Tim Olson
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58 lines
3.3 KiB
Markdown
58 lines
3.3 KiB
Markdown
---
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description: "FX combined strategy that blends the HP-filter moving-average momentum signal with the carry trade using minimum-variance portfolio weights to reduce overall volatility."
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tags: [fx, carry, momentum, combination, minimum-variance]
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---
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# Momentum & Carry Combo
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**Section**: 8.4 | **Asset Class**: FX | **Type**: Combination (Momentum + Carry)
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## Overview
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Both the FX momentum strategy (Section 8.1) and the carry trade (Section 8.2) capture distinct but complementary sources of FX returns. Combining them in an optimally weighted portfolio can reduce overall variance relative to either strategy alone. The weights are derived by minimising the historical variance of the combined return given the sample variances and correlation of the two strategy return series.
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## Construction / Mechanics
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Let R₁(t_s) and R₂(t_s) be the historical return series of the momentum and carry strategies respectively. Define their sample statistics:
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```
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σ₁² = Var(R₁(t_s)) (445)
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σ₂² = Var(R₂(t_s)) (446)
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ρ = Cor(R₁(t_s), R₂(t_s)) (447)
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```
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The combined portfolio return is:
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```
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R(t_s) = w₁ R₁(t_s) + w₂ R₂(t_s) (448)
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w₁ + w₂ = 1 (449)
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Var(R(t_s)) → min (450)
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```
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Solving the constrained minimisation yields the minimum-variance weights:
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```
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w₁ = (σ₂² - σ₁σ₂ρ) / (σ₁² + σ₂² - 2σ₁σ₂ρ) (451)
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w₂ = (σ₁² - σ₁σ₂ρ) / (σ₁² + σ₂² - 2σ₁σ₂ρ) (452)
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```
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## Return Profile
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Profits from both the trend-following component (momentum captures trending FX moves) and the carry component (interest rate differential). The minimum-variance weighting reduces drawdowns when one strategy temporarily underperforms, particularly during carry unwinds when momentum may be positive.
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## Key Parameters / Signals
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| Parameter | Description |
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|-----------|-------------|
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| σ₁, σ₂ | Historical volatilities of momentum and carry strategy returns |
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| ρ | Historical correlation between the two strategy returns |
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| w₁, w₂ | Optimal weights (sum to 1); derived from sample covariance matrix |
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| Estimation window | Rolling historical window for σ₁, σ₂, ρ |
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## Variations
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- **Equal-weighted combo**: w₁ = w₂ = 0.5; simpler but ignores relative volatilities.
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- **Risk-parity combo**: weight each strategy inversely to its volatility (σ₂/(σ₁+σ₂) and σ₁/(σ₁+σ₂)).
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- **Extended combo**: include additional FX strategies (e.g., dollar carry, value) using the full sample covariance matrix (see Sections 3.6 and 4.6 for the general multi-strategy framework).
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## Notes
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- The minimum-variance weights are sensitive to the estimation window; rolling windows introduce parameter instability.
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- If ρ is close to 1, both strategies move together and diversification benefits are limited; if ρ is close to -1, the combo can nearly eliminate variance but requires large offsetting positions.
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- The combo does not necessarily improve the Sharpe ratio compared with each individual strategy; it primarily targets variance reduction.
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- Constraints (e.g., w₁, w₂ ≥ 0) may be applied to avoid short-selling a strategy that has negative expected return.
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