# Calculating confidence: stochastic pricing methods provide more complete information about life settlement risks and returns.

Life settlements can provide policyholders with an amount of cash greater than the surrender value for their policy at a time when they need it. Settlements also can provide the purchaser of the policy with a sound business transaction. How sound that transaction is, however, depends strongly on getting the price of the settlement right.

A life settlement involves purchase by a third party of life insurance policies that might otherwise be lapsed, cash surrendered or continued in force. The range of potential life settlement prices is limited on the low side by policy cash values and on the high side by return on investment expectations of funding entities. Life settlements are not available for most life policies because the market values of the policies, net of acquisition costs, are less than cash values.

Despite the critical nature of price levels to the key parties in settlement transactions, the most commonly used methods for calculating life settlement prices are remarkably unrefined. A sample case described below illustrates the range of calculated settlement prices produced by the three methods employed by life settlement providers. The two most common methods, deterministic and probabilistic, produce single-answer results with no contextual information. The third method, stochastic, provides considerably more pricing information and is explored at greater length.

The range of pricing results using the three methods is dramatic. For the sample case, the same basic inputs produce potential prices that vary depending on the method by as much as 40% of their midpoint.

The fundamental difference between the three life settlement pricing methods is the way they incorporate mortality and underwriting information. Deterministic pricing is the least sophisticated, assuming policy termination and collection of death benefits at the insured's life expectancy. Probabilistic pricing formulas incorporate mortality rates rather than life expectancies and so recognize that death may occur at any future policy duration. A probabilistic settlement price is based on the mortality-weighted cash flow approach used in life insurance pricing and reserving and represents an aggregate of all potential outcomes.

Stochastic pricing, on the other hand, uses Monte Carlo techniques to repeatedly simulate insured lifetimes from which a distribution of policy outcomes is created. A stochastic settlement price is derived from the distribution and may represent a mean, mode, percentile value or some other selected result. A key value of stochastic results is information about the variability of potential outcomes. Though the additional information available may complicate individual policy settlement pricing, it represents decision-related pricing information for both individual policies and portfolios of policies that is not readily derived using other methods.

The Particular Policyholder

In our sample case, the representative insured is an 80-year-old woman determined to be four times as likely to die in each future year as a standard nonsmoking female insured according to the 2001 Valuation Basic Table. The in-force policy is a representative fixed universal life policy maturing at age 95 with a \$1 million level specified face amount and minimum annual premiums after the pricing date beginning at about 7% of face amount. The policy is assumed to have no account value or outstanding policy loan on the pricing date. The representative insured is in considerably poorer health than an individual underwritten as a standard risk, but is not terminally ill. Calculated settlement amounts are gross prices before expenses, commissions, taxes or other reductions.

Before examining the methods in more detail, it is important to note the pivotal role of cash values in evaluating life settlement offers. Issuing insurance companies are currently bound by regulation, contract provision and issues of policy-owner equity to pay policy cash values on surrendering policies that do not vary with the insured's health status. For insureds in good health at the time surrender is contemplated, policy cash values may, in tact, exceed market values and insurance companies effectively overpay for such policies. For insureds in poor health at the time of surrender, policy cash values may be less than market values, making life settlements possible.

Different Settlement Prices

Pricing results in the sample case using deterministic, probabilistic and stochastic methods vary widely:

* Deterministic price--\$121,700;

* Probabilistic price--\$152,200;

* Median stochastic price--\$105,000; and

* Mean stochastic price--\$159,200.

The deterministic settlement price is based on a calculated life expectancy tar the insured of 71 months. The corresponding price is about \$121,700, assuming future premiums and death proceeds are discounted at a 15% annual rate. The discount rate is a risk-adjusted rate of return representative of rates demanded by life settlement buyers. As a percentage of face amount, this price is near the low end of the typical range for life settlements, due in large part to a premium rate near the high end of the typical range.

The probabilistic settlement price, again at a 15% discount rate, is about \$152,200.This is larger than the deterministic price because it incorporates the possibility that death will occur prior to the life expectancy, requiring payment of fewer premiums and providing the death benefit at an earlier duration. Of course, the probabilistic price also incorporates the possibility that the insured will survive beyond the life expectancy, but due to the pattern of expected mortality and the substantial discount rate, such outcomes have less financial impact on the settlement price than early deaths.

The stochastic settlement prices are derived from a distribution of pricing results. These results are based on 1,000 projections of the sample insured's lifetime using a Monte Carlo method and the same mortality rates used to calculate the life expectancy for deterministic pricing and the present values of premiums and benefits for probabilistic pricing.

Many Possibilities

The stochastic process produces a very broad distribution of potential lifetimes as shown in "Lifetime Possibilities" (page 102). Increasing the number of trials would smooth the distribution, but the shape of the distribution would remain essentially the same. Although the deterministic life expectancy of 71 months seems reasonable by inspection, the extremes of the distribution involve material numbers of occurrences. For example, two of the projected lifetimes end in the first month, 78 end in the first year, and nine extend beyond age 95, the maturity age of the policy. The distribution's mean lifetime is 75 months and its median is 73 months.

"Lifetime Possibilities" can also be thought of as showing the distribution of lifetimes that might occur in a portfolio of life settled cases where all of the insureds are identical to the illustrated insured. Since each projected lifetime produces a unique set of future cash flows, the distribution of lifetimes translates into a distribution of present values of cash flows or settlement prices.

The mean stochastic price for this distribution using a 15% discount rate is about \$159,200, an amount slightly higher than the deterministic and probabilistic prices. The median price, however, is only \$105,000.This disparity is the consequence of a relatively large number of negative and small positive stochastically generated prices that pull down the median, and a small number of very large positive prices that pull up the mean.

The relationship between mean and median prices may be counterintuitive given that the mean life expectancy is longer than the median and therefore expected to be associated with a lower price. The statistical measures for life expectancy and price were calculated independently, however, and their relationships are a consequence of the underlying distributions. The stochastically generated distribution of settlement prices is broad and not peaked around the deterministic or probabilistic prices.

In other words, there are many other prices with approximately the same likelihood of occurrence; many of them negative. The broad range of stochastic prices is the consequence of the broad range of likely lifetimes and the fact that the present value of cash flows decreases rapidly as lifetime until death increases.

Two effects drive this rapid decrease. First, the amount of money invested in a policy steadily increases as premiums are paid during the insured's lifetime. Second, the present value of the death benefit decreases as the insured's lifetime lengthens. Perhaps surprisingly, near the deterministic and stochastic life expectancies, calculated settlement prices decrease at a rate of \$8,000 to \$9,000 per additional month of assumed lifetime. Projected lifetimes in excess of 87 months, just one year greater than the stochastic mean, produce negative calculated settlement prices.

In addition to use in individual policy pricing or pricing analysis, these life settlement pricing methods can be adapted for use with portfolios of life settled policies. Such portfolio models facilitate a variety of analyses, including projection of aggregate expected cash flows, calculation of a portfolio value, reserve development and analysis of reinsurance arrangements.

While the deterministic method provides a simple pricing approach, portfolio modeling using deterministic cash flows is fundamentally flawed. As an example, a portfolio of the sample insureds described above would be projected to produce 71 months of negative cash flow from premium payments followed by a large positive flow from death proceeds. A mixed portfolio of individuals and policies would produce a more varied, but still unrealistic, projection of cash flows.

Probabilistic and stochastic method portfolio projections would produce comparable future cash flows, with stochastic results converging on probabilistic results as the size of the portfolio increased.

Rates of Return

An alternate stochastic financial result is the distribution of potential annual rates of return given a fixed settlement price. Assuming the settlement price is the probabilistic price of about \$152,200, internal rates of return can be calculated from the cash flows for each future lifetime. The model distribution of rates of return is shown in "Possible Rates of Return" (page 102).

The result is an extremely skewed distribution. Most of the outcomes are concentrated between -10% and the 15% rote used in calculating the probabilistic price. The remainder extend in a comparatively sparse distribution from 15% upward. The largest returns are for the 78 cases that terminate in the first year with a \$1 million return on an investment equal to the settlement of \$152,200 plus one year's policy premium.

At the other extreme, nine policies mature without paying a death benefit and thereby produce a loss of the entire settlement amount and premiums paid. These nine policies have a rote of return of -100%, which is such an extreme outliner it is not shown on the graph.

Annual rates of return for selected statistical measures are revealing:

* 25th percentile--2.81%;

* 50th percentile (median)--12.09%;

* 75th percentile--36.27%;and

* Mean--62.68%.

It is significant that the 12.09% median return is materially less than the 15% discount rate used to calculate the deterministic price on which these calculations are based. Put another way, for the 1,000 projected lifetimes underlying these results, more often than not the probabilistic price produced a rate of return lower than the discount rate used in its calculation. The mean rote of return has little practical value, influenced as it is by a small number of astronomical returns on early deaths.

Developing Confidence

Regardless of the pricing method used, settlement offers for individual policies are highly dependent upon evaluation of insureds' health, whether in terms of life expectancy or in relation to standard mortality, and estimation of monthly policy cost. The consequences of inexact estimates of either may be significant and can be quantified by scenario testing.

A life settlement provider must develop confidence in its underwriting function. The underwriting performed for a life settlement case is based on available medical information, which does not include a medical examination ordered for the purpose. Variability in the longer life expectancies involved in life settlements produces material uncertainties in prices and cash flows. Experience analysis is essential to improving underwriting results.

The deterministic method was historically the prevalent pricing approach. The deterministic price for a single policy, however, is deficient in that it is an evaluation of the settlement price function at a single point, and deterministic portfolio analysis produces unrealistic results. The probabilistic approach takes advantage of mortality-weighted cash flows, allowing a provider to evaluate the price effects of underwriting uncertainty by scaling up or down the mortality rates underlying a price calculation. Along with stochastic models, probabilistic portfolio models are credible tools for portfolio analysis.

Stochastic simulation, however, clearly provides the most complete information about the expected pattern, amount and variability of life settlement cash flows. The stochastic results calculated for this sample insured are not strongly concentrated around their mean values. The presence of such skewed and fat-tailed distributions reinforces the value of stochastic simulation, allowing providers and investors to better quantify the risks and potential returns associated with individual settlement prices and to project cash flows for portfolios of settled policies.

Key Points

* Correct pricing is critical to life settlement transactions.

* The two most common methods for calculating life settlement prices are remarkably unrefined.

* A third method, stochastic simulation, provides considerably more pricing information, including the expected pattern, amount and variability of life settlement cash flows.

David Cook and Karen Rudolph are consulting actuaries in the Omaha, Neb., office of Milliman.
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