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 P

_{1}, P

_{2}, p

_{1}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 P

_{1}and p

_{1}. That is, P

_{b}= p

_{1}P

_{1}/ (p

_{1}+k(1-p

_{1})). So P

_{b}and P

_{1}, p

_{1}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**

p_{1} | P_{1} | P_{b} | |
---|---|---|---|

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}, P_{1}, p_{1} will converge.

**normal distribution**

p_{1} | P_{1} | P_{b} | |
---|---|---|---|

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}, P_{1}, p_{1} 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**

p_{1} | P_{1} | P_{b} | |
---|---|---|---|

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}, P_{1}, p_{1} will converge (if they ever do).

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