CAPITAL MANAGEMENT TECHNOLOGYSM
Oakley E. Van Slyke, Inc.
243 Avenida La Cuesta
San Clemente, CA 92672
Oakley E. Van Slyke
FCAS, ASA, MAAA
Tel: (949) 361-3619
Fax: (949) 361-3660
leevanslyke@lossreserving.com
“The Cost of Risk” in Ratemaking
Ratemaking, or the pricing of insurance products, serves three audiences:
Senior management, regulators, and stakeholders, including consumers and
investors. Each audience has its own goals, its own constituency, and its own
language. The actuary’s role depends on the audience.
Three Audiences
The primary goals of senior management are growth and share price. This is,
of course, an oversimplification, but it gets to the heart of the matter. To
encourage a strong relationship between share price and earnings, senior
management must communicate to investment analysts and rating
organizations using the language of growth and return on equity (ROE). The
analysts, however, will want to know that serious risks are being dealt with
through capitalization or reinsurance. The actuary’s role, then, is to provide
senior management with risk-adjusted ROE’s that reflect enterprise-wide risk.
The primary goal of the regulatory community is consumer protection.
Regulators seek to balance the divergent goals of protection from insolvencies,
protection from excessive prices, and protection from shortages of market
capacity. The regulator's audience includes voters and political leaders such as
governors, leaders of political parties, and powerful state senators and
assembly members. The language of regulation is "excess," "waste," and
"profit." Note that "profit" has a negative tone in the context of regulatory
communications. What the actuary can provide is documentation that the
profit is fair in light of the competitive marketplace. Even when there a less
than a fully competitive market for a particular insurance product, the actuary
can provide documentation that the profit provision has the qualities of a fair
market price. Judith Mintel has provided a chapter on the subject of legal
considerations in rate regulation in the book Actuarial Considerations
Regarding Risk and Return in Property-Casualty Insurance Pricing.
There is a third audience of stakeholders, the investors in common stock and
consumers of insurance products. The stakeholders are represented by senior
management and regulators only by proxy. The goal of the stakeholders is
value creation in the context of risk. Specifically, the holder of shares of
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common stock hopes to see the stock provide a risk-adjusted return better than
that of other investments. The buyer of an insurance policy, whether an
individual or business, anticipates improving the value of the business by
trading a few current dollars for relief from the risk of significant loss. A
special language has been developed to express the interests of the
stakeholders, including “the invisible hand” of prices in competitive markets,
efficient markets, and the law of supply and demand. What the actuary can
provide for stakeholders is estimates of prices that are consistent with
competitive markets.
Two Perspectives
These different languages and audiences have led to two perspectives regarding
the actuarial approach to the determination of risk and return. These are the
"optimal portfolio" approach and the "optimal price" approach. The optimal
portfolio approach considers the marginal effect on the portfolio from adding a
particular bundle of assets or liabilities. The analysis is done in terms of the
efficient frontier of risk and return. That is, one studies the trade-off between
risk and return and recommends ROE figures for which the greatest return can
be offered for a given level of risk. One underlying assumption of the optimal
portfolio perspective is that returns reflect the market's portfolio of similar
underwriting and investment opportunities. The intellectual giants of the
optimal portfolio perspective include Harry Markowitz, Lintner and Sharpe and
the Capital Asset Pricing Model (CAPM), Roll and Ross and Arbitrage Pricing
Theory (APT), and option pricing theory (OPT) based on the formulas of
Black-Scholes. This school of thought is also broadly known as financial
economics.
The optimal price approach considers the marginal effect of a change in price
on the willingness of insurers to underwrite risks and on the willingness of
consumers to buy insurance products. The trade-off is between price and sales
volume rather than between risk and return. An important assumption of the
optimal price approach is that prices reflect essentially all of the information
available at the time of the transaction. Giants of the optimal price approach
include Adam Smith (who published The Wealth of Nations in 1776), Kenneth
Arrow, Cowles, Borch, and Ross and Roll.
These three audiences and two perspectives lead to two quite different
languages for expressing the profit provision. Each language has associated
with it its own style of estimating the profit provision.
Ratemaking using the return-on-equity perspective has three steps. These are:
1) define the product in the corporate environment; 2) use comparative ROE
data to allocate surplus and set ROE; and 3) adjust as needed for the details of
the rate filing. Returns are measured in percentage terms, rather than in
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dollars. Because returns are measured in percentage terms, the return-onequity way of ratemaking is concerned with both the numerator and the
denominator of the calculation of the rate of return.
The Cost-of-Risk Perspective
Ratemaking using the cost-of-risk perspective also has three steps. These are:
1) define risk in a general way and find the market cost of risk; 2) use the
market cost of risk in product pricing in the corporate environment; and 3)
adjust as needed for the details of the rate filing. Price is measured per unit,
rather than in terms of percentages. Therefore, the cost-of-risk way of
ratemaking is concerned with both the units of risk and the price per unit.
The cost-of-risk perspective focuses attention on the supply of and demand for
risk transfer contracts. The underwriter receives risk premium in consideration
for accepting risk. Note that in this perspective the underwriter of a loan, the
underwriter of an insurance policy, and the underwriter of an initial public
offering of common stock are all doing the same thing, receiving risk premium
in consideration for accepting risk transfer. This perspective relies on the
markets to tell us how to find the units to measure risk and the price per unit of
risk.
D
S
# Units
Price per Unit of Risk
A traditional graph of supply and demand helps illustrate this point. As the
price per unit of risk increases, the supply or underwriting capacity increases.
As the price per unit of risk increases, however, the demand by consumers
decreases. Price theory teaches us that the prices that reflect open competition
lead to the optimal volume of risk transfer at the optimal price.
The following formulas (from Van Slyke (1995)) can be used to determine the
cost of risk of any chosen reference risk. This is a set of three simultaneous
non-linear equations. That is, all three equations are true at the time for the
market cost of risk.
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  E  x   P x 
n





t 0
 x 

E  x    p ( j )   v (t | j )   xp ( x| t , j ) dt 

1
n



t 0
 x 

 s 

P x    ln  p ( j ) exp   v (t | j ) ln   p ( x| t , j ) exp  x  dt 
s 1
  


These can be read, “The risk premium, or market price of risk, is the difference
between the expected present value of the cash flows and the market price of
the cash flows in light of the market’s aversion to risk. The market price is
lower (positive values denote cash inflows) because the markets are averse to
risk. The expected present value is the sum, over all possible scenarios, of the
cash flows, each weighted by the appropriate present-value factor at the riskfree rate and the estimated probability. The market price of the cash flows is
the weighted average of the cash flows, with each value adjusted by scaling it
by the risk premium and exponentiating it to give additional weight to adverse
outcomes; at each point when the weighted average has been computed, the
exponentiation and dollar scaling are undone to get the result in dollars.
Results over time within a given scenario are consolidated by summing their
present values using the prevailing risk-free rates of return.”
After the cost of risk is estimated from data from the capital markets, the next
task is to estimate the risk per unit of exposure, just as one estimates the
expected frequency and severity of losses per unit of exposure. The formulas
above can be applied to the cash flows of any ratemaking problem. The
number of units of risk in the insurance product is the total cost of risk for
divided by the cost of risk for the reference risk. For example, if the cost of
risk of the reference risk is $100, and the cost of risk is , then the number of
units of risk is  divided by $100.
Once the risk per unit of exposure has been estimated, it is a simple step to
apply the market cost per unit of risk to determine the cost of risk per unit of
exposure. With this number in hand, the premium rate can be determined
using the pure premium method, the loss ratio method, or any other method
that is actuarially sound.
In practice, of course, these estimates are often approximated by looking at the
prevailing price structure and loss experience. Because the actuary knows that
a risk premium per unit of exposure exists for each insurance product, they can
examine premium and loss data to estimate that risk premium. For ratemaking
problems for which the probabilities are difficult to estimate, this market-based
approach may provide better estimates than the detailed calculations. Just as
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the actuary uses price data from the markets to estimate the costs of employee
salaries and claimant medical care, the actuary can directly use price data from
the markets to determine the cost of risk per unit of exposure.
Portfolio Management
Once the premium rate level has been determined, the question shifts to, "How
can my insurance company optimize its portfolio of assets and liabilities in
light of this competitive premium level?" If the firm charges too much and
sells too little, it will fail to maximize its opportunity, but if the firm charges
too little and sells too much, it will unduly increase its exposure to risk and
diminish its economic value. First, one must determine the marginal risk and
return to the firm from selling the product. Dynamic Financial Analysis (DFA)
is a good tool to examine marginal risk and return from an enterprise
perspective. (Of course, this is not the only way to do this.) Once the marginal
risk and return to the firm are quantified, one sets the volume and price targets
so as to maximize the value of the firm, that is, to put the entire firm at the
efficient frontier of risk and return.
Whichever approach is used to determine the profit provision in the rates, the
result must be communicated to management in language that senior
management can use. If the profit provision has been determined using the
optimal portfolio perspective, one merely allocates surplus and selects an ROE
target. In such a calculation, the premium rate is set to provide a return on
allocated surplus at every interval that is consistent with the risks faced by that
surplus.
If the profit provision has been determined using the cost-of-risk approach, the
volume and price targets that have been established can be used to develop a
relationship between allocated surplus and ROE. Specifically, if one knows
the total profit provision in the rates and the cash flows associated with those
losses and expenses, one can find a surplus allocation and ROE that reflects
alternative risk-return opportunities in the capital markets.
Rate Filings
Communicating to regulators is somewhat different, of course. If the regulator
will accept discounted cash flow calculations prepared for management, those
calculations can merely be included in the rate filing. More often, the ROE
methodology required for a rate filing is somewhat different from the
company’s ROE methodology. Therefore, some adjustments are made to get to
the regulator’s desired calculations.
On the other hand, if the regulators will accept profit provisions prepared for
the company’s ratemaking calculations, then it is easy to express the cost of
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risk in the filing. This is usually presented as 1) a cost per unit of exposure, 2)
a provision for profit contingencies in the premium rate, or 3) a risk loading
applied to the losses. When required by regulators, the cost of risk can also be
expressed in terms of a discounted cash flow calculation.
A third way of communicating to regulators used to be much more fashionable.
For many lines of business for many decades, reasonable rate regulation was
achieved merely by insisting on conservative assumptions. There is a reason
that this generally works: The less confidence one has in one’s “best”
estimates, the greater the risk; also, the less confidence one has in one’s “best”
estimates, the greater will be the risk premium arising from conservative
assumptions. For example, for many years the “long tail” lines of business
(those characterized by delays between payment of premium and payment of
claims) were valued in rate filings without any discount for the time value
money. Whatever justification was advanced on theoretical grounds, it was not
that money had no cost. However, ever since the risk-free rate of return
exceeded 10 percent in the years around 1980, the use of undiscounted
numbers to take the place of numbers that reflected the time value money and a
reasonable provision for risk has gone out of favor. (The risks appear to be
greater when interest rates are low, and the time value money appears to be
greater when interest rates are unusually high.) Under current accounting
concepts, conservative assumptions are somewhat frowned upon. In addition,
the CAS’s Principles of Ratemaking suggest that members of the CAS are to
explicitly measure the uncertainty and the cost of risk.
Conclusion
The most brilliant minds in economics have applied their talents to the optimal
management of portfolios and the optimal prices in a competitive economy.
Actuaries, senior management, regulators, stockholders and consumers all
benefit when actuaries understand the implications of their work.
Communications to senior management and communications to regulators
through rate filings will always involve somewhat different approaches. These
approaches can and should be reconciled, however, if the stockholders and
customers are to be served.
Lee Van Slyke, FCAS, ASA, MAAA
March 8, 2001
For Further Reading
Borch, K.(1962), "Equilibrium in a Reinsurance Market," Econometrica, 30,
p.424-444.
Buhlmann, H. and Jewell, W. S. (1979), "Optimal Risk Exchanges," ASTIN
Bulletin, 10, p. 243-262.
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Davis, J. L. (1994), "The Cross-Section of Realized Stock Returns: The PreCOMPUSTAT Evidence," Journal of Finance, Vol. 49, No. 5, Dec. 1994, p.
1579-1593.
Daykin, C. D. and Hey, G. B. (1990) "Managing Uncertainty in a General
Insurance Company," Journal of the Institute of Actuaries, Volume 117, part II,
No. 467 (September 1990), pages 173-277.
Drucker, P. F., (1989) The New Realities, Harper Business, New York.
Drucker, P. F., (1993), Post-Capitalist Society, Harper Business, New York.
Freifelder, L. R. (1976), A Decision Theoretic Approach to Insurance
Ratemaking, Huebner Foundation Monograph No. 4, Wharton School,
University of Pennsylvania.
Gerber, H. U. (1979), An Introduction to Mathematical Risk Theory, Wharton
School, University of Pennsylvania.
Lakonishok, J., Shleifer, A., and Vishny, R. W. (1994), "Contrarian
Investment, Extrapolation, and Risk," Journal of Finance, Vol. 49, No. 5, Dec.
1994, p. 1541-1578.
Lintner, John (1965), “The Valuation of Risky Assets and the Selection of
Risky Investments in Stock Portfolios and at Capital Budgets,” Review of
Economics and Statistics, February 1965
Markowitz, Harry (1952), “Portfolio Selection”, Journal of Finance, v.7, pages
77-91
Pratt, John W. (1964), “Risk Aversion in the Small and in the Large”,
Econometrica, v. 32, no. 1-2, pages 122-136.
Roll, R., and Ross, S. A. (1994), "On the Cross-Sectional Relation between
Expected Returns and Betas," Journal of Finance, Vol. 49, No. 1, p. 101.
Ross, S. A. (1976), "The Arbitrage Theory of Capital Asset Pricing", Journal of
Economic Theory, Vol. 13, p 341-360.
Ross, S. A. (1994), "Survivorship in Finance", Record of the Society of
Actuaries, Vol. 20, No. 1, p.1.
Sharpe, William F. (1964), “Capital Asset Prices: A Theory of Market
Equilibrium under Conditions of Risk,” Journal of Finance, v. 19, pages 425442
Taylor, G. C., "Optimal Reinsurance Structures," XXI ASTIN Colloquium,
1989.
Van Slyke, O. E. (1980), “Is Econometric Modeling Obsolete?," Casualty
Actuarial Society discussion paper, May 1980.
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CAPITAL MANAGEMENT TECHNOLOGYSM
Van Slyke, O. E. (1983), discussion of "REROSHE: The Concept of a RiskFree Equivalent Return on Shareholder's Equity," by Alastair Longley-Cook,
Transactions of the Society of Actuaries XXXV, p. 336.
Van Slyke, O. E. (1985), review of "Reinsuring the Captive/Specialty
Company" by Lee R. Steeneck, Proceedings of the Casualty Actuarial Society,
p. 175.
Van Slyke, O. E. (1987a), “The Casualty Actuary's Role in Risk Management,”
CAS Forum, Fall 1987.
Van Slyke, O. E. (1987b), “Regulatory Standards for Reserves,” Casualty
Actuarial Society discussion paper, 1987.
Van Slyke, O. E. (1991a), “Solvency Standards,” Second AFIR Conference,
April 1991.
Van Slyke, O. E. (1991b), “The Tricky Business of Risk-Adjusted Valuations,”
Contingencies, July/August 1991.
Van Slyke, O. E. (1991c), “Valuation of a Business Based on Its Cash Flow,”
Society of Actuaries, 1991-1992 Reports.
Van Slyke, O. E. (1995a), “A General Theory of Finance,” Fifth AFIR
Conference.
Van Slyke, O. E. (1995b), “A General Framework for Financial Decisions,”
Proceedings of the 29th Actuarial Research Conference, Actuarial Research
Clearing House, vol. 1995.1
Van Slyke, O. E. (1996), “Interpreting Model Output: The California
Earthquake Authority and the Cost of Capital of the Reinsurance Layer,” with
Giuseppe Russo, Ph. D., Casualty Actuarial Society Forum, Spring 1996.
Van Slyke, O. E. (1999a), Ed., Actuarial Considerations Regarding Risk and
Return in Property-Casualty Insurance Pricing, Casualty Actuarial Society and
Actuarial Education and Research Foundation
Van Slyke, O. E. (1999b), “The Cost of Capital: An Axiomatic Approach,”
Actuarial Considerations Regarding Risk and Return in Property-Casualty
Insurance Pricing, Casualty Actuarial Society and Actuarial Education and
Research Foundation, Chapter 11.
Venter, G. G. (1991), "Premium Calculation Implications of Reinsurance
without Arbitrage," ASTIN Bulletin, Vol. 21, No. 2, November 1991
Willet, A. H. (1901) The Economic Theory of Risk and Insurance, republished
by the Huebner Foundation, University of Pennsylvania Press, 1951.
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