Appendix to Chapter 1 Calculation of variance and standard deviation Variance and standard deviation are measures of how actual values differ from the expected values (arithmetic mean) for a given series of values. In this case, we want to measure how rates of return differ from the arithmetic mean value of a series. There are other measures of dispersion, but variance and standard deviation are the best known because they are used in statistics and probability theory. Variance is defined as:
Variance(σ 2 ) = a (Probability ) (Possible Return − Expected Return ) 2 n
i=1 n
= a (Pi)[Ri − E(Ri) ] 2 i=1
Consider the following example, as discussed in Chapter 1: Probability of Possible Return ( Pi)
Possible Return ( Ri)
Pi Ri
0.15
0.20
0.03
0.15
−0.20
−0.03
0.70
0.10
0.07 Σ = 0.07
This gives an expected return [E(Ri)] of 7 per cent. The dispersion of this distribution as measured by variance is: Probability ( Pi )
Return ( R i )
Ri − E( R i )
[Ri − E( R i ) ]2
Pi [R i − E( R i ) ]2
0.15
0.20
0.13
0.0169
0.002535
0.15
−0.20
−0.27
0.0729
0.010935
0.70
0.10
0.03
0.0009
0.000630 Σ = 0.014100
The variance (σ 2) is equal to 0.0141. The standard deviation is equal to the square root of the variance:
Standard Deviation (σ ) =
Åa i=1 n
Pi 3 Ri − E( Ri ) 4 2
Consequently, the standard deviation for the preceding example would be:
σi = !0.0141 = 0.11874
In this example, the standard deviation is approximately 11.87 per cent. Therefore, you could describe this distribution as having an expected value of 7 per cent and a standard deviation of 11.87 per cent. 1
2
chapter appendices
In many instances, you might want to calculate the variance or standard deviation for a historical series in order to evaluate the past performance of the investment. Assume that you are given the following information on annual rates of return (HPY) for common stocks listed on the New York Stock Exchange (NYSE): Year
Annual rate of return
2012
0.07
2013
0.11
2014
−0.04
2015
0.12
2016
−0.06
In this case, we are not examining expected rates of return but actual returns. Therefore, we assume equal probabilities, and the expected value (in this case the mean value, R) of the series is the sum of the individual observations in the series divided by the number of observations, or 0.04 (0.20/5). The variances and standard deviations are: Year
Ri
2012
0.07
– Ri − R 0.03
– ( Ri − R )2 0.0009
2013
0.11
0.07
0.0049
2014
−0.04
−0.08
0.0064
2015
0.12
0.08
0.0064
2016
−0.06
−0.10
0.0110 Σ = 0.0286
σ 2 = 0.0286/5 = 0.00572 σ = !0.00572 = 0.0756
= 7.56%
We can interpret the performance of NYSE common stocks during this period of time by saying that the average rate of return was 4 per cent and the standard deviation of annual rates of return was 7.56 per cent.
Coefficient of variation In some instances, you might want to compare the dispersion of two different series. The variance and standard deviation are absolute measures of dispersion. That is, they can be influenced by the magnitude of the original numbers. To compare series with very different values, you need a relative measure of dispersion. A measure of relative dispersion is the coefficient of variation, which is defined as:
Coefficient of Variation 1 CV 2 =
Standard Deviation of Returns Expected Rate of Return
A larger value indicates greater dispersion relative to the arithmetic mean of the series. For the previous example, the CV would be:
CV1 =
0.0756 = 1.89 0.0400
It is possible to compare this value to a similar figure having a markedly different distribution. As an example, assume you wanted to compare this investment to another investment that had an average rate of return of 10 per cent and a standard deviation of 9 per cent. The standard deviations alone tell you that the second series has greater dispersion (9 per cent versus 7.56 per cent) and might be considered to have higher risk. In fact, the relative dispersion for this second investment is much less.
0.0756 = 1.89 0.0400 0.0900 CV2 = = 0.90 0.1000
CV1 =
APPENDIX TO CHAPTER 1
3
Considering the relative dispersion and the total distribution, most investors would probably prefer the second investment.
Problems 1 Your rate of return expectations for the common stock of Gray Cloud Company during the next year are: Gray cloud co. Possible Rate of Return
Probability
−0.10
0.25
0.00
0.15
0.10
0.35
0.25
0.25
a Calculate the expected return [E(Ri ) ] on this investment, the variance of this return (σ 2) and its standard deviation (σ). b Under what conditions can the standard deviation be used to measure the relative risk of two investments? c Under what conditions must the coefficient of variation be used to measure the relative risk of two investments? 2 Your rate of return expectations for the stock of Kayleigh Cosmetics Company during the next year are: Kayleigh Cosmetics Co. Possible Rate of Return
Probability
−0.60
0.15
−0.30
0.10
−0.10
0.05
0.20
0.40
0.40
0.20
0.80
0.10
a b c d
Calculate the expected return [E(R i ) ] on this stock, the variance (σ2) of this return and its standard deviation (σ). On the basis of expected return [E(R i )] alone, discuss whether Gray Cloud or Kayleigh Cosmetics is preferable. On the basis of standard deviation (σ) alone, discuss whether Gray Cloud or Kayleigh Cosmetics is preferable. Calculate the coefficients of variation (CVs) for Gray Cloud and Kayleigh Cosmetics and discuss which stock return series has the greater relative dispersion. 3 The following are annual rates of return for US government T-bills and UK common stocks. Year
US government T-Bills
UK common stock
2012
0.063
0.150
2013
0.081
0.043
2014
0.076
0.374
2015
0.090
0.192
2016
0.085
0.106
a Calculate the arithmetic mean rate of return and standard deviation of rates of return for the two series. b Discuss these two alternative investments in terms of their arithmetic average rates of return, their absolute risk and their relative risk. c Calculate the geometric mean rate of return for each of these investments. Compare the arithmetic mean return and geometric mean return for each investment and discuss the difference between mean returns as related to the standard deviation of each series.
Appendix to Chapter 2 Covariance Because most students have been exposed to the concepts of covariance and correlation, the following discussion is set forth in intuitive terms with examples. A detailed, rigorous treatment is contained in DeFusco, McLeavey, Pinto and Runkle (2004). Covariance is an absolute measure of the extent to which two sets of numbers move together over time, that is, how often they move up or down together. In this regard, move together means they are generally above their means or below their means at the same time. Covariance between i and j is defined as
(i − i)(j − j) COVij = a n If we define 1i − i 2 as i′ and 1j − j 2 as j′, then
ij COVij = an
′ ′
Obviously, if both numbers are consistently above or below their individual means at the same time, their products will be positive, and the average will be a large positive value. In contrast, if the i value is below its mean when the j value is above its mean consistently or vice versa, their products will be large negative values, giving negative covariance. Exhibit 2A.1 should make this clear. In this example, the two series generally moved together, so they showed positive covariance. As noted, this is an absolute measure of their relationship and, therefore, can range from +∞ to −∞. Note that the covariance of a variable with itself is its variance.
Correlation To obtain a relative measure of a given relationship, we use the correlation coefficient (rij), which is a measure of the relationship:
COVij rij = σ σ i j You will recall from your introductory statistics course that
σi =
4
2 a (i − i) N Ç
5
APPENDIX TO CHAPTER 2
Exhibit 2A.1 Calculation of correlation covariance i
j
i–i
j–j
i′j′
1
3
8
−4
−4
16
2
6
10
−1
−2
2
3
8
14
+1
+2
2
4
5
12
−2
0
0
5
9
13
+2
+1
2
6
11
15
+4
+3
12
42
72
7
12
Observation
g Mean
34
COVij =
34 6
= +5.67
If the two series move completely together, then the covariance would equal σ i σ j and
COVij σiσj = 1.0 The correlation coefficient would equal unity in this case, and we would say the two series are perfectly correlated. Because we know that
COVij rij = σ σ i j we also know that COVij = rij σ i σj. This relationship may be useful when calculating the standard deviation of a portfolio, because in many instances the relationship between two securities is stated in terms of the correlation coefficient rather than the covariance. Continuing the example given in Exhibit 2A.1, the standard deviations are calculated in Exhibit 2A.2, as is the correlation between i and j. As shown, the two standard deviations are rather large and similar but not exactly the same. Finally, when the positive covariance is normalised by the product of the two standard deviations, the results indicate a correlation coefficient of 0.898, which is obviously quite large and close to 1.00. This implies that these two series are highly related.
Exhibit 2A.2 Calculation of correlation coefficient Observation
-a i–i
-2 (i – i )
-a j–j
-2 ( j – j )
1
−4
16
−4
16
2
−1
1
−2
4
3
+4
1
+2
4
4
−2
4
0
0
5
+2
4
+1
1
6
+4
16
+3
9
42 σ 2i = 42/6 = 7.00 σi = !7.00 = 2.65 rij = COVij /σiσj =
34 σ 2j = 34/6 = 5.67 σj = !5.67 = 2.38
5.67 5.67 = = 0.898 (2.65) (2.38) 6.31
6
chapter appendices
Problems 1 As a new analyst, you have calculated the following annual rates of return for the stocks of both Lauren Corporation and Kayleigh Industries. Year
Lauren’s rate of return
Kayleigh’s rate of return
2007
5
5
2008
12
15
2009
−11
5
2010
10
7
2011
12
−10
2 Your manager suggests that because these companies produce similar products, you should continue your analysis by calculating their covariance. Show all calculations. 3 You decide to go an extra step by calculating the coefficient of correlation using the data provided in Problem 1. Prepare a table showing your calculations and explain how to interpret the results. Would the combination of the common stock of Lauren and Kayleigh be good for diversification?
Appendix to Chapter 5 A. Proof that minimum portfolio variance occurs with equal weights when securities have equal variance When σ1 = σ2, we have:
σ2port = w21 (σ1) 2 + (1 − w1) 2 (σ1) 2 − 2w1 (1 − w1)r1, 2 (σ1) 2 = (σ1) 2 [w21 + 1 − 2w1 + w21 + 2w1r1, 2 − 2w21r1, 2] = (σ1) 2 [2w21 + 1 − 2w1 + 2w1r1, 2 − 2w21r1, 2] For this to be a minimum,
𝜕 (σ2port) 𝜕 w1
= 0 = (σ1) 2 [4w1 × 2 + 2r1, 2 × 4w1r1, 2]
Assuming (σ1)2 > 0,
4w1 − 2 + 2r1, 2 − 4w1r1, 2 = 0 4w1 (1 − r1, 2 ) − 2 (1 − r1, 2 ) = 0 from which
w1
2(1 − r1, 2) 4(1 − r1, 2)
=
1 2
regardless of r1,2. Thus, if σ1 = σ2, σ2port will always be minimised by choosing w1 = w2 = 1/2, regardless of the value of r1,2, except when r1,2 = +1 (in which case σport = σ1 = σ2). This can be verified by checking the second-order condition
𝜕 (σ2port) 𝜕 w21
>0
Problem 1 The following information applies to Questions 1a and 1b. The general equation for the weight of the first security to achieve minimum variance (in a two-stock portfolio) is given by
w1 =
(σ2) 2 − r1, 2 (σ1)(σ2) (σ1) 2 + (σ2) 2 − 2r1, 2 (σ1)(σ2)
a Show that w1 = 0.5 when σ1 = σ2. b What is the weight of Security 1 that gives minimum portfolio variance when r1,2 = 0.5, σ1 = 0.04 and σ2 = 0.06? 7
8
chapter appendices
B. Derivation of weights that will give zero variance when correlation equals −1.00 σ2port = w21 (σ1) 2 + (1 − w1) 2 (σ2) 2 + 2w1 (1 − w1)r1, 2 (σ1)(σ2) = w21 (σ1) 2 + (σ2) 2 − 2w1 (σ2) − w21 (σ2) 2 + 2w1r1, 2 (σ1)(σ2) − 2w21r1, 2 (σ1)(σ2) If r1,2 = 1, this can be rearranged and expressed as
σ2port = w21 [(σ1) 2 + 2(σ1)(σ2) + (σ2) 2] − 2w[(σ2) 2 + (σ1)(σ2) ] + (σ2) 2 = w21 [(σ1) + (σ2) ] 2 − 2w1 (σ2)[(σ1) − (σ2) ] + (σ2) 2 = 5 w1 [(σ1) + (σ2) ] − (σ2) 2 6
We want to find the weight, w1, which will reduce (σ2port) to zero; therefore,
w1 [(σ1) + (σ2) ] − (σ2) = 0 which yields
w1 =
(σ2) (σ1) , and w2 = 1 − w1 = (σ1) + (σ2) (σ1) + (σ2)
Problem 1 Given two assets with the following characteristics:
E(R1) = 0.12 E(R2) = 0.16
σ1 = 0.04 σ2 = 0.06
Assume that r1,2 = −1.00. What is the weight that would yield a zero variance for the portfolio?
Appendix to Chapter 8 Exhibit 8A.1 Calculation of present value of lease payments for Walgreen Co. as of September 1, 2010* Present value assuming 15 year spread of lump sum
Present value assuming 12 year spread of lump sum
Year
Payment $ mil.
Present value of payment $ mil.
Payment $ mil.
Present value of payment $ mil.
2011
2301
$ 2150
2301
$ 2150
2012
2329
2034
2329
2034
2013
2296
1874
2296
1874
2014
2248
1715
2248
1715
2015
2188
1560
2188
1560
2016
1695
1130
2119
1412
2017
1695
1056
2119
1320
2018
1695
987
2119
1233
2019
1695
922
2119
1153
2020
1695
862
2119
1077
2021
1695
805
2119
1007
2022
1695
753
2119
941
2023
1695
703
2119
879
2024
1695
657
2119
822
2025
1695
614
2119
768
2026
1695
574
2119
718
2027
1695
537
2119
671
2028
1695
502
2029
1695
469
2030
1695
Total
438
$20 342
$21 333
*7% discount rate.
Exhibit 8A.2 Calculation of operating lease obligations for Walgreen Co. for 2010, 2009, 2008* 2010
2009
2008
Year
Payment $ mil.
PV of payment $ mil.
Year
Payment $ mil.
PV of payment $ mil.
2011
2301
$ 2150
2010
$ 2024
$ 1892
2012
2329
2034
2011
2101
1835
2013
2296
1874
2012
2085
1702
2014
2248
1715
2013
2044
1559
Year
Payment $ mil.
PV of payment $ mil.
2009
1843
$ 1722
2010
1961
1713
2011
1960
1600
2012
1929
1472 (continued) 9
10
CHAPTER APPENDICES
Exhibit 8A.2 Calculation of operating lease obligations for Walgreen Co. for 2010, 2009, 2008* (continued ) 2010 Year
Payment $ mil.
2015
2009 PV of payment $ mil.
Year
Payment $ mil.
2188
1560
2014
2002
2016
1695
1130
2015
2017
1695
1056
2016
2018
1695
987
2019
1695
922
2020
1695
2021 2022
2008 Year
Payment $ mil.
1427
2013
1890
1348
1646
1097
2014
1592
1061
1646
1025
2015
1592
992
2017
1646
958
2016
1592
927
2018
1646
896
2017
1592
866
862
2019
1646
837
2018
1592
809
1695
805
2020
1646
782
2019
1592
756
1695
753
2021
1646
731
2020
1592
707
2023
1695
703
2022
1646
683
2021
1592
661
2024
1695
657
2023
1646
639
2022
1592
617
2025
1695
614
2024
1646
597
2023
1592
577
2026
1695
574
2025
1646
558
2024
1592
539
2027
1695
537
2026
1646
521
2025
1592
504
2028
1695
502
2027
1646
487
2026
1592
471
2029
1695
469
2028
1646
455
2027
1592
440
2030
1695
438
2029
1646
425
2028
1592
Total
$20 342
PV of payment $ mil.
$19 107
PV of payment $ mil.
411 $18 194
*7% discount rate, lump sum amortised over 15 year period.
A. Adjusting volatility measure for growth As shown with the Walgreen example for sales in Exhibit 8A.3, the coefficient of variation based on the standard deviation from the mean with no adjustment for growth indicates a significant level of sales volatility (41 per cent), in contrast to the volatility assuming linear growth, where the relative measure of volatility declines to 5 per cent. Finally, when you calculate the firm’s sales deviation from a compound growth curve of 13 per cent, the relative volatility is 11 per cent. These results indicate fairly low sales volatility when the measurement considers constant linear growth. The results when we measure the volatility of operating earnings (EBIT) are similar. Specifically, the relative volatility is 38 per cent when compared to the overall mean, 9 per cent when examined relative to the linear growth curve and 20 per cent when calculated relative to the compound growth curve of 10 per cent a year. This implies what can be seen in the graphs – the growth rates for sales and operating earnings are fairly high, but the growth is quite stable when measured against a constant linear growth line – this means that there is not the uncertainty (risk) implied if the volatility is measured relative to the overall mean.
B. Measuring operating leverage In addition to examining sales and operating income (EBIT) relative to its growth curve, we also discussed a specific measure of operating leverage that calculates the average of the annual ratios of per cent change in EBIT relative to the per cent change in sales. The understanding is that the higher this average ratio (i.e. the more volatile EBIT is relative to the volatility of sales), the greater the operating leverage for the firm that is caused by fixed operating expenses. Assuming that a firm has only variable operating expense, this ratio should be theoretically equal to unity (1.00). In the case of Walgreens, it is shown in Exhibit 8A.4 that the annual ratio varied from 0.77 to 1.24, and during 5 of 10 years the ratio exceeded 1.00. The overall mean of the annual ratios was 0.99, which implies a low operating leverage value, which is consistent with Walgreen’s overall low business risk as indicated by the low relative volatility of sales and operating earnings measured above.
2000
1999
41%
vs. Mean
1999
0
10000
20000
30000
40000
50000
60000
70000
80000
vs. C.G.
vs. Linear
17 839.0
11%
0
5%
0
22 346.4
17 839.0
Linear
Constant Growth
2000
1158003
1298149
20 130.9
21 207.0
17 839.0
Sales
1 224.0
1 015.0
EBIT
2000
21 207.0
17 839.0
Sales
1999
2001
3631906
4976144
22 717.2
26 853.7
24 623.0
2001
1 398.0
24 623.0
2001
2002
9272764
7182887
25 635.9
31 361.1
28 681.0
2002
1 624.0
28 681.0
2002
Sales
2003
12784273
11312826
28 929.5
35 868.5
32 505.0
2003
1 848.0
32 505.0
2003
2005
28745323
7188736
36 840.5
44 883.2
42 202.0
2005
2 424.0
42 202.0
Linear
2004
2005
Sales Volatility
23636583
8224381
32 646.3
40 375.8
37 508.0
2004
2 143.0
37 508.0
2004
2007
2008
395298
52 942.4
58 405.3
59 034.0
2008
3 430.0
59 034.0
2009
178391
59 744.3
62 912.6
63 335.0
2009
3 164.0
63 335.0
12893419
2008
37107608
2007
46882408
18471
46 914.9
53 897.9
53 762.0
2007
3 151.0
53 762.0
Constant Growth
2006
34050949
3926522
41 573.7
49 390.5
47 409.0
2006
2 702.0
47 409.0
2006
2009
0
0
67 420.0
67 420.0
67 420.0
2010
3 373.0
67 420.0
2010
(continued)
2010
1.13
4507.36
Exhibit 8A.3 Calculation of sales and operating earnings volatility for Walgreen Co. from arithmetic mean, from linear growth rate curve and from compound growth rate curve
APPENDIX TO CHAPTER 8 11
1 015.0
38%
Constant Growth
vs. Mean
0 1999
500.0
1,000.0
1,500.0
2,000.0
2,500.0
3,000.0
3,500.0
4,000.0
vs. C.G.
vs. Linear
1 015.0
Linear
20%
0
9%
0
1 015.0
EBIT
1999
2000
8448
29
1 132.1
1 229.4
1 224.0
2000
2001
18311
2091
1 262.7
1 443.7
1 398.0
2001
2002
46510
1162
1 408.3
1 658.1
1 624.0
2002
EBITA
2003
76840
598
1 570.8
1 872.5
1 848.0
2003
2005
220801
15084
1 954.1
2 301.2
2 424.0
2004 Linear
2005
EBIT Volatility
152880
3156
1 752.0
2 086.8
2 143.0
2004
2007
2007
518477
177318
2 430.9
2 729.9
3 151.0
Constant Growth
2006
272981
34765
2 179.5
2 515.5
2 702.0
2006
516427
235931
2 711.4
2 944.3
3 430.0
2008
2008
19559
29
3 024.1
3 158.6
3 164.0
2009
2009
0
0
3 373.0
3 373.0
3 373.0
2010
Exhibit 8A.3 Calculation of sales and operating earnings volatility for Walgreen Co. from arithmetic mean, from linear growth rate curve and from compound growth rate curve (continued )
2010
1.1
214.4
12 chapter appendices
2001
2002
–10
–5
0
5
10
15
20
2001
0.88
2002
0.98
2003
2003
2004
2004
N=
2005
2006
1.05
12.51
13.13
42 201 600
2 424 000
9.86 = 0.99 10
2005
1.03
15.39
15.93
37 508 200
2 142 700
% ∆ EBIT % ∆ Sales
1.03
13.33
13.80
32 505 400
1 848 300
Mean of Annual Operating Leverage Ratios = ∑
% Change EBIT/ % Change Sales
Ratio:
16.16 16.48
14.23
16.11
% Change EBIT
% Change Sales
1 624 200 28 681 100
1 398 300
24 623 000
Net Sales
EBIT
Exhibit 8A.4 Calculation of operating leverage for Walgreen Co.
2007
2006
2008
0.93
12.34
11.45
47 409 000
2 701 500
2007
2009
1.24
13.40
16.63
53 762 000
3 150 700
2010
2008
2009
−0.77
7.29
(5.64)
63 335 000
3 247 000
% Change Sales
% Change EBIT
0.94
9.81
9.21
59 034 000
3 441 000
2010
1.01
6.45
6.50
67 420 000
3 458 000
APPENDIX TO CHAPTER 8 13
Appendix to Chapter 9 Derivation of constant-growth dividend discount model (DDM) The basic model is:
P0 =
D1 (1 + k)
1
+
D2 (1 + k)
2
+
D3 (1 + k)
3
+ ⋯ +
Dn (1 + k) n
where: P0 = current price Di = expected dividend in Period i k = required rate of return on Asset j If growth rate (g) is constant,
D0 (1 + g) 1
P0 =
(1 + k) 1
+
D0 (1 + g) 2 (1 + k) 2
+ ⋯ +
D0 (1 + g) n (1 + k) n
This can be written as:
P0 = D0 c
(1 + g) n (1 + g) (1 + g) 2 (1 + g) 3 + + + ⋯ + d (1 + k) (1 + k) 2 (1 + k) 3 (1 + k) n
Multiply both sides of the equation by
c
1+k 1+g
(1 + g) (1 + g) 2 (1 + g) n −1 (1 + k) + + ⋯ + d P0 = D0 c 1 + d 2 (1 + g) (1 + k) (1 + k) (1 + k) n −1
Subtract the previous equation from this equation:
(1 + g) n (1 + k) − 1 d P0 = D0 c 1 − d (1 + g) (1 + k) n (1 + k) − (1 + g) (1 + g) n c d P0 = D0 c 1 − d (1 + g) (1 + k) n c
Assuming k > g, as n → ∞, the term in brackets on the right side of the equation goes to 1, leaving:
c 14
(1 + k) − (1 + g) d P0 = D0 (1 + g)
APPENDIX TO CHAPTER 9
This simplifies to:
which equals:
This equals:
c
(1 + k − 1 − g) d P0 = D0 (1 + g)
c
k−g d P = D0 (1 + g) 0
(k − g)P0 = D0 (1 + g) D0 (1 + g) = D1 so:
(k − g)P0 = D1 D1 P0 = k−g Remember, this model assumes: ■■ A constant growth rate ■■ An infinite time period ■■ The required return on the investment (k) is greater than the expected growth rate (g).
15
Appendix to Chapter 11 A. Preparing an industry analysis: what is an industry?1 Identifying a company’s industry can be difficult in today’s business world. Although airlines, railroads and utilities may be easy to categorise, what about manufacturing companies with three different divisions, none of which is dominant? Perhaps the best way to test whether a company fits into an industry grouping is to compare the operating results for the company and an industry. For our purposes, an industry is a group of companies with similar demand, supply and operating characteristics. The following is a set of guidelines for preparing an industry appraisal, including the topics to consider and some specific items to include.
Characteristics to study 1 Price history reveals valuable long-term relationships: a Price/earnings ratios b Common stock yields c Price/book value ratios d Price/cash flow ratios e Price/sales ratios 2 Operating data show comparisons of: a Return on total investment (ROI) b Return on equity (ROE) c Sales growth d Trends in operating profit margin e Evaluation of stage in industrial life cycle f Book-value-per-share growth g Earnings-per-share growth h Profit margin trends (gross, operating and net) i Evaluation of exchange rate risk from foreign sales 3 Comparative results of alternative industries show: a Effects of business cycles on each industry group b Secular trends affecting results c Industry growth compared to other industries d Regulatory changes e Importance of overseas operations
Factors in industry analysis Markets for products 1 Trends in the markets for the industry’s major products: historical and projected 2 Industry growth relative to GDP or other relevant economic series; possible changes from past trends 3 Shares of market for major products among domestic and global producers; changes in market shares in recent years; outlook for market share 4 Effect of imports on industry markets; share of market taken by imports; price and margin changes caused by imports; outlook for imports 5 Effect of exports on their markets; trends in export prices and units exported; outlook for exports 6 Expectations for the exchange rates in major non-US countries; historical volatility of exchange rates; outlook for the level and volatility of exchange rates
1
Reprinted and adapted with permission of Stanley D. Ryals, CFA; Investment Council, La Crescenta, CA 91214.
16
APPENDIX TO CHAPTER 11
17
Financial performance 1 2 3 4
Capitalisation ratios; ability to raise new capital; earnings retention rate; financial leverage Ratio of fixed assets to capital invested; depreciation policies; capital turnover Return on total capital; return on equity capital; components of ROE Return on foreign investments; need for foreign capital
Operations 1 2 3 4 5 6
Degrees of integration; cost advantages of integration; major supply contracts Operating rates as a percentage of capacity; backlogs; new-order trends Trends of industry consolidation Trends in industry competition New-product development; research and development expenditures in dollars and as a percentage of sales Diversification; comparability of product lines
Management 1 2 3 4 5
Management depth and ability to develop from within; organisational structure Board of directors: internal versus external members; compensation package Flexibility to deal with product demand changes; ability to identify and eliminate losing operations Record and outlook regarding labour relations Dividend policy and historical progression
Sources of industry information 1 2 3 4 5 6
Independent industry journals Industry and trade associations Government reports and statistics Independent research organisations Brokerage house research Financial publishers (S&P; Moody’s; Value Line)
B. Insights on analysing industry ROAs Insights on industry ROAs Beyond the normal analysis of ROA as a component of ROE (ROA times total assets/equity equals ROE), an article by Selling and Stickney provides some interesting insights for industry analysis based upon an analysis of the two components of the ROA ratio (profit margin and total asset turnover) and what these two components signal regarding the industry strategy.2 Given the two components of the ROA, it is possible to graph each of these values, as shown in Exhibit 11A.1, and determine what each component contributed to the ROA at the point of intersection. As shown, it is possible to draw a constant ROA curve, which demonstrates that it is possible to achieve an 8 per cent (or 4 per cent) ROA with numerous combinations of profit margin and asset turnover. The particular combination of profit margin and asset turnover is generally dictated by the nature of the industry and the strategy employed by management. For example, many industries necessarily require large capital inputs for equipment (e.g. steel, auto, heavy machinery manufacturers). Therefore, the asset turnover is necessarily low, which means the profit margin must be higher. The firms in such an industry are typically in the upper left segment of the graph (Segment a), and improvements of ROA in these industries are derived by increasing profit margins because it is difficult to increase asset turnover. In contrast, industries that have commodity-type products (e.g. retail, food, paper, industrial chemicals) generally have low profit margins and succeed based upon high asset turnover. These industries are generally in the lower right segment of the graph (Segment c), and they attempt to improve their ROA by increasing their asset turnover rather than the profit margin (i.e. they are constrained regarding the profit margin by price competition). Industries in the middle segment (b) are in a more balanced position and can attempt to improve the ROA by increasing either the profit margin or the asset turnover. It is very important for an analyst to understand the nature of the industry and what contributes to the industry’s ROA as well as what this implies about the constraints and opportunities facing the firms in the industry.
homas Selling and Clyde P. Stickney. ‘The Effects of Business Environment and Strategy on a Firm’s Rate of Return on Assets’, T Financial Analysts Journal 45, no. 1 (January/February 1989): 43–52.
2
18
chapter appendices
Exhibit 11A.1 ROA – The trade-off of profit margin and asset turnover Profit Margin (a)
(b) ROA = 8%
Competitive Constraint
(c) Capacity Constraint
ROA = 4%
Asset Turnover
Source: Copyright 1989, CFA Institute. Reproduced and republished from ‘The Effects of Business Environment and Strategy on a Firm’s Rate of Return on Assets’. Financial Analysts Journal, January/February 1989, with permission from the CFA Institute. All rights reserved.
Appendix to Chapter 15 Exhibit 15A.1 Calculation of duration and convexity for an 8 per cent, 5-year bond selling to yield 6 per cent Period
Cash flow
Discount factor
PV
PV × t
PV × t × (t + 1)
1
40.00
0.9709
38.83
38.83
77.67
2
40.00
0.9426
37.70
75.41
226.22
3
40.00
0.9151
36.61
109.82
439.27
4
40.00
0.8885
35.54
142.16
710.79
5
40.00
0.8626
34.50
172.52
1 035.13
6
40.00
0.8375
33.50
201.00
1 406.97
7
40.00
0.8131
32.52
227.67
1 821.32
8
40.00
0.7894
31.58
252.61
2 273.50
9
40.00
0.7664
30.66
275.91
2 759.10
10
1 040.00
0.7441 Total
773.86
7 738.58
85 124.34
1 085.30
9 234.50
95 874.32
Macaulay Duration =
9 234.50 = 4.25 2 × 1 085.30
Modified Duration =
4.25 = 4.13 1.03
Convexity =
95 874.32 = 20.82 (1.03) 2 × 22 × 1 085.30
19
Appendix to Chapter 16 Bond immunisation and portfolio rebalancing Suppose that you have decided to fund a three-year liability with a portfolio consisting of positions in a two-year zerocoupon bond and a four-year zero-coupon bond. The current interest rate is 10 per cent. Therefore:
Price of 2 -year bond (i.e. Bond 2 ) = (1000 ) ÷ ( 1.1 ) 2 = $826.45 Price of 4 -year bond (i.e. Bond 4 ) = (1000 ) ÷ (1.1 ) 4 = $683.01 In order to form a portfolio with duration of three years, you must purchase identical amounts of Bond 2 and Bond 4 since each of these zero-coupon instruments will have duration equal to its maturity. Consequently, you will need to buy 1.0 units of Bond 2 and (826.45 ÷ 683.01) = 1.21 units of Bond 4 for a total initial investment of $1652.89 [826.45 + (1.21)(683.01)]. The duration of your bond portfolio can then be calculated:
Dp = (826.45 ÷ 1652.89 ) (D2 ) + (826.45 ÷ 1652.89 ) (D4 ) = (.5 ) (2.00 ) + (.5 ) (4.00 ) = 3.00 Immediately after you make your initial purchases, interest rates fall to 8 per cent. If you do not rebalance your portfolio, what is your realised yield after three years? Calculate Terminal Value of Portfolio: (1) Allow two-year bond to mature and reinvest for one year: (1000)(1.08) = $1080.00 (2) Sell 1.21 four-year bonds: (1.21)(1000 ÷ 1.08) = $1120.37 so that your total terminal value is $2200.37, and the realised yield is: 3 2200.37 − 1 = 10.00 per cent Å 1652.89
Therefore, your actual return is equal to the original promised return (i.e. the yield to maturity) of 10 per cent. By investing so that the duration of the portfolio is equal to your horizon date, you have immunised yourself against the first interest rate change and locked in the initial promised yield of 10 per cent. Now, continuing with the assumption that interest rates decline immediately after your initial purchases, suppose that you decide to rebalance your portfolio. To understand this, you need to first establish the new bond prices:
Price of Bond 2 = (1000 ) ÷ (1.08 ) 2 = $857.34 Price of Bond 4 = (1000 ) ÷ (1.08 ) 4 = $735.03 To see why you need to rebalance, calculate the new duration of your portfolio:
20
Value of Investment in Bond 2 =
$ 857.34
Value of Investment in Bond 4 = (1.21)(735.03) =
889.39
Value of Total Investment =
$1746.73
APPENDIX TO CHAPTER 16
21
so:
Dp = (857.34 ÷ 1746.73 ) (D2 ) + (889.39 ÷ 1746.73 ) (D4 ) = (.4908 ) (2.00 ) + (.5092 ) (4.00 ) = 3.02 Even though the change in interest rates didn’t change the duration of the individual bonds, it did increase the duration of the portfolio slightly since it altered the relative market values of Bond 2 and Bond 4. To correct the problem (i.e. to rebalance the portfolio), you need to shorten the overall duration by selling some of Bond 4 and purchasing some more of Bond 2. To accomplish this, you must once again invest equal dollar amounts in each security, or $873.365 [(.5)(1746.73)]. This, in turn, means that you must own the following number of each instrument:
Number of Bond 2 to be held = (873.365 ÷ 857.39 ) = 1.0186 Number of Bond 4 to be held = (873.365 ÷ 735.03 ) = 1.1882 After completing this step, you have rebalanced your portfolio to immunise against future changes in interest rate movements. By once again holding equal amounts of each bond, you have reset the duration of the portfolio to your original investment horizon of three years. An important assumption here is that this rebalancing can be done costlessly. If you must pay brokerage fees, the total dollar value of your portfolio and hence your actual return will be reduced accordingly. By rebalancing your portfolio, you have tampered with the composition of your initial investment. However, by doing so, have you also changed the realised yield that you will receive? Put another way, when you rebalance your portfolio at the new yield to maturity of 8 per cent, will you still end up with the original yield of 10 per cent? To answer this, calculate the terminal value of the rebalanced portfolio: (1) Allow 1.0182 two-year bonds to mature and reinvest for one year: (1.0182)(1000)(1.08) = $1100.12; and (2) sell 1.1882 four-year bonds: (1.1882)(1000 ÷ 1.08) = $1100.19. Thus, your total terminal value of the rebalanced investment is $2200.31, and the realised yield is: 3 2200.31 − 1 = 10.00 per cent Å 1652.89
Therefore, rebalancing your portfolio when interest rates change has two effects: (1) you immunise yourself against the next interest rate change, and (2) your actual return is still equal to the yield to maturity that prevailed at the time of your original investment. Finally, it should be noted that although we assumed that the interest rate changed immediately after the initial purchase, the fundamentals of rebalancing also apply to yield curve shifts that occur at any point in the duration of the investment.
Appendix to Chapter 18 Exhibit 18A.1 Information ratios for six investment styles: 1986:Q1 to 1995:Q4
Note: Midpoints of ranges. Information ratios are on the x-axes; relative frequencies, in percentages, are on the y-axes. Source: Brian D. Singer, Renato Staub and Kevin Terhaar, ‘Determining the Appropriate Allocation to Alternative Investment’, Hedge Fund Management (Charlottesville, VA: CFA Institute, 2002), 10. Copyright © 2002 CFA Institute. Reproduced and republished from Financial Management Journal with permission from the CFA Institute. All Rights Reserved.
22
23
APPENDIX TO CHAPTER 18
A. A closed-form equation for calculating duration To calculate the duration statistic, it helps to think of a bond that pays a fixed coupon for a finite maturity as being just a portfolio of zero coupon cash flows. Duration is then the weighted average of the payment (i.e. maturity) dates of those zero coupon cash flows, its zero coupon equivalent maturity. Consider a nonamortising, five-year bond with a face value of $1 000 making annual coupon payments of $120 (i.e. 12 per cent). Assuming a yield to maturity of 10 per cent, this bond will trade at a premium and its weighted average payment date (i.e. duration) is 4.0740 years, as shown in Exhibit 18A.2. The duration of 4.0740 years is the weighted average maturity of that portfolio, where the weights are the respective shares of market value (e.g. the oneyear zero coupon cash flow is 10.14 per cent of the value of the portfolio; the five-year zero is 64.64 per cent). This five-year coupon bond with a duration of 4.0740 years is equivalent in terms of price risk to a zero coupon bond having a maturity of 4.0740 years. When the interest rate increases by 1 per cent above its original level (i.e. [Δ (1 + i) ÷ (1 + i)] = 0.01), then the price of this bond will decline by about 4.074 per cent. The Macaulay duration can be calculated with the following formula:
C Y 1+Y n + c 1n × T2 a F − n b d 1+Y n 18A.1 D= − n ×T Y C Yb Y + − 1 + c a1 d n n n F
where:
C = the periodic coupon payment F = the face value at maturity T = the number of years until maturity n = the payments per year Y = the yield to maturity In the preceding numerical example, Y = 0.10, n = 1, T = 5, C/F = 0.12 and Y/n = 0.10. The bond’s duration can therefore be solved as:
D=
1 + 0.10 + 5 1 0.12 − 0.10 2 1 + 0.10 − = 4.0740 0.10 1 0.12 2 3 1 1 + 0.10 2 5 − 1 4 + 0.10
As a second example, what is the duration of a 30-year Treasury bond with a 758 per cent coupon and a stated yield to maturity of 7.72 per cent? Recall that T-bonds pay semiannual interest and so the appropriate definitions of the variables are C/F = 0.38125, T = 30, n = 2 and Y/n = 0.0386. Therefore:
or 12.09 years.
1 + 0.0386 + 60 ( 0.038125 + 0.0386 ) D = 1 + 0.0386 − = 24.18 0.0386 (0.038125 ) 3 (1 + 0.0386 ) 60 − 1 4 + 0.0386
Exhibit 18A.2 A duration calculation Year
Cash flow
PV at 10%
PV ÷ Price
Year × (PV ÷ Price)
1
$ 120
$ 109.09
0.1014
0.1014
2
120
99.17
0.0922
0.1844
3
120
90.16
0.0838
0.2514
4
120
81.96
0.0762
0.3047
5
$1 120
695.43
0.6464
Price = $1 075.82
3.2321 Duration = 4.0740 years
24
chapter appendices
B. Calculating money market implied forward rates Implied forward rates are an essential factor in understanding how short-term interest rate futures contracts are priced. In Chapter 18, we discussed that implied forward rates represented the sequence of future short-term rates that were built into the yield to maturity of a longer-term security. However, implied forward rates can have another interpretation. Consider an investor who is deciding between the following strategies for making a two-year investment: (1) buy a single two-year, zero coupon bond yielding 6 per cent per annum; or (2) buy a one-year, zero coupon bond with a 5 per cent yield and replace it at maturity with another one-year instrument. An implied forward rate is the answer to the following question: At what rate must the investor be able to reinvest the interim proceeds from the second strategy to exactly equal the total return from the first investment? In other words, the implied forward rate is a breakeven reinvestment rate. In the notation of Chapter 15, we want to solve for 2r1 in the following equation:
1 1 + 0.06 2 2 = 1 1 + 0.05 2 1 1 + 2r1 2
or
r = 3 1 1 + 0.06 2 2 ÷ 1 1 + 0.05 2 4 − 1 = 7 per cent
2 1
Implied forward money market rates can be interpreted in the same way as bond yields, but they must be calculated differently because of differences in the quotation methods for the various rates. We have seen that LIBOR is a bank add-on yield (AY) that is used to figure out how much money, F, an investor will have at maturity in T days given an initial investment of P (i.e. interest is ‘added on’):
F = P + c P × AY × T d = P c 1 × AY × T d 360 360
LIBOR is based on a presumed 360-day year, the standard US money market practice. Smith (1989) has shown that the implied forward rate between two money market instruments quoted on an add-on basis (e.g. LIBOR) can be calculated as:
B
AYA
= c
1 B × AYB 2 − 1 A × AYA 2 d 18A.2 B−A
where AYA and AYB are add-on yields for A and B days from settlement to maturity, with B > A. The implied forward rate (BAYA) also is on an add-on basis and has maturity of (B – A) days. Consider the following short-term yield curve for LIBOR: Maturity
LIBOR
30 days
4.15%
60 days
4.25
90 days
4.35
What is the implied forward LIBOR between Days 60 and 90? Using AYA = 0.0425, AYB = 0.0435, A = 60 days and B = 90 days for LIBOR, we have:
90
AY60
= c
1 90 × 0.0435 2 − 1 60 × 0.0425 2 d 90 − 60
£
1
§ 1 + a 60 × 0.0425 b 360
= 4.52%
So, investing in a 60-day bank deposit at 4.25 per cent and then a 30-day deposit at 4.52 per cent would have the same total return as the 90-day deposit at 4.35 per cent.
