Political Calculations
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December 13, 2019

Can you quickly tell if a number is divisible by any of the numbers from 1 to 12 without actually doing the division?

Let's say you have the number 1,512 and it be really helpful if you could determine if it could be evenly divided by any of the numbers from 1 to 12. Could you do it without launching the calculator app on your mobile phone and performing the divisions?

You can, but you'll need to apply the divisibility rules you might have learned a long time ago in school. In case you don't remember them, here they are, where we've included one or two you may never have seen before. Try them with 1,512 and see which apply....

1. This is the easiest divisibility test of all, because all whole numbers are divisible by 1.
2. Look at the last digit of your number of interest. If it is even (equal to 0, 2, 4, 6, or 8), then your number of interest is divisible by 2.
3. Add up all the individual digits of your number of interest. If the sum is divisible by 3, so is your number of interest. And if you can't tell right away, repeat this process with the digits of your sum!
4. Look at the last two digits of your number of interest. If those two digits are a multiple of 4, your whole number will be evenly divided by 4.
5. Look at the last digit of the number of interest. If that digit is either 0 or 5, your number will be divisible by 5.
6. You'll need to perform a two-part test to determine if your number is divisible by 6. Specifically, you'll need to perform the divisibility tests for both 2 and 3 on it, where if it passes both tests, then it may be evenly divided by 6.
7. This is the hardest of the divisibility tests. Split off the right-most digit of your number of interest from the rest of it, and multiply it by 5. Then add the result to the remaining left-side of your number. If the result is a multiple of 7, your original number will be evenly divisible by 7. If you can't tell right away, repeat this process with your result until you can.
8. Telling if your number can be evenly divided by 8 requires a two-part test. First, do the divisibility test for 4, where if it passes, identify the factor by which you have to multiply by 4 to get the last two digits of your number. Then look at the digit in the hundreds place (the third digit from the right) of your number of interest. If both of these values are even, or if both of these values are odd, then your number will be wholly divisible by 8.
9. Nine is a multiple of 3, so the divisibility test is a lot like that one. Add up all the individual digits of your number of interest and if the sum is divisible by 9, so is your number of interest. If you can't tell right away, repeat this process with the digits of your sum until you can.
10. This may be the second easiest of the divisibility tests. Look at the last digit of your number. If it is 0, then your number will be evenly divided by 10.
11. This is a fun one. Starting with the left-most digit of your number of interest, alternate subtracting and adding the individual digits as you go from left to right. When you reach the end of the number, if your final result is a multiple of 11, the original number will be divisible by 11.
12. Another two-part test. Because 12 is a multiple of 3 and 4, if your number passes both of the divisibility tests for these two smaller numbers, then it will be evenly divisible by 12.

The divisibility rule for 7 is an example of the right-trim method, which may be applied in performing divisibility tests for larger values, provided you know what to multiply the right-most digit of your test number by in applying that process. In the case of 7, you can also apply it by multiplying the right-most digit of your number by 2 and subtracting that result from the remaining left-side of your number - there's more than one way you can make it work to find out if your test number is divisible by 7.

Going back to the number 1,512, hopefully, you found that it is divisible by 1, 2, 3, 4, 6, 7, 8, 9, and 12. If you're looking for a challenge, try 5,856,519,049,581,039. And if you are looking for more divisibility rules for larger numbers, keep reading....

### Right Trim Multipliers

Now, what about divisibility rules for numbers bigger than 12? For most of these numbers, the right trim method provides a relatively simple and effective means to determine if these numbers can evenly divide into your number of interest. Provided you know what multiplier to apply in using it.

At the same time, you may also recognize from the rules we presented for divisors from 1 to 12 that you don't get a whole lot of extra mileage in having divisibility rules for conjugate numbers, or rather, those values that are already the products of two or more smaller factors. For example, you can simply perform the tests for 2 and 7 to determine if a number is wholly divisible by 14. Or you could perform the tests for 3 and 11 to determine if your number is divisible by 33.

From a practical perspective then, if you're trying to tell if a given value is divisible by a smaller number, you only need to apply the divisibility rules that apply for prime numbers, which you can combine together as you might need to conduct divisibility tests for non-prime number divisors.

That brings us to the following tool, which we've constructed to identify the multiplier that you would need to successfully apply the single digit right trim method with your number of interest. Just enter the prime number for which you want to identify a multiplier, and it will provide you the number you need.

Divisibility Rule Generator
Input Data Values
Prime Number

Divisibility Rule Multiplier
Calculated Results Values
'Single Digit' Right Trim Multiplier

Technically, the tool above will work to identify valid right trim method multipliers for any value you enter whose final digit is 1, 3, 7 or 9, which coincidentally happens to include all prime numbers greater than 5. Speaking of which, here are lists of the first fifty million prime numbers, which should keep you busy for a while for your divisibility tests!

Image credit: Gayatri Malhotra

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December 12, 2019

The Federal Reserve's Open Market Committee (FOMC) concluded its two-day December 2019 meeting on 11 December 2019, and tried very hard to give the impression that it was done adjusting short term interest rates in the U.S. for now, holding the Federal Funds Rate in its current target range of 1-1/2 to 1-3/4 percent after having reduced it several times earlier in the year.

Combined with a generally rising spread in the U.S. Treasury yield curve, the lowered Federal Funds Rate has reduced the probability that a new economic recession will someday be determined to have started in the United States during the twelve months from December 2019 to December 2019. Through 11 December 2019, those odds now stand at 1 in 18, which rounds down to roughly a 5% probability.

Those odds had previously peaked at one in nine back on 9 September 2019, where if the NBER eventually NBER ever does determine that the national U.S. economy entered into recession at some future time, they will most likely identify a month between September 2019 and September 2020 as its starting date.

These probabilities come from a recession forecasting method developed back in 2006 by Jonathan Wright, which uses the level of the effective Federal Funds Rate and the spread between the yields of the 10-Year and 3-Month Constant Maturity U.S. Treasuries to estimate the probability of recession based on historical data.

With the rolling one-quarter average of the spread between the 10-Year and 3-Month U.S. Treasuries having turned positive, we will be reducing the frequency at which we will provide updates to this series to coincide with the FOMC's meeting schedule, which are held at approximately six week intervals. Should events cause the Treasury yield curve to re-invert however, with the yield on the 10-Year constant maturity U.S. Treasury falling below the yield of the 3-Month constant maturity U.S. Treasury, we will resume more frequent updates.

But you don't have to wait for us to analyze the recession odds for yourself! Just take advantage of our recession odds reckoning tool, which is really easy to use. Plug in the most recent data available, or the data that would apply for a future scenario that you would like to consider, and compare the result you get in our tool with what we've shown in the most recent chart we've presented above to get a sense of how the recession odds are changing.

As we get set to close the books on 2019, there are several hanging risks that could prompt such a change, with the potential expansion of the Fed's current QE-like effort into a full-on quantitative easing program to tame a liquidity crisis that has developed in repurchase "repo" markets over the last several months leading the list.

### Previously on Political Calculations

We've been tracking the ebb and flow of heightened recession odds since June 2017 - here are all the posts in our latest recession forecasting series!

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December 11, 2019

Four weeks ago, we took a snapshot of an improving outlook for the dividends investors expect to be paid out in each quarter of 2020. One month later, the CME Group's crystal ball for quarterly S&P 500 dividend futures indicates that those expectations have continued to rise.

The following animated chart shows how the future has changed since 10 September 2019, when we took our first snapshot of the expected future for the S&P 500's dividends in 2020. If you're accessing this article on a site that republishes our RSS news feed and you don't see the chart change every 3-4 seconds, please click through to our site to see the animation below.

Most of the change has occurred over the last seven weeks, since 21 October 2019, with the bulk of the change taking place between 21 October 2019 and 6 November 2019. Since then, the S&P 500's quarterly dividend futures have continued to rise, though at a slower pace. Links to our previous analysis of the future for the S&P 500's dividends through all of 2020 are below....

### Previously on Political Calculations

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December 10, 2019

In October 2018, the value of China's exports to the United States reached an all time record high of \$52.2 billion. Although the tariff war between the two nations had begun in earnest six months earlier, Chinese exporters raced to beat even higher tariffs that would take effect in 2019, artificially inflating the country's exports figures in the final months of 2018.

One year later, measured against that unusually high mark, the value of China's exports to the United States has crashed thanks to those higher tariffs. The total value of China's exports to the U.S. was \$40.1 billion in October 2019, a 23.1% year over year decline.

The large percentage decline in the value of China's exports to the U.S. is very evident in our chart tracking the exchange rate-adjusted, year-over-year growth rate of the two nations' exports to each other.

Curiously, U.S. exports to China are only slightly lower in October 2019 than they were a year earlier, which is attributable to China's strategy of targeting U.S. soybeans early in the tariff war. China's tariffs on U.S. soybeans in 2018 led Chinese importers to effectively boycott the U.S. crop that year, where a year later, very little additional negative effect is being observed from what might be considered to be China's most effective retaliatory tariff to date.

That's despite China waiving its tariffs on both U.S.-produced soybeans and pork in its attempt to undo some of the self-inflicted damage related in part to some of the unintended consequences of its tariff war strategy. Unfortunately, the nature of that damage is such that China's short-term demand for soybeans has been greatly diminished, with at least 41% of its domestic hog population lost to African Swine Fever. China won't need to import the kind of quantities of soybeans it was prior to the tariff war for several years, where it may be able to get by with soybeans exported from the world's second and third-largest soybean producers, Brazil and Argentina.

Our final chart shows how large the combined loss of trade between the U.S. and China is when measured against the counterfactual for how great it could be, if not for the tariff war.

Here's the takeaway comment from the chart: In October 2019, the gap between the pre-trade war trend and the trailing twelve month average of the value of goods exchanged between the U.S. and China expanded to \$14.2 billion. The cumulative gap since March 2018 has grown to \$101.2 billion. As you can see in the chart, the magnitude of actual trade losses from 2018 to 2019, as measured by the rolling 12-month average indicated by the heavy black line, exceeds the actual trade losses that were recorded from 2008 to 2009 during the Great Recession.

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December 9, 2019

The S&P 500 (Index: SPX) continued its record-setting spree in the first week of December 2019, ending the week at a new all-time record closing value of 3,145.91.

Although some market observers are claiming the S&P 500's rising trend is attributable only to rising valuations, with the index' Price/Earnings (P/E) ratio reaching up to its highest levels since 2001 without a proportionate increase in earnings, that's missing the point that earnings aren't what drive stock prices. Expectations for future dividends are what do, and they've been rising throughout much of 2019-Q4, with stock prices coming along for the ride.

As of the close of trading on Friday, 6 December 2019, we find the level of the S&P 500 is right in the middle of the range of the redzone forecast provided by our dividend futures-based model would set the index, assuming that investors are focusing on the distant future quarter of 2020-Q3 as they go about setting current day stock prices.

2019-Q3 has become a focal point for investors because that's when investors are anticipating the Federal Reserve's next action on interest rates, with the CME Group's FedWatch Tool currently projecting at least a quarter point rate cut taking place in September 2020:

The FedWatch tool is also now projecting another quarter point rate cut taking place before the end of 2020, but it wouldn't take much to move the probabilities enough to generate the anticipation of a half point rate cut taking place during 2020-Q3.

That's not to say that investors will maintain their focus on that distant future quarter over all the time between now and its end. They could shift their attention toward another quarter in the future, such as 2020-Q2, which would address the market analysts concerns about high P/E ratios for the S&P 500, since that change in how far into the future investors are looking would be accompanied by a fall in stock prices.

Such a change would be driven by the onset of new information. Speaking of which, here are the major headlines we noted for their market moving potential during the week that was.

Monday, 2 December 2019
Tuesday, 3 December 2019
Wednesday, 4 December 2019
Thursday, 5 December 2019
Friday, 6 December 2019

Over at The Big Picture, Barry Ritholtz succinctly summarizes the positives and negatives he found in the week's economics and market-related news.

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December 6, 2019

Have you been saving for retirement and are now ready to retire? What would happen if you withdrew a fixed percentage of the value of your retirement investment once a year for the rest of your life, if that investment was in the S&P 500? Would that provide enough money to pay for the things you might want to buy with it after you no longer have an income from a job? Or might market volatility force you to reconsider your options?

These are difficult questions to answer, because the future is very much an undiscovered country, where chaos controls both the timing and magnitude for when and how much market volatility might erupt.

But if we assume the unfixed future might be like the past, we can test how well a strategy to only withdraw a fixed rate of money from such an investment would have fared using the S&P 500's rich history of data during its most turbulent episodes.

And that's what we've done with our latest tool, where you can see how well you could count on your investment in the S&P 500 would have fared for up to 40 year long periods if you had chosen to retire in any month between January 1871 and 40 years before the present* while fully reinvesting your dividends along the way. If you're accessing this article on a site that republishes our RSS news feed, please click through to working version of the tool at our site.

S&P 500 Investment Data
Description Value
Initial Investment Value (Before Any Withdrawals)
Annual Withdrawal Percentage
Start Month for Withdrawals
Number of Years of Withdrawals

S&P 500 Withdrawal Estimates
Calculated Results Values
Month of Final Withdrawal
Investment Value Before Withdrawal
Withdrawal Amount
Amount Remaining In Investment
Withdrawal Highlights
Average Annual Withdrawal Amount
Total Amount Withdrawn Over All Years
Largest Annual Withdrawal
Smallest Annual Withdrawal

For our tool's default settings, we've chosen September 1929 as the time of the first withdrawal, because this month immediately precedes what happened with the U.S. stock market at the onset of the Great Depression. If your S&P 500 cash out strategy can survive the sustained series of disruptive events that followed this month and provide sufficient funds to support your needs, you can reasonably expect to weather any lesser event.

For good measure, you might also want to consider how your retirement might fare if a market crash occurred well after you've retired. You might choose a date that includes the years of the Great Depression or you could choose the more recent disruptive event of the Great Recession of 2008-2009 by selecting a starting year in the range of the late 1960s through the late 1970s. The maximum length of time the tool will consider for your S&P 500 withdrawal strategy is 40 years.

The tool's results indicate the value of your investment in the S&P 500 before and after your annual withdrawal, along with our estimates of the total amount you would have withdrawn over the number of years you've selected, the average amount of your annual withdrawals, and also the highest and lowest values of your annual withdrawals along with the years in which they would have occurred. Meanwhile, the tool does not consider things like taxes, commissions, or fees, which would most likely be taken out of any money you withdraw if they apply.

* Like our S&P 500 At Your Fingertips and Investing Through Time tools, we plan to periodically update this tool, with the first update in the first quarter of 2020 after the S&P 500's data for 2019 is finalized, and then once a year afterward, which will allow us to roll in new 40-year long periods.

### References

The tool above really doesn't say anthing about what a "safe withdrawal rate" for you may be. For that kind of insight, do check out the following resources on the topic, which you might find useful.

Hubbard, Carl; Cooley, Philip L.; and Walz, Daniel T. Retirement Savings: Choosing a Withdrawal Rate That Is Sustainable. Journal of the American Association of Individual Investors. [Online Article]. February 1998.

Pfau, Wade. The Trinity Study and Portfolio Success Rates (Updated to 2018). Forbes. [Online Article]. 16 January 2018.

RBC Wealth Management. Sustainable withdrawal rates in retirement. [PDF Document]. 25 March 2019.

Where our tool is concerned, so long as you select an annual withdrawal percentage rate of less than 25.0%, your investment will effectively last forever because there will always be some fraction remaining in your investment from which you can withdraw some percentage of it in future years, even though the withdrawal amounts may become vanishingly small, which can be considered the investment math version of one of Zeno's paradoxes.

In practice, you may find it extraordinarily difficult to resist raising the percentage of your investment that you cash out in years where market volatility might crash the amount you would withdraw if you otherwise maintained a fixed percentage rate of withdrawal, where you would intervene to reset that withdrawal rate because you've come to value having cash today more than whatever potential investment value you might have tomorrow. If you need cash to cover your living expenses in retirement or in times of severe economic distress, where your ability to earn income is very limited, it's very understandable.

Our tool can accommodate that kind of decision making. Just update it with the value of your investment at the starting month where you would need to adjust your withdrawal rate, and see what happens next. Then adjust it again at later dates as you might need.

Image credit: Max Harlynking

### Celebrating Political Calculations' Anniversary

Our anniversary posts typically represent the biggest ideas and celebration of the original work we develop here each year. Here are our landmark posts from previous years:

• A Year's Worth of Tools (2005) - we celebrated our first anniversary by listing all the tools we created in our first year. There were just 48 back then. Today, there are nearly 300....
• The S&P 500 At Your Fingertips (2006) - the most popular tool we've ever created, allowing users to calculate the rate of return for investments in the S&P 500, both with and without the effects of inflation, and with and without the reinvestment of dividends, between any two months since January 1871.
• The Sun, In the Center (2007) - we identify the primary driver of stock prices and describe a whole new way to visualize where they're going (especially in periods of order!)
• Acceleration, Amplification and Shifting Time (2008) - we apply elements of chaos theory to describe and predict how stock prices will change, even in periods of disorder.
• The Trigger Point for Taxes (2009) - we work out both when, and by how much, U.S. politicians are likely to change the top U.S. income tax rate. Sadly, events in recent years have proven us right.
• The Zero Deficit Line (2010) - a whole new way to find out how much federal government spending Americans can really afford and how much Americans cannot really afford!
• Can Increasing the Minimum Wage Boost GDP? (2011) - using data for teens and young adults spanning 1994 and 2010, not only do we demonstrate that increasing the minimum wage fails to increase GDP, we demonstrate that it reduces employment and increases income inequality as well!
• The Discovery of the Unseen (2012) - we go where so-called experts on income inequality fear to tread and reveal that U.S. household income inequality has increased over time mostly because more Americans live alone!

We marked our 2013 anniversary in three parts, since we were telling a story too big to be told in a single blog post! Here they are:

• The Major Trends in U.S. Income Inequality Since 1947 (2013, Part 1) - we revisit the U.S. Census Bureau's income inequality data for American individuals, families and households to see what it really tells us.
• The Widows Peak (2013, Part 2) - we identify when the dramatic increase in the number of Americans living alone really occurred and identify which Americans found themselves in that situation.
• The Men Who Weren't There (2013, Part 3) - our final anniversary post installment explores the lasting impact of the men who died in the service of their country in World War 2 and the hole in society that they left behind, which was felt decades later as the dramatic increase in income inequality for U.S. families and households.

Resuming our list of anniversary posts....

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Welcome to the blogosphere's toolchest! Here, unlike other blogs dedicated to analyzing current events, we create easy-to-use, simple tools to do the math related to them so you can get in on the action too! If you would like to learn more about these tools, or if you would like to contribute ideas to develop for this blog, please e-mail us at:

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