201-Lec06-Port1-2015

6.1

Lecture 6: FNCE20001 Business Finance

Modern Portfolio Theory IAugust 13, 2015

Vincent Grégoire, Ph.D.Senior Lecturer of Finance

Room 12.048, Faculty of Business and Economics9035-3756

[email protected]

6.2

6. Modern Portfolio Theory I

1. Explain the concept of risk diversification2. Compute the covariance and correlation of returns3. Compute and interpret the expected return and

standard deviation of a two security portfolio4. Examine the risk-return tradeoff in a two security

portfolio given different correlations

6.3

Required Readings: Lectures 6 - 9

v Lecture 6t PBEHP (12th or 11th ed.), Ch. 7 (sections 7.4 – 7.5.1)

v Lecture 7t PBEHP (12th or 11th ed.), Ch. 7 (sections 7.5.2 – 7.5.3)t Lamba, A. S., 2013, Teaching Note 3: The Effects of Portfolio

Leveraging and Short Selling on Expected Return and Risk (read afterPBEHP Ch. 7, sections 7.5.1 – 7.5.3)

v Lecture 8t PBEHP (12th or 11th ed.), Ch. 7 (sections 7.5.4 – 7.6.1)

v Lecture 9t PBEHP (12th or 11th ed.), Ch. 7 (sections 7.6.2 – 7.6.4)

6.4

6.1 Portfolios and Risk Diversification

v A risk averse investor’s objective is to…t Minimize the risk of portfolio of investments, given a desired

level of expected return, ort Maximize the expected return of portfolio of investments, given

a desired level of riskv The simplest (and naïve) way to minimize risk is to diversify

across different securities by forming a portfolio of securitiest Portfolio risk falls as the number of securities in the portfolio

increases, but…t Portfolio risk cannot be entirely eliminated using this method

v The risk that cannot be eliminated is called systematic riskt More on this in future lectures…

6.5

Portfolio Risk and Return: Two Securities

v A portfolio’s expected return is the weighted average of the expected returns of its component securities t Note that the weights are percentages of the investor’s original

wealth invested in each securityt It is assumed that all the available funds are invested in the two

securitiesv E(rp) = w1E(r1) + w2E(r2)

t wj = Amount invested in security j / Total amount investedt Note: w1 + w2 = 1 and w1 = 1 – w2 (or w2 = 1 – w1)

6.6


v A portfolio’s variance is the weighted average of the variance of its component securities and the covariance between the securities’ returns…

v Var(rp) = σp2 = w1

2σ12 + w2

2σ22 + 2w1w2σ12

t σ12 = Cov(r1, r2) = Covariance between securities 1 and 2t In the above expression, only the third term can be negative

v The standard deviation of the portfolio is…v SD(rp) = σp = [w1

2σ12 + w2

2σ22 + 2w1w2σ12]1/2

v We first need to define the covariance of returns

6.7

6.2 Covariance Between Security Returnsv The covariance of returns measures the level of comovement

between security returnst σ12 = p1[r11 – E(r1)][r21 – E(r2)] + ... + pn[r1n – E(r1)][r2n – E(r2)]t rjk = Return on security j = 1, 2 in state k = 1, 2, …, n

v σ12 > 0: Above (below) average returns on security 1 tend to coincide with above (below) average returns on security 2

v σ12 < 0: Above (below) average returns on security 1 tend to coincide with below (above) average returns on security 2

v Note that the magnitude of covariance will change depending on how returns are measuredt Percentages versus decimals…

6.8

Correlation Between Security Returnsv The correlation coefficient is a “standardized” measure of

comovement between two securitiesv ρ12 = σ12 /σ1σ2

t Note 1: -1 ≤ ρ12 ≤ +1t Note 2: The sign of the correlation coefficient is the same as the

sign of the return covariancev The covariance of returns can be rewritten as…v σ12 = σ1σ2 ρ12

6.9

Correlation Between Security Returnsv There are two definitions for the portfolio’s variance (and

standard deviation) of returnsv Using the covariance of returns…

t Var(rp) = σp2 = w1

2σ12 + w2

2σ22 + 2w1w2σ12

v Using the correlation of returns…t Var(rp) = σp

2 = w12σ1

2 + w22σ2

2 + 2w1w2 σ1σ2 ρ12

v Which expression should we use?

6.10

Examples of Security Correlations

rj

rk Strong Positive

• •••••••

•••

• ••

E(rk)

E(rj)

rj

rk ModerateNegative

• ••

•• • • •

•

••

••• •

•E(rk)

E(rj)

rj

rk Zero

••

• •

•••

•

•

•E(rk)

E(rj)

rj

rk ModeratePositive

• •• • •• • •

••• ••••

•• •

•

••

E(rk)

E(rj)

6.11

6.3 Portfolio Risk and Return: Two Securities

v Example: Stocks X and Y have the following distributions

State Probability Stock X Stock YPoor 0.25 -20.0% 30.0%Fair 0.50 15.0% 20.0%

Groovy 0.25 20.0% -10.0%

v Compute the covariance of returns for the two securitiesv Compute the correlation coefficient for the two securitiesv Compute the expected return and standard deviation of returns

for a portfolio with $10,000 invested in X and $15,000 in Yv What is (are) the source(s) of diversification benefits?

6.12


v Verify that…t E(rX) = 7.5%, E(rY) = 15.0%, σX = 16.0%, σY = 15.0%

v The covariance of returns is…t σXY = 0.25(-0.20 – 0.075)(0.30 – 0.15) +

0.50(0.15 – 0.075)(0.20 – 0.15) +0.25(0.20 – 0.075)(-0.10 – 0.15)

t σXY = -0.01625v The correlation coefficient of returns is…

t ρXY = -0.01625/(0.16)(0.15) = -0.677

6.13

v Proportion invested in security Xt wX = 10000/(10000 + 15000) = 0.40

v Proportion invested in security Yt wY = 15000/25000 = 1 – wX = 0.60

v Expected return of the portfolio…t E(rp) = 0.4(0.075) + 0.6(0.15)t E(rp) = 12.0%


6.14

v There are two methods to compute the standard deviation of the portfolio

v Using the covariance of returns…t σp = [0.42(0.16)2 + 0.62(0.15)2 + 2(0.4)(0.6)(-0.01625)]1/2

t σp = 6.63%v Using the correlation coefficient of returns…

t σp = [0.42(0.16)2 + 0.62(0.15)2 + 2(0.4)(0.6)(0.16)(0.15)(-0.677)]1/2

t σp = 6.63%

v Source(s) of diversification benefits?

Equals σXY

σX × σY × ρXY


6.15

6.4 Risk-Return Tradeoffs: Two Securitiesv Recall that a two-security portfolio’s expected return is…

t E(rp) = w1E(r1) + (1 – w1)E(r2)t w1 = proportion of funds invested in security 1t 1 – w1 = proportion of funds invested in security 2

v The portfolio’s return variance and standard deviation are…t σp

2 = w12σ1

2 + (1 – w1)2σ22 + 2w1(1 – w1)σ1σ2 ρ12

t σp = [w12σ1

2 + (1 – w1)2σ22 + 2w1(1 – w1)σ1σ2 ρ12]1/2

v We consider the following three casest Case 1: Return correlation of +1 (perfect positive correlation)t Case 2: Return correlation of -1 (perfect negative correlation)t Case 3: Return correlation between +1 and -1 (general case)

6.16

Risk-Return Tradeoffs: Two Securities

v Case 1: Return correlation, ρ12 = +1v There are no gains from diversification in this case as the

portfolio’s risk (standard deviation) is a weighted-average of the risks (standard deviations) of the two securities

v E(rp) = w1E(r1) + (1 – w1)E(r2) (1)v σp

2 = w12σ1

2 + (1 – w1)2σ22 + 2w1(1 – w1)σ1σ2(+1)

v σp2 = [w1σ1 + (1 – w1)σ2]2

v σp = {[w1σ1 + (1 – w1)σ2]2}1/2 (2)

6.17


Security Expected return Standard deviation1 14.0% 20.0%2 10.0% 12.0%

v Example: You’re given the following information on two securities. If the securities’ returns are perfectly positively correlated compute the expected return and standard deviation of returns of portfolios with the following weights in security 1: 0%, 25%, 50%, 75%, 100%

6.18


Weight in 1 Expected return Standard deviation0% 10.0% 12.0%

25% 11.0% 14.0%50% 12.0% 16.0%75% 13.0% 18.0%100% 14.0% 20.0%

v The expected return and standard deviation of returns are…v E(rp) = w1(0.14) + (1 – w1)(0.10)v σp = {[w1(0.20) + (1 – w1)(0.12)]2}1/2

6.19

Risk-Return Tradeoffs: Two Securitiesv Case 2: Return correlation, ρ12 = -1v There are maximum gains from diversification in this casev It is also always possible to construct a zero risk portfolio as

well! v E(rp) = w1E(r1) + (1 – w1)E(r2) (1)v σp

2 = w12σ1

2 + (1 – w1)2σ22 + 2w1(1 – w1)σ1σ2(-1)

v σp2 = w1

2σ12 + (1 – w1)2σ2

2 – 2w1(1 – w1)σ1σ2

v σp2 = [w1σ1 – (1 – w1)σ2]2

v σp = {[w1σ1 – (1 – w1)σ2]2}1/2 (2)

6.20



v Example (continued): Consider the previous example but assume that the two securities returns are now perfectly negatively correlated. Compute the expected return and standard deviation of returns of portfolios with the following weights in security 1: 0%, 25%, 50%, 75%, 100%. Compute the weights in securities 1 and 2 and the expected return of the minimum variance (or standard deviation) portfolio

6.21



25% 11.0% 4.0%50% 12.0% 4.0%75% 13.0% 12.0%100% 14.0% 20.0%

v The expected return and standard deviation of returns are…v E(rp) = w1(0.14) + (1 – w1)(0.10)v σp = {[w1(0.20) – (1 – w1)(0.12)]2}1/2

6.22

Risk-Return Tradeoffs: Two Securitiesv The minimum variance (or standard deviation) portfolio is

where σp = 0 v Set σp = 0 and solve for w1 in…

t σp = {[w1σ1 – (1 – w1)σ2]2}1/2

* * 12 1

1 2

1w w σσ σ

= − =+

* 21

1 2

w σσ σ

=+

v In our example…t w1

* = 0.12/(0.20 + 0.12) = 37.5%t w2

* = 1 – w1* = 62.5%

t Ε(rp) = 0.375(0.14) + 0.625(0.10) = 11.5%

6.23

Risk-Return Tradeoffs: Two Securitiesv Case 3: Return correlation, -1 < ρ12 < +1v Some diversification benefits always existv σp

2 = w12σ1

2 + (1 – w1)2σ22 + 2w1(1 – w1)σ1σ2 ρ12

v σp is minimized when w1 is…

2* 21 2 2

1 2

w σσ σ

=+

v In the special case when ρ12 = 0, w1* simplifies to…

2* 12 2 2

1 2

w σσ σ

=+

2* 2 12 1 21 2 2

1 2 12 1 22w σ ρ σ σ

σ σ ρ σ σ−

=+ −

and

6.24



v Example (continued): Consider the previous example but now assume that the two securities returns are not correlated with each other. Compute the expected return and standard deviation of returns of portfolios with the following weights in security 1: 0%, 25%, 50%, 75%, 100%. Compute the weights in securities 1 and 2 and the expected return and standard deviation of the minimum variance (or standard deviation) portfolio

6.25



25% 11.0% 10.3%50% 12.0% 11.7%75% 13.0% 15.3%100% 14.0% 20.0%

v The expected returns and standard deviations are…v E(rp) = w1(0.14) + (1 – w1)(0.10)v σp = [w1

2(0.20)2 + (1 – w1)2(0.12)2 + 0]1/2

6.26

Risk-Return Tradeoffs: Two Securitiesv The weight in security 1 for the minimum variance portfolio

with zero correlation is…2

* 21 2 2

1 2

w σσ σ

=+

v In our example…2

*1 2 2

0.12 26.5%0.12 0.20

w = =+

w2* = 1 – w1

* = 73.5%v E(rp) = 0.265(0.14) + 0.735(0.10) = 11.06%v σp = [(0.265)2(0.20)2 + (0.735)2(0.12)2 + 0]1/2 = 10.29%

6.27

Summary of Risk-Return Tradeoffs

8

9

10

11

12

13

14

15

0 5 10 15 20 25

Expe

cted

retu

rn (%

)

Standard deviation of return (%)

Security 1

Security 2

ρ12 = -1.0

Zero risk portfolio

ρ12= -0.5

ρ12 = +0.5ρ12= +1.0

ρ12= 0.0

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Note: The red dots indicate the data points that were computed in the previous slides

6.28

Key Concepts

v Risk minimization can be achieved by diversifying across different securities or assets

v A portfolio’s expected return is the weighted average of the returns of its component securities

v A portfolio’s variance is the weighted average of the variances of its component securities and the covariance between pairs of securities

v Diversification benefits exist as long as the correlation between two securities is less than perfect

v When two securities are perfectly negatively correlated it is always possible to form a zero risk portfolio

6.29

What’s Next?

v In lecture 7 we will examine the effects on risk and return of portfolio leveraging and short selling

v In lecture 8 we will examine the risk-return tradeoff in portfolios with more than two securities and, in that context, examine the limitations to the benefits of risk diversification

6.30

Formula Sheetv Expected portfolio return: E(rp) = w1 E(r1) + w2 E(r2)v Portfolio return variance

t σp2 = (w1σ1)2 + (w2σ2)2 + 2w1w2σ12

t σp2 = (w1σ1)2 + (w2σ2)2 + 2w1w2σ1σ2 ρ12

v Covariance of returnst σ12 = p1[r11 – E(r1)][r21 – E(r2)] + ... + pn[r1n – E(r1)][r2n – E(r2)]

v Covariance of returns: σ12 = σ1σ2 ρ12

v Correlation coefficient: ρ12 = σ12 /σ1σ2

v Minimum variance portfolio (ρ = -1): w1* = σ2/(σ1 + σ2)

v Minimum variance portfolio:

(Note: The formula sheets on the mid semester and final exams will contain all the formulas covered in lectures but without the descriptions)

2* 2 12 1 21 2 2

1 2 12 1 22w σ ρ σ σ

σ σ ρ σ σ−

=+ −

201-Lec06-Port1-2015

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