STULZ - PQ Spreadsheet

[Zen]

Hello David,

RE: P1.T1 question 40 spread sheet - the price has normally distributed volatility of 20%.

The question does not give us a confident interval for the normally distributed price therefore we have to calculate the normal deviate and normal CDF.

Can you explain this section of the calculations? By that I mean the formulas for the Normal Deviate (the distance to default) and Normal CDF(5% in left tail). Also what does CDF stands for?

Just want to make sure I understand the theory/meaning behind these formulas.

Thanks

 

[David Harper CFA FRM]

Hi Zen,

Sure. The question (http://forum.bionicturtle.com/threads/l1-t1-40-bankruptcy-costs.3550/) mirrors the logic, FWIW, in Stulz 3.1.1. (Bankruptcy costs and firm value). Also, this is a good use-case because it is basically a "Merton lite" model (although not the same, the key difference is here Stulz assumes the price is normally distributed, but Merton--consistent with BSM etc--assumes log returns are normal and so the firm price is lognormal).

The key assumptions are:

• Expected (mean) future firm value = $1,200
• Face value of debt (i.e., owed in the future also) = $805
• Volatility of firm value (tracks volatility of gold price) = 20%

The normal deviate is the "distance to default:" how many standard deviations is the expected future firm value above the default threshold (the face value of debt)?
The expected value is $395 above the default threshold = 1,200 - 805.
But we must standardize this distance: $395 is 1.65 standard deviations from the mean, because one standard deviation = 1,200*20% = 240. So, $395/240 = 1.65.
We just "standardized" the variable and it's an ESSENTIAL concepts; http://en.wikipedia.org/wiki/Standard_score
... In this case, Z = (1200 - 805)/(1200*20%)
... we could also approach from (%): as -395/1200 = - 33%, Z = -33%/20% ~= -1.65

In words, our model (which relies on potentially unrealistic assumptions: 1. that the future firm price is normally distributed; 2. that default & bankruptcy occur if the asset value falls to the debt face value) says "in the future, the firm asset value must fall 1.65 standard deviations in order to experience bankruptcy"

What is the probability of this? if the price is exactly normal, this probability is the area under the standard (unit) normal to the left of -1.65 standard deviations. That's the essence of a cumulative distribution function (CDF) http://en.wikipedia.org/wiki/Cumulative_distribution_function
In this case, what is the Prob [X < 1.65 standard deviations | normal distribution] <<-- to remind that standard deviation does not imply normal, we are just assuming normal!
... and I designed the question so that it happened to give this result: Z = -1.65 and -2.33 are the two you must know for the exam as they correspond to 5%, and 1% of the normal tail region, respectively; i.e., to the left of 1.65 SD is an area that is 5% of total area under the normal, so 5% is the probability that a unit normal random variable will be less than -1.65

Stulz final step is to compute the expected cost of bankruptcy, which is the weighted average, which in this case = 5%* $50 + 95% * 0 = $2.50; i.e., 5% chance of incurring the $2.5 million cost. Then discounted to PV.

I hope that explains (I am adding a copy of this to the source Q&A below)

 

[Zen]

Thanks David!. I now understand the question and more importantly the theory behind the formulas.