The Geithner put : general formula

Many people have been calculating the effect of Geithner put in the Geithner plan (aka PPIP) . However, while many have shown numerical examples, seldom of them have shown the general formula. So I am going to give one.

Usually, when a fund purchases loan from bank, its gain/loss can be written as
　　　P - P_b
where P is true price of the loan, and P_b is the purchase price.
Therefore, expected return of the fund becomes zero when P_b=E[P]
(E[.] denotes expectation value).

But with Geithner put, the gain/loss of the fund becomes
　　　w(P - P_b)　　　if P >= (1-k)P_b
　　　-wP_b　　　　　 if P < (1-k)P_b
where:
(1-k) is ratio of non-recourse loan offered by FDIC (6/7 in case of Geithner plan),
w is ratio of fund money in investment (1/2 in case of Geithner plan; the rest(=1-w) is funded by Treasury).

In this case, P_b, the break-even purchase price, is not necessarily E[P].

Now let P becomes P₁ with probability p₁, and P₂ with probability 1-p₁.
(Assumption: P₁ >= (1-k)P_b > P₂)

Then the break-even purchase price can be written as
　　　P_b = p₁P₁ / (p₁+k(1-p₁))

RHS does not include P₂, which means one does not have to consider P₂ as for down-side risk. This is because if down-side event really happens, the price P₂ does not affect the loss of the fund; the fund just loses the money it put in.

Let confirm this formula by Krugman's example. In Krugman's case, p₁=0.5, P₁=150, k=0.15, so P_b becomes 130.43, which corresponds to Krugman's calculation.

Jeffrey Sachs also showed numerical example. His setting is p₁=0.2, P₁=1,000,000, k=0.1. Then P_b becomes 714,268, which corresponds to Sachs's calculation.

And in this case, break-even price P_b is always higher than expectation value of P. That result can be shown by the following arithmetic:

　P_b-E[P]
=p₁P₁ / (p₁+k(1-p₁)) - (p₁P₁+(1-p₁)P₂)
=p₁P₁ [1/{p₁+k(1-p₁)} -1] - (1-p₁)P₂
=p₁P₁[(1-k)(1-p₁)/{p₁+k(1-p₁)}] - (1-p₁)P₂
=(1-p₁)[p₁P₁(1-k)/{p₁+k(1-p₁)} - P₂]
=(1-p₁)[(1-k)P_b-P₂]

This value is always positive by assumption. And this value represents the "subsidy" to the bank under the Geithner plan.

And if the purchase price is higher than break-even price P_b by, say, R, it adds to that subsidy to the bank.
In that case, the private invester loss is -wR(p₁+(1-p₁)k), Treasury loss is -(1-w)R(p₁+(1-p₁)k).
And FDIC loss becomes larger by -R(1-(p₁+(1-p₁)k)).
Graph below shows the loss of the three stake holders as function of p₁ (total=100%; k=1/7, a=1/2 as in the Geithner plan).
　

Loss of FDIC+Treasury makes up 80% of total when p₁=0.3, and becomes less than 70% only when p₁ is larger than 0.6.