Author: rvarb

Calendar anomalies in the stock market refer to recurring patterns or anomalies that occur at specific times of the year, month, or week, which cannot be explained by traditional financial theories. These anomalies often defy the efficient market hypothesis and provide opportunities for investors to exploit market inefficiencies. In this post, I will feature some calendar anomalies and discuss whether they work in the current market or not.

Do Calendar Anomalies Still Exist?

Calendar anomalies were discovered long ago. Reference [1] examines whether they still persist in the present-day stock market. Specifically, the author investigates the turn-of-the-month (TOM), turn-of-the-quarter (TOQ), and turn-of-the-year (TOY) effects in the US stock market.

Findings

– The paper identifies the presence of the turn-of-the-month (TOM), turn-of-the-quarter (TOQ), and turn-of-the-year (TOY) effects in the US stock market, with the TOY effect being the most prominent.

-The analysis uses panel regression models on four-day return windows for individual stocks listed on the NYSE, AMEX, and NASDAQ from 1986 to 2021.

– The TOM, TOQ, and TOY effects are found to be present, and their strength varies based on firm characteristics.

– The TOY effect primarily affects small stocks with volatile prices, indicating that individual investors may sell their losses for tax purposes before the year-end.

– Stocks with low momentum are more susceptible to the TOY effect, suggesting that institutional investors may engage in performance hedging by selling underperforming stocks.

– The calendar effects have evolved over time, with the TOM and TOY effects resurfacing in recent decades, while the TOQ effect has diminished, potentially due to increased disclosure regulations.

– Companies with low Google search volumes are significantly more impacted by all three effects, indicating a relationship between information accessibility and the magnitude of calendar anomalies.

-A trading strategy is developed to identify stocks with the highest expected returns over TOM and TOY windows. The return exceeds realistic trading costs, indicating that calendar effects can be used to construct profitable trading strategies.

In summary, calendar anomalies continue to exist in the US stock market. Furthermore, they can be exploited to gain abnormal returns. For instance, every four-day TOY window yields an average profit of 1.66% when holding all stocks exclusively over the TOY windows. Similarly, an average profit of 0.55% is generated every four-day TOM window by exclusively holding all stocks over the TOM windows.

Replication

Vahid Asghari and his team at Academic Quant Lab have replicated the strategy presented in this paper. The results and codes can be found here.

Reference

[1] Idunn Myrvang Hatlemark and Maria Grohshennig, Calendar Effects in the US Stock Market: Are they still present?, 2022, Norwegian University of Science and Technology

How End-of-Month Returns Predict the Next Month’s Performance

Reference [2] introduced a novel calendar anomaly known as the end-of-month reversal effect. The study showed that end-of-month returns, i.e. returns from the fourth Friday to the last trading day of the month, are negatively correlated with returns in the following month.

Findings

-This paper identifies a novel 1-month aggregate market reversal pattern, which is driven by the previous end-of-month market return.

– It demonstrates that end-of-the-month returns of the S&P 500 are negatively correlated with returns one month later.

-The reversal effect is statistically significant both In-Sample and Out-of-Sample, confirming its robustness.

-Unlike traditional cross-sectional reversals, this pattern is stronger in high-priced and liquid stocks and follows an economic cycle.

-A simple rule-based trading strategy and more sophisticated models leveraging this pattern generate significant economic gains. The strategy is cyclical in nature and does not rely on short-selling.

-The reversal effect strengthens over the following month, aligning with pension fund inflows and reinforcing the payment cycle explanation.

In short, a simple trading strategy based on this effect, that is buying if the end-of-month return is negative and selling if it is positive, outperforms the buy-and-hold strategy over a 45-year period.

The author also provides an explanation for this anomaly, attributing it to pension funds’ liquidity trading, as they adjust their portfolios to meet pension payment obligations.

Reference

[2] Graziani, Giuliano, Time Series Reversal: An End-of-the-Month Perspective, 2024, SSRN

Closing Thoughts

In this post, I discussed several calendar anomalies. Some of these patterns were discovered long ago and have proven to be persistent in today’s market. One of them represents a newly identified anomaly with promising characteristics. In all cases, profitable trading strategies were developed to take advantage of these recurring effects, highlighting the continued relevance of calendar-based insights in quantitative investing.

A catastrophe (CAT) bond is a debt instrument designed to transfer extreme event risks from insurers to capital market investors. They’re important for financial institutions, especially insurers and reinsurers, because they offer a way to manage large, low-probability. In this post, I feature research on CAT bonds, how they’re priced, and why they matter more than ever in a world of rising tail risks.

A Pricing Model for Earthquake Bonds

An earthquake bond is a type of catastrophe bond, in which an insurer, reinsurer, or government, transfers a portion or all of the earthquake risk to investors in return for higher yields. Earthquake bonds are crucial in countries prone to earthquakes. However, pricing them presents challenges.

Reference [1] developed a pricing model for pricing earthquake bonds. The authors modeled the risk-free interest rate using the Cox–Ingersoll–Ross model. They accommodated the variable intensity of events with an inhomogeneous Poisson process, while extreme value theory (EVT) was used to model the maximum strength.

Findings

– Earthquake bonds (EBs) connect insurance mechanisms to capital markets, offering a more sustainable funding solution, though pricing them remains a challenge.

– The paper proposes zero-coupon and coupon-paying EB pricing models that incorporate varying earthquake event intensity and maximum strength under a risk-neutral framework.

– The models focus on extreme earthquakes, which simplifies data processing and modeling compared to accounting for continuous earthquake occurrences.

– The earthquake event intensity is modeled using an inhomogeneous Poisson process, while the maximum strength is handled through extreme value theory (EVT).

– The models are tested using earthquake data from Indonesia’s National Disaster Management Authority covering 2008 to 2021.

– Sensitivity analyses show that using variable intensity instead of constant intensity significantly affects EB pricing.

– The proposed pricing model can help EB issuers set appropriate bond prices based on earthquake risk characteristics.

– Investors can use the sensitivity findings to select EBs that align with their individual risk tolerance.

In summary, the authors modeled the risk-free interest rate using the Cox–Ingersoll–Ross model. They accommodated the variable intensity of events with an inhomogeneous Poisson process, while extreme value theory (EVT) was used to model the maximum strength.

Reference

[1] Riza Andrian Ibrahim, Sukono, Herlina Napitupulu and Rose Irnawaty Ibrahim, Earthquake Bond Pricing Model Involving the Inconstant Event Intensity and Maximum Strength, Mathematics 2024, 12, 786

No-arbitrage Model for Pricing CAT Bonds

Pricing models for catastrophic risk-linked securities have primarily followed two methodologies: the theory of equilibrium pricing and the no-arbitrage valuation framework.

Reference [2] proposed a pricing approach based on the no-arbitrage framework. It utilizes the CIR stochastic process model for interest rates and the jump-diffusion stochastic process model for losses.

Findings

– This paper explores the concept of CAT bonds and explains how they are modeled using financial mathematics.

– Through a semi-discretization approach, a PIDE and a first-order differential equation were derived.

– A key component, the market price of risk of damage, was unavailable, so a quadratic term was constructed using market ask and bid prices to estimate this variable.

– By utilizing the Euler-Lagrange equation, a Poisson PDE was derived.

– The paper concludes by presenting an approach and numerical results for determining the market price of risk.

We find the stochastic model, equation (1), to be particularly insightful and effective in describing catastrophic losses.

Last year has witnessed numerous hurricanes across Asia, Europe, and America, leading to significant claims for insurers. This paper represents a contribution to advancing risk-sharing practices in the insurance industry.

Reference

[2] S. Pourmohammad Azizi & Abdolsadeh Neisy, Inverse Problems to Estimate Market Price of Risk in Catastrophe Bonds, Mathematical Methods of Statistics, Vol. 33 No. 3 2024

Closing Thoughts

In this post, I discussed catastrophe bonds and why they matter for investors navigating extreme event risks. The first paper focused on earthquake bonds, which present a challenge to model due to their rare and severe nature. Interestingly, both pricing models in the paper relied on the Cox–Ingersoll–Ross framework for interest rates, a reminder that even in the world of tail-risk instruments, some core quantitative models remain consistent.

Cryptocurrencies are becoming mainstream. In this post, I feature some strategies for trading and managing risks in cryptocurrencies.

Arbitrage Trading in the Cryptocurrency Market

Arbitrage trading takes advantage of price differences in different markets and/or instruments. Reference [1] examined some common and unique arbitrage trading opportunities in cryptocurrency exchanges that are not discussed often in the literature. They are,

-Exchange futures contract funding rate arbitrage

-Exchange futures contract intertemporal arbitrage

-Triangular arbitrage

-Pairs trading

-Order book spread prediction arbitrage

I provide details about the funding rate arbitrage below. Other arbitrage strategies are described in the paper.

Cryptocurrency exchanges use the funding rate to ensure perpetual futures prices align with spot prices, improving liquidity and narrowing bid-ask spreads. This mechanism periodically compensates long or short traders based on price differences. For example, Binance settles funding payments every 8 hours to balance demand between buyers and sellers.

When the perpetual contract trades above the spot price, longs pay shorts, discouraging further price increases and encouraging shorts to push it back down. An arbitrage strategy involves shorting Bitcoin in the perpetual market while holding an equal amount in the spot market, earning funding payments with minimal exposure to price fluctuations—excluding exchange and market risks.

When the perpetual contract trades below the spot price, shorts pay longs, so we buy the futures and short the spot.

Findings

– Research on cryptocurrency price prediction focuses on both time-series and cross-sectional analysis.

– This paper explores arbitrage opportunities in cryptocurrency exchanges that are often overlooked in academic literature.

– These arbitrage strategies can generate high returns with minimal risk.

– However, real market conditions and exchange constraints can reduce their effectiveness in live trading compared to backtesting.

– Incorporating these arbitrage strategies into a portfolio can improve the Sharpe ratio compared to simply holding cryptocurrencies.

In short, arbitrage trading is possible and profitable in the crypto market. However, we note that,

-These trading strategies are not riskless. Drawdown can happen

-Diversification helps smooth out the equity curves greatly

Reference

[1] Tianyu Zhou, Semi Risk-free Arbitrages with Cryptocurrency, 2022 5th International Conference on Financial Management, Education and Social Science (FMESS 2022)

Detecting Trends and Risks in Crypto Using the Hurst Exponent

The Hurst exponent is a statistical measure used to assess the long-term memory and persistence of a time series. It quantifies the tendency of a system to revert to the mean, follow a random walk, or exhibit a trending behavior. A Hurst exponent (H) value between 0 and 0.5 indicates mean-reverting behavior, H = 0.5 suggests a purely random process, and H between 0.5 and 1 signals persistent, trending behavior.

Reference [2] utilized the Detrended Fluctuation Analysis technique to study the Hurst exponent of the five major cryptocurrencies. Its main novelty is the calculation of a weekly time series of the Hurst exponent and its usage.

Findings

-This study examines long-range correlations in the cryptocurrency market using Hurst exponents across multiple time scales. It analyzes the log-returns of the top five cryptocurrencies, covering over 70% of market capitalization from 2017 to 2023.

-Four out of five cryptocurrencies exhibit persistent long-range correlations, while XRP follows a random walk.

-Trend Monitoring: The Hurst exponent (H) can help detect trend continuation or reversal. Cryptocurrencies like XRP showed transitions from short-term persistence to long-term anti-persistence, which could signal trend changes.

-Dynamic Strategy Adjustments: Rolling-window DFA estimates can track shifts in market behavior, aiding in strategy adjustments by identifying when a market moves from trend-following (H>0.5) to mean-reverting (H<0.5).

-Asset-Specific Behavior: Different cryptocurrencies exhibit unique behavioral patterns, suggesting that H-based analysis can inform tailored trading strategies.

-Systemic Risk Monitoring: Synchronization of H values across multiple cryptocurrencies during extreme market events may indicate rising volatility or instability, helping traders implement defensive measures like diversification.

In short, the findings suggest opportunities for using Hurst exponents as tools to monitor trend continuation or reversal, develop asset-specific strategies, and detect systemic risks during extreme market conditions, offering valuable insights for traders and policymakers navigating the cryptocurrency market’s inherent volatility.

Reference

[2] Huy Quoc Bui, Christophe Schinckus and Hamdan Amer Ali Al-Jaifi, Long-Range Correlations in Cryptocurrency Markets: A Multi-Scale DFA Approach, Physica A: Statistical Mechanics and its Applications, (2025), j.physa.2025.130417

Closing Thoughts

We have shown that arbitrage strategies in the crypto market are both possible and profitable. Additionally, risk management, trend detection, and reversal identification can be improved using the Hurst exponent, offering traders a valuable tool to navigate market volatility more effectively.

Portfolio allocation is an important research area. In this issue, we explore not only asset allocation but also the allocation of strategies. Specifically, I discuss tactical asset and trend-following strategy allocation.

Tactical Asset Allocation: From Simple to Advanced Strategies

Tactical Asset Allocation (TAA) is an active investment strategy that involves adjusting the allocation of assets in a portfolio to take advantage of short- to medium-term market opportunities. Unlike strategic asset allocation, which focuses on long-term asset allocation based on a fixed mix, TAA seeks to exploit market inefficiencies by overweighting or underweighting certain asset classes depending on market conditions, economic outlooks, or valuation anomalies. This approach allows investors to be more flexible and responsive to changing market environments, potentially improving returns while managing risk.

Reference [1] examines five approaches to tactical asset allocation. They are,

1. The SMA 200-day strategy, which uses the price of an asset relative to its 200-day moving average.
2. The SMA Plus strategy, which builds on the SMA 200-day by adding a volatility signal to the trend signal, dynamically adjusting allocations between risky assets and cash.
3. The Dynamic Tactical Asset Allocation (DTAA) strategy, which applies the same trend and volatility signals as SMA Plus but across the entire portfolio, rather than on individual assets.
4. The Risk Parity method, popularized by Ray Dalio’s All Weather Portfolio, equalizes the risk contributions of different asset classes.
5. The Maximum Diversification method, which aims to maximize the diversification ratio by balancing individual asset volatilities against overall portfolio volatility.

Findings

– The SMA strategy provides strong risk-adjusted returns by shifting to cash during downturns, though it may miss early recovery phases.

– SMA Plus builds on SMA by adding a more dynamic allocation approach, achieving higher returns but at a slightly increased risk level.

– The DTAA strategy yields the highest returns but experiences significant drawdowns due to aggressive equity exposure and limited risk management.

– Risk Parity and Maximum Diversification focus on stability, offering lower returns with minimal volatility, making them suitable for conservative investors.

In short, TAA based on a simple moving average still delivers the best risk-adjusted return.

This is an interesting and surprising result. Does this prove once again that simpler is better?

Reference

[1] Mohamed Aziz Zardi, Quantitative Methods of Dynamic Tactical Asset Allocation, HEC – Faculty of Business and Economics, University of Lausanne, 2024

Using Trends and Risk Premia in Portfolio Allocation

Trend-following strategies play a crucial role in portfolio management, but constructing an optimal portfolio based on these signals requires a solid theoretical foundation. Reference [2] builds on previous research to develop a unified framework that integrates an autocorrelation model with the covariance structure of trends and risk premia.

Findings

– The paper develops a theoretical framework to derive implementable solutions for trend-following portfolio allocation.

– The optimal portfolio is determined by the covariance matrix of returns, the covariance matrix of trends, and the risk premia.

– The study evaluates five well-established portfolio strategies: Agnostic Risk Parity (ARP), Markowitz, Equally Weighted, Risk Parity (RP), and Trend on Risk Parity (ToRP).

– Using daily futures market data from 1985 to 2020, covering 24 stock indexes, 14 bond indexes, and 9 FX pairs, the authors assess the performance of these portfolios.

– The optimal combination of the three best portfolios—ARP (19.5%), RP (51%), and ToRP (30%)—achieves a Sharpe ratio of 1.37, balancing traditional and alternative approaches.

– The RP portfolio, representing a traditional diversified approach, is a key driver of performance, aligning with recent literature.

– The combination of ARP and ToRP offers the best Sharpe ratio for trend-following strategies, as it minimizes asset correlation.

In the context of a portfolio optimization problem, the article solved the optimal allocation amongst a set of trend-following strategies. It utilized the covariance matrix of returns, trends, and risk premia in its optimization algorithm. The allocation scheme combined both traditional and alternative approaches, offering a better Sharpe ratio than each of the previous methods individually.

Reference

[2] Sébastien Valeyre, Optimal trend following portfolios, (2021), arXiv:2201.06635

Closing Thoughts

We have discussed both asset and strategy allocation, one advocating a relatively simple approach, while the other is more sophisticated. Each method has its advantages, depending on the investor’s objectives and risk tolerance. A well-balanced portfolio may benefit from integrating both approaches to achieve optimal performance and diversification.

In a previous newsletter, I discussed momentum strategies. In this edition, I’ll explore mean-reverting strategies.

Mean reversion is a natural force observed in various areas of life, including sports performance, portfolio performance, volatility, asset prices, etc. In this issue, I specifically examine the mean reversion characteristics of individual stocks and indices.

Long-Run Variances of Trending and Mean-Reverting Assets

Trading strategies are often loosely divided into two categories: trend-following and mean-reverting. They’re designed to exploit the mean-reverting or trending properties of asset prices. Reference [1] provides a different perspective and approach for studying the mean-reverting and trending properties of assets. It compares the long-run variances of mean-reverting and trending assets to that of a random-walk process.

Findings

-The paper provides an alternative perspective on studying mean-reverting and trending properties of assets.

– Long-run variances of mean-reverting and trending assets are compared to a random-walk process. The paper highlights a probabilistic model for investment styles.

– Theoretical analysis indicates the variance’s direct dependence on the probability of consecutive directional movements.

– It suggests that variance may be reduced through mean reverting strategies, capturing instances of assets moving in opposing directions.

-The model is applied to US stock data. It is found that in 97 of the largest stocks, a regime of mean-reversion is prevalent.

-The paper demonstrated that relative to a random walk, the variance of these stocks is reduced due to this behavior.

-It concluded that most large-cap US stocks exhibit mean-reverting behavior.

-Mean-reverting asset prices are deemed more predictable than a random walk.

In short, the paper concluded that most large-cap US stocks are mean-reverting, and the mean reversion resulted in a reduction of the variances of the assets. This means that mean-reverting asset prices are more predictable as compared to a random walk. The opposite is true for trending assets: larger variances and less predictability.

Reference

[1] L. Middleton, J. Dodd, S. Rijavec, Trading styles and long-run variance of asset prices, arXiv:2109.08242

Mean-Reverting Trading Strategies Across Developed Markets

Reference [2] studies the mean reversion strategy of individual stocks across developed markets. It shows that the mean-reversion strategy is not profitable in all markets. However, when we apply filters for stock characteristics, the strategy becomes profitable.

Findings

-This study examined the reversal strategy in the five largest developed markets using portfolio analysis and the Fama–Macbeth (FM) regression method.

-Portfolio analysis revealed that the unconditional reversal strategy is persistent only in Germany and Japan.

-When applied to firms with higher expected liquidity provision costs, the reversal returns became stronger across all markets.

-The FM regression method provided the strongest support for the reversal strategy while accounting for key firm-related characteristics.

-Reversal returns were significantly linked to market volatility, indicating that they are more pronounced during periods of higher market liquidity costs.

-The lack of liquidity in smaller, high book-to-market, high volatility stocks contributes to their higher reversal effect.

-Small, high book-to-market ratio and volatile stocks exhibit a prominent reversal effect based on portfolio analysis.

-Traditional asset pricing models like CAPM, FF-3, and CF-4 fail to explain the observed reversal returns.

Reference

[2] Hilal Anwar Butt and Mohsin Sadaqat, When Is Reversal Strong? Evidence From Developed Markets, The Journal of Portfolio Management, June 2024

Closing Thoughts

We have examined the mean reversion characteristics of stocks and indices in both U.S. and international markets. Gaining insights into this dynamic can lead to better risk-adjusted returns for your portfolio.