# Relationship between risk free rate and credit spread in the Merton model

Based on Merton model of credit risk, I understand that investing in a risky debt is the same as buying a treasury bond and writing a put option on the firm's assets with a strike price equal to the face value of the debt. And the the equity is a call option on the assets of the borrowing firm with a strike price equal to the face value of the debt as shown in the diagrams below.

What i dont understand is why there is a negative relationship between the risk free interest rate and credit spread, which Merton has empirically shown. Intuitively, i would have thought the increasing the risk free interest rate would lower the discounted expected future cash flows and thus lead to higher probability of default and wider credit spreads. I found the possible explanations from some research papers:

"When interest rates rise, the value of the put option granted to the shareholders decreases because of the lowering of the discounted expected future cash flows, which in turn increases the value of the corporate bond since the creditors have a written put option position. This mechanism thus reduces the yield of the bond, making the spread over the yield of an equivalent risk-free security narrower. Another way of interpreting the outcome is to view the risk-neutral growth path of the firm value as being higher following an interest rate increase, leading to a lower probability of default and/or a lower default protection put option value, either way increasing the value of the corporate bond and therefore tightening the credit spread between corporate and risk-free government obligations."

"First, Merton (1974) argues that when the risk free rate increases, the present value of the future expected cash flow discount will de- crease, thus reducing the price of the put option. The investor of the corporate bond shorts put option, and the value of his long corporate bond position will increase. The increase of the price of the corporate bond will decrease the spread."

Can someone breakdown the Merton model and provide an intuitive explanation of why there may be a negative relationship between the risk free rate and credit spread?

By formula (14) in Merton (1974) the difference of the yield to maturity $$R(\tau)$$ of the firm's risky debt and the riskless rate $$r$$ is $$\tag{14} R(\tau)-r=\frac{-1}{\tau}\log\Big\{\Phi[h_2(d,\sigma^2\tau)]+\frac{1}{d}\Phi[h_1(d,\sigma^2\tau)]\Big\}$$ where $$\tau$$ is the time to maturity, and \begin{align} d&=Be^{-r\tau}/V\\ h_1&=-\Big[\frac{1}{2}\sigma^2\tau-\log d\Big]\Big/\sigma\sqrt{\tau}\\ h_2&=-\Big[\frac{1}{2}\sigma^2\tau+\log d\Big]\Big/\sigma\sqrt{\tau}\\ \end{align} and $$B$$ is the face value of the firm's outstanding debt, and $$V$$ is the initial firm value and $$\sigma$$ is the volatility of the firm value in a lognormal model.

The difference $$R(\tau)-r$$ can be viewed as a credit spread. It is straightforward to check that \begin{align} \frac{\partial}{\partial r}\frac{d\Phi[h_2(d,\sigma^2\tau)]+\Phi[h_1(d,\sigma^2\tau)]}{d} &=\frac{\tau\,\Phi[h_1(d,\sigma^2\tau)]}{d} \end{align} holds. Therefore, \begin{align}\tag{A} \frac{\partial}{\partial r}\{R(\tau)-r\}&=-\frac{\Phi[h_1(d,\sigma^2\tau)]}{d}\,e^{\tau(R(\tau)-r)}<0\,. \end{align}

• This negative relationship of riskfree interest rate and credit spread is not empirically derived (where has Merton done that?). It is a simple consequence of Merton's model assumptions and holds in so far as his model is realistic or not.

• The first long quote you gave says "When interest rates rise, [...] making the spread over the yield of an equivalent risk-free security narrower." Is that not the same as (A)?

• The second long quote comes to the same conclusion.

Now to the other statements in those texts:

1. Is the value of the corporate bond increasing when the riskless rates rise?

I don't think so:

The RHS of (A) is greater than $$-1$$ because it can be written as $$-\frac{\Phi[h_1(d,\sigma^2\tau)]}{d}\frac{1}{\Phi[h_2(d,\sigma^2\tau)]+\frac{1}{d}\Phi[h_1(d,\sigma^2\tau)]}>-1\,.$$ Therefore (A) implies $$\tag{B} \frac{\partial}{\partial r}R(\tau)=1-\frac{\Phi[h_1(d,\sigma^2\tau)]}{d}\,e^{\tau(R(\tau)-r)}>0\,.$$ This means that the corporate bond $$\exp(-\tau R(\tau))$$ must be decreasing.

1. Is the value of the put option granted to the shareholders [sic!] decreasing when the interest rates rise? [Note that the bond holders have the put option, not the equity shareholders.]

This is correct:

Eq. (14) can be rearranged to express the price of a risky zero coupon bond (ZCB) that redeems the firm's debt $$B$$ at maturity: \begin{align}\tag{C} e^{-\tau R(\tau)}B &=\underbrace{e^{-r\tau}\,B}_{\text{riskless ZCB}}\quad \underbrace{-\,e^{-r\tau}\,B\,\Phi[-h_2(d,\sigma^2\tau)]+V\,\Phi[h_1(d,\sigma^2\tau)]}_{\text{ short put option on firm value with strike B}}\,. \end{align} In the Black-Scholes model it is known that the long postition of the put $$P(\tau)$$ has the following partial derivative w.r.t. $$r\,$$: $$\tag{D} \frac{\partial}{\partial r}P(\tau)=-\tau\,e^{-\tau r}\,B\,\Phi[-h_2(d,\sigma^2\tau)]<0\,.$$

Conclusion

When riskless interest rates rise if I am not mistaken I have shown that

a) The yield of the risky zero coupon bond is increasing (B).

b) The price of the risky zero coupon bond is decreasing.

c) The price of the put granted to the bond holders is decreasing (for them) (D).

d) The spread of the zero coupon bond yield over the riskless rate is decreasing (A).

Only the statements (c) and (d) are in agreement with your quoted texts.

• I'm sorry im not too well versed in mathematics. I think what I dont understand from the second long quote is why would the decrease of the future expected cash flow, as a result of the rise in risk free rate, reduce the price of the put option? It also says "The investor of the corporate bond shorts put option, and the value of his long corporate bond position will increase." But I thought investing in risky debt is equivalent to buying a treasury bond and selling a put option on the firms asset. What is the long corporate bond position referring to? Mar 8 at 18:12
• In contrast, I am not too well versed reading long and verbose texts :) . I found a few statements I don't agree with. Checked everything in python just to be sure. Please digest and ask further questions. Mar 8 at 20:20