CHAPTER 3
COMMON STOCK: RETURN,
GROWTH, AND RISK
1.
HPR is the holding period return defined as 1  rt 
HPY is the holding period yield defined as rt 
Pt  Ct
Pt 1
Pt  Pt 1  Ct
Pt 1
Therefore, HPR – 1 = HPY.
2.
Pt-1 = $100
Pt = $107
Ct = $5
HPR  1  rt 
HPY 
107  5
 1.12
100
107  100  5
 .12
100
or HPR – 1 = HPY
1.12 – 1 = .12
3.
Pt-1 = $50.375
Pt = $66.00
Ct = 2 + 2.2 + 2.2 + 2.2 + 2.2 + 2.2 + 2.2 = $15.20
4.
HPR 
66.00  15.20
 1.61
50.375
HPY 
66.00  50.375  15.20
 .61
50.375
The normal distribution is defined over the range of values from + to - infinity.
The normal distribution is symmetrical around its mean.
The log normal distribution is defined over the range of values from 0 to + infinity.
The log normal distribution is not symmetrical around its mean.
In finance, when we are dealing with price information, it is more realistic to use
the log normal distribution because prices range from 0 to + infinity.
5.
HPY is the holding period yield discrete
HPY is the holding period yield continuous
6.
The arithmetic mean is the sum of the values of the data points divided by the
number of such data points or
N
X
 Xt
t 1
N
The geometric mean is the nth root of the product of the n values of the data
points or
g  N X1 X 2
XN
The arithmetic mean is appropriate to use with cross-sectional type of information.
Whereas the geometric mean is more appropriate to use for time series data.
7.
The weighted unbiased mean is an average of the arithmetic and geometric
means or
T n
 n 1 
M (W )  
 X 
g
 T 1 
 T 1 
where
T = the number of HPRs used to estimate the historical average returns
n = the number of investment horizon periods for which a particular
investment is to be held
For short holding periods (a small n) the Π(W) approaches X and for long
holding periods (the large n) the Π(W) app roaches g .
8.
T n
 n 1 
 (W )  
 X 
g
 T 1 
 T 1 
T=6
n=6
 6  6
 6 1
 (W )  
 X 
g
 6 1 
 6 1 
 0( X )  1( g )
g
 48  2.20   35.125  2.20   34  2.20   44.50  2.20   64  2.20   66  2.20 
gs  6 






48
34
64
 50.375  
  35.125  
  44.50  

 6 (.9965261)(.7776042)(1.030605)(1.3735294)(1.4876404)(1.065625)
 6 1.7389227
 1.0716
(W )  g  1.0716
9.
Risk is the probability of success or failure. We can measure total risks by the
variance or standard deviation of the distribution of outcomes.
Risk can be decomposed into two parts: Systematic risk and unsystematic risk.
Unsystematic risk can be further divided into business risk and financial risk.
10. Systematic risk is that portion of total risk that relates to all securities which are
traded in a given market. Unsystematic risk is that portion of total risk that
relates to the individual issue and issues of a particular security.
With respect to unsystematic risk, we can reduce or diversify it away by holding
securities in portfolios.
It is very difficult to reduce or diversify away the systematic risk because all
securities are related to it.
11. a)
N
correlation coefficient   AB 
RA 
 ( RAt  RA )( RBt  RB ) N
t 1
 A B
.10  .08  .07  .05
 .075
4
4
 A   ( RAt  RA )2
t 1
 (.1  .075)2  (.08  .075)2  (.07  .075)2  (.05  .075)2
 .000625  .000025  .000025  .000626
 .0013
 A  .0361
RB 
.09  .1  .09  .12
 .1
4
4
 B   ( RBt  RB )2
t 1
 (.09  .1)2  (.1  .1)2  (.09  .1)2  (.12  .1)2
 .0001  0  .0001  .0004
 .0006
 B  .0245
 AB
(.1  .075)(.09  .1)  (.08  .075)(.1  .1)  (.07  .075)(.09  .1)  (.05  .075)(.12  .1)
4

(.0361)(.0245)

(.00025  0  .00005  .0005) 4
.0008845

.000175
 .1979
.0008845
 AB  .1979
( RAt  RA )( RBt  RB )
 .000175
t 1
N
N
Covariance (AB)  
b)
If assets are less than perfectly correlated (ρ ≠ +1) , then when they are
combined together the resulting combination will have less risk than the most
risky of the two. Further, if the correct amounts of each security are combined,
the resulting risk will be less than the least risky.
12. This is not necessarily true.
It is not only the standard deviation or total risk
that investor's dislike, but rather how well this security reduces the overall risk
of a portfolio. That is to say what is the securities correlation with other
securities.
13. The market model measures the relationship between the returns on the market
and returns of an individual security.
Ri = α + βRM
where
Ri is the return on an individual security
RM is the return on the market
Beta is the measure of systematic risk for an individual security or a portfolio of
securities.
The holder of a well diversified portfolio is interested in the sensitivity of his
individual securities or portfolios and how they react to market movements. This
is measured by the beta. Beta can be estimated for the market mode1.
14. Regression analysis is a statistical tool which allows us to measure the
relationship between the variation in onevariable with the variation in another
variable.
Portfolio managers, by using regression and the market model, can make an
estimate of this portfolio beta.
15.
Abbott Laboratories
Year
n
n**2
Sales
1966
1967
1968
1969
1
2
3
4
1
4
9
16
265.8
303.3
351.0
403.9
Forecast Sales
Nonlinear
321.8546
309.8043
317.7665
345.7412
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
5
6
7
8
9
10
11
12
13
14
25
36
49
64
81
100
121
144
169
196
457.5
458.1
521.8
620.4
765.4
940.6
1084.8
1244.9
1445.0
1683.2
393.7285
461.7284
549.7408
657.7658
785.8033
933.8533
1101.915
1289.991
1498.078
1726.179
1980
1981
1982
15
16
17
225
256
289
2038.2
2342.5
2602.4
1974.291
2242.417
2530.555
1983
1984
18
19
324
361
2927.9
3104.0
2838.705
3166.868
1985
1986
1987
20
21
22
400
441
484
3360.3
3807.6
4387.9
3515.044
3883.232
4271.432
Regression Output:
Constant
Std Err of Y Est
R Squared
No. of Observations
Degrees of Freedom
X Coefficient(s)
Std Err of Coef.
353.9175
71.00107
0.997203
22
19
-42.0691
9.990266
10.00627
0.421789
Year
n
ln(sales)
Forecast Sales
Growth Rate
1966
1967
1968
1969
1970
1971
1
2
3
4
5
6
5.582744
5.714722
5.860786
6.001167
6.125776
6.127087
259.4194
298.1424
342.6455
393.7916
452.5721
520.1266
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
7
8
9
10
11
12
13
14
15
16
6.257284
6.430364
6.640398
6.846517
6.989150
7.126810
7.275864
7.428452
7.619822
7.758974
597.7649
686.9921
789.5381
907.3909
1042.835
1198.497
1377.394
1582.995
1819.286
2090.847
1982
1983
1984
1985
1986
1987
17
18
19
20
21
22
7.864189
7.982040
8.040446
8.119785
8.244754
8.386606
2402.944
2761.627
3173.849
3647.604
4192.074
4817.817
Regression Output:
Constant
Std Err of Y Est
R Squared
No. of Observations
Degrees of Freedom
X Coefficient(s)
Std Err of Coef.
16.
5 .419320
0.077251
0.993084
22
20
0.139125
0.002596
a)
Ri
i 1 N
 8  10  4  5.3  12.2  14.3  10.2  19.3  16.4 / 9
N
X 
 71.3 / 9  7.9
b)
g  9 (1.08)(1.1)(.96)(1.053)(1.122)(1.143)(.898)(1.193)(1.164)
 9 1.9205468  1
 7.44
C)
X is larger than g because the time series of returns increases and
decreases.