Content uploaded by Steven Kim

Author content

All content in this area was uploaded by Steven Kim on Dec 09, 2020

Content may be subject to copyright.

Short Summary

The world is full of complex and baffling systems. For instance the stock market

confounds all manner of investors, be they amateurs or professionals. On the upside, a

duplex framework can portray flighty systems with the utmost of simplicity, clarity and

efficacy. An example concerns the monthly waves of the Dow stock index.

Mid Summary

The world abounds with complex and puzzling systems. For instance, the stock market

fuddles all manner of investors ranging from newcomers to veterans. According to the

reigning doctrine of financial economics, every market is an efficient system where the

current price always reflects all the information available to investors. In this sleek

environment, no one can discern any cues for predicting the market in a credible way.

On the upside, though, a duplex framework can debunk the myths of the stock market in

decisive ways. An example involves the seasonal waves behind the monthly moves of

the Dow stock index. The benefits of the wavy model include the ease of acquiring the

information required, the leanness of the dataset employed, the ubiquity of the software

deployed, the universality of the experimental setup, and the strength of the conclusions

at high levels of statistical significance.

3

Long Summary

The world around us is chockful of complex and baffling systems. For example, the

stock market confounds all manner of investors ranging from part-time amateurs to full-

time professionals. According to the reigning doctrine of financial economics, every

market is an efficient system where the current price always reflects all the information

available to investors. In this sleek environment, no one can detect useful clues for

predicting the market with any measure of consistency.

On the bright side, though, a duplex framework can portray dicey systems with the

utmost of simplicity, clarity and efficacy. The merits of the binomial scheme show up, for

instance, in debunking the myths of the stock market. An example involves the seasonal

patterns behind the monthly moves of the Dow Jones Industrial Average.

The case study presented here spotlights the simplicity of the cyclic model from a

conceptual slant as well as a pragmatic stance. Other drawcards include the ease of

acquiring the information required, the leanness of the dataset employed, the ubiquity of

the software deployed, the universality of the experimental setup, and the strength of

the conclusions at high levels of statistical significance.

4

* * *

Ruling Myths and Trampled Facts

We live in a world full of complex and chaotic systems. A good example concerns the

stock market that stymies all manner of investors ranging from casual amateurs to

gung-ho professionals.

According to the Efficient Market Hypothesis, the current price always reflects the

totality of information available to the investing public (Fama, 1965). As a byproduct, no

one can detect any clues for predicting the market in a trusty fashion.

Instead, the market is deemed to move in an utterly erratic way. In particular, a popular

myth known as the Random Walk shuffle contends that the price level shifts with equal

likelihood and to similar extent in either direction, whether to the upside or downside.

At first glance, the image of pure randomness does ring true in practice. For instance,

the average investor is unable to beat the market averages such as the Dow Jones

index. While the lack of success may seem like a letdown, the truth is even worse. In

actuality, the participants in the aggregate lag comfortably behind the benchmarks of the

bourse.

If we look more closely, the lousy performance of the actors springs mostly from their

frantic efforts to beat the competition. Amid the frenzy, the brutes of greed and fear prod

the antsy players into making impulsive moves that are not only groundless and futile

but actually counterproductive and harmful to their cause.

5

On the bright side, though, the market displays a smattering of patterns that can be

exploited by a sober person. An example concerns the seasonal cycle behind the

monthly moves of the Dow benchmark.

To fathom the elusive waves in a stringent fashion, we turn to the duplex method of

modeling shifty systems. The sturdy framework makes use of the binomial test: the

simplest and strongest, as well as safest and surest, way to profile chancy events

regardless of the domain.

To this end, we first transform the conceptual models of the stock market into a trio of

precise templates. The formal blueprints are then converted into R code: the top choice

of programming language and software platform for statistical assays. The trenchant

results serve to debunk the fable of efficiency and confirm the existence of hardy

patterns in the marketplace.

In short, the benefits of the seasonal model lie in simplicity and potency in sundry forms.

The drawcards include the ease of acquiring the information required, the leanness of

the dataset employed, the ubiquity of the software deployed, the universality of the

experimental setup, and the strength of the conclusions at high levels of statistical

significance.

Method

To underscore the gulf between the theory and reality, we will take a minimalist

approach. For starters, we use only a minute fraction of the wealth of information freely

available to all comers at the most popular portal among the investing public. Moreover,

the quantitative analysis employs only the simplest technique in hypothesis testing. In

addition, the code required for the study invokes a tiny subset of the core functions

6

within the R system: the top choice of programming language and software platform for

data science.

Precise Models

According to the Efficient doctrine, every asset displays equal odds of rising or falling at

any stage. In that case, the current price always reflects the best guesstimate of the

price level going forward. We will refer to this parody of the market as the Idle Model.

We now take a step back and look at the big picture of the stock market over the long

range. Anyone can see at once that the price level has a way of climbing higher as the

decades go by. The global trend shows up, for instance, in the ascent of the Dow Jones

Industrial Average since its inception in 1896. The index is also known by the shorthand

of DJIA or simply DJI.

In view of the uprise over the long range, a variation on the theme of ideal markets

asserts that the upward tilt poses the best guess going forward. For instance, the

optimal forecast of the price level a month hence is the current price plus a suitable

increment of the gradual uptrend. We will refer to the trending version of the Efficient

mantra as the Drift Model.

Choice of Dataset

The Dow Jones Industrial Average is the most popular benchmark of the stock market

amongst the general public. For instance, the DJI is the yardstick of choice when the

mass media round the world report on the movements of the U.S. bourse on the

evening news.

7

As we noted earlier, the DJI in its modern form evolved from a prototype dating back to

the late 1800s. Given the long history, the use of a lengthy dataset may seem like a

good idea. That might in fact be the case if the target system happens to be stable and

displays a fixed set of properties.

But that’s not the case for us here. For one thing, the action in the stock market stems in

part from internal outcrops such as the rollout of novel products. A case in point is an

option contract or an inverse fund that may be used to offset the risk of a market crash.

A second driver involves the changing habits of the participants within the financial ring.

An example concerns the swelling throng of fund managers along with their penchant

for window dressing toward the end of the year; namely, the switch-out of downcast

assets with high-flying stocks in an effort to mask the crummy performance in their

annual reports.

A third facet concerns the growing role of the stock market for the mass of humanity. An

exemplar involves the multitude of entrepreneurs and employees who pour a bigger

fraction of their earnings into the bourse around the end of the year.

More generally, the lot of the stock market is tied to the fortunes of the real economy

especially over the medium term and the long range. Moreover, the economy itself has

a way of evolving with the passage of time. An example lies in the dwindling role of the

agrarian sector as well as the industrial branch over the past century.

In this fluid milieu, we ought to balance the richness of a lengthy dataset against the

pertinence of a recent sample. We will use the data from Yahoo Finance, the most

popular portal among the investing public (Yahoo, 2020). At the time of research, the

records available at the site for the Dow index stretched back to 29 January 1985. In

that case, a suitable dataset for our study involves the closing values of the benchmark

at the end of each month starting with the onset of 1985.

8

Since we want to distill the seasonal cycle from the historical record, we will sample the

data on a monthly basis until January 2020. In this way, we can calculate the forward

returns over precisely 35 years beginning in 1985.

Check for Null Returns

The financial forum finds itself in a constant state of flux as every asset swells and

swoons as time goes by. Despite the ceaseless heaving, though, the price level could

settle at the same point from one period to the next.

On the other hand, our dataset does not contain any repeated values in direct

succession. As a result, the monthly return is never zero. So we can ignore the prospect

of a null value in assigning each move into one of two disjoint buckets; namely, to the

upside or downside.

Profiling the Data

As we saw earlier, the Dow index grows over time despite a series of sizable upswings

and downstrokes along the way. Moreover, our dataset in particular also displays these

traits as shown in Figure 1.

On the whole, the benchmark appears to burgeon at an exponential rate over the entire

stretch. That makes sense since the yardstick reflects a simple average of the stock

prices for a rolling roster of 30 hulking companies in the prime of their lives. At a

minimum, then, the value of the ensemble should keep pace with the march of inflation

that tends to advance by a couple of percent or more per year.

9

Fig. 1. Monthly level of DJI from January 1985 to January 2020.

Fig. 2. Log to base 10 of DJI by month over the entire timespan.

10

We now profile the overall trend by plotting the logarithm of the raw readings over time.

In Fig. 2, the vertical axis denotes the log to base 10 of the closing prices. The graphic

suggests that the overall trend is more or less steady. In that case, the original variable

displays an exponential streak.

An exponential rate of growth reflects a steady gain in relative terms; in other words, a

fixed percentage on a periodic basis. Fig. 3 shows the relative shift from one month to

the next throughout the timeline.

Fig. 3. Forward monthly return for DJI from 1985 to 2019.

The next step involves a quantitative assay. As a lead-in, we have just reckoned the

fractional changes in price on a monthly basis over the entire stretch. At each stage, we

11

pinned down the relative gain—also called the forward return—over the next period as a

percentage of the current price.

From the graphic, we can see that the returns vary a great deal. Despite the outspread,

though, the scattering pattern appears to be more or less stable especially after the first

hundred cases or so.

Crafting the Models

Our next step is to convert the verbal descriptions of the stock market into formal

models. Two of the templates represent the Efficient bromide. Meanwhile the third

model depicts the seasonal cycle of monthly returns throughout the year.

We begin with the Idle mockup which declares that the market rises or falls with equal

likelihood and to similar extent. Since the expected moves to the upside and downside

are perfectly balanced, the best forecast of the monthly return at any stage is fixed at

zero percent.

On the other hand, the setup differs somewhat for the Drift Model. The latter send-up

does rely in a small way on the actual behavior of the market; namely, the uptrend over

the long range. For the dataset at hand, the geometric mean of the monthly gains

amounted to some 0.7382 percent. Roughly speaking, then, the Dow index was wont to

rise by nearly three-quarters of a percent on average each month.

The third and last model captures the seasonal patterns in the stock market. We will

refer to this portrait of monthly moves as the Sway Model.

Fig. 4 shows the mean return by month throughout the year. We can see that, more

often than not, the average move had a positive bent. In particular, the biggest gains

12

occurred in March, June, October and November. Moreover, the market on average

hardly moved at all during May. On the downside, the worst performance cropped up in

July and August.

Meanwhile, Fig. 5 offers a richer view of the dataset. Each boxplot depicts the forward

returns for a particular month. For a given column, the thick bar within the rectangle

represents the median level rather than the mean value for the pertinent period.

Furthermore, each circle denotes an outlier.

The boxplot for each month reflects a great deal of dispersion compared to the waver of

median values throughout the year. The largest number of outliers appear in the fifth,

ninth and eleventh thumbnails. More precisely, each of May, September and November

features three extreme values.

Fig. 4. Mean return for each month of the year.

13

Fig. 5. Boxplot of forward monthly returns.

Despite the broad scatter of readings along with the clutter, we can discern some

differences in the average values. For instance, the median return for October appears

to surpass those of September and November. Meanwhile, an example on the flip side

applies to August whose average gain falls below those of July and September. On the

whole, the relative placement of the median values resembles the configuration of mean

returns that we saw in the previous chart.

Choice of Average for the Sway Model

We could use the chain of mean values or the median levels to serve as the backbone

of the Sway Model. To this end, we first check which arrangement leads to higher

accuracy; or more precisely, less variation in the absolute values of the errors against

the entire set of forward returns.

14

We find that the string of median values does a better job of modeling the monthly gains

on average than the chain of mean returns. So we adopt the median array as the

marrow of the Sway Model. That is, the median value of the forward returns for each

month will serve as the implicit forecast for that period.

Graphic Matchup of Results

We now compare the three models for profiling the seasonal patterns. The output in the

form of Fig. 6 presents a boxplot of the errors for each model.

Fig. 6. Boxplots of errors by model.

15

Compared to the full scatter of errors, the average values of the residuals hardly varied

at all. On the other hand, a graphic display can be misleading. For this reason, we turn

to several forms of quantitative analysis.

Precise Peg of Ups and Downs

According to the Efficient chorus along with the Random Walk burlesque, the moves to

the upside or downside occur with equal likelihood. In that case, the null hypothesis

posits the chance of upturns to equal 0.5. In the absence of further information or

supposition, the alternative premise entails a two-tailed test in which the critical regions

for rejection lie on the low side as well as the high end.

In the current study, the number of successes—that is, the turnups in price—came out

to 266 out of 420 trials. The ratio amounted to a little over 0.633 which exceeded the

anticipated value of one-half.

More to the point, the assay rendered a p-value of 5.108x10–8. The latter value fell well

below the usual threshold of 0.05 that serves as the upper bound for rejecting the null

hypothesis. As a result, we could eject the premise that the movements of the market

were thoroughly random. That is, the Idle Model contradicted the data at a high level of

statistical significance.

In the case of the Drift motif, the best forecast at each stage reflects the fixed uptrend

over the long range. Other than that, the sloping template resembles the idling motif. In

particular, the odds of leading or lagging the trendline are deemed to be equal. For this

reason, the null hypothesis presumes the fraction of positive residuals to be one-half.

In actuality, the number of upturns equaled 230 out of 420 cases thus yielding a quotient

of some 0.548. The latter fraction strayed by a moderate amount from the expected

16

value of 0.5. Moreover, the p-value of 0.05691 fell just above the threshold of 0.05 for

rejecting the null hypothesis. As a result, the Drift mockup lodged within the realm of

credibility.

In brief, the market displayed a small tendency to rise rather than fall in relation to the

prolonged uptrend. Despite the imbalance, a binomial test of directional moves showed

the Drift motif to be more or less compatible with the data.

We now proceed to the Sway Model. The number of successes, or miscues to the

upside, equaled 204 out of 420 cases. The resulting quotient came out to 0.486.

Moreover, the p-value of 0.5915 proved the null hypothesis to be wholly credible. In

other words, the Sway Model pinned down the midpoint of the movements in the

following sense: the implicit forecast was about as likely to overrun as to undershoot the

actual return going forward.

We have just assessed the pricing models in terms of the fraction of correct calls. In

particular, we proved that the Idle gospel runs afoul of the reality. Moreover, the Drift

template was somewhat plausible from a statistical stance. Best of all, the Sway Model

turned in trustworthy results; the seasonal template bisected the prospects so well that

the fraction of positive errors came much closer to the expected value of 0.5.

To recap, we examined the fitness of the candidate models based on the frequency of

errors to the upside versus downside regardless of the extent of the flubs. Each

template was checked separately to determine whether or not it conforms to the

historical record. A different way to appraise the models is to compare their performance

against each other.

17

Global Odds of Orientation

The trio of pricing models may be pitted against each other in a pairwise fashion. More

precisely, one template outranks another if its tally of miscues undercuts its rival’s.

According to the null hypothesis, the fraction of wins for any model against any other

should equal one-half. Even so, we may have good reason to believe that a given

model ought to outplay another; in that case, the alternative conjecture posits the

fraction of wins to surpass one-half.

As an initial step, the Drift template was compared against the Idle idol. The null

hypothesis presumed no difference in efficacy thus entailing a win ratio of 0.5.

In the previous section, we saw that the Drift makeshift was wont to outshine the Idle

mockup. In that case, a fitting premise for the alternative thesis avers that the former

should trounce the latter with a probability in excess of one-half.

The Drift Model in fact bested the Idle canard in 244 cases out of 420 trials, thus scoring

a win ratio of around 58.1%. Moreover the p-value of 0.0005272 proved the advantage

of the winner to be highly significant.

The results of the previous section also suggest that the Sway Model should outstrip the

Idle version. For this reason, the null hypothesis assumed no difference in accuracy

while the alternate premise favored a fraction in excess of 0.5. A rundown of the data

showed the Sway template to outclass the Idle motif in 244 cases thus amounting to a

hit rate of 58.1%. Moreover, the p-value of 0.0005272 marked a stout level of statistical

significance.

18

The next checkup involved a similar assay for the Sway framework against the Drift

mockup. The former beat the latter in 55.2% of the cases and yielded a p-value of

0.01788. In this way, the cyclic template outran its trending rival at a respectable level of

statistical clout.

To sum up, the Drift makeshift outranked the Idle icon in terms of the odds of rising and

falling. Better yet, the Sway template outflanked both of the Efficient takeoffs with

remarkable consistency. The advantage of the seasonal framework prevailed at ample

levels of statistical significance throughout the timespan covered by the study.

As we saw earlier, the directional moves of the market were by themselves enough to

expunge the dogma of perfect efficiency. Moreover, this section has rendered similar

results based on the performance of the pricing models against each other.

From a larger stance, these types of appraisals may also be performed on smaller

portions of the annual cycle. An example involves a marked tendency of the market to

falter during the summer then advance in the autumn.

Odds of Intra-year Waver

In the previous two sections, the monthly moves of the stock market were canvassed

across the entire timeline without paying heed to the miscues by month over the year.

On the other hand, we may perform similar assays of the relative performance of a

couple of months within the year.

For a specific example, we return to Fig. 5 which presents the boxplot of returns for

each month of the year. From this chart, we infer that the median value in the tenth

boxplot is higher than that of the eighth thumbnail. In other words, the median return for

October outranks that of August.

19

These two months in fact turned in the best and worst scores on average by way of

median returns. As a result, August and October comprise the top candidates for

confirming the gap in performance between different portions of the year.

Two Samples Without Coupling

For the first test of intra-year movements, we examine the two months independently.

That is, we check whether each month displays a notable tendency to rise or fall

regardless of the behavior of the other month during the same year.

The null hypothesis presumes that August and October display equal odds of rising or

falling. For a change of pace, we consider an inquisitor who is so entranced by the

Efficient scripture as to deny the mass of evidence thus far. The fiend asserts instead

that neither month flaunts any advantage over the other. In that case, the alternate

hypothesis entails a two-tailed test.

We now check the difference in performance by way of a binomial test. For starters, we

find that October climbed higher in 25 out of 35 years thus notching a win ratio of

71.4%. The corresponding value for August was 45.7%. The mismatch between the

foregoing rates displayed a p-value of 0.003298. In this way, October prevailed against

August at a staunch level of significance.

In this assay, we examined the independent prospect of each month rising or falling

regardless of the year of occurrence. The checkup took the form of a test for

proportions.

As an aside, we note that Fisher’s exact test would have been an apt method if we were

checking for the consistency of directional moves. In the latter case, the query would

20

take the following form: Is the market prone to rise in October when August does

likewise, and vice versa?

On the other hand, our concern here is more pragmatic: whether October tends to beat

August rather than move in the same direction. Hence the decision to employ a test for

proportions.

From a different slant, we may also check the ratio of paired returns on a yearly basis.

In other words, we want to determine whether October was likely to outshine August in

the same year.

Paired Samples by Year

The latest workout calls for a binomial test of matched pairs. A rundown of the data

showed October to eclipse August in 25 out of 35 years. The upshot was a win ratio of

71.4%.

A test of paired samples differs somewhat from a checkup of unpaired groups as

described in the previous subsection. In the latest grilling, the p-value for the paired test

came out to 0.008337. While the turnout was not as stringent as the prior result for the

unpaired samples, it was highly compelling even so.

In the larger scheme of things, the ratio of wins notched by a given month against

another may well differ in the case of a paired sample as opposed to an unpaired batch.

Naturally, the distinction also applies to the ensuing levels of statistical significance.

In the foregoing contest of October versus August, the peppy month prevailed in 25

cases on its own; and exactly as often in a pairwise matchup against the droopy month.

Although the number of wins for October happened to equal 25 in both assays, there

21

was no preordained reason for the scores to be identical. In keeping with the distinct

themes and objectives, the significance levels did in fact differ between the tests for the

paired sample versus the unpaired grouping.

To bring up a second factor, the checkup of independent samples involved a two-tailed

test under the premise of perfect efficiency. By contrast, the paired grouping was

consigned to the strictness of a one-tailed test as a way to acknowledge the domain

knowledge uncovered thus far. Both types of appraisal yielded hefty levels of statistical

import in confirming the vantage of October over August.

From a larger stance, other combinations of assumptions may be put to the test in the

future. Given the strength of the conclusions to date, however, it seems safe to surmise

that the Efficient dogma will in general be annulled with ample assurance in other

cogent studies to be conducted down the line.

Special Features of the Assay

The foregoing forms of appraisal differ from the conventional method for vetting the

disparity of a couple of groups. More precisely, the usual practice relies on a rough-and-

ready scheme based on the t-test along with its iffy assumptions. All too often, though,

the constraints behind the latter probe are invalid in the real world. Even so, the disdain

for inconvenient facts is spotlighted by the mass of studies trained on the financial forum

including the stock market.

To be precise, a t-test of unpaired samples stands on a couple of shaky premises: the

data from both groups must be normally distributed as well as display the same

variance. In a similar way, a t-test for a matched sample assumes that the difference

between each pair of observations conforms to a Gaussian ogive.

22

By contrast to the usual spree of hand waving and windy denial, the procedure followed

here has invoked the binomial test. For the current study, the only requirement to speak

of concerns the independence of movements.

From a different angle, the latter condition forms the pith of the Efficient doctrine. For

this reason, the premise of independence is not merely admissible but in fact obligatory

for an assay designed to evaluate the orthodox canon.

In short, the duplex framework does not need extraneous crutches. To cite a

counterexample that runs riot in the literature, the t-test draws on a false image of the

stock market; namely, the hegemony of the normal ogive.

On the bright side, though, the binomial test upholds a sound alternative along with an

honest approach to modeling. In this formula, the lack of arbitrary constraints plays a

vital role in the leanness, elegance and robustness of the procedure.

To wrap up, an appreciable gap in performance arose between certain months of the

year. In particular, October was prone to outpace August of the same year at a

persuasive level of statistical clout.

In the previous subsection, an independent sampling of the divergent months also

confirmed the lopsided odds of rising versus falling when the moves were not paired up

by year. In these ways, the existence of a noteworthy gap between a couple of months

was by itself enough to scrub the whitewash of efficiency at an ample level of

assurance.

23

Conclusion

Duplex models can portray complex systems with the utmost of simplicity, clarity and

efficacy in areas ranging from the dearth of premises to the strength of conclusions. The

advantage of the binomial scheme shows up, for instance, in purging the dreck of myths

and misconceptions in the financial forum. A showcase involves the existence of

seasonal patterns behind the monthly moves of the Dow stock index.

According to the dominant creed of financial economics, the real and financial markets

are so nimble as to absorb every scrap of information with startling speed and react at

once with perfect rationality. A direct outgrowth of the Efficient voodoo is the Random

Walk spoof that pictures the market as an erratic system immune to prediction except

perhaps as an occasional fluke. Another offshoot lies in the mantra of buying and

holding a dicey asset forever as an unbeatable strategy for investment.

Unfortunately, the charade of efficiency flouts the facts of life in plenteous ways. To

underscore the gulf between the illusion and reality, the current study has employed

only a minute fraction of the wealth of resources available to the general public.

If we look at the big picture, the duplex framework showcased here is the plainest way

to model shifty systems regardless of their complexity. The crux of the methodology lies

in the binomial test: the most basic tool in statistics for checking presumptions and

drawing conclusions. The two-tone method offers a medley of benefits ranging from the

clarity of concepts and ease of usage to the soundness of inference and sureness of

conclusions.

24

Further Resources

A useful resource for investors of all stripes, ranging from newcomers to veterans, lies in

a Web site maintained by MintKit Institute (mintkit.com). The portal presents a wealth of

articles, books and other materials dealing with the challenges and opportunities that

await earnest investors bent on sound growth in a global marketplace.

For instance, an article at the gateway provides a compact survey of the fables and

bungles that hamper the bulk of amateurs as well as professionals in the financial

arena. The write-up is titled, “Myths versus Mistakes” (MintKit, 2020).

The duplex framework for modeling complex systems is described in detail in an ebook.

A summary of the guidebook is given in a blog post at Kenwave (2020). The same blurb

provides links to sites where the full publication is available as a PDF file.

Supplementary Materials

The statistical assays reported here were encoded in the R language. The resulting

program has been uploaded to GitHub, an open repository of virtual resources on the

Internet. The R script as well as the raw data are available at the following locale:

“github.com/mintkitcom”.

The program resides in the Code folder as a document in txt format under the title of

DjiSeasonMonthPeg. Meanwhile the dataset abides in the Data node as a file in csv

form under the rubric of DjiMonth1985-2020.

25

References

Fama, E. The Behavior of Stock Market Prices. J. of Business, v. 38(1), Jan. 1965: 34–

105. http://doi.org/10.1086/294743.

Kenwave. Duplex Models of Complex Systems. 2020.

https://w.kenwave.com/2020/08/duplex-models-of-complex-systems.html.

MintKit. Myths versus Mistakes: Riot of Passive Muffs and Active Goofs in Financial

Markets. http://www.mintkit.com/myths-vs-mistakes.

Yahoo Finance. Dow Jones Industrial Average (^DJI) – Historical Data.

https://finance.yahoo.com/quote/%5EDJI/history?p=%5EDJI.

* * *

26