substituting out wA from Formula 16.1 (portfolio P™s mean) and Formula 16.2 (portfolio P™s standard deviation).

The result is

(16.3)

Sdv [ E (˜P ) ] = ± a · E (˜P )2 + b · E (˜P ) + c ,

r r r

where

d’1 · V (˜A ) ’ 2 · Cov(˜A , rB ) + V (˜B )

a≡ ar r r˜ ar r ,

d’1 · 2 · E (˜A ) · [Cov(˜A , rB ) ’ V (˜B )]

b≡ r r˜ ar r

d’1 · 2 · E (˜B ) · [Cov(˜A , rB ) ’ V (˜A )]

+ r r˜ ar r ,

(16.4)

’1 2 2

c≡ · E (˜B ) · V (˜A ) + E (˜A ) · V (˜B )

d r ar r r ar r

d’1 · 2 · E (˜B ) · E (˜A ) · Cov(˜A , rB )

’ r r r˜ ,

2

d≡ E (˜A ) ’ E (˜B )

r r .

This formula states that the variance of the rate of return (Sdv (˜P )2 ) on an arbitrary weighted portfolio P is a

r

quadratic formula in its expected rate of return (E (˜P )): a parabola. Therefore, the MVE Frontier function is a

r

hyperbola in a graph of the expected rate of return against their standard deviations.

Of course, the e¬cient frontier is only the upper arm of the hyperbola. Although this is by no means obvious, it

turns out that this is the case even if there are more than 2 securities: the MVE Frontier is always the upper arm

of a hyperbola”the combination of two particular portfolios. This only breaks down when there are additional

constraints, such as short-sales constraints.

16·1.B. Di¬erent Covariance Scenarios

Table 16.2. Portfolio Performance Under Di¬erent Covariance Scenarios.

Correlation is

’1.00 ’0.75 +0.75 +1.00

0.00

Risk (Sdv ) if Covariance is

E (˜P ) ’0.01 ’0.0075 +0.0075 +0.01

wA wB r 0

1 0 10.0% 10.0% 10.0% 10.0% 10.0% 10.0%

0 1 5.0% 10.0% 10.0% 10.0% 10.0% 10.0%

1/5 4/5 6.0% 6.0% 6.6% 8.2% 9.6% 10.0%

1/3 2/3 6.8% 3.3% 4.7% 7.5% 9.4% 10.0%

1/2 1/2 7.5% 0.0% 3.5% 7.1% 9.3% 10.0%

2/3 1/3 8.3% 4.7%

3.3% 7.5% 9.4% 10.0%

4/5 1/5 9.0% 6.0% 6.6% 8.2% 9.6% 10.0%

Portfolios Involving Shorting:

’1 +2 0.0% 30.0% 28.3% 22.4% 14.1% 10.0%

+2 ’1 15.0% 30.0% 28.3% 22.4% 14.1% 10.0%

This ¬gure shows the risk and reward under di¬erent assumptions about the covariance be-

tween A and B. The ¬rst two lines give the characteristics of the two base securities, A and B.

(wB is always 1 ’ wA , and thus could have been omitted.)

¬le=optimalp¬o-g.tex: LP

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

How do di¬erent asset correlations between securities A and B change the shape of the e¬cient

How do ef¬cient

frontiers look like with frontier. Table 16.2 is really the same as Table 16.1, except it works out di¬erent covariance

different correlations.

scenarios. And Figure 16.1 is really the same as Figure 16.2 on Page 387. It plots the data

into coordinate systems, in which the overall portfolio standard deviation is on the x-Axis, and

the overall portfolio expected rate of return is on the y-Axis. The previous Figure 16.1 is now

graph (b), just stretched due to the di¬erent axes.

Of course, regardless of covariance, when you choose only one or the other security (wA = 0

Higher correlations

“compress” the function. or wA = 1), the portfolio risk is 10% (as was assumed), and here noted with arrows. So, these

points are identical in all graphs. More interestingly, when you repeat the exercise for di¬erent

covariance scenarios, shown in Table 16.2 and Figure 16.2, you can verify your earlier insight

that lower covariance helps diversi¬cation. For example, if the two securities are perfectly

negatively correlated, which implies a covariance of ’0.01, then an equal-weighted portfolio of

the two securities has zero risk. (When one security™s value increases, the other security™s value

decreases by an equal amount, thereby eradicating any risk.) If the two securities are perfectly

positively correlated, diversi¬cation does nothing, and the portfolio standard deviation remains

at 10% no matter what weights are chosen. Thus, adding securities with low covariance to your

existing portfolio lowers your overall portfolio risk particularly well; adding securities with

high covariance to your existing portfolio is less e¬ective.

Solve Now!

Q 16.1 In the example in Table 16.2, if the covariance is ’0.01 (Figure 16.2e), what should you

do?

Q 16.2 Assuming no transaction and shorting costs, does going short in a security and going

long in the same security produce a risk-free investment?

Q 16.3 In the example, with risk of 10% for each security, compute the standard deviations of

various portfolios™ returns if the covariance between the two securities is +0.005. Graph the mean

against the standard deviation.

Q 16.4 Security A has a risk (standard deviation) of 20% and an expected rate of return of 6%;

Security B has a risk of 30% and an expected rate of return of 10%. Assume the two securities

have +0.80 correlation. Draw the MVE Frontier.

16·1.C. The Mean-Variance E¬cient Frontier With Many Risky Securities

The MVE Frontier is easy to compute when you can choose only one investment weight, which

With more securities,

the computations then determines your relative allocation between the two assets. In contrast, if you can choose

become cumbersome...

from three securities, you must consider combinations of two portfolio weights (the third

weight is one minus the other two). This is a two-dimensional choice problem. With four

securities, you have three portfolio weights to optimize, and so on.

The good news is that the portfolio selection principle remains the same: each portfolio is a

...but the principle

remains the same. point in the graph of portfolio means (expected rates of return, i.e., reward) vs. portfolio stan-

dard deviation (i.e., risk). After plotting all possible portfolios, you should choose from those

points (portfolios) that lie on the MVE Frontier”which is still the upper half of the hyperbola.

Try out many random investment portfolios from the three stocks from Page 307”S&P500,

Return to the three

stocks, S&P500, IBM, IBM, Sony”and selecting the best ones. Of course, you would still assume that the historical

and Sony.

annual rates of return from 1991 to 2002 are representative of the future, in the sense that we

expect the historical means, variances, and covariances to apply in the future. (Otherwise, this

would not be an interesting exercise!) These historical statistics were

¬le=optimalp¬o-g.tex: RP

387

Section 16·1. The Mean-Variance E¬cient Frontier.

Figure 16.2. The Risk-Reward Trade-o¬ With Di¬erent Correlations

(a) Correlation = ’1.00 (Cov = ’0.01) (b) Correlation = ’0.75 (Cov = ’0.0075)

0.12

0.12

wA = 1 wA = 1

(wB = 0) (wB = 0)

0.10

0.10

Expected Rate of Return

Expected Rate of Return

wA = 4/5 wA = 4/5

wA = 2/3 wA = 2/3

0.08

0.08

wA = 0.5

wA = 1/2 wA = 1/2

(wB = 0.5)

wA = 1/3 wA = 1/3

0.06

0.06

wA = 1/5 wA = 1/5

wA = 0 wA = 0

0.04

0.04

(wB = 1) (wB = 1)

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.00 0.02 0.04 0.06 0.08 0.10 0.12

Standard Deviation of Rate of Return Standard Deviation of Rate of Return

(c) Covariance = 0.0

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

(d) Correlation = +0.75 (Cov = +0.0075) (e) Correlation = +1.00 (Cov = +0.01)

0.12

0.12

wA = 1 wA = 1

(wB = 0) (wB = 0)

0.10

0.10

Expected Rate of Return

Expected Rate of Return

wA = 4/5 wA = 4/5

wA = 2/3 wA = 2/3

0.08

0.08

wA = 0.5 wA = 0.5

wA = 1/2 wA = 1/2

(wB = 0.5) (wB = 0.5)

wA = 1/3 wA = 1/3

0.06

0.06

wA = 1/5 wA = 1/5

wA = 0 wA = 0

0.04

0.04

(wB = 1) (wB = 1)

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.00 0.02 0.04 0.06 0.08 0.10 0.12

Standard Deviation of Rate of Return Standard Deviation of Rate of Return

These ¬gures repeat Figure 16.1, except that they consider di¬erent covariances between the portfolios. If the

correlation is perfectly negative, you can manufacture a risk-free security by purchasing half of A and half of B. (If

the correlation is perfectly positive, and if you can short B, you could even obtain a virtually in¬nite expected rate

of return. This cannot happen in a reasonable market.)

¬le=optimalp¬o-g.tex: LP

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

Covariance between ri and rj

˜ ˜

E (˜i )

r

Security 1=S&P500 2=IBM 3=Sony

1=S&P500 10.110% 3.6224% 3.2980% 4.7716%

2= IBM 15.379% 3.2980% 15.0345% 2.1842%

3= Sony 24.203% 4.7716% 2.1842% 81.4886%

The diagonal covariance elements are the variances, because of the way each is de¬ned (Cov(x, x) ≡

Var(x)). What would the risk-reward trade-o¬s of these portfolio combinations have looked

like? Recall the portfolio statistics from Chapter 14. With three securities, the formulas to

compute the overall portfolio mean and standard deviation are

E (˜P ) = w1 · E (˜1 ) + w2 · E (˜2 ) + w3 · E (˜3 )

r r r r ,

2 2 2

Var(˜P ) = w1 · Var(˜1 ) + w2 · Var(˜2 ) + w3 · Var(˜3 )

r r r r

(16.5)

+ 2 · w1 · w2 · Cov(˜1 , r2 ) + 2 · w1 · w3 · Cov(˜1 , r3 )

r˜ r˜

+ 2 · w2 · w3 · Cov(˜2 , r3 )

r˜ ,

and w3 ≡ 1 ’ w1 ’ w2 .

Armed with formulas and statistics, you can now determine portfolio means and standard

Pick some portfolios and

use the formulas. deviations for some portfolios (di¬erent weights of wS&P500 , wIBM , wSony ). For example, for the

portfolio that invests 20% in S&P500, 40% in IBM, and 40% in Sony, the expected rate of return

is

E (˜P ) ≈ 20% · 10.110% + 40% · 15.379% + 40% · 24.203% ≈ 17.85% ,

r

(16.6)

E (˜P ) = w1 · E (˜1 ) + w2 · E (˜2 ) + w3 · E (˜3 )

r r r r ,

and the variance is

Var(˜P ) ≈ (20%)2 · 3.6224% + (40%)2 · 15.0345% + (40%)2 · 81.4886%

r

+ 2 · (20%) · (40%) · 3.2980% + 2 · (20%) · (40%) · 4.7716%

+ 2 · (40%) · (40%) · 2.1842% ≈ 0.1758

(16.7)

2 2 2

Var(˜P ) = · Var(˜1 ) + · Var(˜2 ) + · Var(˜3 )

r w1 r w2 r w3 r

+ 2 · w1 · w2 · Cov(˜1 , r2 ) + 2 · w1 · w3 · Cov(˜1 , r3 )

r˜ r˜

+ 2 · w2 · w3 · Cov(˜2 , r3 )

r˜ .

Therefore, the risk of the portfolio is Sdv(˜P ) ≈ 41.93%.

r

Table 16.3 lists some more randomly chosen portfolio combinations, and Figure 16.3 plots the

Plot the mean and

standard deviations of data from this table. Looking at the set of choices, if you were extremely risk-averse, you might

these portfolios.

have chosen just to invest in a portfolio that was all S&P500 (1,0,0). Indeed, from the set you

already know, the portfolio with the absolute lowest variance among these 16 portfolios seems

to be mostly an investment in the S&P500. If you were more risk-tolerant, you might have

chosen a portfolio that invested 20% in the S&P500, 40% in IBM, and 40% in Sony (0.2,0.4,0.4).

However, regardless of your risk aversion, the (0.5,0,0.5) portfolio, which invests 50% in S&P500,

zero in IBM, and 50% in Sony, would have been a poor choice: it would have had a lower portfolio

mean and higher standard deviation than alternatives, such as the (0.2,0.4,0.4) portfolio.

Note also that again, portfolios with very similar portfolio weights have similar means and

Similarity in portfolio

weights means proximity standard deviations, and therefore lie close to one another. For example, the portfolio that

in the Figure.

invests (0.5,0.5,0) is relatively close to the portfolio (0.4,0.4,0.2).

¬le=optimalp¬o-g.tex: RP

389

Section 16·1. The Mean-Variance E¬cient Frontier.

Table 16.3. Risk and Reward of Hypothetical Portfolios Consisting Only of S&P500, IBM, and

Sony

Weights Weights

E (˜P ) Sdv (˜P ) E (˜P ) Sdv (˜P )

(wS&P500 , wIBM , wSony ) r r (wS&P500 , wIBM , wSony ) r r

(1,0,0) 0.1010 0.1903 (-1,1,1) 0.2947 0.9401

(0,1,0) 0.1537 0.3877 (1,-1,1) 0.1893 0.9936

(0,0,1) 0.2420 0.9027 (1,1,-1) 0.0128 0.9635

(0.5,0.5,0) 0.1274 0.2513 (1,0.5,-0.5) 0.0569 0.5018

(0.5,0,0.5) 0.1715 0.4865 (0.5,1,-0.5) 0.0833 0.5919

(0,0.5,0.5) 0.1979 0.5022 (0.5,-0.5,1) 0.2156 0.9332

(0.2,0.4,0.4) 0.1785 0.4193 (-0.5,0.5,1) 0.2683 0.9051

(0.4,0.2,0.4) 0.1680 0.4077 (0.4,0.4,0.2) 0.1503 0.2901

(0.99,0.024, “0.014) 0.1004 0.1896

Portfolio means (E (˜P )) and standard deviations ( Sdv (˜P )) are quoted in percent per annum”and based on historical

r r

data. Of course, your computations are only useful if you presume the history to be representative of the future,

too.

Figure 16.3. The S&P500, IBM, and Sony Portfolio Combinations From Table 16.3

0.20

(0,0.5,0.5)

(1,’1,1)

(0.5,0,0.5)

(0.5,0.5,0)

0.15

(0.5,’0.5,1)

(0,1,0)

Portfolio Mean

(0.25,1,’0.25)

0.10

(0.4,0.4,0.2)

(1,0,0)

(0.5,1,’0.5)

0.05

(1,0.5,’0.5)

(1,1,’1)

0.00

0.0 0.2 0.4 0.6 0.8 1.0

Portfolio Sdv

Numbers in parentheses are the investment weights in the three portfolio assets (S&P500, IBM, Sony). For example,

a $200 portfolio that invests $200 in S&P500, $200 in IBM, and “$200 in Sony is (+1,+1,-1), has a mean of about 1.3%

and a standard deviation of about 96%, which places it in the lower right corner.

¬le=optimalp¬o-g.tex: LP

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

Figure 16.4 plots 10,000 further portfolios, randomly chosen. The possible investment choices

Plotting the mean and

standard deviations of are no longer the single curve of points that they were with two assets, but a cloud of points.

many random portfolios

Nevertheless, the MVE Frontier”the upper left portion of the cloud of points”looks remark-

shows that the set of

ably similar to the hyperbolic shape it had with just two assets. You should never purchase

feasible portfolios

produces a cloud of

a portfolio that is a point inside the cloud but not on the frontier: there are better portfolios

points, with a distinct

with higher rewards and lower risks towards its upper left. The minimum variance portfo-

border: the MVE

lio is the portfolio with the lowest variance. It is mostly invested in the S&P500, although

Frontier.

a magnifying glass reveals that adding a little of IBM and shorting a little of Sony is slightly

better”speci¬cally, 99% of S&P500, 2.4% of IBM, and ’1.4% of Sony could reach as low a risk

as 18.96%.

Side Note: The technique for obtaining the MVE Frontier can remain the same for more than three securities:

select many possible portfolio combinations and plot the outcomes. However, the number of possible portfolio

weights quickly becomes overwhelming. If the portfolio optimization is done through such trial-and-error,

commonly called a Monte-Carlo simulation, it would be better to draw not just totally random portfolio weights,

but to draw portfolio weights that lie closer to the best portfolio combinations already obtained. In this case,

you can use a more convenient way to obtain the MVE Frontier, using matrix algebra. This is done in Nerd

Appendix a.

Figure 16.4. 10,000 Randomly Chosen Portfolios Involving Only S&P500, IBM, and Sony

0.30

(’1,1,1) o

r (’0.5,0.5,1) o

e

0.25

nti

(0.2,0.4,0.4)

Fro o

E

MV (0,0,1)

(0.4,0.2,0.4) o

0.20

o (0,0.5,0.5)

Portfolio Mean

o

o (1,’1,1)

o (0.5,0,0.5)

(0.5,0.5,0) o

0.15

o

o

(0.5,’0.5,1)

(0,1,0)

o

o (0.25,1,’0.25)

0.10

o (0.4,0.4,0.2)

(1,0,0) o (0.5,1,’0.5)

0.05

o

(1,0.5,’0.5)

(1,1,’1)

o

0.00

0.0 0.2 0.4 0.6 0.8 1.0

Portfolio Sdv

Numbers in parentheses are the investment weights in the three portfolio assets (S&P500, IBM, Sony). For example,

a $200 portfolio that invests $100 in S&P500, $200 in IBM, and “$100 in Sony (which is the point marked (0.5,1,“0.5))

has a mean of about 8% and a standard deviation of about 60%, which places it about in the middle of the grey area.

If you randomly drew another trillion portfolios, the entire region inside the hyperbola would be gray. That is, for

any mean/standard deviation combination that is inside the hyperbola, you can ¬nd a set of portfolio weights that

achieve it.

Forgive me for repeating myself, but you should keep in mind that this plot is ex-post, i.e., based

The historical MVE

frontier, usually used, is on historical data. Your real interest is of course not the past, but the future. Unfortunately,

not the same as what

you do not have a much better choice than to rely on history. In the real world, where you

you really want to know:

would use many more securities, the historical MVE Frontier would tend to be an indicator of

the future MVE frontier.

the future MVE Frontier, but not a perfect predictor thereof. Use common sense!

¬le=optimalp¬o-g.tex: RP

391

Section 16·1. The Mean-Variance E¬cient Frontier.

The historical portfolio mean estimates tend to be less reliable than the portfolio standard

Side Note:

deviation estimates. Perhaps not surprisingly, academic research has found that the minimum variance portfolio

(among broadly diversi¬ed portfolios) tends to do rather well”it often outperform other portfolio choices. That

is, not only is its risk often lower than that of other portfolios (as might have been expected), but even its actual

expected rate of return has often beaten its predicted expected rate of return (and other portfolios) out-of-

sample.

Solve Now!

Q 16.5 For the three stocks, in Excel, randomly draw 1,000 random investment weights into the

¬rst security, 1,000 weights into the second security, and compute 1 minus these two weights

to be the investment weight into the third security. Use the formulas and the following table to

compute the risk and reward of each of the 1,000 portfolios in two separate columns.

Covariance between ri and rj

˜ ˜

E (˜i )

r

Security 1 2 3

1 10.0% 0.010 0.001 0.003

2 5.0% 0.001 0.010 0.006

3 7.5% 0.003 0.006 0.010

Now plot the means and standard deviations of each portfolio in an X-Y graph.

Q 16.6 Continued: What is the Minimum Variance Portfolio?

Q 16.7 Continued: What is the best portfolio with an expected rate of return of about 11%?

Q 16.8 Continued: Would any investor purchase the portfolio (“12%,32%,81%) ?

¬le=optimalp¬o-g.tex: LP

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

16·2. Real-World Mean-Variance Ef¬cient Frontier Implemen-

tation Problems

Figure 16.4 also hints at the main practical drawback of MVE portfolios. Many points on the

Unfortunately, many

portfolios on the MVE MVE Frontier require shorting of securities (the weight is negative), but this may or may not be

Frontier require

possible in the real world. Even if modest shorting is possible, the portfolio optimization often

shorting securities, and

recommends strange portfolios that suggest shorting huge amounts in one security in order to

often massive amounts

so.

go long a huge amount in another security.

Chances are that this is not because these are great portfolios, but because the historical co-

Here is why.

variance and mean estimates are not perfect predictors of future covariances and means. The

historical covariance estimates cannot be easily relied upon, not so much because they are bad

in themselves in an absolute sense, but because the optimization technique is very sensitive

to any covariance estimation errors. This can be explained with a hypothetical bet optimizer:

assume that you throw a coin 100 times, and observed 51 heads. A naïve portfolio optimizer

relying on historical realizations would determine that the “heads bet” is much better than the

“tails bet,” and might recommend betting a million dollars on heads and against tails. This is

not because the bet optimizer has made a mistake”after all, if it were truly 51%, this would

be a pretty good bet. It is also not because 51% is a bad estimate of the true probability (of

50% if the coin is fair). Instead, it is the interaction: the bet optimizer is just very sensitive to

historical data and therefore sampling error.

There are some methods which try to address this problem in order to make portfolio opti-

There are two solutions

to the problem. mization a more useful tool. They rely on complicated statistical analysis, but simple ideas:

1. You can improve the estimates of future covariances and means, and not just rely blindly

on their historical equivalents. In essence, these improvements rely on techniques that

try to “pull in” extreme outlier returns. In the end, these techniques usually yield decent

variance and covariance estimates.

2. You can use a model, like the CAPM (to be discussed in the next chapter) to better estimate

means and variances.

3. You can use a portfolio optimizer that restricts the amount that can be shorted. This can

be done by assuming a (high) cost of shorting, or by disallowing short-sales altogether.

(See also the upcoming digging-deeper note.)

However, these techniques often still fail, especially when it comes to reliable expected rate of

return estimates. In any case, you are warned”you should not blindly believe that the historical

mean rates of return are representative of future mean rates of return. No one really knows how

to estimate future expected rates of return well. So, in the real world, you must use your own

judgment when the portfolio optimization result”your e¬cient frontier”seems reasonable

and when it does not.

¬le=optimalp¬o-g.tex: RP

393

Section 16·2. Real-World Mean-Variance E¬cient Frontier Implementation Problems.

Digging Deeper:

0.30

(’1,1,1) o

(’0.5,0.5,1) o

0.25

(0.2,0.4,0.4)

o

(0,0,1)

(0.4,0.2,0.4) o

0.20

o (0,0.5,0.5)

o

Mean

o (1,’1,1)

o (0.5,0,0.5)

(0.5,0.5,0) o

0.15

o

o

(0.5,’0.5,1)

(0,1,0)

o

o (0.25,1,’0.25)

0.10

o (0.4,0.4,0.2)

(1,0,0) o (0.5,1,’0.5)

o

0.05

(1,0.5,’0.5)

0.0 0.2 0.4 0.6 0.8 1.0

Sdv

The portfolios with only positive investment weights (no shorting) are highlighted in grey in the graph on the right.

It shows that an investor who is not permitted to short assets has a di¬erent MVE frontier. The short-constrained

MVE frontier usually lies entirely inside the unconstrained MVE frontier.

There is a closely related other problem with optimization. To obtain meaningful results, you There is not enough

data to estimate a good

need to have at least as many time periods (return observations) as there are terms in the

covariance matrix for

covariance formula, just as you would need at least as many equations as you have unknowns 10,000 securities.

to pin down a system of equations. Alas, with 10,000 securities, there are 50 million terms

in the covariance formula. If you use daily returns, you would have to wait 196 thousand

years to have just one data point per estimated covariance. (Mathematically, you could do

the estimation with 10,000 daily data points”still 40 years. These would be fairly unreliable,

of course.) Therefore, you cannot reliably estimate a good variance-covariance matrix with

historical data for too many assets.

Consequently, mean-variance optimization can only be used when there are just a few portfolios”It is better to use

preferably broad asset-class portfolios”to choose from. Fortunately, such broad asset-class portfolio optimization

on asset classes than on

portfolios also tend to have low and more reliable historical variances and covariances esti- many individual

mates. Unfortunately, you must narrow down your investment choices into a small number securities.

of asset class portfolios before you can use the MVE Frontier toolbox. So, in sum, portfolio

optimization is a very usable technique”as long as you restrict yourself to just a few big asset

classes for which you have good historical data. You should not use mean-variance optimiza-

tion with poor variance-covariance estimates. If you try to use it for individual stocks, and/or

when you do not have long historical returns data, and/or if you apply it blindly, chances are

that your results will not be very satisfying. Enhanced mean-variance optimization techniques,

such as those discussed in this section, are indeed in common use among professional investors.

For example, one class of hedge fund called a fund of funds invests itself in other hedge funds.

Many fund of funds ¬rst determine their hedge fund investment candidates, and then allocate

their money according to an estimated mean-variance frontier among their candidates.

¬le=optimalp¬o-g.tex: LP

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

16·3. Combinations of Portfolios on The Ef¬cient Frontier

If you purchase two di¬erent portfolios on the MVE Frontier, is the resulting portfolio still on

Any combination of MVE

portfolios is itself also the MVE Frontier? Put di¬erently, when two MVE investors marry, is their total portfolio still

MVE.

MVE?

Table 16.4. Two Base MVE Portfolios and Two Portfolio Combinations

E1 E2 E3 E4

Portfolio:

Weight in E1: +100% 0% +50% “25%

Weight in E2 : 0% +100% +50% +125%

S&P500 +99.448% “90.620% +4.414% “138.138%

(net

IBM +2.124% +135.770% +68.910% +169.088%

hol-

Sony “1.573% +54.925% +26.676% +69.049%

dings)

E (˜P )

r 10.000% 25.000% 17.500% 28.750%

Reward

Sdv (˜P )

r 18.969% 67.555% 37.498% 83.271%

Risk

Cov(˜E1 , rE2 ) = 0.03508

r ˜

Four speci¬c MVE portfolios are de¬ned by their relative investment weights into E1 = (99.448%, 2.124%, ’1.573%)

and E2 = (’90.620%, +135.770%, +54.925%) in S&P500, IBM, and Sony, respectively. They are named E1, E2, E3,

and E4 for convenience. P is a variable that can be any of these four portfolios.

Let™s try it with our three security example. I tell you two MVE portfolios, and you get to compute

The speci¬c portfolio

example (Figure 16.5). the expected rates of return and standard deviations of portfolios that are combinations of

these two. My part is in Table 16.4. Portfolio E1 is the minimum variance portfolio”you

already knew that it would be mostly S&P500, and we have already mentioned it on Page 390.

Portfolio E2 has a higher risk (67.6%) but also a higher expected rate of return (25%). Looking at

its components, E2 requires shorting a large amount in the S&P500, but in your perfect world

of zero transaction costs and under the assumption that the historical estimates are correct

estimates of the future, this is not a problem. Now, because you know the investment weights,

you can write down their twelve annual historical rates of return to compute the covariance

between E1 and E2 ”which would come to 0.035. You could also write down the historical

realizations of any weighted combination portfolio between E1 and E2 in order to compute the

new portfolio™s mean and standard deviation”or you can use the portfolio formulas, which is

much quicker. For example, the portfolio that invests half in E1 and half in E2 would have

E (˜E3 ) = 50% · 10% + 50% · 25% =

r 17.5%

V (˜E3 ) = (50%)2 · (18.969%)2 + (50%)2 · (67.555%)2 + 2 · (50%) · (50%) · 0.03408 = 0.1406 (16.8)

ar r

√

Sdv (˜E3 ) = =

r .

0.1406 37.498%

(If you do not recall the middle formula, look back to Formula 14.23 on Page 335.) Please

con¬rm the numbers for the E4 portfolio”or better yet, create a small spreadsheet that allows

you to get quick mean/standard deviation values for any weight wE1 = 1 ’ wE2 that you want

to try.

After you have computed many such portfolio combinations, you can overlay their means and

It appears visually as if

the combinations of two standard deviations onto Figure 16.4. This should give you something like Figure 16.5. Visually,

MVE portfolios is itself

it indeed appears as if combinations of the two MVE portfolios E1 and E2 “trace out” the entire

MVE.

MVE Frontier.

Visual con¬rmation is not a mathematical proof, but you can trust me that this is more general”

the answer to our original question is indeed yes.

¬le=optimalp¬o-g.tex: RP

395

Section 16·3. Combinations of Portfolios on The E¬cient Frontier.

Figure 16.5. S&P500, IBM, and Sony Portfolio Combinations

0.20

’100% in A 200% in B

’50% in A 150% in B

0.15

0% in A 100% in B

Portfolio Mean

50% in A 50% in B

0.10

100% in A 0% in B

0.05

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Portfolio Sdv

Portfolio E1 invests 99.448% in S&P500, 2.124% in IBM, and “1.573% in Sony. Portfolio E2 invests “90.62% in S&P500,

+135.77% in IBM, and +54.925% in Sony. Other portfolios along the e¬cient frontier are combinations of E1 and E2

. Points to the right of the “100% in E2 ” portfolio require shorting portfolio E1 in order to obtain the money to

purchase more of portfolio E2 .

Important: If two investors hold portfolios that are mean-variance e¬cient,

then the merged portfolio is also mean-variance e¬cient.

Unfortunately, it requires Nerd Appendices b and c to prove that the combination of two MVE The non-graphical

explanation.

Frontier portfolios is also MVE. However, although the proof itself is not important, the follow-

ing may give you an idea of how the proof works. You must believe me that the combination of

two securities always forms a nice hyperbola without kinks”this is actually what Formula 16.4

states. There are also only three parameters that pin down the hyperbola, and limit how di¬er-

ent inputs can stretch it. For convenience, my argument will work with the MVE-Frontier from

Figure 16.2. Assume that this MVE-Frontier is the solid line in Figure 16.2, and contains the two

MVE portfolios E1 and E2 . The question is whether combinations of E1 and E2 have to also

lie along the solid line. Let™s presume that they do not. The hyperbola can then extend beyond

the e¬cient frontier, as in graph A”but then, the combination of E1 and E2 would best the

original frontier. Or the hyperbola can lie inside the e¬cient frontier”but then shorting E2 to

purchase E1 would extend the hyperbola above the original MVE frontier and best the original

frontier. The only combinations of E1 and E2 that do not break the original MVE Frontier are

the portfolios that lie on the MVE Frontier. This is what we stated: the combination of two MVE

portfolios must itself be MVE.

¬le=optimalp¬o-g.tex: LP

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

Figure 16.6. Two MVE Portfolios

0.15

0.15

r, r,

ntie ntie

e fro e fro

ianc ianc

lie. lie.

’var ’var

E2 E2

The logical inconsistency: The logical inconsistency:

n n

and and

mea mea

s E1 s E1

true true

portfolios better than the frontier! portfolios better than the frontier!

folio folio

The The

port port

hich hich

on w on w

0.10

0.10

Expected Return

Expected Return

Pfio E 1 Pfio E 1

Possibility 2: Possibility 1:

hypothetical combinations of E 1 and E 2 hypothetical combinations of E 1 and E 2

if they did not lie on the frontier if they did not lie on the frontier

0.05

0.05

Pfio E 2 Pfio E 2

0.00

0.00

0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20

Standard Deviation Standard Deviation

(A) (B)

This ¬gure illustrates that if E1 and E2 lie on the MVE frontier, then their combinations must be MVE. A hyperbola is

de¬ned by three points. Two points are already pinned down by E1 and E2. A hyperbola whose minimum variance

portfolio is to the left of the current MVE would contradict the current MVE. A hyperbola whose minimum variance

portfolio is to the right of the current MVE would contradict the current MVE if E2 is shorted to buy more of E1.

Because the combination of MVE portfolios is MVE, your task of ¬nding the best portfolio”given

It is now easy to

determine other your speci¬c risk tolerance”is a lot easier. For example, if you want to know what the best

portfolios on the MVE

portfolio P is that o¬ers an expected rate of return of 15%, determine how much of portfolio E1

Frontier.

and how much of portfolio E2 you have to purchase in order to expect a rate of return of 15%.

The answer is

w = 2/3

wE1 · 10% + (1 ’ wE1 ) · 25% = 15% ”

(16.9)

wE1 · E (˜E1 ) + (1 ’ wE1 ) · E (˜E2 ) = E (˜P ) .

r r r

So, portfolio P should consist of 2/3 of portfolio E1 and therefore 1/3 of portfolio E2 . If you

wish, you can think of portfolios E1 and E2 as two funds, so you can use the method from

Section 12·3.B to compute the underlying net investment weights of portfolio E3:

E1 E2 E3

Portfolio:

2/3

Weight in E1 100% 0%

1/3

Weight in E2 0% 100%

S&P500 +99.448% “90.620% +36.092%

IBM +2.124% +135.770% +46.648%

Sony “1.573% +54.925% +17.260%

E (˜)

r 10.000% 25.000% 15.000%

Sdv (˜)

r 18.969% 67.555% 28.684%

P, which invests 36% in S&P500, 47% in IBM, and 17% in Sony, is the best portfolio that o¬ers

an expected rate of return of 15% per annum; it requires you to take on 29% risk per year.

Solve Now!

Q 16.9 Among the three stocks, what is the portfolio that yields an expected rate of return of

12.5% per year?

Q 16.10 Among the three stocks, what is the portfolio that yields an expected rate of return of

20% per year?

¬le=optimalp¬o-g.tex: RP

397

Section 16·4. The Mean-Variance E¬cient Frontier With A Risk-Free Security.

16·4. The Mean-Variance Ef¬cient Frontier With A Risk-Free

Security

If a risk-free security is available, it turns out that the portfolio optimization problem becomes Risk-free securities do

exist.

a lot easier. From a practical perspective, you can always assume that you can ¬nd a Treasury

bond that is essentially risk-free. So, you can proceed under the assumption that you have

access to one such security. This also means that you know already one MVE portfolio”after

all, you should not be able to get a higher interest rate for a risk-free investment”or you could

become rich.

Side Note: The special role applies only to the risk-free security, not to any other kind of bond. Although we

did discuss these risky bonds in the preceding parts of the book (Chapters 5), you should look at them the same

way you would look at any equity stock or fund you might purchase. They are just other risky components

inside your larger investment portfolio.

16·4.A. Risk-Reward Combinations of Any Portfolio Plus the Risk-Free Asset

So, what are the achievable risk-reward combinations if there is a risk-free security? For exam- It is very easy to

compute the expected

ple, consider portfolios from the combination of a risk-free security (called F) which pays 6%

rate of return and risk

for sure, with only one risky security (called R) which pays a mean rate of return of 10.56% and for a portfolio in which

has a standard deviation of 12%. The formulas for portfolio mean and standard deviation are one of the two securities

is risk-free.

E (˜P ) = (1 ’ wR ) · 6% + wR · 10.56%

r

= + wR · 4.56%

6%

(16.10)

E (˜P ) = (1 ’ wR ) · E (rF ) + wR · E (˜R )

r r

= + wR · E (˜R ) ’ rF

rF r .

Easy. More interestingly,

2 2 (16.11)

Sdv(˜P ) = wF · Var(rF ) + wR · Var(˜R ) + 2 · wF · wR · Cov(˜R , rF ) .

r r r

But a risk-free rate of return is just a constant rate of return, so it has no variance and no

covariance with anything else”it is what it is. Now combine your portfolio R with the risk-free

rate, purchasing wF in the risk-free asset and wR = (1 ’ wF ) in R.

2 2

Sdv(˜P ) = wF · Var(rF ) + wR · Var(˜R ) + 2 · wF · wR · Cov(˜R , rF )

r r r

(16.12)

2

= · Var(˜R ) = wR · Sdv(˜R ) = wR · 12%

wR r r .

This is nice: the standard deviation of the rate of return on any portfolio is simply the weight The combinations are a

straight line in the

times the standard deviation of the risky asset. This makes the calculations much easier. Fig-

mean-standard deviation

ure 16.7 uses Formulas 16.10 and 16.12 to plot the mean and standard deviation of portfolio graph.

pairs for your particular security R. Note how all portfolio combinations lie on a straight line,

which starts at the risk-free rate (at Sdv=0%, E=6%) and goes through the risky portfolio R (at

Sdv=12%, E=10.56%).

It is easy to show that the combination of portfolios that consist of the risk-free security You can prove the

linearity, too!

and a risky security have risk/reward trade-o¬s that lie on a straight line. Solve for wR

¬le=optimalp¬o-g.tex: LP

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

Figure 16.7. Combinations of the Risk-Free Security (F) and a Risky Asset (R)

1/4 1/2 3/4 3/2

Risky Weight 0 1

Mean (Expected Rate of Return) 6% 7.14% 8.28% 9.42% 10.56% 12.84%

Standard Deviation 0% 3% 6% 9% 12% 18%

0.12

0.10

Expected Rate of Return

0.08

Security R

0.06

0.04

Risk’free Security F

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Standard Deviation of Rate of Return

(= Sdv(˜P )/Sdv(˜R )) in the standard deviation formula 16.12, and substitute it into the ex-

r r

pected return formula 16.10,

Sdv(˜P )

r

E (˜P ) = rF + wR · E (˜R ) ’ rF = rF + · E (˜R ) ’ rF

r r r

Sdv(˜R )

r

E (˜R ) ’ rF

r

= rF + · Sdv(˜P )

r

Sdv(˜R )

r

10.56% ’ 6% (16.13)

= 6% + · Sdv(˜P )

r

12%

= 6% + 38% · Sdv(˜P )

r

=a+ b · Sdv(˜P )

r .

This is the equation for a line, where a (= rF = 6%, the risk-free rate) is the intercept and

b (= [E(˜R ) ’ rF ]/Sdv(˜R ) = 38%) is the slope. You have actually already encountered the

r r

slope in Formula 16.13 on Page 374! It is the Sharpe Ratio,

E (˜R ) ’ rF

r

b = Sharpe Ratio ≡ (16.14)

.

Sdv(˜R )

r

For a given portfolio R, [E(˜R ) ’ rF ]/Sdv(˜R ) is the ratio of the expected reward over the

r r

required risk. (Theoretically, you can also compute the Sharpe ratio by ¬rst subtracting the

risk-free rate from all returns, called excess returns, and then dividing its mean by its standard

deviation, because Sdv(˜R ) = Sdv(˜R ’ rF ). This is the more common practical way to compute

r r

the Sharpe ratio.) Please be aware that although the Sharpe ratio is a common measure of the

¬le=optimalp¬o-g.tex: RP

399

Section 16·4. The Mean-Variance E¬cient Frontier With A Risk-Free Security.

risk-reward trade-o¬ of individual portfolios, it is not a good one.

16·4.B. The Best Risk-Reward Combinations With A Risk-Free Asset

Figure 16.8. Combinations of The Risk-Free Security (F) And a Risky Asset (R)

0.12

0.10

Expected Rate of Return

0.08

Security R

0.06

0.04

Risk’free Security F

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Standard Deviation of Rate of Return

The MVE among risky assets is based on the example from Table 16.2 with a covariance of ’0.0075. (This means

the ¬rst security has a reward of 5% and a risk of 10%; the second has a reward of 10% and a risk of 10%.)

Now return to the e¬cient frontier from Table 16.2 and Figure 16.2, and assume the covariance Would you still purchase

a combination of R, if

is ’0.0075. Portfolio R happens to be among the portfolios that you can purchase, combining

you could instead

E1 and E2 . Figure 16.8 repeats the last graph, but also plots the other securities you could purchase any other

purchase, putting together two base securities, E1 and E2 . This raises an interesting question: combination of E1 and

E2 ?

Presume that you have drawn the MVE Frontier using all risky assets. (It will always looks like

a sideways hyperbola, so the drawn ¬gure is accurate even for the more general case.) Now a

risk-free asset becomes available, so that you can draw a line between the risk-free rate and

whatever portfolio you want to purchase. How does your new achievable MVE Frontier look like?

Would you ever purchase portfolio R in Figure 16.8, which is on the previous MVE Frontier (or

any combination of the risk-free security and portfolio R)?

No! You can do better than this. Figure 16.9 draws the “tangency portfolio,” T. By combining The true MVE Frontier is

a line from the risk-free

portfolio T and the risk-free asset, you can do better than purchasing even a tiny bit of R. In

rate to the “tangency”

fact, the ¬gure shows that you would draw a line from the risk-free asset to this Tangency portfolio.

Portfolio (T)”which is the true MVE frontier”and you would only purchase combinations of

these two. No risky portfolio other than T would ever be purchased by any rational investor.

Important: If there is a risk-free security, every investor would purchase a

combination of the risk-free security and the Tangency Portfolio. No combination

of risky assets other than the Tangency Portfolio T would ever be purchased by

any investor.

¬le=optimalp¬o-g.tex: LP

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

Figure 16.9. The Mean-Variance E¬cient Frontier With a Risk-Free Asset

0.12

0.10

Tangency Portfolio T

Expected Rate of Return

Better than R

0.08

Another Risky

Portfolio R

0.06

0.04

Risk’free Security F

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Standard Deviation of Rate of Return

Note how you can do better if you buy a combination of T and the risk-free asset instead of portfolio R. It follows

that you would never purchase risky assets in any other weight combination than that packed into the tangency

portfolio T.

16·4.C. The Formula to Determine the Tangency Portfolio

It is not di¬cult to ¬gure out the tangency portfolio. It is the portfolio on the MVE Frontier

that has the maximum slope, i.e., the highest Sharpe ratio. Let us ¬nd the tangency portfolio

for our three by-now familiar stocks. Recall the ¬rst three MVE portfolios for S&P500, IBM, and

Sony from Table 16.4 on Page 394. Recall that

E1 E2 E3

Portfolio:

Weight in E1: +100% 0% +50%

Weight in E2 : 0% +100% +50%

S&P500 +99.448% “90.620% +4.414%

IBM +2.124% +135.770% +68.910%

Sony “1.573% +54.925% +26.676%

E (˜P )

r 10.000% 25.000% 17.500%

Sdv (˜P )

r 18.969% 67.555% 37.498%

Sharpe Ratio

10% ’ 6% 25% ’ 6% 17.5% ’ 6%

@ rF = 6% = 0.21 = 0.28 = 0.31

19% 67.6% 37.5%

First, you should determine the covariance between two e¬cient frontier portfolios”here be-

The Agenda. Work out

the Sharpe ratio, so tween E1 and E2 . You can do this either by computing the twelve historical portfolio re-

mean ¬rst, covariance

turns and compute it, or you can read the nerd note below, or you can just trust me that the

next, and standard

Cov(˜E1 , rE2 ) = 0.03505. You also know that you can trace out the MVE Frontier by purchasing

r˜

deviation last.

any combinations of these three portfolios. Your goal is to ¬nd, for a given risk-free interest

¬le=optimalp¬o-g.tex: RP

401

Section 16·4. The Mean-Variance E¬cient Frontier With A Risk-Free Security.

rate, the tangency portfolio. First, write down again the risk and reward of portfolios of E1

and E2. For any portfolio P itself comprised of E1 and E2,

E (˜P ) = wE1 · 10% + (1 ’ wE1 ) · 25% = 25% ’ 15% · wE1

r

(16.15)

= wE1 · E (˜E1 ) + (1 ’ wE1 ) · E (˜E2 ) .

r r