Working Full-Time, Part-Time, and Either in 2012

Below, Corrections depicts the proportion of the population working full-time, part-time, or either in 2012 according to the Current Population Survey (click to enlarge).

Chaotic Systems

Corrections has yet to meet anyone who is good at forecasting much of anything.  Why might this be?  One reason is bad statistical models.  Another reason (that we are not overly sympathetic towards!) might be chaotic systems.  The present may perfectly and completely determine the future, but the near present may not have any power at predicting the future.

One simple example of this is the sequence x(t+1)=4x(t)*(1-x(t)).  The sequence will bounce around for a while between 0 and 1 (given we avoid a few bad starting states like {0, 0.25, 0.5, 0.75, 1}) and be completely deterministic.  Surely it wouldn't be hard to forecast, right?  

Wrong.  If your starting point (initial information used for forecasting) deviates the slightest amount, your sequence soon becomes completely different than if you had used the true starting point.  Below, Corrections depicts two such starting values:  X(0)=0.1 and X(0)=0.24, and plot the series for 100 periods (click to enlarge).

What if we were really, really, really close?  If we start out with a percent error of merely 0.0001%, then shouldn't our forecasts match up?  They do, for a while, but diverge rather quickly for having a one-part-in-ten-million difference (click to enlarge).

Does one series provide any forecast of the other, or have a recognizable pattern?  Below, Corrections depicts the two series against one another after the 20th period:  they no longer have a discernible relationship (click to enlarge).

This is one possible reason why the vast majority of sophisticated forecasts Corrections has heard (that don't suffer from selection) have been wrong.   We don't put much stock in it, however.

The Trends of Federal Receipts and Outlays

Below, Corrections depicts log Federal outlays and log Federal receipts under Reagan, Bush-I, Clinton, Bush-II, and Obama up until August 2013.  We also display the Reagan-Bush I-Clinton trend extrapolated out through Bush and Obama's terms.  We attribute the split January to the outgoing President, as he exits around the end of the third week of that month.

Log outlays tell a clear story:  outlays under Reagan, Bush I, Clinton, and Bush II continued on trend.  They saw a dramatic jump, and then a fairly stark arrest under most of Obama's term (click to enlarge).

Log receipts tell a different story:  while outlays have gone according to trend, receipts were halted under Bush, and again under Obama (click to enlarge).  For both, this was a result in part of tax cuts (or tax cut extensions) and bad economies.

Finally, we depict the two together (click to enlarge):  the short time the blue line was above the red line represents the Clinton surpluses, and the near-zero deficit of the Bush term before the financial crisis ended hopes of a balanced budget.

U.S. Federal Debt: Who Holds it, Who is Buying it?

Since 2007:Q4, over the last 21 quarters, U.S. Federal debt has gone up by about 7.5 trillion dollars.  Three of the most common misconceptions Corrections has heard have been:

1. The Federal Reserve is buying all the debt!
2. Foreigners are buying all the debt!
3. Banks and the public are buying all the debt!

All three are incorrect.  U.S. debt purchases  have been fairly balanced.  Below, Corrections depicts the three owners of U.S. debt (click to enlarge):

It can be helpful to see the proportion of U.S. debt held by each of the entities (click to enlarge):

Neither of these is particularly helpful.  Instead, we depict how the three entities have changed their holdings since 2007:Q4 (click to enlarge).  Of the new debt, the public has purchased 41%, the federal reserve has purchased 14%, and foreign entities have purchased 45%.  

Any dramatic stories you hear about U.S. debt eschew the facts in favor of hyperbole: disbelieve them. Debt has risen sharply, but none of these three entities has purchased more than 50% of new U.S. bonds.

Note: Millions should read billions in the relevant graphs!  (E.g. U.S. debt has been in the 16 trillions range recently.

SNAP Benefits by Income and Household Size

Below, Corrections depicts SNAP (food stamp) benefits by income and household size (click to enlarge).  We assume no elderly individuals in the household, but do assume 3% of income is spent on childcare.  ($300/year for (So an income of $30,000 would spend $900/year on child care).

Note the sudden drop-offs. For a family of two, going from earning $19500 to earning $19,750 (earning $250 more) sees a net decrease of total income (including food stamps) from $19,935 to $19,750:  a decrease of  $185 in total income in return for earning $250 more prebenefit.

Whatever one's beliefs about government programs, everyone can agree it makes little sense to implicitly tax the poor at rates above 100% (end up with less for making more), as programs like SNAP do.  

How Big Were the Payroll Revisions this Month?

The news today was filled with hyperbolic reports of how bad the revisions to payroll growth have been.  Below, Corrections depicts nonfarm payrolls from April 2012-August 2013 with and without August revisions (click to enlarge).

 Below, we offer a more blown up view (click to enlarge).

The revisions represented about half a month's loss in payroll employment growth.  Unfortunate, but not the steep revisions many news reports suggested.

Long-Run Geometric Annual Return by Industry: 1970-2012

Below, Corrections depicts the long-run geometric annual return by industry, from January 1970-December 2012, from Kenneth French's industry data.

Probability a Person Lives with Parents by Age

Below, Corrections takes the cross-sectional data on 2012 from the Current Population Survey and calculates the probability that one is the child of the head of household, by age from 16 to 85 (click to enlarge).  The probability a child is head of household with a parent present is small (single-digit percentages), even as they grow older (not shown).

Additionally, we display the time series evidence for four ages:  23, 25, 27 and 29 years of age (click to enlarge).

Treasury Yield Curves

Below, Corrections plots selected Treasury yield curves: 1 year, 5 year, 10 year, 20 year, and 30 year Treasury yields are all historically quite low (click to enlarge).

Time Series and Distribution of S&P 500 Returns by Timespan

Below, Corrections depicts the time series of net percent return on the S&P 500 from January 1926 to March 2013 by day, month, and yearly observation (click to enlarge).

We can alternatively look at the distribution of returns by day, month, and year (click to enlarge):

Inflation Expectations over Time by Duration

Below, Corrections depicts two different of the 10-year expected inflation rate (that is, the average yearly rate of inflation over the next ten years).  The first comes from the Cleveland Fed, and the second comes from the TIPS break-even rate.

First, we depict three different Cleveland Fed inflation expectations series (click to enlarge):  for the most part, from the 1980's onwards it took time for people's inflation expectations to fall from the highs of the 1970's and they currently range around 1 to 2 percent.

Below, we look at the break-even rate for 10-year TIPS vs. 10-year government bonds (click to enlarge).  Note that TIPS fell dramatically against bonds during Fall 2008, perhaps because of their relative illiquidity during a time when liquidity was highly valued (and are therefore probably not useful as a measure of expected inflation during that period).

 Finally, we look at the two measures together (click to enlarge):  they both suggest that over the next ten years, the yearly inflation rate ranges between 1.5 and 2.5%.

Inflation, Stock Market, and Bond Market Returns

Below, Corrections depicts value-weighted one-year stock market returns (including distributions), one-year Treasury bond returns, and the one-year inflation rate (click to enlarge).  We display each one year lagged return by month, from 1951-2012 (inclusive).

Obviously, unexpected inflation takes away from an already-issued bond's return while having an unclear impact on already-owned stock returns.  Interestingly, simple regression on non-overlapping periods suggests a:

• 3.46% return on one-year bonds with 0.57% increase above and beyond that baseline for each one percent of inflation experienced that year.  
• 15.28% return on stocks with a -.76% loss for each one percent of inflation experienced that year

This may be seen in light of:

• One-year bond's arithmetic (geometric) average return of 5.57% (5.49%) with a standard deviation of 3.76%
• Value-weighted stock market's arithmetic (geometric) average return of 12.49% (11.13%) ( (including distributions) with a standard deviation of 16.36%
• The CPI's arithmetic (geometric) average level of 3.67% (3.68%) with a standard deviation of 3.00%

It is interesting and surprising that while bonds return less than stocks (on average) they return more in periods of higher inflation.  This is counter to the fact that a one log-point fall in unexpected inflation impacts the log return of already purchased bonds by one log-point.

Expected Inflation and Treasury Bond Yields

Below, we plot the Treasury bond yields against the Cleveland Fed's estimates of expected inflation (click to enlarge).  It is important to note that the Cleveland Fed's estimates may be a noisy measure of "true" expected inflation.

A simple model in finance would suggest a one-to-one correlation between expected inflation and interest rates.  More complex models may deviate from this.  For example, they may allow for pricing of uncertainty about inflation (and therefore an inflation risk premium) that correlates with the level of inflation.

Fannie Mae and Freddie Mac: Borrowing and Payments to Treasury

In September 2008 Fannie Mae and Freddie Mac were placed under the Conservatorship of the Federal Housing Finance Agency.  At the time, they owned around $3.4 trillion in mortgage backed securities and $1.6 trillion in debt.  In return for $100 billion (later $200 billion) in capital investments guarantees, the U.S. Treasury received $1 billion in senior preferred stock with a 10% coupon (later changed to sweep all of Mae and Mac's profits).

How bad a deal was this for U.S. taxpayers?  We know that the Maiden Lane I and II transactions (the bailout of AIG) ended up making money for the Treasury and Federal Reserve Bank (while the Treasury is still $24.2 billion in the hole, it owns 53% of the stock of a company with a market cap of $71 billion).  How much did the bailouts of Fannie and Freddie cost U.S. taxpayers?

Below, Corrections depicts the cumulative draws and dividend payments that Fannie has engaged in with the Treasury (click to enlarge).  Often, the companies were drawing on the Treasury's capital in order to pay the required 10% dividend to Treasury (hence, dividends are being paid to Treasury at the same time draws are being made).

We can alternatively look at the simple difference between net loans and net payments (click to enlarge):

Both Fannie and Freddie look to be on track to pay back the Treasury for capital infusions over the next few years.  However, because these payments are simply payments to the government's preferred stock, and not considered paying back loans (the government offered a capital line, not loans) the government is likely to make a profit on this deal, even if we wind down Fannie and Freddie over the next few years, as both the House, Senate, and White House desire (though in different ways).  

Decomposition of the U.S. Federal Deficit: Receipt Shortfall & Expenditure Excess

 Below, Corrections decomposes the reasons behind the U.S. Federal deficit as a percent of GDP.  We attribute a deficit to two reasons:  a shortfall in revenue, or an excess of expenditure.  Because the U.S. Federal Government has run a historical deficit (receipts average 17% and expenditures have averaged 19.9%) we close the historical gap by blaming both receipts and expenditures equally:  the "baseline" for both is therefore 18.7%.  

Our method of decomposition is to take the deviation of each from its historical norm and attribute that portion of the deficit to its deviation, as the two deviations will always sum to the deficit that year.  For instance, if revenues ran at 18.6% while expenditures ran at 19%, then we would have a deficit of 0.4% per year:  0.1% of it would be attributed to revenues, and 0.3% would be attributed to expenditures.

Finally, we graph both the levels and the combined contribution of both (click to enlarge).  The blue and red lines represent the simple contributions of each to the deficit, and add up to the black line, which denotes the deficit.  The blue and red areas depict the stacked expenditure and receipts, and also sum up to the black line.

Our takeaway is that from 2008:Q4 to present, expenditures have been 4.62% above historical norms as a fraction of GDP, while receipts have been 2.32% below, giving the "reason" for deficits to be 33.4% receipt shortfall, and 66.5% expenditure excess.

There are, of course, other decompositions one can offer:  perhaps a more promising one would be to attribute a constant growth rate to the level of GDP, expenditures, and revenues, and decompose the shortfall into three parts:  a fall in the trend growth of GDP, a rise above trend in growth of expenditures, and a fall in the trend growth of receipts.  

A Tutorial on Bond Yields and Returns

Articles like this one can be confusing to someone not familiar with how fixed-income securities work. Bond yields have gone up, and so bond prices have gone down.  Because the Federal Reserve holds a significant position in different types of bonds, some note that it has "caused a mark-to-market loss of $192 billion on the Fed's holding assets, equivalent to approximately all of the unrealized gains that the Fed had accumulated [emphasis mine]."

Mark-to-market can be dramatically misleading when dealing with fixed-income securities.  They are, after all, fixed income.  Bonds are initially priced with one future set of paths of interest rates in mind.  When new information comes, and interest rates change, their current price naturally changes to fit this new interest rate:  otherwise, there would be an arbitrage opportunity.  However, absent any default risk, the end point of bonds remains fixed:  their price path swivels.  Below, Corrections plots the interest rate path:  in blue, the expected interest rate path, a constant 5%, with a "surprise" in the second period (4 periods before bond maturity) in which interest rates rise to 12% (click to enlarge).

The consequence of these interest rate changes is given simply by calculating the bond prices backwards from maturity (click to enlarge).  Notice that while the surprised bond path went down, the slope increased, as it must catch up to its initial path by maturity.

This idea can be seen most easily by taking the log of these two price paths (click to enlarge).  When the shock hits, the bond price is 0.194 log points lower than it would have been otherwise (the vertical distance between the initial shocked path and the unshocked path).  Not-coincidentally, our slope changed from 0.0488 to 0.113:  a change of 0.0645 log points.  Added together three times for the three remaining periods, they exactly make up the difference, summing to 0.194 again.  

The idea, therefore, is that while the price of bonds has declined and the Fed could lose money in a mark-to-market sense, it now holds bonds that have higher yields, and will make money faster.  Consequently, they only have to hold to maturity to not lose money.  This is especially true the change was due to mid-term rates, as it is empirically true that interest rates are long-term mean reverting.  Such a temporary, mid-course change in interest rates (click to enlarge) yields a temporary change in bond prices (click to enlarge).  Such mark-to-market accounting can be particularly misleading when dealing with price changes in fixed-income securities due to interest rate changes, rather than default risk, as is the case currently.   

Employment Falls and Recoveries: By Industry

Below, Corrections depicts the loss and gain of jobs over the 2007-present business cycle.  We measured the industry peak (defined as the maximum employment between May 2007 and April 2009) and the industry trough (defined as minimum employment between May 2007 and December 2010).  This difference is the "millions of jobs gained between Jan-2007 to Industry 2010 Trough," and is negative for all industries, denoting a loss of employment.

We then calculated the gain from that trough by taking the present employment and subtracting the trough employment, and graphed the two against one another (click to enlarge). Finally, we included a -45 degree line.  Being above that line means expansion from trough past industry peak:  mining, leisure and hospitality, education and health, and professional and business services all succeeded in expanding past their old peaks.  Being below that line means failure to expand past your old peak.

Remarkably, only government (Federal, state and local) jobs fell both during the recession and the recovery, though both losses were fairly mild.  A second graph includes the same procedure for the entire economy (click to enlarge).  While we lost 8 million jobs, we have regained 6.4 million jobs, and with average net job growth in the last 12 months at about 182,000, we should reach that peak in 9 months following June 2013, around March 2014.

10-Year Stock Market Return

Corrections has been perennially dissatisfied and unclear about the "average" long-run stock market return.  We've always had in our mind a 7% number, but looking at Center for Research on Security Prices (gated) data, or other online sources, numbers seem to vary quite a bit, from 4% to 13%.  The reasons for variance are generally 1)  inclusion of inflation to get real returns, 2)  use of returns based on price but not dividends, 3)  use of different time periods, and 4)  use of different portfolios.

Consequently, we produced two graphs that show real stock market returns.  They take a 10-year portfolio and ask "what is the compounded rate of return over that 10-year window?"  The two graphs reflect the inclusion of distributions (generally dividends, but including other direct payouts to shareholders).  Within each graph there are four lines:  two nominal returns, and two corresponding real returns, crossed with equally-weighted returns vs. value-weighted returns.  The idea behind value-weighted returns is that it reflects the return if you had to hold the market portfolio:  more money in larger market caps, less money in smaller market caps.  The equal-weighted simply buys $1/(Total Number of Stocks) of every stock, rather than $1*(Company Market Cap)/(Market Market Cap).

The four numbers at the bottom of each graph are the average compounded 10-year growth rate.  So, for instance, if you purchased $1 in stock, split evenly across all companies in the NYSE/NASDAQ/AMEX in March 2003 and held it for 10 years through to March 2013 (our final data point), reinvesting dividends all the way and re-balancing each month, you would have received an compounded yearly return of about 11.7% (click to enlarge).  Without including dividends, it would have been about 5% (see second graph below).  

While a graph excluding distributions is less informative, it helps some understand why their personal calculations based only off price are too low (click to enlarge).

Distribution of Market Equity

Below, Corrections displays the distribution of (log) market equity (price times shares outstanding) of the NYSE, NASDAQ, and AMEX in 2012 (click to enlarge).  As always, log means natural log.  Note that the graph would be essentially unreadable if it were transformed into levels:  the skew in market equity is very large.

The large cap nature of the NYSE is evident:  the breakpoint denoting the NYSE's 20th percentile is around the NASDAQ's 60th, and AMEX's 80th percentile.

Benefits of Diversification

Below, Corrections displays the benefits of diversification.  Returns from the the 1990-1991 stock period, we display the simulated standard deviation from a randomly-chosen portfolio of a given number of stocks and holding it for a year (click to enlarge).

Obviously, choosing a random number of stocks gives the same expected return.  But choosing more stocks reduces the expected noise around that return.  Choosing only one stock will yield a more noisy process than another.  As more stocks are included, the standard deviation of portfolio returns converges down to the market's standard deviation of returns (around 18% for annual returns).