Calculating d1 by Hull and Stulz

[EIA]

Hi David,

I trust you doing great.Please I need clarification and more insight.

In Hull pg 530, the formula used in calculating d1 is

d1 = (Ln(Vo/D) + (r + Sigma^2/2)T)
Sigma * Sqr T

and Stulz calculated d1 thus

d1 = (Ln(Vo/D) + 1 * Sigma * Sqr T
Sigma * Sqr T 2

I saw a question in Shweser's question bank on calculation of Equity Value and Debt vaklue of firm.
I applied the Hull's formula and the didn't get the right answer but when I applied the Stulz formula.

My question is,
what is the difference between the two and why the difference?
In the exams which formula do we apply?
What is the concept behind the difference?
And any other explanation you think will help.

Keep up the good job.

BR

EIA

 

[David Harper CFA FRM]

Hi EIA,

They are the same. As is sometimes the case, the only difference is notation or presentation. (although, given Hull's wide audience, several editions, I'd personally give Hull the benefit of the doubt: the odds of Hull being incorrect on a BSM type equation are extremely low, IMO. Whereas we find new issues with the Stulz text every year, no joke)

See http://forum.bionicturtle.com/threads/merton-formula.5517/#post-15553
Stulz d = ln(V/P(T)F)/[sigma*SQRT(T)] + 1/2*sigma*SQRT(T)
But P(T)F is just the discounted debt face value, the discounted strike price = K*exp(-rt), so:
Stulz d = ln[V/K*exp(-rT)]/[sigma*SQRT(T)] + 1/2*sigma*SQRT(T) =
Stulz d = ln[V/K*exp(-rT)]/[sigma*SQRT(T)] + (1/2*sigma^2*T)/[sigma*SQRT(T)] =
Stulz d = ln[V/K*exp(-rT)] + (1/2*sigma^2*T) /[sigma*SQRT(T)], since LN[V/K*exp(-rT)] = LN(V/K) *rT,
Stulz d = ln(V/K) + rT + (1/2*sigma^2*T) /[sigma*SQRT(T)] = ln(V/K) + T*[r + (1/2*sigma^2)] /[sigma*SQRT(T)] = d1
i.e., you should get the same answer

The most common mistake is here to use the expected asset return (drift), which will be higher than the risk-free rate (I just recorded video on this yesterday, T6.c) but the risk-free rate is used here to price the equity. And it may be the only advantage of Stulz's presentation: his Pt(T)*F is the present value of the face value of the debt, which should encourage the correct use of the risk-free rate (r), which is explicitly parsed in Hull's d1, per your first formula.

This contrast between d2 (and d1 by extension) in Merton to compute equity value versus N(-d2) to compute risk neutal PD is a marvelous illustration of Jorion's VaR Chapter 1 distinction, which feels abstract until you encounter it, between:

• Derivatives valuation which discounts to the present value mean of a risk-neutral distribution and therefore uses the risk-free rate: as above, we are applying derivatives valuation to price equity as a call option on the firm's assets ... versus ....
• Risk measurement which estimates a tail in the future actual (physical) distribution and therefore uses the actual expected return in d2 (aka, distance to default)

I hope that helps,