Question

Please explain Step-by-Step. Also, is there a way to solve without a graph? Excel or calculator?

DT3 Management is uncomfortable stating probabilities for the states of nature for a trade bill passing. Depending on its probability they will invest in one of three countries; A B, or C. Let p denote the probability of the bill passing. Trade Bill Fails Trade Bill Passes 160 210 A 320 260 370 B C a. What does graphical sensitivity analysis tell management about location preferences? b. After further review, management estimated the probability of the trade bill passing at 0.65. Based on the results in part a, which country should be selected and what is the expected value associated with that decision?

Answer 1

Let p be the probability that trade passes and 1-p be the probability that trade bill fails.

The expected value of payoff for investment in each of the countries is

E(A) 320p+ 160(1 - p) 260P 210(1 - p) E(C) 370p

a) We will create the following table in Excel for p=0 to 1, with an increment of 0.05

A C D E( c) -370 A2 1 E(B) -260 A2+210*(1-A2) -260 A3+210*(1-A3) -260 A4+210*(1-A4) -260 A5+210*(1-A5) 260 A6+210* (1-A6) -260 A7+210*(1-A7) -260 A8+210*(1-A8) -260 A9+210*(1-A9) =320*A10+160*(1-A10) =260*A10+210*(1-A10)370*A10 320*A11+160*(1-A11) |=260*A11+210*(1-A11) |=370*A11 320*A12+160*(1-A12) 260*A12+210*(1-A12) |=370*A12 320*A13+160*(1-A13) 260*A13+210*(1-A13) |=370*A13 320*A14+160*(1-A14) =260*A14+210*(1-A14) |=370*A14 320*A15+160*(1-A15) 260*A15+210*(1-A15) |=370*A15 320*A16+160*(1-A16) =260*A16+210*(1-A16) |=370*A16 320*A17+160*(1-A17) =260*A17+210*(1-A17) |=370*A17 320*A18+160*(1-A18)260*A18+210* (1-A18) |=370*A18 320*A19+160*(1-A19) =260*A19+210*(1-A19) |=370*A19 =320*A20+160*(1-A20) =260*A20+210*(1-A20)370*A20 320*A21+160*(1-A21) 260*A21+210*(1-A21) |=370*A21 -320*A22+160*(1-A22)260*A22+210* (1-A22) 370*A22 E(A) p 2 0 -320*A2+160* (1-A2) -320*A3+160*(1-A3) 370 A3 A2+0.05 - 320 A4+160*(1-A4) -320*A5+160* (1-A5) -320*A6+160* (1-A6) -320 A7+160*(1-A7) 4A3+0.05 370 A4 5 A4+0.05 370 A5 6 A5+0.05 370* A6 7 A6+0.05 8 A7+0.05 370 A7 -370 A8 -320*A8+160*(1-A8) 370 A9 -320*A9+160*(1-A9) 9A8+0.05 = 10A9+0.05 11 A10+0.05 12A11+0.05 13 A12+0.05 14A13+0.05 15A14+0.05 16 A15+0.05 17 A16+0.05 18 A17+0.05 19 A18+0.05 20-A19+0.05 21 -A20+0.05 22 A21+0.05

get the following

D E(A) Е(B) E( c) 1 p 2 210 0 160 C 3 0.05 168 212.5 18.5 4 0.1 176 215 37 5 0.15 217.5 55.5 184 6 0.2 192 220 74 7 0.25 200 222.5 92.5 8 0.3 208 225 111 129.5 0.35 216 227.5 10 0.4 224 230 148 11 0.45 232 232.5 166.5 12 0.5 240 235 185 13 203.5 0.55 248 237.5 14 0.6 256 240 222 15 242.5 240.5 0.65 264 16 0.7 272 245 259 17 0.75 280 247.5 277.5 18 296 0.8 288 250 19 0.85 296 252.5 314.5 20 0.9 304 255 333 21 0.95 312 257.5 351.5 22 320 370 1 260

We can see that E(B) is the highest for p<=0.45, E(A) is the highest for p between 0.45 to 0.75 and for p>0.75 it looks like E(C) is the highest. You can graph the above values if needed.

But these are just approximations.

The above table is needed only to get a sense of the range of p where the expectations are the highest and to know that there are 2 points at which the transitions take place.

Now we will use the equations for expectations to find the values of p where these transitions occur.

First equate E(A)=E(B) and get (We compare A to B, because in the range less than 0.45 we can see that B beat A)

E(A) E(B) 320p 160(1 p) 260p 210(1 - p 320p 160p 260p 210p 210 160 p 0.45

that is we can say that for p<=0.45 (rounded to 2 decimals) investment in country B is preferred.

Next we will equate E(A)=E(C) (We compare A to C, because in the range p>0.75 C beat A)

E(A) E(C) 320p1601 - p) 370P 370p 320p160p 160 p 0.76

that is we can say that between 0.45 to 0.76 the investment in country A is preferred

Above p=0.76, investment in country C is preferred.

b) From part A we know that A is preferred when p is between 0.45 to 0.76.

From the table we know the at p=0.65, E(A)=264 (or E(A)=320*0.65+160*(1-0.65) = 264)

Country A should be selected and the expected value associated with that decision is 264 (Units?)