4.7The Mean-Standard Deviation Diagram

It is time to put all this information together. This section focuses on a graph that will

help you understand how investors should view the trade-offs between means and vari-

ances when selecting portfolio weights for their investment decisions. This graph,

known as the mean-standard deviation diagram, plots the means (Y-axis) and the

standard deviations (X-axis) of all feasible portfolios in order to develop an under-

standing of the feasible means and standard deviations that portfolios generate. The

mastery of what you have just learned—namely, how the construction of portfolios

generates alternative mean and standard deviation possibilities—is critical to under-

standing this graph and its implications for investment and corporate financial man-

agement.

Combining a Risk-Free Asset with a Risky Asset in the Mean-Standard Deviation Diagram

When Both Portfolio Weights Are Positive.Results 4.1 and 4.3 imply that both the

mean and the standard deviation of a portfolio of a riskless investment and a risky

investment are, respectively, the portfolio-weighted averages of the means and standard

deviations of the riskless and risky investment when the portfolio weight on the risky

investment is positive. This implies that the positively weighted portfolios of the risk-

less and risky investment lie on a line segment connecting the two investments in the

mean-standard deviation diagram. This is a special case of the following more general

mathematical property:

Grinblatt250Titman: Financial	II. Valuing Financial Assets	4. Portfolio Tools	© The McGraw250Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

116	Part IIValuing Financial Assets

Result 4.6

Whenever the portfolio mean and standard deviations are portfolio-weighted averages of themeans and standard deviations of two investments, the portfolio mean-standard deviationoutcomes are graphed as a straight line connecting the two investments in the mean-standard deviation diagram.

As suggested above, the portfolios that combine a position in a risk-free asset (invest-

ment 1 with return r) with a long position in a risky investment (investment 2 with

f

return˜) have the property needed for Result 4.6 to apply and thus plot as the top-

r

2

most straight line (line AB) in Exhibit 4.3.

To demonstrate this, combine the equation for the mean and standard deviation of

such a portfolio

R xrxr	(4.10)
p1f22
x p22

to express the mean of the return of a portfolio of a riskless and a risky investment as

a function of its standard deviation. When the position is long in investment 2, the risky

investment, we can rearrange the standard deviation equation to read

p

x

²

2

implying that

p

x 1

¹

2

Substituting these expressions for xand xinto the expected return equation, equation

12

(4.10), yields

EXHIBIT4.3Mean-Standard Deviation Diagram: Portfolios of a Risky Asset and a Riskless Asset

Mean
Return
	Short in risk-free
R	B
p	asset

r	Long in both
2

Risk-free
asset
	Risky stock
Investment 1
	Investment 2
r fA

	Standard Deviation
2	of Return p

Short in risky asset

C

Grinblatt252Titman: Financial	II. Valuing Financial Assets	4. Portfolio Tools	© The McGraw252Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 4

Portfolio Tools

117

rr

R r^2f

p^fp

2

This equation is a straight line going from point A,representing the risk-free invest-

ment (at 0), through the point representing the risky investment (at ).

pp2

The line thus has an intercept of rand a slope of (rr)/.

f2f2

When the Risky Investment Has a Negative Portfolio Weight.When the new port-

folio is short in the risky investment, the formula for the standard deviation is

x

p22

implying

p

x

²

2

To make the weights sum to 1

p

x 1

¹

2

When these two portfolio weight expressions are substituted into the formula for the

expected return, equation (4.10), they yield

rr

R r^2f

p^fp

2

This also is the equation of a straight line. This line is graphed as the bottom line (AC)

in Exhibit 4.3. Since the risky investment has a negative weight and the graph has a

mean return for the risky investment that is larger than the return of the risk-free asset,

the slope of this line is negative. That is, the more one shorts the (risky) high expected

return asset, the lower is the portfolio’s expected return and the higher is the portfo-

lio’s standard deviation.

When the Risk-Free Investment Has a Negative Portfolio Weight.Consistent with

our earlier discussions, Exhibit 4.3 shows that when the risk-free security is sold short,

the portfolio will be riskier than a position that is 100 percent invested in ˜. In other

r

2

words:

Result 4.7

When investors employ risk-free borrowing (that is, leverage) to increase their holdings ina risky investment, the risk of the portfolio increases.

Depending on the portfolio weights, risk also may increase when the holdings of a risky

investment are increased by selling short a different risky security.

Portfolios of Two Perfectly Positively Correlated or Perfectly Negatively Correlated Assets

Perfect Positive Correlation.Exhibit 4.4 shows the plotted means and standard devi-

ations obtainable from portfolios of two perfectly positively correlated stocks. Points A

and Bon the line, designated, respectively, as “100% in stock 1” and “100% in stock

2,” correspond to the mean and standard deviation pairings achieved when 100 percent

of an investor’s wealth is held in one of the two investments. Bold segment ABgraphs

the means and standard deviations achieved from portfolios with positive weights on

the two perfectly correlated stocks. Moving up the line from point Atoward point B

Grinblatt254Titman: Financial	II. Valuing Financial Assets	4. Portfolio Tools	© The McGraw254Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

118	Part IIValuing Financial Assets

EXHIBIT4.4

Mean-Standard Deviation Diagram: Portfolios of Two Perfectly Positively Correlated Stocks

Mean Short in

Returnstock 1

R

^pLong in both

stocksB

r

²Short in

stock 2

100% in stock 2

A

r

1

C	100% in stock 1

		Standard
		Deviation
1	2	of Return
Short in stock 2		p

places more weight on the investment with the higher expected return (stock 2). Above

point B,the portfolio is selling short stock 1 and going “extra long” (weight exceeds1)

in stock 2. Below point A,the portfolio is selling short stock 2.

Exhibit 4.4 demonstrates that it is possible to eliminate risk—that is, to achieve

zero variance—with a portfolio of two perfectly positively correlated stocks.⁹To do

this, it is necessary to be long in one investment and short in the other in proportions

that place the portfolio at point C.

The graph of portfolios of two perfectly positively correlated, but risky, investments

has the same shape (a pair of straight lines) as the graph for a portfolio of a riskless and

a risky investment. This should not be surprising because a portfolio of the two perfectly

correlated investments is itself a riskless asset. The feasible means and standard devia-

tions generated from portfolios of, say, the riskless combination of stocks 1 and 2 on the

one hand, and stock 2 on the other, should be identical to those generated by portfolios

of stocks 1 and 2 themselves. Also, when the portfolio weights on both investments are

positive, the mean and the standard deviation of a portfolio of two perfectly correlated

investments are portfolio-weighted averages of the means and standard deviations of the

individual assets. Consistent with Result 4.6, such portfolios are on a line connecting the

two investments whenever the weighted average of the standard deviations is positive.

Perfect Negative Correlation.For similar reasons, a pair of perfectly negatively cor-

related risky assets graphs as a pair of straight lines. In contrast to a pair of perfectly

positively correlated investments, when two investments are perfectly negatively cor-

related, the investor eliminates variance by being long in both investments.

9

This also was shown in Example 4.13.

Grinblatt256Titman: Financial	II. Valuing Financial Assets	4. Portfolio Tools	© The McGraw256Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 4

Portfolio Tools

119

Example 4.16:Forming a Riskless Portfolio from Two Perfectly Negatively

Correlated Securities

Over extremely short time intervals, the return of IBM is perfectly negatively correlated with

the return of a put option on the company.The standard deviation of the return on IBM stock

is 18 percent per year, while the standard deviation of the return on the option is 54 per-

cent per year.What portfolio weights on IBM stock and its put option create a riskless invest-

ment over short time intervals?

Answer:The variance of the portfolio, using equation (4.9b), is

²x²²²²

.18.54(1 x)2x(1 x)(.18)(.54) [.18x.54(1 x)]

This expression is 0 when x .75.Thus, the weight on IBM stock is 3/4 and the weight on

the put option is 1/4.

The Feasible Means and Standard Deviations from Portfolios of Other Pairs of Assets

We noted in the last subsection that with perfect positive correlation, the standard devi-

ation of a portfolio with positive weights on both stocks equals the portfolio-weighted

average of the two standard deviations. Now, consider the case where risky investments

have less than perfect correlation ( 1 ). Result 4.2 states that the lower the correla-

tion, the lower the portfolio variance. Therefore, the standard deviation of a portfolio

with positive weights on both stocks is less than the portfolio-weighted average of the

two standard deviations, which gives the curvature to the left shown in Exhibit 4.5. The

EXHIBIT4.5

Mean Standard Deviation Diagram: Portfolios of Two Risky Securities withArbitrary Correlation,

Mean
Return	100% in stock 2
R
p
	= –1

< 0
	> 0
= 0		= 1

= –1

	Standard
100% in stock 1
	Deviation
	of Return
	p

Grinblatt259Titman: Financial	II. Valuing Financial Assets	4. Portfolio Tools	© The McGraw259Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

120Part IIValuing Financial Assets

degree to which this curvature occurs depends on the correlation between the returns.

Consistent with Result 4.2, the smaller the correlation, ,the more distended the cur-

vature. The ultimate in curvature is the pair of lines generated with perfect negative

correlation, 1, which is the smallest correlation possible.¹⁰

The combination of the two stocks that has minimum variance is a portfolio with

positive weights on both stocks only if the correlation between their returns is not too

large. For sufficiently large correlations, the minimum variance portfolio, the portfo-

lio of risky investments with the lowest variance, requires a long position in one invest-

ment and a short position in the other.¹¹