12.3 Efficient Diversification with Many Assets We have considered –Investments with a single risky, and a single riskless, security –Investments where.

12.3 Efficient Diversification with Many Assets

• We have considered – Investments with a single risky, and a single

riskless, security– Investments where each security shares the

same underlying return statistics

• We will now investigate investments with more than one (heterogeneous) stock

Portfolio of Two Risky Assets

• Recall from statistics, that two random variables, such as two security returns, may be combined to form a new random variable

• A reasonable assumption for returns on different securities is the linear model:

1 with ; 212211 wwrwrwrp

Equations for Two Shares

• The sum of the weights w1 and w2 being 1 is not necessary for the validity of the following equations, for portfolios it happens to be true

• The expected return on the portfolio is the sum of its weighted expectations

2211 wwp

Equations for Two Shares

• Ideally, we would like to have a similar result for risk

– Later we discover a measure of risk with this property, but for standard deviation:

(wrong) 2211 wwp

22

222,12121

21

21

2 2 wwwwp

Mnemonic

• There is a mnemonic that will help you remember the volatility equations for two or more securities

• To obtain the formula, move through each cell in the table, multiplying it by the row heading by the column heading, and summing

Variance with 2 SecuritiesW1*Sig1 W2*Sig2

W1*Sig1 1 Rho(1,2)

W2*Sig2 Rho(2,1) 1

2,1212122

22

21

21

2 2 wwwwp

Variance with 3 SecuritiesW1*Sig1 W2*Sig2 W3*Sig3

W1*Sig1 1 Rho(1,2) Rho(1,3)

W2*Sig2 Rho(2,1) 1 Rho(2,3)

W3*Sig3 Rho(3,1) Rho(3,2) 1

3,232323,13131

2,1212123

23

22

22

21

21

2

22

2

wwww

wwwwwp

Note:

• The correlation of a with b is equal to the correlation of b with a

• For every element in the upper triangle, there is an element in the lower triangle– so compute each upper triangle element once,

and just double it

• This generalizes in the expected manner

Correlated Common Stock

• The next slide shows statistics of two common stock with these statistics:

– mean return 1 = 0.15

– mean return 2 = 0.10

– standard deviation 1 = 0.20

– standard deviation 2 = 0.25

– correlation of returns = 0.90

– initial price 1 = $57.25

– initial price 2 = $72.625

2-Shares: Is One "Better?"

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.05 0.1 0.15 0.2 0.25 0.3

Standard Deviation

Exp

ecte

d R

etu

rn

Observation

• The statistics indicate that one security appears to totally dominate the other– Security 1 has a lower risk and higher return

than security 2– In an efficient market:

• Wouldn’t everybody short 2, and buy 1?

• Wouldn’t supply and demand then cause the relative expected returns to “flip”?

Does it Happen?

• The purpose of selecting two shares with this paradoxical form is to illustrate an important point later

• This kind of relationship does occur in the real world

A Pair of Price Trajectories

• The next graph shows a trajectory of two share prices with the statistics we have selected

Share Prices

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6 7 8 9 10

Years

Val

ue

(ad

just

ed f

or

Sp

lits

)

ShareP_1

ShareP_2

Observation

• If you were to “cut a piece” from one trajectory, re-scale it for relative price differences, and slide it over the other, you would observe that both trajectories behave in a broadly similar manner, but each has independent behavior as well

• Quick confirmation is seen in the region 1 to 4 years where prices are close

Correlation

• The two shares are highly correlated– They track each other closely, but even

adjusting for the different average returns, they have some individual behavior

Portfolio of Two Securities

0.00

0.05

0.10

0.15

0.20

0.25

0.15 0.17 0.19 0.21 0.23 0.25 0.27 0.29

Standard Deviation

Exp

ecte

d R

etu

rn

Share 1

Share 2

Efficient

Sub-optima

l

MinimumVariance

Observation

• Shorting the high-risk, low-return stock, and re-investing in the low-risk, high-return stock, creates efficient portfolios– Shorting high-risk by 80% of the net wealth

crates a portfolio with a volatility of 20% and a return of 19% (c.f. 15% on security 1)

– Shorting by 180% gives a volatility of 25%, and a return of 24% (c.f. 10% on security 2)

Observation

• In order to generate a portfolio that generates the same risk, but with a higher return– Compute the weights of the minimum portfolio,

W1 (min-var), W2 (min-var)

• (Formulae given later)

– Use the relationship • Wi (sub-opt) +Wi (opt) = 2 * Wi (min-var)

Observation

– Another way to generate the two securities is to form two portfolios consisting of a risky and a riskless security that each meet the efficient frontier

– Result: two portfolios that are long the risky security, and short the riskless security

– Short one of the portfolios and invest in the other to generate one of the desired efficient portfolios

– Repeat to generate the other

• Prove that the riskless security becomes irrelevant

Optimal Combination of Risky Assets

• The following slides are samples of the computations used to generate the graphs

Fragments of the Output TableData For two securities

This data has been constructed to produce the mean-varience paradox

mu_1 15.00%mu_2 10.00%sig_1 20.00%sig_2 25.00%rho 90.00%

w_1 w_2 Port_Sig Port_Mu-2.50 3.50 0.4776 -0.0250-2.40 3.40 0.4674 -0.0200-2.30 3.30 0.4573 -0.0150-2.20 3.20 0.4472 -0.0100-2.10 3.10 0.4372 -0.0050-2.00 3.00 0.4272 0.0000-1.90 2.90 0.4173 0.0050-1.80 2.80 0.4074 0.0100-1.70 2.70 0.3976 0.0150

1.30 -0.30 0.1953 0.16501.40 -0.40 0.1949 0.17001.50 -0.50 0.1953 0.17501.60 -0.60 0.1962 0.18001.70 -0.70 0.1978 0.18501.80 -0.80 0.2000 0.19001.90 -0.90 0.2028 0.19502.00 -1.00 0.2062 0.20002.10 -1.10 0.2101 0.20502.20 -1.20 0.2145 0.21002.30 -1.30 0.2194 0.21502.40 -1.40 0.2247 0.22002.50 -1.50 0.2305 0.2250

-0.30 1.30 0.2723 0.0850-0.20 1.20 0.2646 0.0900-0.10 1.10 0.2571 0.09500.00 1.00 0.2500 0.10000.10 0.90 0.2432 0.10500.20 0.80 0.2366 0.11000.30 0.70 0.2305 0.1150

Sample of the Excel Formulae

w_1 w_2 Port_Sig Port_Mu-2.5 =1-A14 =SQRT(w_1^2*sig_1^2 + 2*w_1*w_2*sig_1*sig_2*rho + w_2^2*sig_2^2) =w_1*mu_1 + w_2*mu_2=A14+0.1 =1-A15 =SQRT(w_1^2*sig_1^2 + 2*w_1*w_2*sig_1*sig_2*rho + w_2^2*sig_2^2) =w_1*mu_1 + w_2*mu_2=A15+0.1 =1-A16 =SQRT(w_1^2*sig_1^2 + 2*w_1*w_2*sig_1*sig_2*rho + w_2^2*sig_2^2) =w_1*mu_1 + w_2*mu_2=A16+0.1 =1-A17 =SQRT(w_1^2*sig_1^2 + 2*w_1*w_2*sig_1*sig_2*rho + w_2^2*sig_2^2) =w_1*mu_1 + w_2*mu_2

=SQRT(w_1^2*sig_1^2 + 2*w_1*w_2*sig_1*sig_2*rho + w_2^2*sig_2^2)

=w_1*mu_1 + w_2*mu_2

Formulae for Minimum Variance Portfolio

*1

22212,1

21

212,121*

2

22212,1

21

212,122*

1

1

2

2

w

w

w

Formulae for Tangent Portfolio

32tan

2

32tan

1

22

2tan1

1tan2

221212,121

212

212,12221tan

1

1

2

25.0*10.025.0*20.0*90.0*05.010.020.0*05.0

25.0*20.0*90.0*05.025.0*10.0

1

w

w

w

ww

rrrr

rrw

ffff

ff

Example: What’s the Best Return given a 10% SD?

1261.005.010.02409.0

05.02333.0

2409.0

90.0*25.0*2.0*3

5

3

8225.0

3

520.0

3

8

2

2333.0

10.03

515.0

3

8

tan

tan

tan

22

22

2tan

2,121tan2

tan1

22

2tan2

21

2tan1

2tan

tan

tan

2tan21

tan1tan

ff rr

wwww

ww

Selecting the Preferred Portfolio

• The procedure is as follows– Find the portfolio weights of the tangent

portfolio of the line (CML) through (0, rf)– Determine the standard deviation and

expectation of this point– Construct the equation of the CML– Apply investment criterion

Achieving the Target Expected Return (2): Weights

• Assume that the investment criterion is to generate a 30% return

• This is the weight of the risky portfolio on the CML

3636.105.02333.0

05.030.0

1

1

11

ftangent

fcriterion

ftangentcriterion

r

rw

wrw

Achieving the Target Expected Return (2):Volatility

• Now determine the volatility associated with this portfolio

• This is the volatility of the portfolio we seek

3285.02409.0*3636.11 tangentw

Achieving the Target Expected Return (2): Portfolio Weights

COMPUTATION WEIGHT

RISKLESS -0.3636 -0.3636

ASSET 1 1.3636*2.6667 3.6363

ASSET 2 1.3636*(-1.6667) -2.2727

TOTAL 1.0000

Investment Strategies

• We have examined two strategies in detail when– the volatility is specified– the return is specified

• Additionally, one of the graphs indicated an approach to take when presented with investor’s risk/return preferences

Portfolio of Many Risky Assets

• In order to solve problems with more than two securities requires tools such as quadratic programming

• The “Solve” function in Excel may be used to obtain solutions, but it is generally better to use a software package such as the one that came with this book

Chapter Assumptions

• The theory underlying this chapter is essentially just probability theory, but there are financial assumptions

– We do not have to assume that the generating process of returns is normal, but we do assume that the process has a mean and a variance. This is may not be the case in real life

– We assumed that the process was generated without inter-temporal correlations. Some investors believe that there is valuable information in old price data that has not been incorporated into the current price--this runs counter to many rigorous empirical studies.

– There are no “hidden variables” that explain some of the noise

– Investors make decisions based on mean-variances alone

• statistics such as skewness & kurtosis have been ignored

• We have made the assumption the we can lend at the risk-free rate, and that we can “short” common stock aggressively

Summary

• There is no single investment strategy that is suitable for all investors; nor for a single investor for his whole life

• Time makes risky investments more attractive than safer investments

• In practice, diversification has somewhat limited power to reduce risk