12·2. Equities Transaction Costs

12·2.A. Going Long

The process of buying stocks is familiar to almost everyone: you call up your broker to purchase The costs of buying

shares depend on the

100 shares of a stock (say PepsiCo) with cash sitting in your account, and the shares appear in

stock, but typically are

your account and the cash disappears from your account. When you want to sell your shares, less than 0.5 percent

you call your broker again to sell the shares and the appropriate value of the shares returns as (round-trip) for a large

¬rm™s stock.

cash into your account. There are some transaction costs in the process: the broker collects

a commission (typically ranging from about $8 at a discount broker to $100 at a full-service

broker); and you are most likely to buy your shares at the ask price, which is higher than the

bid price, at which you can sell the shares. For a stock like PepsiCo, trading around $50, the

“bid-ask spread” may be 10 to 20 cents or about 0.2 percent. So, buying and then immediately

selling 1,000 shares of PepsiCo ($50,000), a round-trip transaction, might cost you transaction

costs of around $100 to $200 (lost to the bid-ask spread) plus $16 to $200 (lost to your broker).

Your $50,000 would have turned into about $49,600 to $49,900.

12·2.B. Going Short: The Academic Fiction

But, what if you want to speculate that a stock will be going down rather than up? This is called Idealized shorting, as

used in academia, gives

shorting a stock. (In optional Section 4·7.B, we have already discussed shorting in the context

shortsellers interest on

of Treasury securities and apples.) Optimally, you would want to do the same thing that the the stock that is being

PepsiCo company does: give other investors who want to buy shares in PEP the exact same sold.

payo¬s (including dividends!) that PEP will provide in exchange for them giving you $50. If the

share price declines to $30, upon termination of the short, you would have received their $50

upfront and only repaid them $30”you would have earned $20. In addition, you could have

earned interest on the $50. This is the idealized world of theoretical ¬nance and of this book,

in which borrowing and lending can be done without friction. The upper half of Table 12.1

shows such an example of a particular portfolio that involves idealized, frictionless shorting.

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292 Chapter 12. Securities and Portfolios.

12·2.C. Going Short: The Real World

In the real world, shorting is not so easy. First, there are rules and regulations that the SEC

Real-world shorting

requires more than just imposes on short-selling that you have to follow. Second, you need to credibly guarantee that

offering the same cash

you can give the share purchaser all the cash ¬‚ows that PepsiCo shares o¬er. (What have you

¬‚ows as the stock

committed to if the share price triples? Remember that you have unlimited liability as a short!)

Third, a real investor in PepsiCo also receives the accounting statements of PepsiCo in the mail

and can vote at the annual meetings. How do you o¬er this service? The answer is that you

need to ¬nd an investor who already owns the shares and who is willing to lend them to you, so

that you can sell the shares”real physical shares”to someone on the exchange. You then owe

shares to this lending investor, rather than to the person buying the shares on the exchange.

Digging Deeper: The most important SEC regulation concerning shorting is that the broker must borrow the

shares from a willing owner and then resell them to a third party. It is not enough for another investor to be

willing to take the other side of the short trade: instead, the shares have to be actually physically found from an

investor holding them. For some smaller stocks, it can occasionally be di¬cult to ¬nd someone willing to lend the

shares, which can make shorting di¬cult or impossible. In addition, there is a second SEC rule that states that

shares can only be shorted on an up-tick. The intent is to reduce further short-selling during a stock market crash,

when up-ticks are rare.

All of the details necessary to execute a short can be arranged by your broker. Unlike buying

Real-world shorting loses

the use of (some) shares long, execution of a short is often not instantaneous. But more importantly, the broker™s

proceeds. This is a

service comes at a price. The broker usually does not return to you the $100 paid by the person

“friction.”

buying the shares, so that you can invest the proceeds in bonds. That is, if the stock price

declines to $90, you still made $10, but the interest on the $100 is earned by your broker,

not by you. In addition, as with a purchase of shares, the broker earns commissions and the

bid-ask spread goes against you. The lower panel in Table 12.1 contrasts the idealized version

of shorting (used in this book) to the grittier real-world version of shorting.

Large clients can usually negotiate to receive at least some of the interest earned on the $100,

Large fund investors can

short at more favorable at least for large, liquid stocks. Hypothetically, if such a large investor were both short one

rates than ordinary

share of a ¬rm and long one share of the same ¬rm, she would lose about 100 to 300 basis

investors.

points per year. So, on a $100 share, the cost of being long one share and being short one share

would typically be $1 to $3 per share per year. This money is shared between the brokerage

¬rm and the investor willing to loan out shares to you for shorting (so that you can sell them to

someone else). Nowadays in the real-world, large stock index funds earn most of their pro¬ts

through lending out shares to shortsellers.

Shorting is not ideal in the real world”but it is a whole lot more ideal in ¬nancial markets

Our theories will assume

that shorting is than in non-¬nancial markets. Consider the large long exposure risk that a house purchaser

possible”a model

su¬ers. If the house value drops by 20%, the owner could easily lose more than all his equity

simpli¬cation that is

stake in the house. So to hedge against drops in the value of the house, it would make sense

often but not always

acceptable.

for this purchaser to go short on equivalent housing in the same neighborhood. This way, if

real estate prices were to go down, the short position in the neighborhood would mitigate the

own-house loss. For all practical purposes, this is impossible. Trust me”I have tried and failed

to ¬gure out how to do this. In e¬ect, the costs of shorting are almost in¬nitely high. When

you use the situation in real estate as your benchmark, it indeed seems reasonable to assume

no transaction costs to shorting, after all, at least for our academic purposes.

Anecdote: Eternal Shorts?

The short must make good on all promises that the underlying ¬rms make. There are some rare instances

in which this can cause unexpected problems. For example, when Heartland Industrial Partners acquired Mas-

cotech for about $2 billion in 2000, the latter promised to dispose of some non-operating assets and distribute

the proceeds to the original shareholders. As of 2005, that has not yet happened. Anyone having written a

short on Mascotech”including myself”still has an escrowed obligation as of 2005 that cannot be closed out.

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Section 12·2. Equities Transaction Costs.

Table 12.1. Shorting in an Idealized and in the Real World

Idealized Shorting Example

Your Wealth: $200.

You can sell $100 worth of KO shares (or an equivalent promise) to another investor, who wants to

hold KO shares. This gives you $200 + $100 = $300 of cash, which you can invest into Pepsico.

Portfolio P: wKO = ’$100, wPEP = $300, ’ wKO = ’50%, wPEP = 150%.

Hypothetical Rates of Return: KO = ’10%, PEP = +15% .

’ Portfolio Rate of Return: rP = ’50% · (’10%) + 150% · (+15%) = +27.5%.

$100 KO shares borrowed became a liability of $90, for a gain of $10;

$300 PEP shares invested became an asset of $345, for a gain of $45.

’ Your net portfolio gain is $55 on an original investment of assets worth $200, which

comes to a +27.5% rate of return.

Real World Retail Investor Shorting Example

Your Wealth: $200.

The broker ¬nds another investor to borrow shares from and sells the shares (on your behalf) for $100

to another investor, who wants to hold KO shares. The broker keeps $100, because in our example,

the retail investor is assumed to receive absolutely no shorting proceeds. (Institutional investors can

typically receive some, but not all of the shorting proceeds.) You still have $200 in cash ($100 less than

in the idealized case), which you can invest into Pepsico.

Portfolio P: wKO = ’$100, wPEP = $200, ’ wKO = ’50%, wPEP = 100%.

Hypothetical Rates of Return: KO = ’10%, PEP = +15% .

’ Portfolio Rate of Return: rP = ’50% · (’10%) + 100% · (+15%) = +20.0%.

$100 KO shares borrowed became a liability of $90, for a gain of $10;

$200 PEP shares invested became an asset of $230, for a gain of $30.

’ The net portfolio gain is $40 on an original investment of assets worth $200, which

comes to a +20% rate of return.

Both examples ignore trading costs incurred in buying and selling securities.

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294 Chapter 12. Securities and Portfolios.

Solve Now!

Q 12.6 What are the main di¬erences between academic, theoretical, perfect shorting and real-

world, practical shorting?

Q 12.7 If you simultaneously buy and short $5,000 of IBM at the beginning of the year, and

you terminate these two positions at the end of the year, how much would it cost you in the real

world?

Q 12.8 Assume you believe that stock in KO will go up by 12% and stock in PEP will go up by

15% over the next year. The current risk-free interest rate is 2% per year. You have $300,000 to

invest, and your broker allows you to go short up to $100,000.

(a) How much could you go long in PEP?

(b) If your forecast comes true, how much money would you earn in a ¬ctional world? What

would your rate of return be?

(c) If your forecast comes true, how much money would you earn in the real world?

12·3. Portfolios and Indexes

12·3.A. Portfolio Returns

What exactly is a portfolio? Is it a set of returns? No. The portfolio is a set of investment

weights. When these weights are multiplied by their asset returns, you obtain your overall

portfolio return.

Important: A portfolio is a set of investment weights.

You would usually own not just one security but form a portfolio consisting of many holdings.

Overall portfolio returns

are the Your ultimate goal”and the subject of the area of investments”is to select good portfolios

investment-weighted

with high rates of return. But how do you compute your portfolio rate of return? For example,

average returns of its

say you hold $500 in PEP, $300 in KO, and $200 in CSG. Your total investment is $1,000, and

constituents.

your portfolio investment weights are 50% in PEP, 30% in KO, and 20% in CSG. If the rate of

return on PEP is 5%, the rate of return on KO is 2%, and the rate of return on CSG is “4%, then

the rate of return on your overall portfolio (P) is

$500 $300 $200

rP = · (+5%) + · (+2%) + · (’4%)

˜

$1, 000 $1, 000 $1, 000

(12.1)

(50%) · (+5%) + (30%) · (+2%) + (20%) · (’4%) = +2.3%

= wPEP · rPEP + wKO · rKO + wCSG · rCSG .

So, when you own multiple assets, your overall investment rate of return is the investment-

weighted rate of return on each investment, with the weights being the relative investment

proportions. Let us check this. A $500 investment in PEP at a 5% rate of return gave $25. A

$300 investment in KO at a 2% rate of return gave $6. A $200 investment in CSG at a “4% rate

of return gave “$8. The net dollar return on the $1,000 was therefore $25 + $6 ’ $8 = $23.

The portfolio P rate of return was (rP,t=1 ’ rP,t=0 )/rP,t=0 = $23/$1, 000 ’ 1 = 2.3%. This

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295

Section 12·3. Portfolios and Indexes.

also clari¬es that the portfolio formula also works with absolute dollar investments instead of

relative percentage investments:

rP = ($500) · (+5%) + ($300) · (+2%) + ($200) · (’4%) = +$23

(12.2)

= wPEP · rPEP + wKO · rKO + wCSG · rCSG .

Side Note: When a security has additional payouts (such as dividends) over the measurement period, its rate

of return should really be written as

(Pt + Dividendst’1,t ) ’ Pt’1

rt’1,t = (12.3)

.

Pt’1

Alternatively, you could quote a net price at the end of the period, which includes dividends. Our discussions

will mostly just ignore dividends and stock splits. That is, when I write about returns, I usually mean rates of

returns that take into account all payments to the investor, but I sometimes abbreviate this as (Pt ’ Pt’1 )/Pt’1

for convenience.

The goal of the subject of investments is to evaluate all possible investment choices in order to Notation: number

investments. A portfolio

determine the best portfolio. So, we need to come up with good notation that does not make

is a set of known

discussing this task too cumbersome. Let us use R and r as our designated letters for “rate of weights.

return.” But with thousands of possible investment choices, it is rather inconvenient to work

with ticker symbols (or even full stock names). It would also be tedious to write “average the

returns over all possible stocks (ticker symbols) and other securities” and the name them all.

Therefore, we often change the names of our securities to the numbers 1, 2, 3, . . . , N. We also

usually use the letter P to name a portfolio (or, if we work with multiple portfolios, with a capital

letter close by, such as Q or O). We call the investment weight in security i by the moniker wi ,

N

where i is a number between 1 and N. Finally, we rely on “summation notation”: i=1 f (i)

is the algebraic way of stating that √ compute the sum f (1) + f (2) + ...f (N). For example,

we

√ √

√

4

2 + 3 + 4 ≈ 5.15. (Appendix Chapter 2·3 reviews summations.)

i=2 i is notation for

Yes, notation is a pain, but with the notation we have, we can now write the rate of return on a

portfolio much more easily:

Important: The rate of return R on a portfolio P that consists of N securities

named 1 through N is

N

rP = wi ·ri = w1 ·r1 + w2 ·r2 + . . . + wN ·rN , (12.4)

i=1

where wi is the investment weight in the i-th security (from N choices). If weights

are quoted as a fraction of the overall investment portfolio, their sum must add

up to 100%,

N

wi = 100% . (12.5)

i=1

OK, we are cheating a little on notation: each rate of return R should really have three subscripts: Notation: Abbreviations

one to name the ¬nancial security (e.g., PEP or i or 4), one for the beginning of the period (e.g.,

t), and one for the end of the period (e.g., t + 1)”too many for my taste. When there is no

danger of confusion”or the formula works no matter what periods we choose (as long as we

choose the same period for all securities)”let us omit the time subscripts.

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296 Chapter 12. Securities and Portfolios.

12·3.B. Funds and Net Holdings

One can think of portfolios, consisting of stocks, the same ways as one can think of stocks

Funds are themselves

portfolios of other themselves. Indeed, funds are ¬rms which hold underlying stocks or other ¬nancial assets and

¬nancial assets.

are thus themselves de facto portfolio”and funds can be bought and sold just like any other

stocks. Investors often like buying shares in funds because they believe that a professional

manager can pick securities better than they can, plus funds in e¬ect allow individual investors

to purchase thousands of stocks, even if they only have a small amount of money to invest.

Depending on their legal arrangements, funds may be called exchange-traded funds (bought

and sold on a ¬nancial market), mutual funds (bought and sold by the general public, but not

on an exchange), or hedge funds (not marketed to the broad public, and therefore not subject

to SEC restrictions). Many mutual funds have prices that are listed daily in the Wall Street

Journal. Like exchange funds, they can be purchased easily through most stock brokers. An

ADR (American Depositary Receipt) is another common form of fund. It is a relatively easy

way by which a large foreign company can trade shares on the New York Stock Exchange. Its

domestic shares are put into an escrow, and the U.S. exchange trades the ADR. So, an ADR really

operates like an open-end fund which holds shares only in this one company.

Mutual funds come in two main forms. Open-ended funds allow any investor to exchange the

Open-ended vs.

closed-ended Mutual fund shares for the underlying assets in the appropriate proportion. For example, if a fund

Funds.

has sold 50 shares, and used the money to purchase 200 shares of PepsiCo and 300 shares of

Coca Cola, then each mutual fund share represents 4 PepsiCo shares and 6 Coca Cola shares.

The mutual fund share holder can, at her will, exchange her fund share into 4 PepsiCo and 6

Coca Cola shares. This forces an arbitrage link between the price of the fund and the value

of its assets: if the price of the fund drops too much relative to the underlying assets, then

investors will redeem their mutual fund shares. In a closed-end mutual fund, redemption

is not permitted. If the underlying fund assets are very illiquid (e.g., real-estate in emerging

countries), an open-ended like redemption request would be very expensive or even impossible

to satisfy. Closed-end funds often trade for substantially less than their underlying assets, and

for signi¬cant periods of time. Among the explanations for this closed end fund discount are

the signi¬cant fees collected by the fund managers. At the end of 2002, there were about 7,000

open-end mutual funds with $4 trillion in assets. There were only about 500 closed-end mutual

funds with about $150 billion in assets, and another 4,500 hedge funds with assets of about

$350 billion (most hedge funds are closed-end). Thus, open-ended funds controlled about ten

times more money than closed-ended funds.

Most mutual funds disclose their holdings on a quarterly basis to the SEC (semi-annual is

How to compute net

underlying holdings. mandatory), which makes it easy for investors to compute their net exposures (at least on the

reporting day). For example, assume that Fund FA holds $500,000 of PepsiCo and $1,500,000

of Coca Cola. Assume that Fund FBholds $300,000 of Coca Cola and $700,000 of Cadbury

Schweppes. What is your net portfolio if you put 60% of your wealth into Fund FA and 40% of

into Fund FB? The funds have holdings of

Fund FA Holdings: wFA,PEP = 0.25 , wFA,KO = 0.75 , wFA,CSG = .

0

(12.6)

Fund FB Holdings: wFB,PEP = , wFB,KO = 0.30 , wFB,CSG = 0.70 .

0

Anecdote: More funds or more stocks?

In 1999, U.S. equity funds managed roughly 3 trillion dollars of assets, or about one-third of U.S. stock market

capitalization. More surprisingly, there were more U.S. equity funds than there were U.S. stocks: In 1999, there

were 8,435 equities, but 11,882 equity funds. Source: Harry Mamaysky and Matthew Spiegel.

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297

Section 12·3. Portfolios and Indexes.

You can compute your net exposures by computing the sum of your holdings multiplied by the

fund holdings:

wPEP = 60% · 0.25 + 40% · 0.00 = 15% ,

wKO = 60% · 0.75 + 40% · 0.30 = 57% ,

(12.7)

wCSG = 60% · 0.00 + 40% · 0.70 = 28% ,

= wFA · wFA,i + wFB · wFB,i

wi ,

which adds up to 100%. For example, for a $2,000 investment, you own $300 of shares in

PepsiCo, $1,140 of shares in Coca Cola, and $570 of shares in Cadbury Schweppes. In sum,

you can think of funds and portfolios the same way you think of stocks: they are investment

opportunities representing combinations of assets. You can always compute the underlying

stock holdings represented by the funds. In fact, if you wish, you could even see ordinary ¬rms

as portfolios bundling underlying assets for you.

12·3.C. Some Common Indexes

An index is almost like a fund or portfolio, but it is not something that one can invest in because Although an Index is a

weighted average of

an index is just a number. (There are, however, funds that try to mimick the behavior of indexes.)

stock prices, it is just a

Most commonly, an index is the ¬gure obtained by computing a weighted sum of the prices of a number, not a portfolio.

predetermined basket of securities. It is intended to summarize the performance of a particular

market or market segment. For example, the Dow-Jones 30 index is a weighted average of the

prices of 30 pre-selected “big” stocks. (Table b on Page 353 below lists them.) If you purchase

a portfolio holding the same 30 stocks (or a fund holding the 30 stocks), your investment

rate of return should be fairly close to the percentage change in the index”except for one

di¬erence. When stocks pay dividends, their stock prices decline by just about the amounts

of dividends paid. (If they dropped less on average, you should purchase the stocks, collect

the dividends, and then resell them for a pro¬t. If they dropped more, you should short the

stocks, pay the dividends, and then cover your shorts for a pro¬t.) Your portfolio mimicking

the Dow-Jones 30 should earn these dividends, even though the index would decline by the

percent paid out in dividends. Therefore, in theory, you should be able to easily outperform

an index. Unfortunately, in the real world, most portfolio managers fail to do so, primarily

because of transaction costs and excessive trading.

Most indexes, including the S&P500 and the Dow-Jones 30, are not adjusted for dividends, Some more detail: plain

indexes vs. total return

although they are adjusted for stock splits. In a 3:1 stock split, a ¬rm trading at $120 per

indexes.

old share would henceforth trade at $40 per new share. Each investor who held one old share

would receive three new shares. The “guardians of the index” would adjust the index formulas

by tripling the weight on the stock that split. In contrast to plain price indexes, there are

also total return indexes. For example, the guardians of the formula for the German Dax

Performance Index change the formula to re¬‚ect the return that a portfolio of Dax stocks

would earn through dividends.

Anecdote: The Worst of all Worlds: High Losses plus High Taxes

To prevent tax arbitrage”which are basically transactions that create fake losses to reduce taxable income”the

IRS has instituted special tax rules for mutual funds. The precise treatment of the taxes is complex and beyond

the scope of this book, but the most important aspect is simple: investors must absorb the underlying capital

gains/losses and dividend payments received by the funds, as if they themselves had traded the underlying

shares themselves.

In the second half of 2000, many mutual funds had lost signi¬cant amounts of money in the collapse of the

technology bubble. Although their values had declined (and with them, the wealth of their clients), the funds

had generally not yet realized these capital losses. But they had realized capital gains earlier in the year. The

IRS requires these realized losses to be declared by fund investors as “pass-through” capital gains, which were

therefore taxed. Thus, in 2000, many unlucky investors experienced high losses and still had to pay high taxes.

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298 Chapter 12. Securities and Portfolios.

There are literally hundreds of indexes, created and published by hundreds of companies. The

There are more indexes

than one can count, and Money&Investing Section of the Wall Street Journal lists just a sampling. In the United States,

new ones get invented

the most prominent stock market indexes are the S&P 500 (holding 500 large stocks), the

all the time.

aforementioned Dow-Jones 30 (holding 30 big stocks, selected to cover di¬erent industries),

and the Nasdaq index (holding the largest Nasdaq companies). The Russell 2000 covers 2,000

small-¬rm stocks. There are also other asset class indexes. For example, Lehman Brothers

publishes the MBS (Mortgage Bond Securities) index; Dow Jones also publishes a corporate

bond index; and Morgan Stanley publishes a whole slew of country stock prices indexes (MSCI

EAFE). Furthermore, each country with a stock market has its own domestic index. Some

foreign stock market indexes are familiar even to casual investors: the Financial Times Stock

Exchange Index, spelled FTSE and pronounced “foot-sy” for Great Britain; the Nikkei-225 Index

for Japan; and the DAX index for Germany.

12·3.D. Equal-Weighted and Value-Weighted Portfolios

Two kinds of portfolios deserve special attention, the equal-weighted and the value-weighted

An example of an

equal-weighted and a market portfolio. To see the di¬erence between the two, assume that there are only three

value-weighted portfolio.

securities in the market. The ¬rst is worth $100 million, the second $300 million, and the third

$600 million. An equal-weighted portfolio purchases an equal amount in each security. So,

if you had $30 million, you would invest $10 million into each security. Does it take trading

to maintain an equal-weighted portfolio? Table 12.2 shows what happens when one stock™s

price changes: security i = 1 quadruples in value. If you do not trade, your portfolio holdings

would be too much in security 1 relative to securities 2 and 3. To maintain an equal-weighted

portfolio, you would have to rebalance. In the example, you would have to trade $40 worth of

stock.

Table 12.2. Maintaining an Equal-Weighted Portfolio

Time 0 Time 1

Necessary

Investor, Investor,

Market Investor Market

Rate of

Security i Value P¬o Return Value No Trade Desired Trading

’$20

1 $100 $10 +300% $400 $40 $20

+$10

2 $300 $10 0% $300 $10 $20

+$10

3 $600 $10 0% $600 $10 $20

Sum $1,000 $30 $1,300 $60 $40 $0

Table 12.3. Maintaining a Value-Weighted Portfolio

Time 0 Time 1

Necessary

Investor, Investor,

Market Investor Market

Rate of

Security i Value P¬o Return Value No Trade Desired Trading

1 $100 (10%) $3 +300% $400 (31%) $12 $12 $0

2 $300 (30%) $9 0% $300 (23%) $9 $9 $0

3 $600 (60%) $18 0% $600 (46%) $18 $18 $0

Total $1,000 (100%) $30 $1,300 (100%) $39 $39 $0

Anecdote: The Presidential Election Market

The University of Iowa runs the Iowa Electronic Market, which are indexes measuring the likelihood for each

presidential candidate to win the next presidential election. You can actually trade futures based on these

indexes. This market tends to be a better forecaster of who the next president will be than the press.

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299

Section 12·3. Portfolios and Indexes.

A value-weighted portfolio purchases an amount proportional to the availability of each secu- Value-weighted

portfolios are easier to

rity. In the example, with $30 million, a portfolio that invests $3 million in the ¬rst security

maintain.

(weight: $2/$20 = 10%), $9 million in the second security (weight: 30%), and $18 million in the

third security (weight: 60%) is value-weighted. How di¬cult is it to maintain this value-weighted

portfolio? In Table 12.3, the ¬rst security has again quadrupled in value, increasing in market

capitalization to $400 million. Without trading, your previously value-weighted portfolio has

increased its holdings in this security from $3 million to $12 million. The portfolio weight in

the ¬rst security would therefore have increased to $12/$39 ≈ 31%, the second security would

have dropped to $9/$39 ≈ 23%, and the third security would have dropped to $18/$39 ≈ 46%.

But these are exactly the weights that a value weighted portfolio of $39 million, if initiated at

time 1, would require! The portfolio weights require no adjustment because any changes in

the market values of securities are re¬‚ected both in the overall market capitalizations and the

weight of the securities in your portfolio. In contrast to the earlier equal-weighted portfolio,

a value-weighted portfolio requires no trading. (The only exception are securities that enter

and exit the market altogether.) So, even though it may be easier at the beginning to select

an equal-weighted portfolio (you do not need to know how much of each security is available),

over time, it is easier to maintain a value-weighted portfolio.

Important: To maintain an equal-weighted portfolio, continuous rebalancing

is necessary. To maintain a value-weighted portfolio, no rebalancing is usually

necessary.

There is a second important feature of value-weighted portfolios: it is possible for everyone in It is possible for

everyone in the economy

the economy to hold a value-weighted portfolio, but not possible for everyone in the economy

to hold a value-weighted

to hold an equal-weighted portfolio. Return to the example: with $1 billion in overall market portfolio, but not an

capitalization, presume there are only two investors: the ¬rst has $100 million in wealth, the equal-weighted portfolio.

second has $900 million in wealth. Equal-weighted portfolios would have the ¬rst investor

allocate $33 million to the ¬rst security and have the second investor allocate $300 million to

the ¬rst security. In sum, they would want to purchase $333 million in the ¬rst security”but

there is only $100 million worth of the ¬rst security to go around. The pie is just not big enough.

In contrast, holding value-weighted portfolios, both investors could be fully satis¬ed with their

slices. In the example, for the ¬rst security, the ¬rst investor would allocate $10 million, the

second investor would allocate $90 million, and the sum-total would equal the $100 million

available in the economy.

Important: It is possible for all investors in the economy to hold value-weighted

portfolios. It is impossible for all investors in the economy to hold equal-weighted

portfolios.

Three more points: First, over time, if you do not trade, even a non-value weighted portfolio Time Convergence?

becomes more and more value-weighted. The reason is that stocks that increase in market

value turn into larger and larger fractions of your portfolio, and stocks that decline in market

value turn into smaller and smaller fractions. Eventually, the largest ¬rms in the economy will

be the biggest component of your portfolio. Second, the most popular and important stock

market indexes are more like value-weighted portfolios than equal-weighted portfolios. For

example, the S&P500 index behaves much more like the value-weighted than like the equal-

weighted market index. Third, over short time frames (say a month or even a year), broad stock

market indexes within a country tend to be very highly correlated (say, above 95%), no matter

whether they are equal-weighted, value-weighted, or arbitrary (e.g., the Dow-Jones 30 or the

S&P500). Therefore, if the newspaper reports the return of the S&P500 yesterday, it is a pretty

good estimator either for the return of broader portfolios (like a value-weighted overall stock

market portfolio) or for the return of narrower portfolios (like the Dow-Jones 30). It is rare that

one goes up dramatically, while the other goes down, and vice-versa.

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300 Chapter 12. Securities and Portfolios.

Solve Now!

Q 12.9 An investor™s portfolio P consists of 40% of stock A and 60% of stock B. A has a rate of

return of +4%, B has a rate of return of +6%. What is the overall portfolio rate of return?

Q 12.10 An investor owns $40 in stock A and $60 in stock B. A has a return of $1.60, B has a

return of +$3.60. What is the overall portfolio return?

Q 12.11 An investor owns $40 in stock A and $60 in stock B. The ¬rst stock has a return of +4%,

the second has a return of +6%. What is the overall portfolio rate of return?

Q 12.12 Write down the formula for the return of a portfolio, given individual security returns

and their weights. First use summation notation, then write it out.

Q 12.13 A portfolio consists of $200 invested in PEP, and $600 invested in CSG. If the stock price

per share on PEP increased from $30 to $33, and the stock price per share in CSG declined from

$40 to $38 but CSG paid a dividend of $1 per share, then what was the portfolio™s return and

rate of return?

Q 12.14 Fund FA holds $100,000 of PEP and $600,000 of KO, and $300,000 of CSG. Fund FBholds

$5,000,000 of PEP, $1,000,000 of KO, and $4,000,000 of CSG. You have $500 to invest. Can you

go long and short in the two funds to neutralize your exposure to PEP? (This means having a

net zero exposure to PEP.) How much of each fund would you purchase? What are your de facto

holdings of KO and CSG?

5

j2?

Q 12.15 What is j=1

5

2 · i?

Q 12.16 What is i=1

5

’ 5)?

Q 12.17 What is j=1 (j

Q 12.18 What is the di¬erence between a hedge fund and a mutual fund?

Q 12.19 What is the di¬erence between an open-ended mutual fund and a closed-ended mutual

fund?

Q 12.20 You hold two funds. Fund FA has holdings in stocks 1, 2, and 3 of 0.15, 0.5, and 0.35,

respectively. Fund FBhas holdings in stocks 1, 2, and 3, of 0.4, 0.2, and 0.4, respectively. You

would like to have a portfolio that has a net investment weight of 30% on the ¬rst stock. If

you have decided only to hold funds and not individual stocks, what would your exposure be on

stocks 2 and 3?

Q 12.21 What is the di¬erence between an index and a mutual fund?

Q 12.22 List a few prominent ¬nancial indexes.

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301

Section 12·3. Portfolios and Indexes.

Q 12.23 How does an index di¬er from a portfolio?

Q 12.24 Compute the value-weighted dollar investments of the two investors (with wealths $100

million and $900 million, respectively) for the second and third securities in the example on

Page 299.

Q 12.25 Continuing with this example, what would be the dollar investments and relative in-

vestments if the ¬rst security were to double in value? Does this portfolio require rebalancing to

remain value-weighted?

Q 12.26 There are two stocks: stock 1 has a market capitalization of $100 million, stock 2 has a

market capitalization of $300 million.

(a) What are the investment weights of the equal-weighted portfolio?

(b) What are the investment weights of the value-weighted portfolio?

(c) There are 5 equally wealthy investors in this economy. How much of stock 1 would they

demand if they all held the equal-weighted portfolio? If they held the value weighted port-

folio?

(d) If the ¬rst stock appreciates by 10% and the second stock depreciates by 30%, how much

trading would such an investor have to do to continue holding an equal-weighted portfolio?

(e) Repeat the previous question with a value-weighted portfolio.

Q 12.27 To maintain an equal-weighted portfolio, do you have to sell recent winner stocks or

recent loser stocks? (Is this a bad or a good idea?)

Q 12.28 How di¬erent would the one-month performance of an investment in an S&P500 mim-

icking portfolio be, relative to the performance of the value-weighted market portfolio?

12·3.E. Quo Vadis? Random Returns on Portfolios

Most of your attention in the next few chapters will be devoted to the case where returns are not A Road map.

yet known: they are still “random variables,” denoted with a tilde above the unknown quantity

(e.g., r ). The goal of investments is to select a portfolio P (that is, a set of N investment weights,

˜

w1 , w2 , . . . , wN ) which o¬ers the highest likely future performance with the least risk. Using

both the tilde and our portfolio sum formula, we can write the uncertain future rate of return

to our portfolio as

N

rP = wi ·˜i = w1 ·˜1 + w2 ·˜2 + . . . wN ·˜N .

r r r r

˜ (12.8)

i=1

We now need to ¬nd

1. a good measure for the reward (likely performance) of a portfolio;

2. and a good measure for the risk of a portfolio.

For this, you shall need statistics, the subject of the next chapter.

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302 Chapter 12. Securities and Portfolios.

12·4. Summary

The chapter covered the following major points:

• Securities appear through initial public o¬erings (IPOs) on exchanges, and disappear

through delistings.

• A round-trip transaction is one purchase and one sale of the same security. In the real

world, trades incur both brokerage fees and the bid-ask spread. In addition, going short

(selling without owning) incurs one extra cost”lack of full use of (interest earnings from)

the short-sale proceeds.

• Portfolio returns are a weighted average of individual returns.

• Fund holdings can be deconstructed into individual underlying stock exposures.

• An index is usually computed as the weighted averages of its component price ¬gures.

The index is therefore just a number. In contrast, funds and portfolios are collections of

underlying assets, the value of which are similarly computed as the weighted average of

the underlying component values. Index funds attempt to mimick index percent changes

by purchasing stocks similar to those used in the computation of the index.

• Maintaining an equal-weighted index requires constant rebalancing. Maintaining a value-

weighted index requires no trading.

• Unlike other portfolios, the value-weighted market portfolio can be owned by each and

every investor in equilibrium.

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303

Section 12·4. Summary.

Solutions and Exercises

1. The execute orders and keep track of investors™ portfolios. They also arrange for margin.

2. Prime brokers are usually used by larger investors, and allow these investors to employ their own traders to

execute trades.

3. The specialist is often a monopolist who makes the market on the NYSE. Market-makers are the equivalent

on Nasdaq, but usually compete with one another. Both can see the limit orders placed by other investors.

4. The alternatives are often electronic, and often rely on matching trades”thus, they may not execute trades

that they cannot match.

5. Shares can appear in an IPO or an SEO. Shares can disappear in a delisting or a repurchase.

6. For academic shorting, you just promise the same cash that the shares themselves are paying. For real-world

shorting, you ask your broker to ¬nd a holder of the shares, borrow them (i.e., and promise him the same

payo¬s), and then sell them to someone else. Most importantly, the broker who arranges this will not give

you the cash obtained from selling the borrowed shares”that is, the broker will earn the interest on the cash,

rather than you. Other important di¬erences have to do with the fact that you can be called upon to terminate

your short if the lender of shares wants to sell the shares (you have to return the borrowed shorts), and with

the fact that you have to ¬nd someone willing to lend you the shares.

7. The bid-ask transaction round-trip costs (bid-ask spread and broker commissions) for either the long or

the short would be around 30-60 basis points. In addition, you would have to provide the cash for the share

purchase; the cash from the share short is most likely kept by the broker. The loss of proceeds would cost you

another 300 to 500 basis points per year in lost interest proceeds, depending on who you are and whether you

already have the money or whether you have to borrow the money. So, if the shares cost $5,000, you would

have “buy” transaction costs of around $25 and “sell” transaction costs of around $25, for total transaction

costs of $50; plus interest opportunity costs of around $100 to $250.

8.

1. $400,000.

If you use the KO short proceeds to purchase stock in PEP, then $400, 000·(+15%)’$100, 000·(+12%) =

2.

$48, 000. On $300,000 net investment, your rate of return would be 16%.

$300, 000 · (+15%) ’ $100, 000 · (+12%) = $33, 000. On $300,000 net investment, so your rate of return

3.

would be 11%. You would be better o¬ forgetting about the shorting and earn the 15% on PEP.

9. rP = wA ·rA + wB ·rB = 40%·4% + 60%·6% = 5.2%.

10. rp = rA + rB = $5.20.

11. rP = wA ·rA + wB ·rB = $40·4% + $60·6% = $5.20. Note that the formula works with dollar investments, too.

12. See Formula 12.4 on Page 295.

13. rPEP = 10%.rCSG = ($38 + $1 ’ $40)/$40 = ’2.5%. Therefore, the absolute return and the rate of return on

the portfolio was

rP = $200 · (10%) + $600 · (’2.5%) = +$5 .

(12.9)

$200 $600

rP = · (10%) + · (’2.5%) = +0.625% .

$800 $800

14. The PEP weights in the two funds are 10% and 50%, respectively. To have zero exposure, you solve

(12.10)

wFA · 10% + (1 ’ wFA ) · 50% = 0 wFA = 1.25 , wFB = ’0.25 .

Therefore, your net exposures are

wKO = 1.25 · 60% + (’0.25) · 10% = 72.5% , wCSG = 1.25 · 30% + (’0.25) · 40% = 27.5% .

(12.11)

If you have $500 to invest, you would short $125 in Fund FB, leaving you with $625 to go long in Fund FA.

5 2= 1 + 4 + 9 + 16 + 25 = 55. (Note that i is just a counter name, which can be replaced by any letter,

i=1 i

15.

so this answer is correct.)

5

j=1 (2 · j) = 2 + 4 + 6 + 8 + 10 = 30.

16.

5

’ 5) = ’4 ’ 3 ’ 2 ’ 1 ’ 0 = ’10.

j=1 (j

17.

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304 Chapter 12. Securities and Portfolios.

18. See Page 296.

19. See Page 296.

20.

wFA · 15% + (1 ’ wFA ) · 40% = 30% wFA = 40%

(12.12)

wFA · wFA,1 + wFB · wFB,1 = w1

If you purchase 40% in fund FA, your net holdings in each stock are

Stock 1: 40% · 15% + 60% · 40% = 30% ;

Stock 2: 40% · 50% + 60% · 20% = 32% ;

(12.13)

Stock 3: 40% · 35% + 60% · 40% = 38% .

wFA · wFA,i + wFB · wFB,i = wi .

21. An index is a series of numbers; a mutual fund holds stocks. A mutual fund can hold stocks to mimick the

return on an index.

22. See text for possible indexes to mention.

23. Indexes are numbers, not investments. Their percent change di¬ers from a mimicking portfolio rate of return

in that dividends are ignored.

24. In the second security: $30 and $270, for the ¬rst and second investor, respectively. In the third security:

$60 million and $540 million, respectively.

25. As already computed in the text, the ¬rst security would increase from $100 million to $200 million, and

thus total market capitalization would increase from $1 billion to $1.1 billion. Therefore, the weight of the

three securities would be 18%, 27%, and 55%. (Moreover, the portfolio increased by 10% in value, which means

that the ¬rst investor now has holdings worth $110 million, and the second investor has holdings worth $990

million.) The dollar investments are even simpler: the ¬rst investor started with $10, $30, and $60 million,

respectively, and now holds $20, $30, and $60 million. The second investor now holds $180, $270, and $540

million, respectively.

26.

w1 = 50%, w2 = 50%.

1.

w1 = 25%, w2 = 75%.

2.

3. Each investor would own $80 million worth of securities. If each investor held the equal-weighted

portfolio, the total demand for stock 1 would be $40 · 5 = $200. This is impossible. If each investor held

the value-weighted portfolio, the total demand for stock 1 would be $20 · 5 = $100. This is de¬nitely

possible.

The equal-weighted portfolio would have started out at w1 = $40m, w2 = $40m. The returns would

4.

have made the portfolio w1 = $44m, w2 = $28m for a total value of $72m. The revised portfolio would

have to be w1 = $36m, w2 = $36m. Therefore, the investor would have to sell $8m in security 1 in

order to purchase $8m in security 2.

The value-weighted investor portfolio started out at w1 = $20m, w2 = $60m, and without trading

5.

would have become w1 = $22m, w2 = 42m for a portfolio worth $64m. The investor™s weights would

be w1 = 34.375%, w2 = 65.625%.

In terms of the market, the ¬rst stock would have appreciated in value from $100 million to $110

million, while the second would have depreciated from $300 million to $210 million. The value-weighted

market portfolio would therefore invest w1 = $110/$320 ≈ 34.375% and w2 = $280/$320 ≈ 65.625%.

Therefore, no trading is necessary.

27. Recent winners have to be sold, recent losers have to be bought. The answer to the question in parentheses is

that if the stock market is competitive, past returns should have little predictive power for future returns, so

this trading strategy is not necessarily a good or a bad idea. This will be covered in more detail in Chapter 19.

28. The correlation between these two is high. Therefore, it would make little di¬erence.

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

CHAPTER 13

Statistics

last ¬le change: Jan 18, 2006 (11:55h)

last major edit: Aug 2004

This chapter appears in the Survey text only.

This chapter attempts to distill the essential concepts of a full course in statistics into thirty-

something pages. Thus, it is not an easy chapter, but it is also not as complex and painful as

you might imagine.

305

¬le=statistics-g.tex: LP

306 Chapter 13. Statistics.

13·1. Historical and Future Rates of Return

As an investor, your goal is to ¬nd the best possible investment portfolio. Easier said than done.

Finance is not

philosophy”you have What do you know about how a stock, say, IBM, will perform in the future? Not much. Your

little else but history to

prime source of information about how IBM will perform is how it did perform. If it returned

guide us.

10% per year over the last 10 years on average, maybe it is a good guess that it will return

10% over the next year, too. If it had a risk of plus or minus 20% per year, maybe it is a good

guess that it will have a risk of plus or minus 20% over the next year, too. But, is historical

performance really representative of future performance?

Clearly, it makes no sense to assume that future returns will be exactly the same as past returns.

Don™t take history too

literally. An investment in a particular six lotto numbers may have paid o¬ big last year, but this does

not mean that the exact same investment gamble will work again. More sensibly, you should

look at the risk/reward characteristics of the average six lotto number investment, from which

you would probably conclude that the average six number lotto investment is not a great bet.

Similarly, for stocks, it makes more sense to assume that future returns will be only on average

For stocks, we use

general historical like past returns in terms of risk and reward. It is not that we believe this to be exactly true,

characteristics to

but it is usually our best guess today. Of course, we also know that the future will turn out

indicate future general

di¬erent from the past”some ¬rms will do better, others worse”but we generally have no

characteristics.

better information than history. (If you can systematically estimate future risk and reward

better than others, you are bound to become rich.)

Be warned, though. History is sometimes outright implausible as a predictor of future events.

Sometimes, even general

historical characteristics If you had played the lottery for 100 weeks and then won $1 million just by chance, a simple

(average) are obviously

historical average rate of return would be $10,000 per week. Yes, it is a historical average,

wrong.

but it is not the right average forward-looking. (Of course, if wyouhad played the lottery 100

million weeks, we would almost surely come to the correct conclusion that playing the lottery

is a gamble with a negative expected average rate of return.) Similarly, Microsoft or Wal-Mart

are almost surely not going to repeat the spectacular historical stock return performances

they experienced over the last 20 years. There are many examples when investors, believing

history too much, made spectacularly wrong investment decisions. For example, in 1998-2000,

Internet stocks had increased in value by more than 50% per year, and many investors believed

that it was almost impossible to lose money on them. Of course, these investors, who believed

historical Internet returns were indicative of future Internet returns, lost all their money over

the following two years.

For the most part, the theory of investments”which is the subject of this part of our book”

You have no better

choice”so use it, but makes it easy on itself. It just assumes that you already know a stock™s general risk/reward

remain skeptical.

characteristics, and then proceeds to give you advice about what portfolio to choose, given

that you already have the correct expectations. The problem of estimating means and variances

remains your problem. Despite the problems with historical returns, I can only repeat: in many

cases, stock™s average historical risk-reward behavior is the best guidance you have. You can

use this history, even though you should retain a certain skepticism”and perhaps even use

common sense to adjust historical averages into more sensible forecasts for the future.

Anecdote: Void Where Prohibited

Persons pretending to forecast the future shall be considered disorderly under subdivision 3, section 901 of

the criminal code and liable to a ¬ne of $ 250 and/or six months in prison.

(Section 889, New York State Code of Criminal Procedure.)

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307

Section 13·2. The Data: Twelve Annual Rates of Returns.

13·2. The Data: Twelve Annual Rates of Returns

The goal of this chapter is to explain portfolios and stock returns under uncertainty. This The data example that is

used throughout the

is best done with a concrete example. Table 13.1 contains the actual twelve annual rates of

Investments part of the

returns from 1991 to 2002 for three possible investments: an S&P500 mutual fund, IBM stock, book.

and Sony (ADR) shares.

Table 13.1. Historical Annual Rates of Returns for S&P500, IBM, and Sony

rS&P500 rIBM rSony rS&P500 rIBM rSony

˜ ˜ ˜ ˜ ˜ ˜

Year Year

+0.2631 ’0.2124 ’0.1027 +0.3101 +0.3811 +0.3905

1991 1997

+0.0446 ’0.4336 ’0.0037 +0.2700 +0.7624 ’0.2028

1992 1998

+0.0706 +0.1208 +0.4785 +0.1953 +0.1701 +2.9681

1993 1999

’0.0154 +0.3012 +0.1348 ’0.1014 ’0.2120 ’0.5109

1994 2000

+0.3411 +0.2430 +0.1046 ’0.1304 +0.4231 ’0.3484

1995 2001

+0.2026 +0.6584 +0.0772 ’0.2337 ’0.3570 ’0.0808

1996 2002

Source: Yahoo!Finance. (Sidenote: Sony is the SNE ADR.) Numbers are quoted in percent. You will be working with

these returns throughout the rest of the investments part.

You should ¬rst understand risk and reward, presuming it is now January 1, 2003. Although Pretend that historical

rates of return are

you are really interested in the returns of 2003 (and beyond), unfortunately all you have are

representative of future

these historical rates of return. So, you have to make the common presumption that historical returns. Introduce tilde

returns are good indicators of future returns. Further, let us also presume that each year was an notation.

equally likely outcome drawn from an underlying statistical process, so that each year is equally

informative to us. Formally, future returns are random variables, because their outcomes are

not yet known. Recall from Sections 5·1 and 12·3.E that you can denote a random variable with

a tilde over it, to distinguish it from an ordinary non-random variable, e.g.,

(13.1)

rS&P500 , rIBM , rSony .

˜ ˜ ˜

However, because you only know historical rates of return, you shall use the historical data

series in place of the “tilde-d” future variables.

Conceptually, there is a big di¬erence between average historical realizations and expected Please realize that you

are going to make a leap

future realizations. Just because the long-run historical monthly rate of return average was,

of faith here.

say, 10% does not mean that it will be 10% in the future. But, practically, there is often not

much di¬erence, because you have no choice but to pretend that the historical return series

is representative of the distribution of future returns. To draw the distinction, statisticians

often name the unknown expected value by a greek character, and the historical outcome that

is used to estimate it by its corresponding English character. For example, µ would be the

expected future mean, m would be the historical mean; σ would be the expected future stan-

dard deviation, s would be the historical standard deviation. After having drawn this careful

conceptual distinction, the statisticians then tell you that they will use the historical mean and

historical standard deviation as stand-ins (estimators) for the future mean (the expected value)

and standard deviation (the expected standard deviation). This is how it should be done, but it

can become very cumbersome when dealing with many statistics for many variables. Because

we shall mostly work with historical statistics and then immediately pretend that they are our

expectations for the future, let us use a more casual notation: when we claim to compute E(˜), r

the notation of which would suggest an expected return, we really compute only the historical

mean (unless otherwise stated). That is, in this ¬nance book, we will keep the distinction rather

vague.

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308 Chapter 13. Statistics.

Digging Deeper: As with a history, you can think of the tilde as representing not just one month™s outcome,

but this distribution of historical outcomes. As of today, next month™s rate of return can be anything. (We do not

yet have just one number for it.)

13·3. Univariate Statistics

13·3.A. The Mean

If you had invested in a random year, what would you have expected to earn? The reward is

Everyone knows how to

compute an average. measured by the single most important statistic, the expected rate of return (also called mean

or average rate of return). You surely have computed this at one time or another, so let™s just

state that our means are

(13.2)

E (˜S&P500 ) = 0.101 , E (˜IBM ) = 0.154 , E (˜Sony ) = 0.242 .

r r r

Sony was clearly the best investment over these 12 years (mostly due to its spectacular perfor-

mance in 1999), but IBM and the S&P500 did pretty well, too.

Chapter 4 already showed that the average rate of return is not the annualized rate of

Digging Deeper:

return. An investment in Sony beginning in 1999 for three years would have had a compound three-year rate of

return of (1 + 297%) · (1 ’ 51%) · (1 ’ 35%) ≈ 26.5%, which is 8.1% annualized. Its average annual rate of return

is [297% + (’51%) + (’35%)]/3 ≈ 70.3%.

But, what causes the di¬erence? It is the year-to-year volatility! if the rate of return were 8.1% each year without

variation, the annualized and average rate of return would be the same. The year-to-year volatility negatively

a¬ects the annualized holding rates of return. For a given average rate of return, more volatility means less

compound and thus less annualized rate of return.

For purposes of forecasting a single year™s return, assuming that each historical outcome was equally likely, you

want to work with average rates of returns. For computing long-term holding period performance, you would

want to work with compound rates of return.

13·3.B. The Variance and Standard Deviation

How can you measure portfolio risk? Intuitively, how does the risk of the following investment

A Tale of three

investments. choices compare?

1. An investment in a bond that yields 10% per year for sure.

2. An investment that yields ’15% half the time, and +35% half the time, but only once per

year.

3. An investment in the S&P500.

4. An investment in IBM.

5. An investment in Sony.

You know that the ¬rst three investments have a mean rate of return of about +10% per annum.

But the mean tells you nothing about the risk.

You need a statistic that tells you how variable the outcomes are around the mean. Contestant

A naïve attempt at

measuring spread fails. #1 has no variability, so it is clearly safest. What about your other contestants? For contestant

#2, half the time, the investment outcome is 25% below its mean (of +10%); the other half, it is

25% above its mean. Measuring outcomes relative to their means (as we have just done) is so

common that it has its own name, deviation from the mean. So, can you average the deviations

from the mean to measure typical variability? Try it. With two years, and the assumption that

each year is an equally likely outcome,

(13.3)

Bad Variability Measure = 1/2 · (’25%) + 1/2 · (+25%) = 0 .

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309

Section 13·3. Univariate Statistics.

The average deviation is zero, because the minus and plus cancel. Therefore, the simple average The root cause of the

problem, and better

of the deviations is not a good measure of spread. You need a measure that tells you the typical

alternatives: variance

variability is plus or minus 25%, not plus or minus 0%. Such a better measure must recognize and standard deviation.

that a negative deviation from the mean is the same as a positive deviation from the mean. The

most common solution is to square each deviation from the mean in order to eliminate the

“opposite sign problem.” The average of these squared deviations is called the variance:

Var = 1/2 · (’25%)2 + 1/2 · (+25%)2 = 1/2 · 0.0625 + 1/2 · 0.0625 = 0.0625 . (13.4)

You can think of the variance as the “average squared deviation from the mean.” But the The variance and

standard deviation

variance of 0.0625 looks nothing like the intuitive spread from the mean, which is plus or minus

measure the expected

25%. However, if you take the square root of the variance, you get the standard deviation, spread of a random

variable.

√

(13.5)

Sdv = Var = 0.0625 = 25% ,

which has the intuitively pleasing correct order of magnitude of 25%. Although it is not really

correct, it is often convenient to think of the standard deviation as the “average deviation from

the mean.” (It would be more correct to call it “the square root of the average squared deviation

from the mean,” but this is unwieldy.)

Side Note: Aside from its uninterpretable value of 0.0625, there is a second and more important problem

interpreting the meaning of a variance. (It did not show up in this example, because rates of return are unitless.)

If you are interested in the variability of a variable that has units, like dollars or apples, the units of the variable

are usually uninterpretable. For example, if you receive either $10 or $20, the deviation from the mean is either

’$5 or +$5, and the variance is ($5)2 = $2 25, not $25”the same way by which multiplying 2 meters by 2

meters becomes 4 square-meters, not 4 meters. Square-meters has a good interpretation (area); dollars-squared

does not. The standard deviation takes the square root of the variance, and therefore returns to the same

units (dollars) as the original series, $2 25 = $5 in the example. Note also that I sometimes use “x%%” to denote

x·(%2 )”otherwise, there may be confusion whether x%2 means (x%)2 or x·(%2 ). So, 1%% is 0.01·0.01 = 0.0001,

√

and 1%% = 1%.

Because the standard deviation is just the square root of the variance, if the variance of one Standard deviation and

variance have the same

variable is higher than the variance of another variable, so is its standard deviation.

ordering.

Table 13.2. Deviations From the Mean for S&P500, IBM, and Sony

rS&P500 rIBM rSony rS&P500 rIBM rSony

˜ ˜ ˜ ˜ ˜ ˜

Year Year

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

1991 1997

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

1992 1998

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

1993 1999

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

1994 2000

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

1995 2001

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

1996 2002

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

For contestant #3, the S&P500, you must estimate the variability measures from the historical Sometimes-important

nuisance: For historical

data series. Recall that to compute the variance, you subtract the mean from each outcome,

data, do not divide the

square the deviations, and then average them. To compute the standard deviation, you then squared deviations by N,

take the square-root of the variance. Table 13.2 does most of the hard work for you, giving but by N-1.

you deviations from the mean for the S&P500, IBM, and Sony. Computing variances from these

deviations is now straightforward: square and average. Alas, there is one nuisance complica-

tion: because there is a di¬erence between historical realizations (which you have) and true

expected future outcomes (which you do not have [we pretended to know this perfectly in the

“-15%,+35%” example]), statisticians divide by N ’ 1, not by N. Therefore, the estimated vari-

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310 Chapter 13. Statistics.

ance (divides by N ’ 1) is a little bit higher than the average squared deviation from the mean

(divides by N).

Side Note: The reason for this N ’ 1 adjustment is that the future standard deviation is not actually known,

but only estimated, given the historical realizations. This “extra uncertainty” is re¬‚ected by the smaller divisor,

which in¬‚ates the uncertainty estimate.

The best intuition comes from a sample of only one historical data point: What would you believe the variability

would be if you only know one realization, say 10%? In this case, you know nothing about variability. If you

divided the average squared deviation (0) by N, the variance formula would indicate a zero variability. This is

clearly wrong. If anything, you should be especially worried about variability, for you now know nothing about

it. Dividing by N ’ 1 = 0, i.e., V = 0/0, indicates that estimating variability from one sample point makes no

ar

sense.

The division by N rather than N ’ 1 is not important when there are many historical sample data points, which

is usually the case in ¬nance. Thus, most of the time, you could use either method, although you should remain

consistent. This book uses the N ’ 1 statistical convention, if only because it allows checking computations

against the built-in formulas in Excel and other statistical packages.

So, to obtain the variance of one investment series, square each deviation from the mean, add

Executing the formulas

on historical data and these squared terms, and dividing by 11 (N ’ 1).

dividing by N-1 yields

variances and standard

(+0.1620)2 + (’0.0565)2 + ... + (’0.3348)2

Var(˜S&P500 ) = = 0.0362 ;

r

deviations.

11

(’0.3661)2 + (’0.5874)2 + ... + (’0.5105)2

Var(˜IBM ) = = 0.1503 ;

r

11

(13.6)

(’0.3448)2 + (’0.2458)2 + ... + (’0.3228)2

Var(˜Sony ) = = 0.8149 ;

r

11

[˜t=0 ’ E (˜)]2 + [˜t=1 ’ E (˜)]2 + ... + [˜t=T ’ E (˜)]2

r r r r r r

Var(˜) =

r .

T ’1

The square roots of the variances are the standard deviations:

√

Sdv(˜S&P500 ) = 3.62% = 19.0%

r ;

√

(13.7)

Sdv(˜IBM ) = 15.03% = 38.8% ;

r

√

Sdv(˜Sony ) = 81.49% = 90.3% .

r

So, returning to your original question. If you were to line up your three potential investment

choices”all of which o¬ered about 10% rate of return”it appears that the S&P500 contestant

#3 with its 19% risk is a safer investment than the “-15% or +35%” contestant #2 with its 25% risk.

As for the other two investments, IBM with its higher 15%/year average rate of return is also

riskier, having a standard deviation of “plus or minus” 38.8%/year. (Calling it “plus or minus”

is a common way to express standard deviation.) Finally, Sony was not only the best performer

(with its 24.2%/year mean rate of return), but it also was by far the riskiest investment. It had

a whopping 90.3%/year standard deviation. (Like the mean, the large standard deviation is

primarily caused by one outlier, the +297% rate of return in 1999.)

You will see that the mean and standard deviation play crucial roles in the study of investments”

Preview: Use of risk

measures in ¬nance. your ultimate goal will be to determine the portfolio that o¬ers the highest expected reward

for the lowest amount of risk. But mean and standard deviation make interesting statistics in

themselves. From 1926 to 2002, a period with an in¬‚ation rate of about 3% per year, the annual

risk and reward of some large asset-class investments were approximately

E(˜) Sdv(˜)

r r

Asset Class

Short-Term U.S. Government Treasury Bills 4% 3%

Long-Term U.S. Government Treasury Bonds 5.5% 10%

Long-Term Corporate Bonds 6% 9%

Large Firm Stocks 10% 20%

Small Firm Stocks 15% 30%

¬le=statistics-g.tex: RP

311

Section 13·4. Bivariate Statistics: Covariation Measures.

Corporate bonds had more credit (i.e., default) risk than Treasury bonds, but were typically

shorter-term than long-term government bonds, which explains their lower standard deviation.

For the most part, it seems that higher risk and higher reward went hand-in-hand.

Solve Now!

Q 13.1 Use Excel to con¬rm all numbers computed in this section.

Q 13.2 The annual rates of return on the German DAX index were

+0.1286 ’0.0706 +0.4624 ’0.0754

1991 1994 1997 2000

’0.0209 +0.0699 +0.1842 ’0.1979

1992 1995 1998 2001

+0.4670 +0.2816 +0.3910 ’0.4394

1993 1996 1999 2002

Compute the mean, variance, and standard deviation of the DAX.

13·4. Bivariate Statistics: Covariation Measures

13·4.A. Intuitive Covariation

Before we embark on more number-crunching, let us ¬rst ¬nd some intuitive examples of

variables that tend to move together, variables that have nothing to do with one another, and

variables that tend to move in opposite directions.

Table 13.3. Covariation Examples

Negative Zero (or Low) Positive

Agility vs. Height vs.

IQ vs.

Weight Gender Basketball Scoring

Wealth vs.

Wealth vs. Wealth vs.

Disease Tail when ¬‚ipping coin Longevity

Age vs. Age vs. Age vs.

Flexibility Blood Type Being CEO

Sunspots vs. Grasshoppers vs.

Snow vs.

Temperature Temperature Temperature

Your Net Returns vs. IBM Returns in 1999 vs. Returns on S&P500 vs.

Broker Commissions Paid Exxon Returns in 1986 Returns on IBM

Personal statistics (such as weight) apply only to adults. Returns are rates of return on stock investments, net of

commissions.

Table 13.3 o¬ers some such examples of covariation. For example, it is easier to score in An example of a positive

covariation.

basketball if you are 7 feet tall than if you are 5 feet tall. So, there is a positive covariation

between individuals™ heights and their ability to score. It is not perfect covariation”there

are short individuals who can score a lot (witness John Stockton, the Utah Jazz basketball

player, who despite a height of “only” 6-1 would almost surely score more points than the

tallest students in my class), and there are many tall people who could not score if their lives

depended on it. It is only on average that taller players score more. In this example, the

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312 Chapter 13. Statistics.

reason for the positive covariation is direct causality”it is easier to hit the basket when you

are as tall as the basket”but correlation need not come from causality. For example, there is

also a positive covariation between shoe size and basketball scoring. It is not because bigger

feet make it easier to score, but because taller people have both bigger shoe sizes and higher

basketball scores. Never forget: causality induces covariation, but not vice-versa.

Zero covariation usually means two variables have nothing to do with one another. For exam-

An example of a zero and

a negative covariation. ple, there is strong evidence that there is practically no covariation between gender and IQ.

Knowing only the gender would not help you a bit in guessing the person™s IQ, and vice-versa.

(Chauvinists guessing wrong, however, tend to have lower IQs.) An example of negative covari-

ation would be agility vs. weight. It is usually easier for lighter people to overcome the intrinsic

inertia of mass, so they tend to be more agile: therefore, more weight tends to be associated

with less agility.

13·4.B. Covariation: Covariance, Correlation, and Beta

Your goal now is to ¬nd measures of covariation that are positive when two variables tend to

A measure of

dependence should be move together; that are zero when two variables have nothing to do with one another; and that

positive when two

are negative when one variable tends to be lower when the other variable tends to be higher. We

variables tend to move

will consider three possible measures of covariation: covariance, correlation, and beta. Each

together, negative if

they tend to move in

has its advantages and disadvantages.

opposite directions.

Illustrative Data Series

Start with the nine data series in Table 13.4, the returns on nine assets that I have made up. Let

Work with nine made-up

data series. us use asset A as our base asset and consider how assets C through J relate to A. You want to

ask such questions as “if data series A were the rate of return on the S&P500, and data series C

were the rate of return on IBM, then how do the two return series covary?” This question is

important, because it will help you determine investment opportunities that have lower risk.

So try to determine how each series covaries with A. Doing this graphically makes it easier, so

Covariation is best to

understand with an Figure 13.1 plots the points. If you look at it, you can see that you shall need more than just one

extreme, albeit absurd

covariation statistic: you need one statistic that tells you how much two variables are related

example. But you need