RESEARCH LETTER. What is the Best Before Date for data used for estimating volatility and correlation? IN BRIEF. March 2016 AUTHORS:

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1 March 2016 What is the Best Before Date for data used for estimating volatility and correlation? AUTHORS: In this research letter, we turn our attention back to the time horizon upon which the volatility and correlation are calculated by Finaltis EfficientBeta TM strategy RÉMY CROISILLE Head of Research Senior Fund Manager IN BRIEF Although the Finaltis EfficientBeta TM model uses a proprietary estimation method for the volatility and correlation parameters which is different from the classic formulas, it is nevertheless confronted with the same problem of choosing an optimal time horizon upon which the calculation is made. Because we are often questioned on our choice of 65 days, we would like to explain the reasons behind this in one of our monthly research letters. CHRISTOPHE OLIVIER CIO 1. Using a one year time horizon for estimating the variance of a stock is inappropriate 90% of the time. In order to correctly interpret the variance estimator 85% of the time, one should use a time horizon of three months or less. 2. Each time the sample size is 4 times smaller, the variance of the estimator doubles. Less than two months, the imprecision is greater than 20%. NICOLAS RENAUD Head of Risk 3. A time horizon of three months results from the following compromise between quality (recent pertinent information) and volume: the variance parameter is statistically justified more than 85% of the time, and leads to a level of uncertainty of the parameter of 20%. LEWIS MEREDITH SMITH, CFA Senior Fund Manager

2 THE PERTINENCE / VOLUME DILEMMA The statistician who wishes to estimate a parameter from a sample which is susceptible to changing through time is confronted with a classic dilemma between the volume of data he uses and its pertinence. If he only uses recent data, it is very probable that the parameter he wishes to estimate is the same within the entire sample, which is coherent with his wish to estimate a unique parameter for the near future. On the other hand, the sample size is likely to be on the small side, and the uncertainty (or variability) of the estimation for such a small sample size is high. On the other hand, he could favour the volume of data by including a longer time horizon into his sample, for which the large sample size would imply small estimation uncertainty. But he runs the risk of including old data for which the estimation parameter is not the same. Let s consider these two examples: 1. Suppose a polling organisation wishes to estimate a Yes vote in a referendum. It can only question one single representative sample of 1,000 people each week. What is the best estimate? Is it simply the result of the most recent poll prior to the vote or rather the average of the last 12 polls over three months prior to the vote? Of course, the polling organisation will chose to base its prediction on the most recent poll. Indeed, the varying opinions of voters during the campaign period has a greater impact than the uncertainty attached to using a sample of just 1,000 people compared to using the 12,000 people questioned over 3 months: priority is given to pertinence over volume. 2. Suppose now that a statistician wishes to evaluate the probability that a coin lands heads up given that he can only do 10 coin tosses per week for 3 months. Should he use just the last ten tosses from the most recent week or the average from the 120 coin tosses made of the last three months? This time, he will of course favour the maximum possible number of tosses. The reason for this choice resides in the fact that the coin is essentially the same during the three months, its probability of showing heads was the same three months ago as today. The accumulation of coin tosses allows us to reduce the uncertainty related to the sample size: priority is given to volume. Coming back to our practical estimation problem of the variance (the volatility squared) and the classic formula: σ 2 = 1 n X i X 2 i=1 Where: σ 2 is the variance estimation; X i is the observed relative change for a stock on the i th day; X is the average of observed relative change for a stock between the first day and the n th day. 2

3 % of shares where the variances are significantly different RESEARCH LETTER MEASURING THE PERTINENCE This estimator only makes sense if the variance is the same for the sample s whole time horizon. In order to form a judgement regarding the consistency of such a hypothesis while varying the time horizon, we have conducted a simple statistical test on a given period. For each time horizon from one month to 12 months, we tested the hypothesis that the variance V 0 estimated using the first half of the data (the first six months when the horizon is one year) is equal to V 1 the variance estimated using the second half of the sample (the last six months when the horizon is one year). 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Equality observed in more than 85% of the case Equality test of variances on each half Length of the estimation (months) Using a one year time horizon for estimating the variance of a stock is inappropriate 90% of the time, as the numbers demonstrate that the stock has experienced two (at least) distinct variance regimes across the time horizon. In order to correctly interpret the variance estimator 85% of the time, one should use a time horizon of three months or less. MEASURING THE VOLUME EFFECT In order to measure the volume effect, we define the uncertainty attached to the estimator by: φ = Standard Deviation 1 n i=1 X i X 2 With Standard Deviation defined by: Standard Deviation(X) = E X 2 E 2 [X] 3

4 standard deviation of the esimator / parameter RESEARCH LETTER One can indeed demonstrate (1) that the uncertainty of the estimator can be written as: φ = Where: V stock variance; X i is the observed relative change for a stock on the i th day; X is the average of observed relative change for a stock between the first day and the n th day. So, each time the sample size is 4 times smaller, the variance of the estimator doubles. If we plot this relationship between the estimator s uncertainty against the sample size, we get the following graph: 60% Estimator uncertainty 50% 40% 30% Estimator uncertainty below than 20% 20% 10% 0% Sample size in months Less than two months, the imprecision is greater than 20%. Beyond 9 months, the estimator s imprecision is less than 10%. Note 1: We supposed that the daily change for stocks are Gaussians and the volatilities are constant. The demonstration steps and a presentation of equality test of variance used page 2 are available on statistic books, like Méthodes Statistiques, from Tassi P., edition Economica. Note 2: Suppose two samples composed of k 1 and k 2 = 4k 1 realisations. The standard deviations of the estimator are: φ 1 = k 1 1 φ 2 = k 2 1 = So, the second sample, four times bigger, implies a standard deviation two times smaller (or a sample four times smaller implies a standard deviation two times bigger). 2 k φ 1 4

5 CONCLUSIONS We have consequently chosen the following compromise between quality (recent pertinent information) and volume: a time horizon of three months to estimate the variance parameter is statistically justified more than 85% of the time, and leads to a level of uncertainty of the parameter of 20%. This quarterly time horizon also has the advantage of being in phase with our holding period, which in turn is based on the quarterly rebalancing rhythm of the Finaltis EfficientBeta TM Euro portfolio, like its benchmark, the Euro Stoxx Index. CONTACTS 63, Avenue des Champs Elysées Paris France Denis Beaudoin Tel: +33 (0) dbeaudoin@finaltis.com DISCLAIMER Thierry Rigoulet Tel: +33 (0) trigoulet@finaltis.com Mark Grobien Tel: +33 (0) mgrobien@finaltis.com This document does not constitute a recommendation or investment proposal. It was issued for information purposes only. It has no contractual value and may contain errors and/or omissions. Nothing contained herein shall in any way constitute an offer by Finaltis to provide any service or product, or an offer or solicitation of an offer to buy or sell any securities or other investment product. Finaltis will not be liable for the content of these pages, nor for any use thereof or reliance placed thereupon by any person. Product(s) described herein is/are not available to all persons in all geographic locations. It is/ they are of a speculative nature so that its/their success may be affected by a fall in assets value as well as a total loss. It/they may invest in OTC instruments which can be volatile and render difficult to predict or anticipate any fluctuation. The institutions with which it/they will contract may encounter financial difficulties impairing the value of the product(s). There is no guarantee of capital preservation and minimum return. There can be no assurance that the investment objectives shown are achieved. Despite Finaltis EfficientBeta TM Euro not being restricted to professional investors as per the MIFID, this document contains information dedicated to professional investors. As such, if you are not an investment professional, Finaltis recommends you stop reading this document and turn to your financial advisor for proper analysis and advice. 5