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April 2016 Question 6

P

Polina Hadjipanayiotou

Member
Really struggling to understand the derivation of the first two parts of this question. Also watched the YouTube video but still didn't make things clear to me. could someone explain this to me?

Thank you in advance
 
For the decision function you have to choose long or short - and this is based on the observation of the company outperforming or underperforming.
So you should now be able to work through the 2*2 = 4 possible decision functions.
Post back you solution and then I'll take you through the next step which requires me knowing which function is which.
 
John Lee said:
For the decision function you have to choose long or short - and this is based on the observation of the company outperforming or underperforming.
So you should now be able to work through the 2*2 = 4 possible decision functions.
Post back you solution and then I'll take you through the next step which requires me knowing which function is which.
Click to expand...

thank you very much for the reply. my decision functions are:

θ1: outperform - θ2:underperform
d1 long - long
d2 long - short
d3 short - short
d4 short - long
 
Apologies for the severe delay.

Next we need to calculate the risk function. This is just the expected return:

E(return) = probability * return

We are told in the question the expected returns based on our decision and the state of nature (success, failure):

company is success company is failure
we choose long +100% -75%
we choose short -50% +50%

So let's work logically through every combination.

Suppose that the company is a success.
Then there is a 60% chance that it outperformed and thus a 40% chance it underperformed.

So for d1 we have E(return) = 60% * +100% + 40% * +100% = 100%
(as in each case we choose long and the return is 100% for long when the company is a success)

For d2 we have E(return) = 60% * +100% + 40% * -50% = 40%
(60% of the time the successful company outperforms and we choose long which gives 100% return, 40% of the time the successful company underperforms and we choose short which gives -50% return).

Similarly we get:
For d3 we have E(return) = 60% * -50% + 40% * +100% = 10%
For d4 we have E(return) = 60% * -50% + 40% * -50% = -50%

Suppose that the company is a failure.
Then there is 40% chance that it outperformed and thus a 60% chance it underperformed.

So for d1 we have E(return) = 60% * -75% + 40% * -75% = -75%
(as in each case we choose long and the return is -75% for long when the company is a failure)

For d2 we have E(return) = 60% * -75% + 40% * +50% = -25%
(60% of the time the successful company outperforms and we choose long which gives 100% return, 40% of the time the successful company underperforms and we choose short which gives -50% return).

Similarly we get:
For d3 we have E(return) = 60% * +50% + 40% * -75% = 0%
For d4 we have E(return) = 60% * +50% + 40% * +50% = +50%

Put these E(returns) in a table of decision (long/short) against states of nature (success/failure) and then you can apply minimax or Bayes like in other questions.
 
John Lee said:
Apologies for the severe delay.

Next we need to calculate the risk function. This is just the expected return:

E(return) = probability * return

We are told in the question the expected returns based on our decision and the state of nature (success, failure):

company is success company is failure
we choose long +100% -75%
we choose short -50% +50%

So let's work logically through every combination.

Suppose that the company is a success.
Then there is a 60% chance that it outperformed and thus a 40% chance it underperformed.

So for d1 we have E(return) = 60% * +100% + 40% * +100% = 100%
(as in each case we choose long and the return is 100% for long when the company is a success)

For d2 we have E(return) = 60% * +100% + 40% * -50% = 40%
(60% of the time the successful company outperforms and we choose long which gives 100% return, 40% of the time the successful company underperforms and we choose short which gives -50% return).

Similarly we get:
For d3 we have E(return) = 60% * -50% + 40% * +100% = 10%
For d4 we have E(return) = 60% * -50% + 40% * -50% = -50%

Suppose that the company is a failure.
Then there is 40% chance that it outperformed and thus a 60% chance it underperformed.

So for d1 we have E(return) = 60% * -75% + 40% * -75% = -75%
(as in each case we choose long and the return is -75% for long when the company is a failure)

For d2 we have E(return) = 60% * -75% + 40% * +50% = -25%
(60% of the time the successful company outperforms and we choose long which gives 100% return, 40% of the time the successful company underperforms and we choose short which gives -50% return).

Similarly we get:
For d3 we have E(return) = 60% * +50% + 40% * -75% = 0%
For d4 we have E(return) = 60% * +50% + 40% * +50% = +50%

Put these E(returns) in a table of decision (long/short) against states of nature (success/failure) and then you can apply minimax or Bayes like in other questions.
Click to expand...

Hey John - this doesn't match the model answers given in the examiners' report. Can you check?
 
Our d3 and d4 are labelled differently. Also we have given answers as % rather than euros.
 
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