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Hand in Problem Set 3: 4.5, 5.4, 6.3, and 6.4.
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E26 September, Tuesday. Risk I have posted some notes on the math of defining risk aversion.My new notes are on my weblog and at risk.htm.I have other weblog entries that I might talk about on prospect-theory-and-loss-aversion, on Gollier's table-to-check-your-own-level of risk aversion, and on Knightian-risk-and-uncertainty discussed using two-card-bets. For reference: "When Does Extra Risk Strictly Increase the Value of Options?" Eric Rasmusen, forthcoming in The Review of Financial Studies.
Due: Problem Set 3: 4.5, 5.4, 6.3, and 6.4. Answers are (or will be) here. 8FR08$! 1 4?{|J006}J0n{~J0J0J0#J0*gJ0xJ0?DH
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The idea of risk aversion is that people like to keep their consumption at a steady level rather than up and down. They prefer to consume $50,000 per year rather than $10,000 one year and $90,000 the next.
The conventional way to represent this is with a strictly concave utility function, such as
U = log(C)
If the utility function is concave like this, then from any starting point a loss of $1 hurts more than a gain of $1 helps.|+||@8XH
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RLOSS AVERSION Suppose makes $500/month payments for his car.
If he makes $10/month less, he is in trouble.
If he makes $20/month less, that isn't much worse.
Hence, his behavior will seem to violate the assumption of concave utility. Suppose he has a 5% chance of losing $10 and a 5% chance of losing $20.
He would prefer to have a 1% chance of losing $10 and an 8% chance of losing $20. He is positively risk loving!
On the other hand, suppose this person starts to make $10/month more In the long run, when his car needs replacing, he will smoothly change to $510/month. S
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ZApex stock: $0 or $3 with equal probability
Brydox stock: $0 or $3 with equal probability
A share of stock costs $1.
You have $6 to invest.
(X) If you invest in 6 shares of Brydox, your end wealth is either $0 or $18, an expected value of $9.
(Y) If you invest in 3 shares of Apex and 3 of Brydox, your end wealth is either $0, $9, $9, or $18, with equal probabilities, an expected value of $9.
The diversified strategy is less risky.
Thus, if there are lots of independent assets, you should spread your investment across all of them. The risk from any 1 company's behavior would be trivial.
Thus, you would be happy for that company to ignore risk and simply to maximize its expected value. 6">.pH
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Suppose you could invest in the stock of Apex, which yields either $0 or $3 with equal probability, or the stock of Brydox, which also yields either $0 or $3 with equal probability, but with independent probability.
A share of stock costs $1, and you have $6 to invest. All this is the same as in my previous example.
There, you were going to invest either $6 in Brydox or $3 in Brydox and $3 in Apex.
Now, however, you are thinking of investing either $3 in Apex and $3 in cash, or $3 in Apex and $3 in Brydox. Which is less risky?
If you invest in both companies, will the risks tend to cancel out?
The risks will indeed *tend* to cancel out, but the risk is neither higher nor lower--- the change in wealth pattern is ambiguous.*Pu
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ySuppose Jack might earn $100, $200, $300, or $400, with equal probabilities, an expected wealth of $250. His information partition is (100,200,300,400)-- he can't rule out anything.
Jack is very risk averse, so his utilities from known wealths are
U(100) = 0 - eU(200) = 100 - eU(300) = 120 - eU(400) = 128 - e,
where e is the number of hours he works painting a house. zxH
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1Tom offers to tell Jack whether his wealth will either low or high, but not the precise level, if in exchange Jack will spend 2 hours painting Tom's house.
Thus, Jack can get the information partition (100,200), (300,400).
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Originally, Jack s expected utility is
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If he pays the 2 hours, his expected utility is
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which equals
-2 + .25U(100) + .25 U(200) .25U(300) + .25 U(400)
which equals
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D Resolving subjective uncertainty in beliefs is different from reducing risk. Suppose Jack could pay 2 hours to change the probability distribution to
(.5 probability of $200, .5 probability of $300)
His expected utility would be
-2 + .5 (100) + .5 (120) = -2 + 110 = 108 (better than 81)
But that is not just a change in his beliefs (though they would change too).
It is a change in the probabilities of various amounts of cash he might receive.
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) Another way to look at all this is that using Expected Utility Theory, Jack cares only about end states, not about his beliefs along the way. His utility from getting $400 on Friday is entirely pleasure that he gets on Friday when he receives the cash (or, better, when he spends the cash). He get no pleasure on Tuesday, Wednesday, or Thursday from anticipating his happiness on Friday.
We could model anxiety earlier in the week about the uncertainty, but it would take a different model. We could, in the example earlier, say that today, before Jack knows his future wealth, he gets immediate utility of -20*(4-N) if his information partition has N information sets. Then, it would make sense for Jack to pay 20 hours of labor to increase N from 1 to 2, because it would cost 2 in utility and gain 20. L*$c H
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In addition to the Friday utilities above, Jack might have Tuesday utility of
V(100) = -400V(200) = 100V(300) = 120V(400) = 126
from perfect information on Tuesday, and from coarser info:
V(100,200) = -300V(300,400) = 122
V(100,200,300,400) = 0
The expected utility gain as a result of getting better information would then be
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Eric Rasmusen, erasmuse@indiana.edu
26 September 2006
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Hand in Problem Set 3: 4.5, 5.4, 6.3, and 6.4.
(" 4 " 4PPa b :pDiversification reduces risk ( <qThe Fallacy of Large Numbers Hw=Combination of risk does not reduce risk: Only division does.>*((( d, Caring about Probabilities of Probabilities--( >rThe Decision ?sExpected Utility Calculations @t
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T (A) One strategy is to invest $3 in Apex and $3 in cash. This will have a return of $3 or $12, with equal probability. the expected return will be $7.50.
(B) The other strategy is to invest $3 in Apex and $3 in Brydox. This will have a return of $0, $9, $9, or $18, with equal probabilities, as we found earlier. The expected return will be $9.
If you were risk neutral or slightly risk averse, you'd choose strategy (B)
But suppose your utility function was the following strictly concave function:
U = 100 + log(C) if C greater than 2
U = log (C) if C less than or equal to 2
That's discontinuous, but there's nothing wrong with that.
The main thing for this person is to avoid consumption of less than 2. Thus, he would prefer strategy (A).,+H
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@ (A) One strategy is to invest $3 in Apex and $3 in cash. This will have a return of $3 or $12, with equal probability. the expected return will be $7.50.
(B) The other strategy is to invest $3 in Apex and $3 in Brydox. This will have a return of $0, $9, $9, or $18, with equal probabilities, as we found earlier. The expected return will be $9.
If you were risk neutral, you'd choose strategy (B) because of the higher expected return. But its risk is neither higher nor lower than A's, so which strategy you would prefer if you were risk averse depends on your utility function-- on the third derivative of it, actually. See Douglas W. Diamond, "Financial Intermediation and Delegated Monitoring," The Review of Economic Studies, Vol. 51, No. 3. (July 1984), pp. 393-414; Paul Samuelson (1963) "Risk and Uncertainty: A Fallacy of Large Numbers," Scientia, 98: 108-113.
It's easy to see why. First, if you were close to risk neutral, you'd still prefer (B), because of its higher expected value. So all we have to do is understand why anyone would ever prefer strategy (A). The reason is that somebody might really hate a return of $0. Suppose your utility function was the following strictly concave function:
U = 100 + log(C) if C greater than 2
U = log (C) if Cless than or equal to 2
That's discontinuous, but there's nothing wrong with that.
The main thing for this person is to avoid consumption of less than 2. Thus, he would prefer strategy (A).Pvn%H
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$On-screen ShowKelley School of Businessx[dArialTahomaDefault DesignSlide 1 Readings Handouts - Caring about Probabilities of Probabilities
The DecisionExpected Utility CalculationsReducing RiskAnxietyAnxiety Diversification reduces risk >Combination of risk does not reduce risk: Only division doesThe Fallacy of Large Numbers
ApplicationsHousing as Investment Slide 17Course WebsiteFonts UsedDesign Template
Slide Titles 8@_PID_HLINKSAd6(http://www.rasmusen.org/g601/0.g601.htm/http://rasmusen.org/g601/overheads/09-risk.pdfohttp://www.rasmusen.org/x/2006/09/25/housing-as-an-investment-and-the-international-nondiversification-puzzle/+http://rasmusen.org/g601/readings/risk.htmHhttp://www.rasmusen.org/x/2005/03/29/prospect-theory-and-loss-aversion/Zhttp://www.rasmusen.org/x/2004/11/17/risk-aversion-gollier-table-to-check-your-own-level/Shttp://www.rasmusen.org/x/2004/09/15/knightian-risk-and-uncertainty-two-card-bets/4http://www.rasmusen.org/papers/options_rasmusen.pdf'http://rasmusen.or01/readings/risk.htmThttp://rasmusen.org/g601/readings/risk.htm8http://www.rasmusen.org/x/2005/03/29/prospect-theory-and-loss-aversion/http://www.rasmusen.org/x/2005/03/29/prospect-theory-and-loss-aversion/http://www.rasmu@wʚ;l!k8ʚ;g41d1d;T0ppp@<4dddd,!0`,.<4BdBd,O$0`80___PPT10
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Eric Rasmusen, erasmuse@indiana.edu
26 September 2006
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:eZZ0%,#&^@Readings $P$eDHandouts@
Hand in Problem Set 3: 4.5, 5.4, 6.3, and 6.4.
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(A) One strategy is to invest $3 in Apex and $3 in cash. This will have a return of $3 or $12, with equal probability. the expected return will be $7.50.
(B) The other strategy is to invest $3 in Apex and $3 in Brydox. This will have a return of $0, $9, $9, or $18, with equal probabilities, as we found earlier. The expected return will be $9.
If you were risk neutral or slightly risk averse, you'd choose strategy (B)
But suppose your utility function was the following strictly concave function:
U = 100 + log(C) if C greater than 2
U = log (C) if C less than or equal to 2
That's discontinuous, but there's nothing wrong with that.
The main thing for this person is to avoid consumption of less than 2. Thus, he would prefer strategy (A). (Doug Diamond, Paul Samuelson)>+"H
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Eric Rasmusen, erasmuse@indiana.edu
26 September 2006
RISK
:eZZ0%,#&^@Readings $P$eDHandouts@
Hand in Problem Set 3: 4.5, 5.4, 6.3, and 6.4.
(" 4 " 4PPa b :pDiversification reduces risk ( <qThe Fallacy of Large Numbers Hw=Combination of risk does not reduce risk: Only division does.>*((( d, Caring about Probabilities of Probabilities--( >rThe Decision ?sExpected Utility Calculations @t
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Suppose you could invest in the stock of Brydox, which yields either $0 or $3 with equal probability, or the stock of Brydox, which also yields either $0 or $3 with equal probability, but with independent probability.
A share of stock costs $1, and you have $6 to invest.
All this is the same as in my previous example.
Now, however, you are thinking of investing either
(1) $3 in Apex and $3 in cash,
OR (2) $3 in Apex and $3 in Brydox.
Which is less risky? If you invest in both companies, will the risks tend to cancel out?
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f G601, IO I
Eric Rasmusen, erasmuse@indiana.edu
26 September 2006
RISK
:eZZ0%,#&^@Readings $P$eDHandouts@
Hand in Problem Set 3: 4.5, 5.4, 6.3, and 6.4.
(" 4 " 4PPa b :pDiversification reduces risk ( <qThe Fallacy of Large Numbers Hw=Combination of risk does not reduce risk: Only division does.>*((( d, Caring about Probabilities of Probabilities--( >rThe Decision ?sExpected Utility Calculations @t
Reducing Risk DuAnxiety EvAnxiety aAbBCourse Website/_@`AcBdCkF ,.;=ABCFGIrAXϐI1-gs/risk.htm8http://www.rasmusen.org/x/2005/03/29/prospect-theory-and-loss-aversion/http://www.rasmusen.org/x/2005/03/29/prospect-theory-and-loss-aversion/http://www.rasmusen.org/x/2004/11/17/risk-aversion-gollier-table-to-check-your-own-level/http://www.rasmusen.org/x/2004/11/17/risk-aversion-gollier-table-to-check-your-own-level/dhttp://www.rasmusen.org/x/2004/09/15/knightian-risk-and-uncertainty-two-card-bets/http://www.rasmusen.org/x/2004/09/15/knightian-risk-and-uncertainty-two-card-bets/fhttp://www.rasmusen.org/papers/options_rasmusen.pdffhttp://www.rasmusen.org/papers/options_rasmusen.pdfLhttp://rasmusen.org/g601/tasks/ps3.pdfLhttp://rasmusen.org/g601/tasks/ps3.pdf`/0DArial6T0ȃ0``P.T0DTahoma6T0ȃ0``P.T0"@.
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| G601, IO I
Eric Rasmusen, erasmuse@indiana.edu
26 September 2006 (revised after class)
RISK
N{PP0+( ,#<^@Readings $P$eDHandouts@
Hand in Problem Set 3: 4.5, 5.4, 6.3, and 6.4.
(" 4 4PPa d, Caring about Probabilities of Probabilities--( JxThe Decision KyExpected Utility Calculations Lz
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1Tom offers to tell Jack whether his wealth will either low or high, but not the precise level, if in exchange Jack will spend 2 hours painting Tom's house.
Thus, Jack can get the information partition (100,200), (300,400).
Should he accept? H
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Originally, Jack s expected utility is
.25(0) + .25(100) + .25(120) + .25 (128) = 81
If he pays the 2 hours, his expected utility is
-2 + .5[.5U(100)+ .5 U(200) ]+ .5[.5U(300)+ .5 U(400) ]
which equals
-2 + .25U(100) + .25 U(200) .25U(300) + .25 U(400)
which equals
-2 + .25(0) + .25(100) + .25(120) + .25 (128) = -2 + 81 = 79. 6XUH
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D Resolving subjective uncertainty in beliefs is different from reducing risk. Suppose Jack could pay 2 hours to change the probability distribution to
(.5 probability of $200, .5 probability of $300)
His expected utility would be
-2 + .5 (100) + .5 (120) = -2 + 110 = 108 (better than 81)
But that is not just a change in his beliefs (though they would change too).
It is a change in the probabilities of various amounts of cash he might receive.
He has ruled out getting a mere $100. 6H
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) Another way to look at all this is that using Expected Utility Theory, Jack cares only about end states, not about his beliefs along the way. His utility from getting $400 on Friday is entirely pleasure that he gets on Friday when he receives the cash (or, better, when he spends the cash). He get no pleasure on Tuesday, Wednesday, or Thursday from anticipating his happiness on Friday.
We could model anxiety earlier in the week about the uncertainty, but it would take a different model. We could, in the example earlier, say that today, before Jack knows his future wealth, he gets immediate utility of -20*(4-N) if his information partition has N information sets. Then, it would make sense for Jack to pay 20 hours of labor to increase N from 1 to 2, because it would cost 2 in utility and gain 20. L*$c H
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In addition to the Friday utilities above, Jack might have Tuesday utility of
V(100) = -400V(200) = 100V(300) = 120V(400) = 126
from perfect information on Tuesday, and from coarser info:
V(100,200) = -300V(300,400) = 122
V(100,200,300,400) = 0
The expected utility gain as a result of getting better information would then be
0 + .5(-300) + .5 (122) = -150+ 61 = -91. H
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LOSS AVERSION Suppose Joe makes $500/month payments for his car.
If he makes $10/month less, he is in trouble.
If he makes $20/month less, that isn't much worse.
Hence, his behavior will seem to violate the assumption of concave utility. Suppose he has a 5% chance of losing $10 and a 5% chance of losing $20.
He would prefer to have a 1% chance of losing $10 and an 8% chance of losing $20. He is positively risk loving!
On the other hand, suppose this person starts to make $10/month more In the long run, when his car needs replacing, he will smoothly change to $510/month. (here, use my old weblog entry on prospect theory)
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Insurance companies don t work by adding up risks that cancel out. They work by dividing risk among lots of policyholders or stockholders.
Diversification across time is a misleading idea. It is not true that it s safe to invest in stocks so long as you keep your money in for 50 years, because historically no 50-year period has ever had negative stock returns. We d expect that over 50 years the return would be very close to the average per decade say, a 90% chance of that. But there would be some chance of a truly horrendous loss from being unlucky in each of those 50 years. And we re not likely to observe that bad luck in many 50-year periods a peso problem .
This is also related to the Gambler s Ruin paradox that if you keep taking gambles with better than fair returns, you ll go bankrupt eventually with probability one. OOH
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| G601, IO I
Eric Rasmusen, erasmuse@indiana.edu
27 September 2006 (revised after class)
RISK
N{PP0+( ,#<^@Readings $P$eDHandouts@
Hand in Problem Set 3: 4.5, 5.4, 6.3, and 6.4.
(" 4 4PPa d, Caring about Probabilities of Probabilities--( JxThe Decision KyExpected Utility Calculations Lz
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b Suppose I have $40,000 per year in labor income and $100,000 in wealth. This year, I have two choices.
I could invest $100,000 in bonds for a safe 10% return, and pay $10,000 in rent for a house.
B. I could invest $100,000 in a house and not pay rent. House prices swing wildly, between $80,000 and $120,000, with an expected future price of $100,000.Bi^i ^ 8XH
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