Sobre las Ps
While I am waiting for the “ASAP” to happen, I entertained myself with a little quant game to answer the following questions: What is the market view of the probability that K and R get fully paid off? And what is the expected payment date? What I am sharing here is outside any modern asset pricing theory so take it as a grain of salt.
First of all, the market price of $85.6 for PQ does not mean the probability of total payoff is 8.56%. That is a risk neutral probability. We actually live in a (mostly) risk aversion world. Furthermore, there is no perfect hedge to the return volatility of PQ (My broker does not allow me to short K. There are no WMI options traded either) so that no-arbitrage-pricing does not work in this case.
The current market price is the present value of the expected future price. In this case, it is reasonable to use a bimodal distribution to model the payoff, i.e, we are either get paid in full or belly up. The expected price relates to the probability of getting paid. And the present value relates to a discount factor which is determined by the risk of returns and risk aversion of investors.
Modern asset pricing theories tell people do not pay premium for risks that can be diversified, which are unnecessary risks, and only pay for the covariance to systematic risks. Well, last time I check, modern asset pricing theories are only good for portfolio of assets; they are quantitatively terrible for individual assets. So we are not getting help from there.
Assume we “vultures” or “electronic bloggers” actually have some brains and we are rational. Besides we want to see the justice in the end, we also want to get paid for the risks we are taking. We are here seeking for a risk adjusted return no less than our alternative of holding the market, which is historically 0.4 times return volatility.
How about the pay off date? Assuming the probability of payoff has not been changed since WMI filed Ch11 (A bad assumption. EC saved us. But bear with me, it's a game anyway), we can use historical price to calibrate the guess of payoff date. We know PQ was traded about $3 one year ago and about $20 six months ago. Both of them give similar results so I pick one year ago price. Now all needed information is at place.
Solving the following equation set:
Expected Price: E[P]=p*1000+(1-p)*0
Current Price: $85=E[P]*exp(-r*T), where T is the payoff time from now
One Year Ago Price: $3=E[P]*exp(-r*(T+1))
Return Variance: var=p*(1000-E[P])^2+(1-p)*(E[P]-0)^2
Expected Return: r=0.4*sqrt(var) (risk free rate is 0 anyway)
we have:
Probability of payoff, p = 0.99993
Payoff date, T = 0.74 year = 9 months from now
Cheers.
