http://www.soa.org/library/newsletters/ ... 9903.pdfAn
Actuary Looks at Financial Insurance By Richard Q. Wendt With the development of new variations of equity indexed annuities and complex pension benefit design, new forms of return guarantees have been created in both insurance and pension. These guarantees create a form of "financial insurance" for investors and plan participants, as the expected returns are protected against market downturns. Today, both insurance and pension actuaries are determining the cost of return guarantees and appropriate investment strategies to support those guarantees. Some actuaries may be surprised to find that the conditions necessary for the use of traditional actuarial analysis may not exist in financial insurance and that the costs developed by traditional actuarial analysis may significantly understate the "true" cost. This article discusses the differences between traditional actuarial analysis of financial insurance and an option based analysis. In 1995, Professor Zvi Bodie published his controversial article, "On the Risk of Stocks in the Long Run" in the Financial Analysts Journal. That article showed that the price of a put option on the S&P 500, with a strike price based on the risk-free rate of return, would increase with time horizon. That, Professor Bodie argued, indicates that stocks are indeed risky in the long run. Since his conclusion was contrary to the popular view of the long-term benefits of stock investment, his article generated quite a few responses. Bodie's article showed that the price of a 20-year put option, under stated assumptions, would be 34.5 percent of the initial investment. Several commentators (using the traditional analysis) stated that a premium of that level could not possibly be correct, since the probability of a twenty-year shortfall was given as only 4 percent. Option pricing concepts tend to puzzle and perplex actuaries and other analysts who are familiar with the more traditional insurance or probability-based approach. Thinking of a put option as a form of financial insurance, the traditional actuary would likely attack the problem of pricing a put option by determining the distribution of future results, applying an interest discount factor and then summing the probability-weighted present values of all outcomes. It often comes as a shock that the financial insurance premium is quite different than the option price calculated by the well-known Black-Scholes option formula. Many analysts are aware that the Black-Scholes option price is primarily determined by the volatility of the security, and know, but may not fully understand why, that the expected return on the security is not an element of the formula. On the other hand, the traditional insurance methodology primarily references the expected return, while the volatility of the results is not directly referenced. The two methodologies are quite different, so it should not be surprising that they may produce different results. First, some background on the Black-Scholes option pricing model. The B-S model makes the following key assumptions (in simplified form): ? The underlying security is continuously traded and can be bought or sold, in any amount. ? Cash can be borrowed or loaned at the risk-free rate, without limit. ? There are no taxes or transactions costs. ? Security prices are given by geometric Brownian motion. ? There is no arbitrage. Next, consider the key assumptions underlying an ideal insurance transaction: ? Large number of individual insured event ? Independence of insured events, in one year and over time. ? Predictable risk There are two key differences in the underlying assumptions, risk and a related issue, independence. Risk?the option transaction is priced to be arbitrage free, in the sense that an investor selling an option can, on the basis of the B-S model's stated assumptions, make continuous adjusting transactions in the underlying security to provide a guaranteed risk-free result. On the other hand, the financial insurance transaction is not completely risk free. To expand on the issue of risk in the financial insurance arena, recall that in Professor Bodie's model the risk of stocks performing below the risk-free rate is about 4 percent. One commentator suggested that, even if all value were assumed to be lost when there is underperformance, the cost of insurance would still be less than 4 percent (compared to the put option premium of 34.5 percent). Assume that an investor puts $100 in the stock market and also spends $4 for financial insurance. ? The financial insurance provides a benefit only if the 20-year stock return is less than the risk free rate. ? If the 20-year risk-free spot rate is assumed to be 6 percent, then the guaranteed amount would be $100 x 1.0620, which is $320.71. ? If the $4 insurance premium were to be invested at the risk-free rate, then it would accumulate to $12.83. How can $12.83 support a potential loss of $320.71? Only if, say, 100 investors buy the insurance and no more than four receive benefits. But if all 100 investors are investing in the same stock market at the same time, then either they all receive a large benefit or all receive no benefit. Given a stochastic future, the insurer would have a profit 96 percent of the time, but the other 4 percent of occurrences would be financial disasters. Whenever performance is poor and benefits are payable, the insurer would have only $13 of assets and a required payment of $321. Assuming no other resources for the insurer, there would be a 4 percent probability of bankruptcy. In fact, the risk-free insurance premium (i.e., the premium required to be 100 percent confident of covering all claims) would be 100 percent of the potential exposure (ignoring interest discount factors). The wonder of the B-S option pricing formula is that the true risk-free cost is much less than 100 percent. Some analysts might suggest that the insurance premium could be reduced by investing the premium in higher return equities. That would not improve the financial results, since the accumulated premium would itself be valueless in the 4 percent of scenarios where benefits were required. Independence?Suppose the insurer sells financial insurance policies every year, with each year's policies having a four percent expected loss. What premium would the insurer need to charge in order to assure solvency? Consider this simple example: ? 1000 policies sold each year for 25 years. ? The investment made under each policy is $1000 for a total of $1,000,000 in each year. ? The probability of loss for each policy after 20 years is four percent. ? The premium charged is four percent of the investment or a total of $40,000 per year. At the end of the first 20 years, the determination of potential claims for the very first purchasers is made. If there are claims (four percent probability), then $1,000,000 will be paid out, but total premiums only amount to $800,000 over 20 years of sales. Assuming no additional resources, the insurer can make only a partial payment to the initial purchasers, but is now insolvent and cannot provide any benefits to purchasers from years two through 20. It would seem clear that a four percent premium would be insufficient to provide reasonable assurances of solvency. But what level of premium would be required for a higher degree of security? Assuming that each issue year's results are independent for 25 years of issue, an eight percent premium would be required to cover the 92 percent probability of zero to two payout years in a 25-year period (assuming a binomial distribution). There would still be a remaining 8 percent chance of three or more payout years. For 98 percent certainty of covering payouts (up to three payout years), a 12 percent premium would be required. Notice that, even though thousands of investors are buying financial insurance, there are only 25 different insurable events, not 25,000. The traditional insurance requirement of large numbers of independent events does not hold. Furthermore, the payout years for this financial insurance would not actually be independent, since the investment periods overlap. For example, the admittedly extreme result of 100 percent loss in year 20 would wipe out investments for all purchasers in the first 20 years and trigger 20 years of payouts. In other words, $800,000 would be available to pay claims of $20,000,000 (ignoring the risk-free accumulation of both premiums and claims). In order to evaluate the impact of overlapping periods, the Towers Perrin CAP:Link economic simulation model generated 500 stochastic scenarios covering 25 years of sales. That analysis indicated that a premium of 16 percent would be required for 90 percent certainty and a premium of 32 percent would be required for 98 percent certainty of covering all losses. When we adjust the financial insurance premium for the risk of insolvency, it is easy to see that an expected loss of four percent can be translated to a premium that approaches, or even exceeds, the Black-Scholes risk-free cost of 34.5 percent. This analysis shows that the traditional actuarial analysis of financial insurance may significantly understate the risk-free cost of the insurance and that more advanced techniques are needed to properly price the cost of guaranteed returns for equity-linked products or pension plans. Richard Q. Wendt, FSA, CFA is Principal at Towers Perrin in Philadelphia, PA. ________________________________________ References 1. Bodie, Zvi. 1995. "On the Risk of Stocks in the Long Run." Financial Analysts Journal, vol. 51, no. 3 (May/June):18-22. 2. Dempsey, Mike, Robert Hudson, Kevin Littler and Kevin Keasey. 1996. "On the Risk of Stocks in the Long Run: A Resolution to the Debate?" Financial Analysts 3. Journal, vol. 52, no. 5 (September/October):57-62. 4. Campbell, John Y., Andrew W. Lo and A. Craig MacKinlay. 1997. The Econometrics of Financial Markets. Princeton, NJ: Princeton University Press. 5. Ferguson, Robert and Dean Leistikow. 1996. "On the Risk of Stocks in the Long Run: A Comment." Financial Analysts Journal, vol. 52, no. 2 (March/April):67-68. 6. Financial Analysts Journal. 1996. "Letters to the Editor" from George M. Cohen, P. de Fontenay, Gordon L. Gould, and (response) Zvi Bodie, vol. 52, no. 2 (March/April):72-76. 7. Mulvey, John M. 1996. "Generating Scenarios for the Towers Perrin Investment System." Interfaces, (March/April):1ff. 8. Mitchell, G. Thomas and John Slater, Jr. 1996. "Equity-Indexed Annuities?New Territory on the Efficient Frontier." Product Development News. Issue 39 (January):1ff. 9. Ruark, T. J. 1996. "Variable Annuities: What Goes Up, Must Come Down." Contingencies, (March/April):35-37. 10. Siegel, Jeremy J. 1994. Stocks for the Long Run: A Guide to Selecting Markets for Long-Term Growth; Burr Ridge, IL: Irwin 11. Taylor, Richard and Donald J. Brown. 1996. "On the Risk of Stocks in the Long Run: A Note." Financial Analysts Journal, vol. 52, no. 2 (March/April):69-71.