What Hath Man Roth?

In which I model the certainties of death, taxes and exponential growth.

Soliciting Inputs

Seven posts in, and I still haven't managed to chase everyone away!  One of the most satisfying things about sharing these blogs with you is that I get the chance to talk to people offline about the different models.  I mean, I'd probably be making spreadsheets for many of these things anyways, but now instead of relegating them to a random folder on my desktop I get to chat with people who have different (and sometimes better) ways of looking at these things.  By all means, let me know if there is something that you think a simple overly-complicated Excel model would solve.

To that point, one of the first things my friend Ryan asked me was, "How much should I contribute to my IRA?  Can you model that?"

The answer is yes.  Yes I can.

First off, I guess I should reiterate that I am not a financial planner, and you should make your own financial decisions, keeping in mind what is best for you.  Secondly, you should totally invest in bytecoins. By definition they should be 8 times more valuable than bitcoins.

Thirdly, fill up your IRAs if it makes sense to do so.  I could (and will) go through a ton of math and modeling, but all of that pales in comparison to one piece of logic: if something has limits to check its power, it is likely to be very powerful indeed.  The limits on IRA contributions tell me that, in the absence of these limits, people would abuse the system for their own gain.  This means that there is likely no "sweet spot" where you should only partially fill up your IRA, but instead you should take it to the limit each year if you can.

To Roth or Not to Roth

I'm not going to go into the advantages of Traditional vs Roth IRAs too much, but one way to think of it is this:

• If you are going to be taxed more now than you are later, choose a Traditional IRA.  They let you evade some tax now in exchange for taxing your distribution later.

• If you are going to be taxed more later than you are now, choose a Roth IRA.  The contributions are taxed normally, but the distributions are not taxed.

So yeah, I'm a young guy.  I hope to make more money later than I do now.  I know our nation has a lot of stuff it needs to pay for and I don't predict that tax rates will go down in the future.  I choose Roth.

One funny side benefit of the Roth IRA, and the reason I will use it for today's post, is that it is much easier to model.  Unless you were going to spend your money on something tax deductible, investing in a Roth IRA basically allows you to ignore tax all together, which takes out an extremely complicated variable.  I'll just assume that if you have $5500 to contribute to an IRA, it has already been taxed, and would otherwise have been spent on something fun like a year's worth of coffee or a 3-D printer (and not, for instance, your pain medication, last month's mortgage payment, or a donation to your church).

Competing Interests, aka Saver's Remorse

Before we get to the modeling, lets first predict what we want the model output to look like.  For most people, what they want their IRA investment to pay for is, wait for it... retirement.  (I know, it seems silly because it is right in the name "Individual Retirement Account", but you can take out money for your first house or pay penalties to take it out before age 60.  I'm saying that that isn't what we're modeling.)  Basically, every year you want to add a little money to your investment and let it grow.  At or after age 60 you will make withdrawals such that hopefully the rate of money growing matches the rate you are withdrawing and you live out your golden years in financial independence.  This is what it looks like if you grew your retirement account at 7% and withdrew at 24x your contribution rate at age 65 ($5.5k contributions and $130k withdrawals):

But one menace is waiting for you... the same one that saved our hypothetical country last week. Inflation.  Inflation is basically good if you are in debt and bad you are invested. That same graph with inflation of 2.5% (which is usually considered the achievable target):

So a model of our Roth IRA has to start with a model of inflation.  Inflation stacks year over year, so lets make our first variable age.  Other important things are the age you would retire at, and the inflation rate.

Follow my work!

Current Age = B1
Age at Retirement = B2
Inflation Rate = B3

I will assume you can figure out your age.  (I'm this many!)  Age of retirement is a bit tricksy because it may be way in the future and it is hard to plan that far ahead.  There are Roth penalties if you withdraw before 60 so lets make that the lower limit and let people choose 60, 62, 65, 67, 75 or... 100?  I guess some people never retire, but for modeling purposes we'll say that our age list goes to 100.  Lets do that now. Make a column from C11:C93 with the numbers 18:100.  The inflation rate is variable, but is targeted at 2.5% and has historically averaged 3.37% in the US (including spikes where it went over 5%).  Lets make all those values possible.  We can define the inflation factor (typing the [In Brackets] portion) as:

(Cell D11) [=B$3^(C11-B$1)]

Complete the column by dragging the cell down to D93 (or double click the bottom right corner).  If you put in an age like 30, that should correspond to an inflation factor of 1.  Ages older than it should have values greater than 1, and ages less than it should have values less than 1.  As you can see graphically, a higher inflation rate can get exponentially out of control pretty fast.

Now we can use that inflation factor to make all our other numbers "real" by dividing by an inflation factor to equate to today's buying power.

Fertilizing Our Nest Egg

The next thing we need is some way to personalize the model.  You may be 20 years old and just starting out or 50 years old and several thousand dollars invested.  Lets define some more terms:

Starting Nest Egg = B4

"Real" Contributions (in 2013 dollars) = B5

"Real" Withdrawals (in 2013 dollars) = B6

Interest Rate = B7

Contributions for Roth IRAs are limited at $5500 per year per person right now, but they started out in 1998 with a limit of $2000.  That works out to an increase of ~7% for the limit raising.  Lets be cautious and say that in the future they will only raise the limit at the rate of inflation.  That means that even 30 years from now your "real" contribution would still be limited at $5500 factored at today's dollar value.  Also, you could choose $11000 for a couple, although thats really just two individuals.

The same considerations go into determining withdrawals.  Once you turn 60 you could take out as much as you like!  But really you just want to take out some to let the rest keep growing.  This variable should be how much you would take out in today's dollars.  Keep in mind that you are only replacing expenses and not adding more to your savings at this point (although technically with a Roth you could).  You might not have a mortgage any more either.  Make your own assumptions as to social security and medicare offsetting your needs, but a conservative estimate for withdrawals would be your all your current income minus investments and housing costs.

The long term interest rate for the stock market has been 7%, but is quite volatile and not guaranteed to be 7% in the future.  You could also have a Roth IRA that holds other assets like Bonds, Cash, or CDs.  (At this point would investing in Nickleback albums count as "buying low"?)

Oh, and if you haven't contributed yet, enter "0" for your starting nest egg.  Otherwise set it accordingly.

Cool!  Now, in the interest of making things more clear, I'll separate the function temporally.  There are four periods in your investing life.

• Time before now, which we don't care about.
• Right now, where you have the nest egg you have right now.
• Between now and retirement, where your nest egg grows.
• After retirement, where your nest egg may grow or shrink, depending on the size of the distribution and rate of growth.

To model this I'll use nested "if" statements.  (Note: Omitted "nested if / nest egg" joke because the CS majors out there are already cringing.)  It should look something like this:

Complete the column.  Now Make those nominal values real by dividing by the  inflation factor:

 (Cell F11) [=E11/D11]

Complete the column.  I also want to know if the money has run out yet.  I use nested if statements to check if the real value of the IRA is above 0.  If it is, we should check last year.  I want it to report the earliest year that the money runs out, thus:

(Cell G11) [="not yet"]

(Cell G12) [=if(F12>0,"not yet",if(F11>0,C12,G11))]

Complete the column and we are good to go!  Lets test it out!

Running with the Devil

I guess the simplest way to run the model is to check and see what an 18 year old could do if they routinely filled up their IRA until age 60, got the market rate of 7% during a period of low inflation, and took distributions at, lets say, twice the US minimum wage (~$30k/yr):

Thats how it is supposed to work.  A Roth IRA should allow someone of modest means to save for a modest retirement.  I'm guessing that is how they determined the contribution limits too, as the money starts to decay precipitously right at the end of life.  Some would say that if you haven't used up the last dollar then you didn't really optimize things perfectly.

But what if two 18 year old newlyweds routinely filled up their IRAs until age 60, got the market rate of 7% during a period of low inflation, and took distributions at the average income level of $51k.

Now the IRA has taken on a life of its own.  It isn't decaying, but instead it is still growing.  The government allows this and even after you die your Roth IRA can be passed down to your children who must take distributions (tax free) according to their age.

In Case of Emergency, Break IRA Glass

But what if that couple waits until they are 30 to start up their IRA.  

Technically they have only lost 12 years, but that is more than the doubling period of 7% interest.  They have less time to save, and the years they put in compile to give a nest egg only half the size (even though they invest for 71% of the time).  They run out at age 77, which isn't horrible (as the US life expectancy is about 80), but they would worry about finances.  Perhaps they keep working until 65.  Perhaps they take out only $35k a year.  (Both of these let the money last until age 95.)  Either way, what I learned from this is that time is very important.  

What else is important?

Fees.  That 1% mutual fund fee seems small, but it effectively takes you from 7% returns to 6% returns. Big deal?  Our happy newlyweds in example two with the self-sustaining portfolio?  With 6% returns they suddenly run out of money at age 90.  The closest thing to investment advice I'll give is that you should be extra-aware of fees.  There are funds available that have tons of diversity and only charge 0.06 to 0.1%.  If you are being charged more than that they had better make up the difference with performance.

Inflation.  Just going from the optimal inflation (2.5%) to historically average inflation (3.37%) makes our happy newlywed self-sustaining portfolio run out at age 93.

Contribution.  To answer the initial question, what if we only half-fill our IRA?  The happy newlywed self-sustaining portfolio runs out at age 77.  If you have them work until age 63 and only take out $30k/yr, you can get it self-sustaining again, but that is a huge decrease in standard of living.  Even filling it up only 90% takes it from being self-sustaining to a slow draw-down.  Again this comes back to time.  A dollar you didn't invest at age 18 is worth about two dollars at age 28, four at age 38, and sixteen at age 58.  The money you put in the last couple years of saving is barely going to grow at all.  And this doesn't even count the volatility of the markets!

I encourage you to definitely take the model for a test run!

Take Home Message

You basically want to invest early and often.  Don't take out your investment to buy something shiny, because replacing your time is only possible through pushing off retirement.  Also, hope that the Fed knows what it's doing with monetary policy.

I made this model much simpler than it could be.  Perhaps you stop contributing at age 60 but leave the money in for another few years.  Perhaps you shift the interest rate from risky to conservative as you age.  Feel free to suggest improvements.

I didn't mean for this post to be a downer (although you wouldn't know it from all the "things I learned"), so lets look at one rosy scenerio.  One thing to consider (and possibly post on later) is that if you look at all of human history, life-spans are increasing at a crazy rate.  What does that mean for investing?  If instead of "retire at 60, enjoy 20 years of retirement", the average human condition was "retire at 75, enjoy 50 years of retirement", we would get that precious commodity (time) that is the key to all investing.  At that point, our happy (not-so) newlyweds will have an IRA that is nearly indistinguishable from exponential growth.

-----

As always, you can subscribe to get new posts by sending an email to:
overly-complicated-excel+subscribe@googlegroups.com

Growing Pains

In which I let the masses determine their own level of stimulation.  I've honestly reworked this sentence like ten times and this is the least dirty sounding version.

Previously on OCE

If you caught my last posting and are back for more... thank you!  I'm really happy with how many returning visitors the blog is getting.  If you didn't catch my last post, this one is basically an addition to it. You are welcome to go back and check it out in full, or be happy with this short synopsis:

• An influential set of researchers (we'll call them RR) poured over historical data and claimed to observe a correlation between the amount of debt a country had (as a percent of their GDP), and what kind of growth (increase in GDP) they experienced.  As the debt got to be greater than 90% of GDP, there was a precipitous drop off in average growth, going from the 3-4% range down to negative 0.1% growth (recession). 
• A second set of researchers (that we'll call HAP) looked at the same data and found many flaws, from the actual data, to the averages, to the typos in their Excel worksheets, which made HAP's version of the growth number 2.2% under the high debt conditions.

So why is this important?  Again stealing from last week:

• In 2010, when the first set of research came out, the US was nearing that magical 90% debt mark, and had just tried to stimulate its economy.  Parts of Europe wanted to as well.  Some RR believers argued that stimulus was a bad idea as it would lead to even more debt.  Others even went the opposite direction and called for austerity in hopes of balancing the budgets and lowering debt. 

• The idea behind stimulus is that a government can borrow money (adding to debt) and spend it right away on improving the economy (adding it to GDP). Because it is affecting both the numerator and the denominator of debt/GDP it has a dampened effect on the ratio. The bonus is that all that money you paid this year to build bridges, give tax refunds, or even landscape the national mall goes to people who can spend it again, increasing GDP next year. 

The hope is that you can "grow your way out of debt" with the debt/GDP denominator gradually increasing and the debt becoming less burdensome as inflation eats away at it. But will it work? That is today's test.  We are going to model a high-debt national economy and see how different conditions affect it.

Stimulation Simulation

Right off the bat I should point out that I'm making this model as simple as possible.  We will have three columns (lets call them F, G, and H) that will be the year, GDP, and Debt, respectively.  We'll add one extra column that we can define right now as Debt as a percent of GDP (by typing the [In Brackets] portion):

(Cell I2) [=H2/G2*100]

Oh, and you can follow my work.

Great, we're done!  Except for that pesky NAME! error we get for not having put anything into G2.  Ok, lets fill it.  For the year column, make a series of numbers F2:F32 that are 0 through 30.  I'm including a year "0" so that at year "1" we can type in an equation that affects the previous year and can be dragged down to complete the column.  Keeping with the "simple" theme, lets define some terms so that we can use them to fill our Debt and GDP columns.

Initial Values:

Initial GDP (in dollars) = C2 = 1000

Initial Debt (% of GDP) = C3

I think these are easy to justify.  I'm going to use an arbitrary value of 1000 for the initial GDP because actual GDP values make for extra large cells, and honestly, we're going simple.  The "% of GDP" for Initial Debt means we will have to use "C3*0.01*C2" quite a bit (which is annoying), but it makes the presentation look prettier and is more intuitive for some folks.  I'll let people play around with the initial debt levels, keeping in mind that we were at 80ish % right before the crisis.  As a background, we averaged 60ish % over the 1990s and 2000s.

Crisis and Stimulation:

Financial Crisis Hit (% of GDP) = C6

Stimulus (% of GDP) = C7

Percent of Stimulus Respent = C8

The idea here is that the financial crisis basically made about 5% of our GDP evaporate.  I say "about" because it is tough to determine, as we did a lot of mitigation and don't actually have any real-time measurements.  You could make a good argument for any number from 2-10 (and probably some bad ones too) so lets set this as a movable target as well.  This will also let us set up scenarios in the absence of Crisis Hit.  Same thing with the Stimulus.  Technically there are still some remnants of stimulus dollars out there, but for the most part they were contained in the 5% of GDP injection of 2009.  Some people thought we needed more, some less.  Heck, lets make all the numbers from here on out movable values.

Speaking of values, when I say stimulus it means either tax breaks or spending.  They are (basically) the same.  Both are debts the government takes on so that spending in the economy will rise.  Once spent, that money goes to someone else who can choose to spend it again or save it.  The savings rate in the US is about 5%, making the Percent of Stimulus Respent theoretically about 95%.  But since the European savings rate is higher (10-20%) and Asian rates higher still we might as well let this be anything.

A Two State Solution:

Interest Rate on Debt (%)               for Low Debt = C11, for High Debt = D11
Inflation Rate (%)                           for Low Debt = C12, for High Debt = D12
Deficit (-) or Surplus (+ % GDP)      for Low Debt = C13, for High Debt = D13
Real GDP Growth Rate (%)             for Low Debt = C14, for High Debt = D14

For this model we'll set up two states (where we look at the year before).  If the previous year's debt is less than (or equal to) 90% of GDP we'll use the low debt values.  Alternatively for greater than 90% of GDP we'll use the high debt values.

The interest rate on debt is really tough to pin down.  In trying to be simple we are now confounded by the fact that you can hold a 1-year treasury bill which pays you for what you thought of the government's riskiness this year or hold a 30-year treasury bond which pays you for what you thought of the government waaaaaay in the past.  An alternative, all-encompasing, way to think about the interest on our debt is that in 2011 we spent $251 Billion in interest on somewhere around $16 Trillion which is only... like 1.5% interest?  And don't forget, that is nominal interest.  If inflation right now is at 2% then loaning the US government money for 1 year gets you a negative return.  But imagine you are an Greece, which as recently as last year had some 10-year notes as high as 30%, and now issues at around 8%.  What do you pay for your debt?  I'm going to leave this up to the reader and let them insert values for interest between 0 - 5%.

Inflation is a bit easier to track.  For one thing, we have good numbers:

Long-term, the inflation rate has been semi-volitile (averaging 3.37%), but if you had to pin down one or two numbers I'd go with periods of 2.5%, which is about what the Fed targets, and periods of >5% when people get very unhappy. Looking back up at the debt graph again, it is tough to say that high inflation really correlates with debt (and if anything our lowest recent debt was when we had highest inflation).  Lets give people the chance to make up their own minds and go with a range of 2.5%-10%.

Running a deficit or surplus is completely out of the scope of this experiment, but it happens so I guess we should model it.  Whether or not the current level of debt encourages deficits or surpluses (surpli?) is an argument that I'd encourage in the comments.  As a reference, our deficits in the 80s were about -2% GDP, and in the mid 00s neared -3%.  Our surplus in the late 90s was near +2%.

Finally, we get to the point of this: GDP growth.  Our real GDP growth has averaged about 3% since 1945, with an average of about 3.8% in the good times.  As I mentioned earlier, RR and HAP differed on their value for countries at > 90% debt (finding -0.1 and +2.2% respectively).

Nominally Keeping it Real

Because we pulled out inflation as a separate variable, we need to make sure that we change the real GDP growth to nominal growth (otherwise we'd be losing growth to inflation twice!)  The readers won't be able to change this, but it will be listed so that they can see it.

Nominal Growth for Low Debt: (C15) [=((C14*.01+1)*(C12*.01+1)-1)*100]
Nominal Growth for High Debt: (D15) [=((D14*.01+1)*(D12*.01+1)-1)*100]

Time For Sum Fun

For year zero, the GDP value will be the initial GDP minus whatever financial hit we take from the crisis, plus any additional amount from a stimulus. The debt value will be initial debt plus any additional stimulus. While I'm at it lets make a second model with no stimulus and a third with no crisis.

For year 1, the GDP will have three components. There is the GDP from last year plus the GDP growth plus respent stimulus (which is dependent on the year number so it decays exponentially). Divide by the inflation.  Set it up as an "if" statement so it uses the correct variables.

For debt, we will add last year's debt plus interest on last year's debt plus (minus) any deficits (surpluses) and divided by inflation. Again, we will set it up as an "if" statement.

Now Complete the columns.  I put a little ratio in to see which scenario has the larger GDP after 30 years, but other than that the model is pretty much made.  Lets take it for a test drive!

Check Please

The first thing we should check for is what happens to the output during a normal amount of time.  Lets take the years 1955-1970.  We had decent growth, declining debt, and if we cut off in '70 we don't have to deal with the inflation spike:

With that info, I put in these inputs and got these results:

So we have about a 75% increase in real GDP and a drop in debt (as a % of GDP) from the low 60s to the low 30s.  This matches the actual data very closely.  This also tells us that in good times (if we are actually interested in it) we can decrease the debt (without running a surplus) by about 2% a year.  This is the idea in "growing out of debt".

Crisis and Stimulus

So lets look at 2008 in the model.  These are the starting points I'll assume.

If you replace the GDP lost in crisis with debt fueled stimulus (that is not propagated by re-spending it), you end up with the same amount of GDP as if you had no crisis at all.  This is kind of cheating though because the debt never went above 90%.  Lets try it again with a little nudge to the original debt.

  

While the stimulus replaced the GDP lost in the crisis, we also lost some GDP growth when we were above 90% debt for two years.  Importantly, we are still much better off than if we had crisis but no stimulus.  Because the crisis removed some GDP from the denominator of our debt/GDP ratio, even our no-stimulus model ended up going over 90% debt.  Stimulus becomes progressively more needed as the crises get worse.  This is my best guess at what the real conditions might be:

Here the stimulus definitely helped, but we still feel the pain.  Interestingly in this scenario the stimulus model experienced 3 years above 90% debt but the no-stimulus model experienced 6 years.  One big question (and possible flaw) with these circumstances is what the "re-spend" is on stimulus dollars.  If it is set to 95%, like I'd expect from our savings rate, you end up with impossibly rosy situations where we are better off than if there was no crisis.  Granted this takes 5-10 years to notice:

So yeah, I think I'll keep the re-spend toned down.  Lets see what changing to the RR high debt growth rate and higher inflation would do to the model:

Not Pretty.  In this instance the inflation actually helps with the debt (which otherwise goes spiraling out of control), and you can see that while it doesn't get us back to no-crisis, the stimulus definitely is better than no-stimulus.  Anyways, you are probably tired looking at my tables.  

I suggest you go play around with the model on your own!!!1!1!!

Please do keep in mind that this is a communal google spreadsheet, and other people may be working on it at the same time.  Don't mess up someone else's work until they have had a chance to see their answer (unless you really really know that they are a doo doo face that you want to mess with).  Many of the cells are locked, and the variable ones only accept certain values.  If you would like to recommend a different value please submit your idea in the comments and I will add it.

Take Home Message

So what have we learned?  Well, for the most part, stimulus is a good way to squeeze out extra GDP from your economy.  It takes very specific characteristics when you have high debt (high interest rates, low inflation, low growth, little re-spending) for stimulus to be worse than no stimulus.  Can we grow out of debt (regardless of stimulus)?  Absolutely.  The big helper here is inflation.  With moderate inflation and low interest it is possible to continually run a deficit and still decrease your real debt.  That said, I'd much rather we were at a debt level where no one even questioned our credit-worthyness.  At 2% decrease a year we could even get there in my lifetime!

-----

If you want to subscribe, send an email to overly-complicated-excel+subscribe@googlegroups.com and you will be added to the list. The blog will email that list whenever it posts are published. 

90% Wrong

In which we see if incorrectly using the wrong methodology to answer the wrong question is justified by allowing future generations of graduate students write their PhD theses on it.

And Bad Mistakes, I've Made a Few

When I first heard about the Reinhart/Rogoff Excel fiasco my first thoughts were not, "How might they have changed the course of European history?"  They probably should have been, but they weren't.  My first thoughts were, "I'd better go back and check my equations on that model of the unemployment rate so I don't get made fun of as well."  It's healthy for smart people to have their thoughts challenged, because in the end it makes us trust science, and math, and statistics.  (Just kidding, no one trusts statistics.)  Today we'll take a look at a report that was in the news for all the wrong reasons.  Twice.

In 2010, Carmen Reinhart and Kenneth Rogoff (aka RR, both of Harvard) wrote a paper (non-peer reviewed) in which they looked at historical data for developed countries' GDP and National Debt, and determined that:

A Debt/GDP ratio greater than 90% was associated with Negative GDP Growth.

(I don't know if I need to mention it but negative GDP growth is bad.  Two or more quarters of it is called a recession.)  This finding was important because in 2010 the US was nearing that magical 90% mark, and several European countries were already well over it.  The paper got lots of media coverage and was cited everywhere from NPR's Planet Money to the Paul Ryan Budget.  The US had already set it's stimulus in motion, but the EU used the study to justify linking bailout money to austerity measures with the ultimate goal of bringing down European debt/GDP ratios.  The implication (sound byte) of the paper was that this 90% cutoff marked an unstable tipping point after which borrowing became difficult, resulting in inflation that decreased real GDP growth.

Economic Warfare

Flash forward to last month when Thomas Herndon, Michael Ash, and Robert Pollin (aka HAP, U Mass) came out with a paper refuting RR on three different points. The most hilarious of these, and the one which made it on to twitter and the Colbert Report is the typo in the Excel spreadsheet that removed a chunk of data from an "=AVERAGE( )" that conveniently contradicted their theory. This week RR released a correction to account for their typo. This stuff happens. As a researcher it is your worst nightmare that something you do by accident could invalidate your results. I am not going to model their typos… it offends my aesthetic. (Just kidding, I’m not doing it because it’s been done before so it feels like cheating.)

Instead, let’s forget the typos (I know, how is that possible?) and focus on the methodology critiques and the questions they were trying to answer. The first methodology critique is quite small; they didn’t include all their data. They claim the data wasn’t available when they did their analysis (which is reasonable) and with over a thousand data points you’d think missing a handful wouldn’t matter. The problem is that while there are a ton of data overall, there are only about a hundred points where a country went above 90% debt/GDP for the year, so when the few you miss are “high debt – high growth” ones, it will skew your results.  We now have all that data (provided by HAP).

Less forgivable is their “irregular” way of taking averages. RR had many years of data for some countries ("many-year-countries"), and only one year of data for others ("one-year-countries"). How would you tackle this problem? You could:

A.  Ignore the “one-year-countries”, or treat them as a separate group. (Too lazy?) 

B.  Treat each year from each country as a separate data point. (Too simplified?) 

C.  Assume the “many-year-countries” have phases of high debt and low debt. Determine when these phases are and treat each phase as a data point to plot with the "one-year-countries". (Did I just write your thesis premis for you?)

D.  Average all your years of data for each "many-year-country" into one data point. Weight each country equally and take an average these data points within your arbitrary categories. (Take a nap. Have a nightmare about averaging your average of averages.)

If you chose anything but D, congratulations, you can justify your findings and submit your paper for peer review.

As you might guess, RR chose D.

In the HAP rebuttal, they use method B, and find that instead of a 0.1% decrease in GDP, countries with >90% debt actually had 2.2% growth.  Seeing as we can't trust anyone else's Excel spreadsheets any more, lets make our own.  If you download their data, you get 5 columns:

(Follow my work.)

A = Country
B = Year
C = Debt/GDP ratio (%)
D = Category (< 30, 30-60, 60-90, > 90%)
E = Change in Real GDP (%)

Lets not bother ourselves with "in what country" or "when" our data happened, and lets not prejudice our findings with the RR/HAP categories.  Delete columns A, B, and D (Shift cells left so you have two columns A and B).  We are going simple here.  We just want to know if the GDP growth for the "> 90" is different than the "< 90", and is it a negative or positive GDP growth.  Make four new columns to separate the data (typing the [In Brackets] part):

Debt < 90:                                (Cell C2) [=IF(A2<90,A2," ")]

GDP Growth for Debt < 90:        (Cell D2) [=IF(A2<90,B2," ")]

Debt >= 90:                              (Cell E2) [=IF(A2<90," ",A2)]

GDP Growth for Debt <=90:       (Cell F2) [=IF(A2<90," ",B2)]

Complete the columns.  Find the averages [=AVERAGE(D2:D1276)], [=AVERAGE(F2:F1276)] and you can see that indeed there is a lower growth for the >90% debt years (2.15%) than the <90% debt years (3.66%), but also the standard deviation [=STDEV(D2:D1276)], [=STDEV(F2:F1276)] for both (4.3 for low debt, 3.5 for high debt) is huge!  On average, low debt is associated with better growth, but  in no way does it ensure prosperity.  On the flip side, 2.15% growth for the high debt countries is not fantastic (it barely keeps up with population growth), but it won't feel like a recession (-0.1%) that RR predict.  It feels like... well, now.

Asking the Wrong Questions

Conceptually, the bigger problem I have is that the answer RR found (even if it was wrong) was to a meaningless question.  Are there countries that suddenly jump from < 90% debt to > 90% debt?  Certainly!  But if a country does pass that threshold, I'm sure there is a year that they are at 83% or 88% and a following year they are at 92%, so the interesting question is, "What happens to these countries?"

This has actual policy implications if you are a country right at that 90% debt/GDP cusp (like we were), and are hoping to stimulate your economy (like we did).

Stimulating Discussion

The idea behind stimulus is that a government can borrow money (adding to debt) and spend it right away on improving the economy (adding it to GDP).  Because it is affecting both the numerator and the denominator of debt/GDP it has a dampened effect on the ratio.  The bonus is that all that money you paid this year to build bridges, give tax refunds, or even landscape the national mall goes to people who can spend it again, increasing GDP next year.  The hope is that you can "grow your way out of debt" with the debt/GDP denominator gradually increasing and the debt becoming less burdensome as inflation eats away at it.  (In fact, over the last few years the US government is borrowing at a Negative inflation adjusted interest rate.  People are paying the government for the ability to lend them money!)  But will it work?  If GDP growth is slowed enough by our debt load, you might lose out on all that you have gained from the stimulus.  That's what we should be testing.  In addition to the >90% vs <90% average growth, there are three things I'd like to know if I were, say, the US government trying to get out of a recession.

1.  Compare to all countries:  What happens at the 90% debt/GDP ratio?  Is there no effect?  Does the curve flatten out?  Is it an inflection point at which we hit a death spiral of ever-decreasing GDP growth?

2.  Compare to a case study:  Is there a country that has made a similar set of debt choices as the US, but after a certain point made different ones?  Did this affect their GDP?

3.  Compare to our own history:  When have our recessions come, and might it have been our debt load that caused them?

Aaaaand, go.

Thinking Linearly

I absolutely know that the best fit will not be linear, but as this is Overly Complicated Excel (and not PRISM or GraphPad) we'll go the simple route.  We'll take our data from before, copy cells C2:F1276 and paste them "special: values" into a new sheet.  Sort reverse-alphabetically by column C and we now have continuous data for plotting.  Plot it!

Not great R-squared's.  Its almost like GDP growth is caused by other factors than debt!  Anyways, the slope of the < 90% debt decreases faster than that of the > 90% debt.  This means that while being at 90% is bad, going up to 95% is not that much worse.  HAP find similar results with their fancy plotting program:

A Case Study from Down Under

We share a lot with our Australian friends:  language, cultural values, and up until the mid 70s, our debt policy.  You can see this by downloading HAP's data again and plotting just the US and Australia debt/GDP ratios:

Right around 1973 we bottomed out our debt/GDP ratio, while the Aussies kept working to lower theirs.  All other things being equal, if our debt choices had a significant effect on GDP growth you would expect Australia to be clobbering us.  Instead, plotting our % GDP growth gives us this:

From the mid 70s on, we had more debt but roughly the same GDP growth.  Now, I'd be the first to point out that a case study is not proof, but this is one more piece of evidence that there must be other factors influencing GDP growth that play a much larger role.

All About US

Historically, the United States has had periods with no debt, and periods with huge debt.  We also have had many recessions.  If the correlation held true, we'd expect that those recessions might have come during our periods of high debt.  Lets see if that is the case.  RR/HAP actually had a lot of data for the US, so lets use all of it to go back in time and look at our historical debt/GDP ratio and our % GDP growth:

The first thing that jumps out at me is how much GDP growth jumps around.  In the 40s it goes from dangerously high (> 15%) to depressingly low (< -10%) in just a few years.  Also, you can see that the worst time for perpetual recessions in the late 1800s and early 1900s, when debt levels were actually quite low.  Lets do a simple 10 year moving average for GDP growth (like I've done before for U3) and compare that:

Once again, I'm not seeing the correlation based on our own history.

Conclusions

Overall it looks like there is some correlation between debt load and GDP growth, but it isn't strong, and it certainly isn't evident the US-based examples I've shown.  Maybe we're different, though.  We have a very strong and trusted currency, and perhaps investors overlook our debt because they don't envision a world in which the US actually stops paying its debts.  Even for other countries though, the repercussions (a 1% drop in our GDP growth) of a transiently high debt load might not be that bad.  Thats not to say that countries should load up on debt without careful consideration.  I certainly want to see the US debt edge back down over time.  But if you are trying to jump-start the economy with a debt fueled stimulus, a few years of > 90% debt/GDP might be an ok stepping stone.

By the way, I know I didn't include too much modeling in this post.  I'm working on an interactive model for next week, though, where we'll try to answer the question, "If RR or HAP are right, can we still grow our way out of debt?" Stay tuned.

-----

My humor may be low, but I'm trying to keep the quality of my posts high. That means updating only once a week, or whenever I get a really good idea. This, in combination with the imminent demise of google reader, has led people to ask if I could email them when the blog is updated. A compromise that I think will work well is a subscription to this google group (an email list). Send an email to that link (overly-complicated-excel+subscribe@googlegroups.com) and you will be added to the list. The blog will email that list whenever it posts are published. If you want your money back you can always unsubscribe the same way.

Sell in May and Go Away?

Firstly, I am not a certified financial planner, and you should make your own financial decisions, keeping in mind what is best for you.  Secondly, the future is definitely in plastics.

The Market as a Tool

Chances are, you have some money in the stock market.  Over half of America does.  If you don’t, that’s cool too.  But even if you don’t, you still might have exposure to the markets.  Do you have a 401K through your work?  Stock exposure.  Retirement pension?  Stock exposure.  Work in a soup kitchen?  Stock exposure.  The point is that you should care about the stock markets.  Not the day-to-day, week-to-week, or even month-to-month fluctuations, but how you can use markets as (one of many) tools to grow wealth.  

On average (depending on how you calculate it), money in the stock market has returned about +7% per year.  If you invested today, your return could be more positive.  It could be negative.  It could be really really negative, canceling out 100% of your investment.  But if you just invest in the whole stock market (which you can do via various mutual funds or approximate with the S&P 500) over a long period of time, you should get the average return, which may or may not be 7% in the future, and that would ideally look something like this:

That blue line up there is enticing, turning $1 into $30 over 50 years (and more if you invest longer), but the real market is never that smooth.  People (myself included) are often greedy, and would rather limit their risk (smoother line), or increase their reward (steeper slopes) compared to the market alone.  To that end, there are many strategies (tricks) people espouse to get their yield.  Some of these are smart.  Some are not.  Given the month we’re in, lets tackle one of my (least) favorites.

“Sell in May and Go Away, Buy Back in St. Ledger's Day”

In the last few days I've been doing an informal poll of my friends and about 20% had heard this saying.  The idea is that the stock market does better in the winter and worse in the summer.  The hope is that you catch the good times and miss the bad times, moving your portfolio to cash or equivalents when the market isn't growing, and back to stocks when it is.  Last Wednesday was May 1st, and coincidently the stock market dropped 1% (and has since gone up 2%), which spurred a lot of articles about this.

What is St. Ledger's day?  It is basically the British Kentucky Derby, and it takes place on varying days in September or October.  (That alone might be enough to tell you about the people who made up the saying.)  The varying date thing is weird, and most traders treat Halloween as the buy-back date, giving you 6 months on (November - April) and 6 months off (May - October).

So... does it work?

Rate of the Month

Lets make a worksheet with dates (Column A) and S&P 500 values (Column B).  We can use the first value of the month as a proxy for "The Month," and the S&P 500 value on that day as a proxy for "The Market." (If you follow my work, I am adding a few empty rows at the top and putting the data [A6:B766] as increasing date with increasing row number.)

To figure out how a certain month did, make a new column dividing next month's start value by this month's (type the portion [in brackets]).

(C6) [=B7/B6]

Complete the column and you will notice the last value (C766) is 0.  This is because we don't know next month's value yet and it distorts our numerator, so just assume that the S&P will be flatish this month and put in the value of 1.  If you take the average of this column [=AVERAGE(C6:C766)] you get an average monthly increase of 0.69% which equates to a maximum annual increase of [=1.0069^12] about 8.5%.  

As a math aside (and skip this if you hate the maths) I say maximum, because it matters how close to average the months are.  Two months at 10% give you more than one at 5% and one at 15%.  Think of it like maximizing the area of a rectangle (to get a square).  You can prove it to yourself by using the two months at 10% example and trying to add and remove a percentage "x" to each month:

1.21 = 1.10 * 1.10

vs

y = (1.10 - x)(1.10 + x)

which simplifies to

y = 1.21 - x^2

For real values, y is maximized when x is zero.  Another takeaway from this is that order of the months is not important.  Your bad month (or year) can be in 2009 or in 1955, and it affects you exactly the same (percentage-wise).

A Timely Separation

Now that we have rates for every month we can separate just the ones we want to invest in (Jan, Feb, Mar, Apr, Nov, Dec) and see what their rates are.

(Cell D6) [=IF(OR(MONTH(A6)=1,MONTH(A6)=2,MONTH(A6)=3,MONTH(A6)=4,MONTH(A6)=11,MONTH(A6)=12),C6," ")]

Completing the column and taking the average [=AVERAGE(D6:D766)] shows that these months had a higher average increase of 0.94% than the collection of all months (0.69% calculated earlier).  But what does this mean?  What would happen if we actually invested a dollar?  For that, lets assume we get the monthly increase in our pre-May months, and get no increase in our post-May months because our investment is in cash.  As a comparison, we'll also do the inverse.  (Buy in May and here to stay?)

To make this easy, I assigned E5 and F5 to equal [1].

(Cell E6) [=IF(OR(MONTH(A6)=1,MONTH(A6)=2,MONTH(A6)=3,MONTH(A6)=4,MONTH(A6)=11,MONTH(A6)=12),E5*C6,E5)]

(Cell F6) [=IF(OR(MONTH(A6)=1,MONTH(A6)=2,MONTH(A6)=3,MONTH(A6)=4,MONTH(A6)=11,MONTH(A6)=12),F5,F5*C6)]

If you complete the columns and plot versus time you get a pretty striking picture:

Sometime in the mid 80's, the "Sell in May" strategy started working much better than the "Buy in May" strategy.  So does this answer our question?  Well, not really.  No one does the "Buy in May" strategy, so it isn't a good comparison.

Its time to bring out our Champion.  In the green shorts, weighing in at 700 basis points, a strategy people actually use: Buy and Hold.  New column and graph:

(Cell G5) = [1]

(Cell G6) [=G5*C6]

WOW!  Buy and hold knocks the pants off the Sell in May strategy.  How can this be?  Well, while the Buy in May strategy looked crummy, it still wasn't negative.  Over 63 years it still turned $1 into $3.56, and it didn't take any extra effort or years of investing (because the years were built in).  As an investor, your biggest asset is time, so wasting half of every year seems silly!  Myth Busted!

Right?  Is there ever a time when it would make sense to not be invested in the markets?  A time to be in cash because the market is not just slow, but down?  One way to address this is to find the average rate for each month of the year.  We'll need 12 columns:

(Cell H3) = [1], (Cell I3) = [2], (Cell J3) = [3] ... (Cell S3) = [12]

(Cell H4) = [Jan], (Cell I4) = [Feb], (Cell J4) = [Mar] ... (Cell S4) = [Dec]

(Cell H6) [=IF(MONTH($A6)=H$3,$C6," ")]

First complete the row.  It may look like nothing happened because H6 has a value and all the "False" statements (I6:S6) are blank.  Then complete the columns (which can be done all at once).  Now we have pretty data separated by month.  If you are following along, the data table looks like a series of backslashes, but if we take the average of each column [=average(H6:H766), etc] we can see that the months are indeed different:

Most of the months averaged positive (ranging from +0.2% to 1.7%), although four months averaged negative.  While for January, May, and July the result is less than -0.1%, the real standout here is August at greater than -0.5%.  You can see how the Sell in May strategy got started though, as three of the four losing months are in the proscribed cash period.

Remember Remember Buy Back in September?

Does this mean we shouldn't have money in the market in August?  We'll, lets see what would have happened in the past.  We'll make 12 more columns:

(Cell T3) = [1], (Cell U3) = [2], (Cell V3) = [3] ... (Cell AE3) = [12]

(Cell T4) = [Jan], (Cell U4) = [Feb], (Cell V4) = [Mar] ... (Cell AE4) = [Dec]

(Cell T5) = [1], (Cell U5) = [1], (Cell V5) = [1] ... (Cell AE5) = [1]

(Cell T6) [=IF(MONTH($A6)=T$3,T5,T5*$C6)]

Complete the row.  Complete the column.  Plot the individually excluded months with the Buy and Hold control and viola:

Instead of $1 turning into $92, removing a month produces a wide array of possibilities.  Many months, when removed, produced a staggering drop of ~50% in returns (marked red on the graph).  Some months, when removed, would have shifted returns a smidge up (teal) or down (orange).  Again, the one real standout is August (green), which (if removed) increased returns by ~50%.

Covering the Spread (of data)

So I'm recommending that you convert to cash in August?  NO.  What I have been glossing over in this "average growth for the month" is that there is an associated standard deviation.  For every isolated month the standard deviation [=stdev(H6:H766), etc] is WAY more than the average.  The average for August was a 0.5 % decrease, but for that same month, the standard deviation was 4.5%.  This is because much of the market is driven by single events.  August of 1974 and 2002 each had a 12% decrease, without which the August average is only a 0.1% decrease (in line with January, May and July).

Even within these single-event months there are single-event days that cause large changes.  August 1974 had three events where the market dropped by over 2% in a day.  August 2002 had two events where the market dropped by over 4% a day.  If you missed those events, you beat the market handily.  The same works in reverse, though.  If you missed the last three years, you lost out on a bunch of recovery.

Take Home Message

This all gets to the point that past performance does not guarantee future performance, and any deviation from a diversified portfolio requires some type of thesis.  While it is true that in the past May through October (and especially August) have been months with less market growth, deciding to remove diversity from your portfolio (by converting to cash) carries more risk to long term yields than doing nothing.  If we couldn't predict the series of random events that would turn our $1 to $140, maybe we should just be happy that it turned $1 into $90.  Let's not get greedy.

Of course there are many things ignored in this model, from the effects of inflation to the bonus of dividends.  The only omission I see that would help the "Sell in May" strategy would be converting to a higher yield May-October "safety" instrument (such as a 6-month CD), and I seriously doubt there is one that has returned stock-like performance.  For now "Buy and Hold" remains champion, but I'm sure we'll find another challenger ready to go toe-to-toe.

-----

My humor may be low, but I'm trying to keep the quality of my posts high. That means updating only once a week, or whenever I get a really good idea. This, in combination with the imminent demise of google reader, has led people to ask if I could email them when the blog is updated. A compromise that I think will work well is a subscription to this google group (an email list). Send an email to that link (overly-complicated-excel+subscribe@googlegroups.com) and you will be added to the list. The blog will email that list whenever it posts are published. If you want your money back you can always unsubscribe the same way.