Calculating Credit Spread from Debt Formula

[papillonring]

Hi David

In the Stulz reading, Formula 1 should be used to solve Credit Spread (CS):

Formula 1:
CS = -[1/(T) x ln(D/F)] - rf

This formula is actually taken from forumla 2:

Formula 2:
D = F x exp-(rf + cs)T.

However, in the earlier chapter, I learnt that to calculate current Debt, I should use formula 3:

Formula 3:
D = (F x exp-rT) - Put

Why is put value not considered in Formula 1 when solving CS?
Apologies if you have to repeat this again. tried searching for similar question but I was returned with a number of posts.

 

[David Harper CFA FRM]

Hi Papillonring,

It is a interesting direct connection that I have not seen made before (!). If you are interested, in order to verifying the reconciliation, I entered into modified Merton model (see rows 57-65, at the bottom) @ http://db.tt/AUhK5Rk

This uses the same De Servigny example of a Merton model. Specifically,
Face value of Debt (F) = $13,
Time (T) = 1.0 years,
Riskfree rate = 4.0%

And then, per Merton type approach:
Price of risky debt (D) = price of risk free debt - put option = $12.49 - 0.35 = $12.14

And using that price we can solve for an implied spread (Stulz, not really a Merton approach)
= LN(13/12.14) - 4.0% = 2.86% credit spread (s)

It's a long way to show that these two approaches are not in conflict; they are just different approaches. The first uses asset volatility and option pricing (and therefore no real view on future expectation), the second (by discounting a risky return) does incorporate an expectation. I might summarize the difference as the first is based on volatility, the second based on on discounting (and therefore future expectation).

Math-wise,

Discounted price = Merton-based (OPM) price:
F*exp[-(r+s)T] = F*exp(-rT) - p, where p = put option value, such that
F*exp(-rT)*(-sT) = F*exp(-rT) - p, and
F*exp(-rT)*exp(-sT) = F*exp(-rT) * [ 1 - p/F*exp(-rT)], and
exp(-sT) = [ 1 - p/F*exp(-rT)]

… I maybe didn't find the most elegant relationship, but notice this last equation equates the spread with the put value (and the XLS verifies). Hope that helps rather than confuses, like I said, it's really interesting to me b/c nobody has drawn the connection before!

David