and

2

Var(˜P ) = wE1 ·(18.969%)2 + wE2 2 ·(67.555%)2 + 2·wE1 ·wE2 ·3.505%

r .

(16.16)

2 2

Var(˜P ) = wE1 ·Var(˜E1 ) + wE2 ·Var(˜E2 ) + 2·wE1 ·wE2 ·Cov(˜E1 , rE2 ) .

r r r r˜

With a risk-free rate of 6%, the Sharpe Ratio for P is therefore de¬ned to be

E (˜P ) ’ rF

r 25% ’ 15% · wE1 ’ 6%

SR(wE1 ) = = . (16.17)

Sdv (˜P )

r 2

wE1 ·(18.969%)2 + (1 ’ wE1 )2 ·(67.555%)2 + 2·wE1 ·(1 ’ wE1 )·3.505%

You can now use Excel (or calculus) to determine the wE1 that maximizes this fraction. The

solution is wE1 ≈ 68%, which means that the net portfolio weights in this tangency portfolio

are

wS&P500 = 0.68 · (+99.448%) + 0.32 · (’90.620%) = 38.791% ,

(16.18)

= 0.68 · (+2.124%) + 0.32 · (135.770%) = 44.751% ,

wIBM

= 0.68 · (’1.573%) + 0.32 · (54.925%) = 16.458% .

wSony

The mean rate of return of this tangency portfolio is 14.8%, the standard deviation is 28%, and

the Sharpe ratio is 0.3138. So, you should henceforth only purchase a combination of the risk-

free security and this tangency portfolio, with relative weights determined by your taste for

risk.

There are two methods to extract the covariance: one is brute force to manipulate the

Digging Deeper:

component security covariances. The more clever approach recognizes that you already know

(18.969%)2 = 12 ·Var(˜E1 ) 02 ·Var(˜E2 )

+ +

r r 2·1·0·Cov(˜E1 , rE2 )

r˜

(67.555%)2 = 02 ·Var(˜E1 ) 12 ·Var(˜E2 )

+ +

r r 2·0·1·Cov(˜E1 , rE2 )

r˜

(16.19)

2 2 2

= 0.5 ·Var(˜E1 ) + 0.5 ·Var(˜E2 ) +

(37.498%) r r 2·0.5·0.5·Cov(˜E1 , rE2 )

r˜

2

= wE1 ·Var(˜E1 ) + wE2 2 ·Var(˜E2 ) + 2·wE1 ·wE2 ·Cov(˜E1 , rE2 ) ,

Var(˜P )

r r r r˜

which simpli¬es into

(18.969%)2 = Var(˜E1 )

r

(67.555%)2 = Var(˜E2 )

r

(16.20)

2

= + + 2·0.5·0.5·Cov(˜E1 , rE2 )

(37.498%) 0.25·Var(˜E1 )

r 0.25·Var(˜E2 )

r r˜

= 0.25·(18.969%)2 + 0.25·(67.555%)2 + 0.5·Cov(˜E1 , rE2 )

r˜ .

Solving, you ¬nd that Cov(˜E1 , rE2 ) = 0.03505.

r ˜

¬le=optimalp¬o-g.tex: LP

402 Chapter 16. The E¬cient Frontier”Optimally Diversi¬ed Portfolios.

16·4.D. Combining The Risk-Free Security And the Tangency Portfolio

If a risk-free security exists, so that the “true” MVE Frontier is the straight line tracing out

The special case with a

risk-free security. all combinations of the risk-free security and the tangency portfolio T, then the fact that the

combination of two MVE portfolios is still MVE is easy to understand. Say, we want to marry

two portfolios. The groom owns $100 in the tangency portfolio T, and $900 in the risk-free

asset. The bride owns $2,000 in T, purchased partly with $500 worth of debt. The couple then

owns $100 + $2, 000 = $2, 100 in T and $900 ’ $500 = $400 in the risk-free asset. This is still

an MVE portfolio, because it consists only of the risk-free asset and the risky tangency portfolio

T. The fact that the combination of MVE portfolios is itself MVE is the key to the CAPM, as you

shall learn in the next section.

However, the opposite statement”if the merged portfolio is mean-variance e¬cient, then

Digging Deeper:

the individual components are mean-variance e¬cient”is not true. It is possible that E1 and E2 do not invest in T,

but that their merged investment-weighted stock market portfolio is T. For example, allow the tangency portfolio

to consist itself of two securities, X and Y , say in proportions 1/3 and 2/3, respectively. If the groom holds $300 in

X and the bride holds $600 in Y , then the couple™s portfolio is MVE. But each individual™s portfolio was not. Who

says marriage does not pay?

Solve Now!

Q 16.11 For the three securities, S&P500, IBM, and Sony, what is the Sharpe ratio of a portfolio

that invests 50% in the risk-free security and 50% in the tangency portfolio T?

Q 16.12 What is the tangency portfolio if the risk-free rate were not 6%, but 3%? Before solving

it, think where it should lie geometrically.

Q 16.13 What is the tangency portfolio if the risk-free rate were not 6%, but 9%? Before solving

it, think where it should lie geometrically.

¬le=optimalp¬o-g.tex: RP

403

Section 16·5. What does a Security need to o¬er to be in an E¬cient Frontier Portfolio?.

16·5. What does a Security need to offer to be in an Ef¬cient

Frontier Portfolio?

16·5.A. What if the Risk-Reward Relationship is Non-Linear?

Return to Figure 16.4 on Page 390, which graphs the risk and reward for many portfolios. Some Portfolios.

Among them are

E(˜P ) Sdv(˜P )

wS&P500 wIBM wSony r r

Portfolio

G 0.4000 0.2000 0.4000 16.80% 40.77%

E3 0.0414 0.6891 0.2668 17.50% 37.50%

I 0.000 1.000 0.000 15.38% 38.77%

The G portfolio is clearly ine¬cient: it has a lower expected rate of return and a higher standard Portfolio E3 is better

than G. The investor

deviation than the E3 portfolio. Comparing the two portfolios, it appears as if the G portfolio

should buy less in

has too much of the S&P500, too little of IBM, and too much of Sony. Put di¬erently, if you “expensive” stocks and

were holding portfolio G, then S&P500 and Sony would really be too expensive for you, while buy more of “cheap”

stocks.

IBM would be a bargain. You would be better o¬ rebalancing away from the S&P500 and Sony

securities, and towards IBM.

Whether a stock is too expensive depends on your current portfolio. You can choose an example Whether a stock is too

expensive or too cheap

of a portfolio in which IBM appears too expensive, instead. For example, say that you are

depends not just on the

holding portfolio I”IBM only. Again, you would be better o¬ with portfolio E3 instead, but it stock itself, but on the

has smaller holdings in IBM (compared to I), not more. So, from your perspective as an owner portfolio for which it is

begin considered.

of portfolio I who wants to get the better portfolio E3, the S&P500 and Sony would look like

bargains, because they would help you to reduce your portfolio risk, while IBM would look too

expensive.

Important: The risk contribution of a security depends on the particular port-

folio. For some portfolios, a security may appear like a bargain, for others like

a gouge. The process by which you can improve your portfolio is to purchase

securities that appear like a bargain”high reward given their risk contribution

to your portfolio; and to divest securities that appear too expecsive”low reward

given their risk contribution to your portfolio.

It turns out that this insight is the basis for an important method to measure the risk/reward The annual portfolio

returns, including

characteristics of an individual security, i.e., whether this individual security is a bargain, given

portfolio E1:

your speci¬c overall portfolio. Assume you are holding portfolio G. Its annual rates of return

would have been

P¬o G P¬o G

Year S&P500 IBM Sony Year S&P500 IBM Sony

’0.212 ’0.103

1991 0.263 0.022 1997 0.310 0.381 0.391 0.356

’0.435 ’0.004 ’0.070 ’0.203

1992 0.045 1998 0.267 0.762 0.178

1993 0.071 0.121 0.478 0.244 1999 0.195 0.170 2.968 1.299

’0.015 ’0.101 ’0.212 ’0.511 ’0.287

1994 0.301 0.135 0.108 2000

’0.130 ’0.348 ’0.107

1995 0.341 0.243 0.105 0.227 2001 0.423

’0.234 ’0.357 ’0.081 ’0.197

1996 0.203 0.658 0.077 0.244 2002

E 0.101 0.154 0.242 0.168

Var 0.037 0.150 0.815 0.166

Sdv 0.190 0.388 0.903 0.408

¬le=optimalp¬o-g.tex: LP

404 Chapter 16. The E¬cient Frontier”Optimally Diversi¬ed Portfolios.

Now compute your measure of how each security helps diversifying portfolio G”the covariance

Compute the beta of

each stock with respect between each security and portfolio G. These covariances may be tedious to compute, but they

to portfolio G, and

are not di¬cult, and you have done this before (e.g., in Formula 13.18 on Page 320),

graph each beta against

its expected rate of

(+0.2631 ’ 0.101) · (0.0217 ’ 0.168) + ... + (’0.2337 ’ 0.101) · (’0.1971 ’ 0.168)

return.

Cov(˜S&P500 , rG ) =

r ˜

11

= 0.0402

(’0.2124 ’ 0.154) · (0.0217 ’ 0.168) + ... + (’0.3570 ’ 0.154) · (’0.1971 ’ 0.168)

Cov(˜IBM , rG ) =

r ˜

11

= 0.0520

(’0.1027 ’ 0.242) · (0.0217 ’ 0.168) + ... + (’0.0808 ’ 0.242) · (’0.1971 ’ 0.168)

Cov(˜Sony , rG ) =

r ˜

11

= 0.3494

(16.21)

Although you could work directly with covariances, betas are easier to interpret. So, as the

measure for how each stock contributes to the risk of portfolio G, let us adopt the beta of each

stock with respect to portfolio G. To obtain this beta, divide each of these securities™ covariances

by the same number, the variance of G:

Cov(˜S&P500 , rG )

r ˜ 0.0402

β(˜S&P500 , rG ) = = = 0.2417

r ˜

Var(˜G )

r 0.1662

Cov(˜IBM , rG )

r ˜ 0.0520 (16.22)

= = = 0.3129

β(˜IBM , rG )

r ˜

Var(˜G )

r 0.1662

Cov(˜Sony , rG )

r ˜ 0.349

= = = 2.1023

β(˜Sony , rG )

r ˜

Var(˜G )

r 0.1662

Now, plot the expected rates of return against the betas (for portfolio G, of course), and try to

draw a line through these three points. Figure 16.10 shows that the three points do not lie on

one line. IBM seems to have too high an expected rate of return for its beta (it lies above the

line), while the S&P500 has too low an expected rate of return. Therefore, as you already knew

from comparing portfolio G and E3, it appears that good advice to portfolio G owners would be

to rebalance away from the expensive S&P500 and towards the cheaper IBM. (Thereafter, you

should recheck the relative expected pricing of these securities relative to your new portfolio.)

Indeed, as it turns out, the fact that the three securities do not perfectly lie on one line is

MVE of a p¬o means its

securities must lie on a proof-positive that portfolio G can be improved upon, i.e., that portfolio G is not on the MVE

line in a beta vs. exp.ret.

Frontier.

graph.

16·5.B. What if the Risk-Reward Relationships is Linear?

If you repeat the same exercise for portfolio E3, which you will con¬rm in Exercise 16.14, you

For the MVE portfolio E3,

when graphing expected

will ¬nd that needs to be changed; earlier H=Y p¬o was at 17%, not 17.5%

rate of return against

beta (with respect to

Cov(˜S&P500 , rE3 )

r ˜

portfolio E3), all stocks 0.0370

β(˜S&P500 , rE3 ) = = = 0.2909 ,

r ˜

lie exactly on one

Var(˜E3 )

r 0.1275

straight line.

Cov(˜IBM , rE3 )

r ˜ 0.1059 (16.23)

= = = 0.8330 ,

β(˜IBM , rE3 )

r ˜

Var(˜E3 )

r 0.1275

Cov(˜Sony , rE3 )

r ˜ 0.2213

= = = 1.7403 .

β(˜Sony , rE3 )

r ˜

Var(˜E3 )

r 0.1275

¬le=optimalp¬o-g.tex: RP

405

Section 16·5. What does a Security need to o¬er to be in an E¬cient Frontier Portfolio?.

Figure 16.10. Means of Stocks vs. Betas with respect to non-MVE G portfolio

sony

0.22

Too much return

for risk contribution

==> BUY MORE IBM

0.18

Mean

ibm

0.14

Too little return for risk contribution

==> BUY LESS S&P

s&p

0.10

0.5 1.0 1.5 2.0

Beta with P

G is a poor portfolio, because it bought too little of IBM, given its risk contribution to G; and too much of S&P500,

given its risk contribution to G. Portfolios that have either higher mean and the same risk, or the same mean and

lower risk, could have been achieved by buying more IBM and less of S&P500.

Figure 16.11 plots the beta of each stock with respect to portfolio E3 against the expected rate An example of an MVE

portfolio: each stock

of return of each stock. All securities lie on a single line. This actually means that if you hold

offers the right expected

portfolio E3 , each security o¬ers an expected rate of return (reward) that is appropriate for its return for its beta.

beta (risk). There is no way to improve upon portfolio E3 by changing these portfolio weights.

Therefore, portfolio E3 is MVE.

This linear relationship between the expected rate of return of each security with its beta with Preview: This will be the

CAPM equation.

a portfolio, if the portfolio is mean-variance e¬cient, will play a profound role in the CAPM,

which will be discussed in the next chapters.

Digging Deeper: If you had worked with covariances instead of betas, the slope of the line would change, but

not the fact that all points either lie on a line or do not lie on a line.

Important: If all securities™ expected rates of return lie on a line when plotted

against the beta with respect to some portfolio, then this portfolio is MVE. If they

do not lie on a line, then this portfolio is not MVE.

Re¬‚ect on the example more generally. You know that the security holdings are optimized for To enter the tangency

portfolio, each stock

each security i in a mean-variance e¬cient portfolio E”or else E would not be MVE! The exact

must follow a particular

relationship if all the components in Portfolio E are optimized is relationship

Portfolio E is

⇐’ E (˜i ) = a + b · βri ,˜E ,

r

MVE (16.24)

˜r

¬le=optimalp¬o-g.tex: LP

406 Chapter 16. The E¬cient Frontier”Optimally Diversi¬ed Portfolios.

Figure 16.11. Means of Stocks vs. Betas with respect to MVE E3 portfolio

sony

0.22

0.18

Mean

ibm

0.14

s&p

0.10

0.4 0.6 0.8 1.0 1.2 1.4 1.6

Beta with P

E3 is an MVE portfolio. You cannot do better by purchasing more or less of S&P500, IBM, or Sony in terms of your

mean/variance trade-o¬. (This also means that to obtain more mean, you would also have to accept more risk.)

where i is an index naming each and every stock. (Nerd Appendix d proves Formula 16.24.)

The formula contains two constants, a and b, which are numbers that depend on the portfo-

lio E. Formula 16.24 relates how the expected rate of return for each and every single available

security must be related to its beta with respect to E, i.e., lie along a straight line.

Here is a di¬erent way to think of the relationship. If you purchase an MVE portfolio, the

A different way to think

about portfolio relationship between each security™s expected rate of return and its beta with respect to your

optimization: given a

own portfolio comes about automatically. It could even be stated that you purchase a best

portfolio, rebalance

possible (MVE) portfolio, if and only if you always add more of stocks that seem relatively cheap

from more expensive

towards cheaper stocks.

for your portfolio (securities with too high an expected rate of return for their betas, i.e., that lie

above your portfolio™s line); and always reduce stocks that seem relatively expensive (securities

with too low an expected rate of return for their betas, i.e., that lie below your portfolio™s line).

16·5.C. The Line Parameters

You can yet learn more about the “Line Formula 16.24”: you can even determine the two con-

You can determine the

numbers a and b. stant numbers a and b. If βi,E (the covariance) is zero for a particular investment i, then a is

the expected rate of return of this investment. If a risk-free security is available it would have

such zero covariance, so

Portfolio E is

(16.25)

⇐ E (˜i ) = a + b · 0 = rF .

’ r

MVE

So you can replace a with rF ,

Portfolio E is

(16.26)

⇐ E (˜i ) = rF + b · βri ,˜E .

’ r

MVE ˜r

¬le=optimalp¬o-g.tex: RP

407

Section 16·5. What does a Security need to o¬er to be in an E¬cient Frontier Portfolio?.

You can also consider an investment in portfolio E itself. (Naturally, buying more or less of it

does not improve the portfolio performance.) The beta of portfolio E with respect to itself is 1.

Therefore

Portfolio E is

(16.27)

⇐ E (˜E ) = rF + b · 1 = rF + b .

’ r

MVE

You can solve Formula 16.27 for b and ¬nd b = E(˜E ’ rF ). Putting this all together, the line

r

equation must be

Portfolio E is

(16.28)

⇐ E (˜E ) = rF +

’ E (˜E ) ’ rF · βi,E

r r

MVE

Now, for any portfolio on the (upper half) of the MVE frontier, b is positive. This gives the The formula states that

inside any MVE portfolio,

line formula in 16.28 a nice intuitive interpretation. In order to ¬nd its way into your MVE

each stock must offer an

portfolio, each security has to o¬er a reward (expected rate of return) that is appropriate for appropriate expected

its contribution to your portfolio™s risk: rate of return for its

risk.

• A stock that ¬‚uctuates a lot with your E portfolio itself (having high rates of return when

your portfolio E has high rates of return) has a high beta with respect to your portfolio.

Such a stock does not help much in diversifying away the risk inside E. Therefore, this

stock must o¬er you a relatively high expected rate of return to enter your E portfolio.

• A stock whose returns tend to move in the opposite direction from E™s returns has a

negative beta. Therefore, this stock can o¬er a relatively low expected rate of return and

it would still be su¬ciently desirable to enter into your MVE portfolio.

This “appropriateness” relationship between risk and reward is a peculiar property of MVE This is not necessarily

the case for non-MVE

portfolios. If you hold a portfolio that is not MVE, some of your stocks o¬er too low or too high

portfolios.

an expected rate of return for their risk contributions to your portfolio.

Important: All securities inside an MVE portfolio have to o¬er a fair reward

for their risk-contribution. If even a single security o¬ered either too much or too

little reward for its risk, there would be a better portfolio, and the original portfolio

would not have been MVE to begin with.

O¬ering a fair risk/reward means that when the expected rates of return are

graphed against the beta (with respect to the MVE portfolio E) for each and every

security, the relationship must be a straight line,

(16.29)

E (˜i ) = a + b · βri ,˜E

r ,

˜r

where a is the risk-free rate rF and b is E(˜E ) ’ rF , so

r

(16.30)

E (˜i ) = rF + [E (˜E ) ’ rF ] · βi,E

r r .

Be warned: we have covered a lot of variances and covariances in many di¬erent aspects now. Keep The Forest in Mind:

The goal is to ¬nd the

Do not confuse them. To recap, the e¬cient frontier graphs in Figures 16.2“16.4 had stan-

best overall portfolio.

dard deviation on the x-Axis. The lines in Figures 16.10 and 16.11 had β on the x-Axes. It is

important to remember that you are only interested in minimizing your overall portfolio vari-

ance, given your desired overall portfolio expected rate of return. You are not interested in the

expected rate of return of an individual stock per se. You are not interested in the standard

deviation of an individual stock per se. You are not interested in the beta of an individual stock

per se. The individual stock statistics”means, betas, etc.”are useful information only insofar

as they help you in determining your best overall portfolio.

¬le=optimalp¬o-g.tex: LP

408 Chapter 16. The E¬cient Frontier”Optimally Diversi¬ed Portfolios.

Digging Deeper: If there is no risk-free security, then rF is simply replaced by the expected rate of return of

a zero-beta stock. For Portfolio E from Figure 16.11, this means that you could only state that the line is

(16.31)

E (˜E ) = a + [E (˜E ) ’ a] · βi,E .

r r

How can you use the linear risk-reward relationship? Easy. Say, you own a mean-variance

An Example to Illustrate

This Formula. e¬cient portfolio E that has a mean E(˜E ) of 8% and a standard deviation of 4% (variance

r

of 0.0016). Further, the risk-free rate of return is 6%. What is the relationship between the

expected rate of return of each security and the beta of each security that is available to you?

The answer is that Formula 16.24 states that every security should have an expected rate of

return of

(16.32)

E (˜i ) = 6% + [8% ’ 6%] · βi,E .

r

You did not need to use the information about the 4% risk, although it might come in handy

when you have to compute βi,E . For example, if a particular security”call it Oscar”has an

expected rate of return of 10%, then it must have a beta with respect to E of 2. Another

security, Meyer, which has a beta with respect to E of 3, must have an expected rate of return

of 12%.

Solve Now!

Q 16.14 Portfolio E3 contains 4.41% S&P500, 68.9% IBM, and 26.7% Sony. Con¬rm Figure 16.11”

that is, ¬rst compute and then plot the beta of each security with respect to E3 against each secu-

rity™s expected rate of return. Note that you can either put together a historical data table, similar

to how you computed it on Page 403; or you can use the covariance tools from Table 14.5 on

Page 350, and the covariances themselves, e.g., from Page 386 or Table 14.1. Use the standard

deviation of the rate of return on E3 as given on Page 403 as oldG 35.65%.

Q 16.15 Assume that you are told that portfolio E is MVE. Further, you are told that E has an

expected rate of return of 14%. Finally, you are told that security 1 has an expected rate of return

of 8% and a beta with respect to E of 0.5. What expected rate of return would a security i with

a beta with respect to E of βi,E = 2 have to have, in order for portfolio E to be MVE (i.e., not able

to be improved upon)?

Q 16.16 A new security, E5, has appeared. Recall also the mean-variance e¬cient portfolio E2 :

E(˜P ) Sdv(˜P )

wS&P500 wIBM wSony r r

Portfolio

E5 “0.146 0.734 0.286 18.00% 39.38%

E2 +99.448% +2.124% “1.573% 10.00% 18.969%

E5™s variance-covariance is

rS&P500 rIBM rSony rE5 rE2

˜ ˜ ˜ ˜ ˜

rS&P500

˜ 3.622% 3.298% 4.772% 4.362% 3.698%

rIBM

˜ 3.298% 15.034% 2.184% 4.866% 10.587%

rSony

˜ 4.772% 2.184% 81.489% 57.956% 22.124%

rE5

˜ 4.362% 4.866% 57.956% 41.978% 17.974%

rE2

˜ 3.698% 10.587% 22.124% 17.974% 12.707%

The new stock has a mean rate of return of 21.029%. Recall that the mean rate of return of the

MVE portfolio E2 was 17%.

¬le=optimalp¬o-g.tex: RP

409

Section 16·6. Summary.

(a) Compute the beta of portfolio E5 with respect to portfolio E2 .

(b) Plot this new stock into Figure 16.11. Does it lie on the line? What does this mean?

(c) Add a little bit of stock E5 to portfolio E2 . That is, buy 99% of E2 and 1% of E5. Compute

the risk and reward of this combination portfolio. Does this combination portfolio o¬er

superior risk/reward characteristics than portfolio E2 alone?

(d) Subtract a little bit of stock E5 from portfolio E2 . That is, buy 101% of E2 and “1% of E5.

Compute the risk and reward of this combination portfolio. Does this combination portfolio

o¬er superior risk/reward characteristics than portfolio E2 alone?

Q 16.17 Repeat this question, but assume that the expected rate of return on stock E5 is only

10%.

16·6. Summary

This was a long and complex chapter, so this summary is more detailed than our usual chapter

summaries:

1. Each portfolio is a point in a graph of overall portfolio expected rate of return (reward)

against overall portfolio standard deviation (risk).

Portfolios with similar compositions are close to one another.

2. For two securities, the possible risk vs. reward choices are a curve (called a hyperbola).

For more than two securities, the possible risk vs. reward choices are a cloud of points.

3. The mean-variance e¬cient frontier is the upper left envelope of this cloud of points. You

should not purchase a portfolio that is not on the MVE frontier.

4. In real life, computing the historical MVE Frontier is useful especially when choosing

among a small set of asset classes. Otherwise, the historical standard deviations and

expected rates of return are too unreliable to produce good forecasts of the future MVE

Frontier.

5. If there is a risk-free security, the combination of the risk-free security and any other

portfolio is a straight line.

6. If there is a risk-free security, the MVE Frontier is the line between the risk-free rate and

the Tangency portfolio. You should not purchase a portfolio that is not on this line.

7. The combination of two MVE portfolios is also MVE.

8. If two MVE portfolios are known, ¬nding other MVE portfolios is a matter of taking

weighted combinations of these two MVE portfolios.

9. MVE portfolios have an important property: Each security inside the MVE portfolio o¬ers

a fair reward for its risk contribution.

• If any single security were too cheap, you could do better by buying more of it”which

means that the portfolio would not have been MVE to begin with.

¬le=optimalp¬o-g.tex: LP

410 Chapter 16. The E¬cient Frontier”Optimally Diversi¬ed Portfolios.

• If any single security were too expensive, you could do better by buying less of it”

which means that the portfolio would not have been MVE to begin with.

10. The reward of each security is suitably measured by its expected rate of return. The risk

of each security inside the MVE portfolio is suitably measured by its beta with respect to

your speci¬c MVE portfolio.

To be fairly priced, every security in the portfolio must lie on a straight line when the

expected rates of return are plotted against these betas. Algebraically, this means that

the relation is

(16.33)

E (˜i ) = a + [E (˜E ) ’ a] · βi,E ,

r r

for every security i inside the MVE portfolio E. If there is a risk-free security o¬ering an

expected (actual) rate of return rF , then this line becomes

(16.34)

E (˜i ) = rF + E (˜E ) ’ rF · βi,E .

r r

This relationship will play a prominent role in the subsequent CAPM chapters.

¬le=optimalp¬o-g.tex: RP

411

Section A. Advanced Appendix: Excessive Proofs.

Appendix

A. Advanced Appendix: Excessive Proofs

a. The Optimal Portfolio Weights Formula

This section relies on algebra (matrices and vectors). This is necessary, because it is simply too Warning: Linear Algebra.

messy to deal with 10,000 individually named securities, with their requisite 10,000 expected

rates of return and the requisite 50 million covariance terms. Unless you already know linear

algebra inside out, and/or you want to write a computer program to determine the MVE Frontier,

please ignore this section. It is not necessary to an understanding of this book.

Using matrix notation, the portfolio mean and variance can be expressed as Variable De¬nitions and

the Problem Setup.

N

′

E (˜P ) ≡ wm ≡ wi ·Exp(˜i ) ,

r r

i=1

(16.35)

NN

Var(˜P ) ≡ w ′ Sw ≡ wi ·wj ·σi,j ,

r

i=1 j=1

where S is the matrix of variance-covariances, m is the vector of expected rates of return on

each security (which is easier to write than E(˜) ), and 1 is a vector containing only the scalar 1

r

in each of N positions. A prime denotes a row rather than a column vector. The problem to

solve is therefore

Var(˜P ) ≡ w ′ Sw

r

min minimize portfolio variance,

w

1′ w (16.36)

= 1 portfolio weights add to 1,

subject to

≡ m′ w

E (˜P )

r user wants mean return E (˜P ).

r

This problem is usually solved via Lagrangian Optimization techniques. Introduce » and γ as

two new Lagrangian multiplier parameters, and optimize

(16.37)

min L(w, », γ) ≡ w ′ Sw ’ »· 1 ’ 1′ w ’ γ· E (˜P ) ’ m′ w

r .

w,»,γ

The MVE set is obtained by di¬erentiating with respect to the weights w and the two Lagrangian

constants.

If you are not familiar with Lagrangian techniques, di¬erentiating with respect to the

Digging Deeper:

two Lagrangian multipliers produces the two original constraints, thereby ensuring that the optimization really

works across the constrained set only. For example, ‚L/‚» = 1 ’ 1′ w, which if set to zero (as it is in the ¬rst-order

condition) reimposes the constraint 1 = 1′ w.

After some manipulation, the solution is The solution are the

optimal portfolio

weights, w — .

w — = »— · S’1 1 + γ — · S’1 m

C ’ E (˜P )·B

r (16.38)

»— =

D

E (˜P )·A ’ B

r

γ— = ,

D

where A, B, CF, and D are simple numerical constants (scalars):

1′ S’1 1 1′ S’1 m

A≡ B≡

(16.39)

′ ’1 2

C ≡ mS D ≡ A·C ’ B

m .

¬le=optimalp¬o-g.tex: LP

412 Chapter 16. The E¬cient Frontier”Optimally Diversi¬ed Portfolios.

Using this formula, you can obtain the formula relating the mean and variance, the parabola

w ′ — Sw — = w ′ — S »— S’1 1 + γ — S’1 m

Var(˜P ) =

r

(16.40)

2

A·E (˜P ) + 2·B·E (˜P ) + C

r r

= »— + γ — · E (˜P ) =

r .

D

This formula generalizes Formula 16.4 on Page 385.

The weights of the minimum variance portfolio are worth writing down. They are the solution

The minimum variance

when the mean constraint m′ w = E(˜P ) is not imposed. The minimum variance portfolio is

portfolio. r

then

S’1 1

—

wmin = (16.41)

.

A

var

Note that this portfolio does not depend on any security™s expected rate of return.

b. The Combination of MVE Portfolios is MVE ” With Risk-Free Security.

It is di¬cult to prove that the combination of MVE portfolios is MVE”except in the special case

Combining two MVE

portfolios is MVE again. in which a risk-free security exists. Call the MVE tangency portfolio “T.” Because T is MVE,

(16.42)

E (˜i ) = rF + E (˜T ) ’ rF · βi,T ∀i .

r r

Now invest wF in the risk-free security (F) and (1 ’ wF ) in T. Call this portfolio E4.

(16.43)

rE4 = wF ·rF + (1 ’ wF )·˜T .

r

˜

Both F and T are MVE portfolios. The question is whether portfolio E4 is also MVE. The expected

rate of return of E4 is

(16.44)

E (˜E4 ) = wF ·rF + (1 ’ wF )·E (˜T ) .

r r

The beta of any security i with respect to this portfolio E4 is

Cov[˜i , wF ·rF + (1 ’ wF )·˜T ]

Cov(˜i , rE4 ) r r

r˜

βi,E4 = =

Var(˜E4 ) Var[wF ·rF + (1 ’ wF )·˜T ]

r r

(1 ’ wF ) · Cov(˜i , rT ) Cov(˜i , rT )

r˜ r˜

1 (16.45)

= = ·

(1 ’ wF ) 2 ·Var(˜ ) 1 ’ wF Var(˜T )

rT r

βi,T

= ,

1 ’ wF

because the risk-free rate has neither a variance nor a covariance with any other return series.

The new portfolio E4 is also MVE, if and only if

(16.46)

E (˜i ) = rF + E (˜E4 ) ’ rF · βi,E4 ∀i .

r r

(A security i can have weight zero in the portfolio, so this really means for all securities.)

Substitute Formula 16.44 for E(˜E4 ), and Formula 16.45 for βi,E4 :

r

βi,T

E (˜i ) = rF + wF ·rF + (1 ’ wF )·E (˜E4 ) ’ rF ·

r r

1 ’ wF

βi,T (16.47)

= rF + (1 ’ wF )·E (˜E4 ) ’ (1 ’ wF )·rF ·

r

1 ’ wF

= rF + E (˜E4 ) ’ rF · βi,T ∀i .

r

This is true if and only if a portfolio is MVE. Therefore, the new portfolio E4 is also MVE.

¬le=optimalp¬o-g.tex: RP

413

Section A. Advanced Appendix: Excessive Proofs.

c. The Combination of Mean-Variance E¬cient Portfolios is Mean-Variance E¬cient ” With-

out Risk-Free Security.

There are many ways to prove that the combination of two MVE portfolios is also MVE. The The combination of any

two risky portfolios is a

brute-force method is the linear algebra solution from the previous section. In Formula 16.38,

parabola in

S’1 1 is one portfolio (a vector of [unnormalized] portfolio weights), S’1 m is another portfolio mean-variance space.

(another vector of portfolio weights). » and µ determine the relative investment proportions

in each.

The rest of this subsection is an alternative conceptual argument which may or may not help Start with an “intuitive”

proof.

you to get some more intuition. Ignore it if you wish. Return to Formula 16.4, which shows

that the relationship between two risky investment assets is

V [ E (˜P ) ] = a · E (˜P )2 + b · E (˜P ) + c , (16.48)

r r r

ar

where a, b, and c are constants that depend on the particular expected rate of return between

two securities, their individual variances, and their mutual covariance.

The question you should ponder is whether the combination of two MVE portfolios is itself MVE. Being only a parabola

means that the

You already saw the argument in Figure 16.6: it must be so. Imagine two portfolios on the MVE

combination portfolio of

frontier”call them portfolios E1 and E2. Portfolio E1 could be on the upper half of the parabola, any two MVE Frontier

portfolio E2 could be its equal-variance counterpart on the lower part of the parabola. Name the portfolios has to curve

at the same rate as the

actual minimum variance portfolio by the letter E0. The question is whether the combination

MVE Frontier.

of the E1 and E2 portfolios traces out the MVE Frontier. You already know that the combination

of E1 and E2 is a quadratic equation. The previous section showed that the MVE Frontier is

a quadratic equation, too. The question is whether the quadratic frontier between E1 and E2

can de¬ne a set other than the MVE Frontier. In a quadratic equation, three points de¬ne the

parabola. Thus, if E1 and E2 are already chosen as two portfolios with equal risk, then all

possible combinations of these portfolios would have to lie on a parabola, and the minimum

variance portfolio based only on E1 and E2 would have to lie on the same horizontal line that

the actual minimum variance portfolio E0 lies on. What would happen if the combination

of E1 and E2 produced a minimum variance portfolio with lower variance than E0? Then the

original statement that E0 is the minimum variance portfolio would be false. What would

happen if the combination of E1 and E2 produced a minimum variance portfolio with higher

variance than E0? Then, the parabola would overshoot the MVE Frontier when either E1 or E2

is shorted. However, then the real MVE Frontier would not be a simple quadratic equation, but

one quadratic equation to the left of E1 and E2 and another quadratic equation to the right

of E1 and E2. You already know that this cannot be the case, either.

Therefore, the combinations of E1 and E2 trace out the MVE frontier. In other words, the Therefore, any

combination of any two

combination of the two MVE risky portfolios E1 and E2 is the full and sole MVE. Any other

risky MVE portfolios

portfolio, say E4, that is a weighted combination of E1 and E4, can be represented by a unique traces out the MVE

weight on E1 vs. E4. Combining many arbitrary portfolios consisting of E1 and E2 is still a Frontier.

weighted combination of E1 and E2, and therefore still MVE.

d. Proof of the Linear Beta vs. Expected Rate of Return Relationship for MVE Frontier Port-

folios

Our intent is to prove that all securities in an MVE portfolio must follow a linear beta-expected Consider the

combination of the MVE

return relationship. It is the mathematical relationship that is required for every stock to be

Portfolio and a new

at just its optimal level, so that you are indi¬erent between purchasing one more or one less security.

cent of this security. The derivation relies on the envelope theorem, which states that adding a

little, little bit or subtracting a little, little bit at the optimal investment choice wi⋆ should make

only a minimal di¬erence for the optimal portfolio. If the expected rate of return were too high

or too low for a security, given its beta, you could do better than E”either buying a little bit

more or a little bit less of this security. But then, E would not be MVE.

¬le=optimalp¬o-g.tex: LP

414 Chapter 16. The E¬cient Frontier”Optimally Diversi¬ed Portfolios.

Figure 16.12. Possible and Impossible Combinations of a New Security with a Mean-Variance

E¬cient Portfolio

Impossible Combinations of a New Security with a Mean-Variance E¬cient Portfolio:

¨¨

B

¨

T

E (˜)

r

¨¨

Impossible Combinations

¨

c ¨¨bE

¨

E (˜E )

r

¨¨

¨

¨ r Security i

¨

¨

¨¨

¨

¨¨

¨

E (˜0 )

r

Var

E

V (˜0 ) V (˜E )

ar r ar r

Possible Combinations of a New Security with a Mean-Variance E¬cient Portfolio:

¨

B

¨¨

T

E (˜)

r

¨

Possible Combinations: Tangent!

¨¨

d

b¨

¨E

d ¨

E (˜E )

r

d¨

¨

¨¨ d‚

r Security i

¨¨

¨

¨¨

¨

¨¨

E (˜0 )

r

Var

E

V (˜0 ) V (˜E )

ar r ar r

¬le=optimalp¬o-g.tex: RP

415

Section A. Advanced Appendix: Excessive Proofs.

Figures 16.12 shows a MVE portfolio called E. You are considering adding wi of a security i If you combine an

MVEportfolio with a little

(security i alone does not have to be MVE). Could it be that the combination portfolio were not

bit of another portfolio,

tangent to the frontier, i.e. where wi ≈ 0 and wE ≈ 1? The answer must be no”or it would be you should not be able

possible to purchase a combination portfolio that outperforms the MVE Frontier. Perturbing to perform better.

the weight in the E by adding security i (where wi = 0) has to result in a mean-variance trade-o¬

that is tangent to the MVE Frontier, as shown in the graph in Figure 16.12. You can use this

insight to prove the linear relationship between the expected rate of return on security i and

its beta with respect to E.

1. By inspection, the slope of the MVE Frontier at E is

E (˜E ) ’ E (˜0 ) E (˜E ) ’ E (˜0 )

r r r r

Slope = = (16.49)

,

Sdv(˜E ) ’ Sdv(˜0 ) Sdv(˜E )

r r r

because Sdv(˜0 ) is 0.

r

2. Consider an arbitrary combination portfolio (call it P). It is de¬ned by

(16.50)

rP = wi ·˜i + (1 ’ wi )·˜E .

r r

˜

The characteristics of this portfolio P are

E (˜P ) = wi ·E (˜i ) + (1 ’ wi )·E (˜E )

r r r

(16.51)

wi2 ·Var(˜i )

Sdv(˜P ) = + (1 ’ wi )2 ·Var(˜E ) + 2·wi ·(1 ’ wi )·Cov(˜i , rE ) .

r r r r˜

How do the mean and standard deviation of P change with wi ?

‚E (˜P )

r

= E (˜i ) ’ E (˜E )

r r ,

‚wi ±

Cov(˜ , r ) ’ V (˜ ) + w ·[V (˜ ) ’ 2·Cov(˜ , r ) + V (˜ )]

(16.52)

ar rE

‚Sdv (˜P )

r ri ˜E ar rE ar ri ri ˜E

i

= .

w 2 ·V (˜i ) + (1 ’ wi )2 ·V (˜E ) + 2·wi ·(1 ’ wi )·Cov(˜i , rE )

‚wi r˜

ar r ar r

i

Now, compute the slope of the curve that de¬nes the sets of combination portfolios P,

i.e., how its expected rate of return changes as its standard deviation is changed. Because

you can only change wi , you need to compute this slope by dividing the two derivatives

in Formula 16.52:

‚E (˜P )

r

‚E (˜P )

r ‚wi

= . (16.53)

‚Sdv (˜P )

r

‚Sdv(˜P )

r

‚wi

What you really need to know is the slope when wi = 0, i.e., when the weight is solely on

the MVE Frontier portfolio E. So, substitute wi = 0 into this messy expression, and you

get ‚E (˜P )

r

limwi ’0

‚E (˜P )

r ‚wi

=

lim ‚Sdv (˜P )

r

‚Sdv(˜P )

r

wi ’0 limwi ’0 ‚wi

(16.54)

E (˜i ) ’ E (˜E )

r r

= .

Cov(˜i ,˜E )’V (˜E )

rr ar r

V (˜E )

ar r

3. You can now use the fact that the slope of portfolio P at wi = 0 has to be the same slope

as the MVE Frontier. That is, the line de¬ned in equation 16.49 and the curve de¬ned in

equation 16.54 must be tangent.

Slope of Tangent Line = Slope of Curve at wi = 0

(16.55)

E (˜E ) ’ E (˜0 ) E (˜i ) ’ E (˜E )

r r r r

= .

Cov(˜i ,˜E )’V (˜E )

rr ar r

Sdv(˜E )

r

V (˜E )

ar r

¬le=optimalp¬o-g.tex: LP

416 Chapter 16. The E¬cient Frontier”Optimally Diversi¬ed Portfolios.

Use Var(˜E ) = Sdv(˜E )·Sdv(˜E ) to simplify this equation to

r r r

Cov(˜i , rE )

r˜

E (˜i ) = E (˜0 ) + · [E (˜E ) ’ E (˜0 )]

r r r r

Var(˜E )

r

(16.56)

= E (˜0 ) + βi,M · [E (˜E ) ’ E (˜0 )]

r r r ,

which is Formula 16.24 that we wanted to prove! This formula has to hold for every

available security i. Indeed, a portfolio is MVE if and only if this equation holds for all

available investment opportunities i.

Figure 16.12 is drawn for better illustration, not for accuracy. To avoid eventually inter-

Digging Deeper:

secting the MVE Frontier, the possible combinations would necessarily have to intersect E from below, not from

above (as drawn).

¬le=optimalp¬o-g.tex: RP

417

Section A. Advanced Appendix: Excessive Proofs.

Solutions and Exercises

1. The risk is always 10%. By shorting the security with lower expected rate of return and buying the security

with higher expected rate of return, you can invest in an opportunity with an in¬nite expected rate of return.

This is expected, but it is not certain. (If you are curious: by choice of portfolio, you can guarantee yourself

any positive rate of return, but you have still standard deviation, because you do not know whether you will

earn a very large amount or a very large amount plus or minus a little. The only risk is now how much you

will earn, not whether you will earn a positive or a negative return.)

2. Indeed, going long and going short is risk-free. However, it is also a zero-investment and zero-payo¬ strategy

(except for frictions, of course).

3. The portfolios have the following risks:

1/3 1/2 2/3

Weight wA “1 0 1

Risk (σP ) 17.3% 10% 8.8% 8.7% 8.8% 10%

A graph of these portfolios (and weights in between) is:

0.12

wA = 1

(wB = 0)

0.10

Expected Rate of Return

wA = 4/5

wA = 2/3

0.08

wA = 0.5 wA = 1/2

(wB = 0.5)

wA = 1/3

0.06

wA = 1/5

wA = 0

0.04

(wB = 1)

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Standard Deviation of Rate of Return

4. First, compute the covariance: Correlation A,B = Cov A,B /(Sdv A ·Sdv B ) ’ Cov(A, B) = 20%·30%·80% = 0.048.

The MVE frontier is:

0.11

wA = 0

(wB = 1)

Expected Rate of Return

wA = 1/5

0.09

wA = 1/3

wA = 0.5

(wB = 0.5) wA = 1/2

wA = 2/3

0.07

wA = 4/5

wA = 1

(wB = 0)

0.05

0.0 0.1 0.2 0.3

Standard Deviation of Rate of Return

0.15

(0.2,0.4,0.4)

(0.4,0.2,0.4) o

(1,’1,1)

(0.5,0.5,0)

0.10

oo

Portfolio Mean

(1,0,0)

o (0.5,0,0.5)

o

(1,1,’1)

(1,0.5,’0.5)

o

o

o o o

o (0,0,1)

oo (0.5,1,’0.5)

(0,0.5,0.5)

o (0.25,1,’0.25)

0.05

o (’0.5,0.5,1) o

(0,1,0)

(0.5,’0.5,1)

(0.4,0.4,0.2)

0.00

0.00 0.05 0.10 0.15

5. Portfolio Sdv

6. The Minimum Variance Portfolio in this world is a portfolio that invests about 45% into security 1, about 38%

into security 2, and the rest (about 17%) into security 3.

7. An investor with more taste for risk would choose a portfolio higher up along the MVE Frontier. The least

risky portfolio with an expected rate of return of 11% invests about 97% into security 1, shorts 42% in security

2, and invests 45% into security 3.

¬le=optimalp¬o-g.tex: LP

418 Chapter 16. The E¬cient Frontier”Optimally Diversi¬ed Portfolios.

8. No, because it is inside the main “cloud” of points. There are better portfolios that o¬er higher expected

rates of return for the same amount of risk. The three “pure” portfolios, which invest in only one stock are

not on the MVE Frontier, either. However, the portfolio that invests 100% in security 1 and nothing in the

other two securities comes fairly close by accident.

9. You can save time if you work directly o¬ combinations of portfolio E1 and portfolio E3.

1/2·E1 + 1/2·E3

E1 E3

S&P500 99.5% 36.1% 67.770%

IBM 2.1% 46.6% 24.386%

Sony “1.6% 17.2% 7.844%

E (˜)

r 10.0% 15.0% 12.500%

Sdv (˜)

r 19.0% 67.6% 30.949%

10. You can save time if you work directly o¬ combinations of portfolio E2 and portfolio E3.

1/2·E1 + 1/2·E3

E2 E3

S&P500 “90.6% 36.1% “27.264%

IBM 135.7% 46.6% 91.172%

Sony 54.9% 17.2% 36.092%

E (˜)

r 25.0% 15.0% 20.000%

Sdv (˜)

r 19.0% 67.6% 59.266%