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.
Interest rate risk management for banks under Basel II, presentation by Christine Brown, Department of Finance , The University of Melbourne, Shanghai, December 8-12, 2008
Formula Plan in Securities Analysis and Port folio ManagementSuryadipta Dutta
Formula Plan in Securities Analysis and Port folio Management INCLUDING introduction,need, types, advantages with constant rupee value plan, constant ratio plan, Variable Ratio Plan, limitations and with every notes.
Risk and Return: Portfolio Theory and Assets Pricing ModelsPANKAJ PANDEY
Discuss the concepts of portfolio risk and return.
Determine the relationship between risk and return of portfolios.
Highlight the difference between systematic and unsystematic risks.
Examine the logic of portfolio theory .
Show the use of capital asset pricing model (CAPM) in the valuation of securities.
Explain the features and modus operandi of the arbitrage pricing theory (APT).
Interest rate risk management for banks under Basel II, presentation by Christine Brown, Department of Finance , The University of Melbourne, Shanghai, December 8-12, 2008
Formula Plan in Securities Analysis and Port folio ManagementSuryadipta Dutta
Formula Plan in Securities Analysis and Port folio Management INCLUDING introduction,need, types, advantages with constant rupee value plan, constant ratio plan, Variable Ratio Plan, limitations and with every notes.
Risk and Return: Portfolio Theory and Assets Pricing ModelsPANKAJ PANDEY
Discuss the concepts of portfolio risk and return.
Determine the relationship between risk and return of portfolios.
Highlight the difference between systematic and unsystematic risks.
Examine the logic of portfolio theory .
Show the use of capital asset pricing model (CAPM) in the valuation of securities.
Explain the features and modus operandi of the arbitrage pricing theory (APT).
Chapter 06 - Efficient Diversification
Chapter 06
Efficient Diversification
1. So long as the correlation coefficient is below 1.0, the portfolio will benefit from diversification because returns on component securities will not move in perfect lockstep. The portfolio standard deviation will be less than a weighted average of the standard deviations of the component securities.
2. The covariance with the other assets is more important. Diversification is accomplished via correlation with other assets. Covariance helps determine that number.
3. a and b will have the same impact of increasing the Sharpe ratio from .40 to .45.
4. The expected return of the portfolio will be impacted if the asset allocation is changed. Since the expected return of the portfolio is the first item in the numerator of the Sharpe ratio, the ratio will be changed.
5. Total variance = Systematic variance + Residual variance = β2 Var(rM) + Var(e)
When β = 1.5 and σ(e) = .3, variance = 1.52 × .22 + .32 = .18. In the other scenarios:
a. Both will have the same impact. Total variance will increase from .18 to .1989.
b. Even though the increase in the total variability of the stock is the same in either scenario, the increase in residual risk will have less impact on portfolio volatility. This is because residual risk is diversifiable. In contrast, the increase in beta increases systematic risk, which is perfectly correlated with the market-index portfolio and therefore has a greater impact on portfolio risk.
6.
a. Without doing any math, the severe recession is worse and the boom is better. Thus, there appears to be a higher variance, yet the mean is probably the same since the spread is equally large on both the high and low side. The mean return, however, should be higher since there is higher probability given to the higher returns.
b.
Calculation of mean return and variance for the stock fund:
c. Calculation of covariance:
Covariance has increased because the stock returns are more extreme in the recession and boom periods. This makes the tendency for stock returns to be poor when bond returns are good (and vice versa) even more dramatic.
7.
a. One would expect variance to increase because the probabilities of the extreme outcomes are now higher.
b.
Calculation of mean return and variance for the stock fund:
c. Calculation of covariance
Covariance has decreased because the probabilities of the more extreme returns in the recession and boom periods are now higher. This gives more weight to the extremes in the mean calculation, thus making their deviation from the mean less pronounced.
8. The parameters of the opportunity set are:
E(rS) = 15%, E(rB) = 9%, S = 32%, B = 23%, = 0.15, rf = 5.5%
From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(rS, rB) = SB]:
Bonds
StocksBonds
529.0
110.4Stocks
110.4
1024.0
The minimum-variance portfolio proportions are:
wMin(S) = = ...
Return is the amount of gain or loss of an Investment for a particular period of time.
The future is uncertain. When we are dealing with the future, we assign probabilities to future returns. The Expected rate of return on an investment represents the mean probability distribution of possible future returns.
Risk reflects the chance that the actual return on an investment may be different than the expected return.
One way to measure risk is to calculate the variance and standard deviation of the distribution of returns.
We will once again use a probability distribution in our calculations.
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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
SAN LIO 2
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%
SAN LIO 4
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
SAN LIO 5
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
SAN LIO 6
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%
SAN LIO 7
8. Assume that both A and B have expected return
of 10%
REQUIRED
Calculate the Covariance (ALL TO DO)
SAN LIO 8
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
SAN LIO 9
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
SAN LIO 10
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,
SAN LIO 11
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
SAN LIO 12
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
SAN LIO 13
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 14
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 15
16. SECURITY MARKET LINE
ER
SML
Assets here are under-priced
RFR
Assets here are over-priced
COV
SAN LIO 16
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 17
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 18
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 19
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 20
21. Capital markets are in equilibrium i.e. we
begin with all investments properly priced in
line with their risk levels
SAN LIO 21
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
SAN LIO 22
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
SAN LIO 23
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 24
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%
SAN LIO 25
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
SAN LIO 26
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)
SAN LIO 27
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 28
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 29
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 30
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
SAN LIO 31
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
SAN LIO 32
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 33
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%
SAN LIO 35
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
SAN LIO 36