On the Relationship between Arithmetic and Geometric Returns

Transcription

1 O the Relatioship betwee Arithmetic ad Geometric Returs Dimitry Midli, ASA, MAAA, PhD Presidet CDI Advisors LLC August 14, 011 Copyright 011, CDI Advisors LLC

2 Arithmetic ad geometric averages are importat ad somewhat cotroversial measuremets of the past ad future ivestmet returs. Numerous publicatios have discussed the pros ad cos of these measuremets as well as relatioships betwee them. Yet, the cotroversy surroudig arithmetic ad geometric averages appears to persist. Vital decisios for pesio plas, for example, are ofte based o estimates of future ivestmet returs. It is imperative to utilize appropriate measuremets of returs ad apply them properly i a forward-lookig maer. I particular, it is importat to distiguish arithmetic ad geometric averages for asset classes ad portfolios as well as specify the relatioships betwee these averages. A popular formula preseted i several publicatios stipulates that the geometric average is approximately equal to the arithmetic average mius half of the variace. However, a proper justificatio for this formula ad the assessmet of the quality of this approximatio are hard to fid. Moreover, this popular formula may sigificatly uderestimate the geometric retur i practical applicatios. Recogizig the eed for clarity i this area ad the desirability of alterative solutios, this paper presets three additioal formulas for approximate calculatios of geometric averages ad provides simple quatitative explaatios for all four formulas. The results of these formulas are compared to historic geometric averages ad to each other. The paper shows i particular, that the three other formulas are ofte superior to the popular oe. This author hopes that this paper would be useful to practitioers i clarifyig the relatioship betwee arithmetic ad geometric averages as well as their pros ad cos. Arithmetic ad geometric averages are some of the most commoly utilized measuremets of ivestmet returs. 1 Despite their extesive utilizatio, however, there has bee a great deal of cotroversy ad cofusio surroudig these measuremets. A umber of publicatios have attempted to clarify the issues related to arithmetic ad geometric averages, but the cotroversy ad cofusio appear to persist. Accordig to de La Gradville [1998], A umber of serious, widely held errors ad miscoceptios about the log-term expected rate of retur eed to be dispelled. Oe of these miscoceptios is related to the calculatio of the geometric retur of a portfolio. Accordig to several publicatios, the geometric average is approximately equal to the arithmetic average mius half of the variace. Despite the popularity of this formula, few publicatios attempt to justify this approximatio ad gauge its quality. As demostrated i this paper, this formula is the result of a couple of relatively crude approximatios. More troublig, this formula teds to uderestimate the geometric average. This tedecy, i particular, should be of cocer to pesio plas that employ geometric portfolio returs to determie their discout rates. Arithmetic vs. Geometric Returs 8/14/011

3 Calculatios of geometric returs ca have a sigificat impact o asset allocatio decisios. A pesio pla, for example, may select the lowest risk portfolio with the geometric retur equal to the pla s discout rate (by itself, a idea of questioable utility). A calculatio that uderestimates the geometric retur would force the pla to eedlessly icrease the riskiess of the portfolio i order to hit the target retur. I other words, the pla would take additioal risk solely due to questioable math. Ulike geometric averages, arithmetic averages are relatively easy to use. I particular, the arithmetic retur for a portfolio is equal to the weighted average of the arithmetic returs of uderlyig asset classes. This rule, however, does ot work for geometric returs a weighted average of asset classes geometric returs is ot equal to the geometric retur of the correspodig portfolio. Therefore, there is a eed to covert arithmetic portfolio returs to the geometric oes, ad vice versa. Attemptig to establish a better uderstadig of the relatioship betwee arithmetic ad geometric averages, this paper provides a simple quatitative explaatio for the abovemetioed popular formula; presets three more formulas that coect arithmetic ad geometric returs; develops coectios betwee all four formulas; demostrates that the popular formula teds to produce sub-optimal results; idetifies the formula that should be expected to produce better results. Geometric ad Arithmetic Averages: Retur Series For a series of returs, this sectio develops four formulas that coect arithmetic ad geometric averages. The arithmetic average A of the series of returs r 1, of the series:, r is defied simply as the average value 1 rk k 1 A (1) Oe of the mai advatages of the arithmetic average is it is a ubiased estimate of the retur. Oe of the mai disadvatages of the arithmetic average is the probability of achievig the arithmetic average retur may be usatisfactory. I other words, as a predictio of future returs, the arithmetic average may be too optimistic. Aother disadvatage of the arithmetic retur is its icompatibility with the startig ad edig asset values. Specifically, the startig asset value multiplied by the compouded arithmetic retur factor 1 A is greater tha the edig asset value. 3 The cocept of geometric average is specifically desiged to correct this problem. If A 0 ad A are the startig ad edig asset values correspodigly, the, by defiitio, Arithmetic vs. Geometric Returs 3 8/14/011

4 A0 1 r1 1 r A () The geometric average G is defied as the rate of retur that coects the startig ad edig asset values if assumed i all periods. Namely, the startig asset value multiplied by the compouded retur factor 1 G is equal to the edig asset value: A G A (3) 0 1 Combiig () ad (3), we get a stadard textbook defiitio of the geometric average G: G rk (4) k1 Let us try to determie how the arithmetic ad geometric averages relate to each other. Firstly, it is well-kow that the arithmetic average is always greater or equal to the arithmetic average: 4 A G (5) Followig a log-established traditio, oly the first two momets of the uderlyig variables will be used i developig relatioships betwee A ad G. Therefore, the relatioships betwee A ad G cosidered i this paper also ivolve variace V. Let us preset four formulas that coect arithmetic ad geometric returs ad specify the required approximatios to derive each formula. 5 Formula # 1 (A1) Let us make the followig two approximatios o the right side of (4). 1. For all k, replace each factor 1 1 r k by its Maclauri series expasio up to the secod degree.. I the resultig product, igore all summads of the third degree ad higher. See the Appedix for more details. After these approximatios, the right side of (4) becomes A V, where V is the sample variace defied as 6 1 k k 1 (6) V r A Therefore, we get the followig relatioship (deoted as (A1) throughout this paper): Arithmetic vs. Geometric Returs 4 8/14/011

5 G A V (A1) Relatioship (A1) is the popular formula discussed above; it is well-kow amog practitioers. 7 Formula # (A) Note that (4) implies k (7) k1 1 G 1 r Let us make the followig two approximatios o the right side of (7). 1. For all k, replace each factor 1 r k by its Maclauri series expasio up to the secod degree.. I the resultig product, igore all summads of the third degree ad higher. After these approximatios, the right side of (7) becomes 1A V, where V is defied i (6). Therefore, we get the followig relatioship (deoted as (A) throughout this paper): 1 1 G A V (A) Relatioship (A) is ot as well-kow as (A1) amog practitioers, eve though it has bee kow for a log time. 8 Iterestigly, formula (A) is exact whe the retur series has just two poits (see the Appedix for more details). Formula # 3 (A3) Note that (4) implies 1 l 1 G l 1 rk (8) k 1 O the right side of (8), let us replace each summad l 1 r aroud A up to the secod degree. See the Appedix for more details. k by its Taylor series expasio After this approximatio, the right side of (8) becomes l 1 A 1 V 1 A. Therefore, we get the followig relatioship (deoted as (A3) throughout this paper): Arithmetic vs. Geometric Returs 5 8/14/011

6 or, equivaletly, l 1 G l 1 A 1 V 1 A (A3) 1 G 1 Aexp 1 V 1 A (A3) I this author s experiece, relatioship (A3) is little kow amog practitioers, eve though it has bee preseted i some publicatios. 9 Formula # 4 (A4) I (A3), usig approximatio l 1x x, let us replace V1 A with l 1 V1 a result, we get the followig relatioship (deoted as (A4) throughout this paper): 1 1 G 1 A 1V 1 A (A4) Arithmetic vs. Geometric Returs 6 8/14/011 A. As As demostrated i the ext sectio, this relatioship is exact whe arithmetic ad geometric averages (meas) are defied for a logormal distributio. It should be oted that there is a sequece of approximatios ad simplificatios that tur (A3) ito (A4), as preseted above, the tur (A4) ito (A), ad the tur (A) ito (A1) (see the Appedix for more details). It is also worth oticig that the geometric average estimate (A4) is always greater tha (A3), which i tur is always greater tha (A). 10 Loosely speakig, (A) < (A3) < (A4) Iterestigly, the geometric average estimate (A) is ot ecessarily greater tha (A1), although this is true for most practical examples. 11 See the Appedix for more details. To recap, formulas (A1) (A4), which work for ay retur sample, establish approximate relatioships betwee the geometric ad arithmetic averages ad the variace. These formulas are based o Taylor series expasios up to the secod degree. Geometric ad Arithmetic Meas: Retur Distributios The previous sectio developed the relatioships betwee the arithmetic ad geometric averages defied for a series of returs. This sectio, i cotrast, develops similar results whe the distributio of retur is give. To avoid cofusio with the previous sectio, this sectio defies arithmetic ad geometric meas (rather tha averages), which are deoted as E ad M correspodigly (as opposed to averages A ad G i the previous sectio). I this case, the arithmetic mea E of retur R is defied as the expected value of R: 1

7 E E R The geometric mea M of retur R is defied as follows: M exp E l 1 R 1 (9) The primary motivatio for these defiitios comes from the fact that the arithmetic ad geometric meas are the limits of appropriately selected series of arithmetic ad geometric averages, as demostrated below. Specifically, let us defie arithmetic averages idepedet idetically distributed retursr k : A ad geometric averages G for a series of A 1 rk k 1 (10) G rk (11) k1 Accordig to the Law of Large Numbers (LLN), k 1 A coverges to E. Also, from (11) we have 1 l 1 G l 1 rk (1) 1 r Agai, accordig to LLN, l 1 k ad, therefore, l 1 G value E l 1 R To recap, k 1. Cosequetly, G coverges to E R coverge to the expected exp l 1 1, which is equal to M. A coverges to E ad G coverges to M whe teds to ifiity. As discussed i the previous sectio, relatioships (A1) (A4) are true for A ad G, where sample variace defied similar to (6): 1 k k 1 (13) V r A V is Sice series V coverges to the variace of returs V whe teds to ifiity, relatioships (A1) (A4) are true for E ad M as well: Arithmetic vs. Geometric Returs 7 8/14/011

8 M E V (A1) 1 1 M E V (A) 1 M 1 Eexp 1 V 1 E (A3) 1 1 M 1 E 1V 1 E (A4) It should be emphasized that, as a geeral priciple, oe should avoid approximatios wheever direct calculatios are possible. 13 As demostrated below, (A4) represets the exact relatioship betwee the arithmetic ad geometric meas uder commo assumptios. Let us assume that the retur factor 1 R is logormally distributed, which meas l 1 R is ormally distributed with parameters ad. Uder this assumptio, the followig formulas are well-kow: E exp (14) 1M exp (15) V exp exp 1 (16) It easily follows from (14)-(16) that the geometric mea is calculated as (A4): 1 1 M 1 E 1V 1 E (A4) Thus, the relatioship (A4) is exact uder the logormal assumptio. If there is a eed to calculate the arithmetic mea whe the geometric mea ad the variace are give, the, from (A4), the arithmetic mea is calculated as follows: 1 1 4V 1 E 1 M 1 (17) 1 M Which formula amog (A1) (A4) should work better? The utilizatio of idepedet idetically distributed logormal retur factors may be a reasoable forward-lookig assumptio. Arithmetic vs. Geometric Returs 8 8/14/011

9 Therefore, formula (A4) may be the right choice for forward-lookig aalysis. A priori, however, this is ot ecessarily the case for historical data. The ext sectio explores this issue. Historical Arithmetic ad Geometric Averages This sectio presets the arithmetic ad geometric averages for historical data ad aalyzes the quality of the approximatios discussed i prior sectios. The sectio aalyzes three sets of historical data: equity real rates of retur (Exhibit 1), 15 equity premium relative to bills (Exhibit ), ad equity premium relative to bods (Exhibit 3) from 1900 to 005. Each dataset cotais the arithmetic averages, geometric averages ad stadard deviatios calculated exactly. For each dataset, we calculate four approximatios of the geometric averages (A1) (A4) ad compare the approximatios to the actual values. Exhibit 1 Equity Real Rates of Retur, Data Geometric Average Approximatio Arithmetic Average Stadard Deviatio Geometric Average A1 A A3 A4 Best Worst Australia 9.1% 17.64% 7.70% 7.65% 7.78% 7.79% 7.81% A1 A4 Belgium 4.58%.10%.40%.14%.%.7%.3% A4 A1 Caada 7.56% 16.77% 6.4% 6.15% 6.4% 6.6% 6.8% A A1 Demark 6.91% 0.6% 5.5% 4.86% 4.97% 5.01% 5.04% A4 A1 Frace 6.08% 3.16% 3.60% 3.40% 3.5% 3.58% 3.64% A3 A1 Germay* 8.1% 3.53% 3.09%.9% 3.0% 3.43% 3.63% A A4 Irelad 7.0%.10% 4.79% 4.58% 4.71% 4.76% 4.81% A4 A1 Italy 6.49% 9.07%.46%.6%.45%.60%.73% A A4 Japa 9.6% 30.05% 4.51% 4.74% 5.05% 5.0% 5.35% A1 A4 Netherlads 7.% 1.9% 5.6% 4.95% 5.09% 5.13% 5.17% A4 A1 Norway 7.08% 6.96% 4.8% 3.45% 3.63% 3.74% 3.84% A4 A1 South Africa 9.46%.57% 7.5% 6.91% 7.11% 7.16% 7.0% A4 A1 Spai 5.90% 1.88% 3.74% 3.51% 3.6% 3.66% 3.71% A4 A1 Swede 10.07%.6% 7.80% 7.51% 7.7% 7.77% 7.8% A4 A1 Switzerlad 6.8% 19.73% 4.48% 4.33% 4.43% 4.46% 4.49% A4 A1 U.K. 7.36% 19.96% 5.50% 5.37% 5.49% 5.5% 5.55% A A1 U.S. 8.50% 0.19% 6.5% 6.46% 6.60% 6.64% 6.67% A1 A4 World 7.16% 17.3% 5.75% 5.68% 5.77% 5.78% 5.80% A A1 World ex-u.s. 7.0% 19.79% 5.3% 5.06% 5.17% 5.1% 5.4% A4 A1 * excludes Source: Dimso, Marsh, ad Stauto (006). Arithmetic vs. Geometric Returs 9 8/14/011

10 Exhibit Equity Premium Relative to Bills, Data Geometric Average Approximatio Arithmetic Average Stadard Deviatio Geometric Average A1 A A3 A4 Best Worst Australia 8.49% 17.00% 7.08% 7.05% 7.15% 7.17% 7.18% A1 A4 Belgium 4.99% 3.06%.80%.33%.43%.49%.55% A4 A1 Caada 5.88% 16.71% 4.54% 4.48% 4.55% 4.57% 4.59% A A1 Demark 4.51% 19.85%.87%.54%.61%.64%.67% A4 A1 Frace 9.7% 4.19% 6.79% 6.34% 6.56% 6.6% 6.69% A4 A1 Germay* 9.07% 33.49% 3.83% 3.46% 3.80% 4.05% 4.7% A A4 Irelad 5.98% 0.33% 4.09% 3.91% 4.01% 4.05% 4.08% A4 A1 Italy 10.46% 3.09% 6.55% 5.31% 5.70% 5.90% 6.07% A4 A1 Japa 9.84% 7.8% 6.67% 5.97% 6.6% 6.37% 6.48% A4 A1 Netherlads 6.61%.36% 4.55% 4.11% 4.4% 4.9% 4.34% A4 A1 Norway 5.70% 5.90% 3.07%.35%.48%.57%.66% A4 A1 South Africa 8.5%.09% 6.0% 5.81% 5.97% 6.0% 6.06% A4 A1 Spai 5.46% 1.45% 3.40% 3.16% 3.6% 3.30% 3.34% A4 A1 Swede 7.98%.09% 5.73% 5.54% 5.70% 5.74% 5.79% A3 A1 Switzerlad 5.9% 18.79% 3.63% 3.5% 3.60% 3.63% 3.65% A3 A1 U.K. 6.14% 19.84% 4.43% 4.17% 4.7% 4.30% 4.33% A4 A1 U.S. 7.41% 19.64% 5.51% 5.48% 5.60% 5.63% 5.66% A1 A4 World 5.93% 19.33% 4.3% 4.06% 4.15% 4.18% 4.1% A4 A1 World ex-u.s. 6.07% 16.65% 4.74% 4.68% 4.76% 4.77% 4.79% A A1 * excludes Source: Dimso, Marsh, ad Stauto (006). Arithmetic vs. Geometric Returs 10 8/14/011

11 Exhibit 3 Equity Premium Relative to Bods, Data Geometric Average Approximatio Arithmetic Average Stadard Deviatio Geometric Average A1 A A3 A4 Best Worst Australia 7.81% 18.80% 6.% 6.04% 6.16% 6.18% 6.1% A4 A1 Belgium 4.37% 0.10%.57%.35%.4%.45%.49% A4 A1 Caada 5.67% 17.95% 4.15% 4.06% 4.13% 4.16% 4.18% A3 A1 Demark 3.7% 16.18%.07% 1.96% 1.99%.01%.03% A4 A1 Frace 6.03%.9% 3.86% 3.55% 3.66% 3.71% 3.76% A4 A1 Germay* 8.35% 7.41% 5.8% 4.59% 4.83% 4.94% 5.04% A4 A1 Irelad 5.18% 18.37% 3.6% 3.49% 3.56% 3.59% 3.61% A4 A1 Italy 7.68% 9.73% 4.30% 3.6% 3.49% 3.65% 3.80% A4 A1 Japa 9.98% 33.06% 5.91% 4.5% 4.89% 5.1% 5.3% A4 A1 Netherlads 5.95% 1.63% 3.86% 3.61% 3.7% 3.76% 3.81% A4 A1 Norway 5.6% 7.43%.55% 1.50% 1.6% 1.75% 1.86% A4 A1 South Africa 7.03% 19.3% 5.35% 5.16% 5.7% 5.30% 5.33% A4 A1 Spai 4.1% 0.0%.3%.17%.3%.7%.31% A4 A1 Swede 7.51%.34% 5.1% 5.01% 5.16% 5.1% 5.6% A3 A1 Switzerlad 3.8% 17.5% 1.80% 1.75% 1.78% 1.80% 1.83% A3 A1 U.K. 5.9% 16.60% 4.06% 3.91% 3.97% 3.99% 4.01% A4 A1 U.S. 6.49% 0.16% 4.5% 4.46% 4.56% 4.60% 4.63% A A4 World 5.18% 15.19% 4.10% 4.03% 4.08% 4.09% 4.10% A4 A1 World ex-u.s. 5.15% 14.96% 4.04% 4.03% 4.08% 4.09% 4.10% A1 A4 * excludes Source: Dimso, Marsh, ad Stauto (006). Exhibits 1-3 cotai data for 17 coutries plus two totals 19 data series overall. For each data series, we measure the distace betwee approximatios (A1) (A4) of the geometric average ad the actual geometric average. The approximatio that is closest to actual value is raked the best; the farthest is raked the worst. For example, lookig at the data for Australia i Exhibit 3, (A1) is 18 bps away from the actual value (6.04% vs. 6.%), (A) is 6 bps away from the actual value (6.16% vs. 6.%), (A3) is 4 bp away from the actual value (6.18% vs. 6.%), ad (A4) is 1 bp away from the actual value (6.1% vs. 6.%). Therefore, (A4) is raked the best ad (A1) is raked the worst. For each exhibit ad each approximatio, we cout the umber of data series for which the approximatio is the best ad the worst. These couts are preseted i Exhibit 4. Arithmetic vs. Geometric Returs 11 8/14/011

12 Exhibit 4 Approximatio Rakigs A1 A A3 A4 Best Worst Best Worst Best Worst Best Worst Equity Premium Relative to Bods (Exhibit 1 ) Equity Premium Relative to Bills (Exhibit ) Equity Real Rates of Retur (Exhibit 3 ) Total # Total % 11% 8% 16% 0% 11% 0% 63% 18% Exhibit 5 A4 Compared to A1-A3 A4 is better tha A1 A4 is better tha A A4 is better tha A3 Equity Premium Relative to Bods (Exhibit 1 ) Equity Premium Relative to Bills (Exhibit ) Equity Real Rates of Retur (Exhibit 3 ) Total # Total % 8% 67% 63% Arithmetic vs. Geometric Returs 1 8/14/011

13 Overall, (A4) largely looks better tha (A1) (A3), as it is the best approximatios i 63% cases (see Exhibit 4). (A1) largely looks worse tha (A) (A4), as it is the worst approximatio i 8% cases (see Exhibit 4). (A) ad (A3) are mostly i-betwee, ad they are ever the worst. The results withi Exhibits 1-3 are cosistet with this coclusio. Exhibit 5 cotais the results of direct comparisos of (A4) to (A1) (A3) for each data series. (A4) works better tha (A1) i 8% cases, better tha (A) i 67% cases, ad better tha (A3) i 63% cases. The results withi Exhibits 1-3 are cosistet with this coclusio. While the results of (A4) are ot vastly superior, they do demostrate a clear patter. Aother clear patter is the tedecy of (A1) to uderestimate the geometric retur. It happes i 56 out of 57 data series, ad, sometimes, by a sigificat margi. Yet, (A1) should ot be dismissed easily, ot so fast, at least. (A1) provides the best match for the U.S. data i Exhibits 1 ad ; it is a close secod i Exhibit 3, i which it is also the best match for the World ex. U.S. data series. The results of (A1) (A4) ca occasioally be far apart, especially for high volatility portfolios. Let us cosider the followig example. Exhibit 6 shows the data for the U.S. stocks divided ito large ad small stocks (as defied i the source). For the large stocks, (A1) is the best ad (A4) is the worst approximatio. For the small stocks, the opposite is true (A4) is the best ad (A1) is the worst approximatio. But (A1) is ot just the worst approximatio amog the four it is astoudig 16 bps lower tha the actual value. (A) is 100 bps closer, but still disappoitig 6 bps below the actual value. (A3) is aother 37 bps closer, but still 5 bps below the actual value. (A4) is the oly oe that provides a decet approximatio. Exhibit 6 U.S. Large ad Small Stocks Large Stocks Data Geometric Average Approximatio Arithmetic Average 1.49% A1 A A3 A4 Geometric Average 10.51% 10.43% 10.64% 10.67% 10.70% Stadard Deviatio 0.30% Small Stocks Data Geometric Average Approximatio Arithmetic Average 18.9% A1 A A3 A4 Geometric Average 1.19% 10.57% 11.57% 11.94% 1.6% Stadard Deviatio 39.8% Source: Bodie [004], Table 5.3, p Arithmetic vs. Geometric Returs 13 8/14/011

14 Coclusio This paper aalyzes the followig four relatioships betwee arithmetic ad geometric averages (meas) that work for ay retur sample: G A V (A1) 1 1 G A V (A) 1 G 1 Aexp 1 V 1 A (A3) 1 1 G 1 A 1V 1 A (A4) Whe the retur factor is logormally distributed (a commo forward-lookig assumptio), the relatioship (A4) is exact: 1 I this case, there is o eed for approximatios. 1 M 1 E 1V 1 E (A4) Relatioship (A1) is the simplest, popularized i may publicatios, but usually sub-optimal ad teds to uderestimate the geometric retur. Relatioships (A) (A4) are slightly more complicated, but, i most cases, should be expected to produce better results tha (A1). Overall, (A4) looks like a wier it works better i both backward- ad forward lookig settigs. Still, (A1) (A3) should ot be dismissed summarily, ad more research is eeded to determie the coditios uder which a particular formula may work better. For a practitioer, it may be a good idea to compare the results of all four formulas. There may be sigificat disparities amog these approximatios, especially for high volatility portfolios. Both arithmetic averages ad geometric averages are required for a clear uderstadig of ivestmet returs. This author hopes that this paper would be useful to practitioers i clarifyig the relatioships betwee these averages as well as their pros ad cos. Arithmetic vs. Geometric Returs 14 8/14/011

15 APPENDIX: The Developmet of Formulas (A1) (A3) ad Trasitios from (A4) to (A1) This Appedix cotais the techical details of the developmet of formulas (A1) (A4). The arithmetic average A of a series of returs r 1, r is defied as the average value of the series:, 1 rk k 1 A (1) The geometric average G of a series of returs r 1,, r is defied as follows: Sample variace V is defied as G rk (4) k1 Formula # 1 (A1) 1 k k 1 (6) V r A Let us take the Maclauri series expasio for the fuctio f x x 1 degree ad igore the remaider: 1 1 x 1 x x 1 up to the secod 1 1 (18) Substitutig (18) ito (4) ad igorig summads of the third degree ad higher, we get (A1): 1 1 G 11 rk r k k rk r krl r k k 1 k l k 1 AV (19) Formula # (A) From the defiitio of G, we get k (0) k1 1 G 1 r Arithmetic vs. Geometric Returs 15 8/14/011

16 Let us take the Maclauri series expasio for the fuctio degree ad igore the remaider: 1 x 1 x x f x x 1 up to the secod (1) Substitutig (1) ito (0) ad igorig summads of the third degree ad higher, we get (A): Formula # 3 (A3) 1 G 1 xk x k k xk x k xl x k i1 k l k 1 1 A V () Let us take the Taylor series expasio for the fuctio f x l 1 x secod degree ad igore the remaider: aroud poit A up to the x A x A l 1 x l 1 A 1 A 1 A (3) Usig (3) o the right side of (8), we get (A3): 1 l 1 G l 1 rk k l 1 A r A r A 1 A 1 V l 1 A 1 A k k (A3) k1 A k1 Now, below are the sequeces of approximatios ad simplificatios that tur (A3) ito (A4), the tur (A4) ito (A), ad the tur (A) ito (A1) as well as the proof that (A3) (A4) Trasitio (A) < (A3) < (A4) Arithmetic vs. Geometric Returs 16 8/14/011

17 The trasitio from (A3) to (A4) is achieved via replacig V1 A with l 1V1 (usig approximatio l 1 x x ). Notig that 1x exp x, we get Aexp V 1 A 1 A 1V 1 A which meas the geometric average estimate (A4) is o less tha (A3). (A4) (A) Trasitio 1 The trasitio from (A4) to (A) is achieved via replacig 1 V 1A with 1V1A (usig approximatio 1 x 1 1 x). Notig that x x, we get A 1V 1 A 1 A 1V 1 A 1 A V which meas the geometric average estimate (A3) is o less tha (A). (A) (A1) Trasitio The trasitio from (A) to (A1) is achieved via replacig 1 G ad 1 A correspodigly (usig approximatio x 1 G 1 1 x). A with 1 Aad The geometric average estimate (A1) is ot ecessarily less tha (A), although this is true for all data series i Exhibits 1-3 ad most practical applicatios. For example, if r1 99% ad r 100%, the the geometric average estimate (A1) is equal to -49%, ad the geometric average estimate (A) is equal to -86% (which is equal to the actual geometric average of this retur series, see below). Fially, formula (A) is exact whe the retur series cotais just two poits, due to the followig r1 r r1 r r1 r r1 r G 1 r1 1 r 1 r1 r r1 r r1 r 1 A V Arithmetic vs. Geometric Returs 17 8/14/011

18 REFERENCES Bodie, Z., Kae, A., Marcus, A.J. [1999]. Ivestmets, McGraw-Hill, 4 th Ed., Bodie, Z., Kae, A., Marcus, A.J. [004]. Essetials of Ivestmets, McGraw-Hill, d Ed., 004. Booth, D.G., Fama, E.F. [199]. Diversificatio Returs ad Asset Cotributios, Fiacial Aalysts Joural, May/Jue 199, p de La Gradville, O. [1998]. The Log-Term Expected Rate of Retur: Settig It Right, Fiacial Aalysts Joural, November/December 1998, p Dimso, E., Marsh, P., Stauto, M. [006]. The Worldwide Equity Premium: A Smaller Puzzle, Lodo Busiess School, April 006, de La Gradville, O., Pakes, A.G., Tricot, C. [00]. Radom Rates of Growth ad Retur: Itroducig the Expo-Normal Distributio, Applied Stochastic Models i Busiess ad Idustry, 00; 18:3 51. Hughso, E., Stutzer, M., Yug, C. [006]. The Misuse of Expected Returs, Fiacial Aalysts Joural, November/December 006, p Jacquier, E., Kae, A, Marcus, A. [003]. Geometric Mea or Arithmetic Mea: A Recosideratio, Fiacial Aalysts Joural, November/December 003, p Jea, W.H., Helms, W.P. {1983]. Geometric Mea Approximatios, Joural of Fiacial ad Quatitative Aalysis, Vol. 18, No. 3, September, 1983, p Jorda, B. D., Miller, T. W. [008] Fudametals of Ivestmets, McGraw-Hill, 4 th Ed., 008. Klugma, S.A., Pajer, H.H., Willmot, G.E. [1998] Loss Distributios, Wiley, Latae, H.A. [1959]. Criteria for Choice amog Risky Vetures, Joural of Political Ecoomy, Vol. 67, April 1959, p MacBeth, J.D. [1995]. What s the Log-Term Expected Retur to Your Portfolio? Fiacial Aalysts Joural, September/October 1995, p Magi, J.L., Tuttle, D.L., McLeavey, D.W., Pito, J.E. [007]. Maagig Ivestmet Portfolios, Wiley, 3 rd Ed., 007. Markowitz, H.M. [1991]. Portfolio Selectio, d Ed., Blackwell Publishig, Siegel, J. J. [008] Stocks for the Log Ru, McGraw-Hill, 4 th Ed., 008, p.. Pito, J. E., Hery, E., Robiso, T. R., Stowe, J. D. Equity Asset Valuatio, Wiley, d Ed., 010. Arithmetic vs. Geometric Returs 18 8/14/011

19 Importat Iformatio This material is iteded for the exclusive use of the perso to whom it is provided. It may ot be modified, sold or otherwise provided, i whole or i part, to ay other perso or etity. The iformatio cotaied herei has bee obtaied from sources believed to be reliable. CDI Advisors LLC gives o represetatios or warraties as to the accuracy of such iformatio, ad accepts o resposibility or liability (icludig for idirect, cosequetial or icidetal damages) for ay error, omissio or iaccuracy i such iformatio ad for results obtaied from its use. Iformatio ad opiios are as of the date idicated, ad are subject to chage without otice. This material is iteded for iformatioal purposes oly ad should ot be costrued as legal, accoutig, tax, ivestmet, or other professioal advice. Copyright 011, CDI Advisors LLC. All rights reserved. No part of this publicatio may be reproduced or trasmitted i ay form or by ay meas, electroic or mechaical, icludig photocopyig, recordig, or by ay iformatio storage ad retrieval system, without permissio i writig from CDI Advisors LLC. 1 Arithmetic ad geometric averages are two of the three classical Pythagorea meas. The third oe is the harmoic average. For example, see MacBeth [1995], de La Gradville [1998], Jacquier [003], Hughso [006]. 3 That is, obviously, if returs r 1,, r are ot all the same, as assumed i this sectio. If r 1 r r, the the problem is trivial as the arithmetic ad geometric averages are equal. 4 This fact is a corollary of the Jece s iequality. 5 For the purposes of this sectio, the cocers about the quality of these approximatios are set aside. 6 For the purposes of this paper, the cocer that the sample variace as defied i (7) is ot a ubiased estimate is set aside. 7 For example, see Bodie [1999], p. 751, Jorda [008], p. 5, Pito [010], p Accordig to Jea, Helms [1983], formula (A) was origially proposed i Latae [1959]. 9 For example, see Markowitz [1991], p. 1, Jea, Helms [1983], Booth, Fama [199]. 10 Remider: i this sectio, it is assumed that returs r 1,, r are ot all the same (see edote 3). 11 The geometric average estimate (A1) is less tha (A) whe A V 4, which is usually the case. 1 We assume that the first ad the secod momets of the retur distributio are fiite. 13 de La Gradville [1998] ad de La Gradville [00] cotai a similar message. 14 For example, see Klugma [1998], p This data is also preseted i Magi [007], Exhibit 7-, p Arithmetic vs. Geometric Returs 19 8/14/011