Pepsi's (NYSE:PEP) soft drink division has matched Coke (NYSE:KO) in delivering smaller serving sizes. In a move that contradicts basic economics, the smaller cans sport prices that are similar to those for larger cans. It appears the selling point for consumers is: "Since it's too hard to put down a can that isn't empty, we'll help with your diet". Normally in economics we would consider the total amount the consumer is buying, but this evidence suggests that consumers are willing to pay for portion control as a feature. That's not a problem for Coke or Pepsi since it boosts their margins.

It isn't all bad news for America. People are getting less "value" in smaller serving sizes, but how much value was there in drinking a couple extra ounces? From a health perspective, if people poured the can in the sink and recycled the container, we would expect lower rates of obesity. Generally I get annoyed by shrinking serving sizes, but I'll give this one a pass. It's less high fructose corn syrup for the consumer and more profits for the shareholder.

While CSD (Carbonated Soft Drinks) have not been a hot sector, Pepsi is doing pretty well on the snack side of their business. The difference in the two segments of the business has caused activist investor Nelson Peltz to target Pepsi as a company that needs to split up its segments. Pepsi has disagreed with the viewpoints of Mr. Peltz and stated that they had no intention to break up the company. One benefit management has cited from having both segments is the ability to more effectively negotiate with retailers. However, Mr. Peltz has been successful at lobbying other investors to see his point of view. New York Times reported that the California State Teachers' Retirement System has seen Mr. Peltz as adding value to the stock. They wanted Pepsi to add Mr. Peltz to the board of directors.

You may remember Mr. Peltz from his involvement in splitting up Kraft (KRFT) from Mondelez (NASDAQ:MDLZ). It appears reasonable to think that if Pepsi spins off their snack division, it could go into a merger with Mondelez. That would make sense for shareholders since it would reduce competition in the industry. That spin off was recorded October 1st, 2012.

It seems the normal behavior for analysts is to compare Coke and Pepsi. That may not be the best comparison, because Pepsi's snack division can drive the stock as well as the revenue. Consider the following chart on Pepsi. I've added in Mondelez and Kraft.

All charts are courtesy of Yahoo's charting system.

If you start the chart October 3rd, after the effects of the split should be clearly filtered out, you may notice that Pepsi's returns are strongly correlated with the movements of all three stocks. Given the business segments at Pepsi, I would expect on a macroeconomic level that their operating results would reflect the combined statements of Coke, Mondelez, and Kraft. Unfortunately, the chart can't be predictive in the normal sense. To be predictive, the chart would need to predict the returns before they occur rather than moving at the same time. However, the price movements don't always occur at exactly the same time. The biggest divergence in the chart is one that has occurred since late July.

Here's another chart that starts at the beginning of July.

Should Pepsi really be trading at this high of a premium? Here's a very short recap of earnings:

- Coke beat analyst estimates
- Pepsi beat analyst estimates
- Mondelez beat earnings by a penny, and fell short on revenue
- Kraft missed earnings by 3 cents, and fell short on revenue

Did Pepsi really deserve to move up that much on a relative basis? One more chart, this one set to July 24th:

After looking at these charts for a while, I got the itch to test for statistical significance. I downloaded the dividend adjusted close values for each of the four stocks from October 3rd, 2012 through August 8th, 2014. Then I used the values of the stocks to design a hypothetical portfolio. Since I've seen analyst estimates saying 65% of the business value comes from snacks and 70% of revenue comes from snacks, I decided to weight the portfolio as 33.3% each in Coke, Mondelez, and Kraft.

My hypothetical portfolio then, if constructed using those weights today would hold:

.7620907 shares of Coke

.848271 shares of Mondelez

.541794 shares of Kraft

This portfolio also costs exactly 90 dollars and 29 cents, so it perfectly offsets one share of Pepsi stock. Once I had these weights, I assigned each weight to the dividend adjusted close value of the stock for each day and summed the three stocks. This creates the value of a single share of the hypothetical portfolio company at any point. Then I used this portfolio as the independent variable to explain the price of Pepsi stock as the dependent variable. By creating the portfolio, I was able to reduce the number of independent variables from 3 to 1. Using that one independent variable and running a regression analysis in excel brought me to the following ANOVA table:

Clearly the F-statistic is off the chart, which tells you one of two things. Either the test is very statistically meaningful, or an underlying assumption of ordinary least squares has been violated. By tracing prices for securities rather than percentile movements, I changed which problems we would have in the data set. By sticking to prices, I was able to eliminate the problem of markets being slow to react and one price going up before the other. However, using prices instead of percentile returns means the data will have trends in price, meaning the average value of the population is not static. It also means that the values were much more likely to contain positive serial correlation.

If you have never tested for serial correlation in excel, I do not advise you to do it. Unfortunately, excel is the only program I had access to for the analysis. Excel does not natively calculate Durbin-Watson, and the workarounds I found online provided results that were less than credible. Therefore, I recreated the math behind the Durbin-Watson test and built the proper formula into excel. Using the formula the Durbin-Watson statistic on this regression is .115539. In simple terms, the test statistic said that there is very dramatic evidence of serial correlation. That isn't a surprise. If my formula says Pepsi is relatively over-priced on Tuesday, it is unlikely the market will instantly correct to my exact calculations on Wednesday. If it did that, we would've had negative serial correlation. We would prefer no serial correlation, but that's not the case.

To handle a model that presents serial correlation I wanted to use Hansen-White standard errors. This is referred to as the Hansen method, and it can mitigate the impacts of serial correlation and heteroskedasticity. As I write this, I see the red line under heteroskedasticity informing me I have written something that isn't a word, and I can only face palm. Let's move on.

There are a couple ways to create Hansen-White standard errors. The first is to use a real statistics program (not Excel), and the second is to know enough statistics and coding to be able to design modules for Visual Basic in Excel. Borrowing from a statistician that was kind enough to provide their code for VBA online, I was hoping to get a model running tonight that would provide the adjusted test statistics to correct for ARCH (auto regressive conditional heteroskedasticity). However, I am a little out of practice at using VBA in excel and the hour is growing very late. If anyone has access to a good statistics program and would like to calculate the Hansen-White corrected standard errors, I'd be happy to provide my source material.

**Conclusion**

Pepsi is trading at a premium relative to their peers. According to my model, based on the closing prices for Coke, Mondelez, and Kraft, the expected price for Pepsi would be $84.75, rounded to the nearest whole cent. Based on the earnings beat that Pepsi provided, I would expect them to continue to trade at a slight premium (a dollar or two, at most 3) relative to the basket of securities I used to replicate their industry exposure.

The potential trade idea is to go long KO, MDLZ, and KRFT while going short Pepsi.

For everyone 100 shares short on Pepsi, the long position would be:

76.21 shares of KO

84.83 shares of MDLZ

54.18 shares of KRFT

As of writing this, the values of those baskets are identical. I don't know if Pepsi will drop, or if the basket will go up, but I do expect the basket to outperform Pepsi over the next two weeks. When I am able to provide the White-Hansen corrected standard error to establish the statistical relevance and confidence intervals of my model, I will add that. If you would like my original excel documents, please leave a comment requesting them.

**Disclosure: **The author has no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. The author wrote this article themselves, and it expresses their own opinions. The author is not receiving compensation for it (other than from Seeking Alpha). The author has no business relationship with any company whose stock is mentioned in this article. Information in this article represents the opinion of the analyst. All statements are represented as opinions, rather than facts, and should not be construed as advice to buy or sell a security. Ratings of “outperform” and “underperform” reflect the analyst’s estimation of a divergence between the market value for a security and the price that would be appropriate given the potential for risks and returns relative to other securities. The analyst does not know your particular objectives for returns or constraints upon investing. All investors are encouraged to do their own research before making any investment decision. Information is regularly obtained from either Yahoo Finance or the SEC database. If either of these sources contained faulty information, it could be incorporated in our analysis. The analyst holds a diversified portfolio including mutual funds or index funds which may include a small long exposure to the stock. Additionally, the analyst does not hold any degree in statistics. He read very dull statistic books for weeks on end and practiced the problems in them in order to learn statistics.