For a corporate bond, which natuarally has a default risk, the expected yield to maturity (EYTM) is defined as the probability-weighted average of all possible yields. Hence, for a 10-year zero-coupon bond which has expected payoff $E(N)$ and has a price (at time 0) $P$, we have $$P=\frac{E(N)}{(1+EYTM)^n},$$ where $n$ is maturity. This formula tells us that if in numerator we have expected payoff, then we should discount it using EYTM, which is rougly speaking the sum of risk free rate and risk premium (excluding default risk) (for more details see slide 47&48 from the link:https://ocw.mit.edu/courses/sloan-school-of-management/15-401-finance-theory-i-fall-2008/video-lectures-and-slides/MIT15_401F08_lec04.pdf). Ok, this part is intuitive for me. But when I read about Divident Discount Model (DDM) for bond price valuation I observe the following formula: $$P_t=\frac{E_t[D_{t+1}]}{1+r_{t+1}}+...,$$ where $E_t[D_{t+1}]$ is expected divident, and $r_{t+1}$ is risk-adjusted discount rate for cashfow at time $t$. Please note, that $r_{t+1}$ takes into account time value of money, default risk of company as well as market risk (for more details see slide 7 from the link: https://ocw.mit.edu/courses/sloan-school-of-management/15-401-finance-theory-i-fall-2008/video-lectures-and-slides/MIT15_401F08_lec07.pdf).
For both models described above, we have uncertain cash flow in the future, but the first one is discounted using EYTM, which doesn't take into account default risk, but for the DDM the expected divident is discounted using interest rate which takes into account default risk. Is this a contradiction. If no, why? Many thanks in advance!
