Portfolio theory and Market Efficiency 5 -7 Flashcards
What is the difference between a population and sample?
A population is the complete set of all items from a system or process that is being studied.
A sample is an observed subset of population values of manageable size.
How do you calculate Expected return ( the amount of profit or loss an investor can anticipate?
Sum of Probability x return according to state of nature. ( basically like the mean)
How do you calculate the sample mean return?
Sum of R data series returns from our sample / N observable observations.
What is the formula for variance ( measures variability from the average or mean.)?
How do you work out the standard deviation ( allows to get to same units ) ?
How do you work out sample variance and sample standard deviation?
What does covariance tell us and what is formula?
measures the direction of the relationship between two variables
What does Correlation tell us and what is the formula?
measure the strength of the linear relationship between two variables ( it scales covariance)
What is the formula for sample covariance?
What is the formula for sample correlation?
sample covariance / product of sample standard deviations
What does this show?
If x and y return a positive covariance/negative: 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.
If x and y return a negative covariance: 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.
What numbers are correlation inbetween and what if correlation = 1
Correlation = 0 and correlation = -1 ?
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
Correlation = -1 means perfectly negatively correlated.
What do portfolio weights and Expected returns mean?
What is the expected return of a portfolio?
How do you work out variance of a portfolio ( KEY) !!! ( N case)
Explain what the double summation means, in exam we are likely to only have 2 assets?
When we have 2 assets, we have 4 terms to add up, if there was 3 there would be 9 terms ygm.
We are going to break down the variance of portfolio equation, so when n = 1 and m = 1, what is the Cov( R1,R1) and what is the Cov ( R1,R2) equal too?
SO Cov(R1,R2) = Cov (R2,R1) so we just need to find 1 and times by 2.
So how can we rewrite to get a more condensed formula for the variance of portfolio?
Here is a visual representation of Variance of a portfolio N asset case, so answer these questions:
1) In a 10 stock portfolio how many terms would need to be added to calculate the portfolio variance?
2) How many of the terms would be variance terms?
3) How many of the terms would be covariance terms?1
4) How many individual covariance terms would you need?
As tend to have a correlation < +1 what is a smart thing to do?>
Have a diversification of portfolio, because it reduces portfolio risk, it averages out the risk. ( variance)
Lets say 2 stocks have a perfect correlation, and the investor is risk adverse, what can he do? plus what does the perfectly correlated line mean?
Moving through the line, changes the weight of the assets in portfolio. If you are risk adverse, you might invest in A because less variance, but the wise thing to do would be to combine stock A and B into a portfolio, as there is a point ( minimum variance portfolio, that gives you the least amount of risk, from these 2 stocks, hence you don’t give up E(R), you get more with less risk.
So when we have 10 assets e.g. in a portfolio and all equal weight, what is the good thing about it, but what is the risk and what shall we think about when investing e.g. say the 11 asset?
The good thing about having 10 assets is that variance decreases, however you cannot diversify away from systematic risk. there are economy wide risk factors that generally result in a positive correlation/covariance between stock terms. So when investing in the 11 asset, its important to not only thing about variance of asset, but how it covaries with the rest of assets in portfolio?>
What is Beta?
It is a measure of the contribution of one particular stock to a portfolio’s overall risk ( measure of systematic risk, that cannot be diversified away)
What is the formula for Beta, for your portfolio’s overall risk and when it states market?
Covariance between return on asset and return on the portfolio / varaince of the portfolio ( its like adding the rows in the visual diagram )
What does a higher beta mean?
A higher beta means more risk coming from stock i ( The higher the
correlation that stock i has with the overall portfolio return, the greater the
stock’s contribution to risk.)
What does a negative beta mean?
A negative beta stock hedges portfolio risk. If you increase the weight on
such a stock then portfolio variance will fall (as the stock’s return tends to
move the opposite way to the market). ( kind of like insurance, when the market goes bad)
Lets interpret beta in relation to market, if the market goes up and beta is 1.46( GM) and another company with beta 0.3( tesco) what does it mean?
The company with beta 1.46 goes up when market goes up more than the company with beta 0.3, but when market goes down, the company with beta 1.46 goes down further than the company with 0.3 beta
THis makes sense because if market goes bad, people are not going to stop buying food from tesco whereas General motors it will.
What is the difference between sysematic and unsystematic risk e.g. if we are focusing on amazon? What is votailtiy?
Systematic risk - affect overall economy, e.g. economic slowdown, so less global demand.
Unsystematic risk - only affects amazon ( ceo retiring, firm specific)
Votality is the total risk ( Systematic and unsystematic risk)
How do we work out the beta of a portfolio?
What are we going to assume about investors ?
1) When comparing two portfolios of equal expected return, investors prefer
the portfolio with the smallest return standard deviation (or variance).
2)When comparing two portfolios of equal return standard deviation, investors
prefer the one with the higher expected return.
3)Investors like reward (mean return) and dislike risk (standard deviation or
variance). Investors are mean-variance optimisers.
What is a portfolio frontier and what is the minimum variance portfolio?
Portfolio frontier = 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.
Minimum variance portfolio = The portfolio that gives the lowest possible variance ( SD)
What is the Efficient frontier?
It is the section of the portfolio frontier that lies above and to the right of the minimum variance portfolio.
So Identify individual stocks, portfolio frontier, Minimum variance portfolio and Efficient frontier
1) individual stocks are identified by dots, for a given level of SD and E(R).
2) Portfolio frontier, for a given level of E(R) they want a portfolio that gives mimium risk( which is the bold black line)
3) MVP, gives the least SD of the PORFTOLIO FRONTIER.
4) EF - is the upside of the the MVP. ( for the same Risk, there is one point with have E(R) AND LOW E(R), investors will pick the one with high e(R)
Show on diagram assuming CORR = 1
So using the same 2 assets draw, when corr = 1 , 0 and -1,
Show the MVP
MVP is the risk free portfolio, ( sd = 0
Why does forming portfolio’s of stocks reduce risk?
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.
Which portfolio does an investor choose on the efficient frontier?
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).
SO here what is it important to remember?
This is the efficient frontier where there are only risky assets only
When the risk-free asset is combined with a risky asset/portfolio the risk return
opportunity set is always what?
When the risk-free asset is combined with a risky asset/portfolio the risk return
opportunity set is always LINEAR.
What is the expected return of the portfolio consisting of risk free asset and risky asset/portfolio A and variance?
What is the line called?
The capital allocation line = 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
If the risky asset/portfolio A is the market portfolio, combining market portfolio with the risk free rate gives the exact same line but called what?
Capital market line.
What does this section show ?
Positive weights for both assets
What does this show?
Here investor using own money and borrowed money to invest more as they are less risk adverse ( one asset will have a over positive weight and one will have a negative weight.
Combine the efficient frontier with N risky assets and the Rf asset?
Which portfolio will a mean-variance optimising investor combine with the risk- free asset?
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.
Why is the line which is tangent to the original efficient portfolio ( tangency portfolio) when risky assets combined with risk free asset, the mean variance optimising point? To explain is there a new efficient frontier?
The line which is tangent to the original efficient frontier when you have risky assets and the risk free asset is now the efficient frontier, as it dominates any points on the original efficient frontier. ( for some level of risk, you get higher e(r) on the new efficient frontier)
You cant have allocation on the left of tangency line because there is no asset to combine with and to the right its not optimising.
wHAT IS The optimal investment of every mean-variance investor?
a combination of two portfolios (or funds) – the tangency portfolio and the risk-free asset. This result is called two-fund separation.
What is the Sharpe ratio and its formula?
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).
The tangency portfolio maximises the sharpe ratio.
What are CAPM assumptions?
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 are rational mean variance optimisers).
4)Investors have the same information (beliefs/homogeneous expectations).
5) Investors have a one period time horizon.
We are going to show how assumptions of CAPM are derived/understood? EXPLAIN THIS
If all investors have access to the same info, thus have homogeneous expectations for all assets every investor will plot a given risky asset at the same point.
Explain this part 2?
As investors are MV optimisers they are not going to hold individual assets they form portfolios, they want efficient frontier. So they look at assets and choose level of E(R), using the formula, we work out what portfolio gives you the minimum deviation
Explain this part 3
Since all investors can borrow at risk free rate, they will combine the risk free asset and portfolios and derive new efficient frontier, were we can get the tangency portfolio. The weights of these 2 assets depends on level of risk of investor. If all investors hold the same tangency portfolio then the tangency portfolio = market portfolio. Hence everyone should should risk free asset and market portfolio.
CAPM in Equilibrium characteristics (4)?
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.
How do we measure the level of systematic risk of a given stock in CAPM?
Measured by beta ( so essentially CAPM, is a formula for calculating the expected return of a security an investor expect giving the level of systematic risk they are bearing.
What is the formula for CAPM and what are some implications
1) An asset’s expected return depends on risk only through beta.
2) An asset’s expected return is linear and increasing in beta.
What is beta of a market portfolio and risk free asset?
If CAPM holds in equibirum this means that the cAPITAL MARKET LINE IS EFFICIENT = all combinations of the risk free asset and market portfolio are efficient because for any level of E(R) the point of the line gives the least risk in terms of SD. So lets translate this in the model of CAPM, what is the y axis or diagram and x axis and what is the line called. What is the gradient of the point that maximises the sharpe ratio. What does CAPM help us do?
GRAIDENT IS THE MARKET RISK PREMIUM
It allows us to evaluate whether a stock is overpriced or underpriced, as if CAPM holds everything plots on the SML( telling us the E(R) a stock should earn based on beta)
Show alpha above SML that is not on the SML and what does it mean?
Here the stock has been underpriced alpha, the stock has a higher E(R) then what CAPM predicts. Hence investors want to be putting more weight upon this stock and buy more ( this is called extra expected return stock alpha). The extra demand of stock X will increase price and hence lower expected return until in equibirum its plots on the SML.
What are 3 applications of CAPM?
Valuation of stocks - Think of a stock as just a claim to a stream of future dividends. How should we value the stock? Compute the present value of the future estimated dividend stream.
Valuation of projects - npv discount rate
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. 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.
Why is our confidence in CAPM eroded in reality?
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)
( this is a very common exam question, YOU HAVE 2 RISKY ASSETS, THE CORRELATION IS -1, WHAT IS THE MINIMUM VARIANCE PORTFOLIO, what about if its not 1?
The variance is 0, so its easy to solve.
1) We have the variance equation, we sub in what W1 is ( 1-w2) then we differentiate the function with respect to w2
2) set = 0 to solve for the minimum variance portfolio.
Is holding secreted with positive expected return which will deliver gains in the LT profit ?
No it depends on the risk e.g. if one fund manager had a realised return of 5% and another fund manager had a realised return of 10% which manager did best would depnd on the risk. e.g. equity is more risker than bonds, so if the 1st fund manager holds equity and the other holds bonds comparison is wrong
What are profits in CAPM>?
Profit = abnormal return = Alpha given risk they are taking.
In efficient markets, what cant there be?
In informationally efficient capital markets (efficient markets) , there are no
abnormal returns available from the trading of financial securities.
This is because in an efficient market all relevant information is instantly and
accurately reflected in prices.
Thus you cannot tell in advance which assets make an abnormal return.
A stock has a CAPM beta of 0.6 and due to some research you have carried out you believe its future return will be 10%. If the risk-fee rate is 2% and the
expected excess return on the market is 20%, do you expect this stock to earn
any abnormal return
A stock has a CAPM beta of 0.6 and due to some research you have carried out you believe its future return will be 10%. If the risk-fee rate is 2% and the
expected excess return on the market is 20%, do you expect this stock to earn
any abnormal return
What is the Joint-Hypothesis problem?
Testing market efficiency involves using an asset pricing model to calculate abnormal returns, but the chosen model may be incorrect. All efficiency tests are joint hypotheses assessing both market informational efficiency and the correct asset pricing model usage. Rejecting the null hypothesis could be due to either or both hypothesis components being invalid, leading to uncertainty in interpreting test results.
What is an Efficient Market?
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)
How Does Information Get Reflected In Prices?
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.
Why should we think that markets might be efficient, what economic conditions/assumptions might lead to efficiency? (What are the foundations of efficiency ?
Rationality
Independent deviations from rationality
Dominance of rational professional investors
What is the efficient market hypothesis?
Information about future stock returns quickly gets incorporated into
prices. Hence, there is no return predictability in the market.
Why is there no return predictability according to the efficient market hyphothesis ?
Markets are made efficient because arbitrageurs profit off inefficiencies and therefore correct inefficiencies.These investors have unlimited capital to trade out these inefficiencies and unlimited risk capacity, hence eliminate all stock return predictability.
Economists distinguish between 3 forms of market efficiency depending on the type of information that is reflected in stock prices, which are?
Weak form
Semi strong form
Strong form
What is weak form efficient market hyphothesis?
Weak form: investors cannot predict abnormal returns given observed history of past stock prices, trading volume and other trading data ( so based on data, you can invest in higher beta stocks, which give high return but not abnormal return)
What is Semi-strong form efficient market hyphothesis?
investors cannot predict abnormal returns given publicly available information (management quality, balance sheet composition, accounting performance, corporate news etc)
What is strong form efficient market hyphothesis?
investors cannot predict abnormal returns using publicly and privately
available information.
1) (WEAK FORM ) it were possible to make abnormal returns by simply finding ‘’the pattern’’ in stock price movement (technical analysis), WHAT WOULD IT MEAN?
2) If past information suggested a stock’s price is going to move up (down)…
1) everyone would do it, and the abnormal returns would be competed away.
2) demand for the stock would increase (decrease) affecting the price until an
abnormal return is no longer possible.
How does one test to see if semi-strong form market efficiency holds?
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.
What is a critique of Strong form?
The consistent gain of abnormal returns by such participants would constitute evidence against the strong form of EMH, WHEN PARTICPTANTS HAVE PRIVATE INFO. Although exploitation of private information is illegal in most jurisdictions
(insider trading), and hence pretty difficult to track.
What evidence is there that challanges the EMH?
Stock Price Momentum and Reversal
Post Earnings Announcement Drifts
Twin-Stock Puzzles
Asset Price Bubbles
