Lecture 2 Risk and Uncertainty PDF

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Notes for Lecture 2, Economics 2, taught by Dr. Steven Dieterle...

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Lecture 11: Risk and Uncertainty To understand economic decision making with uncertainty, we need a framework  

How do we measure risk? How can we describe preferences toward risk?

With a framework in place we can consider the implications for behaviour  

How might someone choose among various risky alternatives? Are there behaviours that the classic economic approach fails to describe well?

How do we conceptualise risk in economics? Cast as a problem with many potential outcomes, where we do not know beforehand which will be observed  

Denote the first outcome, X1, the second X2, and so on with n possible outcomes X1, X2….Xn

Example of a picnic in the Meadows next week, but the MU picnic depends on the weather X1 = 20C and sunny X2 = 10C and rainy Consider the following gamble: either you win £20 or you lose £20 X1 = 20 X2 = -20 

If you are offered this gamble, do you take it? What additional information would you like to have?

What if I said that the outcomes depend on the flip of a coin?

Obviously, the probabilities will be important for the analysis of risk and uncertainty

Compare our original gamble X1 = 20 Pr1 = 0.5 X2 = -20 Pr2 = 0.5 To a second gamble X1 = 20 Pr1 = 0.99 X2 = -20 Pr2 = 0.01 Does this change how you think about the gamble? Intuitively, we feel better about gamble B than gamble A, but it is often not so clear How can risk be characterised in a uniform way across gambles? 

Expect value: weighted average of the outcomes

  

weights are simply the probabilities put more weight on outcomes that are more likely gives an idea of what the outcome of the gamble will be on average

Expected value (EV) of gambles: What is the EV of our first gamble? EVA = (0.5) (20) + (0.5) (-20) = 0 And the second gamble? EVB = (0.99) (20) + (0.1) (-20) = 19.6 How does this inform our choice between the two?  

EVA < EVB since the good outcome is much more likely for B difference in EV is closely tied to our preference for B

Choosing between gambles:

Should we always choose the gamble with the highest EV? Consider a third gamble X1 = 20 Pr1 = 0.999 X2 = -200 Pr2 = 0.001 How does this compared to B? 

Let’s compare the EV: EVC = (0.999) (20) + (0.001) (-200) = 19.78 So, EVC > EVB, but which would you prefer?

Variability and risk: Gamble C is more variable  

the difference between the two outcomes is much larger variability also depends on the probabilities

How do we measure the variability of a gamble?  

risk from the variability comes from the fact that the actual outcomes differ from the expected outcome that is, with no variability the expected and actual outcomes are equal

Focus here is on variability, not EV -

let’s change the third gamble slightly

X1 = 20 Pr1 = 0.99 X2 = -380 Pr2 = 0.01 Now, EVB = EVC = 19.6 

difference between B and C is driven by variability, not EV

How can we measure variability in a way that captures the difference in actual and expected outcomes?

Measuring variability:

Start with the difference between the actual (xi) and expected outcomes E (X), or the deviation from the mean Xi - E (X) For gamble B: X1: Dev1B = 20 – 19.6 = 0.4 X2: Dev2B = -20 – 19.6 = -39.6 For gamble C: X1: Dev1C = 20 – 19.6 = 0.4 X2: Dev2C = -380 – 19.6 = -399.6 How should we compare these deviations? What about the EV of the deviations? B: E (Dev) = (0.99) (0.04) + (0.1) (-39.6) = 0 -

by the very nature some deviations are positive and some are negative and they cancel out on average

How is this problem solved?  

if we use the deviations squared, then we have all positive numbers this leads us to consider the variance and standard deviation of each gamble

Variance:

Basically a weighted average of the squared deviations (remember they are weighted just by using their respective probabilities) Scale of variance is large – to get back to a similar scale as the deviations we use the Standard Deviation: σ= -

√σ

2

standard deviation is just the square root of the variance

How does this look with our two gambles?

B:

C:

σC ≈ 399.60

So EVB = EVC, but σC > σB  

here the choice between gambles is simple, same EV and C is much more variable therefore likely prefer gamble B

Preferences toward risk: What if two gambles have different EV and σ? How do consumers make choices in the presence of risk?   

we rely on utility maximisation shift our focus from the EV to the expected utility denote utility as a function of income by: u (X)

Then expected utility can be written as:

Expected Utility:

Consider two new lotteries: A: X1 = 16 Pr1 = 0.5 X2 = 9 Pr2 = 0.5 B: X1 = 25 Pr1 = 0.5 X2 = 0 Pr2 = 0.5 EVA = EVB = 12.5 σA = 3.5 σB= 12.5 How does expected utility help us choose between the two lotteries?

Expected utility and risk: Same EV but B has a higher σ  

Intuitively, A should be preferred, how does E (u) reflect this? Depends on the utility function

What if U (X) =

√X

This person is indifferent between the two

What feature of the utility functions determines the attitudes towards risk?

 

shape of the utility function is important graphing each will help illustrate

Uncertain X  E (u) before the outcome is known

Why does E (u) lie on a line connecting the two outcomes? E (u) is a linear function of u (X1) and u (X2) E (u) = Pr1 u (X1) + Pr2 u (X2) = Pr1 u (X1) + (1 – Pr1) u (X2) = u (X2) + Pr1 (u (X1) – u (X2) -

just the function for a line relating the probabilities to the E (u)

The black line lying above the red line shows that 12.5 for certain is preferred to EV = 12.5 Indifferent between 6.25 for certain and EV = 12.5 -

they are willing to pay to reduce risk

For A, E (V) much closer to (12.5) because E (u) = 3.5 (2.5 for B) -

degree of concavity seems to matter

Not surprising that they were indifferent

Prefer more variable gambles – all else equal (as expected utility lies above actual utility)

E (u) and preferences toward risk: How can we summarise the way the utility function affects preferences toward risk? Three distinct categories: risk averse, neutral and loving   

risk averse – concave u (X) Concave u (X)  decreasing marginal utility MU of income – why is this important? Loss of income reduces utility more than an equivalent gain increases it – losses hurt more

If u (X) =

√X

, start with X = 9 and +- 7 (where X is income)

What about risk neutral and risk loving?  

risk neutral – linear u (X) MU is constant in this case

 

risk loving – convex u (X) MU is increasing

E (u) and choices with uncertainty:

E (u) implicitly tells us about the trade-off between EV and σ Previous examples illustrated preferences for risk – but more interesting to consider choices when EVA doesn’t equal EVB and σA doesn’t equal σB Revisit our two lotteries and make a slight change: A: X1 = 16 Pr1 = 0.5 X2 = 9 Pr2 = 0.5 B: X1 = 25 Pr1 = 0.8 X2 = 0 Pr2 = 0.2 EVA = 12.5 EVB = 20 σA = 3.5 σB= 10 -

now we face an interesting choice as the expected value of B is greater than A however, the deviation of B is greater than A also

How will our risk averse individual (u (X) =

√X

) choose?

A: E (u) = 3.5 B: E (u) = 4 -

so for this person, the extra risk is worth it to have the higher expected pay-out

What if: B: X1 = 25 Pr1 = 0.7 X2 = 0 Pr2 = 0.3 EVB = 17.5 σB= 11.46

Now we have:

A: E (u) = 3.5 B: E (u) = 3.5 Now the consumer is indifferent between the two lotteries -

How can we analyse within a common economic framework?

Two lotteries, characterised by EV and σ Given the same E (u) – so they must lie on the same Expected Utility Indifference Curves

Standard deviation of income

In this setup, what direction represents an improved outcome for the consumer? Inwards What does the slope of the IC tell us about risk aversion? Steeper the slope = more risk averse

We can use this setup to think about the demand for risky assets

Demand for risky assets:

Asset  something that provides a flow of money or services to its owner Risk-free asset  an asset that provides a flow of monetary services with certainty Risky asset  an asset that provides an uncertain flow of money or services Return  total monetary flow from an asset as a fraction of its price 

annual return for a stock with beginning price of p 0, ending price p1, and pays a dividend of d: R = p1 – p0 + d / p0 = Δp + d / p0

With risk averse investors, what do you expect the equilibrium relationship between risk and return is? 

riskier assets provide higher return

Consider the choice for a risk averse person to invest in either the risky stock market or a risk-free asset Rf = return on risk-free asset Rm = expected return on stock market rm = realised return on stock market 

since we have risk averse investors, Rm > RF

Consumer must choose how much of the total investment to place in each asset b = proportion invested in stock market

Characterising Assets: How can we characterise the choice facing the investor? Portfolio: the total “bundle” of investments What is the expected value of the portfolio (RP) E (Rp) = E [brm + (1-b) rf] = bRm + (1 –b) Rf Rf = rf as asset is risk-free

What about the standard deviation?

rm is the only uncertain part, so it drives the variance of rp -

note if both assets varied, then the covariance would also show up

Choosing b: How does the investor choose b?   

start with a budget constraint tells us the rate the consumer can trade R p for σp need to derive an expression for the BC – start with the expected return RP = bRm + (1 – b) Rf

This can be rewritten as:

Now we know the trade-off between expected pay-out and risk What does the budget constraint look like? 

linear function relating Rp and σp

Since Rf, Rm and σm are constants: Intercept = Rf Slope = Rm – Rf / σm -

slope can be thought of as the price of risk: the increase in return needed to offset the higher risk

Expected utility maximising: Why might different people choose different portfolios? 

more risk averse investors will choose a safer portfolio with lower expected return

Consider two people:

How do the consumers compare in terms of MRS? Greater for risk averse investor

Reducing risk: Most people tend to be risk averse; therefore there is an interest in reducing risk when possible  

recall the risk premium – suggests that people would be willing to pay to lower the risk they face How might risk reduction take place?

We will consider three risk reducing mechanisms:   

diversification insurance buying information

Diversification:  

reducing risk by allocating resources to a number of separate risky activities that are not closely and positively correlated intuitively, the realised gains and losses from risky activities may offset, thereby reducing risk a priori motivate this with an example:

You have £100 to invest in two different stocks, A and B The price of each stock is £10, so you can purchase a total of 10 shares

Two possible outcomes: X1: Profit per share A = £100 and B = £20 Pr1 = 0.5 X2: Profit per share A = £20 and B = £100 Pr2 = 0.5 What is the EV if you invest only in stock A? EVA = (0.5) (10) (100) + 0.5 (10) (20) = 600 What about investing all in B? EVB = (0.5) (10) (20) + (0.5) (10) (100) = 600 What is the risk involved in either investment? σA = σB = 400 (1000 (10 x 100) – 600) = 400

How can diversification help reduce risk?  

What if you purchase 5 shares each of A and B? What is the EV now? EVAB = 0.5 [(5) (100) + (5 (20)] + 0.5 [(5) (20) + (5) (100)] = 600 So EVA = EVB = EVAB However, σAB = 0 -

any risk averse person would prefer the diversified portfolio

In this simple example, it was easy to use diversification to eliminate the risk – is it always so simple?  

of course not several features of this example were important:

A and B had the same EV for a share -

with different EV for the shares it would not be possible to keep the same EV while lowering risk by diversifying recall our discussion of risky and risk-free assets

Perfect negative correlation in outcomes for A and B

-

What if A and B both perform well or poorly at the same time? Does diversification still help? Important with stocks as they tend to move with the economy more generally

Diversification may help to reduce but not eliminate risk

Insurance: Purchasing insurance as another way to reduce risk -

buyers of insurance pay a premium in exchange for compensation in the case of a bad outcome

Example: cow insurance  

livestock as a major and risky investment for farmers even more so in the developing world as it is harder to “diversify” with a large herd

Kenya: a healthy dairy cow is valued at 40000 Kenyan Shillings (Ksh) or around £300 GDP per capita was 70000Ksh in 2011 

if the cow were to become ill or die, this would be a major loss for the farmer

Cow insurance: Recently, there has been an expansion of the availability of livestock in Kenya -

let’s cast the problem in the potential outcomes framework:

X1: Healthy cow valued at 36000Ksh Pr1 = 0.75 X2: Dead/ill cow valued at 4000Ksh Pr2 = 0.25 What does the uninsured risk look like? EVU = (0.75) (36000) + (0.25) (4000) = 28000 σU = 13856

Now consider the option to purchase insurance to cover the loss if the bad state occurs (X 2) in exchange for a premium p How does this change the problem facing the farmer? X1: 36000 – p Pr1 0.75 X2: 4000 – p + 32000 Pr2 0.25

What does the insured risk look like? EVI = (.75) (36000 – p) + (.25) (4000 – p + 32000) = 36000 – p σI = 0 (i.e. no deviation so no risk)

Choice of p: Insurance eliminates risk, in exchange for the premium How will p affect a person’s choice to purchase insurance? 

↑p = ↓EVI, so it depends on the person’s level of risk aversion

For now, let’s consider the largest p that would make any risk averse person willing to purchase insurance  

What condition will determine this? since with insurance σI = 0, any risk averse person will want to purchase insurance if EV I > OR EQUAL to EVU 36000 – p = 28000 p = 8000

Same p that would make a risk neutral person just indifferent between having insurance or not

Actual fairness: Premium that sets EVI = EVU is called the actuarially fair premium – why is this? Consider the problem facing the insurance provider  

assume they set the actuarially fair premium, p = 8000 With one insured cow, what are the expected profits to the insurance provider?

E (π) = (Pr1) (p) + (Pr2) (p – (X1 – X2)) = (0.75) (8000) + (0.25) (8000 – 32000) = 0 What is the standard deviation of profits? σπ = 13856

Law of large numbers:

Risk has been shifted to the insurance provider – how is it “actuarially fair” for the provider to take over the risk in exchange for E (π) = 0?  

typically there is more than one cow to insure insurance provider can effectively diversify by pooling the risk across many cows

(Extra weak) law of large numbers: while a single realisation of a random process can be highly variable, across many realisations the average of the realisations is much less variable and becomes more predictable with more realisations (realisations a.k.a probabilities)  

100 cows, around 75 will be healthy and 25 will be sick pooling across many similar but independent cases the average pay-out is offset by the total premiums collected

WTP for insurance: willingness to pay Actuarially fair premium can be determined by considering the maximum premium any risk averse person would be willing to pay for the insurance; it does not however tell us the WTP for all risk averse people 

recall, we found it by setting EVU = EVI

Why might the WTP differ for a risk averse person? 

willing to trade EV to eliminate risk – that is for most risk averse people, p can be set so that EVI < EVU

What determines an individual’s WTP for insurance? 

preferences of course

Kenyan Dairy Cow Insurance has an annual premium of around 2000Ksh

What might we have overlooked with our simple example?  

 

Probabilities of having a healthy cow were made up – How could you change the probabilities, holding everything else fixed, to get p = 2000? Value of a healthy cow was altered to make the maths nice ( √ 36 is much nicer than √ 40 ) and completely made up the value of a sick/dead cow – how do these choices affect our analysis? Premiums may be partially subsidised by aid organisations Other issues in insurance markets

Choice of insurance amount: Until now, we have considered the discrete choice to buy insurance or not  

in particular, we considered Full Insurance that completely offsets the loss from the bad outcome What if we allow people to choose different amount of insurance, to partially offset the loss?

Now how will risk averse people choose the amount of insurance they wish to purchase?

Partial insurance: Consider a very general case with two states of the world: one good and one bad Bad: Receive XUB with probability pB Good: Receive XUG with probability pG XU G > XU B Here the “U” superscript denotes the “uninsured” outcomes -

indifference curves can again be used to characterise the consumers preferences over different bundles of consumption in the two states

How should we think of the premium now? Before it was the total paid for full insurance  

now we can think of the price per £ of insurance For the cow insurance, the actuarially fair premium was 8000Ksh to purchase 32000 in insurance, or a rate of .25Ksh for each Ksh of insurance

Denote the premium rate by r, the amount of insurance purchased by K, and actual consumption in the two states by XG and XB -

total cost to consumer is rK

What condition characterises full insurance in this setup? Take r as given, how does the consumer choose K thereby choosing (X G, XB)? XG = XUG – rK XB = XUB – rK + K = XUB + (1 – r) K How can the consumer trade between consumption in the Good state and the Bad state? Start from the uninsured point, where K = 0 -

If they purchase £1 of insurance, how does this change their consumption in both states? XG = XU G – r XB = XUB + (1 – r)

So consumption in the good state goes down by r while consumption in the bad state increases by (1 – r)

BC has slope: – r / (1 – r) = (δXG / δK) / (δXB / δK)

Partial insurance Full insurance No insurance

Maximum consumption in the good state when there is no insurance Maximum consumption in the bad state when there is ful...