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FINC4101 Investment Analysis

Instructor: Dr. Leng Ling

Topic: Portfolio Theory II

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Learning objectives1. Explain the benefit of diversification and identify the

source of this benefit.

2. Understand the concepts of covariance and correlation.

3. Compute covariance and correlation between two risky assets.

4. Compute the expected return and standard deviation of a portfolio of two risky assets.

5. Define the minimum variance portfolio and compute its weights in the 2-risky asset case.

6. Define the investment opportunity set and the efficient frontier.

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Learning objectives7. Compute the expected return and standard deviation of

a portfolio consisting of a risky asset/portfolio and a risk-free asset.

8. Define the Capital Allocation Line.

9. Define and compute the reward-to-variability (Sharpe) ratio.

10. Define the tangency portfolio and compute its weights in a 2-risky asset case.

11. Describe the process of efficient diversification with many risky asset and a risk-free asset.

12. Understand the idea of the separation property.

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Portfolio Theory II: Concept Map

Portfolio Theory

II

Diversification

Portfolio

Covariance,Correlation

Efficient frontier

Separation Property

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Why diversify?

To reduce risk.

This is the basic benefit of diversification.

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How does diversification work? By exploiting the way different assets relate to

each other. We use a statistical measure called covariance

to characterize the tendency of two assets to “move” with each other. Covariance: A measure of the extent to which the

returns of two assets tend to vary with each other. An equivalent and easier to interpret measure is

the correlation coefficient.

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Simple example Consider two assets: a stock and a bond. There

are 3 possible scenarios over the coming year:

HPR (%)

Scenario Probability Stock Bond

Recession 0.3 -11 16

Normal 0.4 13 6

Boom 0.3 27 -4

Expected return 10 6

S.D. 14.92 7.75

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Portfolio of 60% in stock & 40% in bond What is the expected return and S.D. of this

portfolio?

Scenario Prob.

HPR%

Prob. X HPR

Deviation from E( r )

Squared deviation

Prob. X squared deviation

Recession

0.3 -0.2 -0.06 -8.60 73.96 22.188

Normal 0.4 10.2 4.08 1.80 3.24 1.296

Boom 0.3 14.6 4.38 6.20 38.44 11.532

E( r )

8.40 Variance 35.016

S.D. 5.92

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Diversification lowers risk! Compare the standard deviations of the stock, bond and

portfolio:

The portfolio is actually less risky than either of the two assets. Diversification lowers investment risk.

Why? Stock and bond returns tend to move in opposition to each other.

How do we measure this tendency? -- Covariance

Stock Bond Portfolio

14.92% 7.75% 5.92%

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Covariance Deviation from

mean returnCovariance

Scenario Prob.

Stock Bond Product of Dev

Prob. x Product of Dev

Recession

0.3 -21 10 -210 -63

Normal 0.4 3 0 0 0

Boom 0.3 17 -10 -170 -51

Covariance -114

HPR (%)

Scenario Probability Stock Bond

Recession 0.3 -11 16

Normal 0.4 13 6

Boom 0.3 27 -4

Expected return 10 6

S.D. 14.92 7.75

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Covariance formula Suppose there are S possible scenarios

(i = 1,2,…,S), and the probability of scenario i is p(i), then covariance between assets A and B is

Asset A’s return in scenario i

Asset B’s return in scenario i

1

( , ) ( ) ( ) ( )S

S B S S B Bi

Cov r r p i r i r r i r

S

iBBAABA rirririprrCov

1

])(][)()[(),(

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Apply covariance formula Go back to the example, but now suppose the returns on

the stock are -14%, 13% and +30%. Compute the stock’s expected return, variance, and covariance with bond.

HPR (%)

Scenario Probability Stock Bond

Recession 0.3 -14 16

Normal 0.4 13 6

Boom 0.3 30 -4

Expected return ? 6

S.D. ? 7.75

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Correlation between assets A & B, rAB , is

Given this formula, we can write covariance as

Going back to our example, the correlation between stock and bond = -114/(14.92 x 7.75) = -0.99

BA

BAAB

rrCov

)( ,

BAABBA rrCov )( ,

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Interpreting the correlation coefficient

Correlations range from -1 to 1.

Correlation Meaning

-1, Perfect negative correlation Strongest tendency for two assets to vary inversely/ move in opposite direction.

+1, Perfect positive correlation Strongest tendency for two assets to move in tandem.

0, Uncorrelated Returns on the two assets are unrelated to each other.

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Three rules of two-risky asset portfolio

Suppose you have a portfolio consisting of two risky assets, A and B. You invest the proportion, wA, of the portfolio in A and the proportion, wB , in B.

You know the following: The return on asset A is rA ; the return on asset B is rB. The expected return on asset A is E(rA) ; the expected

return on asset B is E(rB). The standard deviation of A is sA; the standard deviation

of B is sB. The correlation coefficient between the returns of A and

B is rAB.

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Rule 1: Portfolio rate of returnA portfolio’s rate of return (rP) is a weighted

average of the returns on the component assets, with the investment proportions as weights,

rP = wA rA + wB rB

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Rule 2: Portfolio expected returnA portfolio’s expected return is a weighted

average of the expected returns on the component assets with the same portfolio proportions as weights,

E(rP) = wA E(rA) + wB E(rB)

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Rule 3: Portfolio varianceA portfolio’s variance (sP

2) is

sP2 = (wAsA)2 + (wBsB)2 + 2(wAsA) (wBsB)rAB

Since cov(rA,rB) = sAsBrAB , we can also write portfolio variance as

sP2 = (wAsA)2 + (wBsB)2 + 2wAwBcov(rA,rB)

Taking the square root of sP2 gives us the

portfolio standard deviation.

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Applying these rules Suppose you have two assets (A,B) with the

following details:

E(rA)=6%, E(rB)=10%, sA=12%, sB=25%, rAB=0

wA = 0.5

Verify that: portfolio expected return is 8% portfolio SD is 13.87%

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Discussion By investing in two assets, the portfolio

volatility is smaller than the average of volatilities of the two assets.

Why not use SD=(12+25)/2=18.5%?

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Minimum variance portfolio Weight on asset A is:

Weight on asset B is = 1 – wA. Using this formula, verify that the min variance

portfolio has wA=81.27%, wB=18.73%.

BAABBA

BAABBAW

222

2

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Investment opportunity set The set of all attainable combinations of risk and

return offered by portfolios formed using the available assets in differing proportions.

B

A

Z

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Mean variance criterion & the efficient frontier

Mean-variance criterion: choosing assets/portfolios based on the expected return and variance (or SD) of portfolios.

Using this criterion, we prefer portfolio A to portfolio B (or A dominates B) if:

E(rA) E(rB) and sA sB Efficient frontier: portfolios on the upward sloping

portion of the investment opportunity set.

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Practice 2Chapter 5: CFA problems: 4,7,8,9,11.Chapter 6: CFA problems: 1, 3(a) only.

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Homework 21. If required risk premium= , where A=4, and risk free rate = 1%, what are the annual required rates of return for the following investments? What will be the value of these investments after 2 years, if these investments have achieved the required return rates?

2. You are considering investing in 2 assets. A asset has an expected return of 15% and standard deviation of 32%. B asset has an expected return of 9% and standard deviation of 23%. The correlation between A and B is 0.15.

a. What is the covariance between A and B?

b. What is the weight of A and B in the minimum variance portfolio?

c. What is the expected return and variance of the minimum variance portfolio?

2

2

1 A

Investment standard deviation value today

1 0.4 $1,000

2 0.5 $1,000

3 0.6 $1,000

4 0.2 $1,000

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Extreme casesSuppose rAB= 1,

sP2 = (wAsA + wBsB)2

sP = wAsA + wBsB

No gains from diversification only in this case! Whenever r < 1, there are gains from diversification.

Suppose rAB= -1,

sP2 = (wAsA - wBsB)2

sP = ABS[ wAsA – wBsB ]

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Investment opportunity set with different correlations

A

B

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Problem Jane Marple has an $800,000 fully diversified portfolio. She

subsequently inherits Rafael Aerospace Inc. stock worth $200,000. Her financial advisor provided her with the following information for the coming year:

The correlation between the original portfolio and Rafael Aerospace is 0.3.

Jane will keep the new stock. (a) Calculate the expected return and standard deviation of the new portfolio. (b) Calculate the covariance between the original portfolio and Rafael Aerospace.

Expected annual HPR (%)

Std Dev of annual return (%)

Original portfolio 6.7 9.3

Rafael Aerospace 8.2 15.6

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Follow-up problems1. Jane decides to keep Rafael Aerospace but

wants to rebalance the new portfolio so that risk is reduced to the minimum. What are the weights, the expected return and std dev of the minimum variance portfolio?

2. Continuing with the previous problem, if Jane keeps the original composition, what is the portfolio expected return and SD if the correlation is (a) 1, (b) -1.

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Asset allocation involving risky assets and a risk-free asset.

What’s a risk-free asset? An asset that provides a sure (for certain)

nominal rate of return. It is a default free asset. Examples: U.S. Treasury securities (Treasury bills, Treasury

notes, Treasury bonds), Portfolios invested in these securities, e.g.,

T-bill money market fund. In practice, money market instruments like bank

certificate of deposits (CDs) and commercial paper are treated as effectively risk-free.

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Properties of risk-free assetThere is no uncertainty in return.

Variance of risk-free asset is 0 Likewise, standard deviation is 0

The risk-free asset does not vary with a risky asset/ portfolio. So, Cov(risk-free asset, risky asset) = 0 Corr(risk-free asset, risky asset) = 0

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3 rules when there is one risk-free asset & one risky asset (1)

P: risky asset/ portfolio (wP)

+

Risk-free asset (wf)

What is the correlation between P and the risk-free asset?

Complete portfolio, C

P Risk-free

Return rP rf

Expected return E(rP) rf

S.D. sP 0

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3 rules when there is one risk-free asset & one risky asset (2)

1) Portfolio return, rC = wP rP + wf rf

2) Portfolio expected return, E(rC) = wP E(rP) + wf rf

Portfolio risk premium, E(rC) - rf = wP [ E(rP) – rf ]

3) Portfolio variance, sC2 = (wPsP)2

Portfolio standard deviation, sC = wPsP

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Capital Allocation Line (CAL) If we plot the expected return and standard deviation

combinations of a complete portfolio, we get the Capital Allocation Line (CAL).

CAL: The plot of risk-return combinations available by varying portfolio allocation between a risk-free asset and a risky asset/portfolio.

CAL is just the investment opportunity set when we have a risk-free asset and a risky asset/portfolio!

Capital Market Line (CML): the CAL when the market index portfolio is used as the risky portfolio.

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Capital Allocation Line (CAL)

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Apply these rules Suppose the risky portfolio P has an expected

return of 15% and a standard deviation of 22%. The risk-free asset promises a return of 7%.

Compute the expected return, risk premium and standard deviation of the complete portfolio if 100% of the portfolio is invested in P 100% of the portfolio is invested in the risk-free asset Equal weight is placed on P and the risk-free asset.

Draw the CAL.

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Results

Expected return Risk premium Portfolio SD

100% in P (1 x 15) + (0 x 7) = 15 1 x [15 – 7] = 8 1 x 22 = 22

100% in risk-free (0 x 15) + (1 x 7) = 7 0 x [15 – 7] = 0 0 x 22 = 0

Equal weight (0.5 x 15) + (0.5 x 7) = 11 0.5 x [15 – 7] = 4 0.5 x 22 = 11

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Capital Allocation Line (CAL)

Slope of CAL = increase in expected return per unit of additional standard deviation (SD), i.e., the extra return per extra risk.

The slope is also called the Sharpe ratio, reward-to-variability ratio.

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Sharpe ratio, S Sharpe ratio is the slope, so use “rise” over “run”.

For our example, Sharpe ratio= (15 – 7)/(22 – 0)

= 8/22

= 0.36

In general, Sharpe ratio of portfolio

S = portfolio risk premium / portfolio std dev Sharpe ratio is the same for all portfolios that plot on

the CAL.

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Problems involving risk-free and risky assets

Assume you manage a risky portfolio with an expected rate of return of 17% and a standard deviation of 27%. The T-bill rate is 7%.

a) Your client invests 70% of his portfolio in your fund and 30% in a T-bill money market fund. What is the expected return and standard deviation of your client’s portfolio?

b) Suppose your risky portfolio includes the following investments in the given proportions:

What are the investment proportions of your client’s overall portfolio, including the position in the T-bills?

c) What is the Sharpe ratio of your risky portfolio and your client’s overall portfolio?

d) Draw the CAL and point the positions of your and your client’s portfolios.

Stock A 27%

Stock B 33%

Stock C 40%

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Optimal risky portfolio with a risk-free asset

When we have risky assets and a risk-free asset, we can identify one single risky portfolio that gives us the highest possible Sharpe (reward-to-variability) ratio.

This risky portfolio is the ‘optimal’ or ‘tangency’ portfolio.

The CAL formed using the optimal risky portfolio has the steepest slope.

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Go back to the 2 risky assets: A and B but assume correlation is 0.2. Now add a risk-free asset with 5% return

A

B

CAL1

CAL2

Recall: E(rA)=6%, E(rB)=10%, sA=12%, sB=25%, rAB=0.2.

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Optimal risky portfolio weights

Using the formulas, the weights in the optimal portfolio (O):

wA = 32.99%, wB = 67.01%

Expected return, SD, Sharpe ratio:

E(rO) = 8.68%

sO = 17.97%

SharpeO = (8.68 – 5)/17.97 = 0.20

AB

ABBAfBfAAfBBfA

ABBAfBBfAA

WW

rrErrErrErrE

rrErrEW

1

])()([])([])([

])([])([22

2

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Risk aversion and portfolio choicePreferred complete portfolio: 55% in Portfolio O, 45% in risk-free asset.

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Consider the following A pension fund manager is considering 3 mutual funds.

The first is a stock fund, the second is a corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.5%. The expected return and sigma of the risky funds are:

The correlation between the risky fund returns is 0.25.

Expected return Std Dev

Stock fund (S)

14% 30%

Bond fund (B) 8 20

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Answer the following:1)Compute the expected return and standard deviation of

the minimum variance portfolio.

2)Compute the expected return and standard deviation of the tangency portfolio.

3)What is the Sharpe ratio of the tangency portfolio?

4)Suppose you want to form a complete portfolio on the CAL. The portfolio must yield an expected return of 12%.

What is the standard deviation of the portfolio? What is the proportion invested in the T-bill fund and each

of the two risky funds?

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Efficient diversification with many risky assets & a risk-free asset (1)

Efficient diversification entails 3 separate steps:

1) Form efficient frontier of risky assets Efficient frontier: the collection of portfolios

that maximizes expected return at each level of portfolio risk/ standard deviation. Northwestern-most portfolios Inputs: expected return and SD of every risky

asset, plus correlation coefficients between each pair of assets.

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Efficient frontier of risky assets

Inefficiently diversified.

Portfolios are discarded. Dominated by portfolios on efficient frontier.

N

W E

S

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Efficient diversification with many risky assets & a risk-free asset (2)

2. Use the risk-free rate and efficient frontier to find the tangency portfolio or optimal risky portfolio

Recall that the tangency portfolio gives us the CAL with the highest Sharpe ratio

3. Investor chooses the preferred complete portfolio based on his/her risk aversion.

Each investors will use tangency portfolio in forming his/her complete portfolio

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Separation PropertyThe act of choosing the appropriate portfolio can

be separated into two independent tasks. 1. Find the optimal risky portfolio.

This consists of steps 1 and 2 and is a purely technical problem.

Given the input data, the best risky portfolio is the same for all clients regardless of risk aversion.

2. Investor constructs complete portfolio using risk-free asset and optimal risky portfolio. Depends on the investor’s risk/personal preference.

Here the client is the decision maker.

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Summary Efficient diversification reduces risk. Benefit of diversification depends on how assets

co-vary with each other. Efficient frontier is the collection of portfolios

offering the highest expected return for each level of risk.

Introducing a risk-free asset allows us to identify the tangency portfolio and the CAL.

Tangency portfolio has the highest Sharpe ratio and therefore is most desirable.

Implications of the separation property.

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Practice 3Chapter 5: 13,14,18,19.Chapter 6: 8,9,10,11,12,19.

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Homework 31. Assume you manage a risky portfolio with an expected rate of return of

20% and a standard deviation of 30%. The T-bill rate is 5%. Your client invests 60% of his portfolio in your fund and 40% in a T-bill money market fund. What is the expected return and standard deviation of your client’s portfolio? What is the Sharpe ratio of your client’s portfolio?

2. A asset has an expected return of 20% and standard deviation of 30%. B asset has an expected return of 10% and standard deviation of 23%. C asset is risk-free with a rate of 5%. The correlation between A and B is 0.15. a) What is the expected return and standard deviation of the optimal risky portfolio? b) Suppose your complete portfolio must yield an expected return of 15% and be efficient. What is the standard deviation of your portfolio? c) What is the proportion of your portfolio invested in A and B, respectively?