## Friday, November 07, 2008

### Catastrophe bonds and the investor's choice problem

Consider the following proposition. You put up an amount of capital X for one year. There is a small probability p (e.g., p = .01) that you will lose the entire amount. With probability (1-p) you get the entire amount back. What interest rate (fee) should you charge to participate?

What I've just described is a catastrophe bond. A catastrophe bond allows an insurer to transfer the tail risk from a natural disaster (hurricane, earthquake, fire, etc.) to an investor who is paid appropriately. How can we decide the appropriate fee for taking on this risk? It's an example of the fundamental investor's choice problem. That is, what is the value of a gamble specified by a given probability distribution over a set of payoffs? (Which of two distributions do you prefer?) One would think that the answer depends on individual risk preferences or utility functions.

Our colloquium speaker last week was John Seo of Fermat Capital, a hedge fund that trades catastrophe bonds. Actually, John pioneered the business at Lehman Brothers before starting Fermat. He's yet another deep thinking physicist who ended up in finance. Indeed, he claims to have made some fundamental progress on the investor's choice problem. His approach involves a kind of discounting in probability space, as opposed to the now familiar discounting of cash flows in time. I won't discuss the details further, since they are slightly proprietary.

I can discuss aspects of the cat bond market. Apparently the global insurance industry cannot self-insure against 1 in 100 year risks. That is, disasters which have occurred historically with that frequency are capable of taking down the whole industry (e.g., huge earthquakes in Japan or California). Therefore, it is sensible for insurers to sell some of that risk. Who wants to buy a cat bond? Well, pension funds, which manage the largest pools of capital on the planet, are always on the lookout for sources of return whose risks are uncorrelated with those of stocks, bonds and other existing financial instruments. Portfolio theory suggests that a pension fund should put a few percent of its capital into cat bonds, and that's how John has raised the \$2 billion he currently has under management. The market answer to the question I posed in the first paragraph is roughly LIBOR plus (4-6) times the expected loss. For a once in a century disaster, this return is LIBOR plus (4-6) percent or so. Sounds like a good trade for the pension fund as long as the event risk is realistically evaluated.

Note there is no leverage or counterparty risk in these transactions. An independent vehicle is created which holds the capital X, invested in AAA securities (no CDOs, please :-). If the conditions of the contract are triggered, this entity turns the capital over to the insurance company. Otherwise, the assets are returned at the end of the term.

In the colloquium, John reviewed the origins of present value analysis, going back to Fibonacci, Fermat and Pascal. See Mark Thoma, who also attended, for more discussion.

Anonymous said...

Professor Hsu

it seems to me that the quants and the Investment Banksters failed to adequately understand the risk they intended to model in that they computed probabilities that an SIV or a tranch or a discrete number of securities would fail.

this is analogous to an insurance company figuring the odds that an individual policy holder may die or have their house burn down in any given year.

but the tail risk...the black swan event that they failed to consider was one where the ENTIRE SYSTEM FAILS.

how can an insurance company compute risk and issue life or home owner insurance policies that hedge risk when an invading army sweeps across the entire landscape burning and killing all in its path?

the financial system would not be teetering on the brink of failure if banks, mortgage companies, etc held the mortgages that they issued, and remained in a position to connect an individual default with a particular security issued.

but they can not and so the entire system risks destruction just as one toppled domino can bring down a thousand others if they are in a chain.

because of the way the system is built and connected, the risk was never .01...it was either zero or 100%

Steve Hsu said...

The current financial crisis was not a black swan event, at least in the sense that many people thought it was a serious possibility (including me).

The frequency of natural disasters (e.g., earthquakes) is a bit easier to predict from historical data because the underlying system is probably not changing. (Hurricanes might be a particular problem due to global warming.)

Financial crises involve humans, and consequently are far less predictable.

milkshake said...

I would feel personaly more comfortable making bets on markets rather than trying to calculate odds and severinity of droughts, pandemics, poisonous algal blooms or volcanic eruptions.

Historical record alone is not a sound basis for risk analysis when the underlying mechanism is poorly understood. NASA made once made a curve-fitting study on speed of O-ring seal charring in solid rocket boosters depending on temperature and pressure - and from this and the historic data record concluded it was safe to keep flying with the faulty design "because seals were likely to burn at most only one third of their thickness through".

Anonymous said...

I am afraid there is most definitely counterparty risk:
"Lehman Brothers was one such counterparty. And, as we all now know it declared Chapter 11 bankruptcy on September 15, 2008. It was swap counterparty to four ILS – Ajax, Carillon A-1, Newton Re 2008 A-1 and Willow Re B 2007. Immediately the swaps were called into question. The rating agencies placed the four on watch on September 15th and downgraded all four on September 30th."
This from the webpage of Lane Financial,the best source for info on these bonds

Steve Hsu said...

Nice catch. They did have some trouble with Lehman bonds, but I was speaking theoretically in the post -- one should stick to real AAA securities in collateralizing a cat bond!

Carl said...

We may be able to estimate the probability of a large earthquake in California or Japan, but the extent of resultant economic losses remains uncertain. The efficacy of architectural countermeasures, the ability of authorities to prevent massive fires from raging out of control, and similar factors add major uncertainty.

Steve Hsu said...

I should make clear that the bond pays off a fixed amount based on a legal definition of the event (e.g., magnitude 7 earthquake in Tokyo region), the damage calculation (and therefore how many bonds to sell) is done by the insurer.

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