may get cheaper as they no longer have to contribute towards the cost of holding capital for the longer tailed business
3.2 Commercial and personal ratings
One subtlety in the current method is that the RSA group chooses its level of capital to maintain an S&P A rating. The reason for this is that commercial policyholders will only consider insurance companies which have such an A rating. This is understandable since commercial claims are likely to be considerably higher. If a large company has an insurance policy against its head office burning down, then they want to be sure any claim will get paid.
This desire to insure with an A-rating insurer is probably much less relevant for personal insurance where people are likely to be ignorant to what an insurance company’s rating is or even what it means. For personal car insurance in particular it is believable that most consumers would chose a lower premium rather than a more secure company. However like the rest of the group personal car insurance is currently being allocated capital on the basis of maintaining an A-rating. It could be argued that this A-rating is higher than necessary and so the personal car insurance line holds more capital that it needs to.
To raise all the necessary capital the line must charge higher premiums than it needs to. To try and lower premiums and allocate capital to personal insurance on the basis of a BBB-rating say, it would be necessary to form this as a separate company. Then we might expect that this separate personal company could hold less capital and charge lower premiums that it did as a part of the group. However this isn’t necessarily true.
Diversification of insurance companies reduces risk. By splitting the group in two, each half of the group faces more risk than it did as part of the combined group. Now that personal insurance faces more risk that it did, it might have to hold more capital and charge higher premiums, even to maintain a lower rating. Even if personal insurance did manage to lower its premiums then we’d certainly expect commercial premiums to go up.
The reason for this that on its own commercial business is now more risky than it was as part of a group, but it still must maintain an A-rating. To do this it must hold more capital than it previously did and therefore must also charge higher premiums. So splitting the group into a commercial group and a personal group could be good for the personal group, but it almost certainly going to be bad for the commercial group.
4 Hedging and Financial Risks: The Froot and Stein model
In this section we suggest that investment managers should heed the advice of the Froot and Stein model that states than an insurance company should hedge all tradable financial risks. This will often involve investing in risk free bonds with low expected rates of return – and low financial risk.
The model from Froot and Stein (1998) aims to characterise the optimal risk strategy for financial institutions. The main principle is that companies should not be exposed to any financial risk at all if they operate with the shareholders’ interest at heart. Formally this can be written as:
Proposition The firm maximizes shareholder value by i) fully hedging its exposure to tradable risks, and ii) not taking any other financial risk.
Structurally, the model comprises of three stages. Firstly the firm must choose its capital structure, then it chooses its financial investments/hedging position before finally realizing its investments and investing the proceeds in a subsequent project. The firm has a hedge demand for financial assets caused by shareholders’ risk-aversion, but no speculative demand, because shareholders can just as well do that themselves without will having to face the possible restructuring costs of going through an intermediary.
The Froot-Stein principle implies that underwriting rather than exposure to financial risks is what must create value for insurance companies, as perfect hedging eliminates all financial risks.
An immediate lesson to draw from the Froot and Stein result is that insurance companies should shun financial markets, since it is “the natural assumption” is that claims are unrelated to financial markets; the drivers do not get better or worse depending on what the NASDAQ does. But there are exceptions, as we discuss next.
4.1 Long-tailed insurance.
Here the present value of the future claims is subject to interest rate risk, so it would be wise to off-set this risk with a duration matched bond portfolio, ie. not just investing in short term bonds. (This is not as easy as it sounds; the insurance side – the liabilities – may be very long-dated and bonds with similar duration hard to come by, and the liabilities are often inflation-linked). When hedging is expensive or not immediately possible we enter the grey area described below.
4.2 Currencies and multinational companies
Exchange rate risk (if unhedged) will make subdivisions that conduct their business in “other currencies” look more volatile than they need be. However, the Froot-Stein approach to hedging gives a clear answer to the relationship between currency risk and capital. All currencies should be hedged according to the main currency in which the capital calculation is being done. For example, in the case of RSA and its Danish subsidiary. Here, RSA should forecast the payment stream in DKK and hedge it in sterling. This eliminates the currency risk and allows the amount of capital in the Danish subsidiary to be at a relevant and fair level.
4.3 “Grey areas” and practical issues
In the real world it is often not possible to hedge perfectly, as the relevant products do not exist. Even when the products are available it can sometimes be too expensive to purchase them and we must trade off security for cost. Actuaries could, in theory, technically calculate the point when products become too expensive but a good guide is when shareholders would prefer not to buy. In this case a non-perfect hedging strategy is decided upon. This means that some financial risk has to be juggled alongside the inevitable insurance risk that insurance companies incur. We argue that this financial risk has to be taken care of by the investment department and we formally agree with Froot and Stein (spage 67) when they suggest to centralize financial decisions based on models, however, we also sympathize with added remark of Schiro (2005) that we should not forget to use practical experience as well.
Hedging requires dynamic trading and monitoring. “If you’re big, you get better deals.” This is yet another argument for a centralised investment department hedging the financial risks of as the entire company. There is, off course, this great difference between actuarial and financial risk: The latter scales linearly (portfolios across assets and time); the former doesn’t (inhomogeneity in the word of Mildenhall (2004)). The different nature of actuarial and financial risk suggest that financial risk can not be measured in terms of the same ROC as the rest of the insurance risk. If the full consequence of the Froot-Stein approach is taken, the performance of the finance department should simply be based on how well they manage – in particular hedge - the financial risks of the company, not on some ability to get a high return on some investments.
5 Conclusion
We have argued that the Froot-Stein approach to risk is relevant to a number of important problems in the daily management of an insurance portfolio and that – taken to its consequence – this approach will lead to an increased co-operation between insurance, actuarial and financial departments with a particular view to the ability to separate financial risk from insurance risk and the ability of the finance department to hedge the right amount of their complicated financial risk at a good price.
Further, we have introduced a new capital allocation principle based on the solvency of the entire company and we claim that this principle is more robust than traditional methods, further we have introduced a way of combining capital allocation and pricing of insurance policies. Through a performance measure of return on captital, we have suggested a way of evaluating the price of capital for a single policy, that this money is hold as a reserve and that this reserve should run off in such a way that old underwriting does not harm the performance of any given business line.
References
Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999), Coherent Measures of Risk, Mathematical Finance, vol. 9(3), pp. 203-228.
Cummins, J. D. (2000), Allocation of Capital in the Insurance Industry, Risk management and Insurance Review, vol. 3, pp. 7-28.
Froot, K. A. and Stein, J. C. (1998), Risk management, capital budgeting and capital allocation for financial institutions: an integrated approach, Journal of Financial Economics, vol. 47(1), pp. 55-82.
Kreps, R. (2005), Riskiness Leverage Values, to appear in Proceedings of the Casualty Actuarial Society, http://www.casact.org/pubs/corponweb/papers.htm
Høgh, N., Linton, O. and Nielsen J. P. (2004), The Froot-Stein Model Revisited, to appear in British Actuarial Journal.
Kaufmann, R., Gadmer, A. and Klett, R. (2001), Introduction to Dynamic Financial Analysis, ASTIN Bulletin, vol. 31(1), pp. 213-249.
Klueppelberg, C., Urban, M., Dittrich, J. and Stoelting, R. (2004), Allocation of risk capital to insurance portfolios, Blaetter der DGVFM, vol 26., pp. 389-406.
Kreps, R. (2005), Riskiness Leverage Values, to appear in Proceedings of the Casualty Actuarial Society, http://www.casact.org/pubs/corponweb/papers.htm
Laeven, R. J. A. and Goovaerts, M. J. (2004), An optimization approach to the dynamic allocation of economic capital, Insurance: Mathematics and Economics, vol. 35(1), pp. 299-319.
Leippold, M and Vanini, P (2003), Half as many cheers – The Multiplier reviewed, Wilmott Magazine, pp. 104-107, http://www.wilmott.com/detail.cfm?articleID=178
Mildenhall, S (2004), “A Note on the Myers and Read Capital Allocation Formula”, North American Actuarial Journal, vol. 8(2, April), pp. 32-44.
Myers, S and Read, J. (2001), “Capital Allocation for Insurance Companies”, Journal of Risk and Insurance, vol. 65(4), pp. 597-636.
Schiro, J. J. (2005), Successful Risk and Capital Management, The Geneva Papers, vol. 30(1), pp. 60-64.
Venter G. G. (2004), Capital Allocation Survey with Commentary, North American Actuarial Journal, vol. 8(2, April), pp.96-107.
RSA Group
UK
Scandinavia
Rest of World
Household
Personal
Property
Motor
Personal
Commercial
Motor
Property
Motor
� This version December 5, 2005. Mumford and Nielsen are from Royal&SunAlliance and the corresponding author Poulsen (� HYPERLINK "mailto:[email protected]" ��[email protected]�) is from the Department of Applied Mathematics and Statistics, University of Copenhagen, Universitetsparken 5, DK- 2100 Denmark. We acknowledge helpful discussion with the two Oxford students - James Schofieldtewart and Sarah Child – that helped us on the preliminaries of this paper during their internship in RSA.
� This celebrated model is widely accepted and although the analysis in the original paper is not fully correct, Høgh, Linton and Nielsen (2004) suggests that their conclusion is to strong and derive a theorem based on the assumptions of Froot-Stein, that leads to a slightly weaker conclusion. However, recent research shows that it is rather the assumptions than the conclusions of Froot and Stein that have to be corrected.
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