# Relationship between default risk and value of bond

In particular Mortgage-backed securities. I've read from a book that underestimating the risk of default is tantamount to overestimating the value of the bond, giving rise to a bubble in MBS. I'm unclear of how that works. My guess is that since the probability of default risk is low, thus banks give out lower coupon rates. Investors buy MBS up because they are safer, demand for MBS increase, price of these MBS pushed up.

[...] underestimating the risk of default is tantamount to overestimating the value of the bond.

The reason behind this is simply due to how bonds are priced.

$$P = \sum_{n=1}^N \frac{C + M[{I_{n=N}}]}{(1+i)^n}$$

Where $$i$$ is the contractual interest rate, $$C$$ is the coupon payment, $$N$$ is the number of payments, $$I_{n=N}$$ is an indicator function taking the value of $$1$$ when $$n=N$$ is $$true$$ or $$0$$ otherwise, $$P$$ is the market price of the bond, $$M$$ is the value of the bond at maturity.

When $$i$$ goes up, $$P$$ goes down and vice versa, see $$\rightarrow$$ if $$i$$ is underestimated, $$P$$ will be overestimated. Indeed, it is $$i$$ which, via a premium, internalizes the default risk consideration and is an increasing quantity of it.

I like @keepAlive's answer, but I thought it might be more accessible to discuss this question in a simpler setting. Imagine that a \\$100 face amount bond that lasts for one period. If the bond does not default than at the end of the period then it pays a coupon $$c$$ and returns the face amount. The bond defaults with probability $$P_{d}$$ and then returns zero. Assume the appropriate discount rate is $$r$$. What is this bond worth?

$$P_{bond} = \frac{0}{(1+r)^0} + (1- P_{d})\cdot \frac{100+c}{(1+r)^1} + (P_{d})\cdot \frac{0}{(1+r)^1}$$ $$= (1- P_{d})\cdot \frac{100+c}{(1+r)}$$ But assume that you are mistaken, and you estimate a default rate that is too small (that is $$\hat{P}_{d} < P_{d}$$). You should infer the bond is worth: $$\hat{P}_{bond} = (1- \hat{P}_{d})\cdot \frac{100+c}{(1+r)}$$ Which implies the difference in price is: $$\hat{P}_{bond} - P_{bond} = (1- \hat{P}_{d})\cdot \frac{100+c}{(1+r)} - (1- P_{d})\cdot \frac{100+c}{(1+r)}$$ $$= (P_{d} - \hat{P}_{d})\cdot \frac{100+c}{(1+r)}$$, which by construction is positive. Therefore, assuming that the default rate is too low implies that you will overestimate the value of the bond all other things equal (even if you get the other parameters correct).