1. Show that R = 1.039 M is the appropriate risk tolerance such that 0.5(U(M) + U(?M?2))…

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1. Show that R = 1.039 M is the appropriate risk tolerance such that 0.5(U(M) + U(−M∕2)) = 0 (as discussed in Section 5.8).

2. Consider the four truss designs described in Example 5.15. For each design and both safety factors, determine the smallest and largest values of each safety factor across the range of the error. How does the range of the safety factors vary? Which design has the least range of the safety factors?

Example 5.15

Suppose Rose has inherited the option described in Exercise 5.14. Use the appropriate risk tolerance value from Exercise 5.13 to determine what Rose should do.

Exercise 5.14

(This problem is adapted from one in Pratt et al., 1995.) Joe has inherited an option on a plot of land and must decide whether to drill on the site before the option expires or abandon the rights. (If he abandons the rights, there is no gain and no loss.) He is not sure if there is oil or not. Drilling will cost $100,000 whether or not there is oil. If oil is found, then it will generate $450,000 in revenue. The likelihood of finding depends on the subsurface structure. If the subsurface structure is type A, then there is certainly oil. If the subsurface structure is type B, then the probability of finding oil is only 10%. In that area, the probability of a type A structure is 80%; the probability of a type B structure is 20%. Before deciding to drill, Joe can decide to pay $10,000 for a seismic sounding that will reveal whether the subsurface structure is type A or type B. He can get the results in time to review them before making the drilling decision. (However, he does not have to get the seismic sounding.) Draw a decision tree that includes the above decisions and uncertainties. Use it to find the optimal policy. In this situation, Joe is risk neutral.