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Friday, March 2, 2018

Bibliography: Financial Turbulence and Absorption Ratio by Mark Kritzman

[1] Financial Turbulence


Skulls, Financial Turbulence, and Risk Management
Mark Kritzman, CFA, and Yuanzhen Li
https://www.cfapubs.org/doi/abs/10.2469/faj.v66.n5.3

- An original paper dealing with the concepts and equation of the Financial Turbulence
- A brief introduction, not a research paper

- "The only problem with diversification is that it's never been tried," said Mark Kritzman, (because volatility and correlation are not stable at all)

Turbulence vs. VIX
- “Turbulence is a measure of statistical unusualness that takes into account both the magnitude of returns and how they interact with one another,” said Kritzman. “A period is deemed turbulent if either the returns are different in magnitude from their norm or if the assets interact in an uncharacteristic way.”
- In other words, turbulence is a statistical measure of both volatility and correlation. It differs from
narrower metrics such as the VIX index, which measure only one asset class (the S&P 500 in the case of the VIX) and don’t take into account correlations across asset classes.
- By incorporating information about correlations, turbulence indicates—in a way that volatility cannot—whether markets are decoupling or converging. This is important because when assets act uncharacteristically, hedging and other investment strategies that rely on consistent correlations may not work.
- Measures such as VIX have additional shortcomings, according to Kritzman. They are only available for asset classes that have liquid option markets, and they are forward-looking measures, so they don’t measure what’s actually going on now.

Turbulence Suggests When to Trim Risky Assets
- Turbulence tends to be persistent. “Even if you can’t forecast when turbulence will begin, you know it will continue for a while. The other indicators are close to random, so you don’t know whether they’ll disappear,” said Kritzman.
- During turbulent periods investors flee to safety, and risk-adjusted returns are substantially lower than in non-turbulent periods. So it makes sense to scale back on risky investments in favor of safe assets like U.S. Treasury bonds when turbulence is afoot.

More Implications for Advisors
- With portfolios that rely on traditional measures of volatility and correlation, “You’re getting diversification exactly when you don’t need it, and you’re getting unification when you don’t want it,” Kritzman said.



[2] Absorption Ratio


Principal Components as a Measure of Systemic Risk
MARK KRITZMAN, YUANZHEN LI, SEBASTIEN PAGE, AND ROBERTO RIGOBON
http://www.iinews.com/site/pdfs/JPM_Summer2011_StateStreet.pdf

- An original paper dealing with the concepts and equation of the Absorption Ratio



[3] Financial Turbulence & Absorption Ratio

Risk Regime Investing

Mark Kritzman
https://www.top1000funds.com/wp-content/uploads/2012/10/4_Mark-Kritzman-Windham.pdf
- Risk Regime Investing is an investment process which is driven by proprietary measures of financial turbulence and systemic risk(absorption ratio).

- Financial turbulence measures the statistical unusualness of a set of returns given their historical pattern of behavior.
- Implied systemic risk, through a statistic called the absorption ratio, measures the fraction of market variability that is explained by a subset of the most important factors.
- Financial turbulence is the statistical unusualness of a set of returns given their historical pattern of behavior, including (1) extreme price moves and (2) decoupling of correlated assets and/or convergence of uncorrelated assets.
- Important empirical features of financial turbulence: (1) returns to risk are lower during turbulent periods, (2) although it may arrive unexpectedly, turbulence tends to be persistent

- Systemic risk (absorption ratio): a measure of the extent to which markets are unified or tightly coupled, which we call the absorption ratio. When markets are tightly coupled, they are more fragile in the sense that negative shocks propagate more quickly and broadly than when markets are loosely linked.
- Important empirical features of systemic risk: (1) systemic risk (absorption ratio) tends to rise prior to the most significant market drawdowns, (2) systemic risk (absorption ratio) can be a leading indicator of financial turbulence.


REGIMES, TURBULENCE AND BRITTLENESS – A NEW WAY OF LOOKING AT RISK
Andrew Harma
http://www.cfsgam.com.au/uploadedFiles/Content/Funds_-_Investment_strategies/Asset_Class_overview/Multi_asset_solutions/Multi_asset_research_papers/MAS%20Research%20Paper%20Issue%208_%20Regimes%20Turbulence%20and%20Brittleness%20-%20A%20new%20way%20of%20looking%20at%20risk.pdf

- A brief introduction, not a research paper, which covers both Financial Turbulence and Absorption Ratio

- Over time financial market relationships evolve and change. These changes, or shifts, can have significant impact on the expected risk and returns of various investments. Traditional risk measures, such as volatility, correlation and betas, have fallen short in timely identification of these shifts.
Allocating capital in a changing financial world
- The investment challenge is to monitor and understand the changing financial landscape and allocate capital accordingly. Shifts in market behavior where relationships change or no long hold, which we would describe as a regime shift, provide significant challenges to asset allocation.
- As asset allocators we constantly monitor a large range of risk measures and indicators with turbulence and absorption ratio (Although the author used brittleness instead of the original word "absorption ratio," the original word is used here.) being two measures of capital market structure.
Do you have the time?
- While there is clearly no silver bullet, a "set-and-forget" asset allocation approach often falls short due to the timing of harmful capital market events relative to a particular investor's time horizon.

Risk is evolutionary, not stationary
- The traditional measures of risk, such as betas, volatilities, correlations, and Value-at-Risk, provide forward-looking or historical point estimates of portfolio characteristics.
- Large changes in the input assumptions can have a meaningful impact on the results and resulting interpretations, or indeed if the future is unlike the past.
- Correlations are not stationary and exhibit asymmetry.
- There is a large body of evidence showing that downside correlations for equity markets are much higher than upside correlations.
- Extreme movements, which focus on the tails of the return distribution, large negative movements result in a much higher correlation than large positive movements.
- The rationale for dynamic reallocation of capital is reinforced by Longin and Solnik's (2001) findings that correlations asymmetries are exacerbated amongst small, value and negative momentum stocks.
Which 'normal'? Old nomal, new normal or abnormal?
- A significant challenge to portfolio construction and risk management is the concept of regime shifts. For example, a shift could be caused by unanticipated changes in growth, inflation, monetary policy, regulation, brittleness of the financial system or other secular shifts. The challenge for investors is to be able to identify regime shifts and determine the appropriate action, which could be a conscious decision of no-action.
- Regime shifts result in different asset returns, volatilities and asset correlations (not to mention autocorrelations). To help us monitor regime shifts, we first need to determine an appropriate measure of abnormality in data or process for the identification of outliners. Mahalanobis (1927, 1936) defines a distance measure that was prompted by requirement to compare the similarilities, or lack therefof, of human skulls. We use this measure to calculate the degree of uncharacteristic behavior within financial markets, capturing extreme price movements and changing relationships. Kritzman and Li (2010) coined the application of this measure to financial markets 'Turbulence'.
- Although historical identification of crises is informative, it would be more useful to identify when a regime shift occurs. This allows for a timely review of any assumptions, such as volatilities and correlations, and the resulting allocations. A portfolio built on average market assumptions will become inappropriate when market behavior changes.
Contagion: are my exposures unified or diversified?
- The holy grail of asset allocation is the identification of uncorrelated compounding assets. So being able to find uncorrelated return and risk drivers would be significantly meaningful. Or in a risk context, the reverse would be informative: how unified are my exposures?
- We use principal components analysis (PCA) to decompose the variation in returns into uncorrelated factors that explain as much of the variation in returns as possible. This quantifies the degree to which performance is explained by the first n factors. Kritzman et al. (2011) used the moniker of ‘Absorption’ to describe the application of PCA to financial market returns. We prefer ‘Brittleness’ as a more intuitive description. The quantum these n factors can explain is called the brittleness ratio. As the brittleness (absorption) ratio increases capital markets become more fragile, increasing the likelihood of shocks propagating through markets as fewer factors drive returns.
- Taking the brittleness ratio we again use regime shifts to provide us with a quantitative measure of the unification of financial markets.
- When capital markets become coupled, broad portfolio protection strategies and tail risk strategies are likely to become meaningful.

Why do we care?
- Although there may be strong fundamental reasons underlying an investment thesis. At times the macro environment is going to dominate: momentum can grow; cheap assets can get cheaper; and expensive assets can continue to rise in value. It is very difficult to design a portfolio based on irrational behaviour, although, it is possible to ensure that it is resilient.
- The reason why we monitor turbulence and brittleness is to capture an additional dimension of risk. As asset allocators we closely monitor valuations, carry, momentum and other meaningful inputs such as supply and demand; central bank action and the political landscape to name a few. But these are only useful if we can understand them in the current market context.


Wednesday, February 21, 2018

Python Desktop Application Mock-up on macOS

Before you start, please refer to the following page to set up your Python environment.
http://thefjg.blogspot.com/2018/02/python-machine-learning-deep-learning.html

Start Terminal on macOS.

Install a light Web framework Flask:
pip3 install Flask

Install the latest version of node.js:
brew uninstall node
brew install node
brew link --overwrite --dry-run node


Check the installation of node.js.
which node
which npm


Install Electron for desktop applications backed by Web technologies like HTML, CSS, JavaScript.
npm install -g electron-prebuilt

Check the installation of Electron.
which electron

Creat a ElectronApp holder.
Move to the holder on Terminal, by using cd commands.
npm init -y

Now package.json file is created.


Start Atom.

File > Add Project Holder > (Select your ElectronApp holder.)

Update the package.json file as follows:

{
  "name": "ElectronApp",
  "version": "0.0.0",
  "main": "main.js",
  "scripts": {
    "test": "echo \"Error: no test specified\" && exit 1"
  },
  "keywords": [],
  "author": "",
  "license": "ISC",
  "description": ""
}


Create (Atom > File > New File) and save the main.js file (Atom > File > Save As...) as follows:

'use strict';

// Modules for application control
var electron = require('electron');
var app = electron.app;
var BrowserWindow = electron.BrowserWindow;

// to avoid GC (Gabage Collection) of mainWindow, make it global
let mainWindow;

// Terminate if all the windows are closed
app.on('window-all-closed', function() {
  if (process.platform != 'darwin') {
    app.quit();
  }
});

// Execute after the Electron's initialization
app.on('ready', function() {
  // Show main screen
  mainWindow = new BrowserWindow({
    width: 800,
    height: 600
  });
  mainWindow.loadURL('file://' + __dirname + '/index.html');

  //Terminate the application when closing the window
  mainWindow.on('closed', function() {
    mainWindow = null;
  });
});


Create (Atom > File > New File) and save the index.html file (Atom > File > Save As... > index.html) for the desktop application's interface as follows:

Back to your Terminal on macOS.

Move up the directory and select/run the ElectronApp holder.
cd ..
electron ElectronApp/


Python Machine Learning (Deep Learning) Runtime Environment Set-up on macOS

[1] Intended Readers

This article is for:
  • Python beginners
  • Python single user (for yourself, but not for a team/project with virtual environments)

[2] System Environment

macOS High Sierra
Version 10.13.3

[3] Installing Package Manager: Homebrew

  • A package manager enable you to install a package software automatically via the internet.
  • You don't need to search the web, download files, and then install them on your computer.

Access the website:
https://brew.sh

Copy a script:
/usr/bin/ruby -e "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/master/install)"

Start Terminal on your macOS.
Paste the script above and press return.
Enter password of your macOS and press return.

After a few minutes, check whether Homebrew is correctly installed. On Terminal, type and enter:
which brew
It returns:
/usr/local/bin/brew


[4] Installing Python

which python3
If it returns like the following, then you don't need to re-install Python:
/Library/Frameworks/Python.framework/Versions/3.6/bin/python3

If it does not return anything, then type the following Homebrew command on Terminal to (re-)install Python.
brew install python3

Type on Terminal:
which python3

It reruns:
/Library/Frameworks/Python.framework/Versions/3.6/bin/python3

When you install Python via Homebrew, then you also install pip3,
which pip3

It returns:
/Library/Frameworks/Python.framework/Versions/3.6/bin/pip3

pip3 is a tool to manage libraries for Python.


Homebrew manages software such as python3, pip3, on macOS. pip3 manages libraries for python3.

On Terminal, type and return:
python3 --version
Python 3.6.4

On Terminal, install useful libraries for Python:
pip3 install numpy  # linear algebra
pip3 install scipy  # mathematical formula manipulation
pip3 install matplotlib   # drawing, plotting
pip3 install pandas  # data(base) manipulation

pip3 install scikit-learn # machine learning package

pip3 install tensorflow # Google Tensorflow (deep learning libraries )

python
>>> import tensorflow as tf
>>> hello = tf.constant('Hello, TensorFlow!')
>>> sess = tf.Session()
>>> print(sess.run(hello))
b'Hello, TensorFlow!'
# TensorFlow successfully installed.
>>> exit()

pip3 install chainer # Chainer (deep learning libraries made in Japan)

pip3 install jupyter # Easy Python running environment

[5] Installing A Text Editor w/ Runtime Environment: Atom

Access:
https://atom.io

Download an installation file.
Copy the file and paste it to the Applications folder on your macOS.

Open Atom.

Install a Package. Or
Menu > Atom > Preferences > Install


Enter terminal on the "Search packages" textbook and press Packages.
Find "platformio-ide-terminal" and press Install. (Note: You need to install Xcode before this.)

Now you can start Terminal within the Atom text editor; you are able to write codes and run them within Atom. It boosts your productivity.
Press + sign in the bottom left corner of the screen.


Similarly, enter script on the "Search packages" textbook and press Packages.
Find "script" and press Install.

Now you can run your code within the Atom text editor; you are able to write codes and run them within Atom. It boosts your productivity.
Press + sign in the bottom left corner of the screen.


To create a Python executable project folder,
File > Add Project Folder...
Select your folder that contains py script files.


To create a new Python (py) file,
File > New File
File > Save As...
test.py

Write a very simple code:
print('Hello, Python!')

To run the script above,
Packages > Script > Run Script

You can see the result:
Hello, Python!
[Finished in 0.084s]


[AQR] Momentum Crashes



  • Despite their strong positive average returns across numerous asset classes, momentum strategies can experience infrequent and persistent strings of negative returns. These momentum crashes are partly forecastable. They occur in panic states, following market declines and when market volatility is high, and are contemporaneous with market rebounds.The low ex ante expected returns in panic states are consistent with a conditionally high premium attached to the option like payoffs of past losers. An implementable dynamic momentum strategy based on forecasts of momentum’s mean and variance approximately doubles the alpha and Sharpe ratio of a static momentum strategy and is not explained by other factors. These results are robust across multiple time periods, international equity markets, and other asset classes.
  • A momentum strategy is a bet on past returns predicting the cross section of future returns, typically implemented by buying past winners and selling past losers.
  • Momentum is pervasive: the academic literature shows the efficacy of momentum strategies across multiple time periods, in many markets, and in numerous asset classes.
  • In a bear market, the loser portfolio implies that momentum strategies behave like written (=short) call options on the market; that is, when the market falls, they gain a little, but when the market rises, they lose much.
  • In normal environments, consistent price momentum is both statistically and economically strong and manifest itself across numerous equity markets and a wide range of diverse asset classes.
  • However, in panic states, following multi year market drawdowns and in periods of high market volatility, the prices of past losers embody a high premium. When poor marketconditions ameliorate and the market starts to rebound, the losers experience strong gains, resulting in a momentum crash as momentum strategies short these assets.
  • We find that, in bear market states, and in particular when market volatility is high, the down-market betas of the past losers are low, but the up-market betas are very large. (like short call options) Consequently, the expected returns of the past losers are very high, and the momentum effect is reversed during these times.
    • This feature does not apply equally to winners during good times, however, resulting in any asymmetry in the winner and loser exposure to market returns during extreme times. 
  • These crash periods are predictable. We use bear market indicators and ex ante volatility estimates to forecast the conditional mean and variance of momentum signals.
  • What can explain these findings?
    • Equity: a Merton '1974) story for the option-like payoffs of equities could make sense, but the existence of the same phenomena and option-like features for futures, bonds, currencies, and commodities makes this story more challenging.
    • Alternatively, these effects can be loosely consistent with several behavioralfindings, in which in extreme situations individuals tend to be fearful and appear to focus on losses, largely ignoring probabilities.



Source
https://www.aqr.com/library/journal-articles/momentum-crashes

[AQR] Market Timing: Sin a Little



  • Is market timing a useful source of added value or a sin to be avoided?
  • We propose an interpretation that offers a practical enhancement to value timing strategies: adding a dose of momentum.
  • By that, one can improve the risk-adjusted performance of contrarian market timing strategies.
  • Valuations can drift higher or lower for years or decades, making it difficult to categorize the current market confidently as "cheap" or "expensive" without hindsight calibration, and therefore difficult to profit from such categorizations.
  • Combining value and momentum has been shown to be very effective in stock selection and cross-sectional strategies, and the combined signal intuitively represents "value with a catalyst," or patient contrarian investing, with a supportive momentum signal potentially reducing the risk of value traps or premium signals.
  • If market timing is a sin, perhaps you could--or even should--sin a little.
  • Value investors looking for a less explicit way to reduce negative exposure to short-term momentum could simply rebalance their tilts less frequently.
  • A better variant of this approach is to "slow down" the value signal using a moving average, which is equivalent to making a sequence of overlapping value bets that are each "locked in" for a long period
  • In this paper, we have examined how drifting valuations contribute to the initially puzzling gap between encouraging in-sample evidence and disappointing out-of-sample performance of value timing strategies. We have shown that the challenge of such long-term drift has been compounded by persistent negative exposure to a shorter-term time series momentum factor, which offers a practical and intuitive avenue for improving value-based tactical timing signals.
  • Our results suggest that even a small does of momentum, whether applied explicitly or just by rebalancing less frequently or smoothing valuation signals, may improve market timing decisions.
  • In summary, attractive predictive correlations do not always translate to successful timing out of sample. This has occurred both because of long-drifts in average weight (as stocks become more or less cheap for very long periods) and because a pure value strategy effectively "shorts" the successful momentum strategy. We have no cure for long drifts in average weights other than our view it's unlikely to be as bad for valuation timing in the next half century as in the last. Adding a momentum component may help investors to become more patient and successful long-term value investors.
  • Finally, we note that even a completely blend of value and momentum timing signals produces only modest long-term outperformance. Thus: sin (only) a little.
  • Timing two assets is a broader and potentially more promising strategy, as well as providing independent supporting evidence.



Source

[AQR] A Century of Evidence on Trend-Following Investing


  • Momentum (trend-following) strategy
    • Time-series (directional, asset-by-asset)
      • Going long a market with recent positive returns
      • Going short a market with recent negative returns
    • Cross-sectional (non-directional / market-neutral, assets as a whole)
      • Going long markets with recent relatively better returns
      • Going short markets with recent relatively worse returns
  • Time-series momentum strategy all the way back to 1880
    • Consistently profitable throughout the past 137 years.
    • How the strategy performs across various macroeconomic environments:
      • recession versus booms
      • war versus peacetime
      • high- versus low-interest-rate regimes
      • high- versus low-volatility periods
      • high- versus low-inflation periods
      • high- versus low-correlation periods
        • The characteristics that appears to have affected the performance the most is correlation -- the strategy has performed the best during low-correlation environments.
  • Constructing the time-series momentum strategy
    • Equal-weighted combination of 1, 3, and 12-month time-series momentum strategies for the 67 markets (four asset classes) from January 1800 to December 2016.
    • Rebalanced each month as follows.
      • Each strategy always holds either a long or short position in every market.
      • Each position is sized to target the same amount of volatility, both to provide diversification and to limit the portfolio risk from any one market.
      • The position across the three strategies are aggregated each month and scaled such that the combined portfolio has an annualized ex ante volatility target of 10% annualized.
      • Transaction costs (not for costs of "rolling" futures contracts) and fees (2% management, 20% performance subject to HWM) subtracted
  • Long-term out-of-sample evidence suggests that it is unlikely that the existence of price trends in markets is a product of statistical randomness or data mining.
  • Price trends exists in part because of long-standing behavioral biases exhibited by investors, such as anchoring and herding, as well as the trading activity of non-profit-seeking participants, such as central banks and corporate hedging programs.
  • The time-series momentum strategy has performed the best during low-correlationenvironments. Said differently, when correlations are high, there are fewer truly different trends to bet on.
  • Conclusion: Trend-following investing has performed well in each decade for more than a century, as far back as we can get reliable return data for several markets. Our analysis provides significant out-of-sample evidence across markets and asset classes beyond the substantial evidence already in the literature. Further, we find that a trend-following strategy performed relatively similarly across a variety of economic environments and provided significant diversification benefits to a traditional allocation. This consistent long-term evidence suggests that trends are pervasive features of global markets.



Source

Sunday, February 18, 2018

[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.


Sources

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

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