![\"\"](\"https:\/\/www.portfolioprobe.com\/wp-content\/uploads\/2012\/05\/geotagged-folkstreet-julepstreet-92997-crop3-520x673.jpg\" "\\"geotagged-folkstreet-julepstreet-92997-crop\\"")
<\/a><\/p>\nLong ago and not so far away, I needed to compute the variance matrix of the returns<\/a> of a few thousand stocks.\u00a0 The machine that I had could hold three copies of the matrix but not much more.<\/p>\n An exponential decay model worked wonderfully in this situation.\u00a0 The data I needed were:<\/p>\n The operations were:<\/p>\n Compact and simple.<\/p>\n There are better ways, it’s just that the time was wrong.<\/p>\n The “right” way would be to use a long history of the returns of the stocks and fit a realistic model.\u00a0 Even if that computer could have held the history of returns (probably not), it is unlikely it would have had room to work with it in order to come up with the answer.\u00a0 Plus there would have been lots of data complications to work through with a more complex model.<\/p>\n “The shadows and light of models”<\/a> used a different metaphor of light in relation to models.\u00a0 If we can only use an exponential decay model, measuring ignorance is going to be a bit dodgy.\u00a0 We won’t know how much of the ignorance is due to the model and how much is inherent.<\/p>\n We can do exponential smoothing of the daily returns of the S&P 500 as an example.\u00a0 Figure 1 shows the unsmoothed returns.\u00a0 Figure 2 shows the exponential smooth with lambda equal to 0.97 — that is 97% weight on the previous smooth and 3% weight on the current point.\u00a0 Figure 3 shows the exponential smooth with lambda equal to 1%.<\/p>\n Note<\/strong>: Often what is called lambda here is one minus lambda elsewhere.<\/p>\n Figure 1: Log returns of the S&P 500. Figure 2: Exponential smooth of the log returns of the S&P 500 with lambda equal to 0.97. Figure 3: Exponential smooth of the log returns of the S&P 500 with lambda equal to 0.99. Notice that the start of the smooth in Figures 2 and 3 is a little strange.\u00a0 The first point of the smooth is the actual first datapoint, which happened to be a little over 1%.\u00a0 That problem can be solved by allowing a burn-in period — dropping the first 20, 50, 100 points in the smooth.<\/p>\n The simple process of always doing a weighted sum of the previous smooth and the new data means that the smooth is actually a weighted sum of all the previous data with weights that decrease exponentially.\u00a0 Figure 4 shows the weights that result from two choices of lambda.<\/p>\n Figure 4: Weights of lagged observations for lambda equal to 0.97 (blue) and 0.99 (gold). What we are likely to really want is some compromise between these extremes.<\/p>\n The half-life of an exponential decay is often given.\u00a0 This is the number of lags at which the weight falls to half of the weight for the current observation.\u00a0 Figure 5 shows the half-lives for our two example lambdas.<\/p>\n Figure 5: Half-lives and weights of lagged observations for lambda equal to 0.97 (blue) and 0.99 (gold). Generally the half-life is presented as if it is an intuitive value.\u00a0 Well, sounds cool — I’m not sure it tells me<\/strong> much of anything.<\/p>\n When an exponential decay model is being used, you should ask:<\/p>\n Is there a good reason to use exponential decay, or is it only used because it has always been done like that?<\/p>\n Advances in hardware means that some of what could only be modeled with exponential decay in the past can now be modeled better.\u00a0 It also means that what could not be done at all before can now be done with exponential decay<\/a>.<\/p>\n He finds a convenient street light, steps out of the shade from “Romeo and Juliet” by Mark Knopfler\n
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Exponential smoothing<\/h2>\n
![\"\"](\"https:\/\/www.portfolioprobe.com\/wp-content\/uploads\/2012\/05\/spxreturn.png\" "\\"spxreturn\\"")
<\/a><\/p>\n![\"\"](\"https:\/\/www.portfolioprobe.com\/wp-content\/uploads\/2012\/05\/spxretes97.png\" "\\"spxretes97\\"")
<\/a><\/p>\n![\"\"](\"https:\/\/www.portfolioprobe.com\/wp-content\/uploads\/2012\/05\/spxretes99.png\" "\\"spxretes99\\"")
<\/a><\/p>\nChains of weights<\/h2>\n
![\"\"](\"https:\/\/www.portfolioprobe.com\/wp-content\/uploads\/2012\/05\/expweight.png\" "\\"expweight\\"")
<\/a>The weights drop off quite fast.\u00a0 If the process were stable, we would want the observations to be equally weighted.\u00a0 It is easy to do that compactly as well:<\/p>\n\n
Half-life<\/h2>\n
![\"\"](\"https:\/\/www.portfolioprobe.com\/wp-content\/uploads\/2012\/05\/exphalflife.png\" "\\"exphalflife\\"")
<\/a><\/p>\nSummary<\/h2>\n
Epilogue<\/h2>\n
\nSays something like, “You and me babe, how about it?”<\/p><\/blockquote>\n
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