We develop a theoretical model to assess the merits of principal-write down contingent convertible (CoCo) bonds. The conversion risk is the key feature of CoCo bonds. Because of this conversion risk, CoCo bonds are hard to price and an equilibrium price does not necessarily exist. In our model, for such a price to exist, the bank needs to hold a minimum amount of equity and/or the expected return associated with its asset portfolio needs to be large enough. When an equilibrium price exists, it is a decreasing function in the amount of equity held by the bank. Well-capitalized banks can thus issue CoCo bonds at a lower price than least-capitalized banks. This is the reason why CoCo bonds are to be thought of more as a complement to equity than as a substitute. In addition, because of the conversion risk, self-fulfilling panics may occur in the CoCo bonds' market. We indeed define a game between CoCo bonds' holders and the Central Bank that allows us to exhibit situations where a panic occurs in the CoCo bonds' market. Using the global game technique, we show that the probability of crisis can be expressed as a function of the return associated with the asset portfolio of the bank. The probability of crisis is shown to be sensitive to the precision of the information available to CoCo bonds' holders. Taken together, our results call for cautiousness when assessing the relevance of regulatory requirements in CoCo bonds, especially concerning their systemic impact.