1.8 Decision making with risk Flashcards
What is the difference between risk and uncertainty and what are we going to deal with?
risk: facing an uncertain but quantifiable future ( e.g. you don’t know if its going to rain tomo or be sunny but a meteorologist can assign probabilities to these events.
uncertainty: facing an uncertain and unquantifiable future
What do agents use which allows them to choose between risky alternatives?
Lotteries
What is a lottery and show the mathematical distrubtion of it?
A lottery is a random variable with a probability distribution:
• List of different outcomes that might happen (in different ‘states of nature’) ( x1, x2 … xn)
• Probability associated with each outcome (or state) ( Pie1 )
Use this to make a lottery, if you flip a coin and get heads you win £10, if you get tails you lose £1
Example 2: you have £100 and are considering paying £1 for a lottery ticket.
There is a one in a thousand chance of winning, in which case you win £1000.
Otherwise, you get nothing! ( showcase this in a lottery)
How do we rank lotteries, which allow us to determine whether someone takes on a gamble? ( KEY)
We find out the Expected Utility the lotteries or the initial wealth, which gives the highest Expected utility is the one you choose.
What is the mathematical formula for expected utlitiy and what is expected utility theory? If there is no utility function then what?
We cant use the expected utility formula, hence we don’t know, we will need to find out what type of risk appetite the person has.
There is no right or wrong answer…which you prefer depends on your
preferences and your initial level of wealth.
Although we could look at the spread of wealth of the 2 random variables and deduce from there.
Lets say initial wealth is £100, what is the state of nature for each lottery in the good state and bad state?
1) good state it is 121 bad state 19.
2) good state it is 169 bad state is 64
3) good state is 225 bad state is 16.
As the individual will pick the lottery that maximises his expected utility, we find the expected utility of each one. Remember we do the same thing for his initial wealth, this is if he takes none of the lotteries.
If the income rises, will the answer change e..g to £1000?
Yes it will, actually lottery 3 ranks the highest.
Why is expected utility subjective and when is it not?
The probabilities often based on individual beliefs about the future.
Sometimes probability is based on an objective process, where there isn’t much scope for psychology. e.g. flipping a coin the probability is half half.
What is a really important property of expected utility?
It is cardinal
What does it mean that Expected Utility is cardinal?
You cannot do a monotonic transformation of utility functions can keep the preference ordering across lotteries the same. e.g. if i square the utlitiy function, i change my risk appeite.
So what does the utility function tell us?
It tells us the shape of the function.
What does the 3 attitudes to risk and what are their shapes?
Risk averse- individual has a concave shape from origin its slope gets flatter as wealth increases.
Risk loving - individual has a convex shape from origin its slope gets steeper as wealth increases.
Risk neutral - has a linear utility function
What is the difference between Expected utility and Expected value?
The Expected value for an outcome = sum of values of each outcome, weighted by the probaility of each outcome ( like mean)
The Expected utility = probability of each outcome, weighted by utility of each outcome
What does the expected value mean in words compared to expected utlitiy?
TBA
For each case what is higher the Expected value or expected utility?
Risk adverse -the utility of expected value of a lottery is
higher than the expected utility of playing the lottery for a risk averse person.
Risk loving - the utility expected value of a lottery is
lower than the expected utility of playing the lottery for a risk loving person.
Risk natural = the utility expected value of lottery = expected utility of the lottery.
We will see later on.
Why is it for a risk adverse person the shape is concave utility function?
Diminishing marginal utility of income - the additional utility of money, gets lower and lower, as you get more money. Because a risk averse person obtains more utility from certain income than from risky income, it follows that a smaller amount of certain income generates the same utility as the risky income.
Why is it for a risk loving person has a convex utility function?
Increasing marginal utility of wealth - the additional utility of money, gets higher and higher as you get more money. Because a risk loving person obtains less utility from certain income than from risky income.
Why does a risk neutral person have a linear utility function?
Constant marginal utility of income. if he is indifferent between a certain given income and an uncertain income with the same expected value. so his marginal utility remains constant with an increase in income.
Show some examples of utility functions of a risk adverse person?
U(w) = log(w)
Show some an example of a risk loving and risk neutral utility function?
So what are the 2 ways we know if a persons preferences are risk adverse, risk loving or risk neutral?
1) looking at utility function
2) Compare Expected utility vs Expected value. If Expected value > Expected utility of playing lottery = risk adverse.
If expected value < expected utility of playing lottery = risk loving
if expected value = expected utility of playing lotterey = risk neutral.
What is certainty equivalent?
Every lottery of uncertain income has associated with it a fixed amount of money which is equivalent to you in utility terms. e.g. this portfolio of shares is equivalent to a cheque of £1000. ( trying to find what certain amount of money gives you the same utility as uncertain amount of money). We compare to the Expected value of the lottery( it must give same utility.
If your a risk neutral, what would be your CE in relation to the expected payoff of the risky income?
Do the same for a risk neutral and risk loving?
1) Risk adverse - you don’t like risk so you are willing to accept less wealth
2) Risk loving- you like risk so you want to accept more money on average to take on risk.
3) You don’t care about risk.
The answer here depends on risk appetite but first what would the certainty equivalent be here? What be the answer for a risk neutral, loving and averse?
- If you chose less than £116.5, you are risk averse CE(l)E(l)
- If you chose exactly £116.5, you are risk neutral CE = E(l)
How would we illustrate this Lottery with initial wealth of 100 and utility function U(w) = SQUARE ROOT W?
Can both outcomes happen?
Y axis - Utility, X axis is wealth
We know that the utility function is concave so we draw that shape and we draw the 2 states of nature and the utility of the 2 outcomes on y axis.
ITS IMPORTANT TO NOTE THAT ONLY ONE OF THE 2 OUTCOMES CAN HAPPEN.
How can i portray the expected value of the 2 states of nature and why the case ( its a weighted average) ( if it the probabilities were (1,0) what would it mean. What is the utility of this the expected value?
We draw a line connecting the 2 points where the states of nature are. ( its not on the utility function line because its uncertain income)
As probabilities are half half ( you know the E(l) will be in the middle, or you can just find the expected value at put on line.
If the probabilities were (1,0) then you know for certain you are getting 69 more .
The utility expected value is the same as the lottery .
We said for a risk averse person the the utility of expected value > expected utility of playing the lottery. Show this on the diagram. Y and x axis.
That is the utility the consumer would get from receiving the 116.5 with certainty. > the expected utility of lottery which is uncertain.
What does the gap between the utlitiy of expected value of the lottery ( certain income> utility of the lottery
Its called the Jensons inequality. If this holds you always have a risk adverse person. ( in other words the utility of getting 116.5 for certain is greater than the
utility expected payoff of 116.5 which is uncertain.)
How would we illustrate the Certainty equivalent?
CE - the the fixed amount of money which is as good as the lottery ( same utlitiy) , remember for a risk adverse person the CE is less than Expected payoff of uncertain income.
How would you calculate the CE?
You would equate the utility function to the Expected utility, then solve for w
We know it must be less than Expected value.
In a question they might ask you to to find CE and explain why it means the person is risk averse so do it here? ( THE QUESTIONS HAS TOLD YOU TO DO THIS WAY, SO DO NOT USE JENSONS INEQUALITY
(CE) is a fixed amount that makes Pedro as well off as the lottery:Since CE is less than the expected value of the lottery, he is willing to sacrifice wealth to
eliminate risk. SO 110.25<116.5
What do we call the gap between the certainty equivalent and the expected value? and how is it similar to idea in finance?
Risk premium= similar to the idea of risk return ( you have to have bigger expected return on lottery as the person is taking on more risk.
If the person was risk loving what would the value of risk premium be?
It would be negative. ( they will be willing to take on risk, with less money as they enjoy it so much)
What does it imply for different levels of…..
B. NO they could have the same utility function but choose differently because it depends on initial wealth.
As people get wealthier, they become less risk averse, even for the same ulitiy function. ( low income your incredibly risk averse, the gradient is steep compared to higher income)
So we can say people are risk averse through our techniques, e.g. Jensens inequality, Risk premium and utility function, but how do you show that one person is more risk averse than someone else? What do we know about concave FOC derivatives and SOC. If the person is risk loving what is the SOC derative?
Yes its called the Arrow-Pratt coefficient of absolute risk aversion, you take the second derartive and divide by the first.
We know for concave functions, the first derivative is positive but the second is negative so that means the ARA will be positive.
We know for convex functions, you have negative second derative ( risk loving ARA is negative.
ARA is 0 iF RISK NEUTRAL , AS SECOND DERATIVE FOR LINEAR LINE IS 0 .
What is another way to say someone is more risk averse than someone else? ( HINT it gets rid of W)
FIND ARA( what happens when wealth increases AND RPA Speak generally when we use ARA and when we use RPA.
Mutiply absolute risk aversion by W.
1/2w tells us as wealth goes up, the risk aversion comes down.
So exam questions could ask you
For ARA we use if wealth level is different.
RPA is for a given level of wealth. ( if we took 2 people with same functions and we looked at relative risk aversion and 1 had a half and the other 2/3, the person with 2/3 is more risk averse. than person with half.
Which of the 2 lotteries do you prefer?
As we don’t have a utility function and are not told initial wealth, originally we wouldn’t know but looking at it intuitively we could derive an answer.
It’s clear its lottery B because the states of nature are the same but lottery 2 shows that you are more likely to get higher payoff.
The expected payoffs are the same
show on a diagram.
Both lotteries have same excepted value but lottery 2 has higher variance.
Originially we wouldn’t know as we don’t have an utility function or initial wealth, but we know one is more spread out than the other. A risk averse person would prefer lottery 1 as its less spread out. A risk loving would prefer lottery 2 and risk neutral wouldnt care.
Now sketch the diagram for a risk loving person,
What are the steps?
Show Excepted value, show CE and show risk premium what is the value of this.
Also show that the utility from the expected utility of playing lottery > utility of the expected value of lottery .
1) Draw a convex utility function, then identify the 2 states of nature from lottery and the utility from this. Then after draw 2 points and connect them from the 2 states of nature.
2) Highlight the CE > E(l), meaning you need more money than Expected value to accept certainty, thus we can highlight risk premium which is negative. as E(l) - CE< 0.
3) Show that the utility from the expected value of lottery < expected utility of playing the lottery.
Now sketch the diagram for a risk neutral person,
What are the steps?
Imagine the utility function
1) We know the shape is linear so draw a linear line from origin
2) Identify the 2 states of nature on the line and connect them. Identify their utility.
3) The CE = E(l) hence risk premium = 0.
4) The utlity from expected value of lottery = the expected utility of playing the lottery.
ii and iii)
iv)
(V)
This implies that if Agnes prefers lottery C then so will Antonio (provided they have the same wealth level initially).
b
REMEMBER SAY WHAT JENSENS INEQUALITY IS.
1) To start to illustrate the State contingent income space, what goes on the y axis and x axis, can they happen at same time like consumer theory? .2) What is the certainty line and what does it mean if we have a lottery on certainty line. 3) What does it mean is if lottery favour of state of nature 1 and not 2?
Finally show on graph?
1) There are 2 possible states of nature, but both cannot happen at the same time, is either or. On either axis you can put whatever state so good or bad one e.g. ( nothing happens( good state) and get robbed ( bad state).
2) tells me whatever happens in the economy, i will get same profit of good state and bad state so no risk. x1 = x2.
3) If the lottery is in favour of state of nature 1 it means the lottery is to the right the lottery is to the right of the certainty line
Now how do we illustrate how a consumer would feel about these bundles on the state contingent income space? illustrate on diagram?
Indifference curves show the preference of consumers.
The higher the indifference curve, the higher the utility.
What type of consumers are these indifference curves for, what assumptions must it hold and does it hold for both x1 and x2?
Consumers with standard convex to the origin IC’s are risk averse.
Downward sloping normal convex to the orgin IC illustrate non-saitation ( more is better, but not together, its either or.
What does this show e.g. each lottery on the indifference curve ( HINT, remember to evaluate a lottery we use the expected utility theory). What happens if a lottery is on Certainty line?
The indifference curves show different combinations of x1 and x2 keeping probabilities constant that give the same expected utility of the indifference curve. e.g. a bundle to the up right of the certainty line would mean State of nature 2 is better than 1 keeping probabilities the same, so x2 > x1, you get a better outcome.. ( you are indifferent between these lotteries).
2) if lottery on certainty line then state of nature 1 = state of nature 2 keeping probabilities the same.
What is the formula for EU(l) given the certainty line is straight and also why is a lottery a combination of x1 and x2 but you only get one outcome.
Lets say the lottery was ( 500, 500; 1/2 , 1/2) where would this be? and does it have the same CE as all the lotteries on the CE? Illustrate CE
Its true that one will happen out of the 2 states of nature, but at the time od deciding you don’t know e,g, if i flip a coin you will get heads or tails( if heads you get £1000 but tails you get £100), but one or the other happens, but before you flip coin the heads its expected to happen 1/2 the times and tails too.
It would be on the certainty line as no matter what you get 500. It will have the same CE as the other lotteries on the indifference curve
What would happen if the x1 and x2 quantities stay the same but the probability of good 1 rises and good 2 falls
Expected utility would rise, going through lottery 1. Lottery 3 wouldn’t be so good because your putting mass on a probability that wouldnt be so likely. IC become steeper.
What is the MRS for the indifference curve on a state contingent income space and what is it at the CE. WHAT DOES SHIFTING PROBABILTIES SUGGEST e.g. probability of state 1 gets higher and 2 gets lower and the reverse.( remember this is on condition that the lottery is on the right of the CL, so x1 is already weighted higher than x2).
MRS is the same as consumer theory but weighted by its probabilities So MU of good 1/ MU of getting good 2.
At CE MU OF GOOD 1 ( multiplied by chance of getting that utility) = MU OF GOOD 2 ( multiplied by the chance of getting that utility) SO IT CANCELS OUT AND WE JUST GET THE PROBABILITES OVER EACH OTHER.
So shifting probabilities around make IC shallower or steeper. if probability of state 1 goes up and state 2 goes down it becomes steeper and it will be shallower for reverse.
Another important characteristic of a lottery is the expected payoff? what is the formula of Expected payoff,(value of lottery 1), rearrange the formula and what is the shape of it.
It is the different outcomes of the lottery X the different probabilities of lottery, when we rearrange this make x2 to the subject and we get a linear line.
The expected payoff line gives all the combinations of x1 and x2 that give the same payoff, keeping probailites the same.
Now how do we know this person is risk averse define CE in the diagram and Expected payoff and compare ? What about if Risk person is risk loving and risk neutral, what would the shape of IC be?
Remember how CE(all of the combinations of x1 and x2, that give same EU(l) and Expected payoff compare( all of the combinations of x1 and x2 for which the payoff of the lottery is the same) to each other measures preference over risk. If CE < Expected payoff of lottery, you are risk averse.
If person was risk loving indifference curves would be concave hence CE > Expected payoff of lottery.
If person was risk neutral, the indifference would be a straight line so CE = E(L1)
Show a state contingent income space for a risk loving person, show the risk premium which proves its risk loving. ( HINT CE> E(L)
Show a state contingent income space for a risk neutral person, show the risk premium which proves its risk neutral . ( HINT CE= E(L)
Lets use an example, write out her lottery?
the loss is that she is burgled. W
Now we are going to mathematically represent insurance? Now show the the lottery of what happens if things go well and when things go bad. Does she have to buy insurance exactly = to the loss.
What is key here?
p = premium for every unit of insurance purchased X( units of insurance she buys) ( for each 1£ payout you pay p premium.)
X( units of insurance she buys) doesn’t have to equal L(loss) e.g. if you lose 1 million worth necklace, and x is how much you are insured if you are partially insured than X< L e.g. you buy x = 500000 worth of insurance.
1) If things go well she has her wealth but she still paid the insurance so its w- pX
If things go badly her wealth takes a hit by L, she pays the insurance but receives x units of insurance she paid for.
If she fully insures, show here lottery?
Remember X = units of insurance she buys
P =premium she pays.
So X=L( thing value of the thing she loses)
Where would Full insurance be on the State contingent income space?
If things go well w - pL
If things go badly w- L - pL + L = w-pL
So lottery is the same x1 and x2 the same
Full insurance would be on the certainty line.
Now we want to translate buying insurance fully or not on graph, but first show the state contingent income space, with a lottery we disused l = (w,w-1;1-pie, pie) the good state being greater than bad state here. X2 = you get robbed x1 = you don’t get robbed.
Show an indifference curve going through that point, if Julie is risk averse.
Now show on the diagram the good state where nothing happens so she still has to pay premium w-pX, but in the bad state she she pays premium but gets paid X back. Repeat what is x determinant on. Show the new point( lottery) she moves to. What does it mean?
Buying X units of insurance, moves lottery closer to the certainty line( it gives you higher utility and more confidence. X is determinant on the price of insurance.
Now imagine there is a budget line going through lottery and bundle B, Draw this line. You can buy X units of insurance, more than x units of insurance or less. Given the price of insurance what is the gradient of that line.
Height divided by the base ( change in y/ change in x)
The premium pins down the gradient of the line
If the green line represents a budget line, will Julie stay being on the blue indifference curve, what can she choose?
No as we trying to maxmise her utility, we are looking for a tangency. The budget line is not up to her( price of insurance pins down the budget line set by the insurance company) . She chooses to go to that black dot.
How do we know she is partially insured here also talk about gradient?
We know she is partially insured because she isn’t on the certainty line that means that the lottery is not ( w-pL, w-pL;1-pie, pie), no matter the outcome she will have w-pL. This must mean she buys less insurance than the actual lose. Also it means that P> pie.
Gradient of budget line = gradient of indifference curve, remember p > n hence if the gradient of the budget line which is pinned down by price is -1- p/p and the gradient of the indifference curve is -1-pie/pie ( as its the probability of good state divided by bad state) as p is not equal the same.
Draw the state contingent income space when, what is special about it?
Gradient of budget line = gradient of indifference curve, remember p = n hence if the gradient of the budget line which is pinned down by price is -1- p/p and the gradient of the indifference curve is -1-pie/pie ( as its the probability of good state divided by bad state) they must be equal at the certainty line.
We are going to prove that when p=n the insurance company breaks even?
So Julie Now what is the lottey of the insurance company given Julie fully insures?
Now the insurance company has to be less risk averse than Julie, the insurance company has to decide whether it is a good lottery, suppose the utility function was u(W) = 2W ( linear) risk neutral. What do we have ot calculate and calculate it.
If p = n then the EU(l,c) = 0 - actuariality fair
If p>n EU(l,c) > 0 there is a gain, but not fair.
PROBLEM SET
She is an expected utility maximiser, she will chose the lottery which provides her the highest EU(l,c)
The original lottery if she didn’t invest would be ( 100,100;0.5,0.5) - this is on the certainty line.
Remember the one with the highest expected utility, the IC is higher.
ii)
iii)
iv)
The answer is D. Johnny would turn down any bet but surely he would be insane to turn down a gamble where he stands to gain £2.5 billion dollars?
Why is the answer D?
Risk aversion comes solely from the concavity of U(W).
The fact that Johnny rejects the small gamble, means his marginal utility for
wealth decreases very rapidly. So even the chance for enormous wealth gains provide such little marginal utility that johnny wouldn’t risk anything significant to get gains.
2 separate questions.
A and D and i prefer F
Now lets look at this again What does it mean if i pick A and D and if i pick B and C, and what does it mean that i should of picked the this poll?
What is the problem here? and give another example?
General concern : people don’t acculumate risks and understand them together, you treat decisions as if they are in isolation. e.g. if you were asked would you like to invest in shares of this company or would you like to buy bonds, human tendencies will make you treat this decision separately, you don’t think about a portfolio of risk.
What is this an anomaly of in decision making?
Framing of questions and loss aversion. The 2 questions are the exact same thing, but framed in a certain way, certain trigger words evoke different response, ‘saved’ is nicer to ‘die’.
What did Tversky and Kahnerman talk about loss aversion?
Give an example of loss aversion
It is an individuals tendency to avoid losses rather than acquiring equivalent gains. e.g. its better not to lose £20 than gain £20.
In this example in an event of loss people want to be compensated more than lose as they hate the loss. e.g. £100>
What does framing of decisions do to your reference point
It changes e.g. here it would change.
We are going to show prospect theory mathematically, how is it different to expected utility theory?
EUT = we took the outcomes, mutplied the utility from outcomes X probabilities.
Prospect theory is different, they assume n different outcomes( we obvi been looking at 2)
Probabilities are multiplied by peoples perceived probabilities ( so objective probabilities X perceive probabilities, here is where overconfidence could come in). - WE WILL NOT FOCUS ON THIS THO.
This is multiplied by something called a value function ( different to utlitiy)it is the value you get from outcome but - reference point. Meaning whats in the brackets can be gains or losses.
Usually with EUT when we have function and we are deciding on the risky project, we used find EU and if positive we go ahead. In prospect theory how is it DIFFERENT ( HINT SHOW WHEN GAIN OR LOSS AND WITH LOSS WHERE WILL BE LANDER which can be greater or less than one depending on risk appeptie ( define it )
We have a value function where are are 2 outcomes X is greater than or equal to R, when r > x, the lander > 1 scales up losses
If lander < 1 they would be loss loving.
How would be plot the value function? ( HINT value function on Y axis, what goes on x axis? Assuming individual is risk neutral
The region of losses are steeper because of lander ( we feel the loss more than the gain.
In expected utility theory would you say yes to this and what about in prospect theory? what would it mean if Lander = 1? WHAT DOES LOSS AVERSION CREATE??
1) Under EUT as that EXPECTED PAYOFF = 50 = EXPECTED UTLITY, AS THE PERSON IS RISK NETURAL SO THEY WOULD SAY YES TO THIS LOTTERY. IF RISK NEUTRAL
Under Prospect theory it would be a no for a risk neutral person once we factor they are loss averse. ( loss aversion creates risk aversion)
Remember if Lander > 1, if it was equal to 1 it means there is no loss aversion hence as in equal to 1, they will be indifferent.
Show this on diagram?
b)
Key concepts to refer to and explain here are: framing, loss aversion, reference point.
Is someone who is loss averse necessarily risk averse?
No, loss averse
people can be risk loving over losses.
