Category Archives: Financial Markets

Low Volatility ETFs

The hip new financial product fangirled by every personal finance columnist on the internet is the low volatility ETF.  It is pretty much exactly what it sounds like – an ETF that, while tracking whichever index/industry/etc. it is supposed to, attempts to limit the variability of returns.  You can think of it as a stock with a low beta that moves with the trend of the market but not as severely in either direction during business cycle booms and busts.   Methodologies vary, but techniques are employed to limit the variance of individual holdings as well as the correlation between them.  I analyzed the performance of the PowerShares Low Volatility S&P 500 ETF (SPLV) to see how it stacks up against the market as a whole.

Over the past four years, the S&P500 had both a significantly higher maximum and lower minimum return compared to the PowerShares Low Volatility Index.  The S&P experienced many more extreme returns (+/- 1% daily return), suggesting that returns on SPLV fluctuate less than the market.  The S&P also earned a lower average return with higher variance than SPLV.

Period 5/6/11 to 1/6/15

S&P 500 SPLV
Max Daily Return 4.63% 3.75%
Min Daily Return -6.90% -5.18%
Returns less than -1% 98 62
Returns greater than 1% 110 71
Average Daily Return 0.04% 0.06%
Average Annual Return 0.99% 0.75%
Standard Deviation of Daily Return 10.98% 14.03%
Standard Deviation of Annual Return 15.71% 11.85%

The table below is the same analysis for only the year 2014, during which the US equity market posted more gains.

Year 2014

S&P 500 SPLV
Max Daily Return 2.37% 2.00%
Min Daily Return -2.31% -1.99%
Returns less than -1% 19 14
Returns greater than 1% 19 13
Average Daily Return 0.04% 0.06%
Average Annual Return 0.72% 0.60%
Standard Deviation of Daily Return 10.70% 15.80%
Standard Deviation of Annual Return 11.34% 9.55%

The claim that the PowerShares Low Volatility ETF (SPLV) tracks the S&P with less variability in returns  is corroborated by this simple analysis.  The graph of daily close prices and trading volume below also seems to corroborate this – the S&P500 Index (Yellow) fluctuates around the steady-ish path followed by SPLV (Blue).  The ETF misses out on some gains during the summer months, but outperforms later in the year.

Untitled picture2

Interestingly, the fund achieves its low volatility by being overweight in Healthcare and Financials, not the quintessentially low-risk sectors like Telecom or Utilities.



Mortgage Market Update from Calculated Risk

Calculated Risk is a blog that basically aggregates and analyzes up-to-date financial and economic data as it is released, particularly that which applies to the housing market.  The number of economic and financial metrics that are available on the internet is useful in some contexts but often feels more like a confusing, frustrating glut of information that renders answering a pithy question like “What is the rate of foreclosures like in the current housing market relative to pre-crisis times?” difficult to answer.  Trying to get beyond this issue is where I’ve found Calculated Risk really useful – relevant date for a particular issue is laid out, cited, and analyzed clearly in an effective and timely fashion.

I was curious about the housing market after meeting a seemingly overzealous realtor on the train, and here’s what I found via calculated risk.


At the end of Q3 2014, the delinquency rate on 1 to 4 unit residential properties was 5.85% of all loans outstanding, down for the 6th consecutive quarter and the lowest rate since the end of 2007.  The delinquency rate does not include loans in foreclosure, though they as well are at their lowest rate since the 4th quarter of ’07 at just under 2.5%.  Though foreclosures have come down from the stratospheric levels reached during their peak in 2010, they’re still more common than they were before the crisis.  Mortgages that are 30 and 60 days past due, on the other hand, have returned to approximately pre-crisis levels.  

Evernote Camera Roll 20141116 042456

Mortgage Rates 

30-year fixed rate mortgage (FRM) rates are down 1 basis point (.01) from last week at 4.01%, roughly the same level as 2011 but lower than last year’s 4.46%.  Obviously there isn’t “one” mortgage rate – the rate we’re talking about here is the one that applies to the most creditworthy borrowers in the best scenario possibly to receive a loan from the bank.  Though all other mortgages are based on this rate, it’s not exactly a rate one should expect to be offered by a bank.

Evernote Camera Roll 20141116 043840

The relatively small difference between a mortgage quoted at 4.01% and 4.45% has a surprisingly large financial impact on the 30 year FRM.  A $250,000, 30-yr. FRM at a 4.01% nominal annual rate compounded monthly (as is typically the case) necessitates a monthly payment of $1,194.98, whereas the same mortgage at 4.45% would require a monthly payment of $1,259.30.  With the higher payment, the borrower pays an additional $23,155 in interest over the term of the mortgage.

Another post talks about subdued refinancing activity, which I’d guess is the result of relatively static mortgage rates as it’s generally only financially viable to refinance when rates have changed significantly.  Banks could also be offering fewer refinancing options after the crisis, a reasonable assumption given their cautious resumption of lending post-crisis and the role that refinancing options played in exacerbating the housing bubble.  I’m purely speculating, though, and I’ll look into this more later.

Residential Prices

A widespread slowdown in the rate of housing price increases has been steadily taking hold since February of this year.  Residential prices aren’t decreasing, they’re just rising at a slower and slower rate each month, and now sit 20% below their 2006 peak.  This is not to say we should expect or even wish that housing prices should resume at 2006 levels, as such was clearly unsustainable – furthermore, though slow relative to preceding months, the (annualized) 6%+ experienced last month is still pretty strong and obviously outpaces inflation.


Evernote Camera Roll 20141116 050031

Analysis of the U.S. Output Gap by EconBrowser

I mentioned in my previous post that low inflation means substantial output gaps persist in many advanced economies.  Econbrowser’s post analyzing of the U.S. output gap is worth a read; the downside risks borne from the composition of recent economic growth and unjustified inflation concerns are also addressed.

Recent Developments in the World Economy

The first part of the WEO, which gives a broad overview of what’s happened since the previous WEO released in April, is (very) briefly summarized in layman’s terms below.  A technical note: any mention of rates of growth (positive and negative) refers to the annualized rate of growth of output, or GDP, in an economy (GDP isn’t the only measure of output that exists but it is what’s used here).  You can think of output, or GDP, as a measure of aggregate economic activity.  We care about growth in GDP because it leads to more employment (to meet the needs of the expansion of economic activity), and, generally speaking, a higher standard of living.  You can read a more thorough discussion of GDP growth here.

Global growth in the first half of 2014 was lower than the April WEOs projection by 0.4%.  That was the general trend, but the story varies by country:


  • Brazil – Negative growth so far this year (two consecutive quarters, which technically qualifies as a recession) due primarily to a lack of investment and confidence
  • France – No growth in output, reflecting fiscal imbalances and declining competitiveness
  • Italy – Contraction of output, albeit small, for Q1 and Q2, high unemployment (youth unemployment is at its historical peak) issues stemming from tight financial conditions (basically no credit available and thus no investment either)
  • Russia – Lack of growth is, not surprisingly, a result of insufficient investment and confidence


  • China – Relatively strong growth in Q1 despite issues in credit and housing markets that Chinese officials successfully subdued (via lowering required reserves and credit easing aimed at small and mid-size firms) for higher growth in the subsequent quarter
  • India – Stronger growth is resuming after a protracted downturn thanks primarily to much-needed investment
  • United Kingdom – Relatively strong growth (‘strong’ in comparison to what was expected in recent years, but considerably less than growth in China in India in raw number), and a strengthening labor market due to increased business investment

Investment is, unsurprisingly, prevalent in healthy economies and positively related to confidence.  If you’re surprised investors are wary of putting money into Russian markets then you must have been under a rock while Russia invaded Ukraine, and if you’re surprised about Brazil, maybe you didn’t know that it’s run by a feckless imbecile who just (barely) survived reelection.  Just as lack of investment and confidence hampers growth, India proves that  investor-friendly reforms spur investment, and the U.K. has recovered almost completely from the crisis thanks to business investment.

Those were the extremes – the rest of the world falls somewhere in the middle.  The United States economy is strengthening, but expected growth has necessarily been revised downward to adjust for the surprising contraction in the first quarter, largely a reflection of temporary factors (harsh weather, inventory accumulation in Q4 ’13, decline in exports), that won’t affect the future much.  In Japan growth continues along weak yet stable path, as the country’s enormous level of public debt inhibits its ability to grow too much despite good signs elsewhere in the economy.  Output nearly stalled in the Euro area as (mostly periphery) countries struggle to emerge from the recession, while some are achieving modest growth (Spain and Germany mainly).

Inflation is below targets in advanced economies which means they’re operating below their potential; meanwhile, inflation in emerging markets hasn’t changed.  Monetary policy is easy/accomodative in advanced economies and will continue to be as the ECB is slated to implement new policies, including targeted credit easing, and the Fed has made clear that it will aim to keep rates low for some time despite having wrapped up its asset purchase program last month.  In response to the Fed’s plans, financial conditions have eased and long term interest rates have decreased a bit, compared to data in the April WEO.  Risk premiums are low and volatility is low in advanced economies, which has some worried that risk is underpriced – but more on risk and its implications in a separate post.

So the global rate of growth or inflation or any other metric doesn’t convey much useful information because conditions are anything but  uniform across countries.  The story of the recovery is and will remain fragmented, with different problems and strengths contributing to a given market’s recovery.  That being said, all economies can expect to adjust to a level of growth that pales in comparison to the growth of the early 2000s.  Potential output, which has been revised downwards for the past 3 years, is too low for the growth rates of old to materialize.  This is due to the legacy of the recession in advanced economies, but growth-limiting structural issues also plays a role in developing economies.  For more on that, directly from the IMF, watch the short video linked below.

Takeaways from the BLS jobs report

Ben Casselman at FiveThirtyEight provides a detailed breakdown of the BLS jobs report.  248,000 jobs were added in September, and figures for July and August were revised upward by almost 70,000.  These data are the talking points you’ll hear on the news, but they’re deficient measures of labor market health on their own.  Casselman delves into the BLS report to corroborate his stance that the report was, in fact, good news – something raw numbers of jobs added can’t do.  (Side note, why is the font on BLS reports so awful?  The color sucks too – it’s like a “my printer is almost out of ink” light grey.)  Anyway, the good:

1) The number of people who gave up on looking for work because they didn’t think any was available is down considerably – less than 700,000 in September, compared to over a million back in 2010.

2) Layoffs are at a 10 year low.

3) The (slight) majority of the unemployed either voluntarily quit their job or (re)started the job search

(1) and (3) show some confidence in the labor market. Fewer people think that a desirable job is totally unattainable given current labor market conditions, and more people are willing to voluntarily quit their jobs because they think better opportunities are out there.  These are good signs.  There are bad signs, too:

1) Many of the jobs added were in Retail, which tends to be low-paying.  More desirable sectors added relatively few jobs

2) There is still no wage growth

3) Lots of people are working part time only because they can’t find full-time employment

(1) is maybe expected, and stems from an issue that has been brewing in the U.S. economy for a while – structural unemployment.  The U.S. economy needs more people with the right skills in the right geographical areas before it can add a decent number of jobs in higher-paying sectors. (Many economists have echoed this train of thought, suggesting that structural unemployment is the driving force behind persistently high unemployment post-recession. One way to investigate this is to analyze the Beveridge Curve.)

(3) shows us that while employment has accelerated, many of those working are underemployed. (Part time workers generally don’t receive benefits – recent legislation, which you can read about here, is starting to change this, however.)  As the linked article explains, part of the increase in part-time employment could reflect better incentives for part-time work, not underemployment.  Nevertheless, while incentives could have driven the work decision of a portion of part-time workers, many indicated that the only reason they are working part-time is because full-time employment is unavailable – corroborating the underemployment suggestion.

(2) is an issue I wrote about in a previous post, and, I’d argue, the most important of the three.  There will not be sustainable growth until wages grow, and the <2% of the past year simply won’t cut it.  Furthermore, the lack of wage growth implies that there’s still plenty of slack in the labor market.

As a technical aside, below is what I mean by real wage growth, i.e. the wage growth that needs to occur before consumption can rebound and support a robust economy.  When we say real anything in economics, we mean inflation adjusted.  The real rate of wage growth is thus the inflation adjusted rate of growth of wages.  The raw, or nominal, rate of wage growth simply tells us by how much wages increased, ignoring the price level.  This is not all that useful, because wages affect consumption via the purchasing power of consumers – and if we don’t know what the inflation situation is like, we don’t know if consumers’ purchasing power increased, stayed the same, or decreased.

You could easily look up real wage growth (i.e. inflation adjusted wage growth), but for the sake of completeness here is how you can calculate the real growth in wages given the nominal rate of wage growth and a measure of inflation:

Screen Shot 2014-10-08 at 1.29.47 AM

For the nominal rate of wage growth, you could use the % change of Average Hourly Earnings (reported by the fed), and for inflation you could use the CPI % change over the same period.  These aren’t, however, the only metrics that will work – there are plenty of ways to quantify wages and inflation, each suited to a slightly different scenario.

Is the Stock Market a Viable Barometer of Economic Health?

The S&P’s record close of 1992.37 on Thursday begs the following question: what, if anything, does a soaring stock market index, up almost 8% just this year, say about the health of the real economy?  As I’ve mentioned previously, there are quite a few issues in the current U.S. economy that may have to be rectified before the real economy can sustain robust growth – a weak labor force and stagnant wage growth, for example.  If we are to interpret the appreciation in the price of a stock market index as a sign of economic health, as many pundits on TV seem to do, then Thursday’s record close seems to contradict what the assertion that wage growth and a robust labor force are vital to the U.S. economy’s health.  This subject is briefly addressed on page 101 of  Freefall by economist Joseph Stiglitz, an account of the financial crisis, its causes, and aftermath.  He says:

“Unfortunately, an increase in stock market prices may not necessarily indicate that all is well.  Stock market prices may rise because the Fed is flooding the world with liquidity, and interest rates are low, so stocks look much better than bonds.  The flood of liquidity coming from the Fed will find some outlet, hopefully leading to more lending to businesses, but it could also result in a mini-asset price or stock market bubble.  Or rising stock market prices may reflect the success of firms in cutting costs – firing workers and lowering wages.  If so, it’s a harbinger of problems for the overall economy.  If workers’ incomes remain weak, so will consumption, which accounts for 70 percent of GDP.” 

I quoted the preceding passage because it cogently argues that stock market gains are not necessarily emblematic of health in the economy, as the media – particularly on business-oriented news shows – often suggest.  The two scenarios Stiglitz mentions (expansionary monetary policy and firms cutting costs) result in higher stock prices but not a healthier economy.  It is erroneous to conclude that the price of the S&P 500 is a sufficient and reliable barometer of economic health.

Consumption Choices and Student Loan Debt

The New York Fed shed some light on student loan debt numbers reported in the FBRNY consumer credit panel in a few presentations and an article at liberty street economics.

If you’ve ever seen the news you know that aggregate student debt has been increasing rapidly over the past decade, second only to mortgage debt.  Student loan debt dwarfs aggregate auto loan, credit card, and home equity debt balances, but that doesn’t tell the whole story.  While student debt per borrower is at its highest point ever, total debt per capita for student borrowers hasn’t increased relative to total debt per capita for non-student borrowers in recent years.

Until recently, borrowers with a history of student debt were more likely than their counterparts without student debt to take on home equity and auto loan debt – reflecting the high financial returns to higher education and the ability for college-educated people to participate in the housing and auto markets.  However, that trend has reversed, and a widespread decline in participation in all debt markets is prevalent among those with a history of student debt.

The article proposes a few rationales for why this is the case.  First, college grads revised their expectations about future income downward after the financial crisis and subsequently reigned in consumption.  Second, student debt borrowers may not have access to credit; debt to income ratio requirements have increased in all debt markets as a result of stricter underwriting standards, which may render student debtors ineligible to borrow.

I don’t know enough about underwriting standards to make a conjecture, but the article is worth reading as it provides the full story on an issue the media loves to turn into something it is not.


Econ Week in Review: 5/26 – 6/1

This was meant to be a sort of  “week in review”, but I’m (predictably) late, so it’s more like an assortment of articles about housing policy.

On May 13 the Wall Street Journal reported that mortgage behemoths Fannie Mae and Freddie Mac are being encouraged to make more credit available, according to overseer Mel Watt, who communicated the current administration’s stark policy reversal with his statement.  Until now, post-crisis housing policy had focused on restricting credit to prevent another boom and bust.  Efforts to do so included proposals from Washington such as requiring larger down payments on mortgages, which critics argued would unfairly punish creditworthy borrowers.  Read the article for more details on the so-called Johnson-Carpo bill.  An interesting point is that the bill would repeal “Affordable-Housing Goals”, which require Fannie Mae and Freddie Mac to cater to purportedly “underserved” markets (i.e. inner cities and rural areas).   One really interesting proposal (not a part of the Johnson-Carpo bill, just an idea) is that borrowers and lenders share more risk, allowing lenders to cash in if home values rose and take a hit if they fell (via a subsequent fall in the principal mortgage balance).  It’s radical, which the article admits, but an interesting idea nonetheless.

Why is the housing market so important?  The following article explains why the housing crash raped the entire economy, whereas the dot com crash, though slightly larger in terms of dollars of wealth destroyed, had a much smaller effect – and therefore also explains why so many people care so much about the housing market.  The answer has to do with the distribution of losses, according to authors Amir Sufi and Atif Mian at FiveThirtyEight.  On the same website, Andrew Flowers discusses the potential that a bond bubble is forming and how hard the economy would get screwed if this were in fact the case.  The articles aren’t explicitly related, but it’s easy to apply the rationale set forth in the first article, about the distribution of losses and macroeconomic consequences, to what Flowers says about a potential bond bubble.

Modeling Stock Market Behavior

In the finance world, there’s some debate about whether or not the daily closing prices for various stock market indices convey  useful information.  Some financiers subscribe to the belief that the daily close price reflects market trends and impacts the probability of realizing a good return.  Others disagree, claiming that the day-to-day movements in the stock market are completely random and convey no useful information.  If this is true, then the closing price changes in the stock market should mirror a geometric random variable.  In this post I’ll explain why the geometric model would imply that stock market fluctuations are random and then test the validity of the model empirically.

Suppose the outcome of some even is binary, and success occurs with probability p.   Obviously failure must occur with probability 1 – p.  A geometric random variable models the number of trials that take place before the first success.  It takes on the value k when the first success occurs on the kth trial.  Trials are assumed to be independent, so we can write the probability density function of the random variable X as follows:

Screen shot 2014-02-07 at 9.09.52 AM

We used the independence assumption to rewrite the event “k-1 failures and a success on trial k” as the product of two distinct groups of events, namely k -1 failures and then 1 success.  Now we use the fact that success occurs with probability p (and the independence assumption, again) to write the following:

Screen shot 2014-02-07 at 9.10.00 AM

To model the behavior of the stock market as a geometric random variable, assume that on day 1 the market has fallen from the previous day.  We’ll call this fall in the closing price a “failure” that occurs with probability 1 – p.  Let the geometric random variable X represent the number of subsequent failures that occur until the stock market rises (“success”).  For example, if on the second day the stock market rises, the random variable X takes on the value 1, because there was only one decline (failure) until the rise (success) that occurred on the second day.  Similarly, if the market declines on days 2, 3, and 4 and rises on day 5, then it has declined on four occasions before rising on the fifth day and thus the random variable X takes on the value 4.  Keep in mind that it is stipulated in the formulation of the random variable that the market declined on day one, and therefore a fall on days 2, 3, and 4 is a sequence of four failures, not three.

To determine whether a geometric model fits the daily behavior of the stock market, we have to estimate the parameter p.  In our model, we are addressing the question of whether stock market price fluctuations are geometric.  Geometric random variables can take on infinitely many values of p (so long as p is between 0 and 1), so our model doesn’t address the probability with which the stock market rises and falls; the geometric model addresses the behavior of the random variable for a given p.  The value p takes on may be of interest in formulating other questions, but here its job is to create a realistic geometric model that we can compare to empirical stock market data.  If the stock market data fits the geometric model, the implication is that stock markets tend to rise and fall randomly with a constant probability of success.  This suggests that daily stock market quotes are meaningless in that today’s price does not reflect historical prices.  One could say that if this model fits stock markets don’t “remember” yesterday, but that sounds a lot like something called the memoryless property, an important characteristic of the exponential distribution, so we should be careful to not confuse the two.

Once we get some empirical data, we’re going to estimate the probability of success p.  So let’s solve for the general case and then compute an actual value with data afterwards.  There is no one way to estimate the value of a parameter, but one good way to do so is to use the maximum likelihood estimator of the parameter.  The idea is simple, but sometimes computationally difficult.  To estimate the value of p with the maximum likelihood estimator, we find the value of p for which the observed sample is mostly likely to have occurred.  We are basically maximizing the “likelihood” that the sample data comes from a distribution with parameter p.   To do this, we take the likelihood function, which is the product of the probability density function of the underlying distribution evaluated at each sample value:

Screen shot 2014-02-07 at 9.11.29 AM

For our model, we just need to substitute in the pdf of a geometric random variable for the generic pdf above and replace theta with p, the probability of success:

Screen shot 2014-02-07 at 9.11.35 AM

To find the maximum likelihood estimate for p, we maximize the likelihood function with respect to p.  That is, we take its partial derivative with respect to p and set it equal to 0.  However, it’s computationally simpler to work with the natural logarithm of the likelihood function.  This won’t affect the value of p that maximizes L(p), since the natural logarithm of L(p) is a positive, increasing function of L(p).  Sometimes you’ll hear of “Log-likelihood functions”, and this is precisely what they are  – just the log of a likelihood function that facilitates easier calculus.

Screen shot 2014-02-07 at 9.12.56 AM

Taking the derivative of this function is a lot easier than the likelihood function we had before:

Screen shot 2014-02-07 at 9.13.05 AM

So our maximum likelihood estimate of p (the probability of success) is one divided by the sample average, or, equivalently, n divided by the sum of all the k values in our sample.  This gives us the value of p that is most consistent with the n observations k1, …, k.  Below is a table of k values derived from closing data for the Dow Jones over the course of the year 2006-2007.

Recall that the random variable X takes on the value K when K – 1 failures occur (market price decreases) before a success (price increase) on trial k.  For example, X takes on the value k = 1 72 times in our dataset, which means that on 72 occasions over the course of the year there was only one failure before the first success; that is, the market declined on day 1 (by definition) and subsequently rose on day 2.  Similarly, there were 35 occasions where two failures were realized before a success, because the random variable X took on the value k = 2 on 35 occasions.

K Observed Freq.












We now have the necessary data to compute p.  We have 128 observations (values of k), so n = 128.  There are two ways we can compute p.  First, we could take the sample mean of the dataset how we normally would for a discrete random variable and then utilize formula 1 above:

Screen shot 2014-02-07 at 9.14.43 AM

The second formula obviously yields the same result, as you directly compute 128/221 instead of first computing its reciprocal.  So we now have a maximum likelihood estimate for the parameter p.  We can use this to model the stock price movement as a geometric random variable.  First let’s make the assumption that the stock market can in fact be modeled this way.  Given our value of p, what would we expect for the values of k?  that is, what proportion or with what frequency do we expect X to take on the values k = 1, 2, … ? First we’ll compute this, and then compare to the empirical data.

Screen shot 2014-02-07 at 9.15.27 AM

The probability that X takes on the value one is equal to the probability of success, which is to be expected since X = 1 corresponds to the situation in which success is realized on the day immediately following the initial failure.

Screen shot 2014-02-07 at 9.16.03 AM

And the rest are computed the same way.  Now since we have 128 observations, we can multiply each expected percentage by the number of observations to come up with an expected frequency.  Then, we can compare these to the observed frequencies and judge how well the model fits.

K N Expected % Expected Frequency

















5 128



6 128



Now that we know what we should expect if the geometric model is a valid representation of the stock market, let’s compare the expected frequencies to the observed frequencies:

Expected Frequency Observed Frequency













The geometric model appears to be a very good fit, which suggests that daily fluctuations in stock market prices are random.  Furthermore, stock indices don’t ‘remember’ yesterday – the probability of the market rising or falling is constant, and whether it actually rises or falls on a given day is subject to random chance.

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