程序代写代做代考 decision tree graph Risk Preferences

Risk Preferences
and Utility CIS 418
Source: S. Bodily, 2007

A dream of chances
Those dreams are built from losing lottery tickets, by Brooklyn-based artists Adam Eckstrom and Lauren Was and it’s entitled Ghost of a Dream. The tickets were discarded by unlucky patrons. “Chance city” was built by the artist Jean Shin.
Simon Business School CIS-418 Ricky Roet-Green

Are you risk averse?
Suppose you owned a lottery ticket that was equally likely to result in $100 loss and $125 gain.
How much would you accept for this ticket?
Expected Monetary Value (EMV)
50%x$125+50%x(-$100) = $12.5
Certainty Equivalent (CE)
the price at which you’d be willing to sell the ticket.
Risk premium (RP)
how much of the EMV you’d be willing to give up to avoid the risk of losing money.
RP = EMV-CE
If RP>0 you are risk averse. If RP<0 you are risk prone. If RP=0 you are risk neutral. Simon Business School CIS-418 Ricky Roet-Green A monetary utility function translates wealth into utility Does this utility function represents the utility of a risk-averse, risk-neutral or risk-prone decision maker? Explain by showing an example. Simon Business School CIS-418 Ricky Roet-Green 5 Expected Utility vs. Certainty Equivalent Risk Premium Under the expected utility model, decision makers make choices that maximize their expected utility. The same choices also maximize the certainty equivalent. Simon Business School CIS-418 Ricky Roet-Green Various functions can be used to model risk • Risk-averse function: Concave. • Risk-neutral function: Linear. • Risk-prone function: Convex. Simon Business School CIS-418 Ricky Roet-Green Are you risk averse? Here is an explanation by Veritasium: https://www.youtube.com/watch?v=vBX-KulgJ1o Simon Business School CIS-418 Ricky Roet-Green 8 Example 1: A simple portfolio problem There is an investment that for every 1$ invested returns $4.3 or $0 with equal probability. My current wealth is $14,000. How much of $14,000 should I invest? -- The more I invest, the higher my expected net wealth, but my risk goes up as well. Simon Business School CIS-418 Ricky Roet-Green 9 My optimal investment would depend on my risk preferences • A decision maker shows constant risk aversion if she has the same positive risk premium for any two risky opportunities that have respective outcomes that differ only by a constant amount. Therefore, her expected utility would be modeled by a negative exponential function: • A decision maker shows decreasing risk aversion if she has decreasing risk premium for any two risky opportunities that have respective outcomes that differ only by a constant amount. Therefore, her expected utility would be modeled by a logarithmic function: EU 1eCE R EU  ln CE  A • R, A = Risk tolerance. Simon Business School CIS-418 Ricky Roet-Green 10 Risk premium as a function of initial wealth Lower risk premiums = higher risk Simon Business School CIS-418 Ricky Roet-Green 11 Risk as a function of initial wealth With the logarithmic utility the percentage of total wealth invested in the risky investment stays the same as the amount of wealth changes. Simon Business School CIS-418 Ricky Roet-Green 12 Optimize by maximizing EU Simon Business School CIS-418 Ricky Roet-Green 13 Optimize by maximizing CE Certainty equivalent and expected utility graphs (previous slide) have different shapes. But are maximized by the same Simon Business School CIS-418 Ricky Roet-Green 14 investment amount. Calculate the Certainty Equivalent The expected utility of the uncertain investment is equal to the utility of certainty equivalent. Negative Exponential Utility Logarithmic utility Logarithmic Utility EU lnCEA gamble  expEUgamble  CE  A EU gamble  1eCE R utility of certainty equivalent  eCE R 1EUgamble   CE  expEU CE Rln 1EUgamble   CERln 1EUgamble  Simon Business School  A Ricky Roet-Green CIS-418 gamble Calculate the Expected Utility Investment: for every 1$ invested, returns $4.3 or $0 with equal probability. My current wealth is $14,000. Calculate EMV, CE and RP for $2000 investment. Simon Business School CIS-418 Ricky Roet-Green 16 Initial Wealth Investment R (risk tolerance) $ 14,000.00 $ 2,000.00 $ 4,166.00 Probability Expected Lose 0.5 $ 12,000.00 0.943891 Win 0.5 $ 20,600.00 0.99288 CE $ 14,389.93 RP $ 1,910.07 Net Wealth $ 16,300.00 Utility 0.968385 EU 1exp(12,000/R) CE  Rln(10.968) Example 2: Keep the investment? The decision maker owns an investment that will result in personal wealth of either $21,000 or $11,000 in today’s dollars with equal probability Q1. The decision maker can choose to o Keep the investment o Sell this investment now for $14,000 o Sell half of this investment now for $7,000 and keep the other half Which option will be preferred? Q2. What is the minimum price that the decision maker will accept now for the entire investment? Assume the decision maker has constant risk aversion with risk tolerance parameter R = $4166 . Simon Business School CIS-418 Ricky Roet-Green Option 1: Write VBA Functions • To increase efficiently, we can write a VBA function to calculate the Negative Exponential Utility and the Certainty Equivalent: Function NEXPEU(CE, R) NEXPEU = 1 - Exp(-CE / R) End Function Function CE_NEXPEU(EU, R) CE_NEXPEU = -R * Log(1 - EU) End Function Simon Business School CIS-418 Ricky Roet-Green Option 2: Compare the expected utilities (or CEs) keep Sell for 14K Decision: Sell 1⁄2 for 7K Simon Business School CIS-418 Ricky Roet-Green 19 Option 3: Use a decision tree 50% Win 21000 Keep 21000 21000 0.993531471 0 13526.0248 50% 0.961100171 Lose 11000 11000 11000 0.928668872 3 Sell 14291 14000 0.967626 14000 14000 0.965283413 50% Win 17500 Sell Half 10500 17500 0.985014498 7000 14291.00156 50% 0.967625663 Lose 12500 5500 12500 0.950236828 Simon Business School CIS-418 Ricky Roet-Green 20 For this investor, the second 50% of this investment are worth less than 7K in CE, but not the first 50% Simon Business School CIS-418 Ricky Roet-Green 21 Q2: The minimum this investor will accept right now for the investment is $13,256 Simon Business School CIS-418 Ricky Roet-Green 22 Example 3: Bet on what horse? I would like to bet $5000 on a horse. The odds, and my beliefs about the probability of winning are given below. So if I place $100 bet on Tea Biscuit, and Tea Biscuit comes in first I would get $1800. That will happen with 15% probability. With 85% probability Tea Biscuit will not come in first, and I will lose my $100. Simon Business School CIS-418 Ricky Roet-Green 23 Sample spreadsheet to compare utilities of betting on different horses EU 1eCE R Simon Business School CIS-418 Ricky Roet-Green 24 What if you could split the bet among the horses? Built a spreadsheet that includes the above data. • What would be your objective? • What are the decision variables? • What are the constraints? Optimize using two different objectives: risk-averse vs. risk-neutral. Compare the results. Simon Business School CIS-418 Ricky Roet-Green 25