# Yield of a risky bond

When working with risky bonds, i.e. corporate bonds, what is usually defined as the yield of such a bond? Is it the yield calculated as if the bond was riskless, or is it calculated by properly taking into account the default risk in this case? And what about (risky and riskless) floating rate notes?

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To find the yield to maturity for zero-coupon or coupon-paying bonds, the calculation is the same whether the bond is default-free or not. –  John Jan 4 '13 at 15:35
@John, I disagree with that. While the translation between a bond price and the yield to maturity is the same, both heavily incorporate risk notions. Telling how a yield to maturity is calculated from a given price is, imho, not really touching on the OP's question. –  Matt Wolf Jan 5 '13 at 2:50
@Freddy I see your point, but then what he really would want to know is, not what the yield is and how to calculate it, but how to price the bond. A broad topic, to be sure, but the OP did not provide sufficient information in the question to be sure that's what he was wondering about. –  John Jan 5 '13 at 3:22
@John, I think we get into the chicken and egg problem, very similarly in the options space to which drives what, options prices -> implied vols, or market perception of implied vols -> prices. I would say that the market prices risk in general and comes up with a total discount factor which is pretty much what ytm is, and hence a price for the bond. What the market does not know (and what analysts are paid to do ) is to back out individual risk components to price, e.g. CDS contracts. –  Matt Wolf Jan 5 '13 at 3:55
@John, my point being that risk is very much priced into ytm as well as the price of each and every single bond in this universe (and I do not know of a single bond series trading free of risk). We do not know how risk breaks down into, e.g. risk of corporate or sovereign default (heck ISDA does not even know how to define and interpret default, judging from the many court cases). But I think OP asked a straight forward question which is "how is risk embedded in the yield and prices of bonds". At least that is my take of it –  Matt Wolf Jan 5 '13 at 3:59

As John already mentioned the formula for calculating yield to maturity is independent of any risk-related numbers. Its just the connection between coupons, time and price.

In theory, default-risk can be seen as already incorporated in the yield. The yield spread between the bond and a comparable investment without default risk is a measure for the default risk premium.

For FRNs there is no way to compute a yield since coupon payments are unknown. You can only estimate the price/yield by forecasting the interest rates at the coupon dates. But the notion of yield is the same - one just estimates the coupons first.

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I somewhat disagree with that notion. As part of the bond yield is reflecting the risk embedded in the bond, the yield (also the yield to maturity) is very much dependent on the risk notions of the bond. The price of the bond is derived through its yield, thus both price and yield are a direct function of the inherent riskiness of various components. Many corporate bonds' yield is in fact calculated by implying a "base yield" and then adding risk premia on top –  Matt Wolf Jan 5 '13 at 2:30
@Freddy thats precisely what I said in the second statement about the yield spread. There is no point we disagree on. The textbook way to calculate a yield just depends on the price and the coupons though. Of course the default risk has impact on the price, thus on the yield-to-maturity. –  vanguard2k Jan 7 '13 at 7:30
I disagreed with your first paragraph. The question has nothing to do with how to calculate yield-to-maturity, I think we can all agree on how its done. The question was whether risk premiums are embedded in the yield-to-maturity, and for me it is a resounding yes (through the price), while saying "the notion of yield to maturity is independent of any risk notions" hints at a different interpretation. "Notion" refers to the idea or conception not to how to calculate things. Not trying to be anal, but if you hold others to high standards you may want to be precise yourself. –  Matt Wolf Jan 7 '13 at 8:45
@Freddy I edited my answer according to your suggestion and hope you can agree with me now. Still, in my opinion, yield-to-maturity is not directly risk related but yield spread is. Nevertheless, I think we are all talking about the same things here. –  vanguard2k Jan 7 '13 at 9:17
agree, from our exchange I sense we talk about the same thing (more or less) here after both our adjustments. I was just thinking of newcomers who may get confused. Adjusted my vote. –  Matt Wolf Jan 7 '13 at 9:40

Despite seeing one of the answers as having been chosen as the desired one, I like to offer a different perspective:

Whether the yield to maturity can be derived from a bond's price in a rather identical fashion, regardless of the inherent risks is, imho, not the point of the OP, given I understood the question correctly.

The yield of a bond with risk components can be seen as an "internal rate of return", basically a discount rate at which future cash flows need to be discounted at in order to arrive at the current market price (fair price?) of the bond. Therefore, the discount rate or ytm very much incorporates each and every perceived risk inherent in the bond because the bond price also reflects each and every risk notion investors perceive.

As some users pointed out the yield to maturity is a rather straight forward translation of bond prices. I think the much more useful approach is to start with the yield to maturity (or total internal risk of return), then strip off the risk premiums, in order to arrive at isolated risk attributional factors. This, I think is how many rating agencies approach corporate bond valuations and the derivative of that, CDS valuation of which an important input is the default probabilities (of course they use additional models on top of that to model CDS, most of the time copula probability-of-default related models).

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I am sorry but I disagree on the point where you say simple-to-calculate risk premia. For most corporate bonds for example you have to work hard to separate default risk from other risk factors. Further more, you don't arrive at a "probability measure" but rather at a yield spread as I stated in my answer. The probability of default still remains unclear. You still have to model the default probability for a given yield spread. –  vanguard2k Jan 7 '13 at 7:40
@vanguard2k, fair point, "simple" is subjective, I would not say, though, its "hard". Its standard practice and there is a reason for every bond trader in the market there are most likely 20-30 analysts working on this kind of stuff (buy-sell-side combined), not because its so hard but because they use risk premiums for all kinds of things plus accounting for the sheer number of corporate bonds out there. I equate probability measure (not used in the statistical sense) with yield spread in this case, I meant to mean isolated risk factors.The non-standard lingo most likely confused you, sorry –  Matt Wolf Jan 7 '13 at 8:39
@vanguard2k, I edited my answer –  Matt Wolf Jan 7 '13 at 8:46
Matt, do you know of any study which would quantify the dependency of riskiness on bond yields? –  gt6989b Oct 30 '13 at 14:20

Generally people price bonds as if they were riskless and then the option adjusted spread is used as a measure of credit spread. There are methods for stripping CDS curves to get hazard rates and then solving for the price of the bond given uncertain survival to cash flows, which produces a new yield that ought to take into account the CDS-implied default probabilities. I have worked with them before, but I don't know how popular they are elsewhere.

Floating rate notes don't present any credit-specific problems, one projects the coupons using the relevant forward curve + quoted yield margin. I suppose in theory if a company financed much of its working capital with floating rate debt, there might be a correlation between forward interest rates and the firm's credit quality, which might be an interesting angle, but I don't even know if that's been addressed in literature, let alone in practice.

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No matter what is the risk profile of the issuer the yield is computed as

yield =   Gross (pre-tax) coupon x 100
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Price in market


Price a bond can be found using one of short rate models. e.g Black Derman and Toy (BDT). One of the papers here talks about how a single stochastic factor model can be used. Once the bond is priced the yeild is backed out from it not the other way round.

HTH

*************EDITED ***************

Reading the questions again I agree with Freddy. The risky bond is a risky loan and yield on that is higher to compensate for risk premium. So I think in context of question Freddy response is correct. Now I am researching how to mathematically quantity that risky bonds carry this risk premium in their yield. Cheers Freddy I did not get the question right in first instance .

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quite off the reservation. That is definitely not THE way to calculate a bond yield. –  Matt Wolf Jan 5 '13 at 1:26
by the way you describe the flow is definitely not market practice. Even in the theoretical realm I can show you many models that output yields and not prices. One could even argue that your assertion that yields are derived from prices is not universal. –  Matt Wolf Jan 5 '13 at 4:02
@freddy If you have yield then you can get price by discounting coupon rate. But how do you construct the yield curve ? You use a intrest rate model. That is what I mentioned. –  ash Jan 5 '13 at 18:51
@freddy I am not objecting existence of models that produce yield curve. When I need to get yield and I have price I use what I mentioned above. When I need to construct one I used the "MODELS" you are mentioning depending on which one suits my requirement. The questions what YEILD of RISKY bond as mentioned in accepted answer I mentioned that yield does not carry notion of risk I too mentioned "No matter what is the risk profile of the issuer the yield is computed as". This is not about yield curve construction. –  ash Jan 5 '13 at 18:56