Stulz Chapter 3 - Distance to Default

[trabala38]

Hello David,

In your Excel sheet relative to Stulz 3.1.1 Bankruptcy costs, I noticed that you compute a normal deviate.

The formula you use is : (Forward Price - Debt Face Value) / (sigma * Forward Price) = -1,43.

I was wondering : what is the concept behind the formula ? Do you have any reference text for this ?

Usually, to standardize a variable, we use the following formula Z = (X - u) / sigma (cf. http://ci.columbia.edu/ci/premba_test/c0331/s6/s6_4.html )

The difference is the "Forward Price" used in the denominator...

Also, a minor comment: I would rather use the spot price as the "random variable" to be "standardized"... Since the company has either the choice to hedge (using Forward Price) or to stay un-hedged/exposed to the spot price. But anyway, it does not change the idea of the computation...

Thanks for your feedback.

Regards,

trabala38

 

[David Harper CFA FRM]

Hi trabala38,

It is an exact replication of the Stulz' illustration in 3.1.1. (page 57) of his Risk Management & Derivatives.
(for what the context is worth, here is a sample FRM question from 2009 that employs a similar method: http://www.bionicturtle.com/wiki/FRM2009.E2.11/ )

I think your question is GREAT for what is reminds. Note that Jorion Chapter 1 introduces a distinction between valuation (pricing) and risk measurement, the differences include:

• valuation: exp discounted value; current value; center of distribution, VERSUS
• risk measurement: distribution of future values; tails of distribution; future value

this Stulz exercise nicely illustrates the focus on the tail of a future distribution.

Also, it is essential similar to the Merton Model for credit risk in the sense that both are computing the area of a future normal distribution (there is an less important technical difference: Merton has the forward price as lognormal whereas this exercise has the forward price as normal). This is a key point about Merton, its d2 (distance to default) is not a PV, it the area under a forward asset distribution! Note Stulz brings it back to PV with the final discounting ....
... your reference link to the standardization still applies, this exercise is doing that, only it is doing it to future values of the future distribution (i.e., standardization can be current or future, it doesn't care per se, so i don't think standardization is in conflict)

I hope that's a start, you raise a point that exposes more theory than a few innocent looking calculations would suggest!

David

 

[trabala38]

Thanks for your answer David.

I had a look at Stulz Chapter 3 p57-58, and it does not explain how they got the result (it refers to Chapter where they use an Excel function to get the quantile based on a known mean and know volatilty).

After re-assessing the computation, I realized that actually, we are not dealing with "normal returns" but we are dealing with prices (probably log-normal prices). Hence, it explains why the denominator contains "the price".

In summary, you must first get the asset return that is normally distributed = X= (Expected Value - Current Value) / Current Value = (250 - 350)/350 = -0,2857.

Then, you need to adjust for both the mean and standard deviation of the return using the Z score formula => Z = (X - u)/ sigma.

Here, we are assuming:
- mean = 0
- volatility = 0,20

Hence, the distance to default, in terms of quantile/percentile of a standard normal distribution is = (-0,2857 - 0) / 0,20 = - 1,4285.

Concerning your explanations on the Merton Model, I am not sure to follow you. It is, IMO, a difficult part of the curriculum and I need to revise first to be sure than I get the essence of what you are saying...

Thanks.

Best regards,

trabala38

 

[David Harper CFA FRM]

Trabala,

thanks, i disagree and i do think p57-58 explains it quite in detail; my XLS is based on the explain and the numbers match (the reference to early chapter is just a reference to using the normal/Z quantile. Stulz is always assuming normality).

Re lognormal prices: not exactly, that was my point above: the assumption here is that future PRICES are normally distributed (as opposed to Merton where normal returns --> lognormal prices), so I agree with your characterization that the computation standardizes into a normal (Z), but this is a standardization of prices/values directly (hence the multiplication) not returns. If these asset levels where lognormally distributed, then he'd retreive a lognormal quantile (or convert lognormal prices into normal returns) but this is a direct normal, N(.), on the standardized future normal price distribution.

Thanks, David

 

[trabala38]

Hello David,

Indeed, Stulz assume normal prices in his textbook. It is a point that I had overlooked.

Regarding the information on how to compute the normal deviate, Stulz (Chapter 3; Risk Management and Derivates, p57-58) gives only this information: "To get the present value of bankruptcy costs, we must specify the debt payment and the distribution of the cash flow. Let's say that the bankruptcy costs are $20 million, the face value of the debt is $250 million, the gold price is normally distributed, and its volatility is 20 percent.The firm is bankrupt if the gold price falls below $250. The probability that the gold price will fall below $250 is 0.077 using the approach developed in Chapter 2. Consequently, the expected bankruptcy costs are 0.077 x $20M, or $1.54 million."

So, it does not really explain how to get the normal deviate... We just have :

- normality assumption
- the "distance to bankrupcy" = 250 - 350 (350 is not indicated in the above text quote, but is specificied earlier).
- volatility = 0.20 (but is it return volatility ? or price volatility ?)

The mean of the distribution is missing (but it could be, by default, equal to 350 = forward price = E(S) since gold price is considered not systematic).

The point I don't get is the "scaling" of the volatility. According to you, it 0.20 * 350 = 70. Unfortunately, I don't know where this result comes from. Before using a formula, I like to understand the basics of it, but here, something is not clear to me .

If anybody can help...

Thanks a lot !

Best regards,

trabala38

 

[David Harper CFA FRM]

Hi trabala38

If it helps, he's not really scaling the volatility, just expressing in dollar terms to match the numerator (we are accustomed to seeing this like Merton or BSM d2 where the ratio is a return difference [btwn mean and threshold] divided by a return volatility). Here it never gets to returns. So, it's like: if asset value = $100, volatility = 12%, and we can express the volatility as either 12% or as, in dollar terms, $12 = $100*12%. It's still the same sigma, not so much rescaled as formatted in dollar terms.

It is still the same standardized formula you linked above (i.e., Z = (X - mean)/sigma, except the sigma is dollar terms because the numerator is in dollar terms; if the (X-mean) expressed a return difference in % terms (as the volatility is, i remind, a volatility of returns), then consistency would have the sigma in percentage terms. (ratio consistency is my #1 ratio tip, i take it everywhere!) It's sort of two issues: 1. the "trivial" issue is ratio consistency ($ or %), 2. the non-trivial is that this assumption reflects on, is implied by, a distributional assumption. This is the more subtle point. By assuming the future price levels are normal, this is specifically NOT the lognormal property we are accustomed to. I hope that helps, David

 

[trabala38]

Thanks for your answer David... It makes this clear...

Still got a question (last one for this topic, I promise) :

Imagine we have to estimate the probability that the stock price X will fall below a certain level, let's say, $10.

The distribution is normal with a mean of $120, and standard deviation of 20%. The initial price is $100. After some time, the price has evolved and is now $80.

What formula price should we use the compute the Z deviate given the current price of $80. Which price should we use for the denominator ?

a) the mean => Z = $20 - $80 / 20% * $120
b) the initial price => Z = $20 - $80 / 20% * $100
c) the current price => Z = $20 - $80 / 20% * $80

According to me, b does not make any sense. I hesitate between a and c and my preference would go for the solution a) since mean and standard deviation were given "grouped", hence there strong link between the mean price of 120$ and the sigma of 20%.

What do you think ?

Also, if the mean is not staten as an assumption, I guess we should compute it given what we know from the question :

i.e. if random walk is stated => E(S) = mean = current stock price
i.e if CAPM framework => E(S) = S0 * (1+CAPM rate) => that one is tricky because we are gain facing the issue of PV vs FV...

Thanks ;-)

Best regards,

trabala38

 

[David Harper CFA FRM]

Hi trabala,

I assume by your formulas that you don't mean a threshold of $10 but rather $20?

The problem in your setup, that i find hard to overcome, is that you appear to be mixing unconditional probability with some conditional variants (i.e., given a prior mean, the current parameters are changes). So, my first observation is that your setup begs some conditionality assumptions. Okay, but after that, two answers:

The standardized normal Z = (X - u)/(sigma), technically the sigma is the SQRT(variance) where the variance is sum of squared differences from mean; i.e., the (u) in numerator is also embedded in denominator. Put more simply, the sigma is a standard deviation around the mean in the numerator. Ergo, on the basis of ratio consistency, answer (c) is the only one (IMO) that matches the mean ($80) to the volatility.

Otherwise, as i mentioned above, all of our applications of this concept (e.g., BSM d2, Merton, risk measurement generally) would employ, unconditionally (i.e., at the beginning of your sequence), the standardization to the FORWARD (future) distribution. If, at time 0, the future forward mean is 120, then (IMO) the best unconditional answer (at time = 0) is none of your choices but instead: Z = (threshold $20 - mean $120)/($120*20%)

Where the difference between that and the other is one is unconditional (at time 0) and the other revisits (at time t = x) to compute a new conditional mean, but please don't make me honor the latter statement against your examples (b/c as i suggested, i think the flaw is actually in the question, the way you phrased the question may not have an answer). I don't necessarily defend any of those answer against the strictly flawed phrasing of the question. I am just trying to illustrate the problem with stating this way, thanks, David

 

[jeff-1984]

Hi David,
I have a question related to the same subject but its from the practice questions 2010 excel spreadsheet.
After calculating the normal deviate which is equal to 1.65, i wanted to get the probability of bankruptcy in percentage but i noticed that you used an excel formula NORMSDIST(-1.65) and it gave us a probability of bankruptcy equal to 5%.
Can you tell me how can i get this 5% while using a regular formula without having to use excel? and why is it negative cause as per the formula and the calculations we don't get a negative value.{ ( Forward Price - Debt Face Value) / (sigma*Forward Price) => 1200-805/1200*0.2 = 1.65 }

 

[David Harper CFA FRM]

Hi Jeffers,

The only calculator that will find the normal CDF is the HP 20B (see http://forum.bionicturtle.com/threads/calculator-options.5235/#post-14475). You can't get the answer with a TI BA II+ or HP 12c.

Otherwise, you need a table; e.g., Stock & Watson Appendix Table 1 is meant for this (it's often the first table in any stat book)

We tend not to worry about +/- in this context, due to symmetry of the normal. The formula above could be (debt value - forward price) without loss of meaning. The formula is computing a positive distance to default, and statistically this is just the number of (standardized) standard deviations from the mean (i.e., forward expected value). So, if the default threshold is 1.65 "below" the mean, we are looking for the area in the tail, which is either N(-1.645) or 1 - N(1.645). So, the indifference w.r.t. +/- is less important than, under a (symmetrical) standard normal, N(-Z) = 1 - N(Z). I hope that helps, thanks!

 

[jeff-1984]

Thanks for your explanation David,
However regarding the 1st part of my question, during the exam how will I be able to calculate this if i dont own a HP 20B? We're allowed to bring a copy of the table you mentioned with us ?

Thanks again

 

[David Harper CFA FRM]

Hi jeffers: you can't bring a table to the exam, GARP does not expect you to be able to figure N(Z). They will provide the values, they will say (eg) that "N(2.0) = 0.97725" In most cases, historically, this has been their approach. You do NOT need to use the 20B, but also you cannot bring a table.

Except you do need to have memorized N(-2.33) = 1% and N(-1.645) = 5%, which are the same as N(2.33) = 99% and N(1.645) = 95%. You can expect to be quizzed on these two classic VaR quantiles, but otherwise, they will tell you (or possibly give you a lookup table snippet, so you do want to know how to operation the Z lookup Table). I hope that explains, thanks!

 

[skoh]

Hi David,

For question 44.1, from the excel sheet answer, why is the value of debt 50% ($1000) and 50% ($800)? Shouldn't the debt be $1000 regardless for the beginning ofthe year?

Thanks!

 

[skoh]

Also how do we differentiate between which part is debt and which part is equity? Because I am afraid I'll be confused when it comes to the actual exam.

 

[David Harper CFA FRM]

Hi skoh,

Source (with follow-on conversation) is here at http://forum.bionicturtle.com/threads/stulz-chapter-3-distance-to-default.4892

1. The 50 and 50 are just probabilities given in the question's assumption. I followed Stulz Chapter 3 here. $1,000 is the par value (aka, face, principal) of the debt due at the end of the year. I followed Stulz in computing the (present) value of the debt by discounted expected value: expected future value is average of [1000,800] then discounted. Yes, the principal is 1,000 but the expected discounted value is 900/1.04 = 865.4. Note how we can look at that, as implying a risky yield = 100/865 - 1 = ~16%. The binomial illustrates not only the overhang but informs a risky yield.
2. This Stulz-type problem gives us (for assumptions) the future asset value of 50% = 1,400 or 50% = 800 (where asset value is function solely of gold price). And problem gives us par (face) value of debt = $1,000. So, the controlling formula is equity = asset - debt. Under the binomial tree-like approach (unrealistic!), at the end of the year EITHER $1,400 - $1,000 = $400 equity. Or, $800 - 800 debt = zero equity. In either case, the debt has priority and is paid in full; or is paid until the assets are consumed. So, it's relying on equity = asset - liability. Thanks,

 

[skoh]

Thank you!

 

[southeuro]

Hi David, on the same question skoh asked, I understand the scenarios in the beginning of the year. With 50-50 binomial given, we either have 1400 or 800 and calculate debt and equity accordingly (though these scenarios happen at the end, so my logic tells me this should be values at the end...)

More cruciall, I don't understand the calculations done for "at the end of the year" how come the value of equity is 600 and 0 depending on the 50-50 scenarios? and why do we take the vlaue of debt as 1 billion (PV'd) as given here? why are those values different than the ones calculated at the beginning of the year? What am I missing? Many thanks

 

[David Harper CFA FRM]

Hi southeuro,

can i point you to the source @ http://forum.bionicturtle.com/threads/l1-t1-44-debt-overhang.3558/
... which include this snapshot below

I did have to make some corrections, not my best question, but the $1,000 face value debt is a given assumption. Then the binomial is gold price (= firm value) of either 1,400 or 800 but with the investment that is +200, so firm asset value either = 1,600 or 1,000, such the FV of equity equal to either 1,600 - 1,000 = 600; or 1,000 - 1,000 = zero. Let me know, it's quite possible that i still have a problem with this question, thanks,