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?
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.
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).
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.
No matter what is the risk profile of the issuer the yield is computed as
yield = Gross (pre-tax) coupon x 100 --------------------------- 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.
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 .