Showing posts with label Square Root Rule. Show all posts
Showing posts with label Square Root Rule. Show all posts

## August 27, 2012

### The Square Root Rule

I probably get more questions about the square root rule than anything else I talk about, aside from Gliebers Dresses.

The theory behind the square root rule is that demand from advertising channels does not increase in a linear manner.  Folks who execute email frequency testing know this better than anybody else.  Demand from email campaigns, on a weekly basis, looks something like this:

Pay close attention to the sales generated by about one email campaign per week (\$0.18), and the total sales generated by about five email campaigns per week (\$0.40).

This relationship is approximated by the square root rule.  If you know that you generate \$0.40 in email demand per week by sending five campaigns per week, you can guesstimate what might happen if you send other combinations of campaigns.

• 1 campaign per week = (1/5)^(0.5) * \$0.40 = \$0.18.
• 2 campaigns per week = (2/5)^(0.5) * \$0.40 = \$0.25.
• 3 campaigns per week = (3/5)^(0.5) * \$0.40 = \$0.31.
• 4 campaigns per week = (4/5)^(0.5) * \$0.40 = \$0.36.
• 5 campaigns per week = (5/5)^(0.5) * \$0.40 = \$0.40.
Now we can easily calculate the incremental impact of each contact.
• 1 campaign per week = (1/5)^(0.5) * \$0.40 = \$0.18.
• 2 campaigns per week = (2/5)^(0.5) * \$0.40 = \$0.07.
• 3 campaigns per week = (3/5)^(0.5) * \$0.40 = \$0.06.
• 4 campaigns per week = (4/5)^(0.5) * \$0.40 = \$0.05.
• 5 campaigns per week = (5/5)^(0.5) * \$0.40 = \$0.04.
Clearly, the "right" answer is to actually test the right number of campaigns.  Unfortunately, so few people execute frequency tests that few people even understand the concept of incrementality.

So if your business leaders won't allow you to test the optimal contact frequency because of some odd excuse like "we can't afford to lose sales", then use the square root rule as a proxy for testing.

By the way, the square root rule works really well on estimating the budget for non-branded paid search terms.

## September 06, 2010

### Page Increases

• "Don't you have to mail a big catalog in order to make decisions about whether you can have a small catalog? In other words, you have to mail big catalogs, they allow you to understand how responsive customers truly are, correct?"
Turns out you can use a small catalog as a base.

Pretend you have a 64 page catalog that generates \$2.00 of demand per customer. What would happen if you had a 124 page catalog instead?

Use the square root rule to guesstimate the outcome!

(124 / 64) ^ (0.50) * (\$2.00) = \$2.78.

Now, simply run your profit and loss statement on each outcome, at each circulation depth, and decide for yourself what the right approach is ... 64 pages, 124 pages, or any other page combination!

## May 02, 2009

### Catalog Page Counts And The Square Root Rule

Many of you are looking to trim expense, while maintaining demand. For some, this means reducing the number of pages in a catalog.

Remember, there is a quick formula for estimating the demand impact of fewer pages in your catalog --- just use the Square Root Rule.

Example:
• You have a catalog with 124 pages.
• You expect the catalog to generate \$5,000,000 in demand.
• Your CFO wants you to cut 8 pages, leaving you with 116 pages.
• Demand = (116/124)^0.5 * \$5,000,000 = \$4,836,021.
You run a profit and loss statement on 116 pages generating \$4,836,021 ... is that more profitable than 124 pages generating \$5,000,000. Your printer can help you with printing efficiencies at different page counts, efficiencies that point to optimal page counts.

Also, your "DMPC", demand per thousand pages circulated, improves as pages are reduced ... this means that you can actually circulate deeper into your file as pages are reduced.

## December 11, 2008

### Rule Of Thumb For Marketing ROI / Profit

You've been through this drill before, haven't you? Your CMO or CEO or CFO wants to know "what would happen if we cut twenty percent from the marketing budget?".

This is where you use the square root rule to your advantage, allowing you to give a "back of the envelope" answer to questions that require a lot of time and energy.

Assume you spent \$1,000,000 on marketing last year, generating \$3,000,000 sales. Your profit factor is 35%, yielding profit of \$3,000,000 * 0.35 - \$1,000,000 = \$50,000 .

What would happen if you cut 20% from your marketing budget.

Step 1: Sales = ((\$800,000 / \$1,000,000) ^ 0.5) * \$3,000,000 = (0.894 * \$3,000,000) = \$2,683,282. The "0.5" number is the square root ... you are taking the square root of the ratio in change of marketing spend. In this case, a 20% reduction in spend yields a 10.5% reduction in sales.

Step 2: Profit = \$2,683,282 * 0.35 - \$800,000 = \$139,149.

In other words, you'd lose a little over \$300,000 in sales, but profit would increase by nearly \$90,000.

The square root rule allows you to play these "what if" scenarios ... and the scenarios are important. Sr. Management needs to get directional answers quickly. You don't want to do a ton of work, only to have the CFO tell you to run a new scenario where you cut marketing expense by 33%.

Is the square root rule 100% accurate? Absolutely not. In fact, there are times when it is blatantly inaccurate (affiliate marketing, shopping comparison sites, e-mail marketing).

When your leadership team needs an immediate answer, at an aggregate level, this rule of thumb works well.

## May 26, 2008

### Great Moments In Database Marketing #9: The Square Root Rule

This is the second in a ten part series of key database marketing moments that shaped the focus of The MineThatData Blog.

We go back to 1991 for the origin of The Square Root rule.

"Back in the day", catalogers were experimenting with statistical models. At Lands' End, statistical models had been used to decide who received catalog mailings since at least the mid 1980s.

But statistical models can make deciding how much a segment of customers might spend a real headache.

Consider this example.
• A segment has 10,000 customers.
• Only 8,000 customers were selected by the statistical model to receive the mailing.
• The segment of 8,000 customers spent \$3.00 per customer, \$24,000 total, when mailed the catalog.
How much would the total segment of 10,000 customers have spent, if all customers were mailed the catalog?

It would take nearly seven years to find a simple solution to this problem.

Honestly, one can use "rules of thumb", or statisticians can create unique models for every catalog to solve this problem.

After witnessing the results of maybe 300 catalog mailings over a seven year period of time, a simple solution was created. In 1998 at Eddie Bauer, we developed "The Square Root Rule".
• The 8,000 customers above spent \$3.00 per catalog, \$24,000 total.
• If 10,000 customers would have been mailed, total spend would increase by the following factor:
• (10,000 / 8,000) ^ 0.5 = 1.25 ^ 0.5 = 1.118.
• In other words, 10,000 customers would have spent \$24,000 * 1.118 = \$26,832.
• Dollar per catalog for 10,000 customers = \$26,832 / 10,000 = \$2.68.
• The remaining 2,000 customers would have spent \$2,832 / 2,000 = \$1.42 each.
By simply knowing the percentage of customers mailed in each segment, this simple rule allowed the circulation analyst to "guesstimate" what might have happened if all customers in the segment were mailed.

The Square Root Rule applies to advertising budgets and page counts and items offered per e-mail campaign ... basically any situation where you have limited information and no good history to estimate what might happen.

Of course, the approximation has numerous limitations. Don't use it to extrapolate too far ... if you only mail 10% of the customers in a segment, the equation might fail. If you want to mail 3x as many customers as were mailed last year, the equation will fail.

But for many instances, this equation solves problems, especially if you have limited data and you don't have a statistician sitting next to you, awaiting your beckon call!

## March 19, 2008

### Profit Week: Prospect Catalogs And Page Counts

Since the great postage increase of 2007, much has been made about prospect catalogs and page counts.

Both concepts are similar ... can one generate the same level of phone and web sales with a 64 page catalog as in a 124 page catalog?

Ultimately, the smaller catalog needs to be merchandised with the best product you have. If this is done, the concept has potential. This is no place for experimentation. Go with the best merchandise you have!

Performance estimates by page count can be determined, in lieu of actual test results, via the magic of the square root rule. When the prospect catalog is merchandised really well, folks observe 90% of the demand on half the pages ... not too shabby at all!

Here's a profit and loss statement for best customers, comparing a 64 page catalog to a 124 page catalog.

 124 Pages 64 Pages Increment Demand \$5.00 \$3.59 \$1.41 Net Sales \$4.00 \$2.87 \$1.13 Gross Margin \$2.20 \$1.58 \$0.62 Less Book Cost \$0.80 \$0.50 \$0.30 Less Pick/Pack/Ship \$0.46 \$0.33 \$0.13 Variable Profit \$0.94 \$0.75 \$0.19

Notice that, for best customers, a 124 page catalog is better than 64 pages.

 124 Pages 64 Pages Increment Demand \$3.25 \$2.33 \$0.92 Net Sales \$2.60 \$1.87 \$0.73 Gross Margin \$1.43 \$1.03 \$0.40 Less Book Cost \$0.80 \$0.50 \$0.30 Less Pick/Pack/Ship \$0.30 \$0.21 \$0.08 Variable Profit \$0.33 \$0.31 \$0.02

Here, the larger catalog is only marginally better than the smaller catalog. Let's take a peek at marginal customers, those who shop infrequently or have not ever purchased.

 124 Pages 64 Pages Increment Demand \$2.15 \$1.54 \$0.61 Net Sales \$1.72 \$1.24 \$0.48 Gross Margin \$0.95 \$0.68 \$0.27 Less Book Cost \$0.80 \$0.50 \$0.30 Less Pick/Pack/Ship \$0.20 \$0.14 \$0.06 Variable Profit (\$0.05) \$0.04 (\$0.09)

Ok, now the smaller book works!!

Smaller page counts typically work best among customers who purchase infrequently. For the cataloger feeling the pressure of postage increases and a recessionary environment, the prospect catalog, with fewer pages, offers opportunities to increase total profit.

## December 02, 2007

### Free Spreadsheet: Diminishing Returns By Pages And Circulation

I promised that I would provide a free spreadsheet for simultaneously evaluating diminishing returns (via the square root rule) caused by page count and circulation depth within any catalog.

The spreadsheet does not have a file forecast component, a very important part of any circulation project. The simulation is not meant to be the "official" plan for any one catalog. Instead, you use the tool to find a good combination of pages/circ-depth, then do the real work via file forecasting and RFM to obtain a circulation plan.

## December 01, 2007

### Diminishing Returns, The Square Root Rule, and Page Counts

One of the least understood issues in cataloging is knowing how many pages to have in each catalog. Let's apply the square root rule to this concept.

Assume that last year you mailed a 100 page catalog to 1,000,000 customers, generating \$5,000,000 demand via phone, mail and website. Demand was converted to profit at a rate of 35%, and the catalog cost \$800,000 to mail. Profit = \$5,000,000 * 0.35 - \$800,000 = \$950,000. Mmmmm .... profit!

The table below simulates what might have happened at different page counts.

 Circulation = 1,000,000 Customers Square Pages Root Demand Profit 60 0.775 \$3,872,983 \$875,544 80 0.894 \$4,472,136 \$925,248 100 1.000 \$5,000,000 \$950,000 120 1.095 \$5,477,226 \$957,029 140 1.183 \$5,916,080 \$950,628

If you're wondering, the square root function is calculated as (simulated pages / actual pages) ^ 0.5. At 60 pages, the value is 0.775.

Notice that peak profit occurs at a simulated page count of 120. If enough merchandise is available, at an appropriate presentation density, your page count could increase.

Folks, this stuff is about to become really important. In a few years, the ecological pressures (i.e. cutting down too many trees) on catalogers will be significant enough that intimate knowledge of appropriate page counts will have to be standard knowledge.

As you can see, this isn't rocket science. Combining page counts with circ depth (which can also be simulated in a similar manner), one can develop a circ plan for a catalog in about ninety seconds. An entire year's worth of catalogs can be "simulated" in a half our or an hour. Historically, our industry chose not to take the "simulation" route in figuring out how to configure a catalog.

Of course, you'll want to do the real work required to manage a circulation plan, including file forecasting, housefile vs. acquisition, and RFM profitability. But this is where your work starts.

Next up: A worksheet for combining circ depth and page count.

## November 29, 2007

### Diminishing Returns, The Square Root Rule and Peak Profit

We've previously talked about diminishing returns in marketing.

Understanding how diminishing returns impact profit is something that anybody can do.

You don't have to be accountable for paid search or online marketing or catalog marketing to understand how effective your strategy is at a macro level.

And the level of science is irrelevant. Most statisticians would become paralyzed by all of the assumptions being violated here. You're not trying to be a perfect statistician. You're trying to approximately understand where marketing spend is optimized.

Here's Example #1:
• Your paid search marketing budget is forecast to be \$1,000,000 this year. You also expect to generate \$4,000,000 demand. You convert demand to profit at a rate of 40%.
• Profit = \$4,000,000 * 0.40 - \$1,000,000 = \$600,000.
• Using the square root rule, we can measure where "peak profit" occurs.
 Square Spend Levels Root Demand Profit \$500,000 0.707 \$2,828,427 \$631,371 \$600,000 0.775 \$3,098,387 \$639,355 \$700,000 0.837 \$3,346,640 \$638,656 \$800,000 0.894 \$3,577,709 \$631,084 \$900,000 0.949 \$3,794,733 \$617,893 \$1,000,000 1.000 \$4,000,000 \$600,000 \$1,100,000 1.049 \$4,195,235 \$578,094 \$1,200,000 1.095 \$4,381,780 \$552,712 \$1,300,000 1.140 \$4,560,702 \$524,281 \$1,400,000 1.183 \$4,732,864 \$493,146 \$1,500,000 1.225 \$4,898,979 \$459,592

Here's where you have a series of choices. You are generating \$600,000 of profit at \$1,000,000 of paid search spend. However, "peak profit" occurs at \$600,000 of spend. If you want to achieve better profitability, you spend \$400,000 less, and give up nearly a million dollars of demand.

Folks who measure lifetime value combine short-term and long-term profit, and probably end up spending more than a million dollars based on those findings. Notice how diminishing returns occur, especially after a million dollars of spend.

Also notice how few online marketers measure anything beyond "cost per order" --- there's an opportunity for online marketers to improve their "tool box" with this style of analysis.

Example #2:
• You spend \$40,000,000 on catalog marketing. Although your productivity has been in decline for several years, your catalogs are still profitable. You generate \$170,000,000 of demand across channels, with 40% converted to profit.
• Where does "peak profit" occur?
 Square Spend Levels Root Demand Profit \$20,000,000 0.707 \$120,208,153 \$28,083,261 \$24,000,000 0.775 \$131,681,434 \$28,672,574 \$28,000,000 0.837 \$142,232,205 \$28,892,882 \$32,000,000 0.894 \$152,052,622 \$28,821,049 \$36,000,000 0.949 \$161,276,161 \$28,510,464 \$40,000,000 1.000 \$170,000,000 \$28,000,000 \$44,000,000 1.049 \$178,297,504 \$27,319,002 \$48,000,000 1.095 \$186,225,670 \$26,490,268 \$52,000,000 1.140 \$193,829,822 \$25,531,929 \$56,000,000 1.183 \$201,146,713 \$24,458,685 \$60,000,000 1.225 \$208,206,628 \$23,282,651

Here, we see that "peak profit" occurs at \$28,000,000 of catalog spend. However, the amount of profit difference between \$28,000,000 of spend and \$40,000,000 of spend is not significant. Most marketers would err on the side of spending more, in this instance, due to the obvious benefits of growing the customer file.

Again, you don't have to be responsible for online marketing, catalog marketing, or e-mail marketing to do this type of analysis. Take the initiative to get your hands on some data, partner with your finance team if necessary, and analyze where "peak profit" occurs in your marketing efforts. Find out where diminishing returns take a bite out of profit.

## May 03, 2007

### Honor Roll For May 3, 2007

A few articles worthy of MineThatData Honor Roll consideration.
• Spike at the Brains on Fire Blog makes an interesting observation at the end of this post about spreading good news. How often do we, as e-mail, online and catalog marketers view our craft as "spreading good news"?

## February 12, 2007

### The Square Root Rule: Better Defined

One of the great things about the blogging world occurs when individuals improve upon your simplistic work, developing the concept into something better, something more useful. Many big analytical companies hoard ideas and intellectual capital. Others freely give, in an effort to benefit all.

Enter Friend of MineThatData Alan Rimm-Kaufman and his development of the Square Root Rule --- found on his organization's excellent blog. Please click here for a well-written exploration of the topic of understanding how much money should/could be spent on advertising.

## January 21, 2007

### Optimal Online Marketing Budget

We've previously discussed the importance of the "square root" rule in analyzing marketing campaigns, when solid test-based data is not available to the analyst.

Assume you spent \$20,000 on an online marketing campaign, yielding \$60,000 net sales, and a net loss of \$2,000 (assuming 30% of sales flow-through to profit). You want to know what might have happened, had you spent more or less than \$20,000.

Square Root Rule --- Assume you wanted to only spend \$10,000. Sales will change by the following factor: (\$10,000 / \$20,000) ^ 0.5 = 0.707. Net Sales of \$60,000 will change by 0.707, or \$60,000 * 0.707 = \$42,426. Profit = \$42,426 * 0.30 - \$10,000 = \$2,728.

Again, if you don't have good test-based data to make comparisons with, use this rule as a quick shortcut.

The table below illustrates different spend levels, associated sales, and profit.

 Spending Level Net Sales Estimated Profit \$10,000 \$42,426 \$2,728 \$12,500 \$47,434 \$1,730 \$15,000 \$51,962 \$588 \$17,500 \$56,125 (\$663) \$20,000 \$60,000 (\$2,000) \$22,500 \$63,640 (\$3,408) \$25,000 \$67,082 (\$4,875) \$27,500 \$70,356 (\$6,393) \$30,000 \$73,485 (\$7,955)

## December 11, 2006

### Setting Your Online Marketing Budget

Undoubtedly, many of you are putting the finishing touches on your online marketing, e-mail marketing, or catalog marketing budget for 2007. Oh, the excitement!

Is there anything more enjoyable than sitting across from your Chief Financial Officer, having to defend why it is important to advertise with a certain affiliate at a time when expenses need to be trimmed by ten percent?

CFO's demand rapid, financially-based answers to questions. The humble Chief Marketing Officer or Online Marketing Executive needs to be able to respond in a credible, but timely manner.

Most of the time, when asked a random question, you don't have the appropriate data with you to answer the question quickly. This is where the "square root" function comes into play.

Frequently, sales generated by advertising follow a "square root" function. In other words, if you had the opportunity to increase your marketing budget by twenty percent, your net sales would increase by the square root of 1.2. This number is (1.2 ^ 0.5) = 1.095. In other words, a twenty percent increase in marketing spend yields a 9.5% increase in net sales.

This becomes important when the CFO makes a random statement like,"Please reduce your marketing budget by ten percent, you have no choice in this, everybody must share in the pain."

Look at this example, where the online marketing budget is reduced by ten percent:

 High-Level Online Marketing Scenario Reduce Ex- Incremental Base Case pense by 10% Sales Lost Orders 90,909 86,244 4,665 Average Order Size \$110.00 \$110.00 \$110.00 Cost per Order (CPA) \$22.00 \$20.87 \$42.87 Net Sales \$10,000,000 \$9,486,833 \$513,167 Gross Margin @ 40% \$4,000,000 \$3,794,733 \$205,267 Marketing Cost \$2,000,000 \$1,800,000 \$200,000 Pick/Pack/Ship Expense @ 13% \$1,300,000 \$1,233,288 \$66,712 Variable Operating Profit \$700,000 \$761,445 (\$61,445) Profit as a % of Net Sales 7.0% 8.0% -12.0% Ad to Sales Ratio 20.0% 19.0% 39.0%

Notice how the profit and loss statement changes. In this case, the CFO may have a good suggestion, as the incremental advertising dollars are not yielding a sufficient return on investment. Conversely, the numbers might work out in your favor, giving you the ammunition to actually ask the CFO for more money!

Not every business follows the "square root" rule. Your analyst can help you figure out which relationship makes the most sense to build the scenarios around. But in a pinch, go with the square root function. And then ask your CFO to quickly cost-justify some of her investments!!