# variance co-variance

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Risk and ReturnLearning Module

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Expected Return

The future is uncertain.

Investors do not know with certainty whether theeconomy will be growing rapidly or be in recession.

Investors do not know what rate of return theirinvestments will yield.

Therefore, they base their decisions on theirexpectations concerning the future.

The expected rate of return on a stock representsthe mean of a probability distribution of possiblefuture returns on the stock.

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Expected Return The table below provides a probability distribution for the returns on

stocksA and B

State Probability Return On Return On

Stock A Stock B

1 20% 5% 50%

2 30% 10% 30%

3 30% 15% 10%

4 20% 20% -10%

The state represents the state of the economy one period in the future

i.e. state 1 could represent a recession and state 2 a growth economy.

The probability reflects how likely it is that the state will occur. The sumof the probabilities must equal 100%.

The last two columns present the returns or outcomes for stocksA andB that will occur in each of the four states.

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Expected Return

Given a probability distribution of returns, theexpected return can be calculated using the followingequation:

N

E[R] = S (piRi)i=1

Where:

E[R] = the expected return on the stock

N = the number of states

pi = the probability of state i

Ri = the return on the stock in state i.

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Expected Return

In this example, the expected return for stock Awould be calculated as follows:

E[R]A= .2(5%) + .3(10%) + .3(15%) + .2(20%) = 12.5%

Now you try calculating the expected return forstock B!

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Expected Return

Did you get 20%? If so, you are correct.

If not, here is how to get the correct answer:

E[R]B = .2(50%) + .3(30%) + .3(10%) + .2(-10%) = 20%

So we see that Stock B offers a higher expectedreturn than Stock A.

However, that is only part of the story; we haven'tconsidered risk.

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Measures of Risk

Risk reflects the chance that the actual return on aninvestment may be different than the expectedreturn.

One way to measure risk is to calculate the varianceand standard deviation of the distribution of returns.

We will once again use a probability distribution inour calculations.

The distribution used earlier is provided again forease of use.

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Measures of Risk

Probability Distribution:

State Probability Return On Return OnStock A Stock B

1 20% 5% 50%

2 30% 10% 30%

3 30% 15% 10%4 20% 20% -10%

E[R]A= 12.5%

E[R]B

= 20%

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Measures of Risk

Given an asset's expected return, its variance can becalculated using the following equation:

N

Var(R) = s2 = S pi(Ri E[R])2i=1

Where:

N = the number of states

pi = the probability of state i

Ri = the return on the stock in state i

E[R] = the expected return on the stock

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Measures of Risk

The standard deviation is calculated as the positivesquare root of the variance:

SD(R) = s = s2 = (s2)1/2 = (s2)0.5

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Measures of Risk

The variance and standard deviation for stock A is calculatedas follows:

s2A= .2(.05 -.125)2 + .3(.1 -.125)2 + .3(.15 -.125)2 + .2(.2 -.125)2 =.002625

sA = (.002625)0.5 = .0512 = 5.12%

Now you try the variance and standard deviation for stock B!

If you got .042 and 20.49% you are correct.

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Measures of Risk

If you didnt get the correct answer, here is how to get it:

s2B = .2(.50 -.20)2 + .3(.30 -.20)2 + .3(.10 -.20)2 + .2(-.10 - .20)2 =

.042

sB = (.042)0.5 = .2049 = 20.49%

Although Stock B offers a higher expected return than

Stock A, it also is riskier since its variance and standarddeviation are greater than Stock A's. This, however, is still only part of the picture because most

investors choose to hold securities as part of a diversifiedportfolio.

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Portfolio Risk and Return

Most investors do not hold stocks in isolation.

Instead, they choose to hold a portfolio of severalstocks.

When this is the case, a portion of an individual stock'srisk can be eliminated, i.e., diversified away.

From our previous calculations, we know that: the expected return on Stock A is 12.5%

the expected return on Stock B is 20% the variance on Stock A is .00263

the variance on Stock B is .04200

the standard deviation on Stock A is 5.12%

the standard deviation on Stock B is 20.49%

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Portfolio Risk and Return

The Expected Return on a Portfolio is computed as theweighted average of the expected returns on the stockswhich comprise the portfolio.

The weights reflect the proportion of the portfolio invested inthe stocks.

This can be expressed as follows:N

E[Rp] = S wiE[Ri]i=1

Where: E[Rp] = the expected return on the portfolio

N = the number of stocks in the portfolio

wi = the proportion of the portfolio invested in stock i

E[Ri] = the expected return on stock i

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Portfolio Risk and Return

For a portfolio consisting of two assets, the aboveequation can be expressed as:

E[Rp

] = w1

E[R1

] + w2

E[R2

]

If we have an equally weighted portfolio of stock Aand stock B (50% in each stock), then the expectedreturn of the portfolio is:

E[Rp] = .50(.125) + .50(.20) = 16.25%

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Portfolio Risk and Return

The variance/standard deviation of a portfolio reflects notonly the variance/standard deviation of the stocks that makeup the portfolio but also how the returns on the stocks which

comprise the portfolio vary together. Two measures of how the returns on a pair of stocks vary

together are the covariance and the correlation coefficient. Covariance is a measure that combines the variance of a stocks

returns with the tendency of those returns to move up or down at

the same time other stocks move up or down. Since it is difficult to interpret the magnitude of the covariance terms,

a related statistic, the correlation coefficient, is often used tomeasure the degree of co-movement between two variables. Thecorrelation coefficient simply standardizes the covariance.

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Portfolio Risk and Return

The Covariance between the returns on two stocks can becalculated as follows:

N

Cov(RA,RB) = sA,B = S pi(RAi - E[RA])(RBi - E[RB])i=1

Where: sA,B = the covariance between the returns on stocks A and B

N = the number of states

pi = the probability of state i RAi = the return on stock A in state i

E[RA] = the expected return on stock A

RBi = the return on stock B in state i

E[RB] = the expected return on stock B

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Portfolio Risk and Return

The Correlation Coefficient between the returns on twostocks can be calculated as follows:

sA,B

Cov(RA

,RB

)

Corr(RA,RB) = rA,B = sAsB = SD(RA)SD(RB)

Where:

rA,B=the correlation coefficient between the returns on stocks A and B sA,B=the covariance between the returns on stocks A and B,

sA=the standard deviation on stock A, and

sB=the standard deviation on stock B

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Portfolio Risk and Return

The covariance between stock A and stock B is as follows:

sA,B = .2(.05-.125)(.5-.2) + .3(.1-.125)(.3-.2) +

.3(.15-.125)(.1-.2) +.2(.2-.125)(-.1-.2) = -.0105

The correlation coefficient between stock A and stock B isas follows:

-.0105

rA,B = (.0512)(.2049) = -1.00

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Portfolio Risk and Return

Using either the correlation coefficient or the covariance,theVariance on a Two-Asset Portfolio can becalculated as follows:

s2p = (wA)2s2A+ (wB)

2s2B + 2wAwBrA,B sAsBOR

s2

p = (wA)2

s2

A+ (wB)2

s2

B + 2wAwB sA,B

The Standard Deviation of the Portfolio equals thepositive square root of the the variance.

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Portfolio Risk and Return

Lets calculate the variance and standard deviation of aportfolio comprised of 75% stock A and 25% stock B:

s2p =(.75)2(.0512)2+(.25)2(.2049)2+2(.75)(.25)(-1)(.0512)(.2049)= .00016

sp = .00016 = .0128 = 1.28%

Notice that the portfolio formed by investing 75% in Stock A

and 25% in Stock B has a lower variance and standarddeviation than either Stocks A or B and the portfolio has ahigher expected return than Stock A.

This is the purpose of diversification; by forming portfolios,some of the risk inherent in the individual stocks can be

eliminated.

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Capital Asset Pricing Model

(CAPM)

If investors are mainly concerned with the risk of their portfolio ratherthan the risk of the individual securities in the portfolio, how should therisk of an individual stock be measured?

In important tool is the CAPM.

CAPM concludes that the relevant risk of an individual stock is itscontribution to the risk of a well-diversified portfolio.

CAPM specifies a linear relationship between risk and required return.

The equation used for CAPM is as follows:Ki = Krf+ bi(Km - Krf)

Where:

Ki = the required return for the individual security Krf= the risk-free rate of return bi= the beta of the individual security Km = the expected return on the market portfolio

(Km - Krf) is called the market risk premium

This equation can be used to find any of the variables listed above,

given the rest of the variables are known.

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CAPM Example

Find the required return on a stock given that the risk-freerate is 8%, the expected return on the market portfolio is12%, and the beta of the stock is 2.

Ki = Krf+ bi(Km - Krf)

Ki = 8%+ 2(12% - 8%)

Ki = 16%

Note that you can then compare the required rate of return

to the expected rate of return. You would only invest in

stocks where the expected rate of return exceeded the

required rate of return.

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Another CAPM Example

Find the beta on a stock given that its expected return is12%, the risk-free rate is 4%, and the expected return onthe market portfolio is 10%.

12%= 4%+ bi(10% - 4%)

bi = 12% - 4%

10% - 4% bi = 1.33

Note that beta measures the stocks volatility (or risk)

relative to the market