Relative Rotation Graph (RRG) — US Equity

The Relative Rotation Graph (RRG) maps each stock into a two-dimensional state space relative to a benchmark (S&P 500). The horizontal axis is RS-Ratio (relative-strength level), and the vertical axis is RS-Momentum (change in relative strength). This separates *level* from *direction*, so you can distinguish names that are strong-but-decaying from weak-but-improving.

The four quadrants are interpreted as market states: Leading (RS-Ratio > 100, RS-Momentum > 100), Weakening (RS-Ratio > 100, RS-Momentum < 100), Lagging (RS-Ratio < 100, RS-Momentum < 100), and Improving (RS-Ratio < 100, RS-Momentum > 100). Animated tails provide path dependence, not just point-in-time ranking.

Let $P_{i,t}$ be stock $i$'s price and $P_{b,t}$ be benchmark price at week $t$. The raw relative-strength series is

$$R{S}_{i,t}=100\times \frac{P_{i,t}}{P_{b,t}}$$

To make names comparable, RS is normalized with a rolling z-score over window $w$ (default 14 weeks):

$$z_{i,t}^{RS}=\frac{R{S}_{i,t}-{\mu }_{i,t}^{(w)}(RS)}{{\sigma }_{i,t}^{(w)}(RS)}$$

RRG RS-Ratio is then centered at 100:

$$R{S}_{i,t}^{\mathrm{RRG}}=100+z_{i,t}^{RS}$$

Momentum is computed from first-difference of RS-Ratio:

$$\Delta R{S}_{i,t}^{\mathrm{RRG}}=R{S}_{i,t}^{\mathrm{RRG}}-R{S}_{i,t-1}^{\mathrm{RRG}}$$

and normalized with the same rolling logic:

$$z_{i,t}^{MOM}=\frac{\Delta R{S}_{i,t}^{\mathrm{RRG}}-{\mu }_{i,t}^{(w)}(\Delta R{S}^{\mathrm{RRG}})}{{\sigma }_{i,t}^{(w)}(\Delta R{S}^{\mathrm{RRG}})}$$

Final vertical coordinate:

$$MO{M}_{i,t}^{\mathrm{RRG}}=100+z_{i,t}^{MOM}$$

Each weekly point is therefore $(R{S}_{i,t}^{\mathrm{RRG}},MO{M}_{i,t}^{\mathrm{RRG}})$. A tail of length $k$ is the sequence $\{(x_{t-j},y_{t-j}){\}}_{j=0}^{k-1}$, which encodes rotational behavior over recent history.

Quadrant transitions are often more informative than static quadrant membership. For example, a path moving Lagging → Improving → Leading indicates strengthening relative trend, while Leading → Weakening → Lagging indicates relative deterioration.

A practical decomposition is to separate *selection* and *timing*: use RS-Ratio for cross-sectional ranking, and use RS-Momentum sign changes as timing confirmation. In notation, a simple filter is $R{S}_{i,t}^{\mathrm{RRG}}>100$ with $MO{M}_{i,t}^{\mathrm{RRG}}>100$ for long candidates, and the symmetric condition below 100 for de-risking.

Angular motion can also be tracked: with $x_t=R{S}_t^{\mathrm{RRG}}-100$, $y_t=MO{M}_t^{\mathrm{RRG}}-100$, define ${\theta }_t=atan2(y_t,x_t)$. Changes in ${\theta }_t$ quantify clockwise/counterclockwise rotation speed, while radius $r_t=\sqrt{x_t^2+y_t^2}$ measures signal strength away from neutral.

Data uses weekly closes for US equities and benchmark ${}^{G}SPC$, reducing day-level noise and making rotation arcs easier to interpret. Missing observations are aligned by common timestamps to avoid look-ahead artifacts.

Rolling normalization introduces edge effects in early samples; the first $w$ points are warm-up territory and should be interpreted cautiously. Very low rolling standard deviation can over-amplify z-scores, so practical implementations typically guard with a volatility floor.

RRG is a relative-strength diagnostic, not a standalone execution system. It works best when combined with liquidity filters, regime context, and risk controls (position sizing, stop logic, and turnover constraints).