A worked example shows the differences very clearly. Let’s assume that there are only 100 scenarios that can impact our firm ABC Company. In best practice net present value terms, in 93 scenarios there is no value destruction and capital is unchanged. In seven scenarios, numbered 94 to 100, we have these losses:
We can then easily determine that the average loss over all 100 scenarios is 0.70. The 99th percentile VAR is 17, and the 95th percentile VAR is 3. We need only $3 of capital to survive 95% of the time, but we need $17 to survive 99% of the time. This is simple enough. How does the put option approach compare?
Let’s make it very clear that when we are talking about a put option, we could equivalently call it an insurance policy that reimburses us for any value destruction in our current portfolio. Puts come at various strike prices and insurance policies come with deductibles and caps. The table below shows the payments that an insurance company would make to us in each of the loss scenarios for four policies: three policies where the losses are capped (at 5, 10 and 20) and one policy with no caps:
How much would each of these put options or insurance policies cost? Since our losses are stated in net present value terms, in a competitive market with perfect competition the put/policy should be priced at its expected value over the 100 scenarios. For the policy capped at a maximum loss of 5, the cost is 0.28. For the 10 dollar cap, the cost is 0.44. For the 20 loss cap, the cost is 0.63. For a policy that reimburses us for all losses, the cost of the put is exactly the expected loss on the portfolio: 0.70. How much capital would be needed if we were comfortable with insurance only up to 20 in losses? We issue additional capital of 0.63, and we use the 0.63 to buy the insurance policy.
Why are the VAR and Put Approaches So Different: Self Insurance versus Third Party Insurance
How do we explain the fact that the 99th percentile VAR capital needed is 17 and the put option with a cap at 20, where the company also loses money in only 1 of the 100 scenarios, costs only 0.63, not 17? The answer is EXACTLY the same as the case of our May 11 post on risk management strategy for individual investors. If you have a 1% chance of living to 100, and you want to be able to spend $75,000 in “year 100” dollars, how much money do you need to save? Is the answer 1% times $75,000, a total of $750? Or is it $75,000? If you are handling all of your savings yourself, you are self-insuring—you need to have $75,000 saved in order to live as you wish in the 1 out of 100 scenario that you live to be 100. If you only saved $750 and you live to be 100, you’ll starve to death in the first month after your birthday.
The VAR capital calculation is analogous to saving $75,000—the VAR capital calculation assumes that you are “self insuring” the company against losses. Even though 17 or more in losses happens in only 2 of the 100 scenarios, you issue 17 in capital, which turns out to be unnecessary 98 times out of 100.
If you are planning to live to 100, you have another alternative to managing your own investments—you can buy an annuity from an insurance company which can diversify over 100 other people. A contract which pays you $75,000 if you live to 100 will cost only 1% times $75,000 (in year 100 dollars), or $750. The capital analogy is a direct one—the put premium or insurance cost of protecting your portfolio of losses up to $20 reflects not only the amount of the losses, but also their probability of occurrence. The insurance company can diversify, and a competitive market place will ensure that the benefits of diversification are passed along to the clients of the insurance company. In a nutshell, the VAR approach assumes you are self insured and can’t diversify. The put option approach assumes you buy protection from a third party who is diversified. It’s all very simple.
Other Considerations
When it comes to capital allocation using the put option concept, we don’t even have to make the calculation for bank holding companies that are listed on stock exchanges in the United States. Put options on their common stock will be publically observable, and, for any “deductible” (strike price), there will be a different amount of capital required for that level of protection.
Another approach to protect one’s capital has tremendous moral hazard for your counterparty—you can buy credit default swaps on yourself. Rumor has it that AIG was “sophisticated” enough to be willing to do this…But what if Nassim Taleb is right and the CDS market is the equivalent of buying marine insurance from another passenger on the Titanic? The put/insurance policy can still be valued and purchased. We do a “credit risk cat scan” that looks through all the assets and liabilities of the firm and all of the counterparties to see how they are affected by key macro factors like interest rates, home prices, foreign exchange rates, and stock indices. This is exactly what we describe in an earlier post on “reduced reduced form” models. We then buy the appropriate insurance protection on these macro factors. Almost all of them have exchange traded futures and (in some cases) options, so we don’t have a Titanic problem.
In short, the difference between a VAR approach to capital needs and a put option or insurance approach to capital needs is the difference between self-insurance or insurance in an efficient market that recognizes diversification. In most cases, the latter assumption will be a much more accurate indicator of how much capital protection is necessary.
Donald R. van Deventer
Kamakura Corporation
Honolulu, May 13, 2009