This document discusses covariance and correlation coefficient, which are important concepts for measuring portfolio risk. Covariance measures how asset returns tend to move together, while correlation coefficient standardizes covariance for easier comparison. The document provides formulas and examples to calculate covariance and correlation between two assets. It also discusses how covariance and beta coefficient relate to the Capital Asset Pricing Model (CAPM) and its assumptions regarding the security market line and required rates of return on assets. The Arbitrage Pricing Theory, which allows for multiple factors to influence returns, is introduced as an alternative to the single-factor CAPM.

1. MEASURING PORTFOLIO RISK
COVARIANCE AND CORRELATION
COEFFICIENT
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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
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3. FORMULA
COV(XY)= ∑(PRX-ERX)(PRY-ERY)P
WHERE
PRX= Possible return of stock X
ERX=Expected return of stock X
PRY= Possible return of stock Y
ERY=Expected return of stock Y
P= Probability of the category accordingly
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4. EXAMPLE
You have been provided with two stocks with the
following outcomes as follows
possible return
Probability X Y
0.10 6% 14%
0.20 8% 12%
0.40 10% 10%
0.20 12% 8%
0.10 14% 6%
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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%
REQUIRED
Determine the covariance between the two
stocks
SOLUTION
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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
= -4.80
 This 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
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7. EXAMPLE TWO (ALL TO DO)
Possible return
A B
Probability
0.10 6% 4%
0.20 8% 6%
0.40 10% 8%
0.20 12% 15%
0.10 14% 22%
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8. Assume that both A and B have expected return
of 10%
REQUIRED
Calculate the Covariance (ALL TO DO)
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9. SOLUTION = +10.80
Meaning these assets tend to move together
as indicated by the +ve sign
NOTE 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
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10. CORRELATION COEFFICIENT
This is calculated as the covariance of two assets
divided by their standard deviations thus
FORMULA
CORRELATION C= COV(XY)
σXσY
EXAMPLE
Calculate the coefficient of correlation between X
and Y in our previous example
SOLUTION
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11. = -4.80
2.2*2.2
= -4.80 = -1.0
4.84
MEANING
Since 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,
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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 ALL
If we are given the standard deviation of A as 2.2%
AND of B as 5.3%
REQUIRED
Calculate the coefficient of correlation of the two
assets
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13. SOLUTION
COLL C= 10.8 = 10.8
2.2*5.3 11.66
= 0.92
MEANING
There is a strong positive relationship
between the two assets and therefore these
assets will tend to bear similar risks
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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.
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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
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16. SECURITY MARKET LINE
ER
SML
Assets here are under-priced
RFR
Assets here are over-priced
COV
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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)
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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
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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
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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
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21. Capital markets are in equilibrium i.e. we
begin with all investments properly priced in
line with their risk levels
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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 investment
 b > 1 ... Above Average risk investment
 b < 1 ... Below Average risk investment
 = 0 ... Riskless investment
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23. FACTORS AFFECTING BETA
Line of business
Amount of financial leverage undertaken by
the firm
Dividend payout
Liquidity
Firm size
Rate of growth of the firm
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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
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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 -0.30
Require: calculate expected rates of return assuming an
economy’s RFR of 5% and return on market portfolio (Rm)
to be 9%
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26. SOLUTION
The market risk premium = 9%-5%=4%
THUS
E(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
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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%
MEANING
 These 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)
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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
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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.
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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
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31. In this particular case, required returns would be
a function of TWO factors namely
 Market risk
 Dividend policy
Additionally, 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
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32.  This approach can include any number of risk factors
meaning the required return could be a function of
several factors
EXAMPLE
Lets 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
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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
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34. risk –bearing portfolio.
REQUIRED
Calculate the stocks required rate of return using
the APT approach
SOLUTION
FORMULA
RR= ∑RFR + (SRR-RFR)Sβ
WHERE
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35.  RR= Requires Rate of return
 RFR= Risk Free Rate
 SRR= Subjective Required Rate of Return
 Sβ= Subjective Beta
THUS
RR= 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%
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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
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