Can the Black-Scholes Model Be Used for Captive Insurance Premiums?

Editor's Note: In the following piece, Pinnacle Actuarial Resources' Derek Freihaut and Tim Mosler share their thoughts on the viability of applying a specific mathematical model, Black-Scholes, in estimating the maximum premium appropriate for a captive insurance company. The Black-Scholes model is considered one of the best tools for fair options pricing. But should it be used to estimate captive insurance premiums?

A Pinnacle client recently inquired about a paper regarding an approach to estimating the maximum premium appropriate for a captive insurance company. The author sought to determine a ceiling on the premium a company might pay for insurance by treating insurance as a put option and applying a popular financial tool, the Black-Scholes model.

One generally thinks of put options in terms of stocks and financial assets. Applied to a stock, a put option is a contract that gives its holder the right to sell that stock at an agreed price on or before a specified future date. Think of an asset manager invested in a stock whose price skyrocketed during the year. The manager is on track to post his or her highest-ever annual gains. The problem is that it's only October. The manager doesn't want to sell the stock but is worried about a price reversal prior to year-end. He or she may, for the price of a put option, lock in the stock's gains. Once holding the put option, any decreases in the stock's value can be offset dollar-for-dollar by increases in the ultimate value of the put option. The put option can be thought of as insurance on the stock's gains.

The Black-Scholes model is one method for pricing options. The model is mathematically complex, widely used in financial circles, and is even included on the Casualty Actuarial Society (CAS) exam syllabus. However, like any model, Black-Scholes also has its share of detractors and tends to struggle most when applied to extreme situations. More important to this discussion is whether it is appropriate to use this model in determining a firm's maximum insurance premium.

The paper we reviewed sought to establish a benchmark for the maximum ratio of insurance premium to revenue that a company might reasonably be charged. The author used the Black-Scholes model to price a put option on the firm. Yet using Black-Scholes to price a put option on a company has little to do with what a company could reasonably pay for insurance.

Consider a man purchasing luggage for his new job because it requires more frequent travel. He asks the salesperson, "How much luggage do I need?" She responds by asking him, "How big is your house?" She then proceeds to inventory all of his possessions and calculate exactly how much luggage he would need to move all of his belongings.

This approach obviously has several issues, as follows.

• A traveler usually packs a limited number of items for a trip.
• Not all of the traveler's household members go on every trip, so there is no need to pack everyone's belongings.
• Most importantly, if a traveler needed to pack up all his or her possessions for a trip, it's probably a trip that he or she is better off not taking.

There are analogous issues with using a put-option estimate for insurance premiums, as follows.

• There are many large risks that cannot be insured, such as
  • a new competitor with more efficient processes,
  • declining demand for products and services, and
  • not leveraging new technologies (would Kodak still dominate the photography industry if it had purchased better insurance?).
• There are some risks that are insurable but that a business will instead choose to retain, possibly due to cost or availability constraints.
• If the cost of insurance is an excessive portion of its revenue, a business may cease to be viable.
• Finally, if the owners of a business are seeking to insure every conceivable risk, they might be better off not owning a business in the first place. Taking calculated risks is generally necessary to make a profit.

As with the luggage assessment from the salesperson in our example, the put-option estimate is an unreasonably high upper bound. It contains far more risk than any company could or would reasonably be able to insure, so this type of estimate adds no value in determining a reasonable premium level.     

The paper our client brought to our attention featured an example concluding that the hypothetical company could plausibly pay in excess of 75 percent of a given year's revenue in insurance premium. This result is similar to our luggage salesperson learning that her customer has a four-bedroom home and recommending that he purchase 47 large suitcases in addition to a 26-foot moving truck. Most people would find that a company paying 75 percent of its revenue in insurance premium makes about as much sense as buying a moving truck to go on vacation. 

The Black-Scholes formula, being designed to consider all risks, will produce unreasonable results when used to establish insurance premiums. It is simply not a reliable model for estimating premiums.