Marginalizing the Cost of Capital
1Daniel Isaac FCAS and Nathan Babcock ACAS
Recently, there has been a lot of discussion in the actuarial community about the best way to allocate capital. While this discussion has been very useful, it may be premature. One question that seems to be missing from all of these discussions is perhaps the most basic: “Should we allocate capital at all?” This is particularly important since modern financial theory would seem to suggest that the correct answer is “No.”
This paper describes a new approach that is more in line with financial theory. Specifically, by combining a method for developing a strategy-specific cost of capital with an estimate of the costs of financial distress, we will attempt to show that capital allocation may not be necessary. Several practical examples will be used to help explain the advantages of this approach.
Keywords: Dynamic Financial Analysis (DFA), Cost of Capital, Regulatory Costs
1 Conning Asset Management, 111 S. Calvert St., Suite 1800, Baltimore, MD 21202. Phone: (410) 369-2822, Fax: (410) 369-2922, E-mail: firstname.lastname@example.org.
Actuarial work on the cost of capital has long focused on devising methods or rules for allocating capital across business units within a company. The resulting work has led to a number of interesting, and sometimes heated, discussions about what makes one allocation method “better” than the others. Despite all this focus on the question of “How should capital be allocated?”, there has been relatively little discussion of a more fundamental one: “Should we allocate capital at all?” This particularly surprising since modern financial theory suggests that
the answer to the latter question should be a resounding “No.” Specifically, products produced in a competitive market, like insurance, should be priced based on their marginal cost, including the cost of the extra capital needed to support the “last” item sold. Based on this, actuaries
should focus on determining the marginal cost of the extra capital required, not on allocating the company‟s total cost of capital.
Most actuaries would argue, however, that pricing based on marginal costs would result in an underpricing of the company‟s entire book of business. Three assumptions lie behind this argument. First, as shown in Exhibit 1A, it is assumed that each additional policy a company writes requires less capital than the previous one to support it. This is the basic premise behind insurance: a pool of risks is less risky than each individual risk. Second, every dollar of capital
icosts the company the same (e.g. 15% per year), whether it‟s the first or last. While the cost of
capital can vary based on a company‟s financial structure (e.g. debt to equity ratio), most people assume that it does not change with the amount of business written (see Exhibit 1B). The final assumption is that all other variable costs (e.g. losses and expenses) are a fixed percentage of
iipremium (see Exhibit 1C). Based on these three assumptions, the company‟s marginal cost is a
constantly decreasing function. As a result, the company‟s average cost is always higher than its marginal cost (see Exhibit 1D), so the company must price its product based on the average.
Although average cost pricing is appropriate in a decreasing marginal cost environment, such a situation almost always leads to either a monopoly or oligopoly marketplace. However, numerous studies of the property-casualty insurance market have concluded that it is relatively competitive. Thus, it is difficult to believe that insurers really are operating on a decreasing portion of their marginal cost curve. If, on the other hand, insurers in fact face rising marginal costs, the question of capital allocation becomes less relevant.
This paper reconsiders the typical actuarial assumptions about capital costs. First, using the approach described in “Beyond the Frontier: Using a DFA Model to Derive the Cost of
Capital” (Isaac and Babcock, 2001), we will demonstrate that the cost of capital depends on the amount of business a company writes. Second, we will show that there are certain ancillary costs (e.g. frictional cost of insolvency) that increase with higher volumes of business. Relaxing these assumptions will allow us to reassess whether insurance really has a decreasing marginal cost.
Recap of Earlier Approach
As was described in “Beyond the Frontier: Using a DFA Model to Derive the Cost of Capital,” financial theory introduced the cost of capital into the evaluation of potential business plans, or “corporate strategies,” to reflect the varying levels of risk to shareholders capital. For each strategy, there is an associated benchmark portfolio on the efficient frontier of potential asset holdings, or “asset-only efficient frontier.” When evaluated within a series of DFA
scenarios, a benchmark portfolio will generate a rate of return (hurdle rate) for each scenario. The cost of capital for a scenario is defined as the increase in the benchmark‟s value, expressed as a currency amount.
In the original paper, we argued that a strategy‟s cost of capital should have three basic properties: 1. it should increase with the strategy‟s riskiness, 2. it should be related to the returns
available from other financial instruments and 3. it should be related to the length of the project.
iiiBased on these desired properties, we suggested a four-step approach. First, the asset-only
efficient frontier is determined. Second, we use a DFA model to calculate the financial results, in particular the ending Market Surplus, for the corporate strategy under consideration. Third, for each portfolio on the efficient frontier, we determine the amount that could have been invested to best duplicate the company‟s financial results. For strategies with no interim dividends, this can be seen as solving a linear regression of the form
Y = m * X,
where X and Y are the cumulative return factor for the benchmark under consideration and the ending Market Surplus, respectively, for the by-scenario results and m is the initial investment in
the potential benchmark. Finally, the strategy‟s benchmark is the portfolio which minimizes the resulting error term (i.e. Y – m * X in the above equation).
There was a major problem with this approach: it created a maximum average hurdle rate. Specifically, since each of the portfolios on the efficient frontier is a direct combination of the available investments, the maximum average hurdle rate is the expected return on the single investment with the highest return, typically stocks. The problem with this arises when the company considers “corporate strategies” that invest a larger and larger portion of its assets into this same category. At first, this increase is matched with an increase in the benchmark‟s
allocation to this asset class. At some point, though, this allocation reaches the 100% maximum. Since insurance companies‟ assets are usually several times their available capital, this tends to
happen well before the company is investing exclusively in this asset category. Therefore, any
further increase in the company‟s investment in this category leads to higher returns and associated risk, without a corresponding increase in the cost of capital.
In order to address this concern, we have extended the methodology by allowing an investment into a risk-free asset. Specifically, the methodology now looks at combinations of both 1. the efficient portfolios discussed in the previous section and 2. investing or borrowing at
ivthe risk-free rate. By allowing the investors to borrow money, this change eliminates the maximum hurdle rate problem. It is also useful to note that, for the simple example described above, this portfolio leveraging changes the linear regression to one of the form:
Y = (m * X) + b,
where b is the ending value of funds invested in the risk-free asset by the shareholders at the
vbeginning of the evaluation period and the other terms are as previously defined. The
shareholders are now able to “leverage down” (i.e. b > 0) their investment in the benchmark
portfolio by investing a portion of their capital in the risk-free asset and investing a reduced amount in the benchmark under consideration. Similarly, the shareholders can “leverage up” (i.e.
b < 0) their investment by borrowing at the risk-free rate and investing both their initial capital and the proceeds of the loan into the benchmark portfolio.
Impact of the Cost of Capital
In order to see how this new approach to the cost of capital affects the traditional assumptions underlying the capital allocation debate, a concrete example appears to be in order. Specifically, we will focus on the DFA Insurance Company (DFAIC) that was discussed in the CAS‟ 2001 Dynamic Financial Analysis Call Paper. DFAIC is a privately held property-casualty insurance company operating in all fifty states with business concentrations in the northeast and
mid-west. The company writes personal and “main-street” commercial coverage through
The Call Papers by Burkett et al. (2001) and Philbrick and Painter (2001) discuss a DFA model in which the projected growth rates for written premium vary from -2.5% to +2.5% per year during the five-year projection period. These levels of new business premium are parameters of our model's Baseline projection, which is a “corporate strategy” as was described in the previous section. According to our new theory regarding cost of capital, the Baseline strategy has a cost of capital benchmark: a specific portfolio on the asset-only efficient frontier. To see how this new theory impacts the cost of capital discussion, we also simulated a wide range of business growth strategies with the levels of new business ranging from 0% (i.e. no new business) to 300% of the Baseline projection (i.e. written premium being three times larger than in the Baseline simulation).
This procedure implicitly assumes that the written premium parameters may be scaled up or down without requiring adjustments to any of the other parameters. Let us look in more detail at this assumption. First, we assumed that all of the expected expense and loss ratios were independent of business level. While this is clearly an overly simplistic assumption, it both serves the purposes of this paper and fits with the general actuarial assumptions that were discussed earlier. Second, to generate the different levels of new business, we simply scaled the underwriting results (i.e. written premium, expenses, losses) of the individual scenarios from the Baseline projection. As a result, there is no reduction in loss ratio volatility as more business is written. While we would suggest refining this assumption for actual client work, especially given the large range of business volumes considered, we did not feel that it would materially affect the outcomes of this paper. In addition, one big advantage of this approach is that it
dramatically reduces the impact of process risk on the differences among these business growth strategies. Finally, we assumed that the same level of capital could, at least initially, support all of these strategies. Under this assumption, the standard actuarial assumptions would suggest a fixed total cost of capital. Therefore, this assumption allows us to focus solely on the differences in the cost of capital between strategies.
Based on these assumptions, we ran 11 strategies, each with a different level of new
viibusiness, through a five-year projection using our proprietary DFA model. A number of interesting observations can be made about the results. For example, Exhibit 2 shows that there is a positive average hurdle rate of 7.5% even when DFAIC isn‟t writing any new business (i.e. the “0%” run). Intuitively, this makes sense: DFAIC‟s shareholders are going to require a return
as long as their capital is tied up, not as long as the company is writing new business. Since this cost cannot be passed on to the policyholders now, it should have been built into the initial pricing methodology. Unfortunately, many of the capital allocation methods currently in use would allocate this cost to the company‟s new business. While this may lead to the right solution (e.g. for a very stable company), it is very likely to create some serious distortions for the
viiimajority of companies, which have fluctuating volumes of business.Another important fact
that can be seen in Exhibit 2 is that the hurdle rate is an increasing function of the amount of new business written. Since all of these strategies have the same capital, this means that DFAIC‟s
shareholders are requiring more compensation in order to put their capital dollars more at risk. For most of the strategies, this increase can be directly attributed to an increase in the leverage ratio. To help understand why this makes sense, it maybe useful to consider the insurance transaction in financial terms: the company “borrows” money from the policyholders (i.e. the premiums) and invests the proceeds in an attempt to earn enough money to pay off the loan (i.e.
the expenses and losses). Viewed in this way, writing more business means that the company has to “borrow” more money from its policyholders. In the financial world, this would be similar to a company increasing its debt to equity ratio. As this simple example shows, when a company “borrows” more money, modern financial theory says that the cost of equity needs to go up as well, which is what happens in this simple example.
Despite the nice parallels between this result and financial theory, there is one anomaly that merits further discussion: when the strategy changes from the “100%” run to the “125%” run, the leverage ratio drops from 135% to 90%. This occurs because of the major shift in the composition of the hurdle rate benchmark portfolio. As Exhibit 3 shows, the equity allocation in the benchmark increases from 20% to 35%. When we combine this increase with the drop in the leverage ratio, we observe a smooth increase in the adjusted equity allocation as a percentage of
ixthe initial investment. The bond composition is a very different story. For the “100%” run, the benchmark portfolio only contains short corporate bonds. The “125%” run, on the other hand, uses 70% short and 30% medium corporate bonds. As a result, the benchmark for the “125%” run has a much higher duration (i.e. 3.7 versus 2.7 for the “100%” run) which means that it is more volatile. However, by leveraging the benchmark up the in “100%” run and down in the
x“125%” run, the net result ends up being fairly similar distributions of hurdle rates (Exhibit 4), even if there can be substantial differences for certain scenarios (Exhibit 5).
One big advantage of this approach is that the total cost of capital is an increasing function of the amount of business written. While the discussion explains why this is appropriate, it is NOT a feature that the traditional approach shares. Specifically, the traditional approach assumes that the hurdle rate is some known constant, such as from the Capital Asset Pricing Model. Since all of the strategies start with the same amount of capital, the marginal cost of
xicapital has to be zero. Unfortunately, this increasing total cost does not solve all of our problems. While an increasing total cost guarantees a positive marginal cost, Exhibit 6 shows
xiithat the marginal cost is not increasing. Therefore, we will now attempt to incorporate the cost
of financial distress into our model.
Impact of Regulatory Constraints
Another cost that is often overlooked is the cost of financial distress. The direct costs of financial distress (e.g. the costs of liquidation) tend to be relatively small. We will focus on the indirect costs, specifically, the profits that are lost from future business after a company gets into trouble. It is important to note that a company‟s ability to write business can be hampered well
before it actually declares bankruptcy. Insurance companies are evaluated by a large number of third parties: regulators, rating agencies, insurance brokers and policyholders. A negative review by any of these groups can dramatically reduce a company‟s ability to continue writing profitable business. For example, many large companies will only purchase insurance from carriers that are rated „A‟ or better by one of the major rating agencies.
How can we factor these types of costs into our simple example? The most
straightforward way is to come up with a simple formula that relates the amount of business lost to the company‟s level of capitalization. For this paper, we assumed that DFAIC could continue writing business at the Business Growth Strategy level as long as their Market Surplus was at least one third of the prior year‟s net written premium (i.e. a 3.0 maximum premium to surplus ratio). In any scenario where DFAIC became insolvent (Surplus < 0), it would stop writing business. In between these two extremes, DFAIC‟s new business levels would be reduced by the
xiiisquare root of the ratio of actual to required capital. For example, assume that DFAIC ended
year 1 with $640 million in surplus after writing $3 billion in premium that year. In this case,
xivDFAIC would only write 80% of its planned premiums for years 2 and beyond. It is important
to note that these reductions are assumed to be 1. permanent – premiums stay at the lower level
even if DFAIC‟s capital recovers and 2. cumulative – if DFAIC fails a second time, then it
would suffer another premium reduction. It is also worth noting that these changes in business level are handled in the same way as the different business growth strategies: underwriting results are simply scaled to the revised levels.
While this approach vastly oversimplifies the actual dynamics, it offers several useful characteristics. First, it is very easy to expand to more complex restrictions: the surplus to premium constraint could easily be replaced by Risk Based Capital ratios. Second, it focuses on the portion of most business plans that is the most flexible: future business production. While there are a number of other options available (e.g. purchase of reinsurance or issuance of corporate debt), they typically cannot be applied in all situations. For example, it is unlikely that an insolvent company will be able to raise additional capital. Finally, since the underwriting results are only being scaled, we are able to directly determine the cost of these regulatory restrictions. Specifically, the only difference between the runs in this example and those in the previous section is the level of business actually written by scenario.
The most obvious impact of this change is in the level of premium written. As Exhibit 7 shows, there is almost no impact during the first year for the „0%‟ to „150%‟ strategies. At the highest levels, on the other hand, there is a fairly dramatic reduction in DFAIC‟s ability to continue writing business. In fact, the reduction is so severe, that there is very little difference between the average new business levels in year 2 for the „250%‟ and „300%‟ strategies (i.e. 202% vs. 211% of Baseline projections). Not surprisingly, given the cumulative nature of our