The Geithner put : general PDF case

In the previous two posts, I showed general formula of break-even purchase price and fund's gain/loss under the Geithner plan. I also did some arithmetic using two-state model.

Krugman wrote "the natural way to think about these things is in fact in terms of simple two-state numerical examples." Nemo commented "binomial distribution is oversimplified."

However, two-state model may be not so simple as it seems. It can be described as reduced form of more general case.

Let p(P) the probability density function(PDF) of P. Then P₁, P₂, p₁ in the two-state model I used can be derived from p(P) as follows:



where P_b is break-even purchase price, and (1-k) is ratio of non-recourse loan offered by FDIC.

As derived here, P_b is function of P₁ and p₁. That is, P_b = p₁P₁ / (p₁+k(1-p₁)). So P_b and P₁, p₁ must be calculated simultaneously.

As for general PDF, that calculation can be done by Excel. Below I show how to do the calculation by Excel for three forms of p(P), i.e. uniform distribution, normal distribution, log-normal distribution.

• uniform distribution

	p1	P1	Pb
Excel row/column	A	B	C
2	Initial value	Initial value	=A2*B2/(A2+$G$2*(1-A2))
3	=($I$2-C2*(1-$G$2))/($I$2-$H$2)	=($I$2^2-(C2*(1-$G$2))^2)/2/($I$2-$H$2)/A3	=A3*B3/(A3+$G$2*(1-A3))

Input k in G2 cell, and upper limit of uniform distribution in I2 cell, lower limit in H2 cell.
If you drag fill handle of 3rd row to appropriate row (say, 41st row), then each value of P_b, P₁, p₁ will converge.

• normal distribution

	p1	P1	Pb
Excel row/column	A	B	C
2	Initial value	Initial value	=A2*B2/(A2+$G$2*(1-A2))
3	=1-NORMDIST(C2*(1-$G$2),$H$2,$I$2,TRUE)	=(EXP(-((((1-$G$2)*C2-$H$2)/$I$2))^2/2)*$I$2/SQRT(2*PI())+$H$2*A3)/A3	=A3*B3/(A3+$G$2*(1-A3))

Input k in G2 cell, and mean of normal distribution in H2 cell, standard deviation in I2 cell.
If you drag fill handle of 3rd row to appropriate row (say, 41st row), then each value of P_b, P₁, p₁ will converge (if they ever do).

(Note: Above Excel formula is based on normal distribution without restriction, so price could be negative.)

• log-normal distribution

	p1	P1	Pb
Excel row/column	A	B	C
2	Initial value	Initial value	=A2*B2/(A2+$G$2*(1-A2))
3	=1-LOGNORMDIST(C2*(1-$G$2),$H$2,$I$2)	=(1-NORMDIST( (LN(C2*(1-$G$2))-$H$2)/$I$2,$I$2,1,TRUE))*EXP($I$2^2/2+$H$2)/A3	=A3*B3/(A3+$G$2*(1-A3))

Input k in G2 cell, and mean of log-normal distribution in H2 cell, standard deviation in I2 cell.
If you drag fill handle of 3rd row to appropriate row (say, 41st row), then each value of P_b, P₁, p₁ will converge (if they ever do).