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# Interest rates (Stultz)

#### saurabhpal49

##### New Member
Hi David,
In the credit spread calculation it's mentioned that an increase in interest rates will increase the value of firm and decreases the credit spread

however in the unanticipated change in interest rate section, it's mentioned that an increase in interest rate reduces the value of debt which will lead to decrese in value of firm( v=debt+equity)

could you please let me know the difference in both the sentences as it seems contradictory to me

thanks

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Suara2349 The Stulz reading contains several apparent and actual contradictions which are exacerbated by the writing style and (dis-organization); e.g., model-based inferences are commingled with empirical observations. We've been running interference on this reading for 10+ years. Superficially, I would explain this with the difference between the credit spread (i.e., the component in addition to the riskfree rate that corresponds to the bond's credit risk) and interest rate changes (where the interest rate is basically the sum of the riskfree rate and the credit spread, to omit other factors, and is the rate the reconciles the debt's current price with it's future par value). I don't want to claim that Stulz doesn't contradict, because he does, but it's a rabbit hole. I'll just keep it superficial here to arbitrate the difference, if that's okay.
• Your second point is more important and fundamental, and only the first clause: an increase in interest rate reduces the value of debt. The basic relationship, let's not lose sight of it! And that's our emphasis of our first point on page 13 of our note: "Unanticipated changes in interest rates can affect debt value through two channels. 1. First, an increase in interest rates reduces the present value of promised coupon payments absent credit risk, and reduces the value of the debt. 2. Second, an increase in interest rates can affect firm value. Empirical evidence suggests that stock prices are negatively correlated with interest rates." Importantly, point 2 is an empirical claim which (in my book) implies that it can be true one day and false the next day. But if point 2 is (empirically) true, than these two points together conspire to suggest that an increase in the interest rate implies a decrease in firm value because both the debt and equity value are reduced.
• The first point ("an interest rates will increase the value of firm and decreases the credit spread") is more exotic and, in my opinion, less relevant. We're just parroting Stulz in 18.1 here , but we probably should delete it due to the confusion engendered. It's just a model-based interpretation, not empirical, for one thing. So, we can't really say it contradicts (2) above because (2) above is empirical. This section is based on credit spread, s = -1/T*LN(D/F)*- Rf, where in this model, ceteris paribus, the credit spread generally decreases as either term to maturity (T) increases and/or the risk-free rate, r, increases. So the shallow interpretation, based on context, is simply: if we hold bond price (D) constant, then an increase in Rf rate implies a decrease in the credit spread. However, the specific reference to Helwege & Turner suggests something less shallow: under the Merton view of the firm, the firm's value drifts upward at the risk-free rate [per the risk-neutral valuation Step 1, not the PD step 2 which uses the ROA] such that with higher risk-free rate the ratio of Debt/Assets falls and the spread decreases due to less leverage. In either case, this a a narrowly mechanical application of a model. Although interesting, I would argue, either interpretation suggests that Stulz actually means the risk-free rate here.
So, you can see it's dicey and possibly contradictory. The safest statement is "higher interest rate [i.e., risk-free + spread] implies lower debt price, ceteris paribus." But the other statements can easily lack precision such as to seem contradictory if we don't clarify:
• Is this an inference of a mathematical model (i..e, the spread model above, Merton Step #1 which is risk-free BSM, Merton Step #2] and therefore only true in the narrow, and/or is this simply an empirical statement?
• Is this a statement about the risk-free rate, the credit spread or the summation of them?
• Are we holding other things constant (ceteris paribus), or as in Section 18.1.4, are we deliberately not doing that and complicating the dynamic (adding realism) by including second-order (knock-on) effects. This is non-trivial and realistic with respect to interest rates: the first-order dynamic is "higher rate lowers the debt value," but second- and third-order effects and feeback loops can alter the dynamic so that the end result seems contradictory but contradicts really because more dynamics are included. I hope that's a helpful start to better understanding this, thanks!

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#### saurabhpal49

##### New Member
Thanks David for the clarification