QUESTION 2 Value of information Guide to marks: 20 marks – 4 for a, 8 for b, 2 f
ID: 3301973 • Letter: Q
Question
QUESTION 2 Value of information
Guide to marks: 20 marks – 4 for a, 8 for b, 2 for c, 6 for d
Show all calculations to support your answers. You may follow the methods shown in the mp4 on Value of info for a way to answer this question if you wish, but however you do the calculations you must specifically provide answers to the 4 questions.
DO NOT ROUND probability calculations with Round Function. You may display them to 2 decimal places if you like but do not round in memory.
Jerry is thinking about opening a bicycle shop. He can open a large shop (a1) or a small shop (a2). He believes that a large shop would earn a profit of $80,000 if the market is good (s1) but would lose $40,000 if the market is poor (s2). A small shop would return $30,000 profit in a good market and a loss of $10,000 in a poor market. Jerry believes that there is a 50-50 chance that the market will be good.
(a) What should Jerry do? Show calculations.
A friend would charge him $3,000 for some market research providing.one of two signals, that the market is favourable or unfavourable. His past record is such that 80% of the time he would correctly provide a favourable market prediction when the market is good and 60% of the time he would correctly provide an unfavourable market prediction when the market is poor.
(b) Revise the prior probabilities in light of his friend’s track record.
(c) What is the posterior probability of a good market given that his friend has provided an unfavourable market prediction?
(d) What is the expected net gain or loss from engaging his friend to conduct the market research? Should his friend be engaged? Why?
QUESTION 2 Value of information
Guide to marks: 20 marks – 4 for a, 8 for b, 2 for c, 6 for d
Show all calculations to support your answers. You may follow the methods shown in the mp4 on Value of info for a way to answer this question if you wish, but however you do the calculations you must specifically provide answers to the 4 questions.
DO NOT ROUND probability calculations with Round Function. You may display them to 2 decimal places if you like but do not round in memory.
Jerry is thinking about opening a bicycle shop. He can open a large shop (a1) or a small shop (a2). He believes that a large shop would earn a profit of $80,000 if the market is good (s1) but would lose $40,000 if the market is poor (s2). A small shop would return $30,000 profit in a good market and a loss of $10,000 in a poor market. Jerry believes that there is a 50-50 chance that the market will be good.
(a) What should Jerry do? Show calculations.
A friend would charge him $3,000 for some market research providing.one of two signals, that the market is favourable or unfavourable. His past record is such that 80% of the time he would correctly provide a favourable market prediction when the market is good and 60% of the time he would correctly provide an unfavourable market prediction when the market is poor.
(b) Revise the prior probabilities in light of his friend’s track record.
(c) What is the posterior probability of a good market given that his friend has provided an unfavourable market prediction?
(d) What is the expected net gain or loss from engaging his friend to conduct the market research? Should his friend be engaged? Why?
QUESTION 2 Value of information
Guide to marks: 20 marks – 4 for a, 8 for b, 2 for c, 6 for d
Show all calculations to support your answers. You may follow the methods shown in the mp4 on Value of info for a way to answer this question if you wish, but however you do the calculations you must specifically provide answers to the 4 questions.
DO NOT ROUND probability calculations with Round Function. You may display them to 2 decimal places if you like but do not round in memory.
Jerry is thinking about opening a bicycle shop. He can open a large shop (a1) or a small shop (a2). He believes that a large shop would earn a profit of $80,000 if the market is good (s1) but would lose $40,000 if the market is poor (s2). A small shop would return $30,000 profit in a good market and a loss of $10,000 in a poor market. Jerry believes that there is a 50-50 chance that the market will be good.
(a) What should Jerry do? Show calculations.
A friend would charge him $3,000 for some market research providing.one of two signals, that the market is favourable or unfavourable. His past record is such that 80% of the time he would correctly provide a favourable market prediction when the market is good and 60% of the time he would correctly provide an unfavourable market prediction when the market is poor.
(b) Revise the prior probabilities in light of his friend’s track record.
(c) What is the posterior probability of a good market given that his friend has provided an unfavourable market prediction?
(d) What is the expected net gain or loss from engaging his friend to conduct the market research? Should his friend be engaged? Why?
Explanation / Answer
a) We will find the expected profit for each of the scenarios - opening a large shop and opening a small shop.
Expected profit in case a large shop is opened is computed as:
P(a1) = (80,000)*P(S1) + (-40,000)P(S2) where S 1 is the good condition and S2 is the poor market conditions. Therefore the expected profit is computed as:
P(a1) = (80,000)*P(S1) + (-40,000)P(S2) = 80,000*0.5 - 40,000*0.5 = 20,000
Similarly now for the small shop, the expected profit is computed as:
P(a2) = (30,000)*P(S1) + (-10,000)P(S2) = 30,000*0.5 - 10,000*0.5 = 10,000
Therefore Jerry should open the large shop .
b) Now for 3,000 in this case is a fixed cost, whether or not the large shop is opened or the smaller one.
We already know that P(good market) = P( bad market) = 0.5
Also as 80% of the time he would correctly provide a favourable market prediction when the market is good, therefore P(friend says good | good market) = 0.8 and P(friend says bad | good market) = 0.8.
Also, 60% of the time he would correctly provide an unfavourable market prediction when the market is poor, therefore, P(friend says bad | bad market) = 0.6 and P(friend says good | bad market) = 0.4
Therefore, now
P( friend says good) = P(friend says good | good market)P(good market) + P(friend says good | bad market)P(bad market) = 0.8*0.5 + 0.4*0.5 = 0.6
P( friend says bad) = 1 - 0.6 = 0.4
Using Bayes theorem now we have:
P(friend says good | good market)P(good market) = P( good market | friend says good )P(friend says good)
0.8*0.5 = P( good market | friend says good )*0.6
Therefore, P( good market | friend says good ) = 0.8*0.5/0.6 = 2/3
Ans therefore, P( bad market | friend says good ) = 1 - 2/3 = 1/3
P(friend says good | bad market)P(bad market) = P( good market | friend says bad )P(friend says bad)
0.4*0.5 = P( good market | friend says bad )*0.4
Therefore, P( good market | friend says bad ) = 0.4*0.5/0.4 = 0.5
Ans therefore, P( bad market | friend says bad ) = 0.5 = 0.5
Now using the above probabilities:
Here expected profit in case friends says that the market is good and large shop is made would be:
E ( large shop | market is good by friend ) = 80,000*(2/3) - 40,000*(1/3) = 40,000
Now expected profit in case friends says that the market is good and small shop is made would be:
E ( small shop | market is good by friend ) = 30,000*(2/3) - 10,000*(1/3) = 16,666.67
Therefore if the friend says, that the market is good then Jerry should open the large shop.
Now, expected profit in case friends says that the market is bad and large shop is made would be:
E ( large shop | market is bad by friend ) = 80,000*0.5 - 40,000*0.5 = 20,000
Now expected profit in case friends says that the market is bad and small shop is made would be:
E ( small shop | market is bad by friend ) = 30,000*0.5 - 10,000*0.5 = 10,000
Therefore if the friend says, that the market is bad then also Jerry should open the large shop.
c) The posterior probabilities are computed as:
P(friend says good | bad market)P(bad market) = P( good market | friend says bad )P(friend says bad)
0.4*0.5 = P( good market | friend says bad )*0.4
Therefore, P( good market | friend says bad ) = 0.4*0.5/0.4 = 0.5
Ans therefore, P( bad market | friend says bad ) = 0.5 = 0.5
Therefore 0.5 is the required probability here.
d) Clearly as computed in the part b) we saw that no matter what his friend says, he will always go with the large shop, therefore the friend should not be engaged. And the expected net gain in case the friend is hired would be:
P(a1) - 3000 ( here 3000 is subtracted as it is the fees for the friend )
= 20,000 - 3000 = 17,000
Therefore 17,000 is the required expected value of gain in case friend is hired.