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# stulz ch 18

#### fashepard

##### New Member
I am having problems with the second full paragraph on page 576. It starts with Helwege and Turner. He says that the expected value of a firm increases with an increase in rates. I would think the opposite. Also a dumb question. He uses notation that I don't remember seeing before. I can't work the numbers out when he writes for example D(V,F,T,t)
can you translate for me

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Frank,

I agree, it contradicts. This para is mostly irrelevant to the exam, for what it's worth (I think what he means is simply that, if V = value of firm today, V must grow over the period to maturity). What matters here (barely) is his conclusion ("credit spreads narrow as interest rates increase") because it jives with the formula at 18.2:

credit spread = 1/T*LN(Debt/Firm) - rate

so he is just saying, if you hold everything else equal (ceteris paribus), then a higher rate implies a lower credit spread. And if he were consistent, then higher rate would also decrease Debt and Firm value (with net-net unclear impact). The formula at 18.6 and these paras have been source of much confusion over the years, because it is a very simple model (i.e., omits much) yet he mixes in empirical findings. What matters (IMO) is:

* the formula above (as a simplistic model), and
* Just like you said, per 18.1.4, what he says about interest rates: that higher rates imply (i) lower debt value and (ii) lower firm value

Regarding the notion, good for you for pausing on the notation (how else to understand?). He is applying the Black-Scholes-Merton but per the structural approach, he is using the call option formula to price the value of common equity, so

S() = value of firm equity ~ (analogous to) value of a call option in Black Scholes = c
V = value of firm assets ~ stock price in Black Scholes = S
F = value of debt ~ strike price in Black Scholes = X or K
T,t = just fancy way of saying term (T-t) = term in Black Scholes = t

So, S(V,F,T,t) is just a way of saying, "the call option S() is a function of the inputs V,F,term"

David