The Zephyr K-Ratio. By Thomas Becker, Ph.D. Zephyr Associates, Inc.

Transcription

1 By Thomas Becker, Ph.D. Itroductio I 996, Lars Kester itroduced the K-Ratio as a complemet to the Sharpe Ratio. With Versio 8., Zephyr Associates makes the K-Ratio available to StyleADVISOR users. This article explais the use, the meaig, ad the exact mathematical defiitio of the ratio. I his 2003 book [], Kester made a modificatio to his origial K-Ratio. For reasos that will be explaied i Sectio 4 below, Zephyr strogly recommeds usig the origial K-Ratio of 996. This poses a amig problem: The term K-Ratio would, by default, refer to Kester s ow most recet versio. To express Zephyr s preferece for the origial versio, we refer to the ratio that we use as the Zephyr K-Ratio. This should ot be iterpreted as Zephyr takig credit for the ratio. What we use is Kester s origial 996 defiitio. We merely preset a argumet for preferrig the origial versio over the modified oe. Sectio gives a brief overview of the Zephyr K-Ratio ad explais why a ivestor may wat to complemet the Sharpe Ratio with the Zephyr K-Ratio. No mathematical prerequisites are required. Sectios 2 goes ito more detail about the Zephyr K-Ratio. It gives a largely visual explaatio of the ratio s meaig. Still, o serious mathematics is required. Sectio 3 presets the exact defiitio of the Zephyr K-Ratio. While o higher mathematics is required, it is assumed that the reader is rather mathematically iclied. I Sectio 4, we give our argumet for preferrig Kester s origial K-Ratio over his 2003 modificatio. Explorig these argumets ca actually be helpful to gai more isight ito the meaig of the K-Ratio. We therefore recommed readig this sectio eve if the reader is already coviced that the Zephyr K-Ratio is the oe that should be used. The mathematical prerequisites for this Sectio are mild.. A Ed User s Perspective of the Zephyr K-Ratio Figure below shows the cumulative retur of three portfolios. It is fair to say that these three portfolios are likely to have differet appeal to differet ivestors. Most ivestors would probably prefer the portfolio that is labeled Cosistet over the other two: it icreases i value over time, ad it does so cosistetly. That is reflected i the fact that the cumulative retur graph of this portfolio slopes upward, ad, o a logarithmic scale, it very closely resembles a straight lie. The Zephyr K-Ratio measures precisely that behavior: the more the cumulative retur graph resembles a straight lie, the higher the

2 K-Ratio, ad the more the cumulative retur graph teds upward, the higher the K-Ratio. For example, the portfolio Cosistet i Figure, whose cumulative retur graph is a almost perfect straight lie, has a Zephyr K-Ratio of 80.3, as opposed to 6.43 for the other two portfolios. Figure The Zephyr K-Ratio should be of particular iterest to those who are curretly usig the Sharpe Ratio i their portfolio aalyses. The Sharpe Ratio is the quotiet of a measure of retur ad a measure of risk. It thus measures the amout of retur per uit of risk. For the measure of retur, the Sharpe ratio uses the portfolio s mea retur over the riskless rate. The measure of risk is the stadard deviatio of the portfolio s retur series. Defiitio of the Sharpe Ratio portfolio's mea retur above riskless ivestmet Sharpe Ratio = stadard deviatio of portfolio's retur series For all its usefuless, the Sharpe Ratio has oe limitatio. The three portfolios whose cumulative retur graphs are show i Figure all have the exact same Sharpe Ratio. That s because all three have the exact same set of mothly returs, amely, 48 returs of % ad 48 returs of -.5%. The oly differece is the order i which these mothly returs occur. The Sharpe Ratio does ot capture that differece: it is etirely isesitive to the order of a portfolio s period returs. The Zephyr K-Ratio aims to remedy this shortcomig of the Sharpe Ratio. Just like the Sharpe Ratio, the Zephyr K-Ratio is a quotiet of a measure of reward ad a measure of risk. However, the measures of reward ad risk are chose i such a way that they take ito accout the order of the period returs, ad thus the shape of the cumulative retur graph. To this ed, a tred lie is fitted to the logarithmic cumulative retur graph. The See Figure 7 below for a example of the tred lie. 2

3 Zephyr K-Ratio the uses as its measure of risk is the degree to which the cumulative retur graph deviates from that tred lie. The measure of reward is the slope of that tred lie. Defiitio of the Zephyr K-Ratio Zephyr K - Ratio = slope of a tred lie fitted to the portfolio's cumulative retur series amout of deviatio of the portfolio's cumulative retur series from the tred lie Sice the cumulative retur graph of the portfolio amed Cosistet i Figure resembles a straight lie much more tha the other two, it will hug its tred lie more closely. This will result i a smaller deomiator ad thus a higher Zephyr K-Ratio. Due to the extreme ature of the example, the differece is actually dramatic: as metioed earlier, Cosistet has a Zephyr K-Ratio of 80.3 as opposed to 6.43 for the other two. 2. The Zephyr K-Ratio i More Detail Figure 2 shows the cumulative retur over time of a hypothetical maager, amed Dow- Up, for a time period of 8 years. Of the 96 mothly returs, the first 48 are all equal to -.5%. The remaiig 48 are all equal to %. As ca be see from the chart, this retur series roughly follows the behavior of the SP 500 over the same time period. Figure 2 Next, we ll cosider two retur series that differ from maager Dow-Up oly by the order i which the mothly returs occur. The first oe, called Cosistet, has a mothly retur stream that alterates betwee % ad -.5%. The secod oe, called Up-Dow, is the mirror image of Dow-Up: it has 48 returs of %, followed by 48 returs that are all -.5%. 3

4 Dow-Up 48 returs of %, the 48 returs of -.5% Cosistet alteratig betwee % ad -.5% Up-Dow 48 returs of %, the 48 returs of.5% Figure 3 Figure 3 shows the resultig cumulative retur over time. It is evidet from the chart that may ivestors will ot perceive these three maagers as equally desirable. However, if we look at the classical absolute MPT statistics (absolute meaig ot versus a bechmark ), we see that these statistics all have the same value for the three maagers: Figure 4 4

5 This is due to the fact that these classical MPT statistics are iheretly uable to distiguish betwee portfolios that differ oly by the order i which the period returs occur. If we were to rate a set of maagers by these statistics, the the three maagers Cosistet, Dow-Up, ad Up-Dow would rate exactly the same, that is, they would be see as idistiguishable ad equally desirable. To see ay differece betwee these maagers at all withi the framework of classical MPT, we d have to look at statistics that are relative to a bechmark: Figure 5 From Figure 5, we see that eve the aualized excess retur vs. the bechmark is the same for the three maagers. It would certaily be desirable to have a precise, mathematical way of describig the differet behavior of our three retur series without referece to a bechmark. Ideed, a umber of statistics have bee developed over the past decade or so that do just that. Here are a few examples: 5

6 Figure 6 These statistics are ofte referred to as Hedge fud statistics. They all deal with a specific aspect of a maager s cumulative retur over time, amely, the occurreces of drawdows ad ruups. The growig popularity of these statistics eve outside the hedge fud world ca be see as a idicatio that the behavior of a maager s cumulative retur series does matter to ivestors more tha is ackowledged by classical MPT. The Zephyr K-Ratio (as set forth by Lars Kester i 996) is the successful attempt to create a ratio that is aalogous to the Sharpe Ratio isofar as it measures the reward per uit of risk. At the same time, the Zephyr K-Ratio is able to capture behavior that is caused by the order of returs. The Zephyr K-Ratio is based o the observatio that the cumulative retur graph of a completely riskless ivestmet, that is, oe with a costat retur stream, is a straight lie whe plotted o a logarithmic scale. Therefore, it is plausible to measure the risk of a maager by the degree to which the logarithmic cumulative retur chart deviates from a straight lie. To this ed, a tred lie is fitted to the maager s cumulative retur graph, as show below 2 : 2 Mathematically speakig, this tred lie is the liear regressio lie for the cumulative retur vs. time. 6

7 Figure 7 It is perhaps oteworthy that these tred lies, although ot ofte ecoutered i portfolio theory, are ofte used to show log-term treds of certai markets such as the stock or bod market. Oe reaso for the use of tred lies is that they take the edpoit sesitivity out of the cumulative retur graph. I other words, a sharp upswig or dowtur i a maager s returs ear the begiig or ed of the time period is ot goig to ifluece the tred lie as much as it iflueces the actual cumulative retur. Oe could thus argue that the slope of the tred lie is a better measure of a maager s logterm retur potetial tha the actual, realized cumulative retur. This is ideed the approach of the Zephyr K-Ratio: the umerator of the Zephyr K-Ratio, that is, the measure of reward, is the slope of the tred lie. It remais to defie the measure of risk, to be used as the deomiator of the Zephyr K- Ratio. As metioed earlier, the cumulative retur graph of a perfectly riskless ivestmet o a logarithmic scale is a straight lie. I particular, the tred lie of the cumulative retur graph coicides with the cumulative retur graph itself. Therefore, it is atural to measure the risk of a arbitrary portfolio by the degree to which its cumulative retur chart deviates from the tred lie. I other words, the more ups ad dows away from the tred lie there are, ad the farther away from the tred lie they go, the more risky the ivestmet is cosidered. Fortuately, mathematicias have already developed a statistic that measures this behavior: i the theory of liear regressio, this is kow as the stadard error of the slope of the tred lie. For a ituitive uderstadig of the Zephyr K-Ratio, it is eough to kow that the more closely the actual cumulative retur graph hugs its tred lie, the smaller the stadard error of the slope will be. We thus have the followig defiitio of the Zephyr K-Ratio: 7

8 Defiitio of the Zephyr K-Ratio Step : Fit a tred lie (liear regressio lie) to the portfolio s logarithmic cumulative retur graph. Step 2: Defie the Zephyr K-Ratio as regressio slope slope of the tred lie K - Ratio = = stadard error of regressio slope amout of deviatio from the tred lie The aalogy to the Sharpe Ratio should be evidet: the Zephyr K-Ratio measures a portfolio s reward per uit of risk. The differece is that here, the measures of risk ad reward are chose i such a way that they refer to the shape ad slope of the cumulative retur graph. The Sharpe Ratio, o the other had, refers to the set of portfolio returs regardless of their order. It turs out that the quotiet that is the defiitio of the Zephyr K-Ratio, besides measurig reward per uit of risk, has a secod meaig that has log bee kow to mathematicias. This secod meaig has to do with hypothesis testig based o give sample data. Suppose the slope of the tred lie is positive, that is, the tred lie poits upward. The the Zephyr K-Ratio is a measure of the cofidece with which the give data supports the followig hypothesis: The maager s cumulative retur is, i the log ru, actually positive. I summary, we see that the Zephyr K-Ratio has three characteristics that distiguish it from the Sharpe Ratio:. The Zephyr K-Ratio s measure of reward uses a tred lie to the logarithmic cumulative retur graph, thus elimiatig the ed poit sesitivity of a portfolio s cumulative retur. 2. The Zephyr K-Ratio s measure of risk measures the deviatio of a portfolio s logarithmic cumulative retur from a straight lie, thus reflectig the fact that movemets away from a straight lie are what ivestors perceive as risk. 3. The Zephyr K-Ratio icorporates a mathematically rigorous measure of cofidece. The higher the Zephyr K-Ratio, the higher the cofidece i a maager s ability to geerate positive retur over time. Let us ow retur to our examples ad look at the Zephyr K-Ratios for the series show i Figure 3. 8

9 Series slope of tred lie (measure of reward) stadard error of tred lie (measure of risk) Zephyr K-Ratio (reward / risk) Cosistet Dow-Up Up-Dow S&P Table As was to be expected, the portfolio amed Cosistet, whose cumulative retur graph almost equals a straight lie, has a very large Zephyr K-Ratio. The portfolios Up-Dow ad Dow-Up, by cotrast, have a K-Ratio that resembles that of real world portfolios, such as the S&P 500. It is perhaps oteworthy that due to the symmetry betwee Up- Dow ad Dow-Up, the Zephyr K-Ratio does ot distiguish betwee these two. To capture the differeces betwee these two i a sigle statistic, you d have to look at thigs like drawdows, recovery legths, pai idex, etc. What this goes to show is that while each of these statistics has its merits, o oe umber is the silver bullet that makes the rest superfluous. 3. The Mathematics of the Zephyr K-Ratio The (i our opiio extraordiary) origiality ad igeuity of Lars Kester s K-Ratio lies i the idea of applyig a liear regressio lie to a portfolio s cumulative retur series ad the usig the slope of that lie as a measure of reward ad the stadard error of the slope (that is, the degree to which the data deviates from a straight lie) as a measure of risk: Zephyr K - Ratio = slope stadard error of slope After that, there is o eed to ivet ay ew mathematics: the exact mathematical defiitios of the slope ad its stadard error have log existed i the theory of liear regressio. To calculate the Zephyr K-Ratio, oe should first replace the dates o the horizotal axis of the portfolio s cumulative retur graph with cosecutive itegers startig at 0. Ay other equidistat series of umbers would of course work just as well, but a sequece of itegers will give us aturally ormalized values for slope ad stadard error. With these itegers as idepedet x-values ad the correspodig cumulative retur values as depedet y-values, oe ca ow calculate the slope of the regressio lie, that is, the umerator of the Zephyr K-Ratio, by the well-kow formula ( xi x)( yi y) slope = i= () 2 ( x i x) i= 9

10 where x ad y deote the mea of the x- ad y-values, respectively. Statistics software packages will usually have this as a built-i fuctio; for example, i Microsoft Excel it is the fuctio SLOPE. The stadard error of the slope, that is, the deomiator of the Zephyr K-Ratio, ca be calculated from the x- ad y-values by the formula y) i i 2 i= ( y i y) i= 2 ( xi x) i= stadard error of slope = (2) 2 ( 2) ( xi x) i= ( x x)( y The followig equivalet formulas ca be useful whe usig statistics software that does ot have a fuctio for the stadard error of the slope, but has related statistics such as the stadard error of the estimate. stadard error of estimate stadard error of slope = (3) 2 ( x i x) Formula (3) above is particularly coveiet whe usig Microsoft Excel; i Excel, it traslates ito STEYX(y-values, x-values) / SQRT(DEVSQ(x-values)) The stadard error of the estimate ca also be calculated directly from the y-values ad their estimates: i= 2 ) ( yi yi ) i= stadard error of estimate = ( 2) 2 (4) Where y ) i is the estimated value of, that is, y i yˆ = itercept + slope (5) i x i Formula (4) i cojuctio with formula (3) is thus coveiet if itercept ad slope are available, but oe of the more sophisticated statistics are. Armed with the slope ad the stadard error of the slope, we obtai the Zephyr K-Ratio as Zephyr K - Ratio = slope stadard error of slope It so happes that whe it is o-egative, this quatity is also the t-score for the rejectio of the hypothesis, The true regressio lie has slope less tha or equal to 0. I other words, the Zephyr K-Ratio has two meaigs: o the oe had, it measures the 0

11 portfolio s reward per uit of risk for a certai rather plausible defiitio of risk ad reward. O the other had, it is also a measure of the cofidece with which we may assume that the true tred lie for the portfolio s cumulative retur is ot flat or poitig dowward. The iterpretatio of the Zephyr K-Ratio as a t-score does of course imply a certai assumptio of ormality. The assumptio is that the estimatio error of the liear regressio o the cumulative retur data is ormally distributed. It is a commo assumptio i Moder Portfolio Theory that the raw period returs are ormally distributed. There has bee ample research ito that assumptio; however, we are ot aware of ay research regardig the assumptio that the estimatio errors of the liear regressio are ormally distributed. Oe should perhaps ot speculate before empirical data has bee examied, but it seems plausible that ormality of the estimatio errors is o less likely tha ormality of the logarithmic period returs. 4. The Zephyr K-Ratio vs. the K-Ratio The Zephyr K-Ratio as we have explaied it i the previous three sectios is the origial K-Ratio that Lars Kester itroduced i 996. I his 2003 book ([]), Kester made the followig modificatio to the defiitio of the K-Ratio: Kester's K - Ratio of slope 2000 book = (stadard error of slope) (umber of data poits) The followig ratioale for this modificatio is give o p. 87 of Kester s book: By dividig by the umber of data poits, we ormalize the K-Ratio to be cosistet regardless of the periodicity used to calculate its compoets. I this sectio, we will preset our argumets for usig the origial versio of the K-Ratio. As before, we will refer to the origial versio as the Zephyr K-Ratio. First off, whe makig ay chage to the Zephyr K-Ratio, oe loses the property that the ratio, whe it is o-egative, is equal to the t-score for rejectig the hypothesis the slope of the tred retur lie is ot positive. That property is certaily desirable to have; this is a reaso to be wary of ay suggestio to modify to the ratio, regardless of what it is. We preset two argumets for preferrig the origial K-Ratio over the modified versio. The first oe cocers the iteded effect of the modificatio, amely, makig portfolios comparable across differet periodicities. We will argue that the origial K-Ratio already guaratees that comparability; o modificatio is ecessary i our opiio. The secod argumet cocers the effect of the modificatio o comparisos betwee portfolios with the same periodicity. We will argue that the modificatio has a detrimetal effect i that situatio. To demostrate the effect of the modificatio to the K-Ratio o data with differet periodicity, we will covert the three portfolios ad the market bechmark that we have bee usig throughout (see Figure 3) to quarterly data ad ispect the way the two K- Ratios are affected by this chage i periodicity. Below is the aalog to Figure 3, with all series coverted to quarterly.

12 Figure 8 We see that this is the same picture as Figure 3, except that there is less graularity. More precisely, there are oly a third of the data poits, with o iformatio o what is happeig i betwee. Here are the correspodig K-Ratios: Zephyr K-Ratio K-Ratio Series mothly data quarterly data mothly data quarterly data Cosistet Dow-Up Up-Dow S&P Table 2 We see that the Zephyr K-Ratio (Kester s origial 996 versio) decreases sigificatly as we pass from mothly to quarterly data. Moreover, this is ot just ay decrease: the chage reflects, with the mathematical precisio of Studet s t-test, the loss of sigificace that is caused by the higher graularity of the quarterly data. We caot see aythig wrog with that. The modified, more recet K-Ratio, o the other had, actually icreases as we pass from mothly to quarterly data. The icrease is sigificat isofar as the K-Ratio of the quarterly S&P 500 is higher tha the mothly K-Ratio of the Up- Dow portfolio, while withi the same periodicity, the S&P 500 scores lower tha Up- Dow (see shaded cells i Table 2). If the adjustmet made for the modified K-Ratio were such that mothly ad quarterly data ow gave the same or roughly the same values, the we could see how a argumet for the adjustmet could be made. We would still prefer the Zephyr K-Ratio, where the loss of sigificace is reflected i the value of the ratio, but we could see how someoe else would wat the adjustmet. However, a adjustmet that actually shows 2

13 sigificatly higher values for quarterly data as opposed to mothly data is somethig that we advise agaist. Our secod argumet agaist modifyig the origial K-Ratio cocers its effect o comparisos betwee portfolios with the same periodicity. Remember, the iteded effect of the modificatio cocers the compariso betwee portfolios of differet periodicity. But it is clear that dividig by the umber of data poits has a o-trivial effect o comparisos betwee portfolios with the same periodicity as well, amely, whe comparig portfolios with differet amout of data. Clearly, it is preferable to oly ever compare portfolios with the same amout of data. However, the Zephyr K-Ratio, because of its coectio to the Studet t-test, does allow comparisos across differet amouts of data. Figure 9 shows a example. Figure 9 Here, Fud A has about twice as much data as Fud B. The slopes of the tred lies of the graphs of Fud A ad Fud B are about equal (.22 ad.364, respectively). The graph of Fud A has visibly more deviatio from a straight lie tha that of Fud B. However, because of the fact that Fud A has twice as much data to back up its tred lie, the stadard errors of the tred lies are about the same ( ad.00268, respectively). As a cosequece, the Zephyr K-Ratio clocks i at almost the same value for both fuds, amely, ad 62.94, respectively. All this is mathematically ad statistically perfectly soud, ad it coveys meaigful ad useful iformatio about the fuds cumulative retur graphs. The modified K-Ratios, as show i the legeds of Figure 9, o the other had, are.770 ad I other words, Fud B ow has more tha twice the K-Ratio of Fud A. We simply do ot see how oe could explai such a dramatic chage i the relatioship betwee two mothly fuds o accout of a adjustmet whose itetio was to esure comparability betwee portfolios whose data has differet periodicity. 3

14 Refereces [] Lars Kester, Quatitative Tradig Strategies: Haressig the Power of Quatitative Techiques to Create a Wiig Tradig Program, McGraw-Hill Traders Edge Series,