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!

  • $\begingroup$ Note: you provide no information about asset $B$. All you say is "some three month one B". Also, you say that $A$ is convertible but you don't say what the terms of the conversion are (nor even what $A$ is convertible into). $\endgroup$ – lulu Apr 30 at 23:17
  • $\begingroup$ B is such that "the price for 100 nominal of B is P_B". What do you mean by "what the terms of conversion are and what is A convertible into"? For "convertible into", it should be "invest and then get money from it" or? $\endgroup$ – DesmondMiles Apr 30 at 23:24
  • $\begingroup$ Many assets are priced at par. You need to specify what return characteristics $B$ has. As to the other...what do you think "convertible" means? Usually the term is used to describe debt instruments that can, at the investor's option, be converted into equity. But I think you are using the word in some nonstandard way. $\endgroup$ – lulu Apr 30 at 23:30
  • $\begingroup$ Your question is missing a lot of details. For instance, it is not possible to price a fixed rate bond $A$ (now I'm assuming you used "convertible" without understanding what it meant) without knowing the risk free rate. Please edit your post to include all the necessary information. $\endgroup$ – lulu Apr 30 at 23:32
  • $\begingroup$ I can't edit it :( I have copied the formulation from the past exam paper. Hence there is no better way for me to express it. $\endgroup$ – DesmondMiles Apr 30 at 23:44

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