Script

DECISION MODELS – Making Decisions Under Risk

Slide 1

  • Welcome back.
  • In this module we talk about making decisions when there is aprobability distribution for the states of nature.

Slide 2

  • We have discussed
  • Decision making under uncertainty where we had no idea as to which state of nature will occur.
  • Decision making under risk refers to making decisions when we have some idea as to the probabilities for each of the states of nature.

Slide 3

  • The approach that can be applied when there are probabilities for the states of nature is the expected value approach
  • Expected value of anything is probability times number, probability times number, sum it all up
  • So the expected payoff value for a decision alternative is the sum of the products of the probabilities times their corresponding payoffs
  • Now expected value is a long run average value if the process is repeated over and over again. In this case, the decision with the highest expected value is the one that should be made.
  • But usually we are making a one-time decision – there is no long term repetition. But still the expected value approach can lend additional information for the decision maker.

Slide 4

  • For the Tom Brown investment problem
  • Suppose after consulting with an investment broker, together you estimate that over the coming year the probability of a large rise is .2, a small rise is .3, no change .3, a small fall .1 and a large fall .1
  • Here is the original payoff table
  • So given these probabilities for the states of nature
  • Then the expected value for gold is:
  • .2 times negative 100 plus .3 times 100 plus .3 times 200 plus .1 times 300 plus .1 times 0 or
  • $100. For the bond the expected value is
  • .2 times 250 plus .3 times 200 plus .3 times 150 plus .1 times negative 100 plus .1 times negative 150 or
  • $130. For the stock the expected value is
  • .2 times 500 plus .3 times 250 plus .3 times 100 plus .1 times negative 200 plus .1 times negative 600 or
  • $125.
  • And for the CD the expected value
  • Is $60.
  • The one with the highest expected value is the bond with an expected value of $130.

Slide 5

  • We now investigate the value of additional information
  • We begin by calculating the expected value of perfect information – the added value over choosing the bond of knowing for sure which state of nature will occur. We assume the probabilities of the states of nature do not change, for example the probability of state of nature 1 is .2, for state of nature 2, .3, for state of nature 3, .3, for state of nature 4, .1 and for state of nature 5, .1. But with perfect information you will know for sure if this is one of the 20% of the time that state of nature 1 would occur or is the one of the 30% of the time that state of nature 2 will occur and so forth.
  • Now if we knew state of nature 1 was going to occur, we would choose the stock with a payoff of $500. This is going to happen 20% of the time.
  • And if we knew state of nature 2 was going to occur, again we would choose the stock with a payoff of $250. This is going to happen 30% of the time.
  • And so forth

Slide 6

  • The expected value of perfect information, abbreviated EVPI
  • Is the gain in value above choosing the bond all the time (the one with the highest expected value) by knowing for sure which state of nature will occur each time.
  • Thus this is an upper bound on the maximum benefit of any information.

Slide 7

  • To calculate this expected value of perfect information
  • We refer to the payoff table
  • And we see 20% of the time we would make $500, 30% we would make $250, 30% of the time we would make $200, 10% we would make $300, and 10% of the time we would make $60.
  • So the expected return with perfect information is $271.
  • But if we chose the bond universally, we calculated that this would give an expected value of $130.
  • The expected value of perfect information then is the gain above this $130 value or $141.

Slide 8

  • We can use the decision template to calculate the optimal decision using the expected value criterion and the expected value of perfect information.
  • To do this we enter the probabilities for the states of nature as shown.
  • Then in the Results section we can read the optimal decision of the bond using the expected value criterion, with an expected value of $130 – and the expected value of perfect information of $141.

Slide 9

  • But we never get perfect information.
  • But we can gather additional information, called sample information, that while not perfect, can aid in the decision making process.
  • We still assume that in the long run the probabilities for the states of nature will not change.
  • But when we gather additional information, we will revise the probabilities given this information for the immediate decision and determine a strategy for what to do if the information turns out one way or the other.
  • And we wish to know the long run value of this information, which is called the expected value of sample information.

Slide 10

  • The way we do this
  • Is to revise probabilities given each possible outcome for the additional information using what is called a Bayesian analysis.
  • Then for each possible outcome we again perform the expected value approach using the new set of probabilities – so that we have a decision for each possible outcome of the sample information.

Slide 11

  • Let’s look at a specific example
  • Suppose a noted economist will perform an economic forecast that will either indicate positive or negative economic growth in the coming year.
  • Using a relative frequency approach suppose we have determined
  • When there was a large rise in the Dow in the past, 80% of the time the economist had given a positive economic forecast and 20% of the time he had a negative economic forecast.
  • When there was a small rise in the Dow in the past, 70% of the time the economist had given a positive economic forecast and 30% of the time he had a negative economic forecast.
  • When there Dow experience little or no change in the past, 50% of the time the economist had given a positive economic forecast and 50% of the time he had a negative economic forecast.
  • When there was a small fall in the Dow in the past, 40% of the time the economist had given a positive economic forecast and 60% of the time he had a negative economic forecast.
  • And when there was a large fall in the Dow in the past, he had always given a negative economic forecast.

Slide 12

  • We now show how to revise the probabilities for the states of nature when the economist gives a positive economic forecast using a Bayesian analysis.
  • The probability that he would give a positive forecast is the probability of a large rise and a positive forecast plus the probability of a small rise and a positive forecast plus the probability of no change and a positive forecast plus the probability of a small fall and a positive forecast plus the probability of a large fall and a positive forecast. We don’t have these but they can be gotten by:
  • The probability of a positive forecast given a large rise (which we do have) times the probability of a large rise (which we also have) plus the probability of a positive forecast given a small rise times the probability of a small rise plus the probability of a positive forecast given no change times the probability of no change plus the probability of a positive forecast given a small fall times the probability of a small fall plus the probability of a positive forecast given a large fall rise times the probability of a large fall. This is
  • .80 times
  • .20 plus
  • .70 times
  • .30 plus
  • .50 times
  • .30 plus
  • .40 times
  • .10 plus
  • 0 times
  • .10 which is
  • .56
  • Then the probability of a large rise given a positive forecast equals the probability of a positive forecast given a large rise times the probability of a large rise divided by the probability of a positive forecast which is
  • .80 times .20 divided by .56 or .286
  • The probability of a small rise given a positive forecast equals the probability of a positive forecast given a small rise times the probability of a small rise divided by the probability of a positive forecast which is.70 times .30 divided by .56 or .375
  • The probability of a no change given a positive forecast equals the probability of a positive forecast given no change times the probability of no change divided by the probability of a positive forecast which is.50 times .30 divided by .56 or .268
  • The probability of a small fall given a positive forecast equals the probability of a positive forecast given a small fall times the probability of a small fall divided by the probability of a positive forecast which is .40 times .10 divided by .56 or .071
  • and the probability of a large fall given a positive forecast equals the probability of a positive forecast given a large fall times the probability of a large fall divided by the probability of a positive forecast which is0 times .10 divided by .56 or 0.

Slide 13

  • Now to determine the decision with the best expected value when there is a positive economic forecast
  • We return to the payoff table
  • Record the revised probabilities
  • And calculate the new expected values. These turn out to be
  • $84 for gold
  • $180 for the bond
  • $249 for the stock
  • and $60 for the CD
  • So if the economist gives a positive economic forecast,
  • the recommended decision would be the stock with an expected value of $249.

Slide 14

  • If the economist gives a negative economic forecast, the procedure is the same
  • The probability that he would give a negative forecast is the probability of a large rise and a negative forecast plus the probability of a small rise and a negative forecast plus the probability of no change and a negative forecast plus the probability of a small fall and a negative forecast plus the probability of a large fall and a negative forecast, which in turn equals….
  • The probability of a negative forecast given a large rise times the probability of a large rise plus the probability of a negative forecast given a small rise times the probability of a small rise plus the probability of a negative forecast given no change times the probability of no change plus the probability of a negative forecast given a small fall times the probability of a small fall plus the probability of a negative forecast given a large fall rise times the probability of a large fall.
  • Substituting these probabilities gives the probability of a negative forecast of
  • .44.
  • Then the probability of a large rise given a negative forecast equals the probability of a negative forecast given a large rise times the probability of a large rise divided by the probability of a negative forecast which is
  • .20 times .20 divided by .44 or .091
  • The probability of a small rise given a negative forecast equals the probability of a negative forecast given a small rise times the probability of a small rise divided by the probability of a negative forecast which is .30 times .30 divided by .44 or .205
  • The probability of a no change given a negative forecast equals the probability of a negative forecast given no change times the probability of no change divided by the probability of a negative forecast which is .50 times .30 divided by .44 or .341
  • The probability of a small fall given a negative forecast equals the probability of a negative forecast given a small fall times the probability of a small fall divided by the probability of a negative forecast which is .60 times .10 divided by .44 or .136
  • and the probability of a large fall given a negative forecast equals the probability of a negative forecast given a large fall times the probability of a large fall divided by the probability of a negative forecast which is 1 times .10 divided by .44 or .227.

Slide 15

  • So when the economist gives a negative economic forecast
  • We take the payoffs
  • And multiply by this revised set of probabilities
  • Giving expected values of
  • $120 for gold
  • $67 for the bond
  • Negative $33 for the stock
  • And $60 for the CD
  • So if the economist gives a negative economic forecast,
  • the recommended decision would be gold with an expected value of $120.

Slide 16

  • So if we gather this additional information of the economic forecast
  • The optimal strategy if the forecast is positive is
  • To invest in the stock
  • And if it is negative
  • Invest in gold

Slide 17

  • So how much is this economic information worth? This is called the expected value of sample information.
  • Recall that we found that the probability of a positive forecast is .56 and of a negative forecast is .44
  • And we determined that if we get a positive forecast, we would invest in the stock with an expected value of $249
  • And if we get a negative economic forecast, we would invest in gold with an expected value of $120.
  • So the expected return with sample information is .56 times 249 plus .44 times $120 or $192.50
  • But the strategy with no additional information was to invest in the bond giving an expected value of $130
  • The expected value of sample information is the gain in expected value of $192.50 minus $130 which is $62.50.

Slide 18

  • Efficiency
  • Is a measure of the relative value of sample information – relative to the value of perfect information
  • Hence it is a number between 0 and 1 found by
  • Dividing the expected value of sample information by the expected value of perfect information
  • For our model
  • The efficiency is 62.50 divide by 141 or .44. This number can be used to compare the value of this indicator information with other possible pieces of indicator information.

Slide 19

  • This information can be generated using the decision template.
  • First we go to the Bayesian Analysis worksheet where the original or prior probabilities will already be shown from the input on the Payoff Table worksheet
  • Then for each possible value of the indicator information we enter the conditional probabilities, here in C7 to C11 and I7 to I 11 respectively. The revised or posterior probabilities will now be given in cells E7 to E11 and K7 to K11 respectively.
  • Then click the tab for the Posterior Analysis Worksheet

Slide 20

  • Here
  • We see the
  • Indicator probabilities in cells K13 and L14 respectively
  • The revised probabilities in rows 13 and 14 respectively
  • The optimal strategy and corresponding expected values for each possible value of the indicator information – here is cells C20 through D21
  • and the expected value of sample information, the expected value of perfect information and the efficiency in cells B23, B24, and B25 respectively.

Slide 21

  • Let’s review what we’ve discussed in this module.
  • We introduced the expected value criterion for making decisions when there are probabilities for the states of nature.
  • And we introduced the concept of the expected value of perfect information which gave an upper bound for the value of any additional information.
  • We talked about imperfect or sample information
  • We’ve showed how to revise our probability estimates for the states of nature using a Bayesian approach
  • We showed how to calculate the probability of each piece of indicator information
  • We showed how to develop a strategy for making decisions based on the results of the sample information
  • We showed how to calculate the value of this extra sample information, the expected value of sample information
  • And how to calculate the efficiency of the information
  • Finally we showed how to use the decision template to perform the Bayesian analysis and generate the optimal strategy based on the results of the indicator information.

That’s it for this module. Do any assigned homework and I’ll be back to talk to you again next time.