Q 13.11 If the mean is 10 and the standard deviation is 20, what is the probability that the value

will turn out to be positive?

Q 13.12 If the mean is 50 and the standard deviation is 20, what is the probability that the value

will turn out to be greater than 80?

Q 13.13 If the mean is 50 and the standard deviation is 20, what is the probability that the value

will turn out to be greater than 30?

Anecdote: Long Term Capital Management

Long-Term Capital Management (LTCM), a prominent hedge fund run by top ¬nance professors and Wall Street

traders, collapsed in 1999 in what their quantitative models believed to be a “10 sigma” (i.e., a 10 standard

deviation) event. According to the normal distribution probability table, such an event that has a score of ’10

should occur with a probability of less than 0.0001%, or 1 in 1,000,000 periods of trading.

You can conclude that either their models were wildly over-optimistic, or their assumption of a normal distri-

bution was incorrect, or the 1 in 1,000,000 actually occurred. Chances are that it was a little bit of all three.

(In essence, LTCM™s model believed that it would be exceedingly unlikely that all their bets would go sour at the

same time. Of course, they did all go sour together, so the LTCM principals lost most of their wealth.)

¬le=statistics-g.tex: LP

328 Chapter 13. Statistics.

Solutions and Exercises

1. Do it!

2. The mean was 9.837%. The variance was 7.7%. standard deviation was 27.74%.

3.

˜˜ ar ˜

Cov(X, X) V (X)

(13.23)

Correlation(x, x) = = =1.

˜˜

˜ · Sdv (X)

˜ ar ˜

Sdv (X) V (X)

4.

˜˜ ar ˜

Cov(X, X) V (X)

(13.24)

βx,x = = =1.

˜˜

ar ˜ ar ˜

V (X) V (X)

5.

Cov(˜DAX , rS&P500 ) = 0.0394 ,

r ˜

(13.25)

Cov(˜DAX , rIBM ) = 0.0464 ,

r ˜

Cov(˜DAX , rSony ) = 0.1264 .

r ˜

You have already computed the standard deviation of S&P500, IBM, and Sony as 19.0%, 38.8%, and 90.3%; and

for the DAX, as 27.74%. Therefore,

Correlation(˜DAX , rS&P500 ) = 74.6% ,

r ˜

(13.26)

Correlation(˜DAX , rIBM ) = 43.1% ,

r ˜

Correlation(˜DAX , rSony ) = 50.5% .

r ˜

The beta of the DAX with respect to the S&P500 is

0.0394

βrDAX ,˜S&P500 = = 1.088 .

r

˜

0.0362

0.0464 (13.27)

= = 0.308 .

βrDAX ,˜IBM

r

˜

0.3882

0.1264

= = 0.155 .

βrDAX ,˜Sony

r

˜

0.9032

6. This is to con¬rm the digging-deeper on Page 333.

7. Because you know the true distribution of future die throws. The historical values are measured with errors.

8. No.

9. Yes, reasonably so.

10. The score is (0 ’ 20)/15 = ’1.3. Therefore, the probability is 9.68%, i.e., roughly 10%.

11. The score is (0 ’ 10)/20 = ’0.5. Therefore, the probability is around 30% that the value will be negative, or

70% that it will be positive.

12. The score is (80 ’ 50)/20 = +1.5. Therefore, the probability is around 93% that the value will be below 80,

or 7% that it will be above 80.

13. The score is (30 ’ 50)/20 = ’1.0. Therefore, the probability is around 16% that the value will be 30, and 84%

that it will be above 30.

(All answers should be treated as suspect. They have only been sketched, and not been checked.)

CHAPTER 14

Statistics of Portfolios

last ¬le change: Feb 5, 2006 (17:55h)

last major edit: Aug 2004

This chapter appears in the Survey text only.

The previous chapter explained how to measure risk and reward for an investment”standard

deviation and expected rate of return. This chapter explains how to measure these statistics

in the context of portfolios. It may be the most tedious chapter in the book. But it is also

important: these formulas will be used in subsequent chapters, where you will want to ¬nd

the risks and rewards of many portfolios. You cannot understand investments without having

read this chapter”without understanding the rules for working with statistics in a portfolio

context. Your ultimate goal in reading this chapter is to ingest (understand how to use) the

methods described in Table 14.5.

Admittedly, some of this chapter “overdoes” it”it tries to explain where the algebra comes

from, even though in the end you only need to know the rules. Consider the extra pages to be

“useful reference.”

329

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330 Chapter 14. Statistics of Portfolios.

Table 14.1. Historical Rates of Returns and Statistics for S&P500, IBM, Sony, and a portfolio P

Historical Annual Rates of Returns

rS&P500 rIBM rSony rP rS&P500 rIBM rSony rP

˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜

Year Year

+0.263 ’0.212 ’0.103 ’0.1758 +0.310 +0.381 +0.391 +0.3842

1991 1997

+0.045 ’0.434 ’0.004 ’0.2903 +0.270 +0.762 ’0.203 +0.4407

1992 1998

+0.071 +0.121 +0.479 +0.2400 +0.195 +0.170 +2.968 +1.1028

1993 1999

’0.015 +0.301 +0.135 +0.2457 ’0.101 ’0.212 ’0.511 ’0.3116

1994 2000

+0.341 +0.243 +0.105 +0.1969 ’0.130 +0.423 ’0.348 +0.1659

1995 2001

+0.203 +0.658 +0.077 +0.4647 ’0.234 ’0.357 ’0.081 ’0.2647

1996 2002

Mean over all 12 years +0.101 +0.154 +0.183

0.242

Quoted as Deviations from the Mean

rS&P500 rIBM rSony rP rS&P500 rIBM rSony rP

˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜

Year Year

+0.1620 ’0.3661 ’0.3448 ’0.3590 +0.2090 +0.2273 +0.1485 +0.2010

1991 1997

’0.0565 ’0.5874 ’0.2458 ’0.4735 +0.1656 +0.6086 ’0.4448 +0.2575

1992 1998

’0.0305 ’0.0330 +0.2364 +0.0568 +0.0942 +0.0163 +2.7261 +0.9196

1993 1999

’0.1165 +0.1474 ’0.1073 +0.0625 ’0.2025 ’0.3658 ’0.7529 ’0.4948

1994 2000

+0.2400 +0.0892 ’0.1374 +0.0137 ’0.2315 +0.2693 ’0.5904 ’0.0173

1995 2001

+0.1015 +0.5046 ’0.1648 +0.2815 ’0.3348 ’0.5105 ’0.3228 ’0.4479

1996 2002

Mean (of Deviations) over all 12 years 0.0 0.0 0.0 0.0

Statistics

The covariances (and variances) computed in the previous chapter:

V (˜S&P500 ) = 0.0362 , Cov( rS&P500 , rIBM ) = 0.0330 , Cov( rS&P500 , rSony ) = 0.0477 ,

ar r ˜ ˜ ˜ ˜

Cov( rIBM , rS&P500 ) = 0.0330 , V (˜IBM ) = 0.1503 , Cov( rIBM , rSony ) = 0.0218 ,

ar r

˜ ˜ ˜ ˜

Cov( rSony , rS&P500 ) = 0.0477 , Cov( rSony , rIBM ) = 0.0218 , V (˜Sony ) = 0.8149 .

ar r

˜ ˜ ˜ ˜

(14.1)

The new portfolio P covariances, computed from the twelve historical returns above, are

Cov( rS&P500 , rP ) = 0.0379 , Cov( rIBM , rP ) = 0.1075 ,

˜ ˜ ˜ ˜

(14.2)

Cov( rSony , rP ) = 0.2862 , Cov( rP , rP ) = 0.1671 .

˜ ˜ ˜ ˜

The standard deviations are therefore

(14.3)

Sdv (˜S&P500 ) = 19.0% , Sdv (˜IBM ) = 38.8% , Sdv (˜Sony ) = 90.3% , Sdv (˜P ) = 40.9% .

r r r r

The return of portfolio P is rP = 66.7%·˜IBM + 33.3%·˜Sony .

r r

˜ Note: to keep investment weights constant at 66.7%

and 33.3%, you must rebalance the portfolio every year.

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331

Section 14·1. Two Investment Securities.

14·1. Two Investment Securities

Our goal in this section is to explore the properties of a portfolio P that invests twice as much Table 14.1 summarizes

what you did in the

into IBM as it invests into Sony. Table 14.1 begins with all the information you computed in

previous chapter, plus it

the previous chapter: historical rates of return, means, deviations from the means, variances, adds statistics for a

covariances, and standard deviations. The only novelty is that the performance of a portfolio P portfolio P.

in each year is now also in the table”as if it were a stock that you could have purchased. The

table also repeats the calculations for the covariances and variances for P. Please check the

calculations”do not go on until you have convinced yourself both that you can compute the

basic statistics for P yourself, and that the table contains no mistakes.

Solve Now!

Q 14.1 Compute the rP related statistics from the twelve historical rates of returns in Table 14.1.

˜

Q 14.2 Is there an error in Table 14.1? If so, can you ¬nd it?

14·1.A. Expected Rates of Returns

You knew this one even before you ever opened my book. If you expect 20% in your ¬rst stock The expected rate of

return is the

and 30% in your second stock, and you invest half in each, you expect to earn 25%. In our case,

investment-weighted

in which portfolio P is de¬ned by average.

(14.4)

P ≡ (2/3 IBM, 1/3Sony) ,

the expected rate of return on P of 18.3%. You could directly work this out from the 12 obser-

vations in the time-series in the ¬rst panel of Table 14.1, so you should con¬rm now that it is

also the investment-weighted average of the expected rates of return on our portfolio,

E (˜P ) = E wIBM · rIBM + wSony · rSony

r ˜ ˜

(14.5)

= wIBM · E (˜IBM ) + wSony · E (˜Sony )

r r

= 66.7% · 15.4% + 33.3% · 24.2% = 18.3% .

Important: Say your portfolio P consists of an investment w1 in security 1 and

an investment w2 in security 2. Therefore, its rate of return is rP ≡ w1 ·r1 +w2 ·r2 .

You can work with expected rates of return of a portfolio by taking the investment-

weighted average of its constituents, as follows:

E (˜P ) = E w1 · r1 + w2 · r2

r ˜ ˜

(14.6)

= w1 · E (˜1 ) + w2 · E (˜2 ) .

r r

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332 Chapter 14. Statistics of Portfolios.

14·1.B. Covariance

A more interesting question is about the covariance of our portfolio P (2/3 in IBM, 1/3 in Sony)

Compute the covariance

the slow way: from the with the returns on some other portfolio, say, the S&P500. In Table 14.1, the covariance worked

twelve historical returns.

out from the twelve historical returns is

(’0.3590) · (+0.1620) + ... + (’0.4479) · (’0.3348)

(14.7)

Cov( rS&P500 , rP ) = = 0.0379 .

˜ ˜

11

(This computation works with the deviations from the means.) But how does our P portfolio™s

covariance with the S&P500 (0.0379) relate to the covariances of its two portfolio components

with the S&P500 (0.0330 for IBM, and 0.0477 for Sony)?

In the previous section, you learned that the expected rate of return on our portfolio is the

The covariance of the

rate of return with investment-weighted average of the expected rates of the portfolio constituents. Can you do the

something else is the

same for covariance”i.e., is the portfolio covariance equal to the weighted sum of its portfolio

investment-weighted

constituents™ covariances? Yes!

average.

66.7% · 0.0330 + 33.3% · 0.0477 = ,

0.0379

(14.8)

wIBM · Cov( rIBM , rS&P500 ) + wSony · Cov( rSony , rS&P500 ) = Cov( rP , rS&P500 ) .

˜ ˜ ˜ ˜ ˜ ˜

So, if you want to ¬nd out the covariance of our portfolio with the S&P500 (or any other security),

all you need to do is compute the weighted average of its constituents. You do not need

to recompute the covariance from scratch in the tedious multi-step manner if you want to

experiment with di¬erent portfolio weights!

Important: Say your portfolio P consists of an investment w1 in security 1 and

an investment w2 in security 2. Therefore, its rate of return is rP ≡ w1 ·r1 +w2 ·r2 .

You can work with the covariances of your portfolio P with any other portfolio X

by taking the investment-weighted average of its constituents, as follows:

Cov( rP , rX ) = Cov w1 · r1 + w2 · r2 , rX

˜ ˜ ˜ ˜ ˜

(14.9)

= w1 · Cov( r1 , rX ) + w2 · Cov( r2 , rX ) .

˜ ˜ ˜ ˜

If you like our σ notation, you can write this as

(14.10)

σP,X = w1 · σ1,X + w2 · σ2,X .

As usual, we just omit the r in the sigma subscripts, so that we avoid double subscripts. (In-

˜

cidentally, this covariance law is only interesting, because we need it to work out the laws for

variance and beta.)

¬le=statsp¬os-g.tex: RP

333

Section 14·1. Two Investment Securities.

14·1.C. Beta

Our next question is: what is the beta of our portfolio P with respect to another security”here The covariance of the

rate of return with

again the S&P500? That is, how can you express the portfolio beta for P in terms of the betas

something else is the

of its two constituents (2/3 in IBM, 1/3 in Sony). Recall from the previous chapter how beta is investment-weighted

˜ ˜ ˜

de¬ned: you divide the covariance between X and Y by the variance of the X variable, average.

Cov( rIBM , rS&P500 )

˜ ˜ 0.0330

βIBM,S&P500 = = ≈ 0.91 ,

Var(˜S&P500 )

r 0.0362

Cov( rSony , rS&P500 )

˜ ˜ 0.0477

βSony,S&P500 = = ≈ 1.32 ,

Var(˜S&P500 )

r 0.0362

(14.11)

Cov( rP , rS&P500 )

˜ ˜ 0.0379

= = ≈ 1.05 ,

βP,S&P500

Var(˜S&P500 )

r 0.0362

Cov( rX , rY )

˜ ˜

=

βY ,X .

Var(˜X )

r

Note that the second subscript on beta is the variance denominator, and that we again omit the

r in the beta subscripts so as to avoid double subscripts.

˜

In the previous subsections, you learned that both the expected rate of return and the co- The beta of the rate of

return with something

variance of our portfolio with another portfolio are the investment-weighted statistics of the

else is the

portfolio constituents, respectively. Can you do the same for beta”i.e., is the portfolio beta investment-weighted

equal to the weighted sum of its portfolio betas? Yes! average.

66.7% · 0.91 + 33.3% · 1.32 = ,

1.05

(14.12)

wIBM · βIBM,S&P500 + wSony · βSony,S&P500 = βP,S&P500 .

Important: Say your portfolio P consists of an investment w1 in security 1 and

an investment w2 in security 2. Therefore, its rate of return is rP ≡ w1 ·r1 +w2 ·r2 .

You can work with betas of a portfolio by taking the investment-weighted average

of its constituents, as follows:

βP,X = β(w1 ·˜1 +w2 ·˜2

r r , rX )

˜

(14.13)

= w1 · β1,X + w2 · β2,X .

Two points. First, because you know the covariance law, you can easily show this algebraically,

Side Note:

Cov( rP , rS&P500 )

˜ ˜

βrP ,S&P500 =

˜

V (˜S&P500 )

ar r

Cov( wIBM · rIBM + wSony · rSony , rS&P500 )

˜ ˜ ˜

=

V (˜S&P500 )

ar r

wIBM · Cov( rIBM , rS&P500 ) + wSony · Cov( rSony , rS&P500 )

˜ ˜ ˜ ˜ (14.14)

=

V (˜S&P500 )

ar r

Cov( rSony , rS&P500 )

Cov( rIBM , rS&P500 ) ˜ ˜

˜ ˜

= wIBM · + wSony ·

V (˜S&P500 ) V (˜S&P500 )

ar r ar r

= wIBM · βIBM + wSony · βSony .

, S&P500 , S&P500

Second, βX,P ≠ w1 ·βX,1 +w2 ·βX,2 . Weighted averaging works only for the ¬rst beta subscript, not the second.

¬le=statsp¬os-g.tex: LP

334 Chapter 14. Statistics of Portfolios.

14·1.D. Variance

Our next question is: how does the variance of the rate of return of our portfolio relate to the

The variance of the rate

of return is not the covariances of its constituents? This time, our trick does not work. The variance of a portfolio

investment-weighted

is not the investment-weighted average of the portfolio constituents:

average.

66.7% · 0.1503 + 33.3% · 0.8149 = 0.3719 ,

0.1671 ≠

(14.15)

Var(˜P ) ≠ wIBM · Var(˜IBM ) + wSony · Var(˜Sony ) .

r r r

(Incidentally, the variance of the portfolio is much lower than just the average of its constituents;

you will explore this in great detail in later chapters”after you learn how to work with vari-

ances.)

So, to ¬nd out how the variance can be decomposed, you now have to cleverly use our covariance

Eliminate the ¬rst rP

˜

from the law. You already know that the variance of a random variable is the covariance with itself”and

variance...ahem, the

you have even computed it from the twelve historical returns,

covariance with itself.

(14.16)

Var(˜P ) = Cov( rP , rP ) = 0.1671 .

r ˜ ˜

Now, drop in the de¬nition of our portfolio, but just once,

(14.17)

Var(˜P ) = Cov( 66.7% · rIBM + 33.3% · rSony , rP ) = 0.1671 ,

r ˜ ˜ ˜

and use our covariance law to pull out the weights and to create a sum,

Var(˜P ) = Cov( 66.7% · rIBM + 33.3% · rSony , rP )

r ˜ ˜ ˜

(14.18)

= 66.7% · Cov( rIBM , rP ) + 33.3% · Cov( rSony , rP )

˜ ˜ ˜ ˜

Table 14.1 has these covariances, so you can check that we have not committed a mistake yet,

Var(˜P ) = 66.7% · 0.1075 + 33.3% · 0.2862 = 0.1671 ;

r

(14.19)

wIBM · Cov( rIBM , rP ) + wSony · Cov( rSony , rP ) .

˜ ˜ ˜ ˜

You are still getting the same number through algebra that you obtained by direct computation

of the variance (from the twelve historical rates of return in Table 14.1).

Now you must handle each of these two covariance terms by themselves: use the covariance

Eliminate the second rP

˜

from each term. law to pull out the weights and distribute the sum, and substitute in the covariance inputs from

Table 14.1,

Cov( rIBM , rP ) = Cov( rIBM , 66.7% · rIBM + 33.3% · rSony )

˜ ˜ ˜ ˜ ˜

= 66.7% · Cov( rIBM , rIBM ) + 33.3% · Cov( rIBM , rSony )

˜ ˜ ˜ ˜

= 66.7% · 0.1503 + 33.3% · 0.0218 = .

0.1075

(14.20)

Cov( rSony , rP ) = Cov( rSony , 66.7% · rIBM + 33.3% · rSony )

˜ ˜ ˜ ˜ ˜

= 66.7% · Cov( rSony , rIBM ) + 33.3% · Cov( rSony , rSony )

˜ ˜ ˜ ˜

= 66.7% · 0.0218 + 33.3% · 0.8149 = .

0.2862

¬le=statsp¬os-g.tex: RP

335

Section 14·1. Two Investment Securities.

(Again, you can check that the algebra is right, because using the information from Table 14.1

gives you the same answers for Cov( rIBM , rP ) and Cov( rSony , rP ).) Now put it all together,

˜ ˜ ˜ ˜

Var(˜P ) = 0.1671

r

= 66.7% · 0.1075 + 33.3% · 0.2862

= wIBM · Cov( rIBM , rS&P500 ) + wSony · Cov( rSony , rS&P500 )

˜ ˜ ˜ ˜

0.1075

= 66.7% · 66.7% · 0.1503 + 33.3% · 0.0218 (14.21)

+ 33.3% · 66.7% · 0.0218 + 33.3% · 0.8149

0.2862

= wIBM · wIBM · Cov( rIBM , rS&P500 ) + wSony · Cov( rIBM , rSony )

˜ ˜ ˜ ˜

+ wSony · wSony · Cov( rSony , rS&P500 ) + wSony · Cov( rSony , rSony )

˜ ˜ ˜ ˜

Tedious substitutions”but no higher math required. Take the last expression, multiply out

(remembering that the covariance of anything with itself is the variance), realize that the two

terms that contain both Sony and IBM are the same, rearrange the terms, and you get

Var(˜P ) = (66.7%)2 · 0.1503 + (33.3%)2 · 0.8149 + 2 · 66.7% · 33.3% · 0.0218

r

= 0.1671,

2 2

Var(˜P ) = wIBM · Var(˜IBM ) + wSony · Var(˜Sony ) + 2 · wSony · wIBM · Cov( rSony , rIBM ) .

r r r ˜ ˜

(14.22)

You are done: this is how the variance of a portfolio is expressed in terms of the variances

and covariances of its constituent securities. It is the sum of the variances, each multiplied by

its weight squared, plus two times each weight times the pairwise covariance. And you know

this is right, because the answer is still the same 0.1671 that you computed directly from the

historical rates of return for portfolio P in Table 14.2!

Important: Say your portfolio P consists of an investment w1 in security 1 and

an investment w2 in security 2. Therefore, its rate of return is rP ≡ w1 ·r1 +w2 ·r2 .

You can work with variances as follows:

2 2 (14.23)

Var(˜P ) = w1 · Var(˜1 ) + w2 · Var(˜2 ) + 2 · w1 · w2 · Cov( r1 , r2 ) .

r r r ˜ ˜

This is not the investment-weighted average of its constituents.

If you liked our σ notation, you can write this briefer as

2

σP = σP ,P = wX · σX,X + wY · σY ,Y + 2 · wX · wY · σX,Y

(14.24)

2 2

= wX · + wY · + 2 · wX · wY · σX,Y .

σX σY

To compute the standard deviation, you always have to ¬rst compute the variance, and then Standard Deviation

take the square-root. You cannot take any shortcuts here”like computing the average standard

deviation of the portfolio™s constituents.

¬le=statsp¬os-g.tex: LP

336 Chapter 14. Statistics of Portfolios.

Solve Now!

Q 14.3 Assume portfolio S consists of 25% IBM and 75% Sony. Compute the covariance of the

rate of return of S with the rates of return on IBM, Sony, and S&P500. (If you feel shaky, do it

with both a historical time-series in a table, and with the formulas.)

Q 14.4 Continue with portfolio S. Compute the beta of portfolio S with respect to the S&P500,

denoted βS,S&P500 . (If you feel shaky, do it with a historical time-series in a table, then directly

from the covariances, and ¬nally with the beta-combination formula.)

Q 14.5 Continue with portfolio S. Compute its variance. (If you feel shaky, do it with both a

historical time-series in a table, and the formulas.)

14·2. Three and More Investment Securities

Of course, you rarely want a portfolio with just two investments”usually, your portfolio will

We now work with

portfolios of more than have more than two investments. Let me remind you of how summation notation works. As in

two securities, often

Chapter 12, we later use i as a counter to enumerate each and every possible investment, from

expressed in summation

N choices (e.g., stocks). wi are the investment weights that de¬ne the portfolio. The return of

notation.

any portfolio Q de¬ned by these weights in each and every time period is

N

rQ = wi · ri , (14.25)

i=1

but because you do not yet know the returns,

N

rQ = wi · ri .

˜ ˜ (14.26)

i=1

14·2.A. Expected Returns, Covariance, Beta

For the three statistics from the previous section that we could average”the expected return,

Expected returns,

covariances, and betas the covariance, and beta”the generalization from two securities to any number (N) of securities

can be averaged, using

is easy: you can just take a weighted sum of all the components, not just of the ¬rst two

the investment-weighted

components. This is what our next important box expresses.

proportions.

Important: (For a portfolio Q that consists of w1 investment in security 1, w2

investment in security 2, all the way up to wN investment in security N, and which

N

therefore has a rate of return of rQ ≡ w1 · r1 + w2 · r2 + ...wN · rN = wi ri ):

i=1

You can work with expected rates of return, covariances, and betas by taking the

investment-weighted average of its constituents, as follows:

N

E (˜Q ) = w1 · E (˜1 ) + w2 · E (˜2 ) + ... + wN · E (˜N ) = wi · E (˜i ) ;

r r r r r

i=1

N

= w1 · σ1,X + w2 · σ2,X + ... + wN · σN,X = wi · σi,X (14.27)

σP ,X ;

i=1

N

= w1 · β1,X + w2 · β2,X + ... + wN · βN,X = wi · βi,X

βP ,X .

i=1

¬le=statsp¬os-g.tex: RP

337

Section 14·2. Three and More Investment Securities.

So, let™s use these formulas on a new portfolio Q that has 70% invested in the S&P500, 20% Checking the formulas

for a particular

invested in IBM, and 10% invested in Sony. You want to know the expected rate of return of our

portfolio, that we shall

portfolio (E(˜Q )), its covariance with S&P500 (Cov(˜Q , rS&P500 ) ≡ σQ,S&P500 ), and its beta with

r r˜ use later, too.

respect to S&P500 (βQ,S&P500 ). Corresponding to the formulas™ numbering schemes, security 1

is S&P500, security 2 is IBM, and security 3 is Sony. Table 14.1 provided all the necessary

expected returns and covariances; betas were in Formula 14.11; and your portfolio investment

weights are given. So,

= 70% · rS&P500 + 20% · rIBM + 10% · rSony

rQ

˜ ˜ ˜ ˜

E (˜Q ) = 70% · 10.1% + 20% · 15.4% + 10% · 24.2% =

r 12.57%

= wS&P500 · E (˜S&P500 ) + wIBM · E (˜IBM ) + wSony · E (˜Sony )

r r r

3

= wi · E (˜i ) = wi · E (˜i )

r r ;

S&P500,IBM,Sony

i=1

σQ,S&P500 = 70% · 0.03622 + 20% · 0.03298 + 10% · 0.04772 = 0.03672

= wS&P500 · σS&P500,S&P500 + wIBM · σIBM,S&P500 + wSony · σSony,S&P500

3

= wi · σi,S&P500 = wi · σi,S&P500 ;

S&P500,IBM,Sony

i=1

βQ,S&P500 = 70% · 1.000 + 20% · 0.910 + 10% · 1.317 = 1.0138

= wS&P500 · βS&P500,S&P500 + wIBM · βIBM,S&P500 + wSony · βSony,S&P500

3

= wi · βi,S&P500 = wi · βi,S&P500 .

S&P500,IBM,Sony

i=1

(14.28)

Table 14.2 gives the historical rates of return of this portfolio next to its constituents. Please

con¬rm from the twelve historical returns for Q and S&P500 that the above three statistics are

correct.

Expectations, covariances, and betas are so-called linear function, because

Digging Deeper:

(14.29)

f(a + b) = f(a) + f(b) ,

For our expectations, covariances, and beta, a would be de¬ned as w1 · r1 , and b as w2 · r2 . For example,

˜ ˜

E ( w1 · r1 + w2 · r2 ) = E ( w1 · r1 ) + E ( w2 · r2 ).

˜ ˜ ˜ ˜

It is not di¬cult to prove that if a function is linear, then it works for more than two securities in a portfolio. Just

replace b with c + d

(14.30)

f ( a + (c + d) ) = f ( a ) + f (c + d) ,

and apply formula 14.29 again on the c + d term to get

(14.31)

f ( a + (c + d) ) = f ( a ) + f ( (c + d) ) = f ( a ) + f ( c ) + f ( d ) .

Solve Now!

Q 14.6 Con¬rm that the computations for the expected rate of return and the covariance in

Formula 14.28 are correct by directly computing these statistics from the historical timeseries in

Table 14.2.

Q 14.7 Consider a portfolio T that consists of 25% S&P500, 35% IBM, and 40% Sony. What is its

expected rate of return, its covariance with the S&P500, and its beta with respect to the S&P500?

(If you feel shaky, compute this from both the twelve historical rates of return on T , and from

the formulas.)

¬le=statsp¬os-g.tex: LP

338 Chapter 14. Statistics of Portfolios.

Table 14.2. Historical Annual Rates of Returns for portfolio Q

Historical Annual Rates of Returns

rS&P500 rIBM rSony rQ rS&P500 rIBM rSony rQ

˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜

Year Year

+0.263 ’0.212 ’0.103 +0.131 +0.310 +0.381 +0.391 +0.332

1991 1997

+0.045 ’0.434 ’0.004 ’0.056 +0.270 +0.762 ’0.203 +0.319

1992 1998

+0.071 +0.121 +0.479 +0.121 +0.195 +0.170 +2.968 +0.468

1993 1999

’0.015 +0.301 +0.135 +0.063 ’0.101 ’0.212 ’0.511 ’0.165

1994 2000

+0.341 +0.243 +0.105 +0.298 ’0.130 +0.423 ’0.348 ’0.042

1995 2001

+0.203 +0.658 +0.077 +0.281 ’0.234 ’0.357 ’0.081 ’0.243

1996 2002

Mean over all 12 years +0.101 +0.154 +0.1257

0.242

Quoted as Deviations from the Mean

rS&P500 rIBM rSony rQ rS&P500 rIBM rSony rQ

˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜

Year Year

+0.1620 ’0.3661 ’0.3448 ’0.006 +0.2090 +0.2273 +0.1485 +0.207

1991 1997

’0.0565 ’0.5874 ’0.2458 ’0.182 +0.1656 +0.6086 ’0.4448 +0.193

1992 1998

’0.0305 ’0.0330 +0.2364 +0.004 +0.0942 +0.0163 +2.7261 +0.342

1993 1999

’0.1165 +0.1474 ’0.1073 +0.063 ’0.2025 ’0.3658 ’0.7529 ’0.290

1994 2000

+0.2400 +0.0892 ’0.1374 +0.172 ’0.2315 +0.2693 ’0.5904 ’0.167

1995 2001

+0.1015 +0.5046 ’0.1648 +0.156 ’0.3348 ’0.5105 ’0.3228 ’0.369

1996 2002

Mean (of Deviations) over all 12 years 0.0 0.0 0.0 0.0

The return of portfolio Q is rQ = 70%·˜S&P500 + 20%·˜IBM + 10%·˜Sony .

r r r

˜

14·2.B. Variance

Here is where it gets more complicated. What is the variance of portfolio Q, consisting of 70% in

Panic Warning!

S&P500, 20% in IBM, and 10% in Sony? Actually, it will get more tedious, but not more complex.

All you need to do is to apply the covariance law twice.

Recall the portfolio Q that invested 66.7% in IBM and 33.3% in Sony. If you invest 70% in S&P500

I made it easy on us:

portfolio Q is a and 30% in P, you end up with portfolio Q, because the remaining 30% in P are appropriately

combination of S&P500

split (wIBM = 30% · 66.7% = 20% and wSony = 30% · 33.3% = 10%):

and P.

rQ = 70% · rS&P500 + 20% · rIBM + 10% · rSony

˜ ˜ ˜ ˜

(14.32)

= 70% · rS&P500 + 30% · 66.7% · rIBM + 33.3% · rSony

˜ ˜ ˜

= 70% · rS&P500 + 30% · rP .

˜ ˜

With only two securities (S&P500 and P) now, you can use our variance formula 14.23:

You already know how

to work with two

securities”and you

V (˜Q ) = V (wS&P500 · rS&P500 + wP · rP )

ar r ˜ ˜

ar

know the other inputs,

too.

= (wS&P500 )2 · V (˜S&P500 ) + (wP )2 · V (˜P ) + 2 · wS&P500 · wP · Cov( rS&P500 , rP )

ar r ar r ˜ ˜

(70%)2 · V (˜S&P500 ) + (30%)2 · V (˜P ) +

= 2 · 30% · 70% · Cov( rS&P500 , rP )

ar r ar r .

˜ ˜

(14.33)

Actually, you already know all three remaining unknowns: 0.0362 was the variance of the

S&P500, given in Table 14.1. You had worked out Var(˜P ) in Formula 14.22,

r

V (˜P ) = (66.7%)2 · 0.1503 + (33.3%)2 · 0.8149 + 2 · 66.7% · 33.3% · 0.0218 = 0.1671 ,

ar r

(14.34)

2 2

V (˜P ) = wIBM · V (˜IBM ) + wSony · V (˜Sony ) + 2 · wSony · wIBM · Cov( rSony , rIBM ) ,

ar r ar r ar r ˜ ˜

¬le=statsp¬os-g.tex: RP

339

Section 14·2. Three and More Investment Securities.

and Cov( rP , rS&P500 ) in Formula 14.8,

˜ ˜

Cov( rP , rS&P500 ) = 66.7% · 0.0330 + 33.3% · 0.0477 = 0.0379 ;

˜ ˜

wIBM · Cov( rIBM , rS&P500 ) + wSony · Cov( rIBM , rS&P500 ) .

˜ ˜ ˜ ˜

(14.35)

Now just substitute these terms into Formula 14.33 to get

(70%)2 · V (˜IBM ) + (30%)2 · V (˜P ) + 2 · 70% · 30% · Cov( rS&P500 , rP )

V (˜Q ) =

ar r ar r ar r ˜ ˜

(70%)2 · 0.0362

=

+ (30%)2 · [(66.7%)2 · 0.1503 + (33.3%)2 · 0.8149 + 2 · 66.7% · 33.3% · 0.0218]

V (˜P )

ar r

(14.36)

+ 2 · 70% · 30% · [66.7% · 0.0330 + 33.3% · 0.0477]

Cov( rS&P500 , rP )

˜ ˜

(70%)2 · 0.0362 + (30%)2 · 0.1671 + 2 · 70% · 30% · 0.0379

=

= 0.0487 .

Please con¬rm this from the twelve annual rates of return for portfolio Q in Table 14.2.

Although we are done”we have our answer for the variance of our portfolio Q”let™s do some A lengthy detailed

step-by-step rewrite of

more algebra “just for fun.” Take the middle form from the previous formula,

the answer shows us

what the answer really

(70%)2 · 0.0362

V (˜Q ) =

ar r consists of: three

variance terms and

+ (30%)2 · [(66.7%)2 · 0.1503 + (33.3%)2 · 0.8149 + 2 · 66.7% · 33.3% · 0.0218] three pairwise

(14.37) covariance terms.

+ 2 · 70% · 30% · [66.7% · 0.0330 + 33.3% · 0.0477]

= 0.04869 .

Now, multiply the (30%) and (30%)2 terms into the parentheses, and pull the 30% and 30%%

into the adjacent weights,

(70%)2 · 0.0362

=

+ [(30% · 66.7%)2 · 0.1503 + (30% · 33.3%)2 · 0.8149 + 2 · (30% · 66.7%) · (30% · 33.3%) · 0.0218]

+ 2 · 70% · [(30% · 66.7%) · 0.0330 + (30% · 33.3%) · 0.0477]

= 0.04869 .

(14.38)

Execute the 30% multiplication, multiply in the 2·70%, eliminate some parentheses, and reorder

terms,

(70%)2 · 0.0362 + (20%)2 · 0.1503 + (10%)2 · 0.8149

=

(14.39)

+ 2 · (20%) · (10%) · 0.0218 + 2 · (70%) · (20%) · 0.0330 + 2 · (70%) · (10%) · 0.0477

= 0.04869 .

Now stare at this formula, and recall the form of the variance formula in eq:p¬ovar-2secs

on Page 335. What does this formula consist of? Well, it is just

2 2 2

V (˜Q ) = wS&P500 · V (˜S&P500 ) + wIBM · V (˜IBM ) + wSony · V (˜Sony )

ar r ar r ar r ar r

+ 2 · wIBM · wSony · Cov( rS&P500 , rSony )

˜ ˜

(14.40)

+ 2 · wS&P500 · wIBM · Cov( rS&P500 , rIBM )

˜ ˜

+ 2 · wS&P500 · wSony · Cov( rS&P500 , rSony )

˜ ˜

This generalizes, too!

¬le=statsp¬os-g.tex: LP

340 Chapter 14. Statistics of Portfolios.

Important: To obtain the variance of a portfolio that invests w1 , w2 , ... , wN

into N securities, do the following:

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.

For example, for three securities, the formula is

2 2 2

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

r r r r

+ 2 · w1 · w2 · Cov( r1 , r2 )

˜ ˜

(14.41)

+ 2 · w1 · w3 · Cov( r1 , r3 )

˜ ˜

+ 2 · w2 · w3 · Cov( r2 , r3 ) .

˜ ˜

If you do not believe me, feel free to repeat this exercise for four securities. (The principle

For portfolios with many

securities, there are remains the same, but it becomes a lot more messy.) Incidentally, the number of covariance

relatively more

terms increases more rapidly than the number of variance terms. With four securities, you will

covariance than variance

have four variance terms, and six pairwise covariance terms. With ten securities, you will have

terms.

ten variance terms, and forty-¬ve pairwise covariance terms. For one-hundred stocks, there are

one-hundred variance terms and 2,475 covariance terms. With more and more securities in the

portfolio, there are fewer and fewer “own return variance” terms, and more and more “return

covariance terms.” Thus, on ¬rst glance, it seems that the overall portfolio variance could be

driven more by the many covariance terms than the few variance terms. This will play a major

role in the next chapters.

Solve Now!

Q 14.8 Continue with our portfolio T that consists of 25% S&P500, 35% IBM, and 40% Sony. What

is its standard deviation of return? Compute this both from the twelve historical rates of return

on T , and from the formulas.

14·2.C. Advanced Nerd Section: Variance with N Securities and Double Summations

Formula 14.27 used notation to write sums more compactly. The point of this section is

The Formal De¬nitions.

to show how to write the variance formula with two summations signs”a lot more compactly,

and perhaps easier to remember. We shall ¬rst write it down”don™t panic”and then explain

and use it. The variance of the rate of return of a portfolio is

±

«

N N N

Var(˜P ) = Var wi · ri = wj · wk · Cov( rj , rk )

r .

˜ ˜ ˜ (14.42)

i=1 j=1 k=1

¬le=statsp¬os-g.tex: RP

341

Section 14·2. Three and More Investment Securities.

So what does this formula mean? Let us write out the terms in this formula to eliminate the

summation signs. Concentrate on one step at a time. Start with the innermost parentheses: j is

still unknown, so leave j untouched. Just write out the sum for k, which consists of N terms,

N

wj · wk · Cov( rj , rk )

˜ ˜

±

k=1

= wj ·w1 ·Cov( rj , r1 ) + wj ·w2 ·Cov( rj , r2 ) + ... + wj ·wN ·Cov( rj , rN ),

˜ ˜ ˜ ˜ ˜ ˜

our k = 1 term our k = 2 term our k = N term

(14.43)

and plug it back into Formula 14.42,

±

N N

V (˜P ) = wj · wk · Cov( rj , rk )

ar r ˜ ˜

j=1 k=1

(14.44)

N

= wj ·w1 ·Cov( rj , r1 ) + wj ·w2 ·Cov( rj , r2 ) + ... + wj ·wN ·Cov( rj , rN ) .

˜ ˜ ˜ ˜ ˜ ˜

j=1

So far, so good. Now this formula tells you that you have N summation terms, each of which

is itself N summation terms, so you have a total of N 2 summation terms in the variance. Do

what you just did again”write out the sum for j,

N

V (˜P ) = wj ·w1 ·Cov( rj , r1 ) + wj ·w2 ·Cov( rj , r2 ) + ... + wj ·wN ·Cov( rj , rN )

ar r ˜ ˜ ˜ ˜ ˜ ˜ =

j=1

’ w1 ·w1 ·Cov( r1 , r1 ) + w1 ·w2 ·Cov( r1 , r2 ) + ... + w1 ·wN ·Cov( r1 , rN )

˜ ˜ ˜ ˜ ˜ ˜

our j = 1 term

’ + w2 ·w1 ·Cov( r2 , r1 ) + w2 ·w2 ·Cov( r2 , r2 ) + ... + w2 ·wN ·Cov( r2 , rN (14.45)

˜)

˜ ˜ ˜ ˜ ˜

our j = 2 term

’ + w3 ·w1 ·Cov( r3 , r1 ) + w3 ·w2 ·Cov( r3 , r2 ) + ... + w3 ·wN ·Cov( r3 , rN )

˜ ˜ ˜ ˜ ˜ ˜

our j = 3 term

+ + + ... +

... ... ...

’ + wN ·w1 ·Cov( rN , r1 ) + wN ·w2 ·Cov( rN , r2 ) + ... + wN ·wN ·Cov( rN , rN )

˜ ˜ ˜ ˜ ˜ ˜

our j = N term

This has everything written out, and no longer needs scary summation notation”here it is The double summation

formula is just an

the length which makes the formula appear intimidating. So, would you rather memorize

easy-to-remember

the long form in Formula 14.45 or the short double summation sign Formula 14.42? They abbreviation, and is

both mean exactly the same thing”the double summation is merely abbreviated notation. I precise. It says exactly

what you already knew:

¬nd it easier to remember the short form”if need be, I can always expand it into the long

variance and pairwise

form. Before you forget about the long form, though, is this formula really the same as the covariance terms.

step-by-step procedure on Page 340? Look at the terms on the diagonal in Formula 14.45,

which are underlined. These are covariances of variables with themselves”which are just

the variances multiplied by weights squared. Now look at the o¬-diagonal terms”each term

appears twice, because both multiplication and covariances don™t care about order. So, the

big-mess Formula 14.45 can also be expressed as

V (˜P ) =

ar r [sum up N diagonal variance terms]

sum up remaining N 2 ’ N covariance terms

+

(14.46)

2

= for each security i, sum up each times the i-th variance

wi

+ for each possible pair i and j, sum up twice wi times wj times the i vs. j covariance ,

which is exactly what was stated on Page 340: to compute an overall portfolio variance, sum

up all the constituent variances (Var(˜i )), each multiplied by its squared weight (wi2 ); and then

r

add each pairwise covariance (Cov( ri , rj )), multiplied by two times its two weights (2 · wi · wj ).

˜˜

¬le=statsp¬os-g.tex: LP

342 Chapter 14. Statistics of Portfolios.

Important: The mean and variance formulas for portfolios deserve memorizing

if you want to concentrate in investments:

N

E (˜P ) = wi ·E (˜i )

r r ,

i=1

(14.47)

N N

Var(˜P ) = wi · wj · Cov( ri , rj ) .

r ˜ ˜

i=1 j=1

N N

If you prefer sigma notation, this is even shorter: σP ,P = wi ·wj ·σi,j .

i=1 j=1

Before you forget about double summations, let us just con¬rm that the formula gives us the

Application of the

formula makes it clear same variance for our portfolio Q:

that it is the same thing.

±

Var(˜Q ) = wi · wj · Cov( ri , rj )

r ˜ ˜

S&P500 S&P500

i∈ IBM j∈ IBM

Sony Sony

= wS&P500 · wS&P500 · Cov( rS&P500 , rS&P500 ) + wIBM · wS&P500 · Cov( rIBM , rS&P500 )

˜ ˜ ˜ ˜

+ wSony · wS&P500 · Cov( rSony , rS&P500 )

˜ ˜

+ wS&P500 · wIBM · Cov( rS&P500 , rIBM ) + wIBM · wIBM · Cov( rIBM , rIBM )

˜ ˜ ˜ ˜

+ wSony · wIBM · Cov( rSony , rIBM )

˜ ˜

+ wS&P500 · wSony · Cov( rS&P500 , rSony ) + wIBM · wSony · Cov( rIBM , rSony )

˜ ˜ ˜ ˜

+ wSony · wSony · Cov( rSony , rSony )

˜ ˜

= 70% · 70% · 0.0362 + 20% · 70% · 0.0330 + 10% · 70% · 0.0218

+ 70% · 20% · 0.0330 + 20% · 20% · 0.1503 + 10% · 20% · 0.0218

+ 70% · 10% · 0.0477 + 20% · 10% · 0.0218 + 10% · 10% · 0.8149

= 0.04869 ,

and you have the answer you already knew!

14·2.D. Another Variance Example: PepsiCo, CocaCola, and Cadbury

Table 14.3. PepsiCo, Coca Cola, and Cadbury Schweppes Monthly Return Statistics, from

September 1995 to August 2002.

Standard Correlations

Investment Means Deviations PEP KO CSG

1 PEP 0.83% 7.47% 100.0%

2 KO 0.90% 8.35% 53.2% 100.0%

3 CSG 1.19% 6.29% 10.8% 9.9% 100.0%

¬le=statsp¬os-g.tex: RP

343

Section 14·2. Three and More Investment Securities.

Let us now use portfolio formulas on a second example with monthly data, based on the The table provides only

statistics for the rates of

historical means, standard deviations, and correlations of Coca Cola, PepsiCo, and Cadbury

return, not the series

Schweppes, over the 1995-2002 period. This example is deliberately reminiscent of the exam- themselves.

ple from the previous section, but we are now given only the correlations, not the detailed his-

torical return series themselves. Table 14.3 shows that the average return of each of these three

stocks was about 1% per month, or 10%“12% per year. The monthly means were signi¬cantly

lower than the monthly standard deviations. Over the sample period, Cadbury Schweppes had

higher performance and lower risk than either PepsiCo or Coca Cola. Note also how high the

53% correlation between PepsiCo and Coca Cola is, especially relative to the 10.8% and 9.9%

Cadbury Schweppes correlations. Figure 14.1 plots the data points. Coca Cola stock seems to

behave more like PepsiCo stock than like Cadbury Schweppes stock.

Figure 14.1. 1,765 Daily Stock Returns of PepsiCo vs. Coca Cola, and PepsiCo vs. Cadbury

Schweppes from August 1995 to August 2002

0.3

0.3

0.2

0.2

0.1

0.1

PEP

PEP

0.0

0.0

’0.1

’0.1

’0.2

’0.2

’0.3

’0.3

’0.3 ’0.2 ’0.1 0.0 0.1 0.2 0.3 ’0.3 ’0.2 ’0.1 0.0 0.1 0.2 0.3

KO CSG

As always, assume that historical means, standard deviations, and correlations are indicative A real-world example

that computes the risk

of future means, standard deviations, and correlations. Now determine the risk and reward of

and reward of a

a portfolio B de¬ned by portfolio of three stocks.

(14.48)

B ≡ (20% in PEP, 30% in KO, and 50% in CSG) .

The unknown rate of return on this portfolio is

(14.49)

rB = 20% · rPEP + 30% · rKO + 50% · rCSG .

˜ ˜ ˜ ˜

The reward is easy. It is

E (˜B ) = 20% · 0.83% + 30% · 0.90% + 50% · 1.19% = 1.03%

r

(14.50)

E (˜B ) = 20% · E (˜PEP ) + 30% · E (˜KO ) + 50% · E (˜CSG ) .

r r r r

The portfolio risk is more di¬cult. As inputs, you need covariances, not standard deviations

or correlations. The covariances of variables with themselves (i.e., the variances) of the three

stocks can be computed from the standard deviations, by squaring:

σrPEP ,˜PEP = (0.0747)2 = 0.005585 ;

r

˜

= (0.0835)2 = 0.006967 ; (14.51)

σrKO ,˜KO

˜r

σrCSG ,˜CSG = (0.0629)2 = 0.003956 .

r

˜

¬le=statsp¬os-g.tex: LP

344 Chapter 14. Statistics of Portfolios.

The covariances have to be computed from the correlations. Recall Formula 13.13,

σ1,2

ρ1,2 = ⇐ σ1,2 = ρ1,2 · σ1 · σ2 .

’ (14.52)

σ1 · σ2

With this formula, you can compute the covariances,

σrPEP ,˜KO = ρrPEP ,˜KO · σrPEP · σrKO = 0.532 · 0.0747 · 0.0835 = 0.003318 ;

r r

˜ ˜ ˜ ˜

(14.53)

σrPEP ,˜CSG = ρrPEP ,˜CSG · σrPEP · σrCSG = 0.108 · 0.0747 · 0.0629 = 0.000507 ;

r r

˜ ˜ ˜ ˜

σrKO ,˜CSG = ρrKO ,˜CSG · σrKO · σrCSG = 0.099 · 0.0835 · 0.0629 = 0.000522 .

˜r ˜r ˜ ˜

You now have all inputs that you need to compute the portfolio return variance:

V (˜B ) = Var 20% · rPEP + 30% · rKO + 50% · rCSG

ar r ˜ ˜ ˜

(20%)2 · σrPEP ,˜PEP (30%)2 · σrKO ,˜KO (50%)2 · σrCSG ,˜CSG

= + +

r ˜r r

˜ ˜

+ 2 · 20% · 30% · σrPEP ,˜KO + 2 · 30% · 50% · σrPEP ,˜CSG + 2 · 20% · 50% · σrKO ,˜CSG

r r ˜r

˜ ˜

(14.54)

2 2 2

= (20%) · 0.005585 + (30%) · 0.006967 + (50%) · 0.003956

+ 2 · 20% · 30% · 0.003318 + 2 · 30% · 50% · 0.000507 + 2 · 20% · 50% · 0.000522

= .

0.002496

Therefore, the risk of portfolio B is

√