tag:blogger.com,1999:blog-7905515.post5977872612353744538..comments2023-01-28T05:19:44.944-06:00Comments on Falkenblog: Is Arithmetic Return Bias Basis of Low Vol Anomaly?Eric Falkensteinhttp://www.blogger.com/profile/07243687157322033496noreply@blogger.comBlogger12125tag:blogger.com,1999:blog-7905515.post-50225981310009372402012-02-03T22:23:59.101-06:002012-02-03T22:23:59.101-06:00I am a professional in finance and I am staggered ...I am a professional in finance and I am staggered by the number of incredibly smart and educated people who don't know about the relationship between arithmetic return, geometric return, and standard deviation. I call it volatility drag: the higher the volatility for a given arithmetic return, the lower the long run growth rate (which is what the geometric return is). When volatility gets large enough the geometric return starts to decline and for really large volatilities, the geometric return can be negative. This is why leverage can be so dangerous: the high arithmetic return with high volatility is actually a very low geometric return.<br /><br />This is a mathematic relationship, not an economic or financial one. It applies to the growth rate of rabbits or carbon dioxide in the atmosphere as well. <br /><br />Here is my simple example: you start with one dollar that you invest. After one one year the investment is worth $2 dollars giving a 100 percent return. In the second year you you have a 50 percent loss, and you end up with $1. The arithmetic return is (100 + -50)/2 = 25%, an impressive number. However, the geometric return is zero: you started with $1 and ended with $1. <br /><br />There are some other interesting nuances that know of about the mathematics of finance. I have had people ask why this stuff isn't taught. It is a good question.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7905515.post-82350962959665818802012-02-02T01:17:55.032-06:002012-02-02T01:17:55.032-06:00Its like you read my mind! You appear to know so m...Its like you read my mind! You appear to know so much about this, like you wrote the book in it or something. I think that you could do with some pics to drive the message home a little bit, but other than that, this is great blog. A great read. I will certainly be back.tiffany jewelleryhttp://www.tiffanysaleukco.comnoreply@blogger.comtag:blogger.com,1999:blog-7905515.post-51362706698961292942012-01-31T10:05:37.311-06:002012-01-31T10:05:37.311-06:00Why don't you report "excess return"...Why don't you report "excess return" instead of raw return? A high beta portfolio requires less cash to get the same market exposure, and the leftover cash earns interest. It ought to be credited with that interest, as it is in the calculation of the calculation of excess return.<br />http://en.wikipedia.org/wiki/Alpha_(investment)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7905515.post-2993648388927803862012-01-29T21:42:01.625-06:002012-01-29T21:42:01.625-06:00I still don't really understand this. If you ...I still don't really understand this. If you could make more in the highly volatile asset via market timing with a cash account, there would appear to be inevitable arbitrage. <br /><br />I think a key is how to treat the cash account used to sweep profits and bolster the principal in the drawdowns, in terms of the internal rate of return.Eric Falkensteinhttps://www.blogger.com/profile/07243687157322033496noreply@blogger.comtag:blogger.com,1999:blog-7905515.post-72951392956439400582012-01-27T17:38:36.441-06:002012-01-27T17:38:36.441-06:00"Ok, help me understand this because I don..."Ok, help me understand this because I don't. You have $200 at time 0, and are looking at two alternative portfolios: one with high variance, one low. Their monthly returns rank order them different than their annual returns. How you you invest $100 in the highly volatile portfolio to generate a greater total return than the lower volatility portfolio?"<br /><br />Suppose portfolio 1 returns 1% every month. Portfolio 2 returns 10% half the time and -7.5% the other half.<br /><br />If you put $100 in each and let it grow, portfolio 1 gives you $112.68 at the end of a year, portfolio 2 gives you $110.97 if you get 6 up months and 6 down months as expected.<br /><br />If you put $100 in each and take out any profit (or replace any losses) each month, portfolio 1 makes $1 per month, $12 in all. Portfolio 2 makes $60 in 6 up months, loses $45 in 6 down months, for a total profit of $15.<br /><br />However, eventually leaving the money in portfolio 1 will do better than keeping a constant $100 in strategy 2. After 4 years, letting $100 grow in strategy 1 produces $161.22, while keeping a constant $100 at risk in portfolio 2 produces only $160 (assuming exactly 24 up months and 24 down months).Aaron Brownhttp://www.eraider.comnoreply@blogger.comtag:blogger.com,1999:blog-7905515.post-79631736757660042422012-01-27T14:20:31.307-06:002012-01-27T14:20:31.307-06:00I'm assuming neither strategy is forecastable,...I'm assuming neither strategy is forecastable, so market timing wouldn't work from some sort of value effect here (indeed, there is no autocorrelation in monthly returns). <br /><br /> Assume two strategies available to you where returns are r~ln(mu,sigma), geometric brownian motion, with parameters such that the expected return in one year generates a different rank ordering than the expected return over a month. I'm confident that if your goal is to maximize your end-of-year portfolio value, the amount of weight put on the higher volatility strategy is zero. I don't have a proof, but perhaps it is a corollary of Samuelson's paper on the log of wealth, <a href="http://www-stat.wharton.upenn.edu/~steele/Courses/434F2005/Context/Kelly%20Resources/Samuelson1979.pdf" rel="nofollow">Why we should not make mean log of wealth big though years to act are long.</a>Eric Falkensteinhttps://www.blogger.com/profile/07243687157322033496noreply@blogger.comtag:blogger.com,1999:blog-7905515.post-90969836733524566102012-01-27T13:53:38.993-06:002012-01-27T13:53:38.993-06:00Well Eric, you've written clearer paragraphs, ...Well Eric, you've written clearer paragraphs, but I'll try :)<br /><br />Your logic implies a higher vol version (say more leverage) of the same strategy is always a worse return for risk. Not true. That is only viewing the strategy alone (as your whole portfolio). If the strategy is a small part of an overall portfolio, and you rebalance (take away from it when it's up, add to it when it's down) its contribution looks much more like its arithmetic than its geometric average.<br /><br />Another way to view it is imagine you do put all your money in only this strategy, but you're much more risk averese, so you do it at less octane (say by adding cash). The artihmetic and geometric get much closer. Did we solve the conundrum?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7905515.post-3701328147110731072012-01-27T09:47:37.124-06:002012-01-27T09:47:37.124-06:00anon: Ok, help me understand this because I don...anon: Ok, help me understand this because I don't. You have $200 at time 0, and are looking at two alternative portfolios: one with high variance, one low. Their monthly returns rank order them different than their annual returns. How you you invest $100 in the highly volatile portfolio to generate a greater total return than the lower volatility portfolio?Eric Falkensteinhttps://www.blogger.com/profile/07243687157322033496noreply@blogger.comtag:blogger.com,1999:blog-7905515.post-73586462844677197312012-01-27T09:16:32.114-06:002012-01-27T09:16:32.114-06:00In a monthly rebalanced strategy (as a fraction of...In a monthly rebalanced strategy (as a fraction of the portfolio) it's the arithmetic return that counts.<br /><br />This is not a good post, it's a quite silly one. Way below he usually high standard.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7905515.post-13250535162007891022012-01-27T06:18:17.290-06:002012-01-27T06:18:17.290-06:00Good post. One could wonder if 1-year investment h...Good post. One could wonder if 1-year investment horizon is enough. For example high-beta did very poor in 2008 but very good in 2009. Using annual returns would not show how poor high-beta stocks actually did over this 2-year period. Therefore the compounding really kick in with 3-year or even 7-year evaluation horizons. <br /><br />It seems that the market becomes more inefficient, if the evaluation horizon increases.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7905515.post-37784195993292097062012-01-26T09:17:38.131-06:002012-01-26T09:17:38.131-06:00I am not sure this is the right way of looking at ...I am not sure this is the right way of looking at it. <br /><br />At the end of the month, let us say you sell 1 stock and at the beginning of the next month you can buy either one stock or you can buy no of stocks for exact dollar value at which you sold on the previous month end. <br />I think the diff between these two strategies will explain the anamoly.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7905515.post-82662289737141332472012-01-25T20:22:35.538-06:002012-01-25T20:22:35.538-06:00What if you were able to look ahead. For instance,...What if you were able to look ahead. For instance, if you knew at the beginning of the year what the high beta stocks over the upcoming year would be and created a portfolio from those.Anonymousnoreply@blogger.com