<< стр. 20(всего 39)СОДЕРЖАНИЕ >>
Digging Deeper:
substituting out wA from Formula 16.1 (portfolio PвЂ™s mean) and Formula 16.2 (portfolio PвЂ™s standard deviation).
The result is

(16.3)
Sdv [ E (ЛњP ) ] = В± a В· E (ЛњP )2 + b В· E (ЛњP ) + c ,
r r r

where

dв€’1 В· V (ЛњA ) в€’ 2 В· Cov(ЛњA , rB ) + V (ЛњB )
aв‰Ў ar r rЛњ ar r ,

dв€’1 В· 2 В· E (ЛњA ) В· [Cov(ЛњA , rB ) в€’ V (ЛњB )]
bв‰Ў r rЛњ ar r

dв€’1 В· 2 В· E (ЛњB ) В· [Cov(ЛњA , rB ) в€’ V (ЛњA )]
+ r rЛњ ar r ,
(16.4)
в€’1 2 2
cв‰Ў В· E (ЛњB ) В· V (ЛњA ) + E (ЛњA ) В· V (ЛњB )
d r ar r r ar r

dв€’1 В· 2 В· E (ЛњB ) В· E (ЛњA ) В· Cov(ЛњA , rB )
в€’ r r rЛњ ,
2
dв‰Ў E (ЛњA ) в€’ E (ЛњB )
r r .

This formula states that the variance of the rate of return (Sdv (ЛњP )2 ) on an arbitrary weighted portfolio P is a
r
quadratic formula in its expected rate of return (E (ЛњP )): 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 eп¬ѓcient 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 hyperbolaвЂ”the combination of two particular portfolios. This only breaks down when there are additional
constraints, such as short-sales constraints.

16В·1.B. Diп¬Ђerent Covariance Scenarios

Table 16.2. Portfolio Performance Under Diп¬Ђerent Covariance Scenarios.

Correlation is
в€’1.00 в€’0.75 +0.75 +1.00
0.00

Risk (Sdv ) if Covariance is
E (ЛњP ) в€’0.01 в€’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:
в€’1 +2 0.0% 30.0% 28.3% 22.4% 14.1% 10.0%
+2 в€’1 15.0% 30.0% 28.3% 22.4% 14.1% 10.0%

This п¬Ѓgure shows the risk and reward under diп¬Ђerent assumptions about the covariance be-
tween A and B. The п¬Ѓrst two lines give the characteristics of the two base securities, A and B.
(wB is always 1 в€’ wA , and thus could have been omitted.)
п¬Ѓle=optimalpп¬Ѓo-g.tex: LP
386 Chapter 16. The Eп¬ѓcient FrontierвЂ”Optimally Diversiп¬Ѓed Portfolios.

How do diп¬Ђerent asset correlations between securities A and B change the shape of the eп¬ѓcient
How do efп¬Ѓcient
frontiers look like with frontier. Table 16.2 is really the same as Table 16.1, except it works out diп¬Ђerent 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 diп¬Ђerent axes.
Of course, regardless of covariance, when you choose only one or the other security (wA = 0
Higher correlations
вЂњcompressвЂќ 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 diп¬Ђerent
covariance scenarios, shown in Table 16.2 and Figure 16.2, you can verify your earlier insight
that lower covariance helps diversiп¬Ѓcation. For example, if the two securities are perfectly
negatively correlated, which implies a covariance of в€’0.01, then an equal-weighted portfolio of
the two securities has zero risk. (When one securityвЂ™s value increases, the other securityвЂ™s value
decreases by an equal amount, thereby eradicating any risk.) If the two securities are perfectly
positively correlated, diversiп¬Ѓcation 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 eп¬Ђective.
Solve Now!
Q 16.1 In the example in Table 16.2, if the covariance is в€’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 portfoliosвЂ™ 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.

16В·1.C. The Mean-Variance Eп¬ѓcient 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 FrontierвЂ”which is still the upper half of the hyperbola.
Try out many random investment portfolios from the three stocks from Page 307вЂ”S&P500,
stocks, S&P500, IBM, IBM, SonyвЂ”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
п¬Ѓle=optimalpп¬Ѓo-g.tex: RP
387
Section 16В·1. The Mean-Variance Eп¬ѓcient Frontier.

Figure 16.2. The Risk-Reward Trade-oп¬Ђ With Diп¬Ђerent Correlations

(a) Correlation = в€’1.00 (Cov = в€’0.01) (b) Correlation = в€’0.75 (Cov = в€’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 п¬Ѓgures repeat Figure 16.1, except that they consider diп¬Ђerent 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 inп¬Ѓnite expected rate
of return. This cannot happen in a reasonable market.)
п¬Ѓle=optimalpп¬Ѓo-g.tex: LP
388 Chapter 16. The Eп¬ѓcient FrontierвЂ”Optimally Diversiп¬Ѓed Portfolios.

Covariance between ri and rj
Лњ Лњ
E (Лњi )
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 deп¬Ѓned (Cov(x, x) в‰Ў
Var(x)). What would the risk-reward trade-oп¬Ђs 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 (ЛњP ) = w1 В· E (Лњ1 ) + w2 В· E (Лњ2 ) + w3 В· E (Лњ3 )
r r r r ,
2 2 2
Var(ЛњP ) = w1 В· Var(Лњ1 ) + w2 В· Var(Лњ2 ) + w3 В· Var(Лњ3 )
r r r r

(16.5)
+ 2 В· w1 В· w2 В· Cov(Лњ1 , r2 ) + 2 В· w1 В· w3 В· Cov(Лњ1 , r3 )
rЛњ rЛњ

+ 2 В· w2 В· w3 В· Cov(Лњ2 , r3 )
rЛњ ,

and w3 в‰Ў 1 в€’ w1 в€’ 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 (diп¬Ђerent 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 (ЛњP ) в‰€ 20% В· 10.110% + 40% В· 15.379% + 40% В· 24.203% в‰€ 17.85% ,
r
(16.6)
E (ЛњP ) = w1 В· E (Лњ1 ) + w2 В· E (Лњ2 ) + w3 В· E (Лњ3 )
r r r r ,

and the variance is

Var(ЛњP ) в‰€ (20%)2 В· 3.6224% + (40%)2 В· 15.0345% + (40%)2 В· 81.4886%
r

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

+ 2 В· (40%) В· (40%) В· 2.1842% в‰€ 0.1758
(16.7)
2 2 2
Var(ЛњP ) = В· Var(Лњ1 ) + В· Var(Лњ2 ) + В· Var(Лњ3 )
r w1 r w2 r w3 r

+ 2 В· w1 В· w2 В· Cov(Лњ1 , r2 ) + 2 В· w1 В· w3 В· Cov(Лњ1 , r3 )
rЛњ rЛњ

+ 2 В· w2 В· w3 В· Cov(Лњ2 , r3 )
rЛњ .

Therefore, the risk of the portfolio is Sdv(ЛњP ) в‰€ 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).
п¬Ѓle=optimalpп¬Ѓo-g.tex: RP
389
Section 16В·1. The Mean-Variance Eп¬ѓcient Frontier.

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

Weights Weights
E (ЛњP ) Sdv (ЛњP ) E (ЛњP ) Sdv (ЛњP )
(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, вЂ“0.014) 0.1004 0.1896

Portfolio means (E (ЛњP )) and standard deviations ( Sdv (ЛњP )) are quoted in percent per annumвЂ”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,в€’1,1)
(0.5,0,0.5)
(0.5,0.5,0)
0.15

(0.5,в€’0.5,1)
(0,1,0)
Portfolio Mean

(0.25,1,в€’0.25)
0.10

(0.4,0.4,0.2)
(1,0,0)
(0.5,1,в€’0.5)
0.05

(1,0.5,в€’0.5)

(1,1,в€’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 вЂ“\$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.
п¬Ѓle=optimalpп¬Ѓo-g.tex: LP
390 Chapter 16. The Eп¬ѓcient FrontierвЂ”Optimally Diversiп¬Ѓed 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 FrontierвЂ”the upper left portion of the cloud of pointsвЂ”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
betterвЂ”speciп¬Ѓcally, 99% of S&P500, 2.4% of IBM, and в€’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

(в€’1,1,1) o

r (в€’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,в€’1,1)
o (0.5,0,0.5)
(0.5,0.5,0) o
0.15

o
o
(0.5,в€’0.5,1)
(0,1,0)
o
o (0.25,1,в€’0.25)
0.10

o (0.4,0.4,0.2)
(1,0,0) o (0.5,1,в€’0.5)
0.05

o
(1,0.5,в€’0.5)
(1,1,в€’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 вЂ“\$100 in Sony (which is the point marked (0.5,1,вЂ“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 п¬Ѓnd 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!
п¬Ѓle=optimalpп¬Ѓo-g.tex: RP
391
Section 16В·1. The Mean-Variance Eп¬ѓcient 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 diversiп¬Ѓed portfolios) tends to do rather wellвЂ”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
п¬Ѓrst 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
Лњ Лњ
E (Лњi )
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 (вЂ“12%,32%,81%) ?
п¬Ѓle=optimalpп¬Ѓo-g.tex: LP
392 Chapter 16. The Eп¬ѓcient FrontierвЂ”Optimally Diversiп¬Ѓed Portfolios.

16В·2. Real-World Mean-Variance Efп¬Ѓcient 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 naГЇve portfolio optimizer
relying on historical realizations would determine that the вЂњheads betвЂќ is much better than the
вЂњtails bet,вЂќ and might recommend betting a million dollars on heads and against tails. This is
not because the bet optimizer has made a mistakeвЂ”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 вЂњpull inвЂќ 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.

However, these techniques often still fail, especially when it comes to reliable expected rate of
return estimates. In any case, you are warnedвЂ”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 resultвЂ”your eп¬ѓcient frontierвЂ”seems reasonable
and when it does not.
п¬Ѓle=optimalpп¬Ѓo-g.tex: RP
393
Section 16В·2. Real-World Mean-Variance Eп¬ѓcient Frontier Implementation Problems.

Digging Deeper:

0.30
(в€’1,1,1) o

(в€’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,в€’1,1)
o (0.5,0,0.5)
(0.5,0.5,0) o
0.15

o
o
(0.5,в€’0.5,1)
(0,1,0)
o
o (0.25,1,в€’0.25)
0.10

o (0.4,0.4,0.2)
(1,0,0) o (0.5,1,в€’0.5)

o
0.05

(1,0.5,в€’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 diп¬Ђerent 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 pointsвЂ”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 portfoliosвЂ”It is better to use
preferably broad asset-class portfoliosвЂ”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 techniqueвЂ”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 п¬Ѓrst determine their hedge fund investment candidates, and then allocate
their money according to an estimated mean-variance frontier among their candidates.
п¬Ѓle=optimalpп¬Ѓo-g.tex: LP
394 Chapter 16. The Eп¬ѓcient FrontierвЂ”Optimally Diversiп¬Ѓed Portfolios.

16В·3. Combinations of Portfolios on The Efп¬Ѓcient Frontier

If you purchase two diп¬Ђerent portfolios on the MVE Frontier, is the resulting portfolio still on
Any combination of MVE
portfolios is itself also the MVE Frontier? Put diп¬Ђerently, 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% вЂ“25%
Weight in E2 : 0% +100% +50% +125%
S&P500 +99.448% вЂ“90.620% +4.414% вЂ“138.138%
(net

IBM +2.124% +135.770% +68.910% +169.088%
hol-

Sony вЂ“1.573% +54.925% +26.676% +69.049%
dings)

E (ЛњP )
r 10.000% 25.000% 17.500% 28.750%
Reward

Sdv (ЛњP )
r 18.969% 67.555% 37.498% 83.271%
Risk

Cov(ЛњE1 , rE2 ) = 0.03508
r Лњ

Four speciп¬Ѓc MVE portfolios are deп¬Ѓned by their relative investment weights into E1 = (99.448%, 2.124%, в€’1.573%)
and E2 = (в€’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.

LetвЂ™s try it with our three security example. I tell you two MVE portfolios, and you get to compute
The speciп¬Ѓc 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 portfolioвЂ”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 вЂ”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 portfolioвЂ™s mean and standard deviationвЂ”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 (ЛњE3 ) = 50% В· 10% + 50% В· 25% =
r 17.5%

V (ЛњE3 ) = (50%)2 В· (18.969%)2 + (50%)2 В· (67.555%)2 + 2 В· (50%) В· (50%) В· 0.03408 = 0.1406 (16.8)
ar r
в€љ
Sdv (ЛњE3 ) = =
r .
0.1406 37.498%

(If you do not recall the middle formula, look back to Formula 14.23 on Page 335.) Please
conп¬Ѓrm the numbers for the E4 portfolioвЂ”or better yet, create a small spreadsheet that allows
you to get quick mean/standard deviation values for any weight wE1 = 1 в€’ 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 вЂњtrace outвЂќ the entire
MVE.
MVE Frontier.
Visual conп¬Ѓrmation is not a mathematical proof, but you can trust me that this is more generalвЂ”
the answer to our original question is indeed yes.
п¬Ѓle=optimalpп¬Ѓo-g.tex: RP
395
Section 16В·3. Combinations of Portfolios on The Eп¬ѓcient Frontier.

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

0.20
в€’100% in A 200% in B

в€’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 вЂ“1.573% in Sony. Portfolio E2 invests вЂ“90.62% in S&P500,
+135.77% in IBM, and +54.925% in Sony. Other portfolios along the eп¬ѓcient frontier are combinations of E1 and E2
. Points to the right of the вЂњ100% in E2 вЂќ 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 eп¬ѓcient,
then the merged portfolio is also mean-variance eп¬ѓcient.

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 kinksвЂ”this is actually what Formula 16.4
states. There are also only three parameters that pin down the hyperbola, and limit how diп¬Ђer-
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. LetвЂ™s presume that they do not. The hyperbola can then extend beyond
the eп¬ѓcient frontier, as in graph AвЂ”but then, the combination of E1 and E2 would best the
original frontier. Or the hyperbola can lie inside the eп¬ѓcient frontierвЂ”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.
п¬Ѓle=optimalpп¬Ѓo-g.tex: LP
396 Chapter 16. The Eп¬ѓcient FrontierвЂ”Optimally Diversiп¬Ѓed Portfolios.

Figure 16.6. Two MVE Portfolios

0.15

0.15
r, r,
ntie ntie
e fro e fro
ianc ianc
lie. lie.
в€’var в€’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 п¬Ѓgure illustrates that if E1 and E2 lie on the MVE frontier, then their combinations must be MVE. A hyperbola is
deп¬Ѓned 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 п¬Ѓnding the best portfolioвЂ”given
It is now easy to
determine other your speciп¬Ѓc risk toleranceвЂ”is a lot easier. For example, if you want to know what the best
portfolios on the MVE
portfolio P is that oп¬Ђers 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%.
w = 2/3
wE1 В· 10% + (1 в€’ wE1 ) В· 25% = 15% в‡”
(16.9)
wE1 В· E (ЛњE1 ) + (1 в€’ wE1 ) В· E (ЛњE2 ) = E (ЛњP ) .
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 12В·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% вЂ“90.620% +36.092%
IBM +2.124% +135.770% +46.648%
Sony вЂ“1.573% +54.925% +17.260%
E (Лњ)
r 10.000% 25.000% 15.000%
Sdv (Лњ)
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 oп¬Ђers
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?
п¬Ѓle=optimalpп¬Ѓo-g.tex: RP
397
Section 16В·4. The Mean-Variance Eп¬ѓcient Frontier With A Risk-Free Security.

16В·4. The Mean-Variance Efп¬Ѓcient 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 п¬Ѓnd a Treasury
bond that is essentially risk-free. So, you can proceed under the assumption that you have
all, you should not be able to get a higher interest rate for a risk-free investmentвЂ”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

16В·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 (ЛњP ) = (1 в€’ wR ) В· 6% + wR В· 10.56%
r

= + wR В· 4.56%
6%
(16.10)
E (ЛњP ) = (1 в€’ wR ) В· E (rF ) + wR В· E (ЛњR )
r r

= + wR В· E (ЛњR ) в€’ rF
rF r .

Easy. More interestingly,

2 2 (16.11)
Sdv(ЛњP ) = wF В· Var(rF ) + wR В· Var(ЛњR ) + 2 В· wF В· wR В· Cov(ЛњR , 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 elseвЂ”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 в€’ wF ) in R.

2 2
Sdv(ЛњP ) = wF В· Var(rF ) + wR В· Var(ЛњR ) + 2 В· wF В· wR В· Cov(ЛњR , rF )
r r r
(16.12)
2
= В· Var(ЛњR ) = wR В· Sdv(ЛњR ) = wR В· 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-oп¬Ђs that lie on a straight line. Solve for wR
п¬Ѓle=optimalpп¬Ѓo-g.tex: LP
398 Chapter 16. The Eп¬ѓcient FrontierвЂ”Optimally Diversiп¬Ѓed 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

Riskв€’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(ЛњP )/Sdv(ЛњR )) in the standard deviation formula 16.12, and substitute it into the ex-
r r
pected return formula 16.10,
Sdv(ЛњP )
r
E (ЛњP ) = rF + wR В· E (ЛњR ) в€’ rF = rF + В· E (ЛњR ) в€’ rF
r r r
Sdv(ЛњR )
r
E (ЛњR ) в€’ rF
r
= rF + В· Sdv(ЛњP )
r
Sdv(ЛњR )
r
10.56% в€’ 6% (16.13)
= 6% + В· Sdv(ЛњP )
r
12%

= 6% + 38% В· Sdv(ЛњP )
r

=a+ b В· Sdv(ЛњP )
r .

This is the equation for a line, where a (= rF = 6%, the risk-free rate) is the intercept and
b (= [E(ЛњR ) в€’ rF ]/Sdv(ЛњR ) = 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 (ЛњR ) в€’ rF
r
b = Sharpe Ratio в‰Ў (16.14)
.
Sdv(ЛњR )
r

For a given portfolio R, [E(ЛњR ) в€’ rF ]/Sdv(ЛњR ) is the ratio of the expected reward over the
r r
required risk. (Theoretically, you can also compute the Sharpe ratio by п¬Ѓrst subtracting the
risk-free rate from all returns, called excess returns, and then dividing its mean by its standard
deviation, because Sdv(ЛњR ) = Sdv(ЛњR в€’ 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
п¬Ѓle=optimalpп¬Ѓo-g.tex: RP
399
Section 16В·4. The Mean-Variance Eп¬ѓcient Frontier With A Risk-Free Security.

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

16В·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

Riskв€’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 в€’0.0075. (This means
the п¬Ѓrst 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 eп¬ѓcient frontier from Table 16.2 and Figure 16.2, and assume the covariance Would you still purchase
a combination of R, if
is в€’0.0075. Portfolio R happens to be among the portfolios that you can purchase, combining
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 п¬Ѓgure 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 вЂњtangency portfolio,вЂќ 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 вЂњtangencyвЂќ
fact, the п¬Ѓgure shows that you would draw a line from the risk-free asset to this Tangency portfolio.
Portfolio (T)вЂ”which is the true MVE frontierвЂ”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.
п¬Ѓle=optimalpп¬Ѓo-g.tex: LP
400 Chapter 16. The Eп¬ѓcient FrontierвЂ”Optimally Diversiп¬Ѓed Portfolios.

Figure 16.9. The Mean-Variance Eп¬ѓcient 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

Riskв€’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.

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

It is not diп¬ѓcult to п¬Ѓgure 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 п¬Ѓnd the tangency portfolio
for our three by-now familiar stocks. Recall the п¬Ѓrst 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% вЂ“90.620% +4.414%
IBM +2.124% +135.770% +68.910%
Sony вЂ“1.573% +54.925% +26.676%
E (ЛњP )
r 10.000% 25.000% 17.500%
Sdv (ЛњP )
r 18.969% 67.555% 37.498%
Sharpe Ratio
10% в€’ 6% 25% в€’ 6% 17.5% в€’ 6%
@ rF = 6% = 0.21 = 0.28 = 0.31
19% 67.6% 37.5%

First, you should determine the covariance between two eп¬ѓcient frontier portfoliosвЂ”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 п¬Ѓrst, 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(ЛњE1 , rE2 ) = 0.03505. You also know that you can trace out the MVE Frontier by purchasing
rЛњ
deviation last.
any combinations of these three portfolios. Your goal is to п¬Ѓnd, for a given risk-free interest
п¬Ѓle=optimalpп¬Ѓo-g.tex: RP
401
Section 16В·4. The Mean-Variance Eп¬ѓcient 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 (ЛњP ) = wE1 В· 10% + (1 в€’ wE1 ) В· 25% = 25% в€’ 15% В· wE1
r
(16.15)
= wE1 В· E (ЛњE1 ) + (1 в€’ wE1 ) В· E (ЛњE2 ) .
r r
 << стр. 20(всего 39)СОДЕРЖАНИЕ >>