I'm interested to hear thoughts from others about the optimal approach for calculating a hedge ratio (HR) between two trading instruments. HR's are useful for basic hedging, as well as pair and spread trading. There are multiple considerations for futures and other derivative contracts based on their specs. So let's keep this simple by limiting the scope of our discussion to two stocks or ETF's, and assume both are priced in USD. To start this off, here are my thoughts: 1. It's simple enough to calculate dollar equivalence by taking a ratio of the nominal prices of both stocks. There's not much more to say on this point. 2. The HR should also adjust for relative volatility of the two stocks. This is really the focus of our discussion. 3. Common statistics like Beta can be calculated for any two stocks. This is a reasonable approach to adjust for volatility. Here are some considerations for Beta: 3a. Beta is normally reported between a stock and the broad market, specifically the S&P500 index. However, using the Beta formula we can calculate Beta for any two price series to give us a measure of relative co-movement. 3b. The lookback period must be specified to calculate Beta. Most websites that report Beta use a long lookback period of 3 to 5 years. A shorter term hedge may prefer a shorter lookback period, perhaps measured in days or weeks depending on your holding period. 3c. Beta only uses the closing price. If you wanted to capture the volatility of high and low prices, another approach must be used (e.g., ATR). Looking forward to comments from others on this topic.

Attached is a chart showing beta calculated for XLK and QQQ since the beginning of this year. These ETFs share the computer technology sector, are correlated, and consequently are good candidates for hedging or spreading. The beta values shown are for XLK with regard to QQQ as the reference index. The 20-day beta is shown in red, the 60-day in cyan, and the 250-day in yellow. The scale of this beta chart is somewhat narrow to reveal the difference in beta values (if I zoomed out the beta axis from zero to one, the three beta series would appear more closely grouped). But the point of this chart is to illustrate how the lookback period impacts beta. Longer lookbacks have more stable betas, and shorter lookbacks have betas that move around as relative volatility varies over shorter time intervals. Interesting to note that relative volatility appears mean-reverting.

Using beta in conjunction with a dollar equivalence multiplier is an example of a minimum variance technique. An alternative technique suggested by another ET member is to use the ratio of ATR's. Simpler to calculate, the resulting ratio incorporates both dollar adjusment as well as volatility adjustment.

Isn't that basically beta in a two asset case? The residual of the two securities given beta squared is the tracking error of one security vs another; Thus, beta, being the value that minimizes the sum of squared residuals, is equivalent to the value that minimizes the tracking variance?

(1) Why can't you use intraday prices for beta calculations? (2) If you really want to use ATR (and I'm not sure whether this adds any value) as a vol measurement, you can still calculate beta using ATR as vol rather than the usual standard deviation.

(1)You can certainly use intraday prices for beta calculation, and probably should if you are trading intraday. Use whatever timeframe is appropriate for your trading strategy. (2)The benefit of using ATR is that it may be more convenient using out-of-the-box software indicators which normally include ATR. Some software packages do not include a beta indicator, or require you to code your own (not a major obstacle, but another workstep). Thanks for weighing in. I think this is the last thread I will start after listening to crickets chirping for so long.

(1) If you are using intraday beta, it's actually rather more complicated because of intraday seasonality. (2) Convenient doesn't equal to optimal, no? Calculating beta is about the simplest possible function to write.... maybe two steps more than moving averages.

(1) Good callout on seasonality. You'll certainly have seasonality factors for intraday prices based on time-of-day session dynamics. But same can be said for daily/weekly/monthly seasonality. Commodity markets exhibit seasonal cycles, and even simple equities have quarterly earnings cycles. I have not attempted to deal with any of them in my previous postings, so you correctly point out this gap in the analysis. (2) Optimizing a solution generally means you are considering all related factors. Maximizing a solution means you are placing greater emphasis on a single factor. In any case, I acknowledge your point that beta is not difficult to code. In fact, that's what I did for the TradeStation indicator in the chart I attached above.

Not so sure seasonalities matter all that much for calculating daily betas in the context of a two assets since the degree of variation due to inter-day seasonality is likely to be pretty weak compare to the effect of volatility and correlation. Intraday matter a lot more, I think, because the quality of the data is subject to a higher degree of asynchronicity.