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 P

_{1}with probability p

_{1}, and P

_{2}with probability 1-p

_{1}.

(Assumption: P

_{1}>= (1-k)P

_{b}> P

_{2})

Then the break-even purchase price can be written as

P

_{b}= p

_{1}P

_{1}/ (p

_{1}+k(1-p

_{1}))

RHS does not include P

_{2}, which means one does not have to consider P

_{2}as for down-side risk. This is because if down-side event really happens, the price P

_{2}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, p

_{1}=0.5, P

_{1}=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 p

_{1}=0.2, P

_{1}=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]

=p

_{1}P

_{1}/ (p

_{1}+k(1-p

_{1})) - (p

_{1}P

_{1}+(1-p

_{1})P

_{2})

=p

_{1}P

_{1}[1/{p

_{1}+k(1-p

_{1})} -1] - (1-p

_{1})P

_{2}

=p

_{1}P

_{1}[(1-k)(1-p

_{1})/{p

_{1}+k(1-p

_{1})}] - (1-p

_{1})P

_{2}

=(1-p

_{1})[p

_{1}P

_{1}(1-k)/{p

_{1}+k(1-p

_{1})} - P

_{2}]

=(1-p

_{1})[(1-k)P

_{b}-P

_{2}]

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(p

_{1}+(1-p

_{1})k), Treasury loss is -(1-w)R(p

_{1}+(1-p

_{1})k).

And FDIC loss becomes larger by -R(1-(p

_{1}+(1-p

_{1})k)).

Graph below shows the loss of the three stake holders as function of p

_{1}(total=100%; k=1/7, a=1/2 as in the Geithner plan).

Loss of FDIC+Treasury makes up 80% of total when p

_{1}=0.3, and becomes less than 70% only when p

_{1}is larger than 0.6.

## 0 件のコメント:

## コメントを投稿