Carry can be defined as return for unchanged market prices and is easy to calculate in real time across assets. Carry strategies often reap risk premia and implicit subsidies. Historically, they have produced positive returns in FX, commodities, bonds and equity. Carry strategies can also be combined across asset classes to render diversification benefits. Historically, since 1990, the performance of such diversified carry portfolios has been strong, with Sharpe ratios close to 1, limited correlation to benchmark indices and less of a downside skew that FX carry trades.

*The post ties in with SRSV’s lecture on implicit subsidies.*

*The below are excerpts from the paper. Emphasis and cursive text have been added. *

### What is carry?

“The term ‘carry’ is generally associated with an FX trading strategy that borrows from a country with a low interest rate and invests in a country with a high interest rate…Generalising [*this*] concept…we define ‘carry’ as the return that an investor enjoys if all market conditions, including the asset’s price, remain the same. Put differently, __carry measures the value appreciation that accrues to the owner of an asset if there is no expected or unexpected price change__: total return is the sum of carry, expected price appreciation and unexpected price shock. Following from this equation, the estimation of carry becomes readily available and therefore requires no model assumptions.”

“We…formally define carry across multiple asset classes by observing directly the behaviour of the respective futures or forward markets:

- Assets that exhibit a term structure of
__futures in backwardation [__when market conditions remain unchanged. These assets should therefore be generally over-weighted in a carry portfolio.*downward sloped futures price curve*] should generate a positive roll yield and therefore a positive excess return - Assets that exhibit a term structure of futures in contango [
*upward sloped futures price curve*] should generate a negative roll yield and therefore a negative excess return when market conditions remain unchanged. These assets should therefore be generally under-weighted in a carry portfolio.”

“[*The panels below*] explain illustratively the above dynamics. Notice that when the market conditions remain unchanged, the term structure does not move at all and therefore the entirety of asset return is the roll yield…Evidently, __carry is just the slope of the futures curve__.”

### How to calculate carry across asset classes?

“Our first objective is therefore to define and measure carry for each asset class.”

*Carry (C _{t})for each asset class is the difference between spot price (S_{t})and the future price (F_{t}), divided by the future price.*

*C _{t} = (S_{t} – F_{t}) / F_{t}*

“[*The table below*] contains the arbitrage-free future price per asset class, as well as the value of carry…after substituting out the futures price…[*and*] explains in detail the type of yield that we can extract from each asset class:

- For FX markets, the investor receives the interest rate of the foreign country in excess of…the domestic interest rate [
*provided that covered interest parity holds*]. - For equity index markets, the investor receives the expected future dividend yield in excess of the financing cost.
- For commodity markets, the investor receives the convenience yield in excess of any storage and financing costs.
- For government bonds the investor receives two components of return: (a) the yield to maturity in excess of the financing cost (this is the so-called “term premium”) and (b) the “roll-down” of the bond across the yield curve as this approaches maturity.”

“For…our empirical analysis, we collect futures data from Bloomberg for a large cross-section of 52 assets: 20 commodities (constituents of the Bloomberg Commodity index excluding the precious metals, i.e. gold and silver), 8 ten-year government bonds, 9 FX rates (G10 pairs against USD) and 15 equity country indices…__Our simulations start in January 1990__…The sample period ends in January 2016.”

“[*The figure below*] presents the median carry for each asset class at the end of each month over the entire sample period in order to identify asset class shifts between contango and backwardation over time.”

### Why is carry indicative of expected return?

“One might wonder why a carry strategy can generate positive excess returns. What is the underlying economic rationale? Is there any risk that we are compensated for?

**FX**:.Higher Libor rates are generally associated with rising inflation, funding liquidity concerns, or consumption growth risk, which__render the higher-yielding currencies generally more vulnerable__, hence justifying the positive FX carry…**Commodities**…Keynes’s (1930) theory of “normal backwardation” suggests that__commodity producers take short futures positions in order to hedge against price drops and therefore pay a premium__to an investor that offers this insurance and takes a long position in the futures contract; this positive premium comes in the form of the carry premium…Commodity consumers take long futures positions in order to hedge against unexpected future price surges; in this scenario, they should pay a premium to the investor that offers the insurance and takes a short position, in which case contango arises…**Government Bonds**…Holding a bond up to maturity should compensate an investor with the yield-to-maturity in excess of the risk-free rate…Yield curves are typically upward sloping and therefore__the term spread is positive in order to compensate long-term bond investors for potential illiquidity risk tightening monetary policy risk and inflation risk__.**Equity Indices**…If the equity carry strategy turns out to be profitable and if it constitutes compensation for systematic risk, then surely__equity indices with higher expected dividend yield must be fundamentally riskier__…Put differently, the profitability of equity carry can be related to the equity value premium.”

### How have multi-asset carry strategies performed historically?

“__Multi-asset carry allocations benefit from the low correlation between asset-class specific carry portfolios__ and do not exhibit significant downside or volatility risk which have been traditionally associated with the FX carry strategy.”

“In constructing carry portfolios, we explore three different weighting schemes:”

**“Cross-sectional (“XS”) Carry**: the relative strength of the carry of each asset compared to all other assets in the same asset class is used in order to construct a **balanced long-short portfolio** in terms of notional exposure…Broadly speaking, __all asset classes generate positive excess returns with Sharpe ratios ranging from 0.19 for commodities to 0.85 for government bonds__…We explore the diversification benefits from pulling together all XS carry portfolios, in order to construct a multi-asset XS portfolio…The __multi-asset XS carry strategy delivers statistically strong average excess returns, generating a Sharpe ratio of 0.79, albeit at a negative skewness__…and with excess kurtosis. The volatility-targeted version of the strategy…requires an average leverage of 4x, so to achieve the required 7% level of volatility… we find that __all XS strategies (across asset classes and at the multi-asset level) exhibit very low betas against various passive broad market indices__ (MSCI World Index, Bloomberg Commodity Index, JPMorgan Aggregate Bond Index and Trade-weighted USD).”

**“Times-series (“TS”) Absolute Carry**: the sign of the carry of each asset is used to determine the type of position (long or short) in order to construct a **portfolio with explicit directional tilts**; net long when the majority of assets are in backwardation, and net short when in contango…One can think of it as the “momentum of carry”…Similar to the XS analysis, __all asset classes generate positive excess returns with Sharpe ratios ranging from 0.10 for commodities to 0.88 for government bonds__; this constitutes evidence that the level and sign of carry experience some degree of positive serial dependence…As far as the skewness of the strategies is concerned, it is only the FX carry that still exhibits large negative skewness. All other asset classes generate either close to symmetrical return distribution or positive skewness…We…combine the four TS carry portfolios into a multi-asset portfolio using inverse-volatility weights (monthly rebalancing using 100-day volatility estimates)… __The multi-asset TS carry strategy delivers a superior return profile compared to all the asset class TS portfolios__. The average excess return is statistically strong (at 1% confidence) and the Sharpe ratio is 0.99 for our sample period.”

**“Optimised** (“OPT”) Carry: both the relative strength and the sign of the carry are used in order to determine the type (long or short) as well as the gross exposure for each asset. Most importantly, the __optimised carry portfolio additionally accounts for the covariance structure between assets and asset classes__ in a way that risk allocation is optimised…The multi-asset OPT carry strategy, in its unlevered form, delivers positive average excess returns that are statistically strong (at 1% confidence) and a __Sharpe ratio of 0.96. Similar to the TS carry strategy, it is positively skewed, but it also comes along with higher kurtosis__ of 5.32…the OPT carry strategy requires more leverage (on average 7.1x) to achieve the required 7% level of volatility in its levered form.”