**Latest Research**

**by Louise A. Francis **

The Risk Premium Project (RPP) is a team of researchers organized in response to a request for proposal distributed by the Committee on the Theory of Risk to support research on valuing property-liability risks. Members of the team are Robert Butsic,** **David Cummins**, **Richard Derrig** **and** **Richard Phillips. The first two phases of the project encompassed a thorough review of and synthesis of the literature with particular attention to literature on profit loads, risk loads, and risk-adjusted discount rates. In the last phase of the project original research was undertaken on one specific approach used to value by-line property-liability exposures. The research focused on full information and sumbetas (sum of betas from the current and one lagged return), valuation approaches based on recently improved methods for estimating the Capital Asset Pricing Model (CAPM) equity betas. Note that numerous other methods are in common use by actuaries and no one approach was viewed superior to the others when awarding the research project. Some of the other methods of valuation are based on ruin theory, utility theory, and transformed distributions (some of the methods used to compute risk/profit loads are found in the white paper on "Fair Value of Liabilities,").

CAPM may be familiar to those involved in rate filings, as it is often one of the key financial theories used in the regulation of insurance companies to determine a "fair rate of return." However, the new methods of CAPM estimation used in this study may not be widely known to actuaries. The use of CAPM is controversial among actuaries, as it has been used in the past to "prove" that insurance companies are exposed only to undiversifiable underwriting risk and therefore merit little or no return above that supplied by the risk-free rate of return. Usually the "proof" involves demonstrating that insurance industry underwriting profit betas are low, or in some cases negative. In his paper, "Underwriting Betas-The Shadow of Ghosts" (*PCAS* 1994), **Thomas Kozik** gives a good discussion of the shortcomings encountered in practice when CAPM is used to compute risk/profit loads in insurance. Under some idealized financial assumptions, including frictionless markets, CAPM states that

^{~}r_{c} = r_{f }+ ß_{c}(^{~}r_{M }-_{ }^{~}r_{f})

where

^{~}r_{c} is the company's expected return

^{~}r_{f } is the risk-free return for the chosen investment horizon

^{~}r_{M} is the expected return on the entire market of all investments

*ß*_{c } is the company's Beta, based on its covariation with market returns

(^{~}r_{M }-_{ }^{~}r_{f}) is the market risk premium

Instead of ^{~}r_{c}, a single company's return, the return for an industry, such as the property and casualty insurance industry, ^{~}r_{pc} can be computed from the full information CAPM model. In capital budgeting and regulatory applications, the result of a CAPM calculation is sometimes used as a cost of equity capital (the return required by shareholders in order to induce them to invest in the company), which then becomes an input for other calculations such as an internal rate of return pricing methodology or a risk adjusted discount rate. It should be noted that as traditionally applied, the simple one-factor CAPM incorporates a return only for systematic risk (that due to covariance of an entity's returns with the market). That covariance is measured by beta. Thus, from the simple CAPM perspective, beta is the only relevant factor determining a company's required return on equity, and therefore its cost of equity.

The Risk Premium Project team incorporated a number of the most recent research developments into a model based on CAPM but with refinements to address some of the limitations of CAPM. One of these refinements is the Fama-French three-factor model. In 1992 two leading researchers in academic finance, Eugene Fama and Kenneth French, created a storm in the finance community when they published a paper suggesting that beta was not the only relevant factor for predicting a company's return. The authors' research indicated that two factors other than beta were significant in explaining returns. These two factors are a size factor (smaller companies tend to have higher returns) and a distress factor (companies with high book-to-market ratios tend to have higher returns). The three-factor Fama-French model incorporates these two additional factors. The Risk Premium Project's results indicate that use of the three-factor model provides clarification of the single factor results and produces higher costs of equity estimates for property-casualty insurance companies.

The three-factor Fama-French model is one of the better known of recent developments in the CAPM literature. However, other important developments were identified and used by the Risk Premium Project. One of these is the sumbeta approach developed at Ibbotson Associates. They found that the returns on smaller stocks are influenced by less frequent trading and low information flow and as such are related not only to the current return on the market but to past returns on the market. Using sumbeta tends to raisethe overall cost of equity for smaller companies, explaining some of the size effect.

A third development allows analysts to compute line of business betas. Previously, if one wanted to compute a beta for a line of business, say workers compensation, one had to assemble a collection of stock companies that write primarily (and preferably only) workers compensation. Needless to say, few such companies exist and for most lines of business no companies exist. In fact, one of the challenges of estimating a beta for the property/casualty industry is that most companies that sell property/casualty insurance also sell other products, i.e., they are not a pure play on the property/casualty industry. The "full-information" method addresses this issue. The method combines accounting data about the amount of business a company writes in various lines of business (as a proxy for allocated project equity) with financial market return data to derive line-specific betas. Costs of capital were computed using straight averages and market value weighted averages. The value-weighted average can be interpreted as the result for the entire industry, while the straight average is the result for the average insurer. The RPP results based on market-weighted averages indicated that

- the cost of capital was higher for short-tail lines than long tail-lines;
- the cost of capital was higher for commercial lines;
- the cost of capital was higher for workers compensation insurance than automobile insurance, but the difference was not statistically significant;
- the Fama-French estimates show the significant influence of size on the by-line weighted average results.

The first result is unexpected, as long-tailed lines are generally considered to be riskier than short-tailed lines. The authors speculate that the companies that write long-tailed lines may have a "natural hedge" against interest rate risk as their runoff liabilities tend to move in the same direction as interest rates, possibly leading to lower costs of capital. Also, short-tailed lines are more susceptible to catastrophe losses such as hurricanes and earthquakes. It should also be noted that this research represents a first attempt to evaluate differential costs of capital by line and results may have been influenced by the specific time period (1996-2000) of the data.

The team's research has been incorporated into an academic paper "Estimating the Cost of Equity Capital for Property-Liability Insurers," which has been submitted to the *Journal of Risk and Insurance*. Results from the team's research will also be presented at the Ratemaking Seminar in March in Philadelphia and the paper is published in the 2004 Winter *Forum*.