Portfolio risk

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UNIT II

Portfolio risk

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Portfolio Expected Return

n

E(R_P) =  w_iE(R_i) i=1

where E(R_P) = expected portfolio return w_i= weight assigned to security i E(R_i) = expected return on security i

n = number of securities in the portfolio

Example A portfolio consists of four securities with expected returns of 12%, 15%, 18%, and 20% respectively. The proportions of portfolio value invested in these securities are 0.2, 0.3, 0.3,and 0.20 respectively.

The expected return on the portfolio is:

E(R_P) = 0.2(12%) + 0.3(15%) + 0.3(18%) + 0.2(20%)

= 16.3%

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Portfolio Risk

The risk of a portfolio is measured by the variance (or standard deviation) of its return. Although the expected return on a portfolio is the weighted average of the expected returns on the individual securities in the portfolio, portfolio risk is not the weighted average of the risks of the individual securities in the portfolio (except when the returns from the securities are uncorrelated).

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Measurement Of Comovements In Security Returns

• To develop the equation for calculating portfolio risk we need information on weighted individual security risks and weighted comovements between the returns of

securities included in the portfolio.

• Comovements between the returns of securities are measured by covariance (an absolute measure) and coefficient of correlation (a relative measure).

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Computation of Portfolio Variance From the

Covariance Matrix

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Two-Security Portfolio: Risk

= Variance of Security D

= Variance of Security E

= Covariance of returns for Security D and Security E



_D _E



E D

E E

D

w w w Cov r r

w

² ² ² ²

2 ,

2

p

    



2



E 2



D



^r_D ^r_E



Cov ,

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7-7

Covariance



_D,E

= Correlation coefficient of returns

Cov(r

_D,

r

_E

) = 

_DE



_D



_E



_D

= Standard deviation of returns for Security D



_E

= Standard deviation of

returns for Security E

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Example

 Determine the expected return and standard deviation of the

following portfolio consisting of two stocks that have a correlation coefficient of .75.

Portfolio Weight Expected

Return Standard Deviation

ABC .50 .14 .20

XYZ .50 .14 .20

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Calculation of return and risk

Expected Return = .5 (.14) + .5 (.14)= .14 or 14%

Standard deviation

= √ { (.5²x.2²)+(.5²x.2²)+(2x.5x.5x.75x.2x.2)}

= √ .035= .187 or 18.7%

Lower than the weighted average of 20%.

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Portfolio Risk : 2 – Security Cas

_p _{= [w}₁² ₁²_{+ w}₂² ₂² _{+ 2w}₁_w₂ ₁₂ ₁₂_]^½ Example : w₁ = 0.6 , w₂ = 0.4,

₁ _{= 10%,}₂ _{= 16%}

₁₂ _{= 0.5}

_p _{= [0.6}² _{x 10}² _{+ 0.4}² _{x 16}²+2 x 0.6 x 0.4 x 0.5 x 10 x 16]^½

= 10.7%

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Portfolio Proportion of A

w_A Proportion of B

w_B Expected return

E (R_p) Standard deviation

_p

1 (A) 1.00 0.00 12.00% 20.00%

2 0.90 0.10 12.80% 17.64%

3 0.759 0.241 13.93% 16.27%

4 0.50 0.50 16.00% 20.49%

5 0.25 0.75 18.00% 29.41%

6 (B) 0.00 1.00 20.00% 40.00%

Efficient Frontier For A Two Security-Case

Security A Security B

Expected return 12% 20%

Standard deviation 20% 40%

Coefficient of correlation -0.2

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Portfolio Options And The Efficient Frontier

••

12%

20%

20% 40%

Risk, _p Expected

return , E(R_p)

1 (A)

6 (B)

2 3

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Feasible Frontier Under Various Degrees Of Coefficient of Correlation

•

12%

20%

A (WA= 1)

Standard deviation,_p Expected

return , E (R_p)

B (WB= 1)

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The Efficient Set of Portfolios

• According to Markowitz’s approach, investors should evaluate portfolios based on their return and risk as measured by the standard deviation

Efficient portfolio – a portfolio that has the smallest

portfolio risk for a given level of expected return or the largest expected return for a given level of risk

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The Efficient Set of Portfolios

 The construction of efficient portfolios of financial assets requires identification of optimal risk-expected return combinations attainable from the set of risky assets

available

 Efficient portfolios can be identified by specifying an expected portfolio return and minimizing the portfolio risk at this level of return

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The Efficient Set of Portfolios

 Risk averse investors should only be interested in

portfolios with the lowest possible risk for any given level of return

 Efficient set (frontier) – is the segment of the minimum variance frontier above the global minimum variance portfolio that offers the best risk-expected return combinations available to investors

 Portfolios along the efficient frontier are equally “good”

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The Efficient Set of Portfolios

The Attainable Set and the Efficient Set of Portfolios

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7-19

 The amount of possible risk reduction through diversification depends on the correlation.

 The risk reduction potential increases as the correlation approaches -1.

 If  = +1.0, no risk reduction is possible.

 If  = 0, σ_P may be less than the standard deviation of either component asset.

 If  = -1.0, a riskless hedge is possible.

Correlation Effects

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Correlation effect

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Therefore, the standard deviation of the portfolio with perfect positive correlation is jus the weighted average of the component standard deviations. In all other cases, the correlation coefficient is less than 1, making the portfolio standard deviation less than the weighted average of the component standard deviations

The lowest possible value of the correlation coefficient is - 1, representing perfect negative correlation. In this case, equation is

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Correlation Coefficients

7-22

 When ρ_DE = 1, there is no diversification

 When ρ_DE = -1, a perfect hedge is possible

D D

E E

P

w  w 

  

D D E

D

E

w

w  

  1



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Correlation coefficient (-1)

For example, when E (rd)= 12% and E(re)= 20%

SD (d)= 20% SD (e)=40% and correlation coefficient is -1, Find out the weights that will drive down portfolio risk to zero.

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Matrix Used to Calculate the Variance of a Portfolio

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Number of Variance and Covariance Terms as a Function of the Number of Stocks in the Portfolio

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Role of covariance

 Putting together the variance and covariance parts of the general expression for the variance of a portfolio yields.

 The variance of the return on a portfolio with many securities is more dependent on the covariances

between the individual securities than on the variances of the individual securities

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Diversification

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Diversification

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Diversification

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Diversification

 expresses the variance of our special portfolio as a weighted sum of the average security variance and the average

covariance

 Now, let’s increase the number of securities in the portfolio without limit. The variance of the portfolio becomes:

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Diversification

 This example provides an interesting and important result. In our special portfolio, the variances of the individual securities completely vanish as the number of securities becomes large.

However, the covariance terms remain. In fact, the variance of the portfolio becomes the average covariance Cov .

 We often hear that we should diversify. In other words, we should not put all our eggs in one basket. The effect of

diversification on the risk of a portfolio can be illustrated in this example. The variances of the individual

securities are diversified away, but the covariance terms cannot be diversified away

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Diversification

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Total Risk

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Portfolio Risk and Diversification



_{p %}

35

20

0

Number of securities in portfolio

10 20 30 40 ... 100+

Total risk

Systematic Risk

Diversifiable

Risk

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Systemic Risk vs. Nonsystemic Risk

 Systemic Risk

The risk that comes from the individual exposure of assets to their individual risk factors. Can be diversified away.

 Nonsystemic Risk

The risk that comes from the common exposure of assets to economy-wide risk factors. Can’t be diversified away.

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Diversification:

The ‘Free Lunch’ of Finance

An investor can achieve higher returns for a given level of risk through diversification.

Three ways to diversify:

1. Diversification across sectors and industries.

2. Diversification across asset classes.

3. Diversification across regions and countries.