Chapter 10 - Capital markets and the pricing of risk

Question 1

in the very long term, why invest in anything other than small stocks?

Answer 1

Although small stocks may return the highest return in the very long run, they are extremely subject to variations from year to year. Therefore, if you happen to actually need the money in a specific year, you would be fucked if you have it in small stocks more oftne than you would if you invest in the market.

Question 2

Different securities sell for differnet amounts, produce different cash flows, and sell for different future amounts. How can we generlize the performance of the securuties+

Answer 2

We use statistics like returns and volatility

Question 3

Theoretically speaking, how do we view the perofrmance of a security?

Answer 3

Like a probability distribution. It is all about the likelihood of various outcomes (returns). If the probability distributoin that describes the returns of the security is very wide, it is more volatile.

in an ideal world, we have a perfect continuous probability distribution that describes how the stock/security works.

Given the probability distribution, we can easily compute the expected return.

Question 4

First thing that should come to mind when thinking about risk

Answer 4

Variance and standard deviation.

Thing is, we need a measure of risk.

Question 5

What is the variance and standard deviaiton of a risk free investment

Answer 5

0

Question 6

What is volatility

Answer 6

Standard deviation of a stocks return

Question 7

Why is STD more useful than variance?

Answer 7

Same unit as the measurement

Question 8

elaborate on computing historical returns

Answer 8

We refer to this as “realized returns”.

The total realized return from A STOCK is equal to the dividends we have received in the duration AND the capital gain. This is the same formula as earlier chapter.

R = dividend yield + capital gain rate

Note how this is not discounting. We only find the growth rate.

Question 9

elaborate on historical returns when we hold the stock beyond dividend date

Answer 9

We have to determine what we are going to do with the dividends.

It is common to assume that one will immediately reinvest all dividends into more shares in the same investment. if so, we use the regular equiation of realized return = dividend yield + capital gain rate, but we use this formula between all the dividend payments, and then compound them as needed.

Say we get quarterly dividends of APR 4%. This is 1% each quarter. Let us say the stock increase by 5% each quarter. Then the final (annual in this case) realized return would be 6% x 6% … x 6% = 1.06^4 = 26.25%.

The image shows this more clearly. however the basic thing here is to just find the return for a smaller period, and then compound it AS IF it were multiple years.

Question 10

what is empirical distribiton?

Answer 10

Probability distribution when we use the empirical/historical data to create it. It is strictly speaking not the pribability distribution, but one can assume that the empirical distribution takes the shape of the actual underlying probability distribution given enough data

Question 11

What is the formula for empirical variance?

Answer 11

Question 12

What is it typical to assume regarding investors assumption of expected return

Answer 12

Sometihng like past returns equals future returns. If you invest in S&P with past returns of 10%, you’d expect 10% in the later years as well.

Question 13

We can use a stocks historical returns to infer expected returns. But there is a difficulty with this, elaborate on this difficulty

Answer 13

It becomes subject to estimation error. The average return is just an approximation of the true expected return.

Therefore, we want to see what we can do with this estimation error as well.

We measure the estimation error by its “standard error”.
The standard error is a statistical measure, and refer to the standard deviation of the estimated value of the mean of the actual distribution around its true value.

In other words, the standard error is the standard deviation of the average return.

the standard error is calcualted as sigma / sqrt(n)

Question 14

elaborate on why we need standard error

Answer 14

TODO, but it is basically the standard deviation of our sampling

Question 15

how do we set up confidence interval for the stocks historical average returns

Answer 15

We use the expected return that is based on the historical return, and we use the standard error to provide the limits on the interval.

KEY: We do not use the regular standard deviaiton here, but we use the standard error. This is becasue we are dealign with the empirical shite, not the actual probability distributon.

Question 16

What is hte key limitation of the expected return estimates?

Answer 16

Firstly, stocks may not follow hirstoircal patterns.

Secondly, stocks may not have that much data. For instance, if a firm is relatively new, it may only have a couple years of data, which is not enough to be considered reliable.

Question 17

How can we verify the fact that the expected return estimates sucks with little data?

Answer 17

The standard error will be extremely large. Because it shirnks with the square root of the number of samples, it is basically up to this factor to do the tihngs.

Question 18

Invstors are assumed to be risk averse. What general implications does this have?

Answer 18

THey will not invest in something that is more risky unless if they are being compensated for it by expecting a higher return

Question 19

Define excess return

Answer 19

Excess return is the difference between the average return on an investment and the average return on the risk free investment (typically treasury bills).

Question 20

Tf do we use excess return for

Answer 20

Excess return is a measure of the risk premium that investors require from an investment. Because of this, if we have it, we can compute how they value the additional risk.

Question 21

empirically using lots of data, what is the relationship between larger vs smaller stocks in terms of expected return

Answer 21

Smaller stocks tend to be more volatile. Therefore, investors require a larger risk premium for small stocks. Because of this, the expected return should be larger for the smaller stocks. This is what the evidence shows at least.

Question 22

This image shows what appears to be a proportional relationship between volatility (risk) and expected return. Does it hold for individual stocks?

Answer 22

No.

The key question is really: Why not? Why wouldnt investors expect the same level of returns correpsonding to the volatility of the investment?

The image shows how when looking at individual stocks, the risk premium is much smaller than that of the portfolios. The answer is basically about what risk actually is.

Question 23

Do expected return for well diversified portfolios of stocks appear to increase with volatility?

Answer 23

Yes

Question 24

Do expected return for individual stocks appear to increase with volatility?

Answer 24

no

Question 25

what is excess return?

Answer 25

The difference between the expected/required return from an investment and the expected/requires/average return from the risk free ivnestment, typically US treasury bills. It is the same as risk premium.

Excess return is a great measure because it tells us how the investors consider the return of the investment as a response to the riskiness of it.

Question 26

The wohle idea of this chapter is that the risk of an individual security cannot be treated the same way as the risk of a portfolio. Elaborate on why this is

Answer 26

The classic case is theft vs earthquake insurance. While both cases have the same expected value in terms of executions of the insurance policies, there are levels to this.

Earthquake scenario happens 1% of the time, and effect everyone. It is very difficult to estimate when this will occur.
Theft, on the other hand, occur 1% of the time as well, but each home has this probability all the time (sort of) which results in the fact that it becomes very predictable to give an interval of how many insurance claims of theft we can expect. The outcome here is basically that theft insurance has no risk at all, because the insurance firm knows to a very precise degree how many claims will be filed each eyear.

the reason why these scenarios differ, is because they are describing different types of risks.
1) Independent risk
2) Common risk

Common risk refers to shit that affects everyone and everything, like an earthquake. or like covid.
Independent risk is specific to each individual case. Manager of a firm could suddenly die.

Question 27

how can we tell whether two risk are common or independent?

Answer 27

It is about whether they are correlated or not.

Question 28

Define diversification

Answer 28

Diversificaiton is the process of averaging out the independent risks in a large portfolio.

Question 29

if we have an average value, and we want to find its standard deviation or volatility, what do we call it?

Answer 29

Standard error. It is actually quite simple as well, because we just use the volatility of the individual case, and divides by the square root of the number of cases.

In the insurance case, where there are 100 000 claims possible, the theft case has a standard error / volatility of 0.03% which is redicoulsly small.

Question 30

The confusing part here is about the expected return, volatility/SD of the return, and then the standard deviation of the expected return, and how they mix. elaborate on clear the confusion

Answer 30

We typically first start by finding the expected return using historical evidence or probability distribution or whatever.

Then we compute the empirical variance, which gives us the standard deviation of the returns. Now, we have the risk from the perspective of the individual case.

IF the events are not correlated, then the appropriate SD of the expected return is the standard error.

The thing is that the expected return is some value that is relatively fixed. When computing the standard deviation of a single case, it will typically be quite large.
IF the risks are independent, then we can diversify them away when doing a large number of them. This introduce the standard deviation of the average return.

Question 31

Define the general implicaitons of risk in terms of stocks

Answer 31

Risk implies that the dividends and the final capital gain will be higher or lower than expected, which makes the return risky/unknown.

Question 32

there are two types of news that can affect securities

Answer 32

firm specific news

market wide news

Question 33

firm specific news are alos referred to as …

Answer 33

idiosyncratic, unique, diversifiable

Question 34

market wide news are also referred to as…

Answer 34

systematic, undiversifiable, market risk

Question 35

Can we cateogrize firms into S or I firms?

Answer 35

No, it is better to consider it as a spectrum. All firms carry both types of risks, but in differnet amounts

Question 36

Consider a bunch of firms that have both types of risk, and we make a portfolio of them. As we increase the number of firms, what happens

Answer 36

The idiosyncratic risk will be more and more negligable because this risk will be diversified away. As the number of firms in the portfolio increase, we approach a portofolio that is only subject to market risk/systematic risk.

Question 37

now we are able to answer why S&P have lower volatility than the stocks it contains

Answer 37

Only the market risk remains, all (at least most of) the independent idiosyncratic risk has been diversified away.

Question 38

Should investor expect to earn a risk premium from investing in a type I firm?

Answer 38

No. Since the risk “can” be diversified away with no loss in expected value, there will be no risk premium for it. At least in a competitive market

Question 39

why cant investors (in a competitive market) earn a risk premium on type I firms?

Answer 39

It would be an arbitrage opportunity. Because, if investors could invest in type I firms and diversify them, they would be able to earn a higher return than the risk free return, which doesnt make sense.

Question 40

What is the law of one price theorem regarding type i firms

Answer 40

A portfolio of type I firms has no risk (all have been diverisfied away) so the risk is 0, which means that the excess return is 0 as well. In other words, investors are not compensated by holding firm specific risk.

Question 41

are investors compensated by holding firm specific risk

Answer 41

No. It would be arbirtrage oppoortunity, which does not exist in competitive markets

Question 42

What can we say about the risk premium of a securtity

Answer 42

Determiend completely by its systematic risk.

This is because risk averse investors can go for the risk free investments, and the idiosyncratic risk can be diversified away.

Question 43

is volatility a reasonble measure of risk for an individual security?

Answer 43

NO.

Tells us nothing. We need the standard error to indicate how much of the indpendent risk is diverisified away. That is, if we want to say anything about the risk premium.

Question 44

What is the implication of volatility not being a reasonable measure of risk for an individual security?

Answer 44

There should be no relationship between a stocks volatility and the expected return of it. Crucial point.

Question 45

if the idiosyncratic risk provide no bearing on the excess retuyrn or expected return or risk, how the fuck are we supposed to find the expected return of a stock?

Answer 45

We need to use the market risk/systematic risk.

Every stock depends on the market risk, but to varying extents. We can use this to find expected return.

Question 46

How can investors eliminate firm specific risk?

Answer 46

Diversification

Question 47

When evaluating an investment, what risk do we care about?

Answer 47

The market risk/systematic risk. we care about this becasue these risks cannot be diversified away.

Question 48

Why should there be a relationship between systematic risk and expected return?

Answer 48

By taking on more risk, investors require higher expected return. This is a basic assumption, and it shows empircally.

Question 49

What is the first step in dtermining the systematic riks of an indicidual investment?

Answer 49

We need to find out how much the indivial returns fluctuate as a result of movements in the market as a whole. IN other words, how sensitive is the investment to changes in the economy as a whole.

Question 50

Ideally, what do we need to find systematic risk of an individual investment?

Answer 50

We would need a portfolio that only has systematic risk. If we had such a case, we would look for how much the individual security change as a result of 1% change in the systematic-risk-only portfolio. Then we would try to aggregate this as best as possible to get the best measure of the sensitivity.

Question 51

What do we call a portfolio that only has systematic risk

Answer 51

Efficient

Question 52

Can we diverisfy an efficient portfolio?

Answer 52

No, it is fully diversified already

Question 53

What is the natural candidate of the efficient portfolio?

Answer 53

The entire market portfolio. However, this is not practical, because a lot of the smaller shit is difficult to find. We therefore typically use S&P 500 instead as a proxy.

Question 54

Define the beta of a securityu

Answer 54

beta is the expected percentage change in the return of the idnvidiual security that comes as as result of 1% change in the efficient portfolio.

Question 55

What is nice about beta?

Answer 55

It can be approximated fairly well using past data

Question 56

What is the average beta of all stocks?

Answer 56

1

Question 57

Who have high, low betas?

Answer 57

Cyclicals tend to have kind of high, meaning that they are sensitive
Non cyclicals (durables) tend to have low betas.
technology firms have very high betas.

Question 58

How can we estimate the risk premium that investors will require from a risky investment?

Answer 58

Since we know that only the market/systematic risk is relevant, and we know that the beta of the investment capture this sensitivty/risk to market, we can use this to figure out the appropriate cost of capital for the investment.

The risk premium that investors require is equal to the difference between expected return of the investment and the risk free return.

We first need to find the risk premium from holding the market.

Risk premium = E[Rmkt] - rf

This is the return, additional return, that investors require by investing in the market relative to the risk free option.

Now we need to integrate the beta concept. Recall that a beta of 2 indicates twice as much systematic risk. Therefore, investor would require twice as much return.

The result is basically the capital asset pricing model.

r_i = rf + beta (E[Rmkt] - rf)