Chapter 11

Chapter 11

Risk and Return in Capital Markets

Chapter Outline

11.1 A First Look at Risk and Return11.2 Historical Risks and Returns of Stocks 11.3 The Historical Tradeoff Between Risk and

Return11.4 Common Versus Independent Risk 11.5 Diversification in Stock Portfolios

2

Learning Objectives Identify which types of securities have historically had the

highest returns and which have been the most volatile Compute the average return and volatility of returns from a

set of historical asset prices Understand the tradeoff between risk and return for large

portfolios versus individual stocks Describe the difference between common and independent

risk Explain how diversified portfolios remove independent risk,

leaving common risk as the only risk requiring a risk premium

3

11.1 A First Look at Risk and Return

Consider how an investment would have grown if it were invested in each of the following from the end of 1925 until the beginning of 2010: Standard & Poor’s 500 (S&P 500) Small Stocks World Portfolio Corporate Bonds Treasury Bills

4

Figure 11.1 Value of $100 Invested at the End of 1925 in U.S. Large Stocks (S&P 500), Small Stocks, World Stocks, Corporate Bonds, and Treasury Bills

5

Table 11.1 Realized Returns, in Percent (%) for Small Stocks, the S&P 500, Corporate Bonds, and Treasury Bills, Year-End 1925–1935

6

11.2 Historical Risks and Returns of Stocks

Computing Historical Returns Realized Returns: the total return that occurs over

a particular time period. Individual Investment Realized Returns

The realized return from your investment in the stock from t to t+1 is:

1 1 1 11

t t t t t tt

t t t

Div P P Div P PRP P P

Dividend Yield Capital Gain Yield

(Eq. 11.1)

7

Example 11.1 Realized Return

Problem: Microsoft paid a one-time special dividend of $3.08 on

November 15, 2004. Suppose you bought Microsoft stock for $28.08 on November 1, 2004 and sold it immediately after the dividend was paid for $27.39. What was your realized return from holding the stock?

8

Example 11.1 Realized Return

Execute: Using Eq. 11.1, the return from Nov 1, 2004 until Nov 15,

2004 is equal to

This 8.51% can be broken down into the dividend yield and the capital gain yield:

1 11

3.08 (27.39 28.08) 0.0851, or 8.51%28.08

t t tt

t

Div P PR

P

1

1

3.08Dividend Yield = .1097, or 10.97%28.08

27.39 28.08Capital Gain Yield = 0.0246, or 2.46%

28.08

t

t

t t

t

DivP

P PP

9

Example 11.1a Realized Return

Problem: Health Management Associates (HMA) paid a one-time

special dividend of $10.00 on March 2, 2007. Suppose you bought HMA stock for $20.33 on February 15, 2007 and sold it immediately after the dividend was paid for $10.29. What was your realized return from holding the stock?

10

Example 11.1a Realized Return

Execute: Using Eq. 11.1, the return from February 15, 2007 until

March 2, 2007 is equal to

This -0.2% can be broken down into the dividend yield and the capital gain yield:

%2.0or,002.033.20

33.2029.1000.10P

PPDivRt

t1t1t1t

1

1

10.00 0.4919, 49.19%20.33

10.29 20.33 0.4939, 49.39%20.33

t

t

t t

t

DivDividend Yield orP

P PCapital GainYield orP

11

Example 11.1b Realized Return

Problem: Limited Brands paid a one-time special dividend of $3.00 on

December 21, 2010. Suppose you bought LTD stock for $29.35 on October 18, 2010 and sold it immediately after the dividend was paid for $30.16. What was your realized return from holding the stock?

12

Example 11.1b Realized Return

Execute: Using Eq. 11.1, the return from October 18, 2010 until

December 21, 2010 is equal to

This 12.98% can be broken down into the dividend yield and the capital gain yield:

1 11

3.00 30.16 29.35 0.1298,or 0.12.98%29.35

t t tt

t

Div P PR

P

1

1

3.00 0.1022,or10.22%29.35

30.16 29.35 0.0276,or 2.76%29.35

t

t

t t

t

DivDividend YieldP

P PCapital GainYieldP

13


Computing Historical Returns Individual Investment Realized Returns

For quarterly returns (or any four compounding periods that make up an entire year) the annual realized return, Rannual, is found by compounding:

1 2 3 41 (1 )(1 )(1 )(1 )annualR R R R R (Eq. 11.2)

14

Example 11.2 Compounding Realized Returns

Problem: Suppose you purchased Microsoft stock (MSFT) on Nov 1,

2004 and held it for one year, selling on Oct 31, 2005. What was your annual realized return?

15


Plan (cont’d):

Next, compute the realized return between each set of dates using Eq. 11.1. Then determine the annual realized return similarly to Eq. 11.2 by compounding the returns for all of the periods in the year.

16


Execute: In Example 11.1, we already computed the realized return

for Nov 1, 2004 to Nov 15, 2004 as 8.51%. We continue as in that example, using Eq. 11.1 for each period until we have a series of realized returns. For example, from Nov 15, 2004 to Feb 15, 2005, the realized return is

1 11

0.08 (25.93 27.39) 0.0504, or 5.04%27.39

t t tt

t

Div P PR

P

17


Execute (cont’d): The table below includes the realized return at each period.

18


Execute (cont’d): We then determine the one-year return by compounding.

1 2 3 4 51 (1 )(1 )(1 )(1 ) 11 (1.0851)(0.9496)(0.9861)(1.0675)(0.9473) 1.0275

1.0275 1 .0275 or 2.75%

annual

annual

annual

R R R R R RR

R

19


Evaluate: By repeating these steps, we have successfully computed

the realized annual returns for an investor holding MSFT stock over this one-year period. From this exercise we can see that returns are risky. MSFT fluctuated up and down over the year and ended-up only slightly (2.75%) at the end.

20

Example 11.2a Compounding Realized Returns

Problem: Suppose you purchased Health Management Associate’s

stock (HMA) on March 16, 2006 and held it for one year, selling on March 15, 2007. What was your realized return?

21


Plan (cont’d):

Next, compute the realized return between each set of dates using Eq. 11.1. Then determine the annual realized return similarly to Eq. 11.2 by compounding the returns for all of the periods in the year.

Date Price Dividend16-Mar-06 21.1510-May-06 20.70 0.06

9-Aug-06 20.62 0.068-Nov-06 19.39 0.06

15-Feb-07 20.332-Mar-07 10.29 10.00

15-Mar-07 11.07

22


Execute: In Example 11.1a, we already computed the realized return

for February 15, 2007 to March 2, 2007 as -.2%. We continue as in that example, using Eq. 11.1 for each period until we have a series of realized returns. For example, from August 9, 2006 to November 8, 2006, the realized return is

%67.5or,0567.062.20

62.2039.1906.0P

PPDivRt

t1t1t1t

23


Execute (cont’d): The table below includes the realized return at each period.

Date Price Dividend Return16-Mar-06 21.1510-May-06 20.70 0.06 -1.84%

9-Aug-06 20.62 0.06 -0.10%8-Nov-06 19.39 0.06 -5.67%

15-Feb-07 20.33 4.85%2-Mar-07 10.29 10.00 -0.20%

15-Mar-07 11.07 7.58%

24


Execute (cont’d): We then determine the one-year return by compounding.

%11.4or0411.10411.1)076.1)(998.0)(048.1(943.0)999.0)(982.0(1)1)(1)(1(1)1)(1(1

annual

annual

654321annual

RR

RRRRRRR

25


Average Annual Returns Average Annual Return of a Security

1 21 ( ... )TT

R R R R (Eq. 11.3)

26

Figure 11.2 The Distribution of Annual Returns for U.S. Large Company Stocks (S&P 500), Small Stocks, Corporate Bonds, and Treasury Bills, 1926–2009

27

Figure 11.3 Average Annual Returns in the U.S. for Small Stocks, Large Stocks (S&P 500), Corporate Bonds, and Treasury Bills, 1926–2009

28


The Variance and Volatility of Returns: Variance

Standard Deviation

(Eq. 11.4) 2 2 21 2

1( ) ( ) ... ( )

1 TVar R R R R R R RT

( )SD R Var R (Eq. 11.5)

29

Example 11.3 Computing Historical Volatility

Problem: Using the data from Table 11.1, what is the standard

deviation of the S&P 500’s returns for the years 2005-2009?

30


Execute: In the previous section we already computed the average

annual return of the S&P 500 during this period as 3.1%, so we have all of the necessary inputs for the variance calculation:

Applying Eq. 11.4, we have:

2 2 21 2

2 22 2 21

5 1

1( ) ( ) ( ) ... ( )1

(.049 .031) (.158 .031) (.055 .031) 0.370 .031 .265 .031

.058

TVar R R R R R R RT

31


Execute (cont'd): Alternatively, we can break the calculation of this equation

out as follows:

Summing the squared differences in the last row, we get 0.233.

Finally, dividing by (5-1=4) gives us 0.233/4 =0.058 The standard deviation is therefore:

( ) ( ) .058 0.241,or 24.1%SD R Var R

32

Example 11.3a Computing Historical Volatility

Problem: Using the data from Table 11.1, what is the standard

deviation of small stocks’ returns for the years 2005-2009?

33


Solution:Plan:

First, compute the average return using Eq. 11.3 because it is an input to the variance equation. Next, compute the variance using Eq. 11.4 and then take its square root to determine the standard deviation as shown in Eq. 11.5.

Year 2005 2006 2007 2008 2009Small Stocks’ Return 5.69% 16.17% -5.22% -36.72% 28.09%

34


Execute: Using Eq. 11.3, the average annual return for small stocks

during this period is:

We now have all of the necessary inputs for the variance calculation:

Applying Eq. 11.4, we have:

2 2 21 2

2 22 2 21

5 1

1( ) ( ) ( ) ... ( )1

(.0569 .0171) (.1671 .0171) ( .0522 0171) .3672 .0171 .2809 .0171

.0615

TVar R R R R R R RT

1(.0569 .1671 .0522 .3672+.2809) .01715

R

35


Execute (cont'd): Alternatively, we can break the calculation of this equation

out as follows:

Summing the squared differences in the last row, we get 0.2462.

Finally, dividing by (5-1=4) gives us 0.2462/4 =0.0615 The standard deviation is therefore:

( ) ( ) 0.0615 0.2480, 24.80%SD R Var R or

2005 2006 2007 2008 2009Return 0.0569 0.1671 -0.0522 -0.3672 0.2809Average 0.0171 0.0171 0.0171 0.0171 0.0171Difference 0.0398 0.15 -0.0693 -0.3843 0.2638Squared 0.0016 0.0225 0.0048 0.1477 0.0696

36

Figure 11.4 Volatility (Standard Deviation) of U.S. Small Stocks, Large Stocks (S&P 500), Corporate Bonds, and Treasury Bills, 1926–2009

37


The Normal Distribution 95% Prediction Interval

About two-thirds of all possible outcomes fall within one standard deviation above or below the average

Average (2 x standard deviation)R (2 x SD R )

(Eq. 11.6)

38

Figure 11.5 Normal Distribution

39

Example 11.4 Confidence Intervals

Problem: In Example 11.3 we found the average return for the S&P

500 from 2005-2009 to be 3.1% with a standard deviation of 24.1%. What is a 95% confidence interval for 2010’s return?

40


Solution:Plan: We can use Eq. 11.6 to compute the confidence interval.

41


Execute: Using Eq. 11.6, we have: Average ± (2 standard deviation) = 3.1% – (2 24.1%) to

3.1% + (2 24.1% )= –45.1% to 51.3%.

42


Evaluate: Even though the average return from 2005 to 2009 was

3.1%, the S&P 500 was volatile, so if we want to be 95% confident of 2010’s return, the best we can say is that it will lie between –45.1% and +51.3%.

43

Example 11.4a Confidence Intervals

Problem: The average return for small stocks from 2005-2009 was

1.71% with a standard deviation of 24.8%. What is a 95% confidence interval for 2010’s return?

44


Solution:Plan: We can use Eq. 11.6 to calculate the confidence interval.

45


Execute: Using Eq. 11.6, we have:

Average 2 standard deviation 1.71% (2 24.8%) to1.71% (2 24.8%)47.89% to50.77%

46


Evaluate: Even though the average return from 2005-2009 was 1.71%,

small stocks were volatile, so if we want to be 95% confident of 2010’s return, the best we can say is that it will lie between -47.89% and +50.77%.

47

Example 11.4b Confidence Intervals

Problem: The average return for corporate bonds from 2005-2009 was

6.49% with a standard deviation of 7.04%. What is a 95% confidence interval for 2010’s return?

48


Solution:Plan: We can use Eq. 11.6 to calculate the confidence interval.

49


Execute: Using Eq. 11.6, we have:

Average 2 standard deviation 6.49% (2 7.04%) to 6.49% (2 7.04%)7.59% to 20.57%

50


Evaluate: Even though the average return from 2005-2009 was 6.49%,

corporate bonds were volatile, so if we want to be 95% confident of 2010’s return, the best we can say is that it will lie between -7.59% and +20.57%.

51

Table 11.2 Summary of Tools for Working with Historical Returns

52

11.3 Historical Tradeoff between Risk and Return

The Returns of Large Portfolios Investments with higher volatility, as measured by

standard deviation, tend to have higher average returns

53

Figure 11.6 The Historical Tradeoff Between Risk and Return in Large Portfolios, 1926–2010

54

11.3 Historical Tradeoff between Risk and Return

The Returns of Individual Stocks Larger stocks have lower volatility overall Even the largest stocks are typically more volatile

than a portfolio of large stocks The standard deviation of an individual security

doesn’t explain the size of its average return All individual stocks have lower returns and/or

higher risk than the portfolios in Figure 11.6

55

11.4 Common Versus Independent Risk

Types of Risk Common Risk Independent Risk Diversification

56

Table 11.3 Summary of Types of Risk

57

11.5 Diversification in Stock Portfolios

Unsystematic Versus Systematic Risk Stock prices are impacted by two types of news:

1. Company or Industry-Specific News2. Market-Wide News

Unsystematic Risk Systematic Risk

58

Figure 11.8 The Effect of Diversification on Portfolio Volatility

59


Diversifiable Risk and the Risk Premium The risk premium for diversifiable risk is zero

Investors are not compensated for holding unsystematic risk

60

Table 11.4 Systematic Risk Versus Unsystematic Risk

61


The Importance of Systematic Risk The risk premium of a security is determined by its

systematic risk and does not depend on its diversifiable risk

62


The Importance of Systematic Risk There is no relationship between volatility and

average returns for individual securities

63

64

References

65

References Portfolio variance(assume )

p

BA

BAB

BA

AA

BA

B

BBA

BpA

BA

BpBBAAp

BA

BAp

BA

BpA

BBAAp

BAAA

BBAA

BABABBAA

BAABBABBAA

ABBABBA

BBAABA

BBBAAAppp

RERERERE

REREREWREWRE

Wif

W

WWW

WW

WWWW

WWWW

WWW

RERREREWW

REREWREREWRERE

1

0

1

2

2

2W

2

2

2

2222

2222

2222A

222222

1AB

66

ReferencesNo Riskless Portfolio( )Optimal investment weighted as follow: ；

0p

BA

BAW

*

BA

ABW

*

67

References Portfolio variance(assume )

Minimum Variance Portfolio

11 AB

ABBABBA

BBAABA

BBBAAAppp

WWW

RERREREWW

REREWREREWRERE

2W

2 2222

A

222222

ABAABAA

ABBABBA

BBAABA

BBBAAAppp

WWW

WWW

RERREREWW

REREWREREWRERE

121W

2W

2

2222A

2222A

222222

ABBA

ABBA

ABBABBAA

ABAABABAAAA

p

W

W

WWWWW

2

222

0)1(212)1(122

22

2*

222

222

68

References

： non-systematic risk :Systematic risk If portfolio satisfy , then portfolio

variance would reduce to

ijij

ij

ji

N

i

N

jij

N

iij

ji

N

i

N

j

N

i

ijj

ji

N

i

N

jiip

NNN

NNN

NN

NNNN

WW

11N11

N1

111

1111N1

W

22

22

2

1 1

2

1

22

1 1

2

1

2

1 1

2N

1i

2i

2

2

N1

ijN

11

ijijpN

010lim 22

N

Chapter 11

Documents

realized returnproblem

realized returnexecute

dividend yield

common risk

total return

average return

risk premium33

historical risks