Canvas & Ratio
Choose your destination platform format
Layout Template
Choose a content structure for your slides
Preset Themes
Typography & Sizing
Brand Kit Customization
AGENCYConfigure brand assets for headers & footers
Outro Slide CTA
Customize your closing call-to-action slide
Background Pattern
Build Your Carousel
Drag and drop any post card below onto a slide, or use the quick buttons to insert content/images instantly!
It's been a long time since I wrote anything remotely informative on X. I have something minor that sits somewhere in between [finance] signal analysis and statistics, which is also very practical, simple, and with a familiar ending. A loong 🧵 👇. 1/18
Say that you have excess independent asset returns📉📈, i.e., you have removed common components of returns so that your n-dimensional return vector z[t] has independent elements. Also, z-score them so that they have unit volatility. There are T investment epochs/year. 2/18
Your holy grail is to find a signal vector x[t], such that the cross-sectional correlation IC:=corr(x, z) is as high as possible (IC stands for "Information Coefficient"). Then the Sharpe Ratio is SR=IC*sqrt(n*T). Fundamental Law of Active Management. So far, so good. 3/18
However, say that you have a model in mind, where some asset-specific variables determine returns. Most common variable: EPS - baseline, were the baseline is consensus EPS, or something like it (because consensus is a bit too easy to beat!). 4/18
EPS is only one choice. The variable depends a lot on asset class, investment class, etc. It could be drug approval; or credit downgrade, etc. You get the idea. And there might be more than one (but we stick to one). 5/18
The important thing is: the variable, at some point, is observed. We can estimate the relationship variable-returns, after the it's been observed. You estimate IC_ideal := corr(y, z). If you choose the variable well, this correlation can be quite high. If you only knew y. 6/18
... you'd be rich. At least, you have a different problem: forecasting the variable y. You might like this problem, because you can encode a lot of expertise in the forecast. Say that you have forecasts x. The R^2 of the model y ~ x is Rsq. You'd like to use x as signals... 7/18
... as that is your best available information. You know that corr(x, y)=sqrt(Rsq). What you need is to know is the Sharpe Ratio of the strategy: corr(x, z)*sqrt(n). 8/18
The statistics part is this: if I know corr(x,y)=rho_1 and corr(y, z) = rho_2, what is corr(x, z)? 9/18
You have seen this at some point. A necessary condition for the correlation is that the determinant of the correlation matrix be positive definite. 10/18
From which we get a worst-case bound: r:=sqrt[(1-rho_1^2)*(1-rho_2^2)] and |rho_1*rho_2 - rho_xz| <= r. The problem is that they are usually not binding. A realistic example: rho_1=0.1, rho_2=0.6, then rho_1*rho_2=0.06 with a range of 0.8. Too wide! 11/18
An alternative way to obtain the same bound (and more) is via partial correlation. The partial correlation of forecast and returns, given the realized variable, is below, and must be in [-1, 1]. 12/18
Geometrical view: rho_xz = rho_1*rho_2 + r*cos(theta), with theta = cor(x, z|y) for high n, it is reasonable to assume that partial correlation concentrates around 0. Given the realization of the variable, the forecast contains no information about the returns. 13/18
So we have an expectation identity: E[rho_xz]=rho_1*rho_2. Now you can factor the Information Coefficient into the product of the sqrt(Rsq) of your forecast, and the IC_ideal, under knowledge of the key variables. 14/18
The annualized Sharpe Ratio is Sharpe = sqrt(Rsq)*IC_ideal*sqrt(breath)*sqrt(# decisions/yr) So there you have it. 15/18
Sharpe decompositions appear often in quantitative investment, but they have very different meanings. This one generalizes the Transfer Coefficient. Replace y with the *true* expected returns, x with the forecasts, and you get the TC identity in the Clarke et al. paper. 16/18
Last thing: I told the story with y as variables that predict returns from their historical realizations. But y and x can be anything. For example, y could be forecasts and x can be the noisy, corrupted estimates of these forecasts. Worst things can happen. 17/18
The story-time interpretation, though, is useful. We should use Oracles more, and then predict those Oracles, and combine them. People do this without thinking carefully about pitfalls and power-ups. Underrated and overrrated at the same time. And that is the end. 18/18