Annualization

This page discusses how to annualize returns.  It also touches on the topic of how to annualize risk estimates such as tracking error and volatility.

Essentially, annualization is a very simple calculation.  However, there are many other closely related topics, such as monthly returns, continuously compounding returns, and geometric excess returns.  These related topics can, because of all the detail they bring along with them, make it much harder to see clearly that annualization is a simple topic.

The Key Idea Behind Return Annualization

The key idea behind return annualization is very simple indeed.  Starting with a return over a period other than a year (usually more than a year), annualization calculates the equivalent constant annual rate of return.  For example, suppose that a portfolio has returned 21% over two years.  What is the annualized rate of return?  Simply, it is 10%.  The working behind this is that (1+10%) x (1+10%) - 1 = 21%.

Annualizing returns is a simple way of standardizing them — and hence making them comparable.

Consider a person who bought a house for $100,000 in 1986.  Twenty years later, they sold the house for $672,700.  This seems like a very good return on the initial investment.  But exactly how good?  We can answer this by calculating the total percentage return (572.7%), then annualizing it.  This comes to exactly 10% per year.  When viewed in this light, the return is still perfectly respectable, but it is more modest than the dollar amounts may intuitively suggest to many people.

Figure 1 shows the effect of compounding returns at an annual rate of 10% over periods ranging from 1 to 20 years.  This helps to illustrate the basic intuition that compounding and annualization are very closely-related processes.

Arithmetic for Annualizing Periodic Returns

Given a return R(y) which is measured over a period of y years, how does one annualize this return?  Essentially, the answer is just

Mathematically, this is simply calculating what the return would have been if the investment compounded for (1/y) of the actual investment period.  In other words, if one has a three-year return, the annualized result looks at what would have happened if the investment compounded for only one year.

Example: Consider a return of 77.2% which had arisen over a period of six years.  In other words, R(y) = 77.2%, and y = 6.  The solution would be R_annualized = 1.772^(1/6) - 1 = 10.00%.  As we would have expected from Figure 1, the answer is 10%.

This is the basic mathematical principle that you can use for most annualization of returns.  However, some difficult or fiddly little implementation problems can arise from time to time.  We deal with them below. 

Complicated Little Details

To get some feeling for the complexities that can arise in annualization, consider one simple example:

In this example, the period to be annualized runs from 31 January 2002 to 29 February 2004.  Exactly how many long (measured in years) is this period?  One plausible answer would be that the period is exactly 25 months long, and hence that it is exactly 25/12= 2.08333 years.  Another plausible answer would be that the period is 759 days long, and if we consider an "average year" to be 365.25 days, this period would be 2.078029 years.  There is a small difference between these two values.  This difference in the number of years y will lead to differences in the annualized return.  This is a problem, because an investor might potentially receive conflicting values for the annualized return from two different sources (e.g. the custodian and the investment manager).  Valuable time could be wasted trying to explain the discrepancy.

Just to complete the example, suppose that the return over the two-and-a-bit years has been 21.0%.  Using 25/12 as the number of years, the annualized return comes out at 9.58%.  Using 759/365.25 as the number of years, the annualized return comes out at 9.61%.  A diligent investor would want to understand this discrepancy.

The problem here has nothing to do with annualization calculations (as such).  Rather, it has to do with conventions about measuring time.  Do we measure fractions of a year in days, months, or otherwise?  Some issues we might like to consider regarding this are:

• Months are a very irregular yardstick, in the sense that different months are clearly of different lengths.  Some months are 28 days long, while some are 31 days long.  From this perspective, months are not a very accurate way to measure elapsed time.  This doesn't necessarily mean, however, that months are a poor basis for calculating annualized returns.  

• Indeed, the investment performance analysis industry has a long heritage of measuring time in months.  It seems intuitively right that six months should be half a year, and that three months should be a quarter of a year, regardless of whether these measurements are strictly accurate when measured in days, hours, seconds, etc.

• If one uses days to measure periods, how many days should one assume are in the average year? 

• If one was to adopt a convention where one measured periods in months, it is not entirely obvious how one would measure the length of the period between (say) 25 April and 27 May.  One could say that this is a month plus 2 days.  Because the end of the period was in May (a month with 31 days), would one say that the period is 1 + 2/31 months long?  Or would the fact that April has 30 days affect the result?  Would it matter if the current year was (or was not) a leap year)?

It seems clear that the basic annualization calculation itself is actually much simpler than the fiddly issue of how to measure periods of time for use in annualization calculations.  We will describe a specific solution (see below) to these issues of time measurement.

First, however, let us confirm some properties that an ideal annualization solution would have.  The underlying assumption here is that we are working with a system that has the capability to measure daily returns (if we only have monthly returns, it seems very natural to use months as the basis for calculating annualization periods).  Some properties that would be desirable in an annualization method are:

• As just stated, we probably don't want to be restricted to annualizing returns only over whole months.  Even though most regular performance reporting might be for an exact number of months, it seems unduly restrictive to decide that a performance reporting system will simply refuse to annualize returns over irregular periods such as (say) 15th March 2001 to 27th April 2005.  In other words, we seek a method that is based to some extent on counting days.

• For the sake of consistency, we want an annualization method that will be "backward compatible" with the whole-months approach to calculation.  For example, if the period was 31st December 2002 to 31st March 2004, we would want to count it as exactly 1.25 years.  Otherwise, our method would produce results that are inconsistent with the traditional month-based calculations.

• A corollary of the previous point is that returns over a period of exactly one year should be exactly the same whether annualized or not.  For example, if the return from 31st December 2004 to 31st December 2005 is exactly 13%, it should also be exactly 13% when annualized.  This may sound like a laughably obvious requirement.  However, one possible approach to annualizing would be to count the days, and then divide by 365.25.  In this case, the period would seem to be slightly shorter than a year: 365 divided by 365.25.  Consequently, the annualized return would differ from the non-annualized return.  This would be confusing and unintuitive.

Proposed Solution for Measuring Annualization periods

This material will appear in a forthcoming journal article.  We apologize that it is not available right at the moment.  We currently aim to publish by June 2008.

Annualizing Periods Shorter than One Year

Many performance reports adopt a convention that they annualize returns over periods longer than a year, but they leave unchanged returns over shorter periods.  This makes a lot of sense.  For example, if a portfolio returned 3% over a single month, it would be mathematically true to say that the annualized return exceeded 42%.  However, it is normally not a good practice to present performance information this way.  It would be a bit extreme to say that a performance measurement system should not permit the user to annualize returns over periods shorter than one year — after all, this can be a useful way to understand the power of compounding.  However, a well-designed performance measurement system should have a default option where returns are annualized over periods exceeding a year, but not for periods shorter than a year.

Annualizing Continuously Compounding Returns

On the Performance Measurement page, we explained continuously compounding returns in some detail.  You are advised to review that material before moving further forward in the current section.

Continuously compounding returns are very easy to manipulate mathematically.  One can compound them simply by summing them.  Because annualization is so closely-related to compounding, it should come as no surprise that it is also incredibly easy to annualize continuously compounding returns.

Figure 2 illustrates compounding and annualization for both periodic returns (i.e. the "plain vanilla" returns" that we are all accustomed to using), and for continuous returns.

To illustrate compounding and annualization, this table demonstrates returns for the kind of portfolio we would all love to invest in: a portfolio whose annual return increases by 1% every year.

In the first year, the portfolio return is 5%.  Next year, it is 6%.  And so on.  The third column of the table shows what these returns look like when converted to continuously compounding form (by taking the natural logarithm of (1 + the periodic return)).

In the centre columns of the table, you can see the cumulative returns since inception for each kind of return.  For example, after one year, the cumulative periodic return is 5.00%.  This indicates that an initial investment of $100 would be worth $105.00.  After two years, the cumulative periodic return is 11.30%.  This indicates that an initial investment of $100 would now be worth $111.30.  In other words, there is a clear and intuitive correspondence between the cumulative periodic return and the cumulative dollar value.

The columns on the right show the annualized cumulative returns (using both the periodic return convention, and the continuously compounding convention).  For both kinds of return, it is easy to observe that the annualized return over the period since inception steadily increases over time, reflecting the steadily increasing annual returns in the left column.  You may wish to consult the spreadsheet to gain a clearer picture of exactly how the annualization calculations work for periodic returns and continuously compounding returns.

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Aside from the fact that the arithmetic is simpler with continuously compounding returns, there is no major difference between periodic and continuous returns when it comes to doing annualization calculations.  Moreover, they do have one tricky issue in common with one another: exactly how to calculate the number of years in a non-annualized period.  Whether you will be doing annualization using the simpler arithmetic of continuous returns, or the slightly more complicated arithmetic of periodic returns, the fact remains that you will get some strange-looking results if your method for measuring a whole year is not reliably accurate.  You may care to review our Proposed Solution for Measuring Annualization Periods to see how the same issues apply just as much to continuous returns as they do to periodic returns.

Take one very small example, to help make things clear.  Suppose that Year 1 in Figure 2 was in fact the year 2001.  When calculating an annualized return from 31st December 2000 to 31st December 2002 (i.e. over Years 1 and 2 in our example), it would be ideal to find that the number of years was exactly 2.  When annualizing the continuous return, the calculation would then be 10.7059% / 2 = 5.357%.  If one counted the number of days, then divided by 365.25, one would come up with a period of 730/365.25 = 1.9986 years.  The annualization calculation would then become 10.7059% / 1.9986 = 5.353%.  In this case, the error is slightly less than a basis point, and it would not be visible on reports which show results to the nearer basis point.  However, sometimes, the error would be large enough to manifest itself as a single basis point difference in the annualized and non-annualized returns.  This kind of small and apparently transient error can undermine users' confidence in the quality of performance reporting they are receiving.

See cells K5-Q11 in Annualizing_Continuous.xls for the exact numbers and formulae for this example.

To summarize, continuously compounding returns make the arithmetic easier, but the problem of calculating the annualization period y still remains.  Furthermore, continuously compounding returns do not have the same intuitive relationship to cumulative value added (e.g. 115% return means that a $100 investment is now worth $215) that periodic returns do.

Annualizing Active Returns

An active return (sometimes loosely called an "excess return") is simply the difference between the portfolio return and the benchmark return.  A question that arises from time to time is how to obtain annualized active returns.  A related question is whether it is possible to compound periodic active returns and then annualize them to get a correct result.

To illustrate what is involved in this issue, we will work with a simple example, shown in Figure 3.

All of the data and calculations for this example are available in the spreadsheet Annualizing_Active.xls.

There is nothing particularly tricky about this example.  In Year 1, the portfolio return exceeded the benchmark return by 7%.  In Year 2, the difference was 3.0%.  The convention we are using here is known as arithmetic active returns, where one calculates the active return simply by subtracting the benchmark return from the portfolio return.

Figure 3 also shows the performance compounded over both years, and also annualized.  As one would expect intuitively, the 10% benchmark return compounds out to 21% over two years, and annualization reduces this figure back again to exactly 10% per year.

In every row of Figure 3, the active return is the difference between the portfolio and benchmark returns.  Note, in particular, that active returns themselves are neither susceptible to compounding nor annualization (which we have seen is a form of compounding).  This is because compounding the difference between two returns is not the same thing as taking the difference between two compounded returns.

To be very specific, there are two completely incorrect calculations that one could do using the data in Figure 3:

1.  One could compound the active return for Year 1 (7%) with the active return for Year 2 (3%): this would give a result of 10.21%, rather than 11.21%.

2.  One could annualize the two-year active return, rather than taking the difference of the annualized portfolio and benchmark returns.  This would give a result of 5.46%, rather than 4.98%.

This fundamental fact that active returns are neither compoundable nor annualizable has big implications for how people analyze the active performance of portfolios.  In particular, performance attribution is a common method for analyzing active returns.  Some people search in vain for a way to do attribution on arithmetic active returns. This is simply not consistent with the way the math works.  The main alternatives that one has are:

• Do the analysis on non-annualized arithmetic active returns; or

• Do the analysis using geometric active returns (see Figure 4); or

• Do the analysis using continuously compounding returns (see Figure 5).

The first option would simply mean that the attribution analysis would explain 11.21% of active performance over two years.

Figure 4 shows the second option, which is to conceptualize the active returns geometrically rather than arithmetically.  We can easily illustrate arithmetic and geometric calculations for the two-year active return:

Arithmetic active return = 32.21% - 21.0% = 11.21%,

Geometric active return = (1+32.1%) / (1+21.0%) = 9.26%.

Adherents of the geometric approach consider it highly intuitive.  The rest of us find the arithmetic calculation much simpler and more intuitive than the geometric calculation.

Notwithstanding that, one clear benefit of the geometric paradigm is that the active returns do compound and annualize.  For example, in Figure 4, the two-year active return (9.26%) is exactly equal to the result one obtains by compounding the active returns for Year 1 and Year 2.

To summarize the differences between Figure 3 and the geometric active returns shown in Figure 4:

• there are no changes to the portfolio or benchmark returns; and
• the active returns are calculated as a ratio, rather than as a difference.

Another approach is to use continuously compounding returns, as shown in Figure 5.

To convert a normal ("periodic") return into a continuously compounding return, one adds one to the periodic return, then takes the natural logarithm of this value.  For example, in Figure 3, the portfolio return in Year 1 is 17.00%.  To convert this into a continuously compounding return, one calculates ln(1+17%) = 15.70%.  Once all the returns are in continuously compounding form, the tasks of compounding and annualization become laughably easy.  For example, the way to compound returns for Year 1 and Year 2 is simply by adding them.  The way to annualize two year returns is by halving them.  Aside from that, the active return is calculated as a difference.

In Figure 5, you can verify that the active returns do indeed exactly compound (by addition) to the two-year active return.  So an attribution system based on continuously compounding returns would be able to easily show results in annualized form.  Why, then, doesn't everybody use continuously compounding returns?  Because they sacrifice certain intuitions (as we will discuss in the next section).  In the meantime, let's summarize the differences between Figure 4 and the continuously compounding returns shown in Figure 5:

• The continuously compounding returns are different from the periodic returns, because they are logarithmic;

• The continuously compounding returns combine over time by addition (rather than by multiplying);

• The annualized continuously compounding returns are simply half of the two-year returns (rather than the square root calculation that is used for periodic returns; and

• The active returns are still calculated by subtraction.  In this sense, they work the same as the periodic returns.  However, because one is subtracting logarithms, the calculation is also akin to calculating geometric active returns (and this is why the active returns in the continuously compounding framework compound together consistently).

What Intuitions are at Stake in Annualizing Active Returns?

All three of the frameworks presented above provide different tradeoffs in analysing active returns.  By introducing continuously compounding returns or geometric active returns, one is able to conduct analyses in which the active returns can be annualized.  Why, then, would someone prefer one of these frameworks over the other?

In a nutshell, the answer revolves around the intuitive principles that are involved.

• Arithmetic active returns preserve the intuition of the active return as the difference between portfolio and benchmark return.  They also preserve the intuition that a cumulative portfolio return of (say) 50% means that the portfolio's value (in the absence of cashflows) has increased by 50%.  This fundamental correspondence between cumulative return and cumulative value is really quite fundamental to the basic purpose of performance measurement.  But, the drawback is that arithmetic active returns don't permit annualization of the active return.

• Geometric active returns introduce the benefit that they annualize, but this comes by sacrificing the intuitive idea that the active return is the difference between the portfolio return and the benchmark return.

• Continuously compounding returns make all the arithmetic much easier to perform.  However, they sacrifice the intuition between cumulative return and cumulative value (e.g. in Figure 2, after ten years the cumulative return is 146.97%, which means that the cumulative value is $246.97.  But the cumulative continuously compounding return (90.41%) bears no obvious relationship to the cumulative value, except for those rare souls who are able to compute logarithms by mental arithmetic.

None of the three frameworks is objectively superior to the others.  The important thing in deciding which approach to adopt at any time is prioritizing the intuitions that do matter for your audience.

Everybody has a preference about which framework to adopt.  Our preference is to use arithmetic active returns, and simply to learn to live with the fact that they don't annualize.  Is it really so unbearable if one has to do active return analysis (including attribution) on a non-annualized basis?

Annualizing Risk Estimates

The modern concept of investment risk normally corresponds to a standard deviation, or some similar estimate of the statistical dispersion of a set of numbers.  Two very notable examples include:

• volatility of returns (i.e. risk of the investment), and
• tracking error (volatility of active returns).

To estimate the volatility of returns, one might (for example) collect 60 monthly returns for a portfolio, and then calculate the standard deviation (perhaps with the STDEVP function in a spreadsheet).  What does that number tell you?  It is an estimate of the monthly standard deviation of returns, based on the 60 months of data.  Suppose, for example, that the standard deviation of the monthly returns (as calculated by STDEVP) was 3%.  What does that number mean in practice?  Assuming (as one customarily does) that the returns are normally distributed, it means that:

• about 65% of the monthly returns were no further than 3% (one standard deviation) above or below the mean;
• about 95% of the monthly returns were no further than 6% (two standard deviations) above or below the mean; and
• about 99% of the monthly returns were no further than 9% (three standard deviations) above or below the mean.

This is very useful, but can we use this information to get some feeling for the likely variability of annual returns?  Indeed we can.

The annualized standard deviation will simply be the monthly standard deviation multiplied by the square root of 12.

Accordingly, if the monthly standard deviation was 3%, it follows that the annualized standard deviation is approximately 10.39%.

Note that annualizing risk numbers is simpler than annualizing returns in one key respect.  When one has calculated a standard deviation using monthly returns, the annualization simply involves multiplying by root 12.  The number of months one used in calculating the standard deviation makes no difference to the annualization calculation.  This stands in contrast with return annualization, where the length of the period over which a return was calculated can make a big difference to how one would annualize that return.

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