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Within the context of the banking-related literature on contingent convertible bonds, we comprehensively formalise the design and features of a relatively new type of insurance-linked security, called a contingent convertible catastrophe bond (CocoCat). We begin with a discussion of its design and compare its relative merits to catastrophe bonds and catastrophe-equity puts. Subsequently, we derive analytical valuation formulae for index-linked CocoCats under the assumption of independence between natural catastrophe and financial markets risks. We model natural catastrophe losses by a time-inhomogeneous compound Poisson process, with the interest-rate process governed by the Longstaff model. By using an exponential change of measure on the loss process, as well as a Girsanov-like transformation to synthetically remove the correlation between the share and interest-rate processes, we obtain these analytical formulae. Using selected parameter values in line with earlier research, we empirically analyse our valuation formulae for index-linked CocoCats. An analysis of the results reveals that the CocoCat prices are most sensitive to changing interest-rates, conversion fractions and the threshold levels defining the trigger times.