# Some Simple (But Useful) Math

Mar 22, 2012

A member wrote a question which spurred some thinking. Essentially, they wanted to understand not just the calculation of the sharpe ratio -- but why its a good metric.  So this blog starts on that topic and hopefully can serve as a reference -- it then extends into a longer piece which are thoughts that should help a few people that want to pursue this angle in general.

We are of the belief that you do not want to get bogged down in too much detail beyond this type of math below -- understanding the basic math can be very helpful in your ability to understand investing -- just as understanding the game of Bridge or Poker might be better if you have a basic understanding of the math associated with a deck of cards (total number of cards, number in each suit and how this translates to some basic understanding of odds).

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First -- let us state that the Sharpe Ratio is not a perfect measure by any means (hint: there are no perfect measures) --- but it is a useful framework to start with because it factors in volatility well.  High volatility is best associated with large drawdowns, a major concept to keep in mind.  What do funds with huge losses all have in common?  They were all very volatile.

Drawdowns are obviously your enemy.  Trust though that you CAN control your drawdowns somewhat by controlling your volatility.  Sometimes you hear how investors want to pair their trades with short-selling.   Few investors actually do well with this structure, especially when they pay a big fee.  Yes, it's important to hedge your volatility --- but no, you do not need to pay a big fee to a short-selling fund just to reduce your volatility.  You can reduce volatility anytime you want and as much as you want and the cost is (should be) very, very low -- they are called short-term bonds.  (It goes beyond this post but you can do quite well by using some basic short-term credit strategies -- and then pair these with your core equity/risk strategies that are designed to deliver long-term return).

You DO need to take on some volatility in order to generate some return -- but nobody who actually knows what they are doing is going to be impressed with making X% if you do it with massive drawdowns.

So let's look at some basic math:

1) Compounding returns:

Let's start at some very basics, skip this and go to #2 volatility if you so desire.

Let's start with a 10% annual return. To convert 10% to a daily return, you only need to know how many days there are in a year.  Most traders will know that there are 252 equity trading days in most years (fixed-income and foreign markets like China have far fewer trading days due to extra holidays but this actually doesn't matter because the ETFs are open for trading on exchanges even when the underlying market is closed).

So to take 10% down to a daily level in a spreadsheet:  place 0.10 in cell A2 and then in B2 type  =(1+A2)^(1/252)-1.  Now you have a daily figure of approximately 0.0378%.  Then practice by reversing it back to the annual 10%.  It may be easiest like this   =((1+.0378)^252)-1 = 10.000%.

How is this equation useful?  Well, for each \$100,000 in your account, a 10% annual return would be about the equivalent of 40 bucks a day (lower in the early days and higher in the later days due to compounding for an average just under \$40).   \$40 doesn't seem like much does it?  A lousy ~\$40 on my \$100,000?  Yes, that is what 10% a year feels like from a daily compounding perspective (40x250= \$10,000).

Let's move to a monthly figure as we find this to be a very useful timeframe, you will see why when we discuss volatility.  252 trading days in a year means 252 / 12 months = 21 days per month, on average.  Let's take 10% and convert to a 1 month (21-day) figure =((1+.10)^(1/12))-1 = 0.797% per month.

Just to do the math a different way,  let's go from daily to monthly (from above) =  (1+0.0378)^21 = ~0.797% per month.

So that is return basics, now let's look at volatility.  If you can just get these 2 items (return and volatility math), you will have the 2 major Sharpe Ratio components:

2) Volatility is a slightly different animal but isn't as hard to think about as you suspect.  If something goes up a little bit ON AVERAGE each day (like the stock market) but not steadily compounding as in example above, then there is a different dynamic to understand.

Volatility grows not at time -- but at the square root of time.  It sounds scary but its not.  All that means is that even though the market moves around a lot on a day to day basis, if you look at it over a longer time period, much of those days offset each other and so you can't compound volatility like before because volatility relates to the path taken rather than just the final return result.   Think of it as more 3-dimensional analysis.  It's trickier -- but it's very important information.

Whenever you take the square root of a number greater than 1, the number is going to get smaller so  =sqrt(12) is obviously a lot smaller than 12.  This reduction is taking into account the fact that up and down days (volatility) offset each other to a significant extent.  So up +10%, down -10%, down -10%, down -10%, up +30% is an awful lot of movement but only gets you back a little positive.  The final return of +4.2% doesn't tell you about the path taken.

So if we look at the S&P 500, we can say something like -- over the past X years, the annual volatility has been about [16%] (brackets just mean fill in whatever number you would like, this is just an example).

Volatility is always discussed as an annualized amount.  But we can of course convert 16% to a daily figure using an equation like this  =(0.16)/sqrt(252) = ~1%.

So given a forecast of 16% volatility, we would now not be at all surprised to see the market go up or down +/-1% tomorrow.   There are of course many other possibilities --- but we would not 'expect' it to move 5% --- because if it did, that would be too far from our baseline range of expectations.  Yes, it could happen but then that would be associated with rising volatility far above the example 16% used -- our 'forecast' of 16% was wrong in that case.  Remember, we are just trying to understand the mechanics of the Sharpe Ratio, we are not trying to forecast 1-day movement.

Now, what practical use does this all have?  Well for one, don't try to pinpoint exactly why the market went up or down 1%, it's not a significant enough move when you are looking at it in the context of a few months.  Yes, there may be some headline news that day -- but many times the market will do the exact opposite of the nature of the morning headline.  It's just short-term volatility in play and it's normal.

We think it makes a lot of sense to look beyond daily movement and toward longer windows of time -- but not too long.  A lot of modules on the website are set-up with this in mind.   We need a time-period that allows some of that offsetting volatility to work itself out.  There is no single magic time-period but various academic papers that we've been reading for the last 15 years point to 1 to 12 months as a good time-period to study.  Shorter than 1 month is too noisy and at 12 months or longer, the edge evaporates.

So let's take an example.  Let's assume a 2% 1-month return and 16% ANNUAL volatility estimate.

First annualize the return:  =((1+.02)^12)-1 = 26.8%.    A quick comparison of 26.8% to 16% = 1.68,  that is a measure of return divided by a measure of risk.  You are risk-adjusting the return into a pretty straightforward ratio.  That is, you are incrementally penalizing the return if it took a wild path to get there.   If that makes sense to you on a conceptual level then you are 95% of the way there.

You should quickly see that 26.8% is an unsustainably high number.   You may have a great year  where you do far better than 26.8% --- but you won't be able to compound at this number for the long-term.   So here is the logic, since 26.8% annual is unsustainable -- that means 2% a month is unsustainable (it's the same number).   And while you may have many months far better than +2%, in order to get a good return/risk situation, you are going to have to also spend a little time thinking about the volatility -- you won't be able to do a good sharpe ratio based on return alone (over the long-run).  And of course, the real reason to control volatility is so that you don't experience large drawdowns.

Now let's look at a leveraged fund as that is an interesting phenomenon to think about.  A leveraged fund is DESIGNED to have 2x or 3x the daily return of the unlevered fund.  The funds will re-balance daily (if necessary) to achieve the next days targeted change.  As you should be able to see from above, we know it's a bad idea to trade off 1-day movement because historically (and consistent with the concept of volatility), the market doesn't go up at its long-term rate, it goes up and down in offsetting nature and then only over a larger sample period can we see a return figure that is not so highly skewed.

What this means is that leveraged funds rebalancing daily is a serious long-term flaw and should not ever be held for significant periods of time.   This by the way is exactly what the trading in these funds indicates as most of the leveraged funds have low assets relative to their high daily volume amounts.  There is much trading during the day and then a lot of people flee before the close.

Back to the Sharpe Ratio.  The ideas in this blog have gone into some math but should be thought about more conceptually in our opinion.  So to speak from these basic concepts, you could put this into a practical statement in a few different ways (just by re-arranging the key factors):

1.  We target a return of 10% a year.  We target volatility just below that.  (This implies on a conceptual level a reward-risk ratio of 1.0 or better).

2.  Volatility has been about 10%, we think we should be able to do a sharpe ratio of 1.0 so that means we think we will do 10% in return or better.

3.  We target a sharpe ratio of 1 or better -- we are tactical so volatility will fluctuate up and down -- we aren't going to micro-manage it but we can say that we will manage volatility such that it is no higher than the [S&P 500] index volatility.  This means/implies that our drawdowns should always be lower, too.

etc...

Now, let's pretend you were rotating between 3x leveraged funds.  How would you rationalize this if in a client meeting??

"Our funds volatility has been running 45% so hey, if we can just do a 0.5 sharpe ratio -- then expect us to compound at 22.5% per year"

Uhh no, the meeting would end there and you would be laughed out of the room.   Again, you can and should have some big individual years that put up some nice performance numbers if you execute your strategy well.  But we aren't talking about that, we are talking about long-term compounding.

Lastly, let's make sure we end on some common sense.  Don't get too mathematical about all this.  At the end of the day, this is about growing your account balance.  Making new all-time highs in your account balance and not having large drawdowns is efficient money management.  If you do that, who really cares whether you did X% or X+1%?  Don't lose sight of the big picture by getting too bogged down in the exact math.  An understanding of the concepts is needed -- an exact knowledge of every nuance in the calculations is not.

# A Look At Some Bond Market Segments

Mar 14, 2012 in Bonds

Let's take a look at some bond market action.   The bond market it far larger than the stock market so this should not be a topic you should just skip -- this is what 'the world' actually owns with majority of its money -- the bond market.   Even if you are primarily an equity person -- understanding some bond market basics will help you to understand what is going on overall in the markets.

While bonds mature at par and always pay a positive (nominal) return to maturity, intermediate and long-term bonds are very much subject to material drawdowns.

If we define the bond market to be the Barclays Aggregate index, let's take a logical look at what has happened when this key index drops.   That is, let's calculate how each segment does on both a relative and absolute basis when bonds get nailed.

As you can see in the table below, through last nights close, the bond market has lost at least -0.23% in daily total return change 101 times over the past 3 years (we effectively solved for the last 100 here).    Obviously, longer-maturities do worse and investment grade bond ETFs are not immune to this -- LQD has dropped on 89 of those 101 days and show a slightly larger loss than the overall Barclays Aggregate ETF return.

We prefer to show many of our big picture themes in our Allocation Board portfolios -- as talk is ultimately cheap.   On Feb 16, we lowered duration further than we were and mentioned in the dated comment the overvaluation of intermediate and long-duration bonds.  We moved incrementally towards a very defensive bond position.    In both E-ETFRE and E-ETFRS, allocations that both hit new all-time highs yesterday -- we are essentially all junk-bonds with a mix of very low duration fixed-income ETFs.  Junk bonds are down slightly today -- but that shouldn't be a big shock as the correlation of junk bonds to the overall bond market is not strongly negative on big down days.   Indeed, while junk bonds tend to rise on big bond market loss days -- as seen below, it's not a vastly dominant percentage of the time --- and that is certainly reasonable and very much consistent with very low correlated assets.

The screenshot above is from the tools page app called Down Day Stats. It is on the Tools page with a number of other useful apps (some of which are free). Try it for other key markets --- like treasury bonds (TLT) vs Gold (GLD) vs Silver (SLV).   Or see how emerging markets act on days the dollar rallies (UDN goes down).  What is 'normal' behavior?   You need to have facts to understand what is 'not normal' as this can be very helpful in understanding what is actually going on.    If it all seems complex, that is because it is kind of complex.    The Euro, junk bonds, bank stocks, tech stocks -- how do they all interact??   Basing your investing on these key ideas is what money management is about, in our opinion --- not gambling on the next Netflix (NFLX) or Green Mountain Coffee (GMCR) earnings report.

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# PIMCO Total Return Comparison

Mar 01, 2012 in Bonds

2011 was a dreadful year for PIMCO Total Return.  How bad?  Of ten major indexed intermediate bond ETFs (all with significant assets), PIMCO beat none of them.  These are essentially pure-play segments so this indicates very poor rotation by Gross during 2011.   Clearly the main problem was during Q3 when Treasuries (and the US Dollar) rallied very hard against them.

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# Fee Layering And The ETF Era

Feb 08, 2012

Is there anything more ridiculous in the financial services industry than the complicated fee structure of the products?

An outline from a SEC report regarding mutual funds -- (you can stop reading when you get the idea):

"The typical structure for a multiple class fund includes A, B and C class shares, along with an occasional institutional or retirement class.  Class A shares often include a high front-end load with a nominal 12b-1 fee.  Class B shares have a contingent deferred sales load, plus a large 12b-1 fee.  The load decreases with each year in which the investor continues in the fund, until eventually decreasing to zero, typically about 6 years from purchase date.  After about 8 years, Class B shares convert to Class A shares, reducing the 12b-1 fee to Class A levels.  Class C shares usually have a large 12b-1 fee and a small contingent deferred sales load (1%) that is eliminated after a 1-year holding period."

So Class A is mostly a big front-end sales load -- a 5% hit before you start -- but hey, you only have to pay a small 12b-1 fee (sarcasm).  (12b-1 is a fee to re-imburse the fund company for its marketing costs).  Class B simply changes the mix of fees and Class C tends to have lower fees -- but only if you hold it long enough and therefore pay the higher dollar amount of fee that accumulates over time.   Of course, this all just relates to the expense ratios you pay at the fund level  -- it does not yet factor in any incremental fee layering done by intermediaries.  The result of all this complexity and the layering is that the end investor who doesn't take the significant time necessary to figure all this out can't possibly determine how much they are paying in fees -- think that is a coincidence?

Below is a sampling of various fee combinations:

Now while you may just laugh at sales loads and other retail investor rip-offs that much of the mutual fund industry has piled on, have you done the actual fund of hedge funds math?  In order for a hedge fund (if sold through a fund of funds) to generate a basic 7% return, it needs to gross +13.6%.  To get a 25% return, a fully fee'd hedge fund manager would need +38.6%.   (Assuming 1+10% on top of 2+20%).  Those aren't retail folks buying those -- they are financially sophisticated people.

Back to mutual funds --- if you are an institution investing you do have access to the i Class mutual fund, which is usually not that much different than an ETF fee (ultra-low cost).   In fact, the 'i' in the iShares name effectively refers to the fund company offering institutional class fees to the public in the form of an ETF.  Still, in order to get an institutional class -- you have to pay some intermediary for the privledge.  Or just skip all that and buy ETFs, the breadth of product is aplenty.

While some highlight the fact that ETFs are just a different wrapper -- this skips over a very important point.   When you buy a product on an exchange, the only fee the fund company can stick you with is the expense ratio.  They have no ability to assess a sales load, or a purchase or redemption fee.   It is just the expense ratio --  and then whatever your broker charges you in commission (though this has changed so that many banks and brokers are now zero commission).   The point is that the fund company loses power in the ETF structure as the product moves from off-exchange -- to be exchange-traded.

In our view, paying a low fee is just a check-off item, not a decision driver.   Don't worry about paying 59 basis pts vs 41 basis pts, it's not important -- that can be the equivalent of 5 mins of trading differential -- like buying at 9:40am rather than 9:45am after an index move of 18 basis pts during that time.

It is however important if its 40 basis pts vs 3.29% (Advisorshares is charging an egregious 3.29% on HDGE, an ETF it markets -- that fee is worse than a lot of mutual funds).

Don't get us wrong, anyone doing a good job is worth a material fee -- including a number of mutual fund and hedge fund managers.  But that said, never forget that the fees you pay are a direct dollar-for-dollar hit to your returns.  The more fee-layering, the higher the hurdle-rate to getting the return you seek.

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# Snapshot of Current Environment

Jan 19, 2012

Bill Luby at www.VIXandMore.blogspot.com did a post the other day showing how there hasn't been a lot of difference between the Staples ETF (XLP) -- which has been around since mid-1990's --- and the S&P Low Volatility ETF -- which was the top asset gatherer of all the new ETFs launched in 2011.

The chart below shows the industry ETF which has best tracked the S&P High-Beta ETF.

The Low-Volatility (SPLV) and High-Beta (SPHB) ETFs were launched in tandem by PowerShares and ETF investors now have over \$1 billion in SPLV, over 30x the assets of SPHB.   Now the interesting twist, the Year-To-Date 2012 performance is a significant recovery from the large underperformance of high-beta during the second half of 2011:

This is somewhat reminiscent of the growth vs value performance divergence that crops up from time to time.   We don't think growth vs value or low-vol vs high-beta are particularly interesting strategies  -- mostly because we believe in testing and quantitative research --- and we don't know of methods that lead to good strategies with these types of things.    The great thing about the markets is that there are lots of ways to skin a cat and you only need to find your method. If you don't test your ideas, then of course it will take a lot longer to discover what works for you. On a somewhat related note to finding a technique that works for you, the equity value of our allocations board portfolios have all hit repeated new all-time highs this month.

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# What Was Range of Available ETF Returns in 2011?

Jan 09, 2012

In this post, we have taken the largest 150 exchange trade products (assets) and grouped the returns into an overall distribution to summarize what was a range of generally available returns in 2011 (all are total return).   We wanted to see how we could summarize the year using a reasonably wide cross-section of widely-owned ETFs.    There are many different kinds of ETFs in the list – from bonds to stocks to MLPs, REITs, Preferreds, small cap, mid cap, country funds, regional funds, developed markets, emerging markets, cash, muni bonds, high-yield, mortgage bonds,  etc etc...  We thought this was an interesting way to do it since these are all liquid securities available to anyone.   While there are a number of useful ETFs that fall below the top 150 – this is a manageable high-level look:

The median for the top 150 in 2011 was….  0.0%.   Yes,  Zero Point Zero.    So the distribution is easy enough to summarize, it is centered around zero where one-half were up and one-half were down.
On the one hand, if you weren’t focused on a small set of niche ETFs -- longer-duration treasuries and/or U.S. utility stocks, there wasn’t a lot of opportunity to do better than the +10 to +15% range.   On the other hand, many of the more mainstream investor themes dropped more than -15% (emerging markets, gold miners, US financials, developed market international equities).

For a point of comparison, let's look at the 2010 distribution using the same scale:

Note that for 2010, the median of this same group of 150 ETFs had a total return of +15.1% (the median ETF in 2010 was actually the S&P 500).   So the entire distribution shifts up very significantly --- note that many ETFs rose in excess of +20% in 2010 and only UNG lost more than -5%  (UNG was also the worst of the lot in 2011).   So this was a very positive year and there were plenty of places to achieve a fairly robust return without any real areas to lose money, everything essentailly went up with many rising a lot.

2011 was a year mostly to just stay out of trouble.    It wasn't an outright bad year, there just weren't that many places to really drive an entire portfolio strongly up.  While we all want to generate high absolute returns --- the fact is that we live in a world where you need to put it all in the context of what was actually available.

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# Hedge Fund Index vs 60-40 Balanced Portfolio

Jan 08, 2012

The issue of which index to use as a performance benchmark is rarely all that clear.  One thing that is clear -- hedge funds had a terrible year in 2011. The Bloomberg hedge fund index was -4.9% in 2011. The HFRX Global Hedge Fund index was -8.9% in 2011.  The HFRX long-short equity index was down a staggering -19.1%.    Meanwhile, a standard 60-40 stock-bond balanced return using index funds was +4.1% for the year.    Below is the breakdown by quarter:

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# Gold ETF Monthly Returns Look

Jan 02, 2012

Since the Gold ETF (GLD) began trading, there have been 85 full months.    December of 2011 represented a  -2 Standard Deviation move down as the month ranked 83rd of the 85 in percentage return.  September 2011 was the second worst (84 / 85) at -11.1%.   GLD  dropped -3.8% for the quarter (Q4) and was +9.6% for 2011, completing its 7th consecutive up year.   The chart below uses the Monthly Returns page to rank all calendar month returns from low to high:

For fun, here is the date CNBC ran its special Gold Vault Visit video report.

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# Extreme Example of Correlation For Some Perspective

Dec 23, 2011

Quick, what is the correlation between a stock-bond balanced portfolio and a portfolio that holds the S&P 500 but is leveraged to 2x its daily moves?

This is an extreme example of correlation which shows how it can often be meaningless in investment analysis. Below is the historical correlation of the plain vanilla Vanguard 60-40 Stock-Bond index mutual fund vs the +2x Leveraged S&P 500 ETF (SSO).

The correlation is 0.99 meaning they both have big moves relative to their own historical movement on the same days.But what does this tell you about ‘risk’? Not much. What does it indicate about return? Not much. The volatility characteristics of these two products are completely different despite correlation of ~1.00. Below is the total return comparison

Another example: Pharmaceuticals vs Financials. The 120-day correlation here is 0.88, which is high --- though there is a 29 percentage point difference in YTD returns (2,900 basis pts).

So when you hear people talk about how high correlations means you can’t differentiate yourself ---- this is not because of correlation. Everyone has times when they are out of sync with the market but that doesn’t mean it wasn’t possible to add material value vs an index. To say so is disingenuous.

While we have a correlation chart on ETFreplay.com, we use it only as peripheral information. In an appreciating market, you WANT high correlation to the appreciating asset. But let's be clear, it is possible to differentiate your portfolio no matter what happens with correlation.

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# Understanding ETF Volatility Part 2

Dec 22, 2011

We are updating our 'Volatility vs Subsequent Drawdown' Chart for 2011.   We did a similar post a year ago and you can review that blog post here: Understanding ETF Volatility Part 1.

The way to read these charts is to simply note the relationship.   By definition, high standard deviation investments offer wider opportunities for both out and underperformance.  The S&P 500 dropped -18.6% off its high during 2011 (using closing total return values).   This was -3.2% worse than the mid-year loss in 2010, when the S&P dropped -15.6%.

Gold had a more normal drawdown in 2011 at -17.5% vs 2010's -7.8%, due mostly to its more aggressive slope up during the summer.   Gold never got particularly volatile in 2010 but in 2011 it traded more typical of a commodity market -- where there is often a momentum-oriented blow-off move.   Still, if you were watching volatility dynamically, you would take note of it becoming riskier during the July-August advance.

The NASDAQ-100 actually dropped slightly less from high to low in 2011 vs 2010.  Large-cap tech -- particulary AAPL have been relatively calm in 2011 -- which was also the case in 2010.   Can it go 3 years with just a mid-teens drawdown?  We don't know but there are certainly other more attractive segments as we close 2011.

It may sound odd but the losses in both 1) small cap US stocks (IWM) and 2) US Energy stocks (XLE) were about the same from high to low as the European ETF (VGK), which dropped -30%.   However, both of the US ETFs have seen strong rebounds in Q4 and are a few percentage points from flat on the year.

Silver (SLV) is pretty consistently the most volatile (unlevered) ETF around and it didn't disappoint with a -41% high to low move in 2011 (and the year isn't quite over).

Treasury bonds ground out fantatstic years.   We've blogged many times about how intermediate and low-duration bonds are the ultimate safe-haven.   You can't count on consistent correlations in times of crisis -- but you can count on low-duration bonds,  the fixed-maturity dates of the underlying securities ensure it.  This is not true of long-term bonds -- which look susceptible to significant drawdowns in 2012.  As much hype as the US downgrade received, it proved yet another meaningless move by the ratings agencies in the United States, which for some antiquated reasons seem to remain relevant to some.

Note that many ETF providers launched 'minimum volatility' ETFs in 2011.  Since starting ETFreplay.com in early 2010 we have always used volatility as 1 of 3 key inputs in our multi-factor screener model -- so we are not surprised to see this development from the ETF providers.   Volatility has and always will be important --- but it's certainly not the only thing that matters.  Myopically focusing on volatility is taking this all too far as building ETFs off a singular concept like this is very limiting.

We view risk-adjusted (volatility-adjusted) relative strength as the cornerstone philosophy of ETFeplay.com and this led to a fixed-income bias throughout most of 2011.  However, a tactical investment process continually looks to adapt to the environment --- and we would look for continue rotations in 2012 --- and every other year for that matter.   There is plenty of room to add value in any year and it's up to you to figure out which group that is and how to manage it.  In 2011, fixed-income ETF showed excellent total returns.  You can't always be right of course --- but you can pursue a process on thoughtful consideration of which markets to be involved with --- and then layering some trend-following techniques on top of that ---  all the while continually striving to protect yourself for the times you are wrong.   If you simply stay consistently loaded in a group of highly volatile securities -- it is just a matter of time before you have a large drawdown.

As can be seen in our tracked Portfolio Allocations, we view equities to be attractive as we end 2011 -- particularly small and midcap US stocks.  We see signs of P/E multiple expansion with rates low and liquidity being added to the problem areas of the global economy.  However, even assuming we enter an intermediate uptrend (which may or may not occur), this does not mean that more rotations within the markets won't occur.  It is never easy in the moment and we will carefully continue to watch and adapt to new potential themes/trends --- while always watching our backs to avoid too much portfolio volatility (and its associated drawdown).