Figure 1: In-sample VoC curves. Each panel plots the normalised in-sample
MSPE against log10(z) for T ∈ {12, 60, 120} months. Curves are
normalised so that all training windows take the value 1.0 at z = zequity = 1, 000
(vertical dashed line). Values below unity indicate lower in-sample
error relative to the equity baseline.
(a) Taylor rule fundamentals
(b) Traditional monetary fundamentals
This figure contains two panels showing normalized in-sample mean squared prediction error (MSPE) as a function of log10 (z) for training windows of 12, 60, and 120 months. Panel (a) displays Taylor rule fundamentals. The x-axis ranges from 0 to 6 on a log10 (z) scale, and the y-axis shows normalized in-sample MSPE from 0 to 1.0. All curves are normalized to equal 1.0 at z = 1,000 (marked by a vertical dashed line at log10 (z) = 3). Three lines represent different training windows: T = 12 months (blue line), T = 60 months (orange line), and T = 120 months (green line). As log10 (z) decreases from 6 toward 0, all three curves decline monotonically. The T = 12 curve shows the steepest decline, approaching zero at log10 (z) ≈ 1. The T = 60 and T = 120 curves decline more gradually, reaching approximately 0.2–0.3 at log10 (z) = 1. Panel (b) shows traditional monetary fundamentals with identical axes and normalization. The pattern is qualitatively similar but with less dramatic variation. The T = 12 curve (blue) again shows the steepest decline but remains above approximately 0.1 at the lowest z values. The T = 60 (orange) and T = 120 (green) curves decline more modestly, converging to approximately 0.4–0.5 at log10 (z) = 1. Note: The figure plots the difference between the fair and amortized cost values of the securities. Values below unity at the equity baseline indicate lower in-sample error relative to z = 1,000. Source: Authors’ calculations based on IMF International Financial Statistics, Haver Analytics, Global Financial Database, and Federal Reserve Economic Data.
Figure 2: OOS VoC curves (no-intercept baseline). Each panel plots the
cross-currency mean OOS R2 against log10(z) for T ∈ {12, 60, 120}. The horizontal
dashed line marks the random walk benchmark (R2 = 0); the vertical
dotted line marks the KMZ equity calibration (zequity = 1, 000).
Triangles indicate the z value
that maximises OOS R2 for each T. The OOS curve lies below zero
across the entire regularisation spectrum for both predictor
sets.
(a) Taylor rule fundamentals
(b) Traditional monetary fundamentals
This figure contains two panels plotting cross-currency mean out-of-sample R2 against log10 (z) for three training window sizes. Panel (a) displays Taylor rule fundamentals. The x-axis ranges from 0 to 6 on a log10 (z) scale, and the y-axis shows out-of-sample R2 from approximately −0.35 to 0.05. A horizontal dashed line at R2 = 0 marks the random walk benchmark, and a vertical dotted line at log10 (z) = 3 marks the equity calibration (z = 1,000). Three lines represent training windows: T = 12 months (blue), T = 60 months (orange), and T = 120 months (green). All three curves remain entirely below the zero line across the full regularization spectrum. Triangle markers indicate the z value that maximizes out-of-sample R2 for each training window. The T = 12 curve shows the most negative values, reaching approximately −0.30 at low z before improving to about −0.02 near log10 (z) = 3–4. The T = 60 and T = 120 curves are less negative, hovering between −0.05 and 0 across most of the range, with optima near log10 (z) = 3–4.
Panel (b) shows traditional monetary fundamentals with similar axes. All curves again remain below zero throughout. The T = 12 curve (blue) is most negative at low z values (approximately −0.30 to −0.35) and improves toward −0.05 at higher z. The T = 60 (orange) and T = 120 (green) curves are more compressed, ranging from approximately −0.15 to −0.02 across the spectrum, with optimal z values (marked by triangles) shifted further right to log10 (z) ≈ 5–6. Note: The OOS curve lies below zero across the entire regularization spectrum for both predictor sets, indicating no level of regularization yields a model that outperforms the random walk in expectation. Source: Authors’ calculations based on exchange rate data and economic fundamentals from multiple sources as detailed in Appendix A.
Figure 3: MSPE ratio of Ridge–RFF relative to OLS (no-intercept
baseline). A ratio below 1 indicates that Ridge–RFF achieves lower
out-of-sample MSPE than OLS. Triangles mark the z value that minimises the MSPE
ratio for each T. The
horizontal dashed line marks parity with OLS (ratio = 1).
(a) Taylor rule fundamentals
(b) Traditional monetary fundamentals
This figure contains two panels showing the ratio MSE(RFF)/MSE(OLS) across the regularization grid. Panel (a) presents Taylor rule fundamentals. The x-axis shows log10 (z) from 0 to 6, and the y-axis displays the MSPE ratio from 0 to approximately 2.5. A horizontal dashed line at ratio = 1 marks parity with OLS. Three lines represent training windows: T = 12 months (blue), T = 60 months (orange), and T = 120 months (green). Triangle markers indicate the z value minimizing the MSPE ratio for each window. The T = 12 curve starts well above 1 at low z (approximately 2.0–2.5), drops sharply, crosses below 1 around log10 (z) = 1, reaches a minimum near 0.15 at log10 (z) ≈ 2–3, then gradually rises toward 1 as z increases. The T = 60 and T = 120 curves show less dramatic variation, starting near 1 at low z, dipping to approximately 0.80–0.90 in the middle range, and converging back toward 1 at high z. Panel (b) shows traditional monetary fundamentals with the same axis structure. The patterns are qualitatively similar but more compressed. The T = 12 curve (blue) ranges from approximately 1.5 at low z to a minimum around 0.5 near log10 (z) = 2, then rises back toward 1. The T = 60 (orange) and T = 120 (green) curves show modest improvements over OLS, with ratios reaching approximately 0.93–0.96 at their optima (marked by triangles) near log10 (z) = 3–4. Note: A ratio below 1 indicates Ridge-RFF achieves lower out-of-sample MSPE than OLS. Ridge regularization meaningfully corrects for the bias-variance trade-off of plain OLS across both predictor sets. Source: Authors’ calculations.
Figure 4: MSPE ratio of Ridge–RFF relative to the driftless random walk
(no-intercept baseline). A ratio below 1 would indicate that Ridge–RFF
achieves lower out-of-sample MSPE than the random walk. No curve crosses
below 1 for either predictor set. Layout and symbols as in Figure 3.
(a) Taylor rule fundamentals
(b) Traditional monetary fundamentals
This figure contains two panels plotting the ratio MSE(RFF)/MSE(RW) across the regularization spectrum. Panel (a) displays Taylor rule fundamentals. The x-axis shows log10 (z) from 0 to 6, and the y-axis shows the MSPE ratio from approximately 1.00 to 1.35. A horizontal dashed line at ratio = 1 marks parity with the random walk. Three lines represent training windows: T = 12 months (blue), T = 60 months (orange), and T = 120 months (green). Triangle markers indicate the z value minimizing each ratio. All three curves remain entirely above the parity line across the full spectrum. The T = 12 curve starts highest at approximately 1.30–1.35 at low z, declines monotonically, and converges to 1 asymptotically as z approaches 6. The T = 60 and T = 120 curves are more compressed, starting near 1.10–1.15 at low z and declining gradually to approximately 1.04–1.06 at log10 (z) = 3 (the equity baseline), then continuing toward 1 at higher z. Panel (b) shows traditional monetary fundamentals with similar axes. All curves again remain above 1 throughout. The T = 12 curve (blue) exhibits the largest gap from the random walk, starting near 1.35 at low z and declining toward 1 at high z. The T = 60 (orange) and T = 120 (green) curves show smaller but persistent gaps, ranging from approximately 1.15–1.25 at low z to 1.07–1.10 at log10 (z) = 3, gradually approaching 1 only as z exceeds 105 . Note: No curve crosses below 1 for either predictor set, confirming that Ridge-RFF never achieves lower MSPE than the random walk at any regularization level. All curves converge to 1 asymptotically as z → ∞. Source: Authors’ calculations.
Figure 5: Effective complexity ratio df(λ)/T as a function of
log10(z). Lines
report the cross-currency mean for each training window T ∈ {12, 60, 120}. The vertical
dashed line marks the Case 1 KMZ calibration zequity = 1, 000. The
horizontal dotted line marks the OLS limit (df/T = 1).
(a) Taylor rule fundamentals
(b) Traditional monetary fundamentals
This figure contains two panels showing cross-currency mean effective complexity ratios. Panel (a) presents Taylor rule fundamentals. The x-axis displays log10 (z) from 0 to 6, and the y-axis shows df(λ)/T from 0 to 1.0. A horizontal dotted line at df/T = 1 marks the OLS limit, and a vertical dashed line at log10 (z) = 3 marks the KMZ calibration (z = 1,000). Three lines represent training windows: T = 12 months (blue), T = 60 months (orange), and T = 120 months (green). All curves decline monotonically from left to right. The T = 12 curve starts near df/T ≈ 0.95 at log10 (z) = 0, crosses approximately 0.24 at the vertical dashed line (z = 1,000), and continues declining toward zero at high z. The T = 60 and T = 120 curves start lower (approximately 0.85 and 0.80 respectively), cross the vertical line at approximately 0.05 and 0.04, and decline toward zero at similar rates. Panel (b) shows traditional monetary fundamentals with identical axes. The pattern is qualitatively similar but with uniformly lower effective complexity. The T = 12 curve (blue) starts near 0.75, crosses the vertical line at approximately 0.14, and declines toward zero. The T = 60 (orange) and T = 120 (green) curves show even lower effective complexity throughout, crossing the vertical line at approximately 0.03 and 0.014 respectively. Note: Under the KMZ specification (z = 1,000), effective complexity ranges from df/T ≈ 0.24 at T = 12 to df/T ≈ 0.04 at T = 120 for Taylor rule fundamentals, confirming the specification operates close to the random walk limit in effective parameter space. Source: Authors’ calculations from Ridge regression effective degrees of freedom at each rolling window.
Figure 6: Average effective complexity ratio $\overline{\mathrm{df}(\lambda)/T}$ by
training window and regularisation case. Blue bars correspond to Case 1
(KMZ, z = zequity = 1, 000
fixed); orange bars to Case 2 (λ = 1, 000 fixed, z = λ/T). Numbers
above each bar report $\overline{\mathrm{df}(\lambda)/T}$ (bold)
and the implied effective degrees of freedom $\overline{\mathrm{df}(\lambda)}$ (in
parentheses). The horizontal dotted lines mark the OLS limit (df/T = 1) and the random walk limit
(df/T = 0).
(a) Taylor rule fundamentals
(b) Traditional monetary fundamentals
This figure contains two panels showing bar charts comparing effective complexity under two regularization cases. Panel (a) displays Taylor rule fundamentals. The x-axis shows three training windows (T = 12, 60, 120 months), and the y-axis shows df(λ)/T from 0 to approximately 0.6. Two horizontal dotted lines mark the OLS limit (df/T = 1) and the random walk limit (df/T = 0). For each training window, two bars are shown: blue bars for Case 1 (KMZ, z = 1,000 fixed) and orange bars for Case 2 (λ = 1,000 fixed). Under Case 1, the bars show declining effective complexity: T = 12 has df/T = 0.239 (2.9 effective df), T = 60 has 0.046 (2.8 effective df), and T = 120 has 0.040 (4.8 effective df). Under Case 2, the bars are substantially higher: T = 12 shows 0.571 (6.9 effective df), T = 60 shows 0.519 (31.1 effective df), and T = 120 shows 0.497 (59.6 effective df). Panel (b) shows traditional monetary fundamentals with the same structure. Under Case 1, the bars indicate T = 12 with df/T = 0.136 (1.6 effective df), T = 60 with 0.024 (1.4 effective df), and T = 120 with 0.014 (1.7 effective df). Under Case 2, T = 12 shows 0.263 (3.2 effective df), T = 60 shows 0.095 (5.7 effective df), and T = 120 shows 0.060 (7.2 effective df). Note: Numbers above each bar report df(λ)/T in bold and the implied effective degrees of freedom df(λ) in parentheses. Case 2 allows considerably more model flexibility, yet the OOS VoC curve shows this added flexibility does not translate into positive out-of-sample R2 at any training window. Source: Authors’ calculations from Gram matrix eigenvalues at each rolling estimation window.
Figure 7: Signal strength and implied optimal regularisation across the
z grid. Left panels:
time-series mean of b̂* (see equation (13)) with cross-currency
confidence bands. Right panels: implied log10(ẑ*)
across the z grid. The
vertical dashed lines mark zequity = 1, 000. The
horizontal dotted line in the right panels marks log10(zequity) = 3
for reference.
(a) Taylor rule fundamentals
(b) Traditional monetary fundamentals
This figure contains two main sections (a and b), each with two panels showing signal strength and implied optimal regularization for different predictor sets. Section (a) presents Taylor rule fundamentals. The left panel shows the time-series mean of signal strength (b̂∗ ) on the y-axis (ranging from 0 to approximately 4 × 10−8 ) plotted against log10 (z) on the x-axis
(0 to 6). Three lines represent training windows: T = 12 (blue), T = 60 (orange), and T = 120 (green), with shaded confidence bands around each line. A vertical dashed line marks z = 1,000. All three curves start at approximately 3–4 × 10−8 at low z and decline monotonically toward zero as z increases, converging near zero by log10 (z) = 6. The confidence bands are narrow throughout. The right panel shows log10 (ẑ ∗ ) on the y-axis (ranging from approximately 10 to 18) against log10 (z) on the x-axis. A horizontal dotted line at log10 (z ∗ ) = 3 provides reference. All three curves increase steeply from left to right, starting near 11–12 at log10 (z) = 0 and reaching approximately 15–16 by log10 (z) = 3, continuing to rise beyond 17 at the highest z values. Section (b) shows traditional monetary fundamentals with identical panel structures. The left panel displays b̂∗ curves that are uniformly lower than panel (a), starting near 1–2 × 10−8 at low z and declining to near zero by log10 (z) = 4–5. The right panel shows log10 (ẑ ∗ ) curves that rise more steeply than in panel (a), starting near 14–15 at log10 (z) = 0, crossing approximately 16–17 at the vertical dashed line (z = 1,000), and continuing toward 18–19 at the highest z values. Note: The average predictive signal per RFF feature is of order 10−8 and monotonically approaches zero as regularization increases. The implied optimal regularization exceeds log10 (ẑ ∗ ) ≈ 15 for both predictor sets, approximately 1012 times larger than the equity baseline. Source: Authors’ calculations using Stein-corrected estimator from Ridge solution at each rolling window.
Figure 8: Intercept comparison of OOS VoC curves. Each panel shows the
no-intercept baseline (left sub-panel) and the with-intercept
specification Buncic, 2025
(right sub-panel). The horizontal dashed line marks the random walk
(R2 = 0); the
vertical dotted line marks zequity. The OOS curve is
bounded above by zero under both specifications, confirming that the
intercept choice is inconsequential for the qualitative conclusions in
FX.
(a) Taylor rule fundamentals
(b) Traditional monetary fundamentals
This figure contains two sections (a and b), each with two sub-panels comparing no-intercept and withintercept specifications. Section (a) presents Taylor rule fundamentals. The left sub-panel shows the no-intercept baseline with log10 (z) on the x-axis (0 to 6) and out-of-sample R2 on the y-axis (approximately −0.35 to 0.05). A horizontal dashed line at R2 = 0 marks the random walk benchmark, and a vertical dotted line marks z = 1,000. Three lines represent training windows: T = 12 (blue), T = 60 (orange), and T = 120 (green). All curves remain below zero across the full spectrum. The right sub-panel shows the with-intercept specification with identical axes and virtually indistinguishable curve patterns. The curves overlap almost perfectly between the two specifications, confirming the intercept choice has no qualitative effect on the out-of-sample VoC curve for Taylor rule fundamentals. Section (b) shows traditional monetary fundamentals with the same two-panel structure. The left subpanel (no-intercept) displays curves ranging from approximately −0.35 to 0 across the z grid. The right sub-panel (with-intercept) shows nearly identical patterns. All curves remain below zero throughout both panels, and the positioning of curves relative to each other is preserved across specifications. Note: The OOS curve is bounded above by zero under both specifications for both predictor sets, confirming that the intercept choice is inconsequential for the qualitative statistical conclusions in FX. Source: Authors’ calculations comparing driftless (no-intercept) and drifting (with-intercept) random walk benchmarks.
Figure 9: Annualised Sharpe ratio of the equal-weighted cross-currency
portfolio as a function of log10(z), Taylor-rule
fundamentals. Lines report T ∈ {12, 60, 120} months. Triangles
mark the z value that
maximises the Sharpe ratio for each T. The horizontal dashed line marks
SR = 0; the vertical dotted line marks
zequity = 1, 000.
Left panel: no-intercept baseline. Right panel: with-intercept
specification Buncic, 2025, in
which the drift random walk serves as the benchmark. The contrast
between panels isolates the role of drift capture in generating positive
Sharpe ratios at longer training windows.
(a) No intercept (baseline)
(b) With intercept Buncic, 2025
This figure contains two panels showing Sharpe ratios as a function of log10 (z). Panel (a) displays the no-intercept baseline. The x-axis shows log10 (z) from 0 to 6, and the y-axis shows annualized Sharpe ratio from approximately −0.1 to 0.5. A horizontal dashed line at SR = 0 provides reference, and a vertical dotted line marks z = 1,000. Three lines represent training windows: T = 12 (blue), T = 60 (orange), and T = 120 (green). Triangle markers indicate the z value maximizing the Sharpe ratio for each window. All three curves show positive Sharpe ratios across most of the spectrum. The T = 12 curve rises from approximately 0.20 at low z to peak near 0.28 at log10 (z) ≈ 4, then flattens. The T = 60 curve increases from approximately 0.10 at low z to approximately 0.37 at log10 (z) = 3–4, then flattens. The
T = 120 curve shows similar behavior, rising from near 0.15 to approximately 0.38 and remaining stable at higher z. Panel (b) shows the with-intercept specification with identical axes. The pattern differs materially, especially at longer windows. The T = 12 curve (blue) remains broadly positive throughout, ranging from approximately 0.17 to 0.23. The T = 60 curve (orange) peaks at approximately 0.17 near log10 (z) = 2, declines to approximately 0.11 at z = 1,000, and falls below zero beyond log10 (z) ≈ 4–5. The T = 120 curve (green) shows the most dramatic reversal: it peaks at approximately 0.22 near log10 (z) = 2, remains near 0.21 at z = 1,000, but collapses to approximately −0.10 by log10 (z) = 4 and remains negative thereafter. Note: The contrast between panels isolates the role of drift capture in generating positive Sharpe ratios at longer training windows under the no-intercept specification. The collapse at higher z under with-intercept reflects insufficient directional signal once unconditional drift is removed. Source: Authors’ calculations from magnitude-proportional market-timing strategies.
Figure 10: Annualised Sharpe ratio of the equal-weighted cross-currency
portfolio as a function of log10(z), traditional
monetary fundamentals. Layout as in Figure 9. The
near-identical profiles across intercept specifications confirm that the
positive T = 12 Sharpe ratio
is not a drift-capture artefact, and that the weak performance at longer
windows is structural rather than specification-specific.
(a) No intercept (baseline)
(b) With intercept Buncic, 2025
This figure contains two panels showing Sharpe ratios as a function of log10 (z). Panel (a) displays the no-intercept baseline. The x-axis shows log10 (z) from 0 to 6, and the y-axis shows annualized Sharpe ratio from approximately −0.3 to 0.4. A horizontal dashed line at SR = 0 provides reference, and a vertical dotted line marks z = 1,000. Three lines represent training windows: T = 12 (blue), T = 60 (orange), and T = 120 (green). Triangle markers indicate optimal z values. The T = 12 curve is the only consistently positive series, starting near 0.31 at low z, declining to approximately 0.18 at z = 1,000, then flattening near 0.19 at higher z. The T = 60 curve remains negative throughout, ranging from approximately −0.23 at log10 (z) = 1 to approximately −0.06 at z = 1,000. The T = 120 curve starts negative at low z, crosses zero near log10 (z) = 2–3, and achieves a marginal positive value of approximately 0.05 near z = 1,000 and beyond. Panel (b) shows the with-intercept specification with similar axes. The patterns are remarkably similar to panel (a). The T = 12 curve (blue) remains positive across the full grid, peaking at approximately 0.34 near log10 (z) = 1, declining to approximately 0.18 at z = 1,000, and flattening near 0.16–0.17 thereafter. The T = 60 (orange) and T = 120 (green) curves remain near zero or negative throughout, with T = 60 reaching a maximum of approximately −0.07 at z = 1,000 and T = 120 achieving only a marginal positive value of approximately 0.01 near z = 1,000. Note: The near-identical profiles across intercept specifications confirm that the positive T = 12 Sharpe ratio is not a drift-capture artifact, and that weak performance at longer windows is structural rather than specification-specific. Source: Authors’ calculations from magnitude-proportional market-timing strategies under traditional monetary fundamentals.
Figure D1: Single-currency annualised Sharpe ratios as a function
of \(\log_{10}(z)\), Taylor-rule
fundamentals, no-intercept baseline. Each panel corresponds to a
training window \(T \in \{12, 60,
120\}\) (left to right). Thin coloured lines show individual
currencies; the thick black line is the cross-currency mean RFF Sharpe;
the dashed gold line is the OLS mean; the dotted green line is the
random walk mean. The vertical dotted line marks \(z_{\text{equity}} =
1{,}000\).
This figure contains three panels showing annualized Sharpe ratios plotted against log10 (z) for individual currencies under Taylor-rule fundamentals without an intercept. Each panel corresponds to a different training window: T = 12 months (left panel), T = 60 months (middle panel), and T = 120 months (right panel). The x-axis in all panels ranges from 0 to 6 on a log10 (z) scale, and the y-axis shows annualized Sharpe ratio ranging from approximately −0.4 to 0.6. A vertical dotted line in each panel marks the equity baseline at zequity = 1,000 (log10 (z) = 3). In the left panel (T = 12), thin colored lines represent the eight individual currencies (AUD, CAD, CHF, EUR, GBP, JPY, NOK, SEK), showing considerable cross-currency dispersion. The thick black line
represents the cross-currency mean Ridge-RFF Sharpe ratio, the dashed gold line shows the OLS mean, and the dotted green line indicates the random walk mean. The individual currency lines exhibit substantial variation, with some currencies (notably JPY and EUR) reaching Sharpe ratios near 0.4–0.5 at higher z values, while others (such as CAD and NOK) remain negative throughout most of the spectrum. The GBP line shows particularly large swings, with a pronounced trough around log10 (z) ≈ 2–3. The cross-currency mean RFF line (thick black) remains consistently above both the OLS mean (approximately 0.07) and random walk mean (approximately 0.06), ranging from about 0.11 to 0.14 across the full grid. In the middle panel (T = 60), the individual-currency dispersion narrows somewhat compared to T = 12. Several currencies maintain positive Sharpe ratios across most of the spectrum, while others remain near zero or negative. The cross-currency mean RFF line rises monotonically from slightly negative values at low z (around log10 (z) = 0–1) to converge with the OLS mean of approximately 0.17–0.18 at and beyond the equity baseline. The random walk mean remains lower at approximately 0.17. The pattern shows that stronger regularization at this window compresses individual-currency variation and brings the mean closer to the OLS benchmark. In the right panel (T = 120), the cross-currency dispersion is further compressed relative to shorter windows. The individual currency lines cluster more tightly around the mean, though some currencies (JPY, CHF) maintain notably positive Sharpe ratios while others (EUR, CAD) remain negative. The crosscurrency mean RFF line tracks closely with the OLS mean (approximately 0.17–0.18) throughout most of the spectrum, particularly at higher z values beyond the equity baseline. The convergence of the mean RFF curve to the OLS benchmark at high z confirms that the equal-weighted portfolio result reflects a genuine cross-currency average rather than outlier-driven performance. Note: Each panel plots single-currency Sharpe ratios (thin colored lines) alongside cross-currency means for Ridge-RFF (thick black), OLS (dashed gold), and random walk (dotted green). The vertical dotted line marks zequity = 1,000. Source: Authors’ calculations from magnitude-proportional market-timing strategies.
Figure D2: Single-currency annualised Sharpe ratios as a function
of \(\log_{10}(z)\), traditional
monetary fundamentals, no-intercept baseline. Layout as in Figure D1.
This figure contains three panels showing annualized Sharpe ratios plotted against log10 (z) for individual currencies under traditional monetary fundamentals without an intercept. The panel structure and axis ranges are identical to Figure D1: three panels for T ∈ {12, 60, 120} months (left to right), with x-axis ranging from 0 to 6 on a log10 (z) scale and y-axis showing annualized Sharpe ratio from approximately −0.6 to 0.6. A vertical dotted line marks zequity = 1,000 in each panel. In the left panel (T = 12), the cross-currency dispersion is substantially wider and more erratic than under Taylor-rule fundamentals. Individual currency lines (thin colored) show extreme variation, with some currencies exhibiting Sharpe ratios exceeding 0.5 at certain z values while others plunge below −0.4. The cross-currency mean RFF line (thick black) starts at approximately 0.18 at low z, declines monotonically across the grid to approximately 0.07 at the equity baseline (log10 (z) = 3), and remains near that level for higher z values. The OLS mean (dashed gold) remains approximately 0.02 throughout, well below the RFF mean at this window. The random walk mean (dotted green) is similarly low. The pattern indicates that traditional monetary fundamentals generate positive mean returns at T = 12 primarily through crosscurrency aggregation rather than consistent individual-currency performance. In the middle panel (T = 60), the mean RFF line (thick black) is negative across most of the spectrum, with a pronounced trough near log10 (z) ≈ 1–2 where it reaches approximately −0.2 to −0.25. This trough is driven in part by GBP, which exhibits an extreme negative excursion at intermediate z values. The mean RFF rises toward zero by the equity baseline but does not cross zero convincingly within the plotted grid. Individual currency lines show substantial dispersion, with several currencies maintaining negative Sharpe ratios throughout. The OLS mean is near zero, and the random walk mean is similarly positioned. The pattern confirms that traditional monetary fundamentals offer little reliable directional accuracy at this window. In the right panel (T = 120), the mean RFF line starts near zero at low z, gradually rises as z increases, and approaches the OLS mean of approximately 0.12 only at very high z values beyond log10 (z) = 5–6. The mean RFF remains below the OLS benchmark throughout most of the spectrum. Individual currency
dispersion remains wide, with several currencies maintaining negative Sharpe ratios while others achieve modest positive values. The random walk mean is lower than both RFF and OLS throughout. Note: Layout as in Figure D1. The substantially wider cross-currency dispersion and more erratic individual-currency profiles under traditional monetary fundamentals, relative to Taylor-rule fundamentals, confirm that traditional predictors offer weaker and less stable directional signal. Source: Authors’ calculations from magnitude-proportional market-timing strategies.
Figure D3: Single-currency annualised Sharpe ratios as a function
of \(\log_{10}(z)\), Taylor-rule
fundamentals, with-intercept specification. Layout as in Figure D1. Compare with Figure D1 (no-intercept
baseline).
This figure contains three panels showing annualized Sharpe ratios plotted against log10 (z) for individual currencies under Taylor-rule fundamentals with an intercept included. The panel structure matches Figures D1 and D2: three panels for T ∈ {12, 60, 120} months, with x-axis from 0 to 6 on a log10 (z) scale and y-axis showing annualized Sharpe ratio from approximately −0.4 to 0.6. A vertical dotted line marks zequity = 1,000 in each panel. In the left panel (T = 12), the individual-currency profiles (thin colored lines) remain broadly positive across the full z grid, similar to the no-intercept baseline. The cross-currency mean RFF line (thick black) fluctuates near 0.07–0.10, marginally above the OLS mean (dashed gold, approximately 0.05) and the random walk mean (dotted green, approximately 0.05). Individual currencies such as JPY and some others maintain positive Sharpe ratios throughout, while GBP and CAD are occasionally negative. The near-invariance of the short-window profiles between intercept specifications confirms that the positive T = 12 performance reflects genuine volatility-timing behavior rather than drift capture. In the middle panel (T = 60), the mean RFF line is near zero or slightly positive at low z but declines toward zero and below as z rises past the equity baseline. The OLS mean (approximately 0.08) remains consistently above the RFF mean for most of the grid. Individual currency dispersion is notable, with some currencies maintaining positive Sharpe ratios while others turn negative at higher z values. The decline in mean RFF Sharpe with increasing z at T = 60 contrasts sharply with the positive, rising pattern under the no-intercept specification, exposing the drift-capture mechanism operative at intermediate windows. In the right panel (T = 120), the mean RFF line starts near 0.05 at low z but declines through zero and turns negative around the equity baseline, lying clearly below the OLS mean (approximately 0.10) across the grid. Individual currencies show mixed performance, with substantial dispersion. Several currencies maintain negative Sharpe ratios throughout higher z values. The collapse of aggregate performance at T = 120 under the with-intercept specification, relative to the positive and rising pattern under no-intercept, confirms that the no-intercept performance at this window was substantially driven by drift capture rather than genuine conditional directional signal. Note: Compare with Figure D1 (no-intercept baseline). The sharp deterioration of mean RFF Sharpe ratios at T = 60 and T = 120 under the with-intercept specification, relative to no-intercept, isolates the drift-capture mechanism at longer training windows. Source: Authors’ calculations from magnitude-proportional market-timing strategies under with-intercept specification.
Figure D4: Single-currency annualised Sharpe ratios as a function
of \(\log_{10}(z)\), traditional
monetary fundamentals, with-intercept specification. Layout as in
Figure D1. Compare with Figure D2 (no-intercept
baseline).
This figure contains three panels showing annualized Sharpe ratios plotted against log10 (z) for individual currencies under traditional monetary fundamentals with an intercept included. The panel structure is identical to Figures D1–D3: three panels for T ∈ {12, 60, 120} months, with x-axis from 0 to 6 on a log10 (z) scale and y-axis showing annualized Sharpe ratio from approximately −0.6 to 0.6. A vertical dotted line marks zequity = 1,000 in each panel. In the left panel (T = 12), the with-intercept individual-currency profiles are qualitatively similar to the no-intercept baseline. The cross-currency mean RFF line (thick black) is positive in the range 0.05–0.10 and broadly stable across the z grid, slightly above the OLS mean (dashed gold, approximately 0.05). Individual currencies (thin colored lines) show substantial dispersion, with several maintaining positive Sharpe ratios while others remain negative. The random walk mean (dotted green) is similarly low. The relative stability
of the T = 12 profiles between intercept specifications confirms that the short-window positive returns under traditional fundamentals are not primarily driven by drift capture. In the middle panel (T = 60), the mean RFF line hovers near zero or slightly below across most of the spectrum, while the OLS mean (approximately 0.08) lies clearly above the RFF mean. Individual currency lines show wide dispersion, with several currencies exhibiting substantial negative excursions at certain z values. The overall pattern is qualitatively similar to the no-intercept baseline at this window, indicating that the weak performance of traditional fundamentals at T = 60 is structural rather than specificationdependent. In the right panel (T = 120), the mean RFF line remains near zero or slightly negative throughout the spectrum, while the OLS mean (approximately 0.14) is consistently higher. Individual currency dispersion remains wide, with mixed performance across currencies. The mean RFF approaches the OLS mean only at very high z values beyond log10 (z) = 5–6. The similarity between intercept specifications at T = 60 and T = 120 reflects the weaker unconditional drift content of traditional monetary predictors relative to Taylor-rule fundamentals, which limits the drift-capture channel through which the zero-intercept restriction generates spurious positive performance. Note: Compare with Figure D2 (no-intercept baseline). The near-identical profiles across intercept specifications confirm that traditional monetary fundamentals’ weak performance at T = 60 and T = 120 is structural rather than an artifact of the intercept choice. Source: Authors’ calculations from magnitude-proportional market-timing strategies under with-intercept specification.
Figure F1: Time-series of effective complexity ratio df(λ)/T. Each figure shows
Case 1 (z = 1, 000 fixed; left
sub-panel) and Case 2 (λ = 1, 000 fixed; right sub-panel).
Lines correspond to T ∈ {12, 60, 120} months (blue,
orange, green respectively). Shaded bands report cross-currency standard
deviations. The horizontal dotted line marks the OLS limit (df/T = 1).
(a) Taylor rule fundamentals
(b) Traditional monetary fundamentals
This figure contains two main sections (a and b), each showing the time-series evolution of the effective complexity ratio df(λ)/T at each rolling estimation window. Each section contains two sub-panels: the left sub-panel displays Case 1 (KMZ specification with z = 1,000 fixed), and the right sub-panel displays Case 2 (fixed-λ specification with λ = 1,000 fixed). Section (a) presents Taylor rule fundamentals. In the left sub-panel (Case 1), the x-axis spans the time period from approximately 1975 to 2024, and the y-axis shows df(λ)/T from 0 to 1.0. Three lines represent training windows: T = 12 months (blue), T = 60 months (orange), and T = 120 months (green), with shaded bands around each line indicating cross-currency standard deviations. A horizontal dotted line marks the OLS limit at df/T = 1. Under Case 1, effective complexity is nearly constant across the sample for all three training windows. The T = 12 line (blue) remains approximately 0.25 throughout the sample with a mild downward drift toward the end, declining to approximately 0.20–0.22 by 2024. The T = 60 line (orange) remains approximately 0.05 throughout, and the T = 120 line (green) remains approximately 0.04 throughout. The narrow shaded bands indicate low cross-currency variation. The mild downward drift visible for T = 12 reflects a gradual increase in collinearity among RFF features over time. In the right sub-panel (Case 2), effective complexity is substantially higher and exhibits a marked downward trend across the sample, most pronounced for T = 12. The T = 12 line (blue) begins near 0.85 in the late 1970s and declines progressively to approximately 0.30–0.35 by 2024, with a notable trough around 2008–2009 that coincides with the global financial crisis. The T = 60 line (orange) and T = 120 line (green) start at different levels (approximately 0.60 and 0.55 respectively) but converge toward each other over time, both ending near 0.30–0.40 by 2024. The shaded bands are wider than in Case 1, indicating greater cross-currency variation under the fixed-λ specification. Section (b) shows traditional monetary fundamentals with identical panel structures. In the left subpanel (Case 1), the three training window lines remain approximately constant across the sample, with T = 12 (blue) near 0.14, T = 60 (orange) near 0.024, and T = 120 (green) near 0.014 throughout. The pattern is qualitatively similar to Taylor rule fundamentals but at uniformly lower levels, reflecting the lower dimensionality of the traditional predictor set. In the right sub-panel (Case 2), the downward trend is sharper than under Taylor rule fundamentals. The T = 12 line (blue) peaks above 0.55 in the early 1980s before declining steeply to approximately 0.15– 0.25 by the mid-1990s and remaining subdued thereafter. The T = 60 (orange) and T = 120 (green) lines
show little effective complexity throughout the sample, remaining below 0.15 for most of the period and converging toward 0.05–0.10 by 2024. The secular decline in Case 2 effective complexity for traditional monetary fundamentals is more pronounced than for Taylor rule fundamentals, consistent with the weaker and more rapidly deteriorating signal strength of the traditional predictor set documented in Section 5.4 of the main text. Note: Each section shows Case 1 (z = 1,000 fixed; left sub-panel) and Case 2 (λ = 1,000 fixed; right sub-panel). Lines correspond to T ∈ {12, 60, 120} months (blue, orange, green respectively). Shaded bands report cross-currency standard deviations. The horizontal dotted line marks the OLS limit (df/T = 1). The secular decline in Case 2 effective complexity for both predictor sets is broadly consistent with the documented reduction in FX fundamental predictability over time. Source: Authors’ calculations from Gram matrix eigenvalues at each rolling estimation window.