Lecture 7_Quantifying Uncertainty and Risk

8/18/2019 Lecture 7_Quantifying Uncertainty and Risk

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Lecture 07: Quantifying Uncertainty and Risk

Beginning with this lecture we introduce uncertainty in the world and attempto quantify uncertainty and risk in this new framework. But before we do so, wewill brie‡y revisit some of the basics of mathematical statistics. We will haveto deal now with random variables which have uncertain outcomes, but withwell-de…ned probabilities.

The simplest case of a random variable x (i.e., r.v.) is one where there areonly two states of the world: 1 with probability 1/2, and -1 with probability1/2. Thus:

E (x) = x = 1

21 +

1

2(1) = 0 (1)

V ar(x) = 1

2

(1 x)2 + 1

2

(1 x)2 = 1 (2)

x =p

V ar(x) = 1 (3)

Now suppose that y is 1/3 with probability 0.9 and -3 with probability 0.1.Then:

E (y) = y = (0:9)(1=3) + (0:1)(3) = 0 (4)

V ar(y) = 0:9(1=3 0)2 + 0:1(3 0)2 = 0:1 + 0:9 = 1 (5)

So here we have two random variables which look quite di¤erent, yet themean and standard deviation are the same. Thus, standard deviation and ex-pectation are insu¢cient to characterize two separate r.v.’s. The joint distri-bution provides us with information regarding the likelihood of x and y being

jointly high or low together (x and y being on the main diagonal) or one beinghigh while the other is low (x and y being on the o¤-diagonal). However, re-member that the joint distribution is not determined by the distribution of xalone or y alone.

This leads us to the concept of the covariance of x and y. There are fourpossible combinations: x could turn out to be 1 when y is 1/3, and x could turnout to be 1 when y is -3. Further, x could be -1, and we could get 1/3 for y or-3:

cov(x; y) = P (1; 1=3)(1 x)(1=3 y) + P (1; 1=3)(1 x)(1=3 y) (6)

+P (1; 3)(1 x)(3 y) + P (1; 3)(1 x)(3 y) (7)

Covariance is giving you a sense of whether things are moving together ormoving the opposite way. The covariance is linear in x, because if you double xyou are going to double every term involving x in the expression for covariance.Other useful formulas:

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V ar(x) = Cov(x; x) (8)

V ar(x + y; x + y) = Cov(x + y; x + y) = Cov(x; x) + Cov(y; y) + 2Cov(x; y) (9)

The formula above follows from the linearity of covariance. Assuming thatx and y are independent then P (x = X; y = Y ) = P (x = X )P (y = Y ) orcox(x; y) = 0:

0.0.1 Key Observations about Uncertainty

0.0.2 Diversi…cation

Most investors care about expected returns and dislike risk (i.e., standard

deviation). For instance, assume that E (x) = E (y) = 0 and x = y = 1:Suppose I put half my money into x and half my money into y, x and y areindependent and I get half the payo¤ of each. Thus, I make half a bet and gethalf the outcome. What happens to my expectation? Well, the expectation willequal:

E (1

2x +

1

2y) = E

1

2x

+ E

1

2y

= 0 (10)

But how about the variance of this portfolio?

V ar

1

2x +

1

2y

= cov

1

2x;

1

2x

+ cov

1

2y;

1

2y

+ 2cov

1

2x;

1

2y

(11)

= 14

V ar(x) + 14

V ar(y) + 0 = 12

(12)

At …rst, it seems like a total waste of time to put half my money in each.After all, they give the same standard deviation, but in fact, they do not. If theyare independent you are drastically reducing your standard deviation. Because if they are independent the covariance is 0 and the variance of the portfolio is half the variance of each separate investment. Thus, by diversi…cation you havereduced your standard deviation without a¤ecting your expectation. The keyinsight is to look for independent risks. Thus, this is one lesson in mathematicsthat has big applications in economics.

0.0.3 Adding N independent risks

If you had N independent risks with identical means and variances, (say, E (xi) =x; V ar(xi) = 2x) what would happen to the expectation and variance of theportfolio, respectively?

E

1

N xi

=

1

N N x = x (13)

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V ar 1N

xi = 1

N 2N 2

x =

1

N 2

x (14)

or that portfolio = 1p N

x: Therefore, with many independent portfolio al-

locations you get a lot of o¤ diagonal outcomes, which are canceling each other:if one investment is turning out badly the other one is turning out well. In thisway, you leave the expectation the same, but you reduce the variance.

0.0.4 Sum of N independent variables

Remember that by summing N independent random variables, you get a nor-mally distributed random variable with a corresponding expectation and stan-dard deviation. For instance, if you add one of our random variables from above(say x) to itself a bunch of times that can only generate 1 and -1, it producestotally di¤erent outcomes: say you can get 18 1s and 12 -1s, so that gives you 6.But an outcome of 6 can also be generated by sampling and adding outcomesfrom the distribution of y: Thus even though x and y yield di¤erent outcomes,once you are adding them up you are starting to produce numbers di¤erentfrom 1 and -1, or 1/3 and -3. So surprinsingly, if you add random variablesthat are independent to each other you get something normally distributed thathas a bell shaped distribution. The normal distribution is characterized only bythe mean and standard deviation. The normal distribution is also thin tailedmeaning that the probabilities in the tails go to 0 very fast.

0.1 Iterated Expectations

One more thing that is used in economics all the time, and that we need toknow before we talk about the economics of uncertainty is called the iterated expectations . For instance, if I told you that x and y were positively correlatedand that if x turned out to be 1, that would tell you a lot about what y wasgoing to be. Therefore, knowing x is going to completely change your mindabout the expectation of y .

So conditional expectation simply means re-computing expectation usingupdated probabilities from your information. For instance, if you knew the

joint distribution of x and y and you were furher told that the good outcome of y occured (i.e., y = 1=3), then you can compute the probability that x = 1 basedon this information. For example, suppose that we knew that y = 1=3 alwayswhen x = 1 (i.e., x and y are perfectly dependent). Then, P (x = 1; y = 1=3) =0:5 and P (x = 1; y = 3) = 0: Further, we have that P (x = 1; y = 3) = 0:1and P (x = 1; y = 1=3) = 0:4: The marginal probabilities have to add up to 1.

Therefore,

P (x = 1jy = 1=3) = P (x = 1; y = 1=3)

P (y = 1=3) =

0:5

0:9 =

5

9 (15)

However, if x and y were independent, then P (x = 1; y = 1=3) = 0:45 andP (x = 1jy = 1=3) = 0:5:

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0.1.1 The Odds of Advancing the Europa League group stages

Suppose that someone might try to assess Athletic Bilbao’s chances of advancingto the Europa League playo¤s. If I ask you your opinion after the …rst game,well, obviously if Athletic Bilbao wins the …rst game your opinion is going togo up, so you are going to have a di¤erent opinion. If Athletic Bilbao loses the…rst game your opinion is going to go down, so you’ll have a di¤erent one. Butyou can ask now another question, what is your expected opinion going to be?So the law of iterated expectations is, the expectation of x has to equal theexpected expectation of x given some information.

Suppose Athletic Bilbao is 70 percent likely to win. If Athletic Bilbao winstheir …rst game, one may think it is 80 percent, and after the …rst game if Athletic Bilbao loses, one may think it has gone down to 60 percent (it hadbetter be that the average of one’s opinion after the information is the same asthe number one started with).

Thus, it is not only the expectation of x, but as you learn stu¤ you cananticipate one’s opinion is going to change, but the average opinion has toalways stay the same as x was.

And now let’s do a simple application of this. So in fact, to that veryquestion, suppose that a betor bets on Atheltic Bilabao’s chances of advancingthe group stage. Atletic Bilbao is playing three other teams and let’s supposethat Atletic Bilbao has a 60 percent chance of winning any game. What is thechance that Athletic wins a 3 game round-robin tournament (I am making asimplifying assumption that Athetic plays the three other teams on neutral siteswith no home and away games)? How do you …gure that out?

Exponential Recombining Tree

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they would win. That would be a 1. They would advance for sure (i.e., outcomeis 0.6*1 +0.4*1=1).

What would I think if they lost the …rst two games? I would know it wasover: 0.6*0 + 0.4*0=0. What would my opinion be if they split? Well, if theysplit, my opinion of them winning would be 0.6*1+0.4*0=0.6. So the oddsAthletico would qualify with 1 game left knowing that they win 60 percent of the time it’s .6. But now what do I think if Athletico wins the …rst game?What is my opinion? The odds would be 0.6*1+0.4*0.6=0.84. What would Ithink after Athletico lost the …rst game? I would think it was only a 36% (i.e.,0.6*0.6+0.4*0) chance of the Athletico winning. So at the very beginning, thechance of Athletico qualifying is 0.6*0.84+0.4*36 which is 0.684 just like above.The Excel spreadsheet Trees.xls documents even more complex scenarios.

0.1.2 Impatience and Uncertainty

Let’s suppose our uncertainty is of a di¤erent kind meaning we do not know howimpatient we are. Remember that the most important idea that we have seenso far is impatience. This is the reason why you get an interest rate and theinterest rate is the key to …nding out the value of all assets. Irving Fisher puttremendous weight on impatience. When talking about uncertainty the naturalthing to make uncertain is how impatient you are going to be. Impatience fromIrving Fisher’s point of view is the discount rate. Do we really believe thatpeople just discount the future, 1 year they discount by delta, 2 years discountby delta squared, 3 years by delta cubed, 4 years by delta to the fourth. Is itreally true that every year people think of as delta less important as the yearbefore? The argument for this is you might not live beyond a certain point intime.

But let me tell a story that seems to contradict that. Suppose someone asksyou to clean your room and they give you a choice of doing it. If somebody saysdo it today or do it tomorrow that may makes a huge di¤erence. You may thinkdoing it today is just impossible, doing it tomorrow one can be almost force intoagreeing. So clearly there is a big discount between today and tomorrow, butwhat about between a year from now and a year and a day from now? Do youthink there is any di¤erence in that? The answer is no. This is called hyperbolicdiscounting. Hyperbolic discounting is discounting much less than exponentialdiscounting. This has a tremendous importance for the environment.

If you thought that people exponentially discounted like they thought eachyear was only 95 percent–if the interest rate is 5 percent it sounds like thediscounting is .95, so if next year’s only 95 percent as important as this year,and the year after that is only 95 percent as important as the …rst year, and the

third year is only 95 percent as important as the second year, .95 in 100 yearsto the hundredth is an incredibly small number.So there is no point in doing something today and investing a lot resources in

order to clean up the environment and help people 100 years from now, becauseby discounting it this much nobody could because the future is so unimportant.You shouldn’t be investing resources now to do something that is going to have

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such a small e¤ect later. So in all the reports on the environment a crucial half of the report is devoted to what the discount rate should be.

However, some have never thought of doing the most obvious thing whichis to ask what would happen if the discounting was uncertain. All of these arecertain discount rates. So if you made the discounting uncertain what wouldyou imagine doing? So suppose you discount today at 100 percent, and maybenext period you’re going to discount at 200 percent, this is the interest rate, andthen it might go down to 50 percent. It could go up to 400 percent or it couldgo down to 100 percent again, or it could go down to 25 percent. The discountrate is thus:

= 1

1 + r (16)

Ho-Lee Interest Rate Model

The higher the r, the less you care about the future. So the question is if you ask for a dollar sometime in the future, what will people be willing to payfor it? So you know today that you think the future is only half as importantas the present. And tomorrow it might be that you think the next year is only2/3 as important as that current year, or you might think the future’s only 1/3as important as this year.

Further, two years from now you might think the future is only 1/5 asimportant as the following year. So you don’t know what it is going to be, andif anything this process seems to give you a bias towards getting really highnumbers, high discounts, meaning the future doesn’t matter. This is the mostfamous interest rate process in …nance. This is called the Ho-Lee interest rate

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model where you think today’s interest rate might be 4 percent. Maybe it’llbe 10 percent higher next year or 10 percent lower and it’ll keep going up and

down like that, and that’s the uncertainty about the interest rate. So if we thinkinterest rates are so important, and patience is so important, and we want toadd uncertainty, the …rst place to do it is to the interest rate, and the Ho-Leemodel in …nance does that.

Nobody bothered to compute this interest rate tree more than 30 years.Suppose you get 1 dollar for sure in year 1. How much would you pay for 1dollar in year 1? Well, if your discount is 100 percent, you would pay half adollar. How much would you pay for 1 dollar in year 2?

Well, you know how much more a dollar now is worth than 1 year fromnow, but you don’t know 2 years from now so you have to work by backwardinduction. So for any time I could …gure out d(t) = amount I would pay today for 1 dollar for sure at time t . And d(t) is going to go down as t goes up, and weknow how to compute it by backward induction (i.e., in the Excel spreadsheetyou just put the 1s further and further out and then you go backwards bybackward induction).

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Lecture 7_Quantifying Uncertainty and Risk