Value added from diversification

You have NOK 10M worth of oil reserves. With 50% probability the value decreases by by 16% and with 50% probability the value increases by 24%.

You consider selling NOK 6M worth of oil reserves and invest the proceeds in a global value-weighted index fund tracking the MSCI Global. With 50% probability, the value of the equity portfolio decreases by by 20% and with 50% probability its value increases by 28%. There are some measurement issues and the correlation coefficient between changes in the value of oil reserves and changes in the value of the equity portfolio is either -0.25, 0.0, or 0.25.

Assume that your trade-off between expected return and risk can be described by the following mean-variance trade-of

U = E[w] 1/2 ²_w , where ²_w is relative variance and is 2.

(a) If you keep all your wealth in oil reserves, what is expected wealth, its variance, and your expected utility?

(b) If reallocate NOK 6M of your wealth from oil reserves to a global equity portfolio, what are the four possible outcomes one year from now?

(c) In this case, what is expected wealth and its variance? And expected utility? Compute separately for each of the three possible correlation coefficients.

(d) What is value added from this diversification and how would you compute that? Compute separately for each of the three possible correlation coefficients.