MT (5-7) Portfolio theory and Market Efficiency

Question 1

What is the difference between a population and sample?

Answer 1

A population is the complete set of all items from a system or process that is being studied.

A parameter is a characteristic of a population.

A sample is an observed subset of population values of manageable size.

A statistic is a characteristic of a sample.

Question 2

How do you calculate Expected return?

Answer 2

E(R)=p1R1+p2R2 +……..+pnRn

The expected return on an asset is the weighted average rate of return of all possible outcomes where the weights are the probabilities.

E(R)= Sum of (p(i)R(i) from i=1 to N

Question 3

How do you calculate the sample mean return?

Answer 3

R(bar) = 1/N x sum of R(i) from i=1 to N

N is the number of observations in the sample

Question 4

What is the formula for Varience of Returns?

Answer 4

σ^2 = Sum of p(i)[R(i) −E(R)]^2 from i=1 to N

One measure of the risk of a financial asset is the variance of its returns. The variance is the expected value of the squared deviations of the random variable from its population mean.

Remember that the units of the variance are not in the same units as R because they have been squared.

Question 5

Formula for the standard deviation of returns?

Answer 5

The standard deviation is simply the square root of the variance and hence the units of the standard deviation are in the same units as the R.

σ

Question 6

How does one calculate the sample varience and sample standard deviation?

Answer 6

Sample Varience:

s(R)^2= 1/N-1 x Sum of [R(i) - R(bar)]^2

R(bar)= 1/N x sum of R(i) from i=1 to N

Remember that the units of the sample variance are not in the same units as R because they have been squared.

Sample Standard deviation:

The sample standard deviation is simply the square root of the sample variance and hence the units of the sample standard deviation are in the same units as R.

Question 7

What is the formula for Covarience

Answer 7

σx,y= E[(X −E(X))(Y −E(Y))]

Question 8

What is the formula for Correlation

Answer 8

P= σx,y/σxσy

Question 9

What is the formula for sample co-varience

Answer 9

COV(X,Y)= 1/N-1 sum of (Xi - X bar)(Yi - Y bar) from i=1 to N

Question 10

What is the formula for sample correlation

Answer 10

CORR (X, Y) = Cov(X,Y)/Sample. STDEV (X) x Sample.StDev(Y)

Question 11

If X and Y are returns then a positive covariance means what?

Answer 11

This means that the asset returns generally move together.

The returns on assets X and Y are either both below their sample means or both above their sample means.

Question 12

If X and Y are returns, then a negative covariance means what?

Answer 12

It means that the asset returns generally move in opposite directions. Where one asset’s
return will be above their sample mean return and the other will be below.

Question 13

What does a Correlation of 1 mean? Also, what values is corelation always between?

Answer 13

Correlation is always a number between -1 and 1 which makes it easier to interpret and is unit free.

Correlation is equal to 1 if there is an exact linear relation with positive slope between X and Y (perfectly positively correlated).

Correlation is equal to 0 if X and Y are uncorrelated

Question 14

What is diversifcation?

Answer 14

The reduction in portfolio variance associated with holding different stocks. This works best when correlations are smaller between stocks.

Question 15

What is Beta?

Answer 15

It is a measure of the contribution of one particular stock to a portfolio’s overall risk

Question 16

What is the formula for Beta

Answer 16

The contribution of asset i to the overall portfolio variance is:

B(i)=Cov(R(i), R(p)/ σ^2(Rp)

The contribution of asset i to the market portfolio variance is:

B(i)=Cov(R(i), R(m)/ σ^2(Rm)

A higher beta means more risk coming from stock i

Question 17

What does a higher beta mean?

Answer 17

A higher beta means more risk coming from stock i

Question 18

If the beta of a stock to the market portfolio is 1.4, what do you expect the return of the stock to increase by, if the market portfolio return increases by 1.4%?

Answer 18

When the market portfolio return increases by 1%, the stock return tends to increase by 1.4%.

Question 19

How does one calculate the beta of a portfolio?

Answer 19

In general, in a portfolio of N stocks where the weight on stock i is wi and the beta of stock i is βi, the beta of the portfolio is:

βp= Sum of wiβi, from i =1 to N

Question 20

What is the portfolio frontier?

Answer 20

The set of portfolios that can be formed from a given set of securities that minimise variance (standard deviation) for varying levels of expected return.

Question 21

What is the Minimum Variance Portfolio?

Answer 21

The portfolio that gives the lowest possible variance (standard deviation).

Question 22

What is the efficient frontier?

Answer 22

The set of portfolios that can be formed from a given set of securities with the property that each portfolio has the highest possible expected return for a given variance (standard deviation). It is the section of the portfolio frontier that lies above and to the right of the minimum variance portfolio.

Question 23

Why does forming portfolio’s of stocks reduce risk?

Answer 23

Some of the portfolios on the portfolio frontier have smaller standard deviation than either of the stocks.

The benefit of diversification is that portfolios of securities tend to have much smaller risk (as measured by return standard deviations) than individual securities. Thus risk-averse investors will tend to prefer ‘diversified’ portfolios (i.e. portfolios containing many stocks).

The benefit of diversification is driven by the fact that security returns are not perfectly correlated. If all stocks returns were perfectly correlated, the benefit of diversification would disappear.

Question 24

Which portfolio does an investor choose on the efficient frontier?

Answer 24

This depends on the investor’s level of risk aversion and hence where their indifference curve is tangent to the efficient frontier (as at this point the investor
is getting the highest possible utility).

Question 25

When the risk-free asset is combined with a risky asset/portfolio the risk return
opportunity set is always what?

Answer 25

When the risk-free asset is combined with a risky asset/portfolio the risk return
opportunity set is always LINEAR!

Question 26

The expected return on a portfolio consisting of the risk-free asset and a risky asset/portfolio A is given by

Answer 26

E(Rp) =waE(Ra) + (1−wa) (Rrf)

Where wrf= 1 -wa

After rearranging:

E(Rp)=Rrf+[E(Ra) - Rrf/σRa]σRp

Question 27

The variance of the portfolio consisting of the risk-free asset and a risky asset/portfolio A is given by?

Answer 27

σrp^2 =wa^2σa^2

So σrp =wa x σa

Because the variance of the risk-free asset is zero as well as its covariance with any other asset.

Question 28

What does the Capital allocation line (CAL) ?

Answer 28

simply represents the risk return opportunity set an investor can obtain by varying the weights they invest in the risk-free asset and risky asset/portfolio A

Question 29

Which portfolio will a mean-variance optimising investor combine with the risk- free asset?

Answer 29

The optimal portfolio of risky assets must be the point at which the efficient frontier of risky assets only is tangent to a straight line through the risk-free rate. This portfolio of risky assets, denoted by T, is the tangency portfolio. (pg. 26)

Question 30

wHAT IS The optimal investment of every mean-variance investor?

Answer 30

a combination of two portfolios (or funds) – the tangency portfolio and the risk-free asset. This result is called two-fund separation.

Question 31

What is the Sharpe ratio and its formula?

Answer 31

The Sharpe ratio is the gradient of a capital allocation line (CAL). If the risky asset is the market then it is the gradient of the capital market line (CML).

Sharpe ratio = E(Ra) - Rrf/σRa

Question 32

What are the five assumptions of CAPM?

Answer 32

1) There are N risky assets and a risk-free asset (N+1 assets in total). All investors can borrow/lend at the same risk free rate.
2) Trading of assets is costless (no transaction costs or taxes).
3) Investors care only about mean and variance (investors are rational mean variance optimisers).
4) Investors have the same information (beliefs/homogeneous expectations).
5) Investors have a one period time horizon.

Question 33

CAPM in Equilibrium characteristics (4)?

Answer 33

In equilibrium the tangency portfolio must be exactly the market portfolio of risky assets.

All investors choose the same optimal portfolio of risky assets and this is the market portfolio.

The market portfolio includes all risky securities (in principle it includes stocks, bonds, real estate and so on).

The weight of an asset in the market portfolio is its market capitalization divided by the total market cap of all assets.

Question 34

What is the CAPM in equilibrium equation?

Answer 34

E(R)=Rrf+βi[E(Rm)−rrf]

An asset’s expected return depends on risk only through beta.

An asset’s expected return is linear and increasing in beta.

Question 35

What is a stocks alpha?

Answer 35

Stock X should be up-weighted in your portfolio as it delivers a higher expected return than the CAPM says it should. We call this extra expected return the stock’s alpha. The extra demand for Stock X will increase its price and hence lower its expected return until in equilibrium it plots on the SML.

Question 36

What are the applications for CAPM?

Answer 36

1) Valuation of Stocks

How should we value the stock? Compute the present value of the future estimated dividend stream. What discount rate should we use? Use the CAPM expected return for a company as CAPM tells you what return you should expect from a company with that level of risk – it gives a risk-adjusted discount rate.

2) Valuation of Projects

For an equity financed firm, the CAPM expected return is the return that one should require any project that the firm takes on to earn. Thus use it as the discount rate in a NPV calculation.

3) Portfolio Selection

If the CAPM is an entirely accurate description of the world, then portfolio selection is straightforward. Hold the combination of the risk-free asset and the market portfolio that maximises your personal utility.

If you feel that the CAPM holds for the majority of stocks, but is violated for a few:

Start off with the market portfolio.
Upweight those few stocks that have positive alpha.
Downweight those few stocks with negative alpha.

Combine the resulting risky portfolio with the risk-free asset so as to maximise your utility.

Question 37

How to Empirical Evaluate whether CAPM holds? (4 ways)

Answer 37

If the CAPM holds, estimated values for alpha should be zero for all stocks.

Also, under the CAPM, expected excess returns should be linear in beta.

The slope of the SML should be approximately the excess return on the market portfolio.

Finally, expected returns should only depend on beta.

Question 38

Why is our confidence in CAPM eroded in reality?

Answer 38

The SML we see in the data seems to have a slope less than the excess return on the market portfolio (MRP).

Factors other than beta explain expected returns.
Size - small stocks tend to have higher expected return than big stocks (holding beta constant)
Value - mean returns on stocks with high book-to-market tend to be larger than those on low book-to-market stocks (holding beta equal)

Question 39

Other Asset Pricing Models?

Answer 39

Multi-factor models e.g. Fama and French 3 factor model, and the Arbitrage Pricing theory.

Inter-temporal CAPM.

Consumption-based asset pricing models.

As of the present, when most academic researchers wish to estimate expected returns for stocks, they tend to use a multi-factor model that explains stock returns with the CAPM beta, but also allows expected returns to vary with a firm’s size and value characteristics.

Question 40

How does one measure abonormal return?

Answer 40

If the CAPM is the relevant asset pricing model then a security’s expected abnormal return is the CAPM alpha
Therefore we use the concept of alpha to measure the abnormal return from holding a security.

Question 41

What is the Joint-Hypothesis problem?

Answer 41

Conducting a test of efficiency requires us to use an asset pricing model to calculate abnormal returns. However, the model we use could be the wrong one!

For example, perhaps the CAPM does not reflect the way that expected returns are determined in the real world.

The implication is that all tests of efficiency are a test of a joint hypothesis containing two parts:
The market under analysis is informationally efficient.
The researcher knows and uses the correct asset pricing model to generate abnormal returns.

Question 42

What is an Efficient Market?

Answer 42

An efficient market is a market in which asset prices reflect all past and present information.

In an efficient market, profitable trading opportunities (inefficiencies) are rare.

In an efficient market, asset prices reflect information quickly (the window of opportunity for taking advantage of mispricings is small).

Question 43

How Does Information Get Reflected In Prices?

Answer 43

In an efficient market, prices are expected to react only to the elements of information that have not been fully anticipated by investors.

It’s the ‘’surprise’’ (or ‘’unexpected’’) element of news (new information) that affects prices.

Based on the new information, investors update their expectations which are incorporated into market prices through trades:

Hence, the market price of an asset represents the balance of expectations prevailing in the market.

Question 44

What are the foundations of an efficient market?

Answer 44

Shleifer (2000), argues that any one of the conditions below should lead to efficiency:

Rationality:
All investors are rational, receive information in a timely fashion and all process it quickly. Thus all investors adjust stock valuations in the same way when news arrives and prices adjust instantly.

Independent deviations from rationality:
If not all people are rational, but optimists and pessimists have equal influence on markets, then their irrationality will balance out and stocks will be fairly priced.

Dominance of rational, professional investors:
If there is a large group of investors (maybe working in banks and funds) who are rational and have a great amount of money to invest, then when they see prices that are ‘wrong’, they trade and their trading (and size) pushes prices to the ‘right’ level.

Question 45

What is the efficient market hypothesis?

Answer 45

The Efficient Market Hypothesis (formulated by Eugene Fama in 1970, Nobel Memorial Prize in Economic Science, 2013) states that stock prices accurately reflect all available information.

Information about future stock returns quickly gets incorporated into prices.
Hence, there is no return predictability in the market.

Question 46

What are the three forms of market efficiency?

Answer 46

Weak form: investors cannot predict abnormal returns given observed history of past stock prices, trading volume and other trading data.

Semi-strong form: investors cannot predict abnormal returns given publicly available information (management quality, balance sheet composition, accounting performance, corporate news etc)

Strong form: investors cannot predict abnormal returns using publicly and privately available information.

Question 47

How does one test to see weak-form market efficiency holds?

Answer 47

To test this hypothesis, one can look at the correlations of current returns and past returns at different horizons.

A serial correlation (or autocorrelation) that is insignificantly different from 0 is generally consistent with the weak form of the EMH.

Empirical tests reveal that autocorrelations in market index returns (e.g. S&P 500) are close to zero.

Question 48

How does one test to see if semi-strong form market efficiency holds?

Answer 48

The semi-strong form of the market efficiency suggests that market prices instantly and fully reflect all past and publicly available information.

If the semi-strong form of the EMH holds, then fundamental analysis will not lead to abnormal returns (as long as the analysis is based on publicly available information).

To empirically test the semi-strong form, one can look at stock price responses to various corporate events (event studies), such as earning announcements, dividend initiations/omissions, takeover announcements, etc.

Alternatively, one can also test if professional money managers can earn abnormal returns in the stock market.

Question 49

How does one test to see if strong form market efficiency holds?

Answer 49

The strong form hypothesis implies that corporate insiders cannot make abnormal returns using their private information.

The strong form hypothesis is tested by studies observing market participants who are reasonably expected to have access to private information, or those with the expertise and the resources to generate such.

The consistent gain of abnormal returns by such participants would constitute evidence against the strong form of EMH.

Question 50

What evidence is there that challanges the EMH?

What is an explanation for EMH not holding up?

Answer 50

Some of the confirmed anomalies are:

Stock Price Momentum and Reversal
Post Earnings Announcement Drifts
Twin-Stock Puzzles
Asset Price Bubbles

Limits to Arbitrage

The main assumption of the EMH is that arbitrageurs have unlimited capital and unlimited risk capacity, and that the market is frictionless.

In reality, arbitrage can be severely constrained by various factors:
Implementation costs: commissions, bid-ask spreads, short sales constraints, costs of discovering and exploiting mispricing.

Noise trader risk: mispricing can get worse in the short term before convergence. Arbitrageurs facing capital constraints may be deterred from correcting mispricing.

Question 51

What are the main assumptions of the EMH? (3)

Answer 51

The main assumption of the EMH is that arbitrageurs have unlimited capital and unlimited risk capacity, and that the market is frictionless.