CH 13- Textbook Flashcards
to a diversified investor what sort of risks matter
systematic risk
what are the two types of risks associated with individual assets
systematic risk: affects all asssets in economy
unsystematic risk: affects small number of assets
what is the SML high level
realtionship between risk and return; it is a representation of market equilbrium
what are the states of the eocnomy
boom, recession, etc.
how do you calcualte expected reurn from a stock?
Let the stock be Stock C
E(Rc)= (prob of boom)( stock c returns in boom)+(prob of recession)(stock c return in recession)
or E(R) = sumj Rj x Pj (R=Value of jth outcome, P= prob of jth outcome)
its the sum of the possible returns
multiplied by their probabilities
How do you calculate regular riskpremium?
how do you calc projected/ expected risk preiium?
risky investment- risk free investment
expected return on a risky investment- certain return on a risk free investment
How to calculate the vriance
- get expected return
- return-expected return
- square #2
- ultiply by Probability
and then if you want SD then sqrt #4
Var= Sumj[ Rj - E(R)]^2 x Pj
SD= sqrt(var)
how to do analysis with sd, var and mean?
- which has the higher expected return?
- does this also have higher volatility? (remember you can cover 2/3 probability by adding / subtracitng one standard deviation)
what is a portfolio?
owning several stocks/bonds/assets
Portfolio weights
the amount of the total portfolio value that is invested in each portfolio asset
if you have $50 in asset A and $150 in asset B, portfolio is worht $200, and A has 25% weight and B has 75%
PORTFOLIO WEIGHT MUST ADD UP TO 1
How to calcualte portfolio returns in each state of economy
hwo to calcualte expected return of the portfolio from this?
FOR BOOM= (portfolio weight x return of asset A)+ (portfolio weight x return of Asset B)
FOR RECESSION= (portfolio weight x return of asset A)+ (portfolio weight x return of Asset B)
E(Rp)= Sum FOR BOOM + FOR RECESSION
IS Standard deviaiton of a portfolio the sum of standard deviation of all individual assets?
NO! diversifiying portoflio reduces risk!!!
this is because of correlation
positive correlation
negative correlation
0 correlation
assets A and b move together
aseets a and b move in opp directions
unrelated
what does it mean when portfolios are dominated b the min var portfolio
like a curve on the grpah, there is apoint with min variance and higher expected retunr
though you could invest 100% in these portfolios it wouldr educe the expected return and require greater variance so no one would od it
When the returns on the two assets are perfectly correlated, the portfolio standard
deviation is simply the weighted average of the individual standard deviations. In this
special case:
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With perfect correlation, all possible portfolios lie on a straight line between U and L in
expected return/standard deviation space as shown in Figure 13.4. In this polar case,
there is no benefit from diversification. But, as soon as correlation is less than perfectly
positive (CORR , < +1.0), diversification reduces risk.
As long as CORR is less than +1.0, the standard deviation of a portfolio of two securities is
less than the weighted average of the standard deviations of the individual securities.
Figure 13.4 shows this important result by graphing all possible portfolios of U and L
for the four cases for CORR given in Table 13.7. The portfolios marked 1, 2, 3, and 4
in Figure 13.4 all have an expected return of 20.91%, as calculated in Table 13.7.
*
COVL,U = CORRL,U × σL × σU
TABLE 13.7
Portfolio standard deviation and correlation
Stock L x = 2/11 σ = 45% E(R ) = 25%
Stock U x = 9/11 σ = 10% E(R ) = 20%
E(R ) = (2/11) × 25% + 9/11 × 20% = 20.91%
CORR Portfolio Standard Deviation of Portfolio σ
1. −1.0 0.0000%
2. 0.0 11.5708%
3. +0.7 15.0865%
4. +1.0 16.3636%
L L L
U U U
P
L,U P
(
2
11
× 45%) + (
9
11
× 10%) = 16. 3636%
L U
L,U
Their standard deviations also come from Table 13.7. The other points on the respective
lines or curves are derived by varying the portfolio weights for each value of CORR .
Each line or curve represents all the possible portfolios of U and L for a given correlation.
Each is called an opportunity set or feasible set. The lowest opportunity set representing
CORR = 1.0 always has the largest standard deviation for any return level. Once again,
this shows how diversification reduces risk as long as correlation is less than perfectly
positive.
While higher correlations reduce its advantages, doubts about
diversification were overstated for two reasons. First, although correlations undoubtedly increased during
the crisis, they later returned to more normal levels. Second, even with relatively high positive correlation
between assets, diversification still reduces risk as long as the correlation coefficient is less than 1.0
(perfect positive correlation).
What is total return?
expected return + unexpected return
R= E(R) +U
Announcement = expected part + surpruse
expected part= what we use to form expectation (E(r)
SURPRISE = UNEXPECTED PART
market generally already discounts for the expected part
Systematic risk:
affects lots of assets, market-wide, market risks
unsystematic risk
surprise risk, affects a single asset or a small group, unique/asset speciifc,
break down the return using systematic and unsytsematic risk
Return= exp return + systematic portion + unsystematic portion
R= E(R) + SYSTEMATIC + UNSYSTEMATIC
Systematic = m
USYSTEMATIC= e
the more stocks you have in your pprootfolio
the less risk you face in terms of standard deviation
Principle of diversificaiton
spreading an investment across many assetss elimiates some of the risk!
diversifiabel risk is
the part hta t can be elimiated by diveristiofication
Can a minimum level of risk be elimated?
not possibel simply by diversifying, min level= non diversificable risk
DIVERSIFICATION REDUCES RISK BUT ONLY TO A CERTAIN POINT
Unsystematic risk is essentially eliminated by diversification, so a
relatively large portfolio has almost no unsystematic risk. In fact, the terms diversifiable risk and unsystematic
risk are often used interchangeably.
basically if you hold a large enough portfolio you do not face unsystematic risk since they cancel out
can you elimiate systematic risk by diversificaton
NO! SYSTEMATIC RISK= NON DIVERSIFIABLE
Total risk= systematic risk + usnystematic risk
Systematic risk is also called nondiversifiable risk or market risk. Unsystematic risk is also called
diversifiable risk, unique risk, or asset-specific risk
for a well diversified portoofio what is the type of risk you have
mostly all systematic risk
systematic risk principle
MAIN TEACHING
reward for bearing risk depends only on the systematic risk of an investent
(since you can eliminate all unsystematic risk by diversifying, you dont get rewardsed for this risk)
the expected return on an
asset depends only on that asset’s systematic risk
WHAT DOES BETA TELL US
how much systematic risk a particular asset has relative to an average asset representing the market portfolio
an avg asset has a beta of 1.0 itself
an asset with beta 0.5 has half as much systematic risk as an avg asset
is high or low beta better
LOW!! high=high volatilyty
if you have two assets, one has high sd and low beta and other has high beta low sd what do you know?
low beta one has low systematic risk, so it has greater unsystematic irsk
high beta= high systematic risk so higher premiums, low unsystematic risk
how to calculate portfolio beta
just like expected return
Port Beta= (weight invested in stock A x stock A beta) + (Weight invested in stock B x stock B beta)
multiply each asset’s beta by its
portfolio weight and then add the results to get the portfolio’s beta
what does risk free rate mean
-what does it imply for beta?
no systematic risk
beta=0
for assets that are risk free, they do not have an expected return they have a
risk free rate (no risk so this is the guaranteed return)
lets say you have portfolio w ASSET A and a risk free asset
you invest 25% into asset a and 75% into b
what is your expected return?
what is your portfolio beta?
Exp return= (inv into asset a x exp return asset a) + (1- inv into A x risk free rate)
Beta Portfolio= (inv into asset a x Beta A)+(1- inv into A x 0)
==X 0 BECAUSE THERE IS NO BETA ON RISK FREE ASSETS
==Why do you do 1- inv into A and not just inv into B ? because often you can borrow money at risk free rate and then invest it backk
If you plot portfolioe xpected return (E(rp)) along y axis and Portoflio beta( Bp) along x axis
describe the graph
the y int is the risk free rate
the line itself is the combinations of different %’s being invested into each asset (SECURITY MARKET LINE)
slope is the reward to risk ratio
(E(R(A)) - Rf)/Beta A
The reard to risk ratio tells us the premium per uunit of systematic risk on Asset A
The reard to risk ratio tells us the premium per uunit of systematic risk on Asset A
The reard to risk ratio tells us the premium per uunit of systematic risk on Asset A
1How can we compare if asset investments are better given that we know the reward to risk ratio?
PLOT THE LINES ON GRAPH, the higher reward risk ratio (slope is steeper) means its a bettter investment!!!!
how to know if stocks are correctly priced?
- plug in values to CAPM
- COMPARE to given expected returns (is it higher or lower)?
or use reward to risk ratio
-> calculate it
-> compare to market risk premium! is it too high or too low
What happens if for Asset A the reward to risk is higher than B?
PRICE AND RETURN MOVE IN OPP DIRECTIONS, so Price of A would rise since its better and return would fall; this would create vice versa effect
UNTIL the reard to risk ratio for both is equal (this is what happens in an active market, the reward risk ration on all securtiies is equal)
if one asset has twice as much
systematic risk as another asset, its risk premium is simply twice as large; meaning that the return is lower
visualize market equilbirum of assets returns and asset prices on return and beta chart
If you have an asset above the market equilbrium (meaning that it has higher return) the price will rise until the return falls to the equilbrium, beocming one with the main line again
An asset is said to be overvalued if its expected return is too low given its risk.
how to determine if an assset is overvalued or undervaluesd?
- compute reward to risk ratio for al l idnividual assets
- the higher ratio will be what investors want to go for, and the price of the other would drop, the lower ratios return would rise until market equilbrium
so if prices are going to fall then it is overvalued
How to calcualte beta
Draw a line relating expected return to diff returns in the market, it is the characteristic line of security, and slope= stocks beta
low R^2 means that most of the risk of a firm is unsystematic
What is SML
- equation
(E(Rm)-Rf/BetaM), WHERE E(Rm)= expected return on the market portfolio [portfolio of all the assets in the market], and the beta is 1
SO
E(Rm)-Rf = SML Slope
THIS IS CALLED THE MARKET RISK PREMIUM
Capital asset pricing model CAPM
if you are given a beta of a stock, expected return on the market, and risk free rate, how to calc expceted retunron on this stock?
FORMULA CAVEAT!!
E(Ri) = Rf + [ERm -Rf] x Bi
Expected Return on Stock i= Risk free rate +(Expected return on market - risk free rate) x Stock Beta
CAVEAT!! IF YOU ARE GIVEN MARKET RISK PREMIUM- > that is equal to (ERm-Rf)
ExP RETURN ON asset i is risk free rate + exp return on market portfolio = risk free rate x beta of i
WHAT DOES CAPM SHOW
exp return on an asset depends on 3 things:
- The pure time value of money. As measured by the risk-free rate, R , this is the reward for merely waiting
for your money, without taking any risk.
f - The reward for bearing systematic risk. As measured by the market risk premium [E(R ) − R ], this
component is the reward the market offers for bearing an average amount of systematic risk in addition to
waiting.
M f - The amount of systematic risk. As measured by β, this is the amount of systematic risk present in a
particular asset, relative to an average asset.
arbitrage pricing theory APT
handles multiple factors CAPM ignore,
The three factors F , F , and F represent systematic risk because these factors affect many securities.
The term ε is considered unsystematic risk because it is unique to each individual security.
Under this multifactor APT, we can generalize from three to K factors to express the relationship between
risk and return as:
beta with respect to the second factor, and so on. For example, if the first factor is inflation, β is the
security’s inflation beta. The term E(R ) is the expected return on a security (or portfolio) whose beta with
respect to the first factor is one and whose beta with respect to all other factors is zero. Because the market
compensates for risk, E[(R ) − R )] is positive in the normal case. (An analogous interpretation can be
given to E(R ), E(R ), and so on.)
The equation states that the security’s expected return is related to its factor betas. The argument is that each
factor represents risk that cannot be diversified away. The higher a security’s beta is with regard to a
particular factor, the higher the risk that security bears. In a rational world, the expected return on the
security should compensate for this risk. The preceding equation states that the expected return is a
summation of the risk-free rate plus the compensation for each type of risk the security bears.
As an example, consider a Canadian study where the factors were:
Total return = Expected return + Unexpected return
R = E(R) + U
R = E(R) + βIFI + βGNPFGNP + βrFr + ε
I GNP r
E(R) = RF + E[(R1) − RF]β1 + E[(R2) − RF]β2 + E[(R3) − RF]β3 + … + E[(RK) − RF]βK
1 2
1
1
1 f
12
2 3
1. The rate of growth in industrial production (INDUS).
2. The changes in the slope of the term structure of interest rates (the difference between the yield on longterm and short-term Canada bonds) (TERMS).
Using the period 1970–84, the empirical results of the study indicated that expected monthly returns on a
sample of 100 TSX stocks could be described as a function of the risk premiums associated with these five
factors.
Because many factors appear on the right side of the APT equation, the APT formulation explained
expected returns in this Canadian sample more accurately than did the CAPM. However, as we mentioned
earlier, one cannot easily determine which are the appropriate factors. The factors in this study were
included for reasons of both common sense and convenience. They were not derived from theory, and the
choice of factors varies from study to study. A more recent Canadian study, for example, includes changes
in a U.S. stock index and in exchange rates as factors.
The CAPM and the APT by no means exhaust the models and techniques used in practice to measure the
expected return on risky assets. Both the CAPM and the APT are risk based. They each measure the risk of a
security by its beta(s) on some systematic factor(s), and they each argue that the expected excess return
must be proportional to the beta(s). As we have seen, this is intuitively appealing and has a strong basis in
theory, but there are alternative approaches.
One popular alternative is a multifactor empirical model developed by Fama and French and based less on a
theory of how financial markets work and more on simply looking for regularities and relations in the past
history of market data. In such an approach, the researcher specifies some parameters or attributes
associated with the securities in question and then examines the data directly for a relation between these
attributes and expected returns. Fama and French examine whether the expected return on a firm is related to
its size and market to book ratio in addition to its beta. Is it true that small firms have higher average
returns than large firms? Do growth companies with high market to book ratios have higher average returns
than value companies with low market to book ratios? A well-known extension of the Fama-French
model includes a fourth, momentum, factor measured by last year’s stock return.
Although multifactor models are commonly used in investment performance analysis, they have not become
standard practice in estimating the cost of capital. Surveys of corporate executives show that only one in
three employ multifactor models for this purpose while over 70% rely on the CAPM.
Concept Questions
3
The best thing is when we find that two stocks have a negative correlation; then we can start to
make a portfolio out of them where one goes up and another goes down
diversificaiton can reduce our risk as long as our stocks arent perfectly corealted
p<1
Does expected return have to be a possible return?
NO! it can be an imagined rturn
what is the feasible set
the opportuntiy set/ or the curve that comprises all of the possible portoflio combos
what is the efficient st
the protion of the feasible set that only includes efficient protoflio (max return for x amount of risk or min risk for x amount of return)\
That’s what efficient means: no other portfolio ties on risk and wins on return,
or ties on return and wins on risk, or wins on both
WHAT IS THE MVP on the graph
minimum variance portfolio the posible pportfolio with the least amount of risk
NOT SYAING ITS THE BEST, BUT IF YOU WANT TO AVOID RISK THIS IS THE ONE TO GO WITH
OVERTIME WHAT is the averagge of the unexpected component
it is zero
Efficient markets are a result of investors trading
on the unexpected portion of announcements.
they also involve random price changes bc we cant predict surpirses
what is the slope of the SML
reward to risk ratio
E(Rm)-(Rf)
What is the CAPM
capital asset pricing model
E(ra) = Rf + BetaA(E(Rm)-Rf)
relates risk and return
tells us what return woudld make us happy iwth a fiven stock
capm, if we have a specialc ase when portfolio is equally weighted betwene risk free asset and a stock
sum the returns of each asseet and divide by number of assets
If you are given an assets expected return, beta and the risk free rate, told to make a table for portfolios of asset and a risk free asset, how do you calc portfolio expected return and protfolio beta?
- We know beta of risk free asset is 0, and weight of a risk free asset (with 2 assets in the porftolio only) is 1- weight of stock
- Solve for Beta:
Beta = Weight of stock(Beta of stock) + (1-Weight of stock)(Beta of risk free)
-> note that the second term is 0, now you have to solve using the CAPM
- Use CAPM to solve for the Market Risk Premium=
E(Rm) = Rf +(ERm - Rf)(Beta)
**MRP= (Erm-Rf)
- Now you have created an equation that is a slope, with the x being Beta (plug in the various weights to caluclate the beta), and then fill the chart
if given two stocks and asked to give a risk free rate where the stocks are correctly priced what do you do
1, set each stocks reward to risk ratio = to each other
2. cross mulitply and solve for the risk free rate
