[AQR] Carry

Summary (w/ my own comments)

• An asset’s “carry” - expected return assuming that market conditions, including its price, stays the same
• A security’s expected return = expected “carry” + expected price return, where carry can be measured in advance (i.g., ex-ante) without an asset pricing model, namely model-free.
• Carry predicts returns both in the cross section and time series for a variety of different asset classes that include global equities, global bonds, currencies, commodities, U.S. Treasuries, credit and equity index options.
• This predictability underlies the strong returns to “carry trades” that go long high-carry and short low-carry securities, a strategy applied almost exclusively to currencies but shown here to be a robust feature of many assets.
• We decompose carry returns into static and dynamic components and analyze the economic exposures. Despite unconditionally low correlations across asset classes, we find times when carry strategies across all asset classes do poorly, and show that these episodes coincide with global recessions.
• Carry strategies are commonly exposed to global recession, liquidity, and volatility risks though none fully explains carry's premium.
• A security’s realized return = expected “carry” + expected price return + unexpected price shock = (expected “carry” & price return) + (unexpected price shock)
• Carry is a model-free characteristic directly observable ex ante from futures, synthetic futures, or forward prices makes it special.
• Expected price return must be estimated using an asset pricing model.
• Seemingly unrelated predictors of returns across different assets can be bonded together through the concept of carry.
• For instance, the carry for bond is closely related to the slope of the yield curve studied in the bond literature, plus "roll down" component that captures the price change that occurs as the bond moves along the yield curve as time passes.
• The commodity carry is akin to the basis or convenience yield.
• Equity carry is a forward-looking measure of dividend yields.
• While carry predicts future returns in every asset class with a positive coefficient, the magnitude of the predictive coefficient differs across asset classes, indicating whether carry is positively or negatively related to future price appreciation.
• In global equities, global bonds, and credit markets, the predictive coefficient is greater than one, implying that carry predicts positive future price changes that add to returns, over and above the carry itself.
• In commodity and option markets, the estimated predictive coefficient is less than one, implying that the market takes back part of the carry (although not all, as implied by Uncovered Interest Parity and the Expectation Hypothesis).
• Hence, there are commnly shared features across different carry strategies and also interesting differences.
• Carry timing strategies buy a security when the carry is positive or above its historical mean.
• Theory suggests that expected returns can vary due to macroeconomic risk, limited arbitrage, market liquidity risk, funding liquidity risk, volatility risk, and downside risk exposure. Further, we examine whether carry can be explained by other predictors of returns across global asset classes such as value and momentum.
• The returns to carry strategies cannot be explained other known global return factors such as value, momentum, and time series momentum within each asset class as well as across all asset classes. The relation between carry and these factors varies across asset classes, where carry is positively related to value and momentum in some asset classes and negatively in others. However. none of the carry exposures to value, momentum, or time series momentum is large in any asset class, and carry consistently produces positive alpha with respect to these factors. Hence, carry represents a different return predictor within and across asset classes, adding to the list of factors that drive returns across many markets.
• Crash risk cannot explain the ubiquitous returns to carry strategies as suggested by the literature on currecy carry trades (Brunnermeier et al., 2008). While it is well documented that the currency carry trade has negative skewness, this cannot be said for all carry strategies. In fact, several of the carry strategies have positive skewness and the across-all-asset-class global carry factor has negligible skewness. All carry strategies have excess kurtosis, however, indicating fat-tailed returns with large occasional profits and losses. Crash risk theories (for currencies) are unlikely to explain the general carry premium.
• Carry strategies generally tend to incur losses during times of worsened liquidy and heightened volatility. (US Treasuries of different maturities has the opposite loadings and thus, acts as a hedge against the other carry strategies during these times.) Carry returns tend to be lower during global recessions, which appears to hold uniformly across asset classes.
• Part of return premium earned on average for going long carry could be compensation for exposure that generates large losses during extreme times of global recessions. Whether these extreme times are related to macroeconomic risks and heightened risk aversion or are times of limited capital and arbitrage and funding squeezes remains an open question.
• Carry factor is a set of compensations for macro/economic (global business cycle), margin requirements and funding costs (liquidity), (downside) volatility, and limited arbitrage risks.
• Carry in various asset classes
• Carry across asset classes
• high Sharpe ratio, exposure to recessions, liquidity risk, and volatility risk
• Currency
• Yield spread. The foreign interest rate in excess of the local risk-free rate because the forward contract is a zero-cost instrument whose return is an excess return. The historical positive return to currency carry trade is a well-known violation of the uncovered interest rate parity (UIP). The UIP is based on the simple assumption that all currencies should have the same expected return, but many economic settings would imply differences in expected currency returns could arise from differences in consumption risk, crash risk, liquidity risk, and country size, where a country with more exposure to consumption or liqudity risk could have both a high interest rate and a cheaper currency.
• Only curerncy carry - negative skewness
• Global equity
• The equity carry is simply the expected dividend yield minus the local risk-free rate, multiplied by a scaling factor, which is close to one.
• Commodity
• Basis trade. The commodity carry is the expected convenience yield of the commodity in excess of the risk-free rate (adjusted for a scaling factor that is close to one).
• Global bond
• Bond carry = (yield spread to the risk-free rate) + (roll down) = (the bond's yield spread to the risk-free rate, which is also called the slope of the term structure) + (the "roll down," which captures the price increase due to the fact the bond rolls down the yield curve)
• Credit
• The definition of carry is the credit spread (the yield over the risk-free rate) plus the roll down on the credit curve.
• Option
• Shorting volatility. The size of the carry is driven by the time decay, which often leads to a negative carry, and the "roll down" on the implied volatility curve.
• A carry trade is a trading strategy that goes long high-carry securities and shorts low-carry securities. Various ways exist of choosing the exact carry-trade portfolio weights:
• A top minus bottom x% approach. Rank assets by their carry and go long the top x% of securities and short the bottom x%, with equal weights applied to all securities within the two groups, and ignore (e.g., place zero weights on) the securities in between these two extremes.
• A rank-based weighting scheme. Signal-weighted scheme that can place considerable weight on the extremes (i.e., highest and lowest rankings). Takes a position in all securities weighted by their carry ranking. This weighting scheme accounts for differences across carry signals even within the top and bottom x% and does not ignore the securities in the middle. Yet, by using ranks instead of weights that are linear in the signals, avoid the impact of outliners in the signals.
• Skewness in carry
• Negative: (strong) currency, options; (some) commodity, fixed-income
• Positive: equities, US Treasuries, and credit
• Excess Kurtosis (fat-tailed positive and negative returns) in carry
• All asset classes
• Carry is a strong predictor of expected returns, with consistently positive and statistically significant coefficients on carry, save for the commodity strategy, which can be tainted by strong seasonal effects in carry for commodities, and for call options.
• c in (23)
• c>1 (when carry is high, prices tend to appreciate more than usual)
• equities, global bond levels and slope, and credit
• c~1 (high interest rate currencies neither depreciate nor appreciate on average; hence, the currency investor earns the interest rate differential, on average)
• currencies
• c<1 (when carry is high, prices tend to appreciate more than usual as yields tend to fall)
• US Treasuries, commodities, and options.
• When a commodity has a high spot price relative to its futures price, implying a high carry, the spot price tends to depreciate on average, thus lowering the realized return on average below the carry.
• In every asset class (global equities, global fixed income, US Treasuries, commodities, currencies, credits, call options, and put options), a carry strategy provides abnormal returns above and beyond simple passive exposures to that asset class. Put differently, carry trades offer excess returns over the "local" market return in each asset class. Further, the betas are often not significantly different from zero. Hence, carry strategies provide sizeable return premia without much market exposure to the asset class itself.
• Carry vs Value and Momentum
• Equities
• Carry: positive exposure to Value
• Carry: no exposure to (cross-sectional and time series) Momentum
• Fixed Income
• Carry: positive exposure to (cross-sectional and time series) Momentum
• Commodities
• Carry: negtive exposure to Value
• Carry: positive exposure to cross-sectional Momentum
• Carry: little (almost no) exposure to time series Momentum
• Currencies
• Currency carry strategies exhibit no reliable loading on value, cross-sectional momentum, or time series momentum. Consequently, the alpha of the currency carry portfolio remains large and significant.
• Credit
• Similarly, for credit, no reliable loadings on these other factors are present and, hence, a significant carry alpha remains.
• Options
• For call options, the loadings of the carry strategies on value, momentum, and TSMOM (time series momentum) are all negative, making the alphas even larger.
• For puts, there are no reliable loadings on these other factors.
• Value, cross-sectional momentum, and time series momentum, do not capture the returns to carry.
• Trading costs: Our results cannot be explained by, and are not subsumed by, trading costs.
• Risks
• Crash and downside (business cycle) risk exposure
• some component of global carry returns can be explained by downside risk
• currencies, commodities, call and put options 
• Global liquidity and volatility risk
• Liquidity and volatility risk explain part of the carry premia across asset classes.
• On the contrary to other asset classes, US Treasuries carry strategy provides a hedge against liquidity and volatility risk, suggesting that liquidity and volatility risk are an incomplete explanation for the cross section of carry strategy returns (or, alternatively, this could be due to trandom chance or noise, which investors might not have expected ex ante)
• An aggressive interpretation concludes that carry is unexplained by downside, liquidity, or volatility risks and presents a substantiall asset pricing puzzle that rejects many theories, posibly offering a wildly profitable investment opportunity.
• A cautious interpretation can conclude that carry strategies almost uniformly load significantly on these downside, liquidity, or volatility risks that partially explains their returns and that, perhaps if better measures of these risks were available, carry's exposure to them, and if risk premia estimats were more precise, most of hte returns to carry through risk could be explained.

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Sources

https://www.aqr.com/library/journal-articles/carry
https://eorder.sheridan.com/3_0/app/orders/7342/article.php