4.1Portfolio Weights

To develop the skills to implement mean-variance analysis, we need to develop math-

ematical ways of representing portfolios.

The portfolio weightfor stock j, denoted x, is the fraction of a portfolio’s wealth

j

held in stock j; that is,

Dollars held in stock j

x

^jDollar value of the portfolio

By definition, portfolio weights must sum to 1.

The Two-Stock Portfolio

Example 4.1 illustrates how to compute portfolio weights for a two-stock portfolio.

Example 4.1:Computing Portfolio Weights fora Two-Stock Portfolio

A portfolio consists of $1 million in IBM stock and $3 million in AT&T stock.What are the

portfolio weights of the two stocks?

Answer:The portfolio has a total value of $4 million.The weight on IBM stock

is$1,000,000/$4,000,000 .25 or 25 percent, and the weight on AT&T stock is

$3,000,000/$4,000,000 .75 or 75 percent.

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100Part IIValuing Financial Assets

Short Sales and Portfolio Weights.In Example 4.1, both portfolio weights are pos-

itive. However, investors can sell shortcertain securities, which means that they can

sell investments that they do not currently own. To sell short common stocks or bonds,

the investor must borrow the securities from someone who owns them. This is known

as taking a short positionin a security. To close out the short position, the investor

buys the investment back and returns it to the original owner.

The Mechanics of Short Sales of Common Stock: An Illustration

Consider three investors, Mike, Leslye, and Junior, all three of whom have accounts at

Charles Schwab, the brokerage firm. Mike decides to sell short McGraw-Hill stock, selling

100 shares that he does not own to Junior. Legally, Mike has to deliver 100 shares, a phys-

ical piece of paper, to Junior within three working days. Schwab personnel enter the vault

where shares are kept (in what is known as “street name”) and remove 100 shares of

McGraw-Hill that are owned by Leslye. They don’t even tell Leslye about it. The 100 shares

are deposited in Junior’s Schwab account. Everyone is happy. Mike has sold short McGraw-

Hill and delivered the physical shares to Junior. Junior has shares of McGraw-Hill that he

bought. And Leslye still thinks she owns McGraw-Hill. Because no one is going to tell her

that the shares are gone,she doesn’t care either. From an accounting perspective, even

Schwab is happy. They started out with 100 shares among their customers. They credited

Junior’s and Leslye’s accounts with 100 shares (even though Leslye’s are missing) and gave

Mike a negative 100 share allocation.

Aminor problem arises when a dividend needs to be paid. McGraw-Hill pays dividends

only to holders of the physical shares. Hence, Leslye thinks McGraw-Hill will pay her a

dividend, but her dividend is going to Junior. Schwab solves the problem by taking the div-

idend out of the cash in Mike’s account and depositing it in Leslye’s account. Again, the

accounting adds up. Junior gets a dividend from McGraw-Hill, Mike gets a negative divi-

dend, and Leslye gets a dividend (through Mike).

Has Leslye given up any rights by allowing her shares to be borrowed? The answer is

she has, because, when it comes time to vote as a shareholder, only Junior, the holder of

the physical shares, can vote. Hence, Leslye has to give Schwab permission to borrow her

shares by signing up for a margin account and allowing the shares to be held in Schwab’s

“street name.”

What if Leslye wants to sell her shares? She doesn’t have them anymore. No problem

for Schwab. They’ll simply borrow them from some other customer. However, if there are

too many short sales and not enough customers from whom to borrow shares, Schwab may

fail to execute Leslye’s trade by physically delivering the shares in three days to the per-

son who bought her shares. This is called a short squeeze. In a short squeeze, Schwab has

the right to force Mike to close out his short position by buying physical shares of McGraw-

Hill and delivering them on Leslye’s sale.

To sell short certain other investments, one takes a position in a contract where

money is received up front and paid back at a later date. For example, borrowing from

a bank can be thought of as selling short, or equivalently, taking a negative position in

an investment held by the bank—namely, your loan. For the same reason, a corpora-

tion that issues a security (for example, a bond) can be thought of as having a short

position in the security.

Regardless of the mechanics of selling short, it is only relevant for our purposes

to know that selling short an investment is equivalent to placing a negative portfolio

weight on it.In contrast, a long position, achieved by buying an investment, has a pos-

itive portfolio weight. To compute portfolio weights when some investments are sold

short, sum the dollar amount invested in each asset of the portfolio, treating shorted

(or borrowed) dollars as negative numbers. Then divide each dollar investment by the

sum. For example, a position with $500,000 in a stock and $100,000 borrowed from a

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Chapter 4

Portfolio Tools

101

bank has a total dollar investment of $400,000 ( $500,000 $100,000). Dividing

$500,000 and $100,000 by the total dollar investment of $400,000 yields the portfo-

lio weights of 1.25 and .25 on the stock and the bank investment, respectively. Note

that these sum to 1.

Feasible Portfolios.To decide which portfolio is best, it is important to mathemati-

cally characterize the universe of feasible portfolios, which is the set of portfolios that

one can invest in. For example, if you are able to invest in only two stocks and can-

not sell short either stock, then the feasible portfolios are characterized by all pairs of

nonnegative weights that sum to 1. If short sales are allowed, then any pair of weights

that sums to 1 characterizes a feasible portfolio.

Example 4.2 illustrates the concept of feasible portfolio weights.

Example 4.2:Feasible Portfolio Weights

Suppose the world’s financial markets contain only two stocks, IBM and AT&T.Describe the

feasible portfolios.

Answer:In this two-stock world, the feasible portfolios consist of any two numbers, x

IBM

and x, for which x 1 x.Examples of feasible portfolios include

ATTATTIBM

1.	x	.5	x	.5
	IBM		ATT
2.	x	1	x	0
	IBM		ATT
3.	x	2.5	x	1.5
	IBM		ATT
4.	x	2	x	1 2
	IBM		ATT
5.	x	1/3	x	4/3
	IBM		ATT

An infinite number of such feasible portfolios exist because an infinite number of pairs of

portfolio weights solve xx 1.

ATTIBM

The Many-Stock Portfolio

The universe of securities available to most investors is large. Thus, it is more realis-

tic to consider portfolios of more than two stocks, as in Example 4.3.

Example 4.3:Computing Portfolio Weights fora Portfolio of Many Stocks

Describe the weights of a $40,000 portfolio invested in four stocks.The dollar amounts

invested in each stock are as follows:

Stock:	1	2	3	4
Amount:	$20,000	$5,000	$0	$25,000

Answer:Dividing each of these investment amounts by the total investment amount,

$40,000, gives the weights

x .5	x .125	x 0	x .625
1	2	3	4

For an arbitrary number of assets, we represent securities with algebraic notation (see

Exhibit 4.1). To simplify the language of this discussion, we refer to the risky assets

selected by an investor as stocks. However, the discussion is equally valid for all classes

of assets, including securities like bonds and options or real assets like machines, facto-

ries, and real estate. It is also possible to generalize the “risky assets” to include hedge

funds or mutual funds, in which case the analysis applies to portfolios of portfolios!

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102Part IIValuing Financial Assets

EXHIBIT4.1Notation

Term

Notation

Portfolio return	˜
	R
	p
Expected portfolio return (mean portfolio return)	¯or E(˜)
	RR
	pp
	22˜)
Portfolio return variance	or (R
	pp
Portfolio weight on stock i	x
	i
Stock i’s return	r
	˜
	i
Stock i’s expected return	r¯or Er)
	(˜
	ii
	2
Stock i’s return variance	or var(˜r)
	ii
Covariance of stock i and stock j’s returns	or cov(˜r, r˜)
	ijij
Correlation between stock i and stock j’s returns	or (r˜, r˜)
	ijij

The next few sections elaborate on each of the items in Exhibit 4.1.