# Nominals, prices, utility

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!

• 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). – lulu Apr 30 at 23:17
• 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? – DesmondMiles Apr 30 at 23:24
• 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. – lulu Apr 30 at 23:30
• 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. – lulu Apr 30 at 23:32
• 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. – DesmondMiles Apr 30 at 23:44