## How did Lehman manage that ?

Robert Waldmann

A commenter over at naked capitalism notes that

$110b of senior LEH debt went from trading .95 to .12 in a matter of days …. If you include the less senior debt that is trading at essentially zero, LEH had $110b hole in its balance sheet. And just days before this, the market was being told and was believing that the $10b disposition of Neuberger was going to solve their funding problems.

Now is there a precedent in this history of bankruptcy–excluding cases of accounting fraud–where bonds collapsed like this once a bankruptcy court opened up the books? I’m thinking the answer is ‘no.’ Which then makes you re-evaluate the premise that there wasn’t fraud at LEH in marking the value of their assets.

Now I may be crazy, but I think that the idea that accounting fraud is required to achieve this is old fashioned and out of date. I think it can be achieved without breaking the laws, such as they are.

Lehman could have made their senior debt worth 12 cents on a dollar in case of default by selling CD insurance on their own debt — lots of it. This would not require any false accounting as they are not required to report this fact.

Now I would be reluctant to believe that a bank could insure its own debt if it hadn’t already happened .

The trader then went on to tell me that Commercial Bank of Korea would sell credit default protection on bonds issued by the Commercial Bank of Korea.

Who would buy a CDS on Lehman from Lehman ? Only a fool ? Well I have another candidate — someone who had bought lots of cash settlement CDSs on Lehman debt from a third party. The payout on a CDS depends on par value minus settlement value. A huge amount of Lehman insurance of Lehman is not very useful to someone who wants to hedge, but it is very useful for someone who wants Lehmen to settle for as few cents on the dollar as possible because, he or she has bought Lehman CDSs from a third party.

Now to Lehman, insuring their own debt is a very very attractive proposition. It is money for nothing unless they go bankrupt and if they are bankrupt well they are bankrupt. The whole source of moral hazard and adverse selection in credit markets is that it doesn’t matter to the debtor how much he goes below zero.

A positive price for Lehman insurance of Lehman makes sense (algebra will be after the jump when I type it up). There was money to be made at the expense of holders of Lehman debt who didn’t think of the possibility of over-insurance.

Is this what happened ? I have no idea, but I guess we will find out fairly soon.

update: My claim about reporting requirements has been contested. I should stress *again* that I am not an expert and add that I know jack about accounting standards.

I have learned a lot from comments here and at crooked timber. I reply at tiresome length in comments at crooked timber and at my home blog.

As far as I can tell firms must report the total fair market value of CDS written as liabilities, but this is not what matters to bond holders. To them liabilities matter only to the extent that they cause bankruptcy and/or affect the value of liabilities or assets in case of bankruptcy. Knowing the expected value of a liability which is worth zero the 99.9 % of the time in which bond holders just get interest and principal and a whole lot the 0.1% of the time in which Lehman brothers is liquidated is of little use to bondholders. Also, as John Quiggin notes, accounts are not published continuously and our latest information on Lehman Brothers appears to date from May 31 2008 which was a while ago.

First a CDS on LEH issued by LEH is definitely not worthless. It can’t possible pay out as described in the terms of the contract, because it only pays when Lehman defaults, but it can pay its stated value times the cents on a dollar payout ratio found by a bankruptcy court. Even in the case of LEH,this will be positive.

Now algebra. I will assume all debt is equally senior. The par value of LEH debt is 1 (for simplicity). They go under with assets equal to y (which must be less than 1 for them to be bankrupt). the payout ratio phi is equal to assets over total liabilities. However the liabilities are not just debt. They include self CD insurance for x units of par which, in theory shold pay x(1-phi). Actual payment on the CDSs is .

phi is given by

1) phi = y/(1+x(1-phi))

Note that phi is not zero, so the CDS has positive value — you don’t have to be crazy to buy it.

This gives a quadratic equation with one solution to phi between 0 and 1

2) phi = (1+x – ((1+x)^2-4yx)^0.5)/(2x)

phi is definitely real and positive. 2 can be rearranged to

3) phi = (1+x – ((1-x)^2+4(1-y)x)^0.5)/(2x)

And y

taking a first order approximation alpha is approximately equal to

4) phi is roughly equal to y/(1+x)

Which is positive.

So total payouts on CDSs are

5) x(1-phi)phi= xy(1+x-y)/(1+x)^2

And the ratio of the payout to the face value is

6) payout/x = y(1+x-y)/(1+x)^2

oh this is odd. take the derivative of the payout/x with respect to x

7 d(payout/x)/dx = y[(1+x)^2-2(1+x)(1+x-y)]/(1+x)^4 = y(2y-1-x)/(1+x)^3

So the payout per unit of self CDS increases in units of self CDS outstanding until the number of units, x is equal to 2y-1. For LEH senior debt, imagine the accounts were accurate (as far as they were supposed to go) and y = 0.95, the value of LEH self insurance would increase in LEH self insurance outstanding until the amount was equal to 90% of total LEH senior debt. That seems to me to be an unstable market.

Now how high would it have to go before it stops making sense to buy CDS from LEH ?

Well that depends on the price doesn’t it. If normal investors like Janet Tavakoli won’t touch it, the price could be very low, say one tenth as much as third party insurance. That would make it optimal to a price taker so long as alpha is greater than 0.1. Obviously I chose 0.1 out of my hat, because it is close to current market estimated phi of 0.12.

Could this have happened ? I don’t see why not. As far as I know it was all legal and profitable to both parties in the contract.