Trading with a Perfect Return Model: R², Costs, and Positioning

# Interview question: strategy with perfect R2


# Interview question: strategy with perfect R2












I got a question in an interview, not sure if I got it: ‘’’ We trade stocks and futures, and you can both long and short for futures. Assume you built a perfect quantitative trading model with perfect r2, what trading strategy do you recommend and what are the expected returns of such a strategy? ‘’’ I don’t quite understand it. Appreciate any insight.

## Answer by Arshdeep (score 1)

https://quant.stackexchange.com/a/79695

It is ironic if you know what's going to happen to the market perfectly, why do you need an expected value? You know it already. If you know the die is going to be 6, the "expected value" is 6.

It is like asking what is your expected loss the day you lose a million dollars?

If you build a perfect relationship, you put as much capital as you can when returns are positive and short as much as you can when returns are negative.

## Answer by Dave (score 1)

https://quant.stackexchange.com/a/79697

We can’t know what your interviewer wanted you to say. It might be that the ideal answer would have involved a discussion of how you verify results so you don’t get suckered into a situation that should jump as as too good to be true (unless you can put together compelling evidence that it really is that good).

However, if we take it as a given that your model, perhaps through time travel, is perfect, there is an interesting discussion to be had.

Buying when anticipate an increase and selling when you anticipate a decrease sounds appealing, but I do see some caveats. For instance, if you buy when a tiny gain is expected, your gains might be negated by a trading fee. When it comes time to sell, you might sell at one time to maximize your overall return but push back the sale a bit if that means long-term capital gains taxes instead of short. Sure, the return might be a bit less, but your after-tax return (how much money you get in your pocket) would be greater. You might favor the latter situation as a real asset manager, even if the higher return from freshman finance is the former.

There is also a matter of how you calculate $R^2$. If you just square the Pearson correlation between true and predicted values, you can get a perfect score of $1$ despite the predictions being terrible. Discussing this may have been worthwhile in the interview answer: "There are several ways of calculating $R^2$ that are equivalent in settings like simple linear regression but that are not equivalent in more complicated settings. What calculation yields that perfect value of one?" Being tricked into thinking bad predictions are good could have implications when it comes to trading fees and taxes. For instance, if your model prediction is one where your return would exceed the trading fee, you would want to make that trade. However, if the return is not as great, you might not earn enough on that trade to justify the trading fee.

As far as calculating expected returns, you know exactly what your returns would be, since you know your buy and sell prices. There isn’t really any averaging to do.