Proceedings of 30th International Business Research Conference
20 - 22 April 2015, Flora Grand Hotel, Dubai, UAE, ISBN: 978-1-922069-74-0
Equity Valuation Meets the Sigmoid Growth Equation: The
Gordon Growth Model Revisited
Evelyn Madoroba1* and Jan Kruger1
The valuation of equity is a central topic in finance and accounting from many fronts. However,
equity valuation is still subjective as different analysts obtain target prices that are not similar
despite using the same publicly available information. Growth has a substantial influence on the
value of equities when using Gordon Growth Model. Nevertheless, initial high growth and a
constant growth to perpetuity assumption whereby a company continues to grow indefinitely is
considered to be naive and simplistic. Therefore, this study proposes the use of sigmoid growth
curve equation that is expected to better capture practicalities of firm growth. The sigmoid curve
consists of lag, exponential, stationary and decline phases. The aim of this study was to provide
evidence of sigmoid growth patterns of Johannesburg Securities Exchange listed companies and
to propose a model for valuing equities using principles behind the sigmoid growth curve. This has
implications for improving accuracy when approximating the value of equities, detecting mispriced
stocks and informing buy/ sell decisions. For this purpose, cumulative sustainable growth patterns
of 64 JSE listed companies were examined using data from 1994 to 2014, followed by curve fitting
to determine whether the profiles were sigmoid. Interestingly, 50% of the firms showed typical
sigmoid growth patterns. Ultimately, a model was developed for valuing equities by replacing
growth, g, in the Gordon Growth Model with equivalent parameters in the sigmoid equation. In
conclusion, the findings of this study accentuate that the Gordon Growth Model for equity valuation
must factor in the complex growth parameters of firms in order to circumvent mispricing shares.
Key Words: Equity Valuation; Sigmoid Growth Equation; Gordon Growth Model;
Sustainable Growth Rate; Johannesburg Securities Exchange Listed Companies
Field of Research: Finance; Stock markets, Financial modelling
1. Introduction
The valuation of equity is a central topic in finance and accounting (Schreiner, 2007).
Nevertheless, equity valuation is still subjective as different analysts obtain target prices
that are not similar despite using the same publicly available information. This could be
due to the analysts’ different forecasts or the use of various models where the inputs vary
because of different assumptions.
The growth dynamics of diverse populations follows a sigmoid shape, which consists of
lag, exponential, stationary and decline phases. A typical example is microbial growth
when bacteria are introduced into a closed environment with fresh nutrients under
laboratory conditions. The number of bacterial cells increases at a slow pace initially,
followed by rapid exponential growth, then the growth becomes stable for some time
_______________________________________________________________________
1
University School of Business Leadership, Cnr Janadel and Alexandra Avenues, Midrand, 1686, Gauteng,
South Africa*: Corresponding author, Evelyn Madoroba, University School of Business Leadership, Cnr
Janadel
and
Alexandra
Avenues,
Midrand,
1686,
Gauteng,
South
Africa,
Email:
evelyn.madoroba@gmail.com
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Proceedings of 30th International Business Research Conference
20 - 22 April 2015, Flora Grand Hotel, Dubai, UAE, ISBN: 978-1-922069-74-0
before death of bacterial cells leads to a decline phase (Baranyi, 2010). During the lag
phase, it takes time for bacteria to adjust to the new environment. During the exponential
or logarithmic phase, bacteria divide and grow rapidly and exponentially. The exponential
bacterial growth leads to depletion of nutrients, accumulation of toxic metabolites and
waste, which leads to a stationary phase and subsequent decline phase if there is no
injection of fresh growth media.
There is scope in applying the principles behind the bacterial sigmoid growth curve
dynamics to the business environment. Almost all companies do not start by experiencing
high growth, but instead, they are anticipated to go through the ‘lag phase’ of adaptation,
followed by ‘exponential’, ‘stationary’ phase with or without value addition and a ‘decline
phase’ depending on whether management adjusts company strategy to avoid decline.
The lag phase may be equated to a phase when a business is being set-up or after a new
strategy is introduced. The exponential phase may be akin to the high growth period of
businesses. The stationary phase is reminiscent of stable growth in businesses. The
decline phase during bacterial growth can be equated to the phase when business growth
declines due to market saturation or stiff competition in the market.
Due to the pertinent role that growth rate plays in equity valuation, sufficient attention
should be awarded to the growth parameter, otherwise the resulting equity values might
be misleading. This study argues that equity valuation should be based on sigmoidal
growth depicted by the sigmoid growth equation (Gompertz, 1825). Therefore, the aim of
this study was to provide evidence for firm sigmoid growth and to develop a model for
valuing equities by capturing the complexities of firm growth dynamics through substituting
the growth rate, g, in the Gordon Growth Model (Gordon, 1962), with the growth rate
function in sigmoid equation using the Gompertz equation as an example.
The specific objectives of the study were:
i)
To examine cumulative sustainable growth patterns of 64 Johannesburg Stock
Exchange Listed Companies using data from 1994 to 2014, followed by
comparison with sigmoid curves;
ii)
To develop a model for valuing equities by substituting growth rate, g, in the
Gordon Growth Model with parameters of a sigmoid Gompertz equation.
1.1.
Investigation Approach
In the subsequent sections, we discuss each of the objectives of this study. Section 2
presents empirical evidence of sigmoid growth patterns for 64 JSE listed companies. In
section 3, we propose a ‘new’ model. Section 4 concludes the study findings.
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Proceedings of 30th International Business Research Conference
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2. Sigmoid growth in equity valuation
Despite the success of sigmoid curves in predicting future growth, there is very little
information regarding application of the principles behind the sigmoid growth curve in
equity valuation.
Herbst and Wu (2004)
Herbst and Wu determined the influence of earnings growth on cost of equity capital, and
the optimal time to sell stocks. The authors concluded that in the long run, companies
follow sigmoid growth patterns. Subsequently, the authors fitted a sigmoid curve to
earnings data of Microsoft and measured the magnitude to which investors presumably
miscalculate if they calculate share value based on Gordon formula instead of the method
that shows varying growth. Gap analysis: The study was not aimed at equity valuation,
which is the thrust of our study.
2.1.
Sample Selection and Description
In this section of the study, we sought to determine the cumulative sustainable growth
patterns of JSE listed firms with data for at least 15 consecutive years from 1994 to 2014.
Companies that did not have complete data for sustainable growth rate data for at least 15
consecutive years from 1994 to 2014 were excluded from this study resulting in 64 firms
being assessed. Cumulative sustainable growth was plotted on a time scale for each
company.
2.2.
Curve fitting and Statistical analysis
The cumulative sustainable growth patterns of all the 64 JSE firms were examined for
sigmoid patterns through curve fitting using the DMFit programme, which was developed
by Baranyi and Roberts (1994). Briefly, DMFit is an Excel add-in that uses regression
analysis to enable fitting of sigmoid curves, linear and biphasic functions to the data under
investigation. DMFit calculates the best fit parameters that comply with the input controls.
In addition, other statistical and computational indicators were calculated.
The four main parameters of DMFit are rate, lag, y0, yEnd and two curvature parameters
named mCurv and nCurv. The primary parameter is ‘rate’, which is the potential or
theoretical maximum rate of the model. The initial point of the sigmoid curve is denoted by
y0. The upper asymptote of the sigmoid curve is called yEnd. The curvature parameters at
the beginning and end of the linear phase are respectively described by nCurv and
mCurv.
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Proceedings of 30th International Business Research Conference
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2.3.
Empirical Findings
Figures 1 and 2 show examples of typical sigmoid growth patterns that were obtained from
cumulative sustainable growth of JSE companies. The number of firms that followed
sigmoid patterns after fitting the sustainable growth patterns of 64 JSE listed firms was
50% (n = 32). These firms had the lag, exponential and stationary phases. The values for
the different phases are shown in Table 1. The same proportion of firms that were
analysed in this study did not display sigmoid patterns.
Figure 1: Typical diversity of the sigmoid* curves of cumulative sustainable growth
patterns of JSE listed companies after curve fitting using DMFit software (Baranyi and
Roberts, 1994). The yValue represents cumulative sustainable growth, and Fit 1 shows
the line that was plotted through DMFit. TFG refers to The Foschini Group.
800
TFG
350
700
300
600
250
500
200
yVal
150
Fit 1
100
Cash Build
400
yVal
300
Fit 1
200
50
0
1990
-50
yVal
yVal
400
100
2000
2010
Time in years
2020
0
1990
2000
2010
Time in years
2020
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Proceedings of 30th International Business Research Conference
20 - 22 April 2015, Flora Grand Hotel, Dubai, UAE, ISBN: 978-1-922069-74-0
2500
350
ARGENT
AFROX
300
2000
1500
yVal
1000
yVal
yVal
250
200
yVal
150
Fit 1
Fit 1
100
500
50
0
1990
2000
0
1990
2010
2020
Time in years
2000
2010
Time in years
2020
Figure 2: Typical diversity of the non-sigmoid curves of cumulative sustainable growth
patterns of JSE listed firms after curve fitting using DMFit software. The yValue represents
cumulative sustainable growth, and Fit 1 shows the line that was plotted through DMFit.
300
600
AVI
TRUWORTHS
500
250
400
300
150
yVal
Fit 1
100
200
yVal
100
0
1990
-100
50
0
1990
yVal
yVal
200
Fit 1
2000
2010
2020
-200
2000
2010
time
2020
-300
time
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Proceedings of 30th International Business Research Conference
20 - 22 April 2015, Flora Grand Hotel, Dubai, UAE, ISBN: 978-1-922069-74-0
1000
800
GROUP 5
900
ELB
700
800
600
700
500
500
yVal
400
Fit 1
300
yVal
yVal
600
400
yVal
300
Fit 1
200
200
100
100
0
1990
2000
2010
2020
0
1990
-100
2000
2010
2020
time
time
Table 1: Summary of DMFit results obtained for some JSE listed firms
Firm
rate
Lag
y0
yEnd
se(fit)
R^2_stat
First Rand
33.2467959
3.90780356
9.2563109
395.36707
7.8777468
0.997204159
FPT
133.113293
9.66921833
0.6955976
319.36675
12.980232
0.992994219
INVLTD
23.8172473
7.63881008
28.06562
239.81864
14.594816
0.971531987
PUTPRO
44.6046753
13.013106
16.101247
257.22769
8.7533831
0.990085133
SANTAM
573.074136
8.88302787
54.430004
891.70679
34.852635
0.993030578
STANBANK
39.7999405
4.04288962
56.129875
518.44444
16.503533
0.991404856
SYCOM
59.4488134
45.741419
317.50165
14.19903
0.987061065
AVI
22.4241302
10.0312889
NL
29.520436
21.888975
0.926811061
Crookes
188.310358
14.3185733
122.58341
257.61634
NS
85.48716
0.935538069
Distell
9.89257591
NS
11.581869
0.94247422
19.7896402
6.12752689
NL
15.769882
Illovo
9.5437087
14.274991
0.979029474
Metair
24.6965455
NL
86.722165
294.83307
NS
44.799573
0.920948539
Nuworld
52.7654871
NL
14.267791
418.38905
148.73255
0.408677303
Oceana
82.6386885
NL
556.631
45.506318
0.943860241
SAB
10.7114586
NL
-25.31682
17.895316
NS
7.7522075
0.985243904
Sovfood
22.1751439
25.63889
82.041595
4.9391516
0.910026716
Tongaat
26.1549022
2.5797762
NL
-1.811203
98.489374
31.454805
0.445843563
41.915845
719.21337
13.109105
0.997808593
127.96754
22.889461
0.951738092
11.500592
0.97850978
Cash Build
Caxton
18.8996807
7.15570179
NL
City Lodge
12.5138312
NL
34.70026
434.65709
NS
163.30323
9.69794378
122.4789
563.59491
66.050554
0.910518575
Fam Brands
105.197269
9.04803135
66.322239
623.4228
35.619434
0.980221229
Italtile
8820.00244
7.07166601
215.92084
8314.3541
143.5494
0.998706107
CMH
77.62648
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Proceedings of 30th International Business Research Conference
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JD Group
26.9834998
NL
62.528293
348.23609
14.974797
0.977290366
Mass Mart
571.354978
102.59648
0.97290704
140.71862
776.87792
NS
33.420166
Mr Price
8.94633956
NL
127.44331
0.987518895
ND
NICTUS
23.6796836
10.7274789
14.477805
NS
9.6235741
0.983824942
Rex True
69.1866748
49.497367
483.67875
30.246013
0.973484611
Shoprite
31.9613279
9.85403082
NL
37.74414
606.84552
15.757928
0.992534241
SUNINT
91.2018918
4.01659056
1064.6712
77.937776
0.964396805
TFG
34.4347254
337.30651
11.39365
0.993153485
24.269745
5.209539
NL
10.257066
0.6264769
337.43803
40.869869
0.830409471
Truworths
41.8438726
4.06866277
49.1951
34.927766
464.47088
17.585371
0.987756418
Wollies
18.7198659
2.40887208
431.99889
540.17651
39.613749
0.277978416
Altron
689.071844
6.03053259
88.673581
903.06209
66.665892
0.970791224
Argent
569.753664
7.34320252
2104.0429
173.41658
0.964278506
Barworld
27.9914682
NL
204.9893
9.6006148
259.47696
26.821785
0.920417199
Bowcalf
Firm
61.2330325
rate
NL
Lag
188.56715
y0
1023.774
yEnd
55.305388
se(fit)
0.964300107
R^2_stat
Cargo
142.364893
7.44731133
85.708691
43.505243
0.990815812
ELB
76.0727155
11.0610733
47.264548
1135.0277
NS
41.778335
0.969019974
Grindrod
521.984022
5.82968332
55.934084
0.969357737
54.0429054
6.28845989
171.52936
2027.0182
NS
160.96155
Group 5
27.59099
0.988258391
ILIAD
1044.43963
589.20766
2.32449351
344.08922
5292.2536
NS
494.69899
Imperial
2.53807928
NL
123.98768
0.879137117
ND
Invicta
M&R
Holdings
1454.7947
0.91670998
674.28556
2579.4586
189.32256
0.795774257
14.2415614
NL
61.435421
NS
74.033462
0.52746959
Masnite
12.8150576
NL
32.495681
136.19323
11.077272
0.904731568
6.7913466
NL
0.6889697
4.1397115
0.960715439
Reunert
18.6524805
NL
137.86567
61.569576
NS
36.350301
0.909805878
Sasol
35.4958859
4.77170456
16.49745
439.59867
18.57781
0.981405762
AFROX
31.8059851
3.2970976
0.967644967
28.5294156
38.893939
302.52664
NS
21.842996
ANGLO
3.27559927
NL
78.74222
0.833443057
Anglo Gold
Anglo
Platinum
11.8570351
NL
4.9727895
NS
111.42937
0.232539699
50.6074545
NL
-63.99026
NS
153.64398
0.804952028
ARC Mittal
52.1437893
8.90335511
36.785947
327.58645
15.092221
0.986710595
BHP Billiton
265.572258
10.5072198
NL
192.22416
42.476126
0.988267495
Tsogo
PPC
18.49632
1011.8942
NS
44.877776
0.933316687
GFIELDS
54.6151743
NL
-30.23806
505.29389
39.187829
0.955868901
IMPLATS
35.8979724
6.34903177
315.09523
15.416075
0.985271594
LONMIN
235.157497
1.49472686
14.077197
78.376955
493.77951
218.80699
0.314681569
NOTHAM
10.7336415
9.22162961
14.910801
91.074813
4.1256078
0.983194113
Delta
27.106987
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Proceedings of 30th International Business Research Conference
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21.499758
NS
101.1161
0.939329756
20.919797
3.19684926
NL
17.949368
NS
33.840954
0.930179743
382.577218
8.49589761
89.32693
1961.4092
90.572302
0.989051576
OMNIA
71.7273253
SAPPI
Sentula
Key: Rate refers to the potential or theoretical maximum rate of the model. The 'lag'
represents the lag growth parameter as defined by Baranyi and Roberts (1994). The y0
represents initial point of the sigmoid curve. The yEnd refers to the upper asymptote of
the sigmoid curve. NL, NS and ND indicates that there was no lag phase, no stable phase
and no defined fit respectively; which implies that the growth pattern was not sigmoid.
R^2_stat refers to the Adjusted R-square statistics of the fitting, whereR2 is the coefficient
of determination, which shows the proportion of variability in the data set due to the
statistical model.
2.4.
Discussion of empirical findings
The fact that 50% of the 64 JSE listed firms in this study followed sigmoid growth patterns
may have implications on mispricing of equities during the period studied if the GGM was
used to value equities. This is because the findings clearly demonstrated sigmoid growth.
Similar findings were obtained by Herbst and Wu (2004) who illustrated how investors
would err if they applied constant growth models instead of using growth patterns that are
specific to the company.
The very high R2 values that were obtained in this study demonstrate that the estimates of
regression lines were very close to the actual data points. This implies that more accurate
share values would be obtained if the growth parameter in equity valuation models is
based on the sigmoid equations.
There are a plethora of reasons that give rise to non-sigmoidal growth patterns among
firms. Failure to follow sigmoid growth patterns by firms may be caused by growth
setbacks or high growth as a result of complicated industry, macroeconomic or firm
dynamics. We propose that companies follow sigmoid growth patterns as a simple way of
managing resources and predicting future growth.
3. Linking Gordon Growth Model and the
Development of An Equity Valuation Model
Gompertz
curve:
Due to the theoretical scope of substituting growth in the Gordon Growth Model with
parameters from the Gompertz equation, we developed the equity valuation model by
decomposing both the Gordon Growth formula and the Gompertz equations. The
substitution could also be based on Logistic, and Richards sigmoid equations or their
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Proceedings of 30th International Business Research Conference
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variations depending on the best determined model. This model selection could be done
using other software that specifically defines the type of sigmoid curve, but this was
outside the scope of this study. Nevertheless, model selection constitutes future work.
In the standard Gordon growth model, the constant growth rate is g and the model is
represented by:
Vj= D / (Ke - g) …………………………………………………………………………….....(1)
Where:
Vj = Value of stock j
Ke = The required rate of return on stock j, which is the cost of equity capital,
D = Dividend,
g = Expected long-term sustainable growth rate in cash flow to the investor
We then incorporated growth parameters of a Gompertz sigmoid growth equation in
Gordon growth formula (1). Although the substitution is not expected to fit completely due
to the differences between continuous and discrete values, the growth parameters are
comparable and it is estimated that the continuous growth function is a better
representative of the reality (Herbst and Wu, 2004). Unlike previous one, two and three
stage growth models that were characterized by discrete phases, our model assumes
continuous growth.
The Gompertz curve has diverse mathematical representations and some of the
expressions are shown below:
According to McKellar and Lu, (2004), the Gompertz Model can be represented as:
loge N (t)= A+C exp{exp[−B (t −M)]}………………………………………….................…(2)
where N(t) is the momentary number of cells in the population,
A: represents the logarithm of the initial number, that is, A ≈ logeN0,
C: is the asymptotic logarithmic growth ratio, that is, N(t)/N0 when t →∞ ,
B: is the relative growth rate, and
M: the time that corresponds to the maximum growth rate, that is, it marks the inflection
point location of the sigmoid growth curve.
In a study of mammalian growth curves by Zullinger and co-workers (1984), the Gompertz
equation is represented as:
………………………………………………….............................................................(3)
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Proceedings of 30th International Business Research Conference
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Where A = Asymptomatic mass
M(t) = mass (g) at age t
K = growth rate constant
I = age at inflection point
Using the Gompertz curve in (3), we developed the model as follows:
In order to make K the subject of formula, the following algebraic steps were taken:
Each side of the equation was divided by A resulting in the expression:
M(t)= e^-e^K*(t – I)…………………………………………………………….....................(4)
A
Then the natural logarithm of each side of the equation was considered, which gives the
expression:
Ln
A
M(t)
= ln M(t) – ln A, ……………………………………………………….....(5)
Therefore,
ln M(t) – lnA = -e^-K*(I–t)…………………………………………………………….........(6)
Taking an additional natural logarithm to each side of the equation yields:
ln (ln M(t)–ln A) = K*(t – I)……………………………………………………………........(7)
Dividing both sides by (t – I) gives the expression;
ln (ln M(t) – ln A) = K…………………………………………………………………….....(8)
(t – I)
We then substituted the constant and discreet growth function g in the Gordon Growth
formula with K, the natural growth rate function in the Gompertz growth equation. We
assign K in the Gompertz equation the designation Kappa to avoid confusion with Ke,
which is the cost of equity in the Gordon Growth formula. The assumption is that dividends
at time t, Dt follow the Gompertz growth equation and the firm will not ordinarily pay a
dividend that is greater than the earnings (Herbst and Wu, 2004).
Our proposed model for valuing equities is therefore:
Vj =D …………………………………………………………………………………(9)
Ke – Kappa
Where Kappa = ln (ln M(t) – ln A)
(t – I)
A represents the maximum expected dividends over a chosen time horizon
Mt = dividends at any time t within the chosen time horizon
t = any time within the chosen time horizon
Kappa is the natural growth rate
I = duration of growth at inflection point
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Proceedings of 30th International Business Research Conference
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4. Conclusion
This study has used cumulative sustainable growth data for 64 JSE listed companies over
a 20 year period to provide empirical evidence that the growth of many firms actually
follows sigmoid patterns. Indeed, empirical support for the proposed model was obtained
as 50% of the companies clearly illustrated sigmoid growth patterns with high coefficient of
determination values. The findings of this study can be used to reflect the intricacies of
company growth. Furthermore, managers of companies may use the findings of this study
to predict the optimal time to change company strategies.
A ‘new model’ for valuing equities was then proposed based on the principles behind the
Gordon Growth Model and the sigmoid growth equation. This model is expected to
circumvent the challenges of discrete constant growth rate to perpertuity that are
associated with the GGM. The ‘new model’ is expected improve equity valuation accuracy.
5. Future Work
The quintessential test of any model is the precision of its prediction (Modis, 2007). It is
our intention to use the ‘new’ model for predicting the value on stocks of publicly listed
firms on diverse securities counters.
6. References
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distributed lag time, Doctoral Thesis. School of Informatics University of Szeged,
Hungary.
Baranyi, J. and Roberts, T.A. 1994. ‘A dynamic approach to predicting bacterial growth
in food;’ International Journal of Food Microbiology, vol. 23, pp. 277-294.
Bierman, H. Jr. 2001. ‘Valuation of stocks with extraordinary growth prospects’, The
Journal of Investing, vol. 10, issue 1, pp. 23-26.
Damodaran, A. 2006, Valuation approaches and metrics: A survey of the theory and
evidence, Stern School of Business.
Gompertz, B. 1825. ‘On the nature of the fucntion of the law of human mortality, and a
new mode of determining the value of live contingencies’, Philosophical
Transactions of the Royal Society, vol. 182, pp. 513-585.
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Proceedings of 30th International Business Research Conference
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.
Gordon, M. 1962, The investment financing and valuation of the corporation. Irwin:
Homewood, IL.
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