A trust officer at the Blacksburg National Bank needs to determine how to invest $100,000 in the following collection of bonds to maximize the annual return.

Bond	Yield	Maturity	Risk	Task free
A	9.50%	LONG	0.996	YES
B	8.00%	SHORT	0.55	YES
C	9.00%	LONG	0.368	NO
D	9.00%	SHORT	0.775	NO
E	9.00%	LONG		YES

With the low risk score of bond E being uncertain, The officer wants the average risk score for his investments to be less than 0.50. The officer wants to invest at least 50% of the money in short-term. At least 30% of the funds should go in tax-free investments, and at least 40% of the total annual return should be tax free.

The trust officer needs to determine his investments before realizing the risk in Bond E, whether he chooses to invest in Bond E or not. Formulate the stochastic problem faced by the trust officer assuming that the risk of Bond E follows a normal distribution. Solve the problem under two cases: (1) One where the normal distribution has a mean 0.5 and standard deviation 0.15. (2) Another where the normal distribution has a mean 0.5 and standard deviation 0.01. In both these settings, do the following:

1. In excel find the optimal investment decision.
2. Compare the optimal decisions with the mean value solution and interpret. Specifically, comment on the role of standard deviation of the underlying normal distribution.
3. When the standard deviation is 0.15, change the unit penalty to understand the trade-off between infeasibility and return, and provide recommendations on what value of looks reasonable to use in practice.