Upload
urusha-hada
View
746
Download
1
Embed Size (px)
DESCRIPTION
Citation preview
SAN LIO 1
COVARIANCE AND CORRELATION COEFFICIENT
MEASURING PORTFOLIO RISK
SAN LIO 2
Again these two concepts namely covariance and correlation coefficient are important
Covariance is a measure that combines the variance (volatility) of a stock’s return with the tendency of those returns to move up or down at the same time other stocks move up and down
For instance the covariance between two stocks X and Y tells us whether the returns of the two stocks tend to RISE and FALL together and how large those movements tend to be
SAN LIO 3
FORMULACOV(XY)= ∑(PRX-ERX)(PRY-ERY)P
WHEREPRX= Possible return of stock XERX=Expected return of stock XPRY= Possible return of stock YERY=Expected return of stock YP= Probability of the category accordingly
SAN LIO 4
EXAMPLEYou have been provided with two stocks with the following
outcomes as follows possible returnProbability X Y0.10 6% 14%0.20 8% 12%0.40 10% 10%0.20 12% 8%0.10 14% 6%
SAN LIO 5
The expected returns of the two stocks X and Y are 10% and 10% respectively
The standard deviation for the two stocks is provided as follows for X an d Y respectively 2.2% and 2.2%
REQUIREDDetermine the covariance between the two
stocksSOLUTION
SAN LIO 6
= (6-10)(14-10)(0.10)+(8-10)(12-10)(0.20)+(10-10)(10-10)(0.40)+(12-10)(8-10)(0.20)+(14-10)(6-10)(0.10)
=-1.6+-0.80+0+-0.8+ -1.6= -4.80This negative sign is an indication that the rates of
return on stock X and Y tend to move in opposite directions.
Is this consistent with the figures provided?We can also plot X against Y on a graph and observe
SAN LIO 7
EXAMPLE TWO (ALL TO DO) Possible return A BProbability 0.10 6% 4%0.20 8% 6%0.40 10% 8%0.20 12% 15%0.10 14% 22%
SAN LIO 8
Assume that both A and B have expected return of 10%
REQUIREDCalculate the Covariance (ALL TO DO)
SAN LIO 9
SOLUTION = +10.80Meaning these assets tend to move together
as indicated by the +ve signNOTE if either stock has zero standard
deviation, meaning it is RISKLESS, then all its deviations (PR-ER) will be zero and the covariance will also be zero
SAN LIO 10
CORRELATION COEFFICIENT
This is calculated as the covariance of two assets divided by their standard deviations thus
FORMULACORRELATION C= COV(XY) σXσYEXAMPLECalculate the coefficient of correlation between X
and Y in our previous exampleSOLUTION
SAN LIO 11
= -4.80 2.2*2.2= -4.80 = -1.0 4.84MEANINGSince the sign of correlation of coefficient is the same
as the sign for covariance, i.e. positive sign means the variables move together and negative sign means the variables move in opposite directions,
SAN LIO 12
and that if they are close to zero, the variables are independent of each other;
Then we can observe that stock X and Y are perfectly negatively correlated
EXAMPLE TWO FOR ALLIf we are given the standard deviation of A as 2.2% AND
of B as 5.3%REQUIREDCalculate the coefficient of correlation of the two assets
SAN LIO 13
SOLUTIONCOLL C= 10.8 = 10.8 2.2*5.3 11.66 = 0.92MEANINGThere is a strong positive relationship
between the two assets and therefore these assets will tend to bear similar risks
SAN LIO 14
REMEMBER AGAIN
COVARIANCE- is the measure that combines the variance or the volatility of a stock’s return with the tendency of those returns to move up or down at the same time other stocks move up or down
CORRELATION COEFFICIENT- Is used to measure the degree of co-movement between two variables (stocks) . The correlation coefficient standardizes the covariance by dividing it by a product term, which facilitates comparisons by putting things to a similar scale. NOTE THAT it is difficult to interpret the magnitude of the covariance term.
SAN LIO 15
CAPITAL ASSET PRICING MODEL
Capital Asset Pricing Model basically helps us determine the relationship between RISK and required rates of RETURN on ASSETS when held in a well diversified portfolio.
The attitude of CAPM is the SECURITY MARKET LINE
SEE THE LINE
SAN LIO 16
SECURITY MARKET LINE
ER SML Assets here are under-priced
RFR
Assets here are over-priced COV
SAN LIO 17
NOTE that covariance is the relevant risk measure as discussed earlier.
We shall at this stage introduce the asset Beta (β) Beta is a standardized measure of risk because
it relates this covariance to the variance of the market portfolio.
Consequently, the market portfolio has a beta of 1
The SML tells us that an individual stock’s required rate of return is equal to the RFR PLUS A PREMIUM for bearing risk (the risk premium)
SAN LIO 18
The CAPITAL MARKET LINE which takes the same shape specifies a linear relationship between EXPECTED RETURN and RISK, with the slope of the CML being equal to the expected return on the market portfolio of risky stocks MINUS the risk-free rate (called the market risk premium) , all divided by the standard deviation of returns on the market portfolio
SAN LIO 19
CAPM ASSUMPTIONS
ASSUMPTIONS- Builds on Markowitz portfolio model
All investors are Markowitz efficient investors-risk-return utility function
Investors can borrow and lend any amount of money at risk-free-rate of return
All investors have homogeneous expectations i.e. they estimate intended probability distribution for future rates of return
SAN LIO 20
All investors have the same one-period time horizon e.g. one month, six or one year
All investors are infinitely divisible i.e. it is possible to buy or sell fractional shares of any asset or portfolio
There are no taxes or transaction costs involved in buying or selling assets e.g. churches
There is no inflation or any change in interest rates or inflation fully anticipated
SAN LIO 21
Capital markets are in equilibrium i.e. we begin with all investments properly priced in line with their risk levels
SAN LIO 22
THE BETA COEFFICIENT
This is seen as a standardized measure of systematic risk because it relates the covariance to the variance of the market portfolio
The market portfolio has a beta of 1 (one)Betas are standardized around one.b = 1 ... Average risk investmentb > 1 ... Above Average risk investmentb < 1 ... Below Average risk investment= 0 ... Riskless investment
SAN LIO 23
FACTORS AFFECTING BETA
Line of businessAmount of financial leverage undertaken by
the firmDividend payoutLiquidityFirm sizeRate of growth of the firm
SAN LIO 24
EXPECTED RETURN OF RISKY ASSETS
Determined by the RFR plus a risk premium for the individual asset
The risk premium is determined by the systematic risk of the asset (BETA) and the prevailing MARKET RISK PREMIUMS (Rm- RFR)
EXAMPLE
SAN LIO 25
Assume the betas of the following stocks have been computed (DONE USING REGRESSION LINE)
STOCK BETA A 0.70 B 1.00 C 1.15 D 1.40 E -0.30Require: calculate expected rates of return assuming an economy’s
RFR of 5% and return on market portfolio (Rm) to be 9%
SAN LIO 26
SOLUTION
The market risk premium = 9%-5%=4%THUSE(R1)= RFR +β(Rm-RFR)E(RA)= 0.05+ 0.7(0.09-0.05) = 0.078= 7.8%CALCULATE THE EXPECTED RETURN FOR THE
OTHER ASSETS
SAN LIO 27
B= 9.0%C= 9.6%D= 10.6%E(RE)= 0.05+(-0.30)(0.09-0.05) = 0.05-0.012 = 0.038= 3.8%MEANINGThese are the required rates of return that these stocks
should provide based on their systematic risks and the prevailing SML (Security Market Line-relates E(R1) and CV)
SAN LIO 28
NOTE
At equilibrium all assets and all portfolios of assets should plot on the SML
Means all assets should be priced so that their estimated rates of returns which in effect are the actual holding period rates of return that you anticipate, are in harmony with their levels of systematic risk
Securities with an estimated rate of return above the SML are considered underpriced
SAN LIO 29
Because this means the estimated return is above its required rate of return based on its systematic risk
Assets with estimated rates of return that plot below the SML are considered overpriced because it implies your estimated rate of return is below what you should require based on the asset’s systematic risk.
SAN LIO 30
THE ARBITRAGE PRICING THEORY
Note that CAPM is a single-factor model since it specifies risk as a function of only one factor-the security’s beta coefficient
For example consider a situation where the personal tax rates on capital gains are lower than those on dividends, investors will value capital gains more than dividends
Thus if two stocks had the same market risk, the stock paying the higher dividend would have the higher required rate of return
Why? Due to the prevailing dividend policy
SAN LIO 31
In this particular case, required returns would be a function of TWO factors namely
Market riskDividend policyAdditionally, many factors may be required to
determine the equilibrium risk/return relationship rather than just one or two
Stephen Ross tries to address this problem by introducing the approach called the ARBITRAGE PRICING THEORY
SAN LIO 32
This approach can include any number of risk factors meaning the required return could be a function of several factors
EXAMPLELets assume that all stocks returns depend on three factors;
inflation, industrial production and aggregate degree of risk aversion.
Lets further assume that the risk-free rate is 8%; the required rate of return is 13% on a portfolio with unit sensitivity (β=1) to inflation and ZERO sensitivities
SAN LIO 33
(β=0) to industrial production and degree of risk aversion; the required return is 10% on a portfolio with unit sensitivity to industrial production and ZERO sensitivities to inflation and degree of risk aversion; the required return is 6% on a portfolio (the risk-bearing portfolio) with unit sensitivity to the degree of risk aversion and ZERO sensitivities to inflation and industrial production.
Finally lets assume that the stock has factor sensitivities (betas) of 0.9 to the inflation portfolio, 1.2 to the industrial production portfolio and -0.7 to
SAN LIO 34
risk –bearing portfolio. REQUIREDCalculate the stocks required rate of return using the
APT approach
SOLUTIONFORMULARR= ∑RFR + (SRR-RFR)SβWHERE
SAN LIO 35
RR= Requires Rate of returnRFR= Risk Free RateSRR= Subjective Required Rate of ReturnSβ= Subjective BetaTHUSRR= 8%+(13%-8%)0.9+ (10%-8%)1.2 +(6%-8%)-
0.7= 8% + 4.5+2.4+1.4=16.3%Means investors will not buy the stock if it
earns them LESS than 16.3%
SAN LIO 36
NOTE
This approach is build on very complex mathematical and statistical theories and its practical use has been limited
Usage may increase in the future however and thus the need to be aware of the approach accordingly