# 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.

Mar 24, 2012 08:37 #

Great blog, very informative - Thank you!  If, by extension, I am running (let's say) a SR of 1.2, can I say I am creating for myself an "edge" of 20%?  And if log normally I have a 50 -50 chance of making money on any given trade, am I skewing my odds to 60 - 40?  Can I apply the Kelly Criterion to any of this?  Just wondering...

Mar 25, 2012 14:48 #

You should have also included the formula for the CAGR
Compound Annual Growth Rate.

Mar 26, 2012 05:48 #

Great post.  In the spirit of tax season, for next week's post can you discuss any research you've done on tax implications on a portfolio (ie implementing a relative strength system that may use some of your default models)?

Mar 26, 2012 05:54 #

Great post.  For next week's post, in the spirit of tax season, can you discuss tax implications on a relative strength porfolio that may use one of your default models?  Thanks.

Mar 26, 2012 09:36 #

noebro,   no.  I don't think you can think like that.   you should understand how to do the calculations but I think you are better off not trying to get too mathematical about odds of future.   stay with the themes and trends and watch your back by watching volatility.   I wouldn't try to get too fancy with the math about odds and what exactly your edge is --- those will be very rough guesses and I think its better to just try to get the big picture right.

Mar 26, 2012 09:44 #

hugos,

to make the CAGR calculation easy -- convert everything to days and then apply that.

So 50% over exactly 2 years (204 trading days or 730 calendar days) would be either:

=(((1+.5)^(1/(504/252)))-1
=22.47% CAGR

so if you then held it for X years + a partial year like 525 days:

=((1+.5)^(1/(525/252)))-1
= 21.49% CAGR

you are essentially creating a ratio of the (time held) vs (1 full year).   you can also use calendar days and 365 days in a year and the number will be (almost) the exact same.

Mar 26, 2012 09:58 #

bartmart,

taxes are tricky to discuss because of the way tax brackets work and all the various components to your personal situation.   ie, were your income/FICA withholdings correct on the w-2 income side?  are you filing jointly or single?? etc etc etc....

generally speaking, cap gains will be in a range of 25-35% if short-term and 15% if long-term.  of course, taxes won't be owed in current year if the securities with gains were sold within a retirement account.

if you generate big returns and it's not within a retirement account -- then you will owe a lot of tax.  It shouldn't be viewed as negative any more than if your company paid you a big bonus and you got 35% withheld from it in your pay-stub.

btw, there are some very good tax calculators online if you want to ballpark your numbers -- though you will have to input a number of assumptions regarding the rest of your income and various withholdings.    But basically, I think you will find that if its a small absolute amount, your tax on short-term gains will be closer to 25% and if its a high absolute amount, it will be at 35%.    But then don't just assume that will be the dollars you owe -- it depends on the rest of your 1040.   I am not a tax professional and in no way should this be taken as advice.  consult a tax professional for specific tax advice.

Mar 26, 2012 10:55 #

Thanks Chris,
for your simple example   "50% over exactly 2 years"  =(1+0.5)^(1/2)-1 =22.47% CAGR.
I use 365.2 days a year, then one can use regular dates in a spreadsheet:
(1+gain)^(1/(end date -start date)*365.2)-1