### Buffett's Derivative Loss

Reading the famous Berkshire's annual report, I was striken by Buffett's duality view on derivatives:

Buffett's "WMD" view:

"Derivatives are dangerous. They have dramatically increased the leverage and risks in our financial system. ... They allowed Fannie Mae and Freddie Mac to engage in massive misstatements of earnings for years. So indecipherable were Freddie and Fannie that their federal regulator, OFHEO, whose more than 100 employees had no job except the oversight of these two institutions, totally missed their cooking of the books."

Then here is Buffett's "I-Know-Better" view:

"Considering the ruin I’ve pictured, you may wonder why Berkshire is a party to 251 derivatives contracts... The answer is simple: I believe each contract we own was mispriced at inception, sometimes dramatically so. I both initiated these positions and monitor them, a set of responsibilities consistent with my belief that the CEO of any large financial organization must be the Chief Risk Officer as well. If we lose money on our derivatives, it will be my fault."

We can safely assuming "sometimes dramatically so" means the mispricing is dramatically in his favor in his belief (otherwise there is no point of initiating the trades). Furthermore, his OTC derivative deals, at least for the ones we know and all the index puts I talked about, are ALL selling risks. That means he believes his counterparty dramatically OVERPRICE risks, so all he does is SELLING the protection to them.

What an amazing contradicting view! On one hand, he believes derivatives is too dangerous (and others underestimate the risk) -- for this, he only talks about it; on the other hand, he believes others dramatically overprice risk -- for this, he does a lot about it -- by selling protection (and selling only)!

Details About Buffett's Short Index Puts

Coming to the index puts options I discussed in my last posting. Here is the exact quote from the Annual Report:

"Our put contracts total $37.1 billion (at current exchange rates) and are spread among four major indices: the S&P 500 in the U.S., the FTSE 100 in the U.K., the Euro Stoxx 50 in Europe, and the Nikkei 225 in Japan. Our first contract comes due on September 9, 2019 and our last on January 24, 2028. We have received premiums of $4.9 billion, money we have invested. We, meanwhile, have paid nothing, since all expiration dates are far in the future. Nonetheless, we have used Black-Scholes valuation methods to record a yearend liability of $10 billion, an amount that will change on every reporting date. The two financial items – this estimated loss of $10 billion minus the $4.9 billion in premiums we have received – means that we have so far reported a mark-to-market loss of $5.1 billion from these contracts."

In my last estimate using SPX (since only until this report Berkshire revealed the detail 4 indices in these put contracts), the notional to premium ratio estimate is too high if all puts are assumed to be done on SPX only: for 15-Year SPX put with strike at spot, 4% rate, the ratio is 14.5 at 20 vol, 6.57 at 30 vol. With a mixture of SPX (low vol) and Nikkei/FTSE/Stoxx50 (higher vol), Berkshire's total notional/premium = 37.1/4.9 = 7.57 is right in the middle, indicating my original estimate of his selling SPX around 20 vol/4% rate was pretty right on the target. SPX 15 year 1300 put (strike at spot) at 20 vol/4% rate was valued at about $90. The other 3 indices were probably sold at vol a few point higher than 20, and rate around 4%. (Nikkei might be sold at much lower rate.)

His "mark-to-market" loss of 10 billion loss is not too far from my estimate of 12 billion (if vol mark at 35 and no change in rate). But the key point in my previous analysis -- the sensitive dependence of rate in these 15 year deals, and what's Buffett's assumption in his mark-to-market calculation (when the rate market in last 2 years goes hugely against these positions) is still unanswered in this annual report.

The sensitivity of these 15 year contracts for Rho to Vega at 25 vol/3% rate is about 5, which means 1% in rate change has the same effect as 5 vol point. Interestingly, Rho/Vega ratio increases dramatically when the vol increase: at 35 vol/3% rate, it becomes 6.5, which means in current market condition, it becomes even more sensitive to rate.

Mark-To-Market: Is It Necessary?

Reading Buffett's report, you can't help noticing that he repeatedly indicate his reservation for mark-to-market methodology. Or put it in another way, he follows mark-to-market with a disdain. Is that justified? For these put contracts, since they cannot be early exercised, who care how much loss they would be marked to in the interim, as long as on the settlement day, the index is not below the strike.

That sounds like a very plausible argument against mark-to-market methodology. As someone lives by the market, I believe mark-to-market is the fairest way for everyone. For Buffett's OTC deals, let's look in details two realistic scenarios that would support this methodology as the fairest for all market players:

1. If Buffett would have sold these puts in December 2008 rather than in 2006, he would have fetched 10-12 billion more than his original 4.9 billion, so just as in any market, selling at 4.9 billion for something worth 10-12 billion more is a loss of 10-12 billion, when the liability is still open.

2. From the counterparty's point of view, the buyer of these contracts can hedge and lock in part of these mark-to-market profits. Here is how: 1) he can sell new contracts with the same or earlier expiration dates as the original contracts, at the current market volatility and rate: for example, he can sell a 13-year puts (at 35 vol/4% rate, or 28vol/3% rate) with 1300 strike (while SP500 at 900) for 15 billion or more; 2) He can even just go to exchange traded options and hedge. Since these options are much shorter in lifetime than Buffett's 15 year contracts, this hedging process can be done several times.

The Shocking Number

Here is the most shocking fact: On December 31, 2008 (as Buffett close his 2008 book), SPX Dec2011 put (which you can readily traded on CBOE) was quoted as 459.8 bid/470.4 ask (SPX close at 903.25). That's 5 times of the original put value Buffett would had sold -- in another word, Buffett's counterparty can readily lock-in 15 billion profits with a shorter term option. Even more, when the Dec 2011 option expires, he still has Buffett's contract in place for another 10 years, for free, and he can continue to sell puts to hedge and add more profits! Ouch. Who is "dramatically" mispricing? Is the entire market totally stupid to trade Dec2011 1300 put for about $460, while Buffett was willing to sell a 3 or 4 time longer period puts for 1/5 of the price? Buffett's own estimate of 10 billion loss, or my fixed-rate/35 vol loss of 12 billion estimate are well too low. My other estimate of 18 billion loss with 35vol/2.5% rate is more close to reality.

Thus the mark-to-market P/L is very real and can be realized if you choose to.

So please, Mr Buffett, will you reveal the rate assumption you use for the mark-to-market calculation for these contracts?

"4.9 Billion That Is Free to Invest"

Another point Buffett kept making in his report is his repeated mentions that the premium or cash he received for selling these puts (4.9 billion), he is free to invest. Such concept is nothing new -- in insurance business, that's call "float". The question is, did Buffett getting a different (or better) deal in these OTC puts, than an investor selling 10 contracts of SPY put and collect $10,000? Does the Black-Scholes formula misses such "Free-to-Invest" capability of the cash received from the premium?

The answer is: there is no difference between Buffett and the 10-contract put seller, except on the funding condition the investor would get from his brokerage account might be different from Berkshire would get. For an investor who is treated decently by his broker (translated: he has a big bank account, or he is a professional investor), any cash from short sales (options or stocks) are entitled to interest income -- typically linked to Fed Open rate. Namely, this cash from option short sell works just like insurance "float" invested in CD. Any cash you put up as collateral for this short sale should also receive interests. (Unfortunately, for many small investors, they never realize this, and the brokers simply take advantage of them). There is nothing wrong or new in this part in the option valuation: it also assuming the premium is freely investable, with a rate as part of the inputs. Buffett's fixation on treating option premium as "float" doesn't mean he discovers anything new (or missed) in the option valuation. It is already built-in in Black-Scholes, or any other Arbitrage-Free based methodologies.

On the accounting side, there is some difference though: Buffett uses the standard used by the insurance industry -- the premium of the put options he sold was counted as "income" immediately, namely, on the date of the contracts are signed, he would have a positive P/L of 4.9 billion, the subsequent P/L would just be the interest income from the premium, and then a realized loss at expiration if the index is below strike. The mark-to-market writedown in-between is immaterial to him. For the investor, mark-to-market standard is used: on the day he sold the puts for $10000, his P/L is about 0 (asset of 10000 and liability of 10000), and the $10000 premium, the interest income as well as the eventual settlement loss (if any) is realized slowly through daily mark-to-market process.

Buffett's 100-Year Put Example

At the end of the report, Buffett went on to give a detail explanation of an example to show Black-Scholes formula's "irrational" results (or "dramatic mispricing"): a 100-year put option on SP500 with strike at spot of 903 (This, I guess, is used as a rationale to justify these now famous long-dated short put contracts). He argued that a notional to premium ratio of 400 (1 billion for 2.5 million premium) is well too high (so he would sell the premium), and he felt from fundamental argument (SP profit growth, etc) that the chance of losing money is so low that he would be gladly sell the put and pocket the 2.5 million.

To get this 400 ratio, there are limitless combination of volatility and rate, one set is 26 vol/4.5% rate, another is 20 vol/3.75%.

With such a long dated option, sensitivity on the interest rate is much more than on the volatility. Rho/Vega ratio is 9 (at 26 vol/4.5% rate) - or 10 bps rate change is equivalent to 0.9 vol point change. Thus selling this put is equivalent to longing rate (or betting on inflation?) than selling vol (while delta is negligible). Buffett's argument, which is totally fixed upon the "fundamental", or eventually SP 500 is below or above the strike - or equivalently looking at the delta only -- seems to totally miss the most part of the equation: in the long run, it is the interest rate that determines the outcome. Black-Scholes is totally capable (or equally flawed) here, just as in other shorter dated -- the problem is not in the formula, but with the user of the formula -- if you assume 100 years of fixed interest rate (or fixed volatility), so you will have an irrational results -- as irrational as the assumption itself.

As a comparison, Rho/Vega ratio for the typical exchange-traded options are small, even for the farthest ones (Jan11), it is only around 2.

In Buffett's view, 2.5 million for 1 billion notional (400 to 1) is too expensive. This actually is nothing to do with Black-Scholes at all. In any trading with small probability (or option with tiny premium), any price is too expensive if you believe it will never happen, and the guy who buys it from you must be very dumb. For evaluating small probability event, whatever formula can appear to be irrational. If 400 to 1 is too expensive, what about 500 to 1? or 1000 to 1 (or $1 million for $1 billion notional)?

Still, It's Just Selling Tiny Puts

The 100-year put in Buffett's example, though with strike at spot, has only 0.2 delta, or equivalent to selling tons of Google (at 340) 1 year 100 put for 20c. Similar "fundamental" argument can be made here - based on Google's earnings etc, it is "inconceivable" that Google will be dropped to 100, so go for it, sell as much puts as you can! By the way, as in early March, GOOG Jan10 110P can be sold for $2.25 (at 75 vol due to skew, while using at-the-money vol to evaluate this put it worth only 20c -- By the way, Buffett priced these tiny delta index puts with at-the-money vol, not even considered any skew effect, which would push the vol even more dramatically higher), so if we use Buffett's argument, the market has "dramatically" mispriced (overpriced) the risk. He does not need to go so far to OTC derivatives but just to exchange-traded options for tons of such "mispriced" deals.

(The author can be contacted at huangxinw@gmail.com.)

Buffett's "WMD" view:

"Derivatives are dangerous. They have dramatically increased the leverage and risks in our financial system. ... They allowed Fannie Mae and Freddie Mac to engage in massive misstatements of earnings for years. So indecipherable were Freddie and Fannie that their federal regulator, OFHEO, whose more than 100 employees had no job except the oversight of these two institutions, totally missed their cooking of the books."

Then here is Buffett's "I-Know-Better" view:

"Considering the ruin I’ve pictured, you may wonder why Berkshire is a party to 251 derivatives contracts... The answer is simple: I believe each contract we own was mispriced at inception, sometimes dramatically so. I both initiated these positions and monitor them, a set of responsibilities consistent with my belief that the CEO of any large financial organization must be the Chief Risk Officer as well. If we lose money on our derivatives, it will be my fault."

We can safely assuming "sometimes dramatically so" means the mispricing is dramatically in his favor in his belief (otherwise there is no point of initiating the trades). Furthermore, his OTC derivative deals, at least for the ones we know and all the index puts I talked about, are ALL selling risks. That means he believes his counterparty dramatically OVERPRICE risks, so all he does is SELLING the protection to them.

What an amazing contradicting view! On one hand, he believes derivatives is too dangerous (and others underestimate the risk) -- for this, he only talks about it; on the other hand, he believes others dramatically overprice risk -- for this, he does a lot about it -- by selling protection (and selling only)!

Details About Buffett's Short Index Puts

Coming to the index puts options I discussed in my last posting. Here is the exact quote from the Annual Report:

"Our put contracts total $37.1 billion (at current exchange rates) and are spread among four major indices: the S&P 500 in the U.S., the FTSE 100 in the U.K., the Euro Stoxx 50 in Europe, and the Nikkei 225 in Japan. Our first contract comes due on September 9, 2019 and our last on January 24, 2028. We have received premiums of $4.9 billion, money we have invested. We, meanwhile, have paid nothing, since all expiration dates are far in the future. Nonetheless, we have used Black-Scholes valuation methods to record a yearend liability of $10 billion, an amount that will change on every reporting date. The two financial items – this estimated loss of $10 billion minus the $4.9 billion in premiums we have received – means that we have so far reported a mark-to-market loss of $5.1 billion from these contracts."

In my last estimate using SPX (since only until this report Berkshire revealed the detail 4 indices in these put contracts), the notional to premium ratio estimate is too high if all puts are assumed to be done on SPX only: for 15-Year SPX put with strike at spot, 4% rate, the ratio is 14.5 at 20 vol, 6.57 at 30 vol. With a mixture of SPX (low vol) and Nikkei/FTSE/Stoxx50 (higher vol), Berkshire's total notional/premium = 37.1/4.9 = 7.57 is right in the middle, indicating my original estimate of his selling SPX around 20 vol/4% rate was pretty right on the target. SPX 15 year 1300 put (strike at spot) at 20 vol/4% rate was valued at about $90. The other 3 indices were probably sold at vol a few point higher than 20, and rate around 4%. (Nikkei might be sold at much lower rate.)

His "mark-to-market" loss of 10 billion loss is not too far from my estimate of 12 billion (if vol mark at 35 and no change in rate). But the key point in my previous analysis -- the sensitive dependence of rate in these 15 year deals, and what's Buffett's assumption in his mark-to-market calculation (when the rate market in last 2 years goes hugely against these positions) is still unanswered in this annual report.

The sensitivity of these 15 year contracts for Rho to Vega at 25 vol/3% rate is about 5, which means 1% in rate change has the same effect as 5 vol point. Interestingly, Rho/Vega ratio increases dramatically when the vol increase: at 35 vol/3% rate, it becomes 6.5, which means in current market condition, it becomes even more sensitive to rate.

Mark-To-Market: Is It Necessary?

Reading Buffett's report, you can't help noticing that he repeatedly indicate his reservation for mark-to-market methodology. Or put it in another way, he follows mark-to-market with a disdain. Is that justified? For these put contracts, since they cannot be early exercised, who care how much loss they would be marked to in the interim, as long as on the settlement day, the index is not below the strike.

That sounds like a very plausible argument against mark-to-market methodology. As someone lives by the market, I believe mark-to-market is the fairest way for everyone. For Buffett's OTC deals, let's look in details two realistic scenarios that would support this methodology as the fairest for all market players:

1. If Buffett would have sold these puts in December 2008 rather than in 2006, he would have fetched 10-12 billion more than his original 4.9 billion, so just as in any market, selling at 4.9 billion for something worth 10-12 billion more is a loss of 10-12 billion, when the liability is still open.

2. From the counterparty's point of view, the buyer of these contracts can hedge and lock in part of these mark-to-market profits. Here is how: 1) he can sell new contracts with the same or earlier expiration dates as the original contracts, at the current market volatility and rate: for example, he can sell a 13-year puts (at 35 vol/4% rate, or 28vol/3% rate) with 1300 strike (while SP500 at 900) for 15 billion or more; 2) He can even just go to exchange traded options and hedge. Since these options are much shorter in lifetime than Buffett's 15 year contracts, this hedging process can be done several times.

The Shocking Number

Here is the most shocking fact: On December 31, 2008 (as Buffett close his 2008 book), SPX Dec2011 put (which you can readily traded on CBOE) was quoted as 459.8 bid/470.4 ask (SPX close at 903.25). That's 5 times of the original put value Buffett would had sold -- in another word, Buffett's counterparty can readily lock-in 15 billion profits with a shorter term option. Even more, when the Dec 2011 option expires, he still has Buffett's contract in place for another 10 years, for free, and he can continue to sell puts to hedge and add more profits! Ouch. Who is "dramatically" mispricing? Is the entire market totally stupid to trade Dec2011 1300 put for about $460, while Buffett was willing to sell a 3 or 4 time longer period puts for 1/5 of the price? Buffett's own estimate of 10 billion loss, or my fixed-rate/35 vol loss of 12 billion estimate are well too low. My other estimate of 18 billion loss with 35vol/2.5% rate is more close to reality.

Thus the mark-to-market P/L is very real and can be realized if you choose to.

So please, Mr Buffett, will you reveal the rate assumption you use for the mark-to-market calculation for these contracts?

"4.9 Billion That Is Free to Invest"

Another point Buffett kept making in his report is his repeated mentions that the premium or cash he received for selling these puts (4.9 billion), he is free to invest. Such concept is nothing new -- in insurance business, that's call "float". The question is, did Buffett getting a different (or better) deal in these OTC puts, than an investor selling 10 contracts of SPY put and collect $10,000? Does the Black-Scholes formula misses such "Free-to-Invest" capability of the cash received from the premium?

The answer is: there is no difference between Buffett and the 10-contract put seller, except on the funding condition the investor would get from his brokerage account might be different from Berkshire would get. For an investor who is treated decently by his broker (translated: he has a big bank account, or he is a professional investor), any cash from short sales (options or stocks) are entitled to interest income -- typically linked to Fed Open rate. Namely, this cash from option short sell works just like insurance "float" invested in CD. Any cash you put up as collateral for this short sale should also receive interests. (Unfortunately, for many small investors, they never realize this, and the brokers simply take advantage of them). There is nothing wrong or new in this part in the option valuation: it also assuming the premium is freely investable, with a rate as part of the inputs. Buffett's fixation on treating option premium as "float" doesn't mean he discovers anything new (or missed) in the option valuation. It is already built-in in Black-Scholes, or any other Arbitrage-Free based methodologies.

On the accounting side, there is some difference though: Buffett uses the standard used by the insurance industry -- the premium of the put options he sold was counted as "income" immediately, namely, on the date of the contracts are signed, he would have a positive P/L of 4.9 billion, the subsequent P/L would just be the interest income from the premium, and then a realized loss at expiration if the index is below strike. The mark-to-market writedown in-between is immaterial to him. For the investor, mark-to-market standard is used: on the day he sold the puts for $10000, his P/L is about 0 (asset of 10000 and liability of 10000), and the $10000 premium, the interest income as well as the eventual settlement loss (if any) is realized slowly through daily mark-to-market process.

Buffett's 100-Year Put Example

At the end of the report, Buffett went on to give a detail explanation of an example to show Black-Scholes formula's "irrational" results (or "dramatic mispricing"): a 100-year put option on SP500 with strike at spot of 903 (This, I guess, is used as a rationale to justify these now famous long-dated short put contracts). He argued that a notional to premium ratio of 400 (1 billion for 2.5 million premium) is well too high (so he would sell the premium), and he felt from fundamental argument (SP profit growth, etc) that the chance of losing money is so low that he would be gladly sell the put and pocket the 2.5 million.

To get this 400 ratio, there are limitless combination of volatility and rate, one set is 26 vol/4.5% rate, another is 20 vol/3.75%.

With such a long dated option, sensitivity on the interest rate is much more than on the volatility. Rho/Vega ratio is 9 (at 26 vol/4.5% rate) - or 10 bps rate change is equivalent to 0.9 vol point change. Thus selling this put is equivalent to longing rate (or betting on inflation?) than selling vol (while delta is negligible). Buffett's argument, which is totally fixed upon the "fundamental", or eventually SP 500 is below or above the strike - or equivalently looking at the delta only -- seems to totally miss the most part of the equation: in the long run, it is the interest rate that determines the outcome. Black-Scholes is totally capable (or equally flawed) here, just as in other shorter dated -- the problem is not in the formula, but with the user of the formula -- if you assume 100 years of fixed interest rate (or fixed volatility), so you will have an irrational results -- as irrational as the assumption itself.

As a comparison, Rho/Vega ratio for the typical exchange-traded options are small, even for the farthest ones (Jan11), it is only around 2.

In Buffett's view, 2.5 million for 1 billion notional (400 to 1) is too expensive. This actually is nothing to do with Black-Scholes at all. In any trading with small probability (or option with tiny premium), any price is too expensive if you believe it will never happen, and the guy who buys it from you must be very dumb. For evaluating small probability event, whatever formula can appear to be irrational. If 400 to 1 is too expensive, what about 500 to 1? or 1000 to 1 (or $1 million for $1 billion notional)?

Still, It's Just Selling Tiny Puts

The 100-year put in Buffett's example, though with strike at spot, has only 0.2 delta, or equivalent to selling tons of Google (at 340) 1 year 100 put for 20c. Similar "fundamental" argument can be made here - based on Google's earnings etc, it is "inconceivable" that Google will be dropped to 100, so go for it, sell as much puts as you can! By the way, as in early March, GOOG Jan10 110P can be sold for $2.25 (at 75 vol due to skew, while using at-the-money vol to evaluate this put it worth only 20c -- By the way, Buffett priced these tiny delta index puts with at-the-money vol, not even considered any skew effect, which would push the vol even more dramatically higher), so if we use Buffett's argument, the market has "dramatically" mispriced (overpriced) the risk. He does not need to go so far to OTC derivatives but just to exchange-traded options for tons of such "mispriced" deals.

(The author can be contacted at huangxinw@gmail.com.)