<<

. 18
( 39)



>>

(14.55)
Sdv(˜B ) = Var(˜B ) = 0.002496 = 5.00% .
r r

Note that this standard deviation is lower than the standard deviation of each of the three
stocks by themselves (Table 14.3). This is caused by “diversi¬cation,” which is explored in
great detail in the next chapter.
Solve Now!
Q 14.9 Compute the standard deviation (risk) of portfolio B from Table 14.2 to 4 digits. Is it the
same as what we computed in the text?


Q 14.10 Compute the expected rate of return (reward) and standard deviation (risk) for another
portfolio, called BF, that invests 10.7% in the S&P500, 64.5% in IBM and 24.8% in Sony. Compute
this both from the formulas and from a historical rate of return series of this portfolio.


Q 14.11 Compute the risk and reward for portfolio CF:

(14.56)
wS&P500 = 0.5, wIBM = 0.5, wSony = 0 .




Q 14.12 Compute the risk and reward for portfolio DF:

(14.57)
wS&P500 = 0.614, wIBM = 0.288, wSony = 0.98 .




Q 14.13 What is the risk and reward of a portfolio EF that invests 10% in PEP, 10% in KO, and
80% in CSG?


Q 14.14 What is the slope (beta) of the lines in Figure 14.1?
¬le=statsp¬os-g.tex: RP
345
Section 14·3. Historical Statistics For Some Asset-Class Index Portfolios.

14·3. Historical Statistics For Some Asset-Class Index Port-
folios

Enough with statistical torture! Let™s look at the historical performances of some realistic This section describes
the 1997-2002
investment portfolios. Table 14.4 describes historical rates of return from portfolios managed
investment performance
by Vanguard over the January 1997 to October 2002 period. (Vanguard is a prominent low- of some Vanguard funds.
cost provider of index funds.) Each Vanguard fund purchases a large number of securities,
often simply everything that quali¬es in an asset category (e.g., all suitable bonds, all European
stocks, all real estate investment trusts, etc.), and without much attempt at picking winners
within each class. Naturally, although you really are interested in forward-looking statistics,
standing here today, all you have are historical statistics. So, let us look at the properties of
historical rates of return of these portfolios.

Experience shows that historical means are not good predictors of future means, but histor-
Side Note:
ical standard deviations and betas are good predictors of their future equivalents. See the local Nerd Ap-
pendix 13·6.A.


The ¬rst statistic that this chapter described was the mean. The historical mean (also called Bonds offered only low
rates of return.
sample mean) of the monthly rates of return describes how you would have fared on average.
The second column in Table 14.4 shows that over the sample period, the short-term govern-
ment bond fund earned a rate of return of about 60 basis points (per year ). The intermediate
government bond fund earned 100 basis points; municipal bonds earned 80 basis points; cor-
porate junk bonds earned 180 basis points; corporate convertible bonds earned about 2.4%;
and so on.


Part IV explains corporate bonds in more detail.



Continuing on to pure equity (stock) investments, you can see that the 500 large stocks in Equities offered higher
rates of return.
Vanguard™s S&P500 fund earned about 4.9% per year. The tax-managed version of the same
investment strategy minimizes trading (to minimize capital gains). It did even better than the
unmanaged version, earning a rate of return of 6.0% per year. Value ¬rms are large and unex-
citing companies, and growth ¬rms are small, fast-growing, exciting companies. Yet, following
a long-standing historical trend, value ¬rms earned higher rates of return than growth ¬rms.
(During this particular sample period, small growth ¬rms did best, though.) Neither could
outperform the S&P500 over the sample period. Among industries, health care ¬rms earned
the highest rates of return, and utilities ¬rms earned the lowest rates of return. International
investors fared especially poorly in this sample period: Japanese stocks in particular lost 7.4%
per annum over the sample period.

Side Note: The Vanguard S&P500 index portfolio outperformed the S&P500 index itself (second to last line
in the table). This is presumably [a] because the fund also received dividends, which are not counted in the
S&P index itself; and [b] because the fund could lend out securities to short-sellers and thereby earn some extra
return. Incidentally, to reduce transaction costs, index funds usually do not hold the 500 stocks in the exact
right proportion, which causes further tracking error”a deviation of the performance of the index fund™s rate
of return from the index percent change.


The second set of statistics that this chapter described included the standard deviation. The Equities were riskier
than bonds.
third column in Table 14.4 shows that short-term and medium-term government bonds and
municipal bonds were exceptionally safe. Their monthly rates of return varied only a little
over the sample period. A large cluster of investment strategies had risks of about 15% to 25%
per year, including the overall S&P500 stock market index investment strategy. The riskiest
stock market investment strategy in the sample period would have been U.S. Gold and Metals,
whereas the safest would have been Real Estate Investment Trust (REITs) and Utilities ¬rms.
346
Table 14.4. Asset Class Portfolios, Based on Monthly Rates of Returns, Annualized, January 1997 to October 2002.

Annualized Returns Monthly Returns Ann.Market-Model
±i βi,S&P
Asset Mean StdDev %Neg %Pos Worst Q2 Median Q3 Best
’0.0
govbonds: short-term 0.6% 2.0% 0.7%
’1.5% ’0.3%
43% 51% 0.1% 0.4% 1.2%
govbonds: intermediate 1.0% 4.4% 1.2% -0.0
’3.2% ’0.7%
44% 53% 0.2% 0.8% 2.9%
bond: long-term munis 0.8% 4.5% 1.0% -0.0
’3.1% ’1.1%
43% 54% 0.4% 0.9% 2.7%
’1.5%
bond: high-yield (junk) bonds 1.8% 16.2% 0.7
’15.0% ’2.3%
46% 54% 0.4% 2.9% 9.9%
’0.5%
bond: corporate convertibles 2.4% 15.4% 0.6
’12.8% ’2.7% ’0.2%
50% 49% 3.2% 10.6%
u.s. s&p500 4.9% 18.5% 0.1% 1.0
’14.5% ’3.0%
46% 54% 0.7% 4.7% 9.5%
u.s. tax managed 6.0% 21.3% 0.6% 1.1
’17.6% ’3.4%
46% 54% 0.4% 5.2% 9.6%
’2.7%
u.s. value ¬rms 1.6% 18.1% 0.9
’16.1% ’3.2%
46% 53% 0.3% 3.8% 8.6%
’2.1% ’8.2%
u.s. growth ¬rm 25.7% 1.3
’21.7% ’5.2%
49% 51% 0.8% 5.6% 15.2%




¬le=statsp¬os-g.tex: LP
u.s. small growth ¬rms 7.7% 25.5% 3.1% 1.0
’19.3% ’4.6%
46% 54% 0.3% 6.3% 20.1%
’1.3%
u.s. small-cap ¬rms 2.7% 22.6% 0.8
’19.3% ’4.3%
49% 51% 0.3% 5.2% 16.5%
u.s. energy ¬rms 5.2% 23.7% 2.3% 0.6
’18.0% ’4.1% ’0.5%
53% 47% 2.8% 19.0%
u.s. gold and metals 3.6% 35.1% 1.1% 0.5
’19.9% ’6.1% ’0.8%
51% 47% 6.0% 39.3%
u.s. health care ¬rms 13.6% 14.0% 11.3% 0.5
’10.5% ’1.0%
36% 64% 1.8% 3.3% 10.3%
’0.4% ’1.2%
u.s. real estate inv. trusts 12.6% 0.2
’9.2% ’2.2% ’0.2%
50% 49% 2.0% 9.6%
’2.3% ’4.2%
u.s. utilities ¬rms 13.9% 0.4
’11.9% ’2.5% ’0.7%
57% 43% 2.4% 7.7%




Chapter 14. Statistics of Portfolios.
’4.4% ’9.7%
intl emerging market ¬rms 28.0% 1.1
’26.1% ’6.2% ’0.4%
50% 50% 4.5% 15.4%
’1.7%
intl european ¬rms 1.9% 17.6% 0.7
’13.1% ’3.4%
41% 56% 0.3% 3.2% 9.7%
’7.4% ’10.6%
intl paci¬c ¬rms 21.3% 0.7
’12.1% ’5.3% ’2.0%
59% 40% 4.0% 17.6%
’1.6% ’5.1%
intl growth ¬rms 17.5% 0.7
’13.2% ’3.2%
46% 53% 0.5% 3.1% 9.6%
’4.2% ’7.5%
intl value ¬rms 18.7% 0.7
’15.0% ’3.1% ’0.7%
57% 43% 2.9% 13.0%
s&p 500 index 4.8% 18.5% 0.0% 1.0
’14.6% ’3.1%
46% 54% 0.5% 5.0% 9.7%
dow jones 30 index 6.3% 18.7% 1.8% 0.9
’15.1% ’3.6%
43% 57% 0.9% 4.0% 10.6%
¬le=statsp¬os-g.tex: RP
347
Section 14·3. Historical Statistics For Some Asset-Class Index Portfolios.

The third set of statistics that this chapter described was covariation, which included beta. The An important statistic,
used later again, is the
last column in the Table 14.4 shows the beta of the rate of return of each investment portfolio
covariation of
with the rate of return on the S&P500 index, investments with the
stock market.
Cov( rB , rS&P500 )
˜ ˜
βB, S&P500 = (14.58)
.
Var(˜S&P500 )
r

These “market-betas” tell us how much a particular investment portfolio™s rate of return co-
varied with the rate of return on the S&P500. A beta of 1 tells us that the rate of return of a
portfolio tended to covary one-to-one with the rate of return in the U.S. stock market. A beta
of 0 tells us that the rate of return of a portfolio tended to be unrelated to what happened to
the U.S. stock market.
Side Note: The linear regression by which the beta measure can be obtained is so common that it is called the
market model, and this particular beta is called the market beta. It can be obtained by running the time-series
regression
(14.59)
rB = ± + β · rS&P500 .
˜ ˜
Again, we use a historical beta as estimate for the future beta.


The last column shows that government bonds had practically no covariation with the S&P500. Equities covaried more
with the s&p than bonds.
Corporate bonds, energy stocks, precious metals, health care stocks, real estate investment
trusts (REITs) and utilities had very mild covariation, indicated by betas around 0.5. The next-
most correlated segment are international stocks, having betas of around 0.7. But many other
portfolios varied about 1-to-1 with the overall stock market. Note that U.S. growth ¬rms swung
even more than 1-to-1 with changes in the stock market: The beta of 1.3 tells us that a 10%
increase in the stock market tended to be associated with a 13% increase in the growth ¬rm
portfolio. This is typical.
For your curiosity, there are two more tables with the same statistics: Table a describes the Can you read and
understand tables of
historical performance of non-U.S. stock markets; and Table b describes the historical perfor-
historical performances
mance of the 30 stocks that constitute the Dow-Jones 30 Index. At this point, you can read and now?
interpret the table as well as I can, so enjoy!

Table 14.4 also shows some other statistics, such as the percent of all months that earned a
Side Note:
positive rate of return. (Naturally, one minus this percent are the months in which the portfolio had a negative
return.) The Table further shows the single worst month, the single best month, the median month (half of all
return months were better, half were worse). Finally, although this is beyond what we have covered so far, the
alpha (±) in the Table is sometimes interpreted as a risk-adjusted reward measure”the higher the better.
Some other random observations: The table shows no systematic relationship between risk and rate of return
over the sample period. However, it is the case that the least risky and least covarying investment strategies
(government bonds) provided a very modest, but positive average rate of return. With hindsight, it would have
been terri¬c to invest in U.S. health care stocks: they had the most spectacular return, plus a very modest risk.
Naturally, with hindsight, you could have selected the right six numbers for the lottery. So, which numbers can
you trust to be indicative of their future equivalent? First and foremost, covariation measures. They tend to be
very stable. Next, standard deviations are reasonably stable. Historical means, however, are very untrustworthy
as predictors of the future: it is not especially likely that health care ¬rms will continue to outperform other
stocks, and that Japanese ¬rms will continue to underperform other stocks.

Solve Now!
All questions refer to Table 14.4.

Q 14.15 Which investment class portfolio would have done best over the sample period? Do you
believe this will continue?


Q 14.16 Which investment class portfolio would have done worst over the sample period? Do
you believe this will continue?


Q 14.17 Assuming you had held only one asset class, which investment class portfolio was safest
during the sample period? Do you believe this will continue?
¬le=statsp¬os-g.tex: LP
348 Chapter 14. Statistics of Portfolios.

Q 14.18 Assuming you had held only one asset class, which investment class portfolio was riskiest
during the sample period? Do you believe this will continue?


Q 14.19 Which asset class portfolio had the lowest covariation with the S&P500 index? Do you
believe this will continue?


Q 14.20 Which asset class portfolio had the highest covariation with the S&P500 index? Do you
believe this will continue?
¬le=statsp¬os-g.tex: RP
349
Section 14·4. Summary.




14·4. Summary

The chapter covered the following major points:

• The formulas in this chapter decompose the statistics of a portfolio return in terms of
the statistics of its constituent securities™ portfolio returns.
The formulas are merely alternative computations. You can instead write out the time-
series of the portfolio™s rates of return and compute the portfolio statistics directly from
this distribution.
You shall use these formulas later, because you want to consider portfolios when you vary
the weights. The formulas express the overall portfolio statistics in terms of investment
weights, which will make it easier to choose the best portfolio.

• For three statistics, you can take investment-weighted averages:

1. The portfolio expected rate of return is the investment-weighted average of its com-
ponents™ expected rates of return.
2. The portfolio covariance with anything else is the investment-weighted average of its
components™ covariance with this anything else.
3. The portfolio beta with respect to anything else is the investment-weighted average
of its components™ beta with respect to this anything else.

• The portfolio variance can not be computed as the investment-weighted average of its
components™ variances. Instead, it is computed as follows:

1. For each security, square its weights and multiply it by the variance.
2. For each pair of di¬erent securities, multiply two times the ¬rst weight times the
second weight times the securities™ covariance.
3. Add up all these terms.

There are other ways to compute the variance. In particular, you can instead compute the
historical portfolio rate of return for each time period, and then compute the variance
from this univariate time-series. Or you can use the double summation Formula 14.42.

• For a sense of order-of-magnitude, Table 14.4 on Page 346 provides recent return statistics
for some common asset-class portfolios. The appendix gives equivalent statistics for the
Dow-Jones 30 stocks and for foreign stock markets.
350
Table 14.5. Summary of Portfolio Algebra in the Context of the Chapter Example


Three Input Securities Investment-
Weighted
Statistic Notation S&P500 IBM Sony Average Formula Portfolio of 70% IBM, 20% IBM, and 10% Sony

N
E (˜i ) wi ·E (˜i ) 70% · 19.0% + 20% · 38.8% + 10% · 90.3% = 40.9%
r r
Expected Return 19.0% 38.8% 90.3% Yes
i=1


Covariance, e.g.
N
wi ·Cov(˜i , rx ) 70% · 0.0362 + 20% · 0.0330 + 10% · 0.0477 = 0.03672
σi,S&P500 r˜
with x=S&P500 0.0362 0.0330 0.0477 Yes
i=1


Beta, e.g.




¬le=statsp¬os-g.tex: LP
N
wi ·β(˜i ,˜x ) 70% · 1.00 + 20% · 0.91 + 10% · 1.32 = 1.01
βi,S&P500
with x= S&P500 1.00 0.91 1.32 Yes rr
i=1




Requires three variance terms and the three mutual covariance
N N
σi,i = σi2 wi ·wj ·σi,j
No
Variance 0.036 0.150 0.815 terms. The latter are not provided in this table.
i=1 j=1




Chapter 14. Statistics of Portfolios.
For method, see Page 340, and Formulas 14.36 and 14.48. It is 0.0487 here.


0.0487 = 41% here).
σi No
Standard Deviation 19% 39% 90% Squareroot of variance (which is




The goal of this chapter was to explain these portfolio rules.
Know what they mean and how to use them!
¬le=statsp¬os-g.tex: RP
351
Section A. Appendix: More Historical Statistics.

Appendix




A. Appendix: More Historical Statistics
352
a. Country Fund Rates of Return

Table 14.6. AMEX Country Funds, Based on Monthly Rates of Returns, Annualized, January 1997 to October 2002.

Annualized Returns Monthly Returns Ann.Market-Model
±i βi,S&P
Asset Mean StdDev %Neg %Pos Worst Q2 Median Q3 Best
’0.4% ’4.1%
australia index fund amex 23.1% 0.8
’17.5% ’4.2%
47% 49% 0.0% 3.4% 14.5%
’0.2%
canada index fund amex 5.0% 23.5% 1.1
’22.4% ’4.0%
40% 57% 0.7% 5.5% 11.2%
’4.4%
sweden index fund amex 1.2% 30.3% 1.2
’21.2% ’4.9%
46% 53% 0.6% 5.3% 21.7%
’1.2%
germany index fund amex 3.9% 28.6% 1.1
’24.0% ’4.6%
46% 53% 0.9% 5.1% 24.4%




¬le=statsp¬os-g.tex: LP
’5.7% ’11.7%
hong kong index fund amex 36.0% 1.2
’28.1% ’7.4% ’1.3%
56% 41% 5.7% 38.4%
’8.0% ’11.4%
japan index fund amex 24.2% 0.7
’16.5% ’5.9% ’1.2%
59% 41% 3.7% 21.1%
belgium index fund amex 5.9% 26.6% 2.3% 0.8
’19.0% ’2.7%
44% 56% 0.4% 4.0% 37.9%
’3.9%
netherlands index fund amex 0.2% 22.2% 0.9
’17.3% ’3.3%
46% 53% 0.3% 3.6% 14.4%
’1.0%
austria index fund amex 1.2% 21.0% 0.5
’19.7% ’4.2%
47% 49% 0.0% 4.4% 11.9%
spain index fund amex 6.5% 25.0% 2.0% 0.9
’22.6% ’4.1%
44% 53% 0.3% 4.8% 15.5%
france index fund amex 6.2% 23.5% 2.0% 0.9
’15.0% ’2.5%
43% 57% 0.8% 3.9% 16.8%




Chapter 14. Statistics of Portfolios.
’7.5% ’13.8%
singapore index fund amex 37.7% 1.3
’27.0% ’6.5%
49% 46% 0.0% 3.6% 40.3%
’3.1%
uk index fund amex 0.2% 15.4% 0.7
’11.9% ’3.2%
43% 54% 0.6% 3.1% 8.6%
mexico index fund amex 11.3% 36.1% 4.7% 1.4
’35.3% ’7.0%
44% 53% 3.0% 9.0% 20.6%
s&p 500 index 4.8% 18.5% 0.0% 1.0
’14.6% ’3.1%
46% 54% 0.5% 5.0% 9.7%
Section A. Appendix: More Historical Statistics.
b. Dow-Jones Constituents

Table 14.7. Dow Jones Constituents, Based on Monthly Rates of Returns, Annualized, January 1997 to October 2002.

Annualized Returns Monthly Returns Ann.Market-Model
±i βi,S&P
Asset Mean StdDev %Neg %Pos Worst Q2 Median Q3 Best
alcoa 15.1% 41.3% 8.8% 1.3
’23.9% ’6.2%
49% 51% 0.4% 8.2% 51.1%
american express 17.3% 31.4% 11.3% 1.2
’29.3% ’3.0%
36% 63% 1.7% 8.4% 16.9%
’2.9% ’6.0%
boeing 34.6% 0.7
’34.6% ’7.1%
46% 53% 0.6% 6.6% 20.2%
citigroup 24.9% 37.4% 17.6% 1.5
’34.0% ’5.8%
43% 57% 2.4% 9.0% 25.8%
caterpillar 9.6% 34.0% 5.7% 0.8
’17.4% ’5.9%
46% 54% 0.8% 6.7% 40.8%
du pont 4.4% 29.0% 0.6% 0.8
’17.0% ’6.1% ’0.2%
50% 50% 6.5% 21.7%
’4.0%
disney 0.3% 32.2% 0.9
’26.8% ’6.4%
46% 54% 0.4% 5.0% 24.2%
’5.5% ’8.4%
eastman kodak 34.6% 0.6
’34.4% ’6.6% ’0.5%
53% 47% 5.5% 24.2%
general electric 12.8% 28.4% 7.6% 1.1
’17.7% ’4.1% ’0.2%
50% 49% 5.9% 19.2%
general motors 5.1% 36.7% 0.1% 1.0
’24.1% ’5.0% ’0.5%
50% 49% 5.8% 25.4%




¬le=statsp¬os-g.tex: RP
home depot 22.8% 35.1% 16.8% 1.2
’20.6% ’4.1%
41% 59% 2.8% 8.8% 30.2%
honeywell 6.0% 44.5% 0.1% 1.2
’38.4% ’4.1%
44% 56% 0.7% 6.5% 51.0%
’3.1%
hewlett packard 4.0% 48.2% 1.5
’32.0% ’9.1% ’1.4%
51% 49% 8.7% 35.4%
ibm 20.2% 39.6% 13.2% 1.5
’22.6% ’5.8% ’0.6%
50% 50% 7.8% 35.4%
intel 15.8% 53.1% 7.5% 1.7
’44.5% ’9.6%
46% 54% 0.8% 11.4% 33.9%
international paper 6.0% 36.0% 1.6% 0.9
’22.3% ’6.3%
47% 53% 0.5% 5.3% 27.0%
johnson and johnson 19.5% 26.4% 17.0% 0.5
’16.0% ’2.7%
43% 57% 1.5% 6.2% 17.4%
’1.9%
jp morgan 5.9% 42.5% 1.6
’30.6% ’6.3%
46% 54% 0.7% 8.2% 32.9%
coca-cola 3.7% 30.7% 0.9% 0.6
’19.1% ’4.7%
49% 51% 0.1% 6.7% 22.3%
’3.1%
mcdonalds 0.6% 28.8% 0.8
’25.7% ’5.7%
44% 56% 1.0% 5.9% 17.7%
3m 12.8% 26.0% 10.4% 0.5
’15.8% ’3.9%
44% 56% 1.1% 6.0% 25.8%
philip morris 6.9% 33.0% 5.1% 0.4
’23.7% ’4.6%
41% 57% 0.8% 5.3% 24.4%
merck 12.1% 32.0% 9.3% 0.6
’21.7% ’5.6%
49% 51% 0.1% 6.5% 22.8%
microsoft 28.4% 50.0% 20.0% 1.7
’34.4% ’8.7%
47% 51% 0.1% 11.9% 40.8%
proctor and gamble 13.7% 28.4% 12.6% 0.2
’35.4% ’1.8%
41% 59% 1.4% 5.3% 24.7%
sbc communications 8.1% 34.4% 4.5% 0.8
’18.8% ’7.0% ’0.2%
50% 50% 5.7% 29.3%
’5.7% ’10.3%
att 40.3% 1.0
’23.8% ’8.5% ’2.6%
54% 46% 6.1% 39.1%
united technologies 18.1% 34.1% 12.6% 1.1
’32.0% ’3.4%
40% 60% 1.9% 7.0% 24.6%
wal-mart 31.5% 31.1% 27.2% 0.9
’20.8% ’3.2%
33% 66% 3.0% 8.2% 26.4%
exxon 7.1% 18.3% 4.9% 0.5
’10.2% ’3.5% ’0.1%
50% 50% 4.7% 17.7%
dow jones 30 index 6.3% 18.7% 1.8% 0.9
’15.1% ’3.6%
43% 57% 0.9% 4.0% 10.6%
s&p 500 index 4.8% 18.5% 0.0% 1.0
’14.6% ’3.1%
46% 54% 0.5% 5.0% 9.7%




353
¬le=statsp¬os-g.tex: LP
354 Chapter 14. Statistics of Portfolios.

Solutions and Exercises




1.

(’0.176) + (’0.290) + ... + (’0.265)
E (˜P ) = =
r ,
0.183
12
(’0.359)2 + (’0.474)2 + ... + (’0.448)2
V (˜P ) = =
ar r ,
0.167
11
(+0.1620) · (’0.359) + (’0.0565) · (’0.474) + ... + (’0.3348) · (’0.448)
Cov(˜S&P500 , rP ) = = 0.0379 ,
r ˜
11
(’0.3661) · (’0.359) · + (’0.5874) · (’0.474) + ... + (’0.5105) · (’0.448)
Cov(˜IBM , rP ) = = 0.1075 ,
r ˜
11
(’0.3448) · (’0.359) · + (’0.2458) · (’0.474) + ... + (’0.3228) · (’0.448)
Cov(˜Sony , rP ) = = 0.2862 ,
r ˜
11

Sdv (˜P ) = V (˜P ) = 16.71% = 40.9%
r ar r .
(14.60)

2. There is no error.
3.

Cov( rIBM , rS ) = 0.05397 , Cov( rSony , rS ) = 0.6166 , Cov( rS&P500 , rS ) = 0.04403 .
˜ ˜ ˜ ˜ ˜ ˜
(14.61)

4.
βS,S&P500 = Cov( rS , rS&P500 )/V (˜S&P500 ) = ≈ 1.22 .
ar r
˜ ˜ 0.04403/0.03622

(14.62)
βS,S&P500 = wIBM · βIBM,S&P500 + wSony · βSony,S&P500

= 25% · 0.910 + 75% · 1.317 ≈ 1.22 .

5.

= V (˜P ) = (25%)2 · 0.15035 + (75%)2 · 0.81489
V (˜P ) = Cov( rS , rS )
ar r ar r
˜ ˜
(14.63)
+2 · 25% · 75% · 0.02184 = .
0.4760




6. Do it!
0.0131 + ... + (’0.243)
E (˜Q ) = ≈ 12.57% .
r
12 (14.64)
(’0.006 · 0.1620) + ... + (’0.369 · ’0.3348)
Cov( rQ , rS&P500 )= ≈
˜ ˜ ??.
12

7.

E (˜S ) = 25% · 10.1% + 35% · 15.4% + 40% · 24.2 ≈ 17.59%
r

Cov(˜S , S&P500) = 25% · 0.03622 + 35% · 0.03298 + 40% · 0.04772 ≈
r 0.0397

= 25% · 1.00 + 35% · 0.9104 + 40% · 1.3172 ≈
βS,S&P500 1.096
(14.65)

8. For the historical computation, compute the returns and their deviations:
¬le=statsp¬os-g.tex: RP
355
Section A. Appendix: More Historical Statistics.

Year 1991 1992 1993 1994 1995 1996
’0.04965 ’0.14210 +0.25130 +0.15549 +0.21218 +0.31198
Historical Return
’0.22556 ’0.31801 +0.07539 ’0.02043 +0.03626 +0.13607
Deviation from Mean


Year 1997 1998 1999 2000 2001 2002
+0.36708 +0.25242 +1.29560 ’0.30389 ’0.02390 ’0.21557
Historical Return
+0.19117 +0.07651 +1.11969 ’0.47980 ’0.19981 ’0.39148
Deviation from Mean

because the portfolio mean is 17.59%. The variance is therefore

0.0509 + 0.1011 + ... + 0.1533
(14.66)
Var(˜S ) = ≈ 0.1725 .
r
11
The alternative calculation is

0.252 · 0.03622 + 0.352 · 0.15035 + 0.402 · 0.81849
V (˜S ) =
ar r

+2 · 0.25 · 0.35 · 0.03298

+2 · 0.25 · 0.40 · 0.04772
(14.67)
+2 · 0.35 · 0.45 · 0.02184

= 0.1516 + 0.02220

≈ 0.1738

and the di¬erence is rounding error (of which half is in the Sony variance term).
9. The variance is 0.0487, the standard deviation is 0.2207. (Divide by N ’ 1, not by N.)
10. E (˜BF ) = 17.00%, Sdv (˜BF ) = 35.65%
r r
11. E (˜CF ) = 12.74%, Sdv (˜CF ) = 25.12%.
r r
12. E (˜DF ) = 13.00%, Sdv (˜DF ) = 22.90%.
r r
13.
(14.68)
E (˜EF ) = 10% · 0.83% + 10% · 0.90% + 80% · 1.19% = 1.125%
r


10% · (7.47%)2 + 10% · (8.35%)2 + 80% · (6.29%)2
V (˜EF ) =
ar r

+2·10%·10%·53.2% + 2·10%·80%·10.8% + 2·10%·80%·9.9%
(14.69)
= 0.0004858 ,

’ Sdv (˜EF ) = 6.97%
r .

14. We want the slope of a line where KO is the X variable. Therefore, the slope is

Cov( rPEP , rKO )
˜ ˜ 0.003318
= = ≈ 0.48 .
βPEP,KO
V (˜KO )
ar r 0.006967 (14.70)
Cov( rPEP , rCSG )
˜ ˜ 0.0005
= = ≈ 0.13 .
βPEP,CSG
V (˜CSG )
ar r 0.004



15. Health care ¬rms. Unlikely: You know that historical expected rates of return are not reliable predictors of
future expected rates of return.
16. Paci¬c (Japanese) Firms. Unlikely: You know that historical expected rates of return are not reliable predictors
of future expected rates of return.
17. Government Bonds, short term. Likely: Historical standard deviations tend to be good predictors of future
standard deviations.
18. Gold and Metals, then Emerging Stock Market Investments. (These are stock markets from developing coun-
tries.) Likely: Historical standard deviations tend to be good predictors of future standard deviations.
19. Look at the ¬nal column, βi,S&P500 . Government bonds of all kinds had almost no correlation with the S&P500.
Among more risky securities, REITs were almost uncorrelated, too. Likely: historical covariations tend to be
good predictors of future covariations.
20. Growth ¬rms. These contain many technology ¬rms. Likely: historical covariations tend to be good predictors
of future covariations.
¬le=statsp¬os-g.tex: LP
356 Chapter 14. Statistics of Portfolios.

(All answers should be treated as suspect. They have only been sketched, and not been checked.)
CHAPTER 15
The Principle of Diversi¬cation

Eggs and Baskets
last ¬le change: Feb 17, 2006 (14:52h)

last major edit: Aug 2004




This chapter appears in the Survey text only.


Having the statistical artillery now in place to describe risk (i.e., the standard deviation), you are
ready to abandon the previously maintained assumption of investor risk-neutrality. Henceforth,
you will no longer be assumed to be indi¬erent among investments with the same expected
rates of return. Instead, you can now prefer the less risky investment if two investment options
have the same expected rate of return.




357
¬le=diversi¬cation-g.tex: LP
358 Chapter 15. The Principle of Diversi¬cation.

15·1. What Should You Care About?

For the remainder of this book, we are assuming that you care only about the risk and reward
What we have assumed
about investor of our portfolios, and at one particular point in the future. (You may however reinvest your
preferences?
portfolio at this point to earn more returns.) You care about no other characteristics of your
portfolio, or whether a bigger portfolio at this point in time might cause a lower portfolio at
the next point in time. What does this mean, and how reasonable are these assumptions?
First, you are assuming that you do not care about anything other than ¬nancial returns. In-
What non-¬nancial
preferences have you stead, you could care about whether your portfolio companies invest ethically, e.g., whether a
assumed away?
¬rm in your portfolio produces cigarettes or cancer cures. In real life, few investors care about
what their portfolio ¬rms are actually doing. Even if you care, you are too small to be able to
in¬‚uence companies one way or the other”and other investors stand ready to purchase any
security you may spurn. Aside, if you purchase an ordinary mutual fund, you will hold all sorts
of companies”companies whose behavior you may or may not like.
Second, you are assuming that external in¬‚uences do not matter”you consider your portfolio™s
What have you assumed
away by looking at the outcome by itself without regard for anything else. So you do not seek portfolios that o¬er
portfolio problem in
higher rates of return if you were to lose your job. This would not be a bad idea”you probably
isolation?
should prefer a portfolio with a lower mean (given the same standard deviation), just as long
as the better outcomes occur in recessions when you are more likely to lose your job. You
should de¬nitely consider such investment strategies”but unfortunately few investors do so
in the real world. (If anything, the empirical evidence suggests that many investors seem to
do the exact opposite of what they should do if they wanted to ensure themselves against
employment risk.) Fortunately, our tools would still work with some modi¬cations if you de¬ne
your portfolio return to include your labor income.
Third, you are assuming that risk and reward is all you care about. But it is conceivable that
What return preferences
have you assumed away? you might care about other portfolio characteristics. For example, the following two portfolios
both have a mean return of 20% and a standard deviation of 20%.

P¬o “Symmetric” with 50% probability, a return of 0% with 50% probability, a return of +40%
P¬o “Skewed” with 33% probability, a return of “8% with 67% probability, a return of +34%


Are you really indi¬erent between the two? They are not the same. The symmetric portfolio
cannot lose money, while the skewed portfolio can. On the other hand, the skewed portfolio has
the better return more frequently. By focusing only on mean and standard deviation, you have
assumed away any preference between these two portfolios. Few investors in the real world
actively invest with an eye towards portfolio return skewness, so ignoring it is acceptable.
Fourth, you are assuming that you want to maximize your portfolio value at one speci¬c point
What multi-period
return processes have in time. This could be problematic if, for example, in the symmetric portfolio case it were true
you assumed away?
that a return of 0% was always later followed by a return of 100%, while a return of +40% was
always followed by an unavoidable return of “100% (nuclear war!), then you might not care
about the return at the end of the ¬rst measurement period. This is so unrealistic that you can
ignore this issue for most practical purposes.



Important: The remainder of this book assumes that you care only about the
risk (standard deviation) and reward (expected return) of your portfolio.
¬le=diversi¬cation-g.tex: RP
359
Section 15·2. Diversi¬cation: The Informal Way.

15·2. Diversi¬cation: The Informal Way

Let us assume that you dislike wealth risk. An importent method to reduce this risk”and the Don™t put all your eggs
into one basket (on one
cornerstone of the area of investments”is diversi¬cation, which means investing not only in
bet), if you dislike
one but in many di¬erent assets. We shall expound on it in great detail, but it can be explained variance!
with a simple example. Compare two bets. The ¬rst bet depends on the outcome of a single
coin toss. If heads, the bet pays o¬ $1 (if tails, the bet pays o¬ nothing). The second bet
depends on the outcome of 100 coin tosses. Each coin toss, if heads, pays o¬ 1/100 of a dollar,
zero otherwise. The expected outcome of either bet is 50 cents. But, as Figure 15.1 shows, the
standard deviation of the payo¬”the risk”of the latter bet is much lower.


Figure 15.1. Payo¬s Under Two Bets With Equal Means.
0.6




0.6
0.5




0.5
0.4




0.4
Probability




Probability
0.3




0.3
0.2




0.2
0.1




0.1
0.0




0.0




0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Outcome in $$s Outcome in $$s



The left ¬gure is the distribution of payo¬ where you bet $1 on heads for 1 coin throw. The right ¬gure is the
distribution of payo¬ where you bet $0.01 on heads for each of 100 coin throws.




Anecdote: Risk: Aversion or Las Vegas?
We are assuming that investors dislike risk. Putting money into many di¬erent bets, rather than one big bet,
accomplishes this goal. But, is this a good strategy on the roulette table, too? Should you bet your entire money
on red (one big bet only), or should you bet it one dollar at a time on red? From a purely ¬nancial perspective,
the answer is that the single bet is better: if you bet one dollar at a time, you are indeed likely to have a lower
variance of payo¬s around your expected rate of return. Unfortunately, in roulette, your expected rate of return
is negative. The casino would be perfectly happy to have you pay up your (negative) expected rate of return
without any risk for each roll of the ball.
Indeed, if your strategy is to gamble until you either are bankrupt or have doubled your money, you are more
likely not to go bankrupt if you make fewer but bigger bets.
¬le=diversi¬cation-g.tex: LP
360 Chapter 15. The Principle of Diversi¬cation.

15·3. Diversi¬cation: The Formal Way

15·3.A. Uncorrelated Securities

Recall from Section 12·3 (Page 294) that a portfolio™s return, rP , is
The base case:
independent security
N
returns.
rP ≡ wi ·ri , (15.1)
i=1

where P is the overall portfolio, wi is the investment weight (proportion) in asset i, and i is a
counter that enumerates all assets from 1 to N. If you do not yet know the return outcomes,
then your returns are random variables, so
N
rP ≡ wi ·˜i .
r
˜ (15.2)
i=1

It is now time to put the laws of expectations and standard deviations (from Section 14·2) to
good use. To illustrate how diversi¬cation works, make the following admittedly unrealistic
assumptions:

1. All securities o¬er the same expected rate of return (mean):

(15.3)
E (˜i ) = 5% for all i ;
r



2. All securities have the same risk (standard deviation of return):

(15.4)
Sdv(˜i ) ≡ σ (˜i ) ≡ σi = 40% for all i ,
r r

which is roughly the annual rate of return standard deviation for a typical U.S. stock;
3. All securities have rates of return that are independent from one another. Independence
implies that security returns have zero covariation with one another, so Cov(˜i , rj ) = 0

for any two securities i and j”just as long as i is not j:

(15.5)
Cov(˜i , rj ) = σi,j = 0 for all di¬erent i and j .


(This last assumption is the most unrealistic of the three.)

What are the risk and return characteristics of the portfolio, rP , if it contains N securities? For
˜
The portfolio de¬nition.
an equal-weighted portfolio with N securities, each investment weight is 1/N, so the portfolio
rate of return is
N N
(1/N) · ri = 1/N·˜1 + 1/N·˜2 + ... + 1/N·˜N .
rP = wi · ri = r r r
˜ ˜ ˜ (15.6)
i=1 i=1



Let™s start with one security. In this case, rP ≡ r1 , so
The Expected Rate of
Return.

(15.7)
E (˜P ) = E (˜1 ) = 5% , Sdv(˜P ) = Sdv(˜1 ) = Var(˜1 ) = 40%
r r r r r .

Two securities now. The portfolio consists of a 50-50 investment in securities 1 and 2:

(15.8)
rP = 1/2 · r1 + 1/2 · r2 .
˜ ˜ ˜
¬le=diversi¬cation-g.tex: RP
361
Section 15·3. Diversi¬cation: The Formal Way.

In this portfolio, it is easy to see that the average expected rate of return on a portfolio is the
average of the expected rates of return on its components:

E (1/2 · r1 + 1/2 · r2 )
E (˜P ) =
r ˜ ˜

(15.9)
= 1/2 · E (˜1 ) + 1/2 · E (˜2 )
r r

1/2 · 5% 1/2 · 5%
= + = 5% .

More generally, it is not a big surprise that the rate of return is 5%, no matter how many
securities enter the portfolio:
N N
E (1/N · ri ) = 1/N · E (˜i )
E (˜P ) =
r r
˜
i=1 i=1
(15.10)
N
1/N · 5%
= = .
5%
i=1



It is when you turn to the portfolio risk characteristics that it becomes interesting. The variance The base case: Variance
and Standard Deviation
and standard deviation of the rate of return on the portfolio P are more interesting. Begin with
are lower for 2
two securities: securities.


Var(1/2 · r1 + 1/2 · r2 )
Var(˜P ) =
r ˜ ˜

= (1/2)2 · Var(˜1 ) + (1/2)2 · Var(˜2 ) + 2 · (1/2) · (1/2) · Cov(˜1 , r2 )
r r r˜

1/4 · 0.16 + 1/4 · 0.16 + 2 · 1/2 · 1/2 · 0
=
(15.11)
1/4 · 0.16 + 1/4 · 0.16
=

1/2 · 0.16
=

1/2 · 0.16 = 70.7% · 40% = 28.3%
⇐ Sdv(˜P ) =
’ Var(˜P ) =
r r .

You could drop out the covariance term, because we have assumed security returns to be
independent. Pay close attention to the ¬nal line: the portfolio of one security had a risk of 40%.
The portfolio of two securities has the lower risk of 28.3%. This is important”diversi¬cation
at work!
If you ¬nd it easier to understand the formula if you see some data, below are two sample
Side Note:
series that are consistent with our assumptions: each has 5% mean and 40% standard deviation, and they have
zero mutual covariance. The ¬nal column is the rate of return on the portfolio P that invests half in each
security, thus appropriately rebalanced each year, of course . You can con¬rm that the standard deviation of
this portfolio is indeed the same 28.284% that you have just computed via the formula.


r1 r2 rP r1 r2 rP
˜ ˜ ˜ ˜ ˜ ˜
Year Year
1980 25.420 “11.476 6.972 1985 21.268 “35.560 “7.146
1981 39.376 62.064 50.720 1986 “45.524 30.620 “7.452
1982 41.464 “19.356 11.054 1987 42.976 59.616 51.296
1983 30.500 “41.256 “5.378 1988 “10.324 30.360 10.018
1984 “67.552 15.772 “25.890 1989 “27.604 “40.784“34.194
E 5.000 5.000 5.000
Sdv 40.000 40.000 28.284


Now, compute the standard deviation for an arbitrary number of securities in the portfolio. Variance and Standard
Deviation for N
Recall the variance formula,
securities.
± 
« 
N N 
N
Var(˜P ) = Var  ci · ri  = cj · ck · Cov(˜j , rk )
r r˜
˜
 
i=1 j=1 k=1
¬le=diversi¬cation-g.tex: LP
362 Chapter 15. The Principle of Diversi¬cation.

2
2 2
w1 ·V (˜1 ) + w2 ·V (˜2 ) + ... + wN ·V (˜N )
ar r ar r ar r

+ 2·w1 ·w2 ·Cov(˜1 , r2 ) + 2·w1 ·w3 ·Cov(˜1 , r3 ) + ... + 2·w1 ·wN ·Cov(˜1 , rN )
r˜ r˜ r˜

(15.12)
+ 2·w2 ·w1 ·Cov(˜2 , r1 ) + 2·w2 ·w3 ·Cov(˜2 , r3 ) + ... + 2·w2 ·wN ·Cov(˜2 , rN )
r˜ r˜ r˜

+ + + ... +
... ... ...

+ 2·wN ·w1 ·Cov(˜N , r1 ) + 2·wN ·w3 ·Cov(˜N , r3 ) + ... + 2·wN ·wN-1 ·Cov(˜N , rN-1 ) .
r˜ r˜ r˜

In our example, all the covariance terms are zero, all the weights are 1/N, and all individual
security variances are the same, so this is

(1/N)2 ·V (˜1 ) + (1/N)2 ·V (˜2 ) + ... + (1/N)2 ·V (˜N )
V (˜P ) =
ar r ar r ar r ar r

(1/N)2 ·V (˜i ) + (1/N)2 ·V (˜i ) + ... + (1/N)2 ·V (˜i )
= ar r ar r ar r
(15.13)
2
= N · (1/N) ·V (˜i ) = (1/N)·V (˜i )
ar r ar r .

1/N · V (˜i ) = 1/N · Sdv (˜i ) = 1/N · 40%
⇐’ Sdv (˜P ) =
r ar r r .


This formula states that for 1 security, the risk of the portfolio is 40%; for 2 securities, it is 28.3%
Re¬‚ect on the
effectiveness of
(as also computed in Formula 15.11); for 4 securities, it is 1/4 · 40% = 20%; for 16 securities, it
diversi¬cation: a lot for
is 10%; for 100 securities, it is 4%, and for 10,000 securities, is 0.4%. In a portfolio of in¬nitely
the ¬rst few additions,
then less and less.
many securities, the risk of the portfolio gradually disappears. In other words, you would
practically be certain to earn the expected rate of return (here 5%). Now take a look at the risk
decline in Figure 15.2 to see how more securities help to reduce risk. The square root function
on N declines steeply for the ¬rst few securities, but then progressively less so for subsequent
securities. Going from 1 to 4 securities reduces the risk by 50%. The next 5 securities (going
from 4 to 9 securities) only reduce the risk by 17% (from 1/4 = 50% to 1/9 = 33%). To drop
the risk from 50% to 25% requires 12 extra securities; to drop the risk from 50% to 10% requires
96 extra securities. In other words, if security returns are independent, diversi¬cation works
really well in the beginning, but less and less as more securities are added. It is important to
have, say, a dozen independent securities in the portfolio, which drops the portfolio risk by
two-thirds; additional diversi¬cation through purchasing more securities is nice, but it is not
as important, in relative terms, as these ¬rst dozen securities.


Figure 15.2. Diversi¬cation if security returns were independent.
0.4
0.3
Portfolio Risk

0.2
0.1
0.0




0 20 40 60 80 100

Number of Securities
¬le=diversi¬cation-g.tex: RP
363
Section 15·3. Diversi¬cation: The Formal Way.

15·3.B. Correlated Securities

When does diversi¬cation fail? Recall from Page 319 that the maximum possible correlation of Diversi¬cation fails
when security returns
1 implies that
are perfectly positively
correlated.
˜˜ ˜˜ ˜ ˜ (15.14)
Correlation(X, Y ) = +1 ⇐ Cov(X, Y ) = Sdv(X) · Sdv(Y ) .


Therefore, the covariance of two perfectly correlated securities™ rates of returns (in our example)
is
Cov(˜i , rj ) = 40% · 40% = 0.16 .

(15.15)
Cov(˜i , rj ) = Sdv(˜i ) · Sdv(˜j ) .
r˜ r r

The variance of a portfolio of two such stocks is

Var(1/2 · r1 + 1/2 · r2 )
Var(˜P ) =
r ˜ ˜

= 1/22 · Var(˜1 ) + 1/22 · Var(˜2 ) + 2 · 1/2 · 1/2 · Cov(˜1 , r2 )
r r r˜

(15.16)
1/4 · 0.16 + 1/4 · 0.16 + 2 · 1/2 · 1/2 · 0.16
=

1/2 · 0.16 + 1/2 · 0.16 = 0.16
=

Sdv(˜P ) = Var(˜P ) = 40%
r r .

In other words, when two securities are perfectly correlated, diversi¬cation does not reduce
portfolio risk. It should come as no surprise that you cannot reduce the risk of a portfolio
of one PepsiCo share by buying another PepsiCo share. (The returns of PepsiCo shares are
perfectly correlated.) In general, the smaller or even more negative the covariance term, the
better diversi¬cation works.



Important:

• Diversi¬cation fails when underlying securities are perfectly positively corre-
lated.

• Diversi¬cation reduces portfolio risk better when its underlying securities are
less correlated.

• Diversi¬cation works perfectly (reducing portfolio risk to zero) when under-
lying securities are perfectly negatively correlated.



Question 15.3 at the end of this section asks you to prove the last point.


15·3.C. Measures of Contribution Diversi¬cation: Covariance, Correlation, or Beta?

It seems that diversi¬cation works better when the covariation between investment securities Covariance, Correlation,
and Beta are multiple
is smaller. The correct measure of overall risk remains, of course, the standard deviation of
possible measures of
the portfolio™s rate of return. But, the question you are now interested in is “What is the best covariation.
measure of the contribution of an individual security to the risk of a portfolio?” You need a
measure of the risk contribution of just one security inside the portfolio to the overall portfolio
risk.
¬le=diversi¬cation-g.tex: LP
364 Chapter 15. The Principle of Diversi¬cation.

Let™s presume that you already hold a portfolio Y, and you are now considering adding “a little
Candidates to measure
how a new security helps bit” of security i. How much does this new security help or hurt your portfolio through diver-
portfolio diversi¬cation.
si¬cation? Because you are adding fairly little of this security, it is a reasonable approximation
to assume that the rest of the portfolio remains as it was (even though the new security really
becomes part of portfolio Y and thereby changes Y). The three candidates to measure how
correlated the new investment opportunity i is with the rest of your portfolio Y are

Cov(˜i , rY )

The Covariance: (Uninterpretable)


Cov(˜i , rY )
r˜ (15.17)
The Correlation: (Interpretable)
Sdv(˜i ) · Sdv(˜Y )
r r

Cov(˜i , rY )

The Portfolio Beta: (Interpretable)
Var(˜Y )
r

Although all three candidates share the same sign, each measure has its own unique advantage.
The covariance is used directly in the portfolio formula (e.g., Formula 15.11, Page 361), but its
value is di¬cult to interpret. The correlation is easiest to interpret, because it lies between ’1
and +1. However, its real problem as a measure of risk for a new security (which you want to
add to your portfolio) is that it ignores a security™s relative variability.
This last fact merits an explanation. Consider two securities that both have perfect correlation
Debunking correlation:
it is not a good risk with your portfolio. In our example, there are only two equally likely possible outcomes:
measure.

Security A Security B
Your Portfolio
Outcome 1 +24% +12% +200%
Outcome 2 “12% “6% “100%
E (˜)
r +6% +3% 50%
Sdv (˜)
r +18% +9% 150%


Now assume you had $75 in your portfolio, but you are adding $25 of either A or B. Therefore,
your new combined portfolio rate of return would be

Your Portfolio Y Plus Security A Your Portfolio Plus Security B
75% · (+24%) + 25%·(+12%) = 21.0% 75% · (+24%) + 25%·(+200%) = 68%
Case 1
75% · (’12%) + 25%·(’6%) = ’10.5% 75% · (’12%) + 25%·(’100%) = ’34%
Case 2
E (˜P )
r 5.25% 51.00%
Sdv (˜P )
r 15.75% 153.00%


Adding stock B causes your portfolio risk to go up more. It adds more portfolio risk. Yet,
the correlation between your portfolio and either security A or security B was the same. The
correlation would not have told you that B is the riskier add-on.
In contrast to correlation, our third candidate for measuring risk contribution tells you the right
But the beta of a stock
with your portfolio still thing. The beta of security A with respect to your portfolio Y is 0.5 (which you can compute
works as a good risk
either with the formula Cov(˜A , rY )/Var(˜Y ), or by recognizing that A is always one-half of
r˜ r
measure!
Y); the beta of security B with respect to your portfolio Y is 8.33. Therefore, beta tells you
that adding security B would increase your overall portfolio risk more than adding security A.
Unlike correlation, beta takes into account the scale of investments.
You can also ¬nd this scale problem within the context of our earlier three-investments scenario
A concrete example that
shows that correlation is with the annual returns of the S&P500, IBM, and Sony. On Page 322, you found that, compared
not a great
to IBM, Sony had a higher beta with the market, but a lower correlation. So, does a portfo-
diversi¬cation measure.
lio I consisting of one-half S&P500 and one-half IBM have more or less risk than a portfolio S
consisting of one-half S&P500 and one-half Sony?

Var(˜I ) = (1/2)2 · 3.62% + (1/2)2 · 15.03% + 2 · 1/2 · 1/2 · 3.30% = 5.075%
r ,
(15.18)
2 2
Var(˜S ) = (1/2) · 3.62% + (1/2) · 81.49% + 2 · 1/2 · 1/2 · 4.77% = 21.874% .
r
¬le=diversi¬cation-g.tex: RP
365
Section 15·3. Diversi¬cation: The Formal Way.

Even though Sony has lower correlation with the S&P500 than IBM, Sony™s higher variance
negates this advantage: the S portfolio is riskier than the I portfolio. The covariances re¬‚ect
this accurately: the covariance of the S&P500 with Sony is higher than the covariance of the

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