Fun with uniform distributions

(Update: If there are too many numbers and equations below, Mike at Rortybomb has created a fantastic post illustrating the principles graphically. And he even uses lognormal distributions like a real financial engineer.)

In my earlier post on the “Geithner Put”, some people objected to my model as unrealistic.  Which is true.  So, using ideas from Andrew Foland (via private mail), I decided to grind out the math for a uniform distribution. Yes, a Gaussian might make more sense, but I doubt the answers would be all that different.  And besides, that might not lead to a nice closed-form solution.

Anyway, here is Andrew’s model.  Assume lots of identical assets.  Assume each has an unknown value uniformly and independently distributed between m-a and m+a.  In other words, m is the average value and 2a is the range of possible values, and everything in the range is equally likely.  Let k be the “leverage factor”; i.e., the fraction of the purchase price that consists of equity.  So for 6:1 leverage, k is 1/7.

Finally, let y represent the price the investor pays, and denote by p the average profit per asset.

Thanks to the wonderful site QuickMath, I can share my formula and you can play with it.  For example, if the leverage is 6:1 (set k=1/7), and the possible values are uniformly distributed from $0 to $100 (set m=50, a=50), and the investor pays $60 (set y=60), then the average profit will be $3.22 per asset (solve for p). So in this example, admittedly a very wide range, the fund puts in 1/7 * $60 = $8.57 and thus earns a 37.6% return… While still overpaying by 20%.

Another example: Same leverage factor (k=1/7), but say the assets are worth between $35 and $65 (m=50, a=15), and say the fund simply pays what they are really worth (y=50). By paying only what the assets are really worth, the fund is entitled to no returns at all… But in fact it will earn $1.03. That is a double-digit return on a 1/7 * $50 = $7 investment that should have broken even. Where did the extra return come from? From the FDIC, who ate the cost of the losing bets. I can see why Bill Gross is getting excited.

If you like, you can also turn it around:  Assume leverage of 6:1 (set k=1/7) and assume the range of possible values is $10 to $30 (set m=20, a=10). Then the investor can pay up to $21.94 and still break even (set p=0 and solve for y).  In this case the equation has two solutions, but only one of them is actually less than $30, so that is the answer. (You can try setting k=0 or k=a/m to see that the formula passes the “sniff test”.)

Or use the same assumptions, but instead of setting p=0, set it to $.50 and then solve for y. That tells us the investor can bid up to $21.62 and still receive a $0.50 return (on a 1/7 * $21 = $3 investment).

Adjust the numbers yourself and click “Solve” if you want to experiment. The investments and profits are split with the Treasury, but this does not affect the returns and so it does not enter into the formula.

The big formula at the top is the key, and if someone out there could check my math, I would be much obliged.  All I did was calculate the average profit by integrating it from m-a to m+a and dividing by 2a.  Because the loan provides a “floor” of k*y for the loss, the profit function is piecewise linear with two pieces, which is why the formula is the weighted sum of two parts.  Note that the formula is only valid for (1-k)*m ≥ m-a — or, equivalently, k ≤ a/m — that is, when the “floor” provided by the non-recourse loan is at least the minimum possible value of the asset.  This will be true for reasonably high leverage and wide ranges, which is what we are interested in here.

I will have more to say about the qualititative features of this model later.  But at first glance, it does appear that those claiming my model was simply too extreme may have a point.

Update

Although once you take the leverage into account, it sure looks to me like the private equity folks are going to make out pretty well at FDIC’s expense.

5 comments to Fun with uniform distributions

  • I like it Nemo.

    However, isn’t the loan going to cost something like 1% or 2% and, if the assets are fairly priced (at expected value), this subsidy alone will be sufficient to jack prices?

    Assuming that the bank started out with any subordination at origination and has taken some decent sized haircuts, the leverage plus heavily subsidized interest rate would make this a winning deal.

    Vulture investors want 20% plus returns in this environment. To get it on cash flows with any significant duration and no financing, this requires very low prices. With leverage @ 2%, the same vultures could get the 20% and pay much higher prices.

  • Dimon

    applying your model it seems to be right, that IF investors pay the “real” price they will earn a nice profit, but
    the question is how do they know what the real price is? so isn’t there an incentive to underpay?
    even if they don’t underpay and pay the “real” price, there is still the question will the banks sell
    their assets at the “real” price or will it make them insolvent(if investors underpay it would become an even larger problem)?

  • grr

    I like it too. I echo CapVandal’s comments.

  • One other comment.

    Obviously the investors are getting a subsidy. However, it seems to me that the Treasury is pari passu with the investors for the “equity” piece.

    I think the Treasury needs to offer up their share to the public in small quantities. There is excess back room capacity so this could be easily and cheaply done. Especially since the Treasury sets the rules. Warn people in big red letters that it is risky, but let people buy if they want.

    This would eliminate some of the accusations of big subsidies to speculators. Make it the People’s hedge fund. I would personally buy a little chunk of Maiden Lane III. Call me a sucker, but the Treasury and FRBNY, etc. make money on most deals and offset a big chunk of the few big losers.

  • […] program was correctly identified as the public writing a “put option” on those debts. As such, the public insurance would cause the hedge funds to overbid for the […]

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