Basic Prerequisites: 03 · Factor-Based Allocation and Factor Investing & Factor Timing.
1. Intuition & Practical Objective
“Style” is the cross-asset generalisation of a factor. The four canonical styles that recur across every asset class are value, momentum, carry, and defensive (sometimes “quality” or “low-risk”). This page focuses on the one that is definable in every asset class without a valuation model - carry - because carry is the cleanest bridge between an asset-class world and a factor world: it prices the term structure / roll-down / dividend / interest differential that each instrument pays merely for staying still.
Carry, defined operationally: the return you earn over the next period if prices do not move. It is not a forecast of price; it is the structural yield embedded in the instrument. It generalises:
| Asset class | Carry proxy | What it prices |
|---|---|---|
| Equity | dividend yield (minus financing) | earnings distribution |
| Government bond | yield-to-maturity / roll-down (minus funding) | term premium |
| FX | interest-rate differential (long high-yield vs low-yield) | rate differential |
| Commodities | roll yield = | storage / convenience / scarcity |
| Credit | credit spread (minus expected default loss) | credit risk premium |
The key empirical fact (Koijen, Moskowitz, Pedersen & Vrugt, “Carry,” 2018): carry predicts returns in every asset class, with the same sign. High-carry assets earn more than low-carry assets, on average, because carry is a compensation for risk - and the risk is a carry crash: the strategies earn steadily and lose suddenly when the risk shows up.
The one-sentence essence. “Carry is what an instrument pays you for standing still - the same signal in bonds, FX, commodities, and equities; it is a real premium and a real crash risk, which is why it must be sized, not simply maximised.”
2. Mathematical Ground Truth & Derivations
Carry as a conditional expectation. Let be the spot and the price of a claim on . Define carry as the return under the assumption the term structure stays unchanged:
Equivalently, in a one-factor term-structure model where the future spot equals today’s forward, the expected excess return under unchanged prices is exactly the carry.
Carry in a no-arbitrage framework. Under the futures price is a martingale: . The basis therefore isolates the risk premium that the physical measure adds: . Carry is the deterministic part; the stochastic part is what you are paid to bear.
Carry as a cross-sectional factor. Standardise carry within the cross-section and go long the top, short the bottom:
This is the same construction as a value or momentum factor, which is why carry slots directly into a factor-allocation framework (page 03): compute the factor covariance , then optimise or risk-balance across value / momentum / carry / defensive.
The four styles, cross-asset (the “style” premium family):
| Style | Signal | Economic story |
|---|---|---|
| Value | cheap vs long-run fair value | over-extrapolation reversal |
| Momentum | past 12-month return | under-reaction / flows |
| Carry | yield if prices unchanged | risk premium / term structure |
| Defensive | low beta / low vol | leverage-aversion (betting-against-beta) |
Why carry diversifies the others. Value and momentum are negatively correlated; carry is closest to value (both like “cheap/high-yield”), defensive is closest to nothing. The style block therefore has low average pairwise correlation - the same diversification property that makes factor allocation efficient (page 03).
3. Computational Implementation - carry across asset classes
Runnable (numpy). It ranks six cross-asset carry proxies, simulates a “prices unchanged plus noise” return environment, and shows the carry-tilted portfolio out-earning the equal-weight one. Stdlib + numpy.
The equal-weight portfolio earns a negative annualised return () because it holds the negative-carry leg (gold) and the price noise is unhelpful over a short sample; the carry-tilted long-only portfolio earns and the long-short carry portfolio . The carry signal, not the volatile spot, does the work - and the spot noise is exactly the carry-crash channel that page 05 quantifies.
4. Failure Modes & First-Principles Breakdowns
- Carry is compensation for crash risk. The realised return distribution is negatively skewed: steady gains punctuated by rare large losses (the “carry crash”). A carry-tilted portfolio needs explicit tail hedging or leverage discipline.
- Carry and momentum disagree - by construction. Momentum buys what is rising, carry buys what is yielding; the two can be simultaneously long and short the same asset. Diversification across styles means tolerating being on the wrong side of one.
- Carry changes regime. In a funding crisis, FX carry reverses violently as the high-yielders depreciate; in a commodity shortage, roll yield can flip sign. Carry must be conditioned (page 06).
- Roll yield return. The term is the expected carry if the term structure holds; a persistent backwardation can be a symptom of scarcity, not free money.
5. References
- Koijen, Moskowitz, Pedersen & Vrugt, “Carry,” Journal of Financial Economics 127(2):197–225, 2018
- Asness, Moskowitz & Pedersen, “Value and Momentum Everywhere,” Journal of Finance 68(3):929–985, 2013
- Frazzini & Pedersen, “Betting Against Beta,” Journal of Financial Economics 111(1):1–25, 2014
- Ilmanen, Expected Returns (2011)
- Ang, Asset Management (2014)
6. Connected Graph Bridges
- Back: 03 · Factor-Based Allocation
- Forward: 05 · Failure Modes · Index Hub
- Cross-pillar: The Factor Zoo · Factor Crowding & Capacity