Ratio of Means Versus Mean of Ratios for Relative-Price Signals

# help me compare methods to compute one instrument price from another instrument price


# help me compare methods to compute one instrument price from another instrument price












Assume we have two instruments `A` and `B`. Also time is increasing from 1 to n. Let's say that `A1` is price of instrument `A` at time `1`. Let's assume that `A` and `B` are highly correlated instruments. Then we can try to compute `TruePrice` of stock `B` from stock `A`.

Then if `Bn` is more enough than `TruePrice` we sell `B`, and if `Bn` is less enough than `TruePrice` we buy `B`. I think that would be so-called `statistical arbitrage`.

I would prefer not to discuss pronse and cons of this scheme in general. Cause it proved to work cause I'm using it for a long time and still profitable.

What I really want to discuss is how can we compute `TruePrice` having what we have. Let me describe two algorithms:

- TruePrice = Bn * ( (A1 + A2 + .... + An) / (B1 + B2 + .. + Bn) )

- TruePrice = Bn / n * (A1 / B1 + A2 / B2 + .... An / Bn)

Also please note that `Ai` and `Bi` are measurements of price. This could be `median` `low` `close` or anything else of the certain interval (i'm using `median` now). The last items `An` and `Bn` are `life` i.e. changing while trading to reflect current situation.

The question is what are prons and cons of each of these (`1` and `2`) algorithms? Probably you can suggest something else?

Now I'm using `1` but I'm not satisfied with it. When instrument `A` grow (so `An` increase, other `Ai` are fixed as they in "past") `TruePrice` is not changing enough. And you can see from formula that `TruePrice` of stock `B` depends a lot on `Bn` what is actually `current B price`. So `TruePrice` of stock `B` depends on stock `B` too much what is not good. I think TruePrice of stock `B` should more depends on correlated `An`, but it doesn't.

I'm not sure if `2` would solve this problem.

## Answer by Andrew Cheong (score 2, accepted)

https://quant.stackexchange.com/a/7828

This is probably not a good question for Q.SE—despite its application in your strategy, you're (perhaps unaware) asking a general math question: the difference between the ratio of means and the mean of ratios. There are numerous results online that distinguish the difference, but I assume the terminology wasn't apparent to you at the time. I've come across some elementary books and sites that use one or the other though, seemingly arbitrarily and without thought, e.g. when smoothing the stochastic; so maybe it'll be useful to comment on it:

Your first formula, $$B_n\frac{\sum{A_i}}{\sum{B_i}} = B_n\frac{\frac{1}{n}\sum{A_i}}{\frac{1}{n}\sum{B_i}}$$ is the ratio of means.

Your second formula, $$B_n\frac{1}{n}\sum{\frac{A_i}{B_i}}$$ meanwhile, is the mean of ratios.

It's not really about advantage/disadvantage, but what you seek from an analytical perspective.

The first formula is not sensitive to how the series compare per point in time. For example, both $A'_i$, a wildly fluctuating series whose average is $\mu'$, and $A''_i$, a constant series that is always $\mu'$, would yield the same result.

The second formula, however, is sensitive to each point in time, and averages the various measurements of comparison, over time.

I'm skeptical about your method, to say the least, but as you insist, I'll leave that discussion out. Given your claims, though, I imagine the second formula, the mean of ratios, is what you're after, (1) because you appear to be calculating whether one instrument is an "overestimator" or "underestimator" of the other, and I imagine you'd like to calculate the average of those comparisons (estimations) at each point in time, not the comparison (estimation) of two averages across all points in time, and (2) because the mean of ratios is not sensitive to absolute price levels (only relative price levels per point in time), e.g. if the price jumped up 200% in the middle of your series and you were using the ratio of means, differences in the latter half of the series would dominate in effect over the former half (unless you're using log-normalized values).