Tag Archives: finance

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.

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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.  

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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.

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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.


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Level Payment vs. Sinking Fund Loans

Below is a document explaining how to derive formulas for the most basic level payment and sinking fund loans. This is a simple introduction, as I’m currently working on a more detailed analysis of the benefits/drawbacks to various types of loans (including installment, variable rate, etc.) using empirical data and considering various scenarios, like the option to refinance and varying interest rates.  I used the results from my post on annuity formulas to simplify the derivation, so if you’re confused about how I got from one step to the next, check there!

Level Payment and Sinking Fund Loans

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.


Deriving the Present Value and Future Value of an Annuity Immediate

Below is the derivation of the present and future value of a unit annuity immediate, or a series of $1 cash flows that occur at equal intervals of time at the end of each period.  I originally wrote this document as a review for myself in preparation for actuary exam FM/2.  The majority of questions on the exam, despite the wide array of topics covered, come down to solving for the value of some annuity.  Granted, it likely won’t be a case as simple as the one below, but many problems about loans, bonds, yield rates, and even financial derivatives biol down to an annuity problem.


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.

First Quarter GDP revision and Long Run Growth

In their second estimate, the Bureau of Economic Analysis (BEA) revised Q1 2014 Real GDP down 1.1%, which means that Real GDP contracted by a seasonally adjusted annual rate of 1% in the first quarter of this year.  In comparison, Real GDP grew at 2.6% in the fourth quarter of 2013.  A substantial decline in inventory investment is responsible for much of the decline; in fact, if we leave inventory investment out of the GDP calculation, Real GDP actually rose by 0.6% over the period from Q4 2013 to Q1 2014.

Despite disappointing metrics, the IMF stood behind previous assertions that U.S. growth will pick up in the medium term, but extended the forecasted period of subdued growth by lowering Real GDP in 2014 to 2% from 2.8%.  However, the forecast remained unchanged for years 2015 and beyond (3% in 2015 and a LT rate of 2% thereafter), suggesting that the downward revision in Real GDP isn’t indicative of any economic issues ahead.

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The U.S. economy is expected to grow above trend into 2015.  Why?  Fiscally speaking, consolidation (i.e. efforts to reign in the deficit) will be less severe than last year.  Monetary policy in the U.S. is still very accommodative, which should help the economy pick up some slack until about mid 2015 when the policy rate is expected to increase.  Household wealth in the U.S. has also increased, thanks in part to the Fed propping up the stock market.  The ‘Wealth Effect’ seems to be working via the stock market, encouraging financial asset-holding consumers to consume more.  Keep in mind, though,  that stock market gains disproportionately benefit wealthy consumers, limiting the wealth effect’s impact.  As for credit markets, lending conditions depend on the customer in question.  Business and industrial lending is easy, as is lending to the most credit-worthy customers.  However, credit has been tightening for risky borrowers, even as demand for credit has surged and banks have tons of cash reserves.  So, while total lending activity in dollar terms is increasing, and thus households are incurring more debt in aggregate, it doesn’t follow that banks are willing to take on more risk and lend to consumers who are dependent upon credit.  Wage growth is also slow, further constraining the majority of consumers’  ability to consume.  Personal consumption is 70% of the U.S. GDP, so to see broad-based and sustainable gains in GDP, wages will have to grow and/or credit will have to become more available.

Wage and Inflation

Inflation declined in 2013 which was presumably helped by lower commodity prices (like energy and food).  The issue is (mostly) in what low inflation tells us about where the economy is operating relative to its potential.  The widespread decline in inflation reminds us that the output gap in the US economy persists and is closing only gradually.  Additionally, there are some prices that we want to see increase – like wages!  Unfortunately, that’s happening only very modestly (as shown above).  Wage growth will need to pick up in order to grow consumption, especially given credit conditions for middle and low income consumers.  If you’re looking at unemployment figures and thinking that the labor market looks strong, I’d invite you to take a look at the labor force participation rate as well, as the labor force is at a 35-year low.  The picture isn’t all that good, and unemployment is still high relative to the long term trend.  To see a healthy ish labor market, look at Germany.

To summarize the summary, the good news outweighs the bad news – but remember two things: 1) better economic conditions aren’t necessarily good economic conditions, and 2)  the U.S. economy, like all advanced economies, is heavily integrated with the global economy, whose recovery remains patchy, uneven and fragile in some areas.  I suppose you could say cautious optimism about the U.S. economy as a whole in 2014 and 2015 is warranted given the current state of demand and prices.

Sources: IMF World Economic Outlook, BEA, and WSJ

Immunization: A buzzword-free introduction

In a previous post, I talked about a few popular measures of interest rate risk: Macaulay Duration, Modified Duration, and Convexity.  However, I didn’t mention the practical implementation of these metrics or their relationship with the concept of Immunization.  Broadly, the task of managing the interest rate risk associated with a given portfolio of financial assets comes down to minimizing the impact of a specific case of rate fluctuations: the decrease in asset value that results from an increase in interest rates.  It’s hard to come up with good definition of immunization, and rather than copy and paste cookie cutter bullshit I’ll just say that a portfolio is “immunized” when its value is guarded from said interest rate fluctuations.  It is not difficult to mathematically derive the conditions that are necessary for this to be the case, and once they’ve been derived they can be expressed in terms of the familiar interest risk metrics Macaulay Duration, Modified Duration, and Convexity.

First we need to generalize the concept of the Duration of a single asset to the Duration of a portfolio.  In the simplest case, we have two assets, A and B.  The change in the portfolio value when interest rates change is just the sum of the changes in value to assets A and B individually, assuming that the change in the interest rate is the same for assets A and B.  If you want to be fancy, this uniform interest rate fluctuation across all assets can be called a parallel shift in the yield curve.  Below, an expression is derived for the change in portfolio value:

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This expression doesn’t say a whole lot about the underlying process.  Since we assumed that the assets are affected by the same change in interest rates, we could factor that term (delta*i) out of the above expression.  We could also manipulate the expression so as to express the change in price as the multiplicative product of P and some other term containing the relevant duration and price metrics for each asset.  It takes some algebraic simplification and clever factoring, but you can pretty easily show that the total change in portfolio value is simply the weighted average of the changes in asset A and B, which implies that the Modified Duration of the portfolio is the weighted average of the Modified Durations of the individual assets in the portfolio, where an asset’s weight is its proportion of total portfolio value.  The formula the duration of a portfolio consisting of m investments (X1 ,…, Xm ) is below.

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It’s important to remember that we made a simplifying assumption in deriving this formula, namely that interest rate fluctuations are the same for each individual asset that comprises the portfolio.  In other words, when interest rates change (in our simplified model) it’s due to a parallel shift in the yield curve.  This won’t usually be the case, but the above model is still a useful approximation.

I’m convinced that a concise definition of Immunization doesn’t really exist, but I’ll explain the steps involved in “immunizing” against interest rate fluctuations in mathematical terms and then try to express the result in terms of duration and convexity to bridge the gap between the math and portfolio management.  Suppose at any given time you have some assets and some liabilities.  The liabilities are to be paid out at future dates, but you’ve estimated the present value of said payments given an assumption about the current interest rate.  Ideally you’d like to match those liabilities with assets in order to cover them; for example, you would like to match a liability of $P at time t by purchasing an asset today for some price that yields exactly $P at time t, a situation sometimes referred to as an exact match.  But that definition is useless because the idea that a portfolio of any realistic size could be matched exactly given the infinite number of possible combinations of assets that exist in our society is fucking retarded.  Maybe it’s “possible”, but I’m using the word lightly and what I actually mean is a firm could theoretically (maybe) hire someone, make his job title “exact matcher” and it would take him an incredible amount of time to (maybe) eventually find an answer.  So exact matching is conceptually only mildly retarded, but, in practice it is incredibly and wildly retarded.

So let’s stop thinking in finance buzzwords for a minute and just think about the conditions necessary for you to not get fucked by an increase in interest rates.  Actually, first, let’s figure out if and why you’d get fucked if interest rates change.  You have some liabilities with a present value that you’ve estimated, and we’ll assume that you’ve picked assets so as to match them at the current time.  (Note – this is not “exact matching” because I’m talking about the present).  It’s not unreasonable to say that at the present time assets are equal to liabilities.  Suppose interest rates increase; the present value of your assets will decrease.  This sucks, but won’t the present value of your liabilities decrease too?  Yes.  So what are we worried about, then?  Immunization deals with guarding against an interest rate change that disproportionately affects the present values of assets and liabilities; specifically, the case in which an interest rate variation results in the PV of liabilities exceeding the PV of assets.

Naturally, our first condition to immunize a portfolio is that for small (this is important, but we’ll come back to it later) change in interest rates, which we’ll denote as a change from i0 to some nearby i, causes the PV of assets to exceed the PV of liabilities:

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Inequalities suck so let’s define a new function h(i) as the difference between the PV of assets and the PV of liabilities.  So the statement above is equivalent to saying h(i) is positive, and the condition mentioned earlier, that the present value of assets should equal that of liabilities at time 0, is equivalent to saying h(i0) = 0, or, more generally, h(i) = 0 at i0.  So we have the following:

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What do we know about h(i)?  The first condition is pretty obvious and uninteresting; all that is said in requiring that h(i) = 0 is that PV assets = PV liabilities initially, which, has already been stated more than enough.  The next condition is more interesting; regardless of the direction, any small change in the independent variable must result in a positive increase to the function h.  This is obviously a local minimum, which, by calculus is a stationary point with a positive second derivative.  The result is summarized below in terms of the function h and also in terms of the actual assets and liabilities.

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If we wanted to, we could write the preceding expressions in terms of duration and convexity, since they are derived from the first and second derivative respectively.  Deriving these expressions is computational (involves a lot of substituting and rearranging and dealing with negatives but nothing actually hard, just annoying) so I will leave that part out.  If you don’t believe me you can practice your computational high school algebra skills and try it on your own.

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I think there are finance-y terms for each of the three conditions above, but I can’t remember and a quick Internet search didn’t yield any helpful results.  I don’t think buzzwords matter anyway, but if I had to explain them in words, I’d say something like the following (you can put a buzzword-y spin on it):

1)    The present values of Assets and Liabilities are equal

2)    The Durations of Assets and Liabilities are equal

3)    The Convexity of Assets is greater than the Convexity of Liabilities

4)    You’re Immunized!


Continuously Compounded Returns

I’m in a financial “math” class that I really shouldn’t have taken (it’s in the business school) and our lecture notes the other day included some incredibly basic properties about interest calculations.  Anyway, I told one of my friends that it was super fucking easy to derive the formula for continuous interest by taking the limit as the number of compounding periods approaches infinity.  Upon further review, I stand by the fact that it is conceptually easy to do this, but there are some computational issues that one could run into – for example, I had to use a relatively simple substitution in order to avoid using L’Hopital’s rule or something else messy like the definition of a difference quotient, etc.  But, regardless, below is the derivation, starting with the definition of the effective annual rate for an interest rate, r, compounded n periods per year for t years.  Suppose there are n compounding periods per year and r is the interest rate.

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How do we show that the effective annual rate under continuously compounded interest (i.e. effective interest with arbitrarily large n) is er – 1?  We need to show the equation below, which equates the limit of the EAR as the compounding periods approach infinity to the formula we claimed represents the continuously compounded rate (Exp(rt)-1):

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If I were being tested on this in a math class for whatever reason, I might not show the continuously compounded interest rate this way.  This is more of an intuitive justification than a real proof.  To see a real proof, you can check out notes from Wharton here – it’s not really any “harder” per se, but some of the steps might be less obvious.  For example, the proof involves taking the natural logarithm of both sides of the equation (as shown below) which serves a purpose, though it may not seem like it at first.

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This was done in order to take advantage of a useful property of logarithms: the log of a product is the sum of the logs.  Using this, the right hand side is split up into a sum and the first term no longer depends on the limit because it doesn’t have an n in it.  You’re left with a sort of messy equation that involves logs on both sides (it’s implicitly defined), so to rectify that you just need to remember that e and the natural log are inverses, so e raised to the power ln is 1 and the ln of e is 1.

Note: just so you don’t sound like an asshole, e is Euler’s number, and it is pronounced like “oiler”.

Another note:  There are tons of definitions of e, but the one I think of first when I think of e is the one below:

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In words, e is the positive real number such that the integral from 1 to e of the function 1/t is equal to 1.  Equivalently, the area of the integral of 1/t from 1 to e is 1.  ln(x) crosses the x-axis when x=1 (since ln(1)=0), so the area in between the function ln(x) and the x-axis from its x-intercept to x = e is 1.  Obviously, ln(x) takes on the value 1 when x=e since ln(e)=1.  I’m not good at making fancy graphs online, but below is a heinous picture of the point I’m trying to convey.  Sorry Euler 😦

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Intro to Value at Risk

Value at risk (VaR) is a way to statistically measure the worst expected loss on an investment given certain parameters.  Specifically, the level of confidence (called the alpha level, α) and time horizon must be specified to compute it.  We must also assume that returns on the asset in question are approximately normally distributed, because the value at risk model calculates the worst expected loss over a given horizon at a given level of confidence under normal market conditions1.  Although it’s a well-established fact that, in general, stock prices are log normally distributed while stock returns are normally distributed.  It may be a good idea to verify this first, using a goodness of fit test.  The Anderson-Darling, Shapiro-Wilk, and  Kolmogorov-Smirnov tests are all good examples.  When we say normal, we mean that the data is normally distributed; we are making a judgment regarding the distribution of returns, not the likelihood of realizing a return, as the dictionary definition of normal would suggest.  As we’ll see, the VaR turns out to be a relatively unusual result in most cases, and its rarity is inversely related to the level of confidence we choose.  It follows that a formal definition of the value at risk is a measure of the worst expected loss over a given time frame and level of confidence under normal market conditions.

The actual VaR number should be the worst expected loss on an asset in terms of currency (not probability).  If we let f represent the probability density function of the return2 of the asset and c the confidence interval, we have the following formula:

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In words, this formula equates the confidence level, or degree of certainty with which we can draw a conclusion, with the integral from -∞ to the negative of our VaR value (some real number measured as a return in dollars) of the probability density function (pdf).  The Fundamental Theorem of Calculus tells us that integrating over a range gives us the area between the x-axis and the function.  In the context of a probability density function, whose y-axis is probability, this says that by integrating from A to B, we’re finding the probability that the random variable takes on a value between A and B.  In the context of VaR, when we integrate from negative infinity to –VaR, we’re finding the probability of realizing a value equal between -∞ and –VaR, or, more simply, the probability of realizing a value less than –VaR.  So for a given confidence level c, we’re looking for the value –VaR such that the integral from -∞ to –VaR is equal to (1 – c).

It might be confusing that the negative of the VaR is used in this computation.  This arises because of the symmetry property of the normal distribution.  Each value, however extreme (extreme meaning far from the mean in terms of standard deviations) has a complementary value in the opposite direction.  If we’re talking about the standard normal distribution, we’re just saying that the distribution is normally distributed with a mean of 0 and a standard deviation of 1.  So a relatively extreme value, say 3, has a complementary value, -3, that is just as extreme in the opposite direction.  3 is a value 3 standard deviations greater than the mean, and -3 is 3 standard deviations below the mean.  But we’re only interested in a potential loss in VaR models, as risk is by definition a quantitative measure of realizing a loss, we’re only interested in the –VaR.  The corresponding positive VaR will be a value just as extreme but positive – in this case, a good thing since it means higher returns.

Every potential portfolio value corresponds to a z score, or a standardized measure of how many standard units (standard deviations) a value lies from the mean of the distribution, as was mentioned above.  Ordinarily, we transform a random variable X into a standardized Z score with the formula Z = (x – μ)/σ, where, in this case, x = VaR and Z = Z value corresponding to the confidence level.  However, since we’re only interested in the negative values, we substitute x = -VaR and Z = -Z.  We can rewrite the equation more simply now: VaR = Zσ – μ.  This equation will standardize our VaR value.

So if you’d like to be 95% certain you won’t lose more than the VaR, you set the left hand side of the equation equal to (1 – 95%) = (1 – .95) = .05 and solve for the number VaR, which can then be interpreted as the 95% value at risk.  What you’ve really found here is the portfolio value that represents the spot on the x axis where the total area above the axis and below the pdf on the left hand side all the way to x = negative infinity is equal to .05.

If we use the 99% confidence level, we have c = (1 – .99) = .01.  We have to find a z score that divides the normal distribution into a section with 1% of the area to the left and 99% of the area to the right.  The picture below has 1% of the area in each tail of the distribution, and thus 98% shaded in the middle.

Screen shot 2013-08-13 at 1.48.48 PM

We’re interested in the z score that divides the lower white region from the rest of the graph, because that will correspond to the value that 99% of the x values are greater than.  But since the normal curve is symmetrical, that value is just going to be the negative of z value, which represents the x value with 99% of observations below it.

Screen shot 2013-08-13 at 2.24.26 PM

The pictures reveal that the z value in question is z = -2.33.  To reiterate, standardizeing the value of any random variable x (i.e. assigning it a z value), uses the relationship Z = (x – mean)/(StanDev).  This Z value is the VaR in standardized form.  Therefore, we can substitute –VaR for Z in computations because what we’re really finding when we find the VaR is the left tail (Z score) of the distribution.

If the VaR is 200 at the 99% confidence level, you could say that largest possible loss on your asset at the 99% confidence level is 200, given normal market conditions.  Alternatively, you could be 99% certain your losses won’t exceed 200.

Since the cumulative density function (F) is the indefinite integral of the probability density function (f), we can use it to simplify this expression (sorry for the bullshit above the equation – I had to screenshot it because wordpress hates equations):

Screen shot 2013-08-13 at 4.20.37 PM

An example should make this clearer.  Suppose the daily return on a portfolio follows a normal distribution with a mean of $1000 and a standard deviation of $500.  At the 99% confidence level, we have alpha = .01 which corresponds to a z-score of 2.33, i.e. the CDF of the normal distribution is equal to .99 when the z value is 2.33; F(2.33) = 0.99.  But we want the lower end of the distribution, -2.33.  Therefore:

VaR = ασ – μ = 2.33(500) – 1000 = 165

So with 99% confidence we can assume that our worst possible expected loss in one day is $165.  The value of -$165, i.e. a $165 loss, is 2.33 standard deviations (z scores) below our mean value of $1000.  So although we still expect (on average) to earn $1000 with a standard deviation of $500, we will only lose more than $165 in a single day less than 1% of the time.  The PDF and CDF are below.



So the value -165 is exactly equivalent to the standardized z score of -2.33 as they both represent the value that 1% of observations fall below and 99% exceed.  The only difference is that with the number -2.33 we’re talking about a standardized random variable with a mean of 0 and a standard deviation of 1, whereas the VaR number of -165 represents a portfolio return with a mean of $1000 and a standard deviation of $500.  Alternatively, -$165 is the value 2.33 standard deviations to the left of the $1000 mean portfolio return.


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