Our two posts on capital allocation, comparing the "common practice" of VAR with the Merton and Jarrow put option approach, have prompted an interesting dialogue. "I read the posts," said R in New York, "and I need you to help me understand why actual bank losses were so much larger than their VAR figures at the time. I thought the VAR numbers were supposed to overstate the true capital needs as measured by the put." This post explains the differences between black swans, bad VAR, good VAR, and better puts!
Our blog post on May 13 on www.kamakuraco.com explained that using "true" VAR for capital allocation is implicitly assuming that the organization must self-insure its capital needs with, say, 99% capability. A put option or insurance policy that insures the firm against the same loss profile is the probability weighted present value of all scenarios. Given identical loss projections, this normally means that the "true" VAR will be much higher than the put price (unless the losses are the same in all scenarios) because the put is probability weighted and the VAR is not. Then what about this quote from www.bloomberg.com on January 28, 2008?: "Merrill's highest one-day value at risk in the third quarter was $92 million, indicating that the firm's maximum expected cost during the 63-trading day period would be $5.8 billion. In fact, the firm wrote down $8.4 billion from the value of collateralized debt obligations, subprime mortgages and leveraged finance commitments, 45 percent more than the worst- case scenario." How could that happen? Shouldn't VAR have been less than the cost of insurance, and shouldn't the actual losses been less than VAR?
The answer, in this case, is no. What was reported in the Bloomberg story was "false" VAR, a gross underestimate of risk that has a host of problems that we listed in the April 6, 2009 blog on www.kamakuraco.com, "Is Your Value at Risk from Value-at-Risk? Beware..." The Bloomberg story goes on to state that most of the large U.S. securities firms were using "false" VAR based on 1-4 years of historical data. Given the history of U.S. home prices, a 4 year historical "false" VAR would have predicted minimal losses (as it did) because over the 4 year period ending December 31, 2007, home prices were strongly up. Even though home prices peaked in Los Angeles in September 2006, they did not begin to have a powerful impact on mortgage defaults until the second half of 2007. "False" VAR is based on an average of history. "True" VAR, which is what we assumed in our May 13 post, is a complete and accurate listing of all possible future outcomes and their probabilities, Prof. Kenneth Arrow's famous "state space." If we base "true" VAR on that complete listing of outcomes and probabilities, the put price or cost of insuring risks via a third party than can diversify will almost always be less than the high percentile (say 99th) "true" VAR number. That "true" VAR number is the amount of self-insurance you need to survive with 99 percent probability.
Wasn't the miss with respect to VAR just a "Black Swan" as Nassim Taleb argued? No, it was just bad math. It was an assumption that the world was flat. As we explain in our April 6 post, historical VAR contains an implicit assumption that neither Lehman nor Bear Stearns could fail, because (based on historical equity returns) the implied probability of -100% stock returns within a month for each of them was 0.000000%. Similarly, if the monthly changes in home prices have been strongly positive over the measurement period, the historical VAR approach will implicitly assume away the probability of a decline. The "true" VAR calculation and a rationally priced put will consider all possible future scenarios, not just those that have actually come about in the last 12 or 48 months.
Many thanks, R. Have a great weekend. Hana hou!
Donald R. van Deventer
Kamakura Corporation
Honolulu, May 15, 2009