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

r˜

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 .

r˜

(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 .

r˜

(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 )

r˜

The Covariance: (Uninterpretable)

Cov(˜i , rY )

r˜ (15.17)

The Correlation: (Interpretable)

Sdv(˜i ) · Sdv(˜Y )

r r

Cov(˜i , rY )

r˜

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