Chapter 10

```Chapter 11
Risk and Return in Capital Markets
Chapter Outline
11.1
11.2
11.3
11.4
11.5
A First Look at Risk and Return
Historical Risks and Returns of Stocks
The Historical Tradeoff Between Risk and Return
Common Versus Independent Risk
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:
Divt 1  Pt 1  Pt Divt 1 Pt 1  Pt
Rt 1 


Pt
Pt
Pt
(Eq. 11.1)
 Dividend Yield  Capital Gain Yield
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
Divt 1  Pt 1  Pt 3.08  (27.39  28.08)

 0.0851, or 8.51%
Pt
28.08
This 8.51% can be broken down into the dividend yield and the capital gain
yield:
Rt 1 

Dividend Yield =
Divt 1
3.08

 .1097, or 10.97%
Pt
28.08
Capital Gain Yield =
9
Pt 1  Pt  27.39  28.08 

 0.0246, or  2.46%
Pt
28.08
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
R t 1 

Div t 1  Pt 1  Pt 10.00  10.29  20.33

 0.002, or  0.2%
Pt
20.33
This -0.2% can be broken down into the dividend yield and the capital gain
yield:
Dividend Yield 
Divt 1 10.00

 0.4919, or 49.19%
Pt
20.33
Capital GainYield 
11
Pt 1  Pt 10.29  20.33

 0.4939, or  49.39%
Pt
20.33
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
Rt 1 

Divt 1  Pt 1  Pt 3.00  30.16  29.35

 0.1298, or 0.12.98%
Pt
29.35
This 12.98% can be broken down into the dividend yield and the capital gain
yield:
Dividend Yield 
Divt 1 3.00

 0.1022, or10.22%
Pt
29.35
Capital GainYield 
13
Pt 1  Pt 30.16  29.35

 0.0276, or 2.76%
Pt
29.35
11.2 Historical Risks and Returns of Stocks

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  Rannual  (1  R1 )(1  R2 )(1  R3 )(1  R4 )
14
(Eq. 11.2)
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
Example 11.2 Compounding Realized Returns
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
Example 11.2 Compounding Realized Returns
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
Rt 1
Divt 1  Pt 1  Pt 0.08  (25.93  27.39)


 0.0504, or  5.04%
Pt
27.39
17
Example 11.2 Compounding Realized Returns
Execute (cont’d):

The table below includes the realized return at each period.
18
Example 11.2 Compounding Realized Returns
Execute (cont’d):

We then determine the one-year return by compounding.
1  Rannual  (1  R1 )(1  R2 )(1  R3 )(1  R4 ) 1  R5 
1  Rannual  (1.0851)(0.9496)(0.9861)(1.0675)(0.9473)  1.0275
Rannual  1.0275  1  .0275 or 2.75%
19
Example 11.2 Compounding Realized Returns
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
21
Example 11.2a Compounding Realized
Returns
Plan (cont’d):
Date
Price
16-Mar-06
10-May-06
9-Aug-06
8-Nov-06
15-Feb-07
2-Mar-07
15-Mar-07

Dividend
21.15
20.70
20.62
19.39
20.33
10.29
11.07
0.06
0.06
0.06
10.00
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.
22
Example 11.2a Compounding Realized
Returns
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
R t 1 
23
Div t 1  Pt 1  Pt 0.06  19.39  20.62

 0.0567, or  5.67%
Pt
20.62
Example 11.2a Compounding Realized
Returns
Execute (cont’d):

The table below includes the realized return at each period.
Date
24
Price
Dividend
Return
16-Mar-06
21.15
10-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
2-Mar-07
10.29
15-Mar-07
11.07
4.85%
10.00
-0.20%
7.58%
Example 11.2a Compounding Realized
Returns
Execute (cont’d):

We then determine the one-year return by compounding.
1  Rannual  (1  R1 )(1  R2 )1  R3 (1  R4 )(1  R5 )(1  R6 )
1  Rannual  (0.982)(0.999)0.943(1.048)(0.998)(1.076)
Rannual 1.0411  1  .0411or 4.11%
25
11.2 Historical Risks and Returns of Stocks

Average Annual Returns

Average Annual Return of a Security
1
R  ( R1  R2  ...  RT )
T
26
(Eq. 11.3)
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
11.2 Historical Risks and Returns of Stocks

The Variance and Volatility of Returns:

Variance

1
Var  R  
( R1  R )2  ( R2  R ) 2  ...  ( RT  R ) 2 

T 1
Standard Deviation
SD( R)  Var  R 
29
(Eq. 11.4)
(Eq. 11.5)
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
Example 11.3 Computing Historical Volatility
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:
Var ( R) 

1
( R1  R ) 2  ( R2  R ) 2  ...  ( RT  R ) 2 
T 1
(.049  .031)2  (.158  .031) 2  (.055  .031) 2   0.370  .0312  .265  .0312 

5 1 
1
 .058
31
Example 11.3 Computing Historical Volatility
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:


SD(R)  Var (R)  .058  0.241,or 24.1%
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
Example 11.3a Computing Historical Volatility
Solution:
Plan:
Year
2005
Small Stocks’ Return 5.69%

2006
2007
16.17% -5.22%
2008
2009
-36.72%
28.09%
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.
34
Example 11.3a Computing Historical Volatility
Execute:

Using Eq. 11.3, the average annual return for small stocks during this period
is:

1
R  (.0569  .1671  .0522  .3672+.2809)  .0171
5
We now have all of the necessary inputs for the variance calculation:

Applying Eq. 11.4, we have:
Var ( R) 

1
( R1  R ) 2  ( R2  R ) 2  ...  ( RT  R ) 2 
T 1
(.0569  .0171) 2  (.1671  .0171) 2  (.0522  0171) 2   .3672  .01712  .2809  .01712 

5 1 
1
 .0615
35
Example 11.3a Computing Historical Volatility
Execute (cont'd):

Alternatively, we can break the calculation of this equation out as follows:
Return
Average
Difference
Squared



2005
0.0569
0.0171
0.0398
0.0016
2006
0.1671
0.0171
0.15
0.0225
2007
-0.0522
0.0171
-0.0693
0.0048
2008
-0.3672
0.0171
-0.3843
0.1477
2009
0.2809
0.0171
0.2638
0.0696
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:
SD(R)  Var(R)  0.0615  0.2480, or 24.80%
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
11.2 Historical Risks and Returns of Stocks

The Normal Distribution

95% Prediction Interval
Average  (2 x standard deviation)
R  (2 x SD  R  )

(Eq. 11.6)
About two-thirds of all possible outcomes fall within one standard
deviation above or below the average
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 20052009 to be 3.1% with a standard deviation of 24.1%. What is a 95%
confidence interval for 2010’s return?
40
Example 11.4 Confidence Intervals
Solution:
Plan:

We can use Eq. 11.6 to compute the confidence interval.
41
Example 11.4 Confidence Intervals
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
Example 11.4 Confidence Intervals
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
Example 11.4a Confidence Intervals
Solution:
Plan:

We can use Eq. 11.6 to calculate the confidence interval.
45
Example 11.4a Confidence Intervals
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
Example 11.4a Confidence Intervals
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
Example 11.4b Confidence Intervals
Solution:
Plan:

We can use Eq. 11.6 to calculate the confidence interval.
49
Example 11.4b Confidence Intervals
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
Example 11.4b Confidence Intervals
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.
2.


Company or Industry-Specific News
Market-Wide News
Unsystematic Risk
Systematic Risk
58
Figure 11.8 The Effect of Diversification on
Portfolio Volatility
59
11.5 Diversification in Stock Portfolios

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
11.5 Diversification in Stock Portfolios

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
11.5 Diversification in Stock Portfolios

The Importance of Systematic Risk

There is no relationship between volatility and average returns
for individual securities
63
References
64
References
Portfolio variance(assume  AB  1 )
  E R  E R    W E R  E R   W E R
2
2
p
p
p
2
A
2
A
2
B
A
 E R B  
2
B
2W AWB E R A  E R A RB  E RB 
 WA2 A2  WB2 B2  2W AWB AB
 W A2 A2  WB2 B2  2W AWB  AB A B
 W A2 A2  WB2 B2  2W AWB A B
 W A A  WB B 
2
 W A A  1  W A  B 
2
  p  W A  A   B    B 
 WA 
 p B
 A B
if  p  0  W A 
B
 A B
  p B
 E R A   1 
 A B
 A B

B
A
E R A   E R B 

 E R A  
 E R B  
p
 A B
 A B
 A B
E R p   W A  E R A   WB  E R B  
65
 p B

  E R B 

References
No Riskless Portfolio( p  0)
Optimal investment weighted as follow:
B
； WB*   A
WA* 
 A B
66
 A  B
References
 1   AB  1
Portfolio variance(assume
  E R  E R    W E R
2
p
2
p
p
2
A
)
 E R A   WB2 E RB  E RB  
2
A
2
2W AWB ER A  E R A RB  E RB 
 WA2 A2  WB2 B2  2W AWB AB
Minimum Variance Portfolio
  E R  E R    W E R
2
2
p
p
2
A
p
 E R A   WB2 E RB  E RB  
2
A
2
2W AWB ER A  E R A RB  E RB 
 WA2 A2  WB2 B2  2W AWB AB
 WA2 A2  1  W A   B2  2W A 1  W A  AB
2
 p2
W A
 2W A A2  21  W A (1) B2  21  W A  AB  2W A (1) AB  0

 
 2W A  A2   B2  2 AB  2  B2   AB
W 
*
A
67


  AB
 A2   B2  2 AB

2
B


References
N
N
N
   W    Wi W j  ij
2
p
i 1
2
i
2
i
i 1 j 1
i j
2
 1  2 N N  1  1 
1
          ij   
N
i 1  N 
i 1 j 1  N  N 
N
i j
2
2 N
1
2  

N
i 1
2
N
N
 
i 1 j 1
i j
ij
2
1
1
   N 2    N  N  1 ij
N
N
1
N 1
1
1

 2 
 ij   2  1   ij
N
N
N
N

1 2

N
1

1   ij
 N
：non-systematic risk
:Systematic risk
If portfolio satisfy N   , then portfolio variance
would reduce to lim  p2  0   2  1  0   ij   ij
N 
68
```