So I want to calculate the effect of default risk on the required interest of bonds (not on the price of bonds as that is normalized to one).
I thought of using the consumption capital asset model and calculating the following.
$$ U = u(C_1) + \beta (1-p)E_tu(C_2) + \beta p u(C_2) $$
Where the following applies:
$$ p = no \; default ,\; (1-p) = default $$
and the following budget constraints apply:
$$C_1 = Y_1 - T_1 - S_1$$ $$C_2 = Y_2 - T_2 + (1+r)S_1$$
I got to the following result using Lagrange, such that the consumer maximizes utility by deciding on how much to invest:
$$1 = \frac{\beta E_t[U'(C_2)p(1+r)]}{(u'(C_2)}$$
However, I want to get it ultimately in an equation that uses covariances but I do not know how.