Weight P/E by P*E ?

A while ago, Jeremy Siegel attacked P/E ratio calculated by S&P for overestimating the real value (here and here. HT Greg mankiw's Blog).

S&P are now using the following formula:


where V_i is the market value and E_i is the earning of issue i, and PER denotes P/E ratio.
Siegel asserts that S&P should weight both market value(numerator) and earning(divisor) by market value, such as:



Siegel says that this weighting will alleviate the problem of overestimating the overall P/E ratio, because the huge loss of the company like AIG will be downplayed by its relatively small market value, in comparison with S&P formula .


However, when you transform S&P formula as follows, you can see that it is also a kind of weighted sum:



That is, individual P/E ratio is weighted by earning.


When you look at Seigel's formula, it can be re-written as follows:



That is, individual P/E ratio is weighted by the product of value and earning.

It can also transformed as follows:


Individual P/E ratio is weighted by the product of P/E and squared earning.

From these transformations, you can see that the meaning of Siegel's formula is much harder to grasp than S&P one.


For Siegel's purpose, maybe it's better to weight individual earning by market value, instead of by earning in S&P formula, as follows:




　

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).



　

The Geithner put : option formula

In the previous post , I showed the general formula of break-even purchase price under Geithner plan.

In that post, I showed the gain/loss of the investment fund as follows:
　　　w(P - P_b)　　　if P >= (1-k)P_b
　　　-wP_b　　　　　 if P < (1-k)P_b
where:
P is true price of the loan,
P_b is the purchase price,
(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).

It also can expressed in option formula as follows:
　　　w{Max(P-P_b , -kP_b)}
　　=w{Max(P-(1-k)P_b , 0)-kP_b}

This is the formula of call option with strike price=-(1-k)P_b and option purchase price=kP_b (See image below).


By applying the above option formula, it can be calculates as
　　　0.5*{(0.5*(100-84*6/7)+0.5*0)-84/7}=1
which corresponds to Nemos's result.

As Nemo showed, the money fund put in is wkP_b=0.5*1/7*84=6. So the fund performance reaches 1/6*100=16.67%.

FDIC gain/loss also can be expressed in option formula as follows:
　　　-Max(0 , (1-k)P_b-P)
(In fact, maximum gain is not zero, as there is interest revenue. Here I ignore it for simplicity)

This is the formula of put option with strike price=-(1-k)P_b (See image below).


The FDIC position is short on this put option, and that "Geithner put" is what enables the floor part of the fund's call option shown above.


　

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.