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COVARIANCE AND CORRELATION COEFFICIENT

MEASURING PORTFOLIO RISK

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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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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

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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%

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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

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= (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

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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%

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Assume that both A and B have expected return of 10%

REQUIREDCalculate the Covariance (ALL TO DO)

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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

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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

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= -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,

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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

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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

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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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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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SECURITY MARKET LINE

ER SML Assets here are under-priced

RFR

Assets here are over-priced COV

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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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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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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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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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Capital markets are in equilibrium i.e. we begin with all investments properly priced in line with their risk levels

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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

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FACTORS AFFECTING BETA

Line of businessAmount of financial leverage undertaken by

the firmDividend payoutLiquidityFirm sizeRate of growth of the firm

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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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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%

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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

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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)

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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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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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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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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

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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

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(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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risk –bearing portfolio. REQUIREDCalculate the stocks required rate of return using the

APT approach

SOLUTIONFORMULARR= ∑RFR + (SRR-RFR)SβWHERE

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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%

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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