An investor is considering two possible assets, a three month one A with a yield of $4\%$ convertible quarterly and some three month one B.
a) For a nominal of 100, determine the price of A.
b) The investor believes that A is risk-free but that B has a probability of default $2.5\%$ and a loss given default of $100%$. The utility function is $u(w) = \sqrt{w}$. The initial wealth is $100$ and is to be fully invested so that the expected utility three months from now is maximized. Denote the price for 100 nominal of B by $P_B$. For which values of $P_B$ the investor would choose to invest:
i) Everything in A ii) Half in A, half in B iii) Everything in B
My approach - I am just asking if I have started correctly, as there are some new terms for me.
a) If the price is $P$ then we denote $i^{(4)} = 4\%$ and have $100 - P(1+i)^{-1} = 0$ where $1+i = (1+\frac{i^{(4)}}{4})^4$ from which $P = 104.060401$ (My concern here is whether it is $100 - P(1+i)^{-1} = 0$ or $100 - P(1+i) = 0$ as I am not completely sure what is meant)
b) If we invest $\alpha$ in $A$ and $100 - \alpha$ in $B$, then the realized wealth is $\alpha(1+i) + (100 - \alpha)X$ where $1+i$ is as in a) and $X = 0$ with probability $0.025$ and $P_B/100$ with probability $0.975$. The expected utility is $0.975\sqrt{\alpha(1+i) + (100 - \alpha)P_B/100} + 0.025\sqrt{\alpha(1+i)}$ and we wish to see for which $P_B$ this maximized at $\alpha = 100, 50, 0$.
Thanks for the help!