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Hello, welcome back.

In the last lecture, we talked about preferences, to describe how individuals,

households or firms evaluate trade-offs when they are faced with choices.

We talked about how a utility function can be used to represent those preferences and

measure the decision makers level of satisfaction.

Now an important dimension of decision making in finance and

economics however is uncertainty.

There's probably no decision in economics that does not involve risk,

so how do you feel about risk for example?

Well, if your like most people, you probably don't like risk that much.

In fact, studies of human behavior faced with risk,

strongly suggest that human beings are risk averse.

We're risk averse, we don't typically like risk that much, so for

example, most households will want to ensure their assets.

If an actuarial fair insurance is offered to them or

most investors will not want to purchase risk assets if they are not

delivered an expected return that is larger than the risk re-rate.

So when we think about the optimal portfolio choice problem,

we assume that investors are risk averse.

In this lecture, we're going to talk about what this means,

how we describe risk aversion?

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As human beings, we all come with different genes and

preferences, and that also goes for attitudes towards risk.

Therefore, risk averse is key dimension of how we describe preferences and

how we might construct.

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Individuals with a high degree of risk aversions

will value safety at a high price while others may not.

So somebody with a high risk aversion will be reluctant to

face a situation with a risky outcome and

she might be willing to pay a high insurance premium to avoid that risk.

Or alternatively, a risk adverse individual will require to

be compensated If she needs to bear that risk.

So let me illustrate this point graphically,

we're we are rescuers we don't like risk, right?

That means we prefer the sure outcome instead of the uncertain outcome the bird

and the hand.

Let me illustrate this with the picture for you, okay?

So let suppose that we have the following utility function again I have.

The Y-axis here, right that gives the level of duty and

the X-axis wherever outcomes are, all right?

And suppose there are two outcomes, X and Y and

let suppose you took the function look something like this, right?

Which means that X gives me,

this level of utility, you have X, right?

And Y gives me, right?

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Your Y, right?

And suppose that they each outcome happens with equal probability, right?

Health and health, right?

So what is your expected utility?

Well, clearly, it's one-half times U of X, right?

The level of utility you get from X, plus one-half times U of Y, right?

Where is that on the graph?

Well, it's going to lie on this diagonal line

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I am a terrible drawer, sorry,

which is going to be halfway at the midpoint of tis diagonal line.

All right, so that is the, let me point this,

this is one-half u of x plus one-half u of y.

Now, what if you were getting the expected

outcome for certain, for sure.

What is the expected outcome?

Well, it's going to be the midpoint here one-half X plus one-half Y.

What if instead of having that app insert an outcome you were given that for

sure outcome.

What would your utility level be?

Well, you would be getting this much more utility,

so in fact, you would be getting a higher

utility from that certain outcome, right?

This is U of one-half x plus one-half Y,

which is going to be greater, right?

How much more utility you get, defines how risk averse you are, right?

Individuals who are risk averse will get a much higher utility

out of getting that sure outcome, all right.

And what determines that, well graphically you can see that it's that concavity of

the line that's the difference between that yellow and blue point.

Which tells us how much more you value the sure outcome, so

the more risk averse the investor, the more she wants the sure outcome.

In other words, the more concave is the acuity function,

the more risk averse is the investor.

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So how do we measure an individual's degree of risk aversion?

Do you know what your risk aversion coefficient is?

Of course, I don't expect you to just rattle a number but we can estimate.

Risk aversion indirectly, right?

So, suppose I gave you the following lottery, right?

So suppose you can win $1,000 with 50% probability,

or win $500 with 50% probability.

That is, you get $500 for sure but

you also have the possibility of winning $1,000.

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Doesn't want to take the lottery at all.

On the other hand, if you're willing to pay the fair value of the lottery,

750, right?

50% times 1,000, a 50% times 500 which gives you 750, then you are risk neutral.

You have a risk aversion coefficient of 0.

Now most people are willing to pay somewhere

between 540 to a little over 700,

all right, to enter this lottery.

So that says that most individuals have risk aversions between 1 and

10, 10 being really really rare.

Now, where is this coming from?

This comes from a large body of experiments and survey evidence.

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If you're familiar with this show, the stakes on this show are very high and

only the winning player gets to return back to the show.

So there’s a lot of pressure and

furthermore the participants are very smart so

you can't say that they are not that they don't know what they're doing, right?

And the study finds that the risk aversion levels that

they find are fairly low, near risk neutrality.

Now in the financial world, financial advisers often use

questionnaires to try to infer risk aversions of their clients.

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Risk aversion is the notion that in face of uncertainty or

risk, human beings, we are, generally averse to risk.

That is, faced with two alternatives, we will prefer the one with less risk