Options in the valuation of risky corporate debt.
Try the Course for Free Transcript Well, let's say a few words about options in the valuation of risky corporate debt risks debt and the use of option evaluation in this vast area. Well, first of all as we discussed earlier when we talked about riskless debt, we can say that the overall price for the bond looks like this. Here, C is the coupon of the bond, F is the face value, and r1, rk, and rT, all these are T corresponding correct rates at which we have to discount these cash flows.
This is the right formula.
And as we said before, the good news about that is that this is all right. So it correctly values each coupon, and coupon and principal cash flow. Now the bad news is that unfortunately this formula contains a lot of parameters some of which are not nicely known. So when we talk about all of that was for riskless debt, for government securities.
But now I would like to talk about risky debt.
5.8. Option application (2) – valuing risky debt
And the first big question arises, where does the risk of investing in risky debt come from? Well, first of all we can say that here are the changing rs. Because, for example, if you invest in a bond, you bought it and then if all rates would go up, then the price and the value of the bond will go down.
Just because, well, we can say, this is clearly seen from the formula. But there is a very simple and clear straightforward explanation to that, because the bond produces some kind of return.
And if all the rates now are higher, then to reach these rates, you will have to quote start from a lower base and that is why the price of the bottle go down. Now here there are a couple of areas where we have to take this changing rs into account. I would say that there are two major kinds of risk. This is reinvestment risk.
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And capital gain, Or loss risk. Again, with capital gain and loss, that is what I just said.
But what is reinvestment risk? Let's say if you hold a fixed income security, that sort of provides you with some promised cash flows in the future, so you have a locked-in return in that. But what if for any reason these cash flows can be delivered to you earlier, or maybe at one of the same moment?
Then, you are full of cash, but that most often happens when the rates go down, and you will be able to reinvest this cash only at a lower rate, to reach a lower return, and that's a huge risk. Again, in general, people who manage fixed income portfolios, here we talk about huge amounts of money. And therefore, even minor changes in some of the parameters that influence valuation they can result in significant changes in overall values of these portfolios.
Now, all that is just about changing rs. But then there's another huge area that now we can talk about. And we can say that fixed income investments have a lot of embedded options. And that makes the evaluation a very challenging task. What kind of options are we talking about? For example, I will put like bonds can be, Well, callable, that means that the issuer, I am a corporation, I issue bonds.
But I have the right to buy them back at an earlier moment than the maturity at a pre-specified price. Now this option is available for me, because I can go ahead for that if it's beneficial for me, and clearly this option is bad for the holder of these bonds. Then they can be convertible, convertible bond is the one that can be exchanged in some amount of shares of stock at the specified proportions.
They can be puttable, that's the other case. At some point in options in the valuation of risky corporate debt the option, I'm sorry, the bond holder can put the bond back to the corporation. And when we talk about convertible, in general, they can be exchangeable.
The term exchangeable is used when bonds are exchanged not for the stock of this corporation.
But into some other assets of some other corporations, most often linked to the one issuing one. So basically, you can see that here we deal with a huge universe of various risks and as we will see in just very quickly, that unfortunately the proper valuation of this risk is quite a challenging thing. Well, let me first of all go over some important parameters of risk adapt relation.
You, I think, you have not completely forgotten the idea of the yield to maturity.
So we said that if this is the time, and this is the corresponding r. So you have some curve that is called the term structure, so basically, this is the curve that shows the expected correct rates of return that normally are applied to riskless debt. And all rates for valid risky debt, they are taken or they are produced from this term structure. And the ideal yield to maturity of the use of a uniform, just one number to discount all cash flows goes like this.
So you discount all cash flows at this rate which is yield to maturity. Now what you can see here is that this is a time and there are some cash flows that basically all cash flows of such a bond are mispriced.
Probably with the exception of the one, if that goes exactly here. So the yield to maturity is a great instrument that allows you to use just one number, but that results in a huge compression of information, and therefore you have mispriced cash flows.
This is sometimes called a simple finance approach.
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- Corporate Debt Valuation: The Structural Approach | SpringerLink
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Now, the thing is that the analysis of fixed income securities is actually a vast area, and that in detail options in the valuation of risky corporate debt fundamentally beyond the scope of this course and this specialization.
And this is a long and complex piece, but now I'm just showing some of the important things that are studied there. So here you can see all cash flows are mispriced and there is one idea that is very important here is that if, these rs change, what happens to the value of the bond?
And therefore, so is the value of the bond. And what people do they basically take some point of this interest rate and then going around this we study the small changes of the value of the bond with respect to changes in this r.
Basically as in math, we can always expand that into serious and there are some approximations to that. And one idea is the idea of duration, well, duration is sort of, double quoted, effective maturity.
Model summaries are presented in a continuous-time arbitrage-free economy.
So basically, the idea is as follows what if we put all cash flows of the bond, at one point. So as the PV of newly created security where all coupons and the principal, they come at this point, we discount that at the rate that is observed at this point, and then the PV is the same.
So that is called duration. So the idea is if we put a chart here, again, this is T, and this is a term structure. So we basically put all of them at one point. This is a point D so that the distribution is like delta function.
Corporate Debt Valuation: The Structural Approach
And all cash flows they just come here, sort of artificially. Now, you can say, well, it's kind of a strange way to do that.
Well, this is basically an help online earnings, because the formula for the duration is like this.