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For a risk-free bond such as a US treasury bond, the YTM would be solving for $r$ in the denominator of each ($\frac{coupon payment}{(1+r)^n})$ such that the total equals the given price. And such a YTM is a 'risk-free' YTM.

How would that equation be different if we are dealing with a risky bond (ie, a corporate bond or a risky sovereign bond like from Italy or Greece)? And how should one interpret that YTM ?

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The equation would be the same given that the bond is vanilla, i.e. no exotic coupon types, etc. Otherwise, the cash flow is constructed differently, but the idea is the same.

Yield is used to discount your future cash flows, hence the interpretation is the same. However, in theory the yield of a risky bond should be higher than the yield of a risk-less bond (benchmark treasury for example) that matches the maturity of the risky bond. The additional risk is represented as the yield spread.

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The major difference is that the bond is not default free. In the case of a risk free bond you have the following formula:

\begin{equation} Price_0 = \sum^T_{t=1}\frac{CF_t}{(1+YTM_{rf})^t} \end{equation}

Assume a risky bond with the same Cashflows ($CF_t$) the major difference to the previous one is that there is now a risk of default (or partial default), therefore the formula above needs to be changed to:

\begin{equation} Price_0 = \sum^T_{t=1}\frac{E_0[CF_t]}{(1+YTM_{risky})^t} \end{equation}

where $E_0[CF_t]$ is the expected cashflow. As an example if the bond would default for sure at $t^\star$ then all cashflows between $t^\star$ and $T$ would be zero. In general:

\begin{equation} E_0[CF_t] < CF_t \implies YTM_{rf} < YTM_{risky} \end{equation}

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When I teach beginners about YTM I always make a point to describe it as the Promised Yield to Maturity, which I abbreviate as PYTM.

Mathematically the PYTM is computed in the usual way, as Will Gu said. But for a risky bond the PYTM is not a measure of "how much money I am going to earn on this bond", but rather the best that you can do if the company is able to come through and make the payments it has promised. Obviously for a risky company the probability of this happening is less that 1, perhaps much less than 1 for a seriously troubled company, so you expect to make less than this amount on average.

If you are interested in the expected return on the bond the PYTM is not useful and you have to pursue an approach like the one shown by phdstudent above where you take expected values of the cash flows. You then have a different concept, which you can call Effective YTM or something like this.

Be very careful not to misinterpret the [P]YTM for a risky bond when you see it quoted (and they are quoted all the time). The interpretation is different than for a risk free bond.

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