Junior Sayou, a financial analyst for Chargers Products, a manufacturer of stadi
ID: 2773199 • Letter: J
Question
Junior Sayou, a financial analyst for Chargers Products, a manufacturer of stadium
benches, must evaluate the risk and return of two assets, X and Y. The firm is considering
adding these assets to its diversified asset portfolio. To assess the return and risk
of each asset, Junior gathered data on the annual cash flow and beginning- and end-of year
values of each asset over the immediately preceding 10 years, 2000–2009. Junior’s investigation suggests that both assets, on average, will tend to perform in the future just as they have
during the past 10 years. He therefore believes that the expected annual return can be
estimated by finding the average annual return for each asset over the past 10 years.
Return Data for Assets X and Y, 2000–2009
Asset X Asset Y
Value Value
Year Cash flow Beginning Ending Cash flow Beginning Ending
2000 $1,000 $20,000 $22,000 $1,500 $20,000 $20,000
2001 1,500 22,000 21,000 1,600 20,000 20,000
2002 1,400 21,000 24,000 1,700 20,000 21,000
2003 1,700 24,000 22,000 1,800 21,000 21,000
2004 1,900 22,000 23,000 1,900 21,000 22,000
2005 1,600 23,000 26,000 2,000 22,000 23,000
2006 1,700 26,000 25,000 2,100 23,000 23,000
2007 2,000 25,000 24,000 2,200 23,000 24,000
2008 2,100 24,000 27,000 2,300 24,000 25,000
2009 2,200 27,000 30,000 2,400 25,000 25,000
Junior believes that each asset’s risk can be assessed in two ways: in isolation and
as part of the firm’s diversified portfolio of assets. The risk of the assets in isolation
can be found by using the standard deviation and coefficient of variation of returns
over the past 10 years. The capital asset pricing model (CAPM) can be used to assess
the asset’s risk as part of the firm’s portfolio of assets. Applying some sophisticated
quantitative techniques, Junior estimated betas for assets X and Y of 1.60 and 1.10,
respectively. In addition, he found that the risk-free rate is currently 7% and that the
market return is 10%.
To Do
a. Calculate the annual rate of return for each asset in each of the 10 preceding
years, and use those values to find the average annual return for each asset over
the 10-year period.
b. Use the returns calculated in part a to find (1) the standard deviation and
(2) the coefficient of variation of the returns for each asset over the 10-year
period 2000–2009.
c. Use your findings in parts a and b to evaluate and discuss the return and risk
associated with each asset. Which asset appears to be preferable? Explain.
d. Use the CAPM to find the required return for each asset. Compare this value
with the average annual returns calculated in part a.
e. Compare and contrast your findings in parts c and d. What recommendations
would you give Junior with regard to investing in either of the two assets?
Explain to Junior why he is better off using beta rather than the standard deviation
and coefficient of variation to assess the risk of each asset.
f. Rework parts d and e under each of the following circumstances:
(1) A rise of 1% in inflationary expectations causes the risk-free rate to rise to
8% and the market return to rise to 11%.
(2) As a result of favorable political events, investors suddenly become less riskaverse,
causing the market return to drop by 1%, to 9%.
Explanation / Answer
Answer (a)
Asset X
annual
Asset Y
annual
cash Flow
Values
return
Cash Flow
Values
return
year
Annual
beginning
ending
Average
(%)
Annual
beginning
ending
average
(%)
2000
1000
20000
22000
21000
4.76
1500
20000
20000
20000
7.50
2001
1500
22000
21000
21500
6.98
1600
20000
20000
20000
8.00
2002
1400
21000
24000
22500
6.22
1700
20000
21000
20500
8.29
2003
1700
24000
22000
23000
7.39
1800
21000
21000
21000
8.57
2004
1900
22000
23000
22500
8.44
1900
21000
22000
21500
8.84
2005
1600
23000
26000
24500
6.53
2000
22000
23000
22500
8.89
2006
1700
26000
25000
25500
6.67
2100
23000
23000
23000
9.13
2007
2000
25000
24000
24500
8.16
2200
23000
24000
23500
9.36
2008
2100
24000
27000
25500
8.24
2300
24000
25000
24500
9.39
2009
2200
27000
30000
28500
7.72
2400
25000
25000
25000
9.60
Total
71.11
87.57
Average Annual Return of Asset X = 71.11/10 = 7.11%
Average Annual Return of Asset Y = 87.57/10 = 8.76%
Answer (b)
Asset X
Asset Y
year
r (%)
r-avg rtn
(r-avg rtn)^2
year
r (%)
r-avg rtn
(r-avg rtn)^2
2000
4.76
-2.35
5.5136
2000
7.50
-1.26
1.5876
2001
6.98
-0.13
0.0178
2001
8.00
-0.76
0.5776
2002
6.22
-0.89
0.7881
2002
8.29
-0.47
0.2184
2003
7.39
0.28
0.0791
2003
8.57
-0.19
0.0356
2004
8.44
1.33
1.7807
2004
8.84
0.08
0.0060
2005
6.53
-0.58
0.3357
2005
8.89
0.13
0.0166
2006
6.67
-0.44
0.1965
2006
9.13
0.37
0.1372
2007
8.16
1.05
1.1094
2007
9.36
0.60
0.3620
2008
8.24
1.13
1.2663
2008
9.39
0.63
0.3941
2009
7.72
0.61
0.3712
2009
9.60
0.84
0.7056
Avg return
7.11
Avg return
8.76
Total
11.4585
4.0407
Standard Deviation of Asset X SD(x) = (11.4585)^1/2
SD(x) = 3.385
Standard Deviation of Asset Y SD(y) = (4.0407)^1/2 = 2.0101 or 2.01
Coefficient of Variation for Asset X (x) = SD(x)/Avg return = 3.385/ 7.11 = 0.4761
Coefficient of Variation for Asset Y (y) = SD(y)/Avg return = 2.01/8.76 = 0.2295
Answer (c)
Asset X has an average return of 7.11% with a standard deviation of 3.385 and coefficient of variation of 0.4761
Asset Y has an average return of 8.76% with a standard deviation of 2.01 and coefficient of variation of 0.2295
The coefficient of variation is a measure of extent of variability with reference to mean of the population. That is the lower the coefficient of variation the lower the variability of returns. That is the probability of getting the mean return is higher with lower variation. Based on this Asset Y is more preferable.
Answer (d)
Risk Free Return rf = 7%
Market Return rm = 10%
eta of Asset X (x) = 1.60
Beta of Asset Y (y) = 1.10
Required Rate of return for Asset X r(x) = Risk Free rate + Beta of Asset * (Market Return – Risk free return)
r(x) = 7% + 1.60 * (10% - 7%) = 7% + 1.60 * 3% = 7% + 4.8% = 11.8%
Required rate of return for Asset Y r(y) = 7% + 1.10 *(10% - 7%) = 7% + 1.1 * 3% = 7% + 3.3% = 10.3%
Required return on Asset X = 11.8%
Average rate of return on Asset X = 7.11%
Difference = 11.8% - 7.11% = 4.69%
Required return on Asset Y = 10.3%
Average rate of return on Asset Y = 8.76%
Difference = 10.3% - 8.76% = 1.54%
The difference in required return to average return is lower for Asset Y (1.54%) compared to for Asset X (4.69%).
Also the coefficient of variation of returns is lower for Asset Y compared to Asset X.
However coefficient of variation and standard deviation of a small sample of a total population will give results / decisions which are in contrast to the reality. Also, when both both negative and positive values are present in a population, then coefficient of variation becomes meaningless. Hence it is preferable to use Beta for assessing the risk of an asset.
Answer (f) (1)
Risk Free rate = 8% and Market return = 11%
r(x) = 8% + 1.60 * (11% - 8%) = 8% + 1.60 * 3% = 8% + 4.8% = 12.8%
r(y) = 8% + 1.10 *(11% - 8%) = 8% + 1.1 * 3% = 8% + 3.3% = 11.3%
Required return on Asset X = 12.8%
Average rate of return on Asset X = 7.11%
Difference = 11.8% - 7.11% = 5.69%
Required return on Asset Y = 11.3%
Average rate of return on Asset Y = 8.76%
Difference = 10.3% - 8.76% = 2.54%
Answer (f) (2)
Risk free rate = 7%
Market return = 9%
r(x) = 7% + 1.60 * (9% - 7%) = 7% + 1.60 * 2% = 7% + 2.6% = 9.6%
r(y) = 7% + 1.10 *(9% - 7%) = 7% + 1.1 * 2% = 7% + 2.2% = 9.2%
Required return on Asset X = 9.6%
Average rate of return on Asset X = 7.11%
Difference = 9.6% - 7.11% = 2.49%
Required return on Asset Y = 9.2%
Average rate of return on Asset Y = 8.76%
Difference = 9.2% - 8.76% = 0.44%
Additional Solution as per Request
Answer (a)
Asset X
annual
Asset Y
annual
cash Flow
Values
return
Cash Flow
Values
return
Year
Annual
beginning
ending
Average
(%)
Annual
beginning
ending
average
(%)
2000
1000
20000
22000
21000
4.76
1500
20000
20000
20000
7.5
2001
1500
22000
21000
21500
6.98
1600
20000
20000
20000
8
2002
1400
21000
24000
22500
6.22
1700
20000
21000
20500
8.29
2003
1700
24000
22000
23000
7.39
1800
21000
21000
21000
8.57
2004
1900
22000
23000
22500
8.44
1900
21000
22000
21500
8.84
2005
1600
23000
26000
24500
6.53
2000
22000
23000
22500
8.89
2006
1700
26000
25000
25500
6.67
2100
23000
23000
23000
9.13
2007
2000
25000
24000
24500
8.16
2200
23000
24000
23500
9.36
2008
2100
24000
27000
25500
8.24
2300
24000
25000
24500
9.39
2009
2200
27000
30000
28500
7.72
2400
25000
25000
25000
9.6
2010
2200
30000
30000
30000
7.33
2400
25000
25000
25000
9.6
2011
2200
30000
30000
30000
7.33
2400
25000
25000
25000
9.6
2012
2200
30000
30000
30000
7.33
2400
25000
25000
25000
9.6
Total
94.27
116.37
Average Annual Return of Asset X = 94.27/13 = 7.25%
Average Annual Return of Asset Y = 116.37/13 = 8.95%
Answer (b)
Asset X
Asset Y
year
r (%)
r-avg rtn
(r-avg rtn)^2
Year
r (%)
r-avg rtn
(r-avg rtn)^2
2000
4.76
-2.49
6.2001
2000
7.5
-1.45
2.1025
2001
6.98
-0.27
0.0729
2001
8
-0.95
0.9025
2002
6.22
-1.03
1.0609
2002
8.29
-0.66
0.4356
2003
7.39
0.14
0.0196
2003
8.57
-0.38
0.1444
2004
8.44
1.19
1.4161
2004
8.84
-0.11
0.0121
2005
6.53
-0.72
0.5184
2005
8.89
-0.06
0.0036
2006
6.67
-0.58
0.3364
2006
9.13
0.18
0.0324
2007
8.16
0.91
0.8281
2007
9.36
0.41
0.1681
2008
8.24
0.99
0.9801
2008
9.39
0.44
0.1936
2009
7.72
0.47
0.2209
2009
9.6
0.65
0.4225
2010
7.33
0.08
0.0064
2010
9.6
0.65
0.4225
2011
7.33
0.08
0.0064
2011
9.6
0.65
0.4225
2012
7.33
0.08
0.0064
2012
9.6
0.65
0.4225
Avg return
7.25
Avg return
8.95
Total
11.6727
5.6848
Standard Deviation of Asset X SD(x) = (11.6727)^1/2
SD(x) = 3.4165 or 3.417 (rounded off)
Standard Deviation of Asset Y SD(y) = (5.6848)^1/2 = 2.384
Coefficient of Variation for Asset X (x) = SD(x)/Avg return = 3.417/ 7.25 = 0.4713
Coefficient of Variation for Asset Y (y) = SD(y)/Avg return = 2.384/8.95 = 0.2664
Answer (c)
Asset X has an average return of 7.25% with a standard deviation of 3.417 and coefficient of variation of 0.4713
Asset Y has an average return of 8.95% with a standard deviation of 2.384 and coefficient of variation of 0.2664
The coefficient of variation is a measure of extent of variability with reference to mean of the population. That is the lower the coefficient of variation the lower the variability of returns. That is the probability of getting the mean return is higher with lower variation. Based on this Asset Y is more preferable.
Answer (d)
Risk Free Return rf = 7%
Market Return rm = 10%
eta of Asset X (x) = 1.60
Beta of Asset Y (y) = 1.10
Required Rate of return for Asset X r(x) = Risk Free rate + Beta of Asset * (Market Return – Risk free return)
r(x) = 7% + 1.60 * (10% - 7%) = 7% + 1.60 * 3% = 7% + 4.8% = 11.8%
Required rate of return for Asset Y r(y) = 7% + 1.10 *(10% - 7%) = 7% + 1.1 * 3% = 7% + 3.3% = 10.3%
Required return on Asset X = 11.8%
Average rate of return on Asset X = 7.25%
Difference = 11.8% - 7.25% = 4.55%
Required return on Asset Y = 10.3%
Average rate of return on Asset Y = 8.95%
Difference = 10.3% - 8.76% = 1.35%
The difference in required return to average return is lower for Asset Y (1.35%) compared to for Asset X (4.55%).
Also the coefficient of variation of returns is lower for Asset Y compared to Asset X.
However coefficient of variation and standard deviation of a small sample of a total population will give results / decisions which are in contrast to the reality. Also, when both both negative and positive values are present in a population, then coefficient of variation becomes meaningless. Hence it is preferable to use Beta for assessing the risk of an asset.
Answer (f) (1)
Risk Free rate = 8% and Market return = 11%
r(x) = 8% + 1.60 * (11% - 8%) = 8% + 1.60 * 3% = 8% + 4.8% = 12.8%
r(y) = 8% + 1.10 *(11% - 8%) = 8% + 1.1 * 3% = 8% + 3.3% = 11.3%
Required return on Asset X = 12.8%
Average rate of return on Asset X = 7.25%
Difference = 11.8% - 7.11% = 4.55%
Required return on Asset Y = 11.3%
Average rate of return on Asset Y = 8.95%
Difference = 10.3% - 8.76% = 2.35%
Answer (f) (2)
Risk free rate = 7%
Market return = 9%
r(x) = 7% + 1.60 * (9% - 7%) = 7% + 1.60 * 2% = 7% + 2.6% = 9.6%
r(y) = 7% + 1.10 *(9% - 7%) = 7% + 1.1 * 2% = 7% + 2.2% = 9.2%
Required return on Asset X = 9.6%
Average rate of return on Asset X = 7.25%
Difference = 9.6% - 7.25% = 2.35%
Required return on Asset Y = 9.2%
Average rate of return on Asset Y = 8.95%
Difference = 9.2% - 8.76% = 0.25%
Asset X
annual
Asset Y
annual
cash Flow
Values
return
Cash Flow
Values
return
year
Annual
beginning
ending
Average
(%)
Annual
beginning
ending
average
(%)
2000
1000
20000
22000
21000
4.76
1500
20000
20000
20000
7.50
2001
1500
22000
21000
21500
6.98
1600
20000
20000
20000
8.00
2002
1400
21000
24000
22500
6.22
1700
20000
21000
20500
8.29
2003
1700
24000
22000
23000
7.39
1800
21000
21000
21000
8.57
2004
1900
22000
23000
22500
8.44
1900
21000
22000
21500
8.84
2005
1600
23000
26000
24500
6.53
2000
22000
23000
22500
8.89
2006
1700
26000
25000
25500
6.67
2100
23000
23000
23000
9.13
2007
2000
25000
24000
24500
8.16
2200
23000
24000
23500
9.36
2008
2100
24000
27000
25500
8.24
2300
24000
25000
24500
9.39
2009
2200
27000
30000
28500
7.72
2400
25000
25000
25000
9.60
Total
71.11
87.57
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