Optimal Trend-Following With Transaction Costs
by Valeriy Zakamulin and Javier Giner
In spite of the widespread popularity of trend-following investing, little is still known about optimal trend-following with transaction costs. A few existing studies consider this question using a continuous-time model within the stochastic optimal control theory framework. However, despite being theoretically the most appropriate, this approach makes the problem intractable analytically, and the numerical solutions are extremely complex. In this paper, we suggest a discrete-time model that represents an acceptable compromise between theoretical simplicity and practical relevance. On the one hand, we formulate a partly tractable optimization problem that can be solved numerically using standard and efficient optimization methods. On the other hand, we demonstrate that our model produces a satisfactory solution that substantially reduces trading costs. Under reasonable assumptions about the trend process, we show that the optimal trend-following rule is very much like the popular simple moving average crossover rule. Thus, our model justifies using the crossover rules in practice. Historical simulations show that the empirical evidence is in satisfactory agreement with our theoretical model.
The current version of this paper on SSRN
In spite of the widespread popularity of trend-following investing, little is still known about optimal trend-following with transaction costs. A few existing studies consider this question using a continuous-time model within the stochastic optimal control theory framework. However, despite being theoretically the most appropriate, this approach makes the problem intractable analytically, and the numerical solutions are extremely complex. In this paper, we suggest a discrete-time model that represents an acceptable compromise between theoretical simplicity and practical relevance. On the one hand, we formulate a partly tractable optimization problem that can be solved numerically using standard and efficient optimization methods. On the other hand, we demonstrate that our model produces a satisfactory solution that substantially reduces trading costs. Under reasonable assumptions about the trend process, we show that the optimal trend-following rule is very much like the popular simple moving average crossover rule. Thus, our model justifies using the crossover rules in practice. Historical simulations show that the empirical evidence is in satisfactory agreement with our theoretical model.
The current version of this paper on SSRN
Optimal Trend Following Rules in Two-State Regime-Switching Models
by Valeriy Zakamulin and Javier Giner
Academic research on trend-following investing has almost exclusively been focused on testing the profitability of various trading rules. However, all existing trend-following rules are ad-hoc rules whose optimality has never been justified theoretically. The goal of this paper is to fill this gap in the literature. Specifically, this paper examines the optimal trend-following rules when the returns follow a two-state process that randomly switches between bull and bear markets. We show that if the returns are modeled by a Markov switching model, it is optimal to follow the trend using the Exponential Moving Average. In a more realistic case where the returns are modeled by a semi-Markov switching model (SMSM) where the state duration times exhibit positive duration dependence, the optimal trend-following rule is somewhat similar to the Moving Average Convergence/Divergence rule. We confirm the validity of the SMSM by an empirical study that uses the data on the S\&P 500 and Dow Jones Industrial Average indices. We demonstrate that the theoretically optimal trading rule outperforms the popular 10-month Simple Moving Average and 12-month Momentum rules.
The current version of this paper on SSRN
Academic research on trend-following investing has almost exclusively been focused on testing the profitability of various trading rules. However, all existing trend-following rules are ad-hoc rules whose optimality has never been justified theoretically. The goal of this paper is to fill this gap in the literature. Specifically, this paper examines the optimal trend-following rules when the returns follow a two-state process that randomly switches between bull and bear markets. We show that if the returns are modeled by a Markov switching model, it is optimal to follow the trend using the Exponential Moving Average. In a more realistic case where the returns are modeled by a semi-Markov switching model (SMSM) where the state duration times exhibit positive duration dependence, the optimal trend-following rule is somewhat similar to the Moving Average Convergence/Divergence rule. We confirm the validity of the SMSM by an empirical study that uses the data on the S\&P 500 and Dow Jones Industrial Average indices. We demonstrate that the theoretically optimal trading rule outperforms the popular 10-month Simple Moving Average and 12-month Momentum rules.
The current version of this paper on SSRN
Not All Bull and Bear Markets Are Alike: Insights From a Five-State Hidden Semi-Markov Model
by Valeriy Zakamulin, forthcoming in the Risk Management (Palgrave)
This paper employs the hidden semi-Markov model and a novel model selection procedure to detect different states in the US stock market. The empirical results suggest that the market is switching between five states that can be classified into three bull states and two bear states. The three bull states are categorized as a low volatility bull market, a high volatility bull market, and a stock market bubble. One of the bear states represents a regular bear market, while the other one corresponds to either a stock market crash or a market correction. The paper demonstrates that the five-state model is consistent with a number of stylized facts and provides many valuable insights into the dynamics of the US stock market. Besides, the five-state model has clear implications for the success of some active strategies that aim to enhance returns and reduce losses.
The current version of this paper on SSRN
This paper employs the hidden semi-Markov model and a novel model selection procedure to detect different states in the US stock market. The empirical results suggest that the market is switching between five states that can be classified into three bull states and two bear states. The three bull states are categorized as a low volatility bull market, a high volatility bull market, and a stock market bubble. One of the bear states represents a regular bear market, while the other one corresponds to either a stock market crash or a market correction. The paper demonstrates that the five-state model is consistent with a number of stylized facts and provides many valuable insights into the dynamics of the US stock market. Besides, the five-state model has clear implications for the success of some active strategies that aim to enhance returns and reduce losses.
The current version of this paper on SSRN
Warren Buffett versus Zvi Bodie: Should You Buy Or Sell Put Options?
by Steen Koekebakker and Valeriy Zakamulin, published in the Journal of Wealth Management
Academics and investment professionals often disagree when it comes to investment advice. Legendary investor Warren Buffett is a proponent of time diversification and firmly believes that stocks are less risky in the long run. Therefore, he often sells long-term put options instead of buying them for portfolio protection. By contrast, the famous finance professor Zvi Bodie argues that time diversification is a fallacy and, therefore, his advice to fund managers is to buy long-term portfolio insurance. In this article, we consider the optimal portfolio choice problem for a loss-averse investor. First, we demonstrate that our loss-averse investor subscribes to the principle of time diversification. In particular, our investor allocates more to stocks as the investment horizon lengthens. Second, we allow our investor to trade in stocks and put options. We find that when the investment horizon is short, our investor is better off with portfolio insurance. Conversely, when the investment horizon is long, our investor sells put options. That is, our loss-averse investor prefers Buffett’s investment advice over Bodie’s.
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Academics and investment professionals often disagree when it comes to investment advice. Legendary investor Warren Buffett is a proponent of time diversification and firmly believes that stocks are less risky in the long run. Therefore, he often sells long-term put options instead of buying them for portfolio protection. By contrast, the famous finance professor Zvi Bodie argues that time diversification is a fallacy and, therefore, his advice to fund managers is to buy long-term portfolio insurance. In this article, we consider the optimal portfolio choice problem for a loss-averse investor. First, we demonstrate that our loss-averse investor subscribes to the principle of time diversification. In particular, our investor allocates more to stocks as the investment horizon lengthens. Second, we allow our investor to trade in stocks and put options. We find that when the investment horizon is short, our investor is better off with portfolio insurance. Conversely, when the investment horizon is long, our investor sells put options. That is, our loss-averse investor prefers Buffett’s investment advice over Bodie’s.
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A Regime-Switching Model of Stock Returns with Momentum and Mean Reversion
by Valeriy Zakamulin and Javier Giner, forthcoming in the Economic Modelling
A vast body of empirical literature documents the existence of short-term momentum and medium-term mean reversion in various financial markets. By contrast, there is still a great shortage of theoretical models that explain the presence of these two common phenomena. A Markov model, where the return process randomly switches between bull and bear states, can reproduce many stylized facts of financial asset returns but the mean reversion. An important limitation of the Markov model is that the state termination probability does not depend on age. We develop a semi-Markov model where, following the empirical evidence, the state termination probability increases with age. We demonstrate that this model induces short-term return momentum and subsequent reversal. We calibrate our model to real-world data and show that the empirical results are in satisfactory agreement with our theoretical model.
The current version of this paper on SSRN
A vast body of empirical literature documents the existence of short-term momentum and medium-term mean reversion in various financial markets. By contrast, there is still a great shortage of theoretical models that explain the presence of these two common phenomena. A Markov model, where the return process randomly switches between bull and bear states, can reproduce many stylized facts of financial asset returns but the mean reversion. An important limitation of the Markov model is that the state termination probability does not depend on age. We develop a semi-Markov model where, following the empirical evidence, the state termination probability increases with age. We demonstrate that this model induces short-term return momentum and subsequent reversal. We calibrate our model to real-world data and show that the empirical results are in satisfactory agreement with our theoretical model.
The current version of this paper on SSRN
A New Predictability Pattern in the US Stock Market Returns
by Valeriy Zakamulin, forthcoming in the Journal of Portfolio Management
In this article, we document a new stock market anomaly that seems to have escaped the attention of both investment professionals and academics alike. We find that over more than a century, the monthly market return has been predicted by the monthly market return at lag 5. This predictability is market-wide and is most evident in the returns of portfolios of large and growth stocks. The trading strategy that incorporates this predictability yields superior performance that cannot be attributed to common risk factors. A closer investigation of the new anomaly reveals that not each calendar month possesses predictive ability. Therefore, there is a linkage between the new anomaly and calendar effects in stock returns.
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In this article, we document a new stock market anomaly that seems to have escaped the attention of both investment professionals and academics alike. We find that over more than a century, the monthly market return has been predicted by the monthly market return at lag 5. This predictability is market-wide and is most evident in the returns of portfolios of large and growth stocks. The trading strategy that incorporates this predictability yields superior performance that cannot be attributed to common risk factors. A closer investigation of the new anomaly reveals that not each calendar month possesses predictive ability. Therefore, there is a linkage between the new anomaly and calendar effects in stock returns.
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Optimal Trend-Following in a Markov Switching Model
by Valeriy Zakamulin and Javier Giner
This paper assumes that the market returns follow a two-state Markov process that randomly switches between bull and bear states. We show that in this case, the exponential moving average (EMA) represents the optimal trend-following rule. The paper provides the analytical solution to the optimal window size (decay constant) in the EMA rule. We estimate the optimal window size for timing the S&P 500 stock market index using real-world data. A comparative statics analysis finds that the optimal window size depends mainly on the signal-to-noise ratio of returns and the state transition probabilities.
The current version of this paper on SSRN
This paper assumes that the market returns follow a two-state Markov process that randomly switches between bull and bear states. We show that in this case, the exponential moving average (EMA) represents the optimal trend-following rule. The paper provides the analytical solution to the optimal window size (decay constant) in the EMA rule. We estimate the optimal window size for timing the S&P 500 stock market index using real-world data. A comparative statics analysis finds that the optimal window size depends mainly on the signal-to-noise ratio of returns and the state transition probabilities.
The current version of this paper on SSRN
Revisiting the Duration Dependence in the US Stock Market Cycles
by Valeriy Zakamulin, published in Applied Finance
There is a big controversy among both investment professionals and academics regarding the question of how the probability that a bull or bear market terminates depends on its age. Using more than two centuries of data on the broad US stock market index, in this paper we revisit the duration dependence in bull and bear markets. We find that for both bull and bear markets the duration dependence is a nonlinear function of the state age. Our results suggest that the duration dependence in bear markets is strictly positive. For 93% of bull markets the duration dependence is also positive. Only about 7% of the bull markets, those with the longest durations, do not exhibit positive duration dependence. We also compare a few selected theoretical distributions in describing the duration dependence in bull and bear markets. We find that the gamma distribution most often provides the best fit to both the survivor and hazard functions of bull and bear markets. However, our results reveal that none of the selected distributions correctly describes the right tail of the hazard functions.
This is an open access paper. To read it, click here
There is a big controversy among both investment professionals and academics regarding the question of how the probability that a bull or bear market terminates depends on its age. Using more than two centuries of data on the broad US stock market index, in this paper we revisit the duration dependence in bull and bear markets. We find that for both bull and bear markets the duration dependence is a nonlinear function of the state age. Our results suggest that the duration dependence in bear markets is strictly positive. For 93% of bull markets the duration dependence is also positive. Only about 7% of the bull markets, those with the longest durations, do not exhibit positive duration dependence. We also compare a few selected theoretical distributions in describing the duration dependence in bull and bear markets. We find that the gamma distribution most often provides the best fit to both the survivor and hazard functions of bull and bear markets. However, our results reveal that none of the selected distributions correctly describes the right tail of the hazard functions.
This is an open access paper. To read it, click here
Time Series Momentum in the US Stock Market: Empirical Evidence and Theoretical Analysis
by Valeriy Zakamulin and Javier Giner, published in International Review of Financial Analysis
There is much controversy in the academic literature on the presence of short-term trends in financial markets and the trend-following strategy's profitability. We restrict our attention to studying the time series momentum in the S&P Composite stock price index. Our contributions are both empirical and theoretical. On the empirical side, we present compelling evidence of the presence of short-term momentum. For the first time, we suppose that the returns follow a p-order autoregressive process and evaluate this process's parameters. On the theoretical side, we develop a tractable theoretical model that contributes to our fundamental understanding of the trend-following strategy's risk, return, and performance and helps estimate the power of statistical tests for profitability. Our analysis reveals that the most popular Sharpe ratio test suffers from extremely low statistical power. By contrast, the less prevalent CAPM alpha test exhibits sufficient power in detecting superior performance.
This is an open access paper. To read it, click here
There is much controversy in the academic literature on the presence of short-term trends in financial markets and the trend-following strategy's profitability. We restrict our attention to studying the time series momentum in the S&P Composite stock price index. Our contributions are both empirical and theoretical. On the empirical side, we present compelling evidence of the presence of short-term momentum. For the first time, we suppose that the returns follow a p-order autoregressive process and evaluate this process's parameters. On the theoretical side, we develop a tractable theoretical model that contributes to our fundamental understanding of the trend-following strategy's risk, return, and performance and helps estimate the power of statistical tests for profitability. Our analysis reveals that the most popular Sharpe ratio test suffers from extremely low statistical power. By contrast, the less prevalent CAPM alpha test exhibits sufficient power in detecting superior performance.
This is an open access paper. To read it, click here
Trend Following with Momentum Versus Moving Average: A Tale of Differences
by Valeriy Zakamulin and Javier Giner, published in Quantitative Finance
Despite the ever-growing interest in trend following and a series of publications in academic journals, there is still a great shortage of theoretical results on the properties of trend following rules. Our paper fills this gap by comparing and contrasting the two most popular trend following rules, the Momentum (MOM) and Moving Average (MA) rules, from a theoretical perspective. Our approach is based on the return-based formulation of trading rules and modelling the price trends by an autoregressive return process. We provide theoretical results on the similarity between various trend following rules and the forecast accuracy of trading rules. Our results show that the similarity between the MOM and MA rules is rather high and increases with increasing trend strength. However, as compared to the MOM rule, the MA rules have a more robust forecast accuracy of the future direction of price trends. As a result, under uncertain market dynamics the MA rules tend to gain an advantage over the MOM rule. Overall, the results reported in this paper help traders to understand more deeply the properties of trend following rules as well as the differences and similarities between them.
This is an open access paper. To read it, click here.
Despite the ever-growing interest in trend following and a series of publications in academic journals, there is still a great shortage of theoretical results on the properties of trend following rules. Our paper fills this gap by comparing and contrasting the two most popular trend following rules, the Momentum (MOM) and Moving Average (MA) rules, from a theoretical perspective. Our approach is based on the return-based formulation of trading rules and modelling the price trends by an autoregressive return process. We provide theoretical results on the similarity between various trend following rules and the forecast accuracy of trading rules. Our results show that the similarity between the MOM and MA rules is rather high and increases with increasing trend strength. However, as compared to the MOM rule, the MA rules have a more robust forecast accuracy of the future direction of price trends. As a result, under uncertain market dynamics the MA rules tend to gain an advantage over the MOM rule. Overall, the results reported in this paper help traders to understand more deeply the properties of trend following rules as well as the differences and similarities between them.
This is an open access paper. To read it, click here.
The Term Structure of Volatility Predictability
by Xingyi Li and Valeriy Zakamulin. Published in the International Journal of Forecasting.
Volatility forecasting is crucial for portfolio management, risk management, and pricing of derivative securities. Still, little is known about the accuracy of volatility forecasts and the horizon of volatility predictability. This paper aims to fill these gaps in the literature. We begin this paper by introducing the notions of spot and forward predicted volatilities and propose describing the term structure of volatility predictability by spot and forward forecast accuracy curves. Then, we perform a comprehensive study of the term structure of volatility predictability in stock and foreign exchange markets. Our results quantify the volatility forecast accuracy across horizons in two major markets and suggest that the horizon of volatility predictability is significantly longer than that reported in earlier studies. Nevertheless, the aforesaid horizon is observed to be much shorter than the longest maturity of traded derivative contracts.
This is an open access paper. To read it, click here
This is an open access paper. To read it, click here
Stock Earnings and Bond Yields in the US 1871 - 2016: The Story of a Changing Relationship
by Valeriy Zakamulin and Arngrim Hunnes, published in Quarterly Review of Economics and Finance
Using historical data that spans almost 150 years, we examine whether there is a long-run equilibrium relationship between the stock's earnings and bond yields. The novelty of our econometric methodology consists in using a vector error correction model where we allow multiple structural breaks in the equilibrium relationship. The results of our analysis suggest the existence of equilibrium relationship over 1871-1929 and 1958-2016. On the two historical segments, our analysis finds that the stock's earnings yield followed the bond yield in both the short- and long-run, but not the other way around. Perhaps the most important and surprising finding of our empirical study is that, after the break in 1929, a completely new equilibrium relationship re-emerged in 1958 that was later termed as the "Fed model." Our main argument for the emergence of a new equilibrium relationship is that a major "paradigm shift" in the stock valuation theory occurred in the late 1950s. To support our argument and explain the transition from the old to the new paradigm, we carefully review the evolution of the stock market, fixed-income markets, inflation, and income taxes since 1871. Here we highlight the main historical events that potentially could have caused the transition from one paradigm to another. We continue our review by describing the new developments in the stock valuation theory since the early 1920s. Finally, we identify the primary impetus for the paradigm shift.
This is an open access paper, to read it click here
Using historical data that spans almost 150 years, we examine whether there is a long-run equilibrium relationship between the stock's earnings and bond yields. The novelty of our econometric methodology consists in using a vector error correction model where we allow multiple structural breaks in the equilibrium relationship. The results of our analysis suggest the existence of equilibrium relationship over 1871-1929 and 1958-2016. On the two historical segments, our analysis finds that the stock's earnings yield followed the bond yield in both the short- and long-run, but not the other way around. Perhaps the most important and surprising finding of our empirical study is that, after the break in 1929, a completely new equilibrium relationship re-emerged in 1958 that was later termed as the "Fed model." Our main argument for the emergence of a new equilibrium relationship is that a major "paradigm shift" in the stock valuation theory occurred in the late 1950s. To support our argument and explain the transition from the old to the new paradigm, we carefully review the evolution of the stock market, fixed-income markets, inflation, and income taxes since 1871. Here we highlight the main historical events that potentially could have caused the transition from one paradigm to another. We continue our review by describing the new developments in the stock valuation theory since the early 1920s. Finally, we identify the primary impetus for the paradigm shift.
This is an open access paper, to read it click here
Stock Volatility Predictability in Bull and Bear Markets
by Xingyi Li and Valeriy Zakamulin, publihsed in Quantitative Finance
Recent literature on stock return predictability suggests that it varies substantially across economic states being strongest during bad economic times. In line with this evidence, we document that stock volatility predictability is also state dependent. In particular, using a large data set of high-frequency data on individual stocks and a few popular time-series volatility models, in this paper we comprehensively examine how volatility forecastability varies across bull and bear states of the stock market. We find that the volatility forecast horizon is substantially longer when the market is in a bear state than when it is in a bull state. In addition, over all but the shortest horizons the volatility forecast accuracy is higher when the market is in a bear state. This discrepancy increases as the forecast horizon lengthens. Our study concludes that the stock volatility predictability is strongest during bad economic times proxied by bear market states.
This is an open access paper, to read it click here
This is an open access paper, to read it click here
Superiority of Optimized Portfolios to Naive Diversification: Fact or Fiction?
by Valeriy Zakamulin, published in the Finance Research Letters
DeMiguel, Garlappi, and Uppal (2009) conducted a highly influential study where they demonstrated that none of the optimized portfolios consistently outperformed the naive diversification. This result triggered a heated debate within the academic community on whether portfolio optimization adds value. Nowadays several studies claim to defend the value of portfolio optimization. The commonality in all these studies is that various portfolio optimization methods are implemented using the datasets generously provided by Kenneth French and the performance is measured by means of the Sharpe ratio. This paper aims to provide a cautionary note regarding the use of Kenneth French datasets in portfolio optimization without controlling whether the superior performance appears due to better mean-variance efficiency or due to exposures to established factor premiums. First, we demonstrate that the low-volatility effect is present in virtually all datasets in the Kenneth French online data library. Second, using a few simple portfolio optimization models that are said to outperform the naive diversification, we show that these portfolios are tilted towards assets with lowest volatilities and, after controlling for the low-volatility effect, there is absolutely no evidence of superior performance. The main conclusion that we reach in our paper is that a convincing demonstration of the value of portfolio optimization cannot be made without showing that the superior performance cannot be attributed to profiting from some known anomalies.
The current version of the paper on the SSRN
The current version of the paper on the SSRN
Abnormal Stock Market Returns around Peaks in VIX: The Evidence of Investor Overreaction?
by Valeriy Zakamulin, working paper
Even though the VIX index was intended to be a measure of future volatility of the stock market, researchers argue that in reality VIX measures the investor sentiment. Anecdotal evidence suggests that peaks in VIX coincide with stock market bottoms followed by rallies, yet so far there have been no scientific evidence confirming this casual observation. In this paper we perform an event study of abnormal stock market returns around peaks in VIX and discuss our findings within the framework of behavioral finance theory. First of all, we detect peaks in VIX using formal turning-point identification procedures and provide detailed descriptive statistics of periods of rising and falling VIX. The results of our event study reveal strong evidence of the presence of abnormal stock market returns around peaks in VIX. We argue that the pattern of abnormal returns can be attributed to investor overreaction to bad news with subsequent correction. To validate our conjecture, we test whether the abnormal returns around peaks in VIX satisfy the two properties of overreaction. We find that the results of these empirical tests are consistent with the overreaction hypothesis. To further confirm the idea that the VIX index reflects the investor sentiment, we test the predictions of behavioral finance theory which postulates that investor sentiment affects various types of stocks to different degrees. In agreement with the theoretical predictions, we find evidence that over the event window around a peak in VIX the prices of large and value stocks undergo a relatively small downward correction, while the prices of more speculative small and growth stocks are corrected down to a higher degree. Our additional tests suggest that these cross-sectional differences cannot be explained by a set of standard risk factors.
The current version of the paper on the SSRN
The current version of the paper on the SSRN
Revisiting the Profitability of Market Timing with Moving Averages
by Valeriy Zakamulin, published in the International Review of Finance
In a recent empirical study by Glabadanidis ("Market Timing With Moving Averages" (2015), International Review of Finance, Volume 15, Number 13, Pages 387-425; the paper is also available on the SSRN and has been downloaded more than 7,500 times) the author reports striking evidence of extraordinary good performance of the moving average trading strategy. In this paper we demonstrate that "too good to be true" reported performance of the moving average strategy is due to simulating the trading with look-ahead bias. We perform the simulations without look-ahead bias and report the true performance of the moving average strategy. We find that at best the performance of the moving average strategy is only marginally better than that of the corresponding buy-and-hold strategy. In statistical terms, the performance of the moving average strategy is indistinguishable from the performance of the buy-and-hold strategy. This paper is supplied with R code that allows every interested reader to reproduce the reported results.
The current version of the paper on the SSRN Download the R code and data
The current version of the paper on the SSRN Download the R code and data
Volatility Weighting Over Time in the Presence of Transaction Costs
by Valeriy Zakamulin, published in the Journal of Wealth Management
Numerous empirical studies demonstrate the superiority of dynamic strategies with volatility weighting over time mechanism. These strategies control the portfolio risk over time by adjusting the risk exposure according to updated volatility forecasts. Yet, in order to reap all benefits promised by volatility weighting over time, the composition of the active portfolio must be revised rather frequently. Transaction costs represent a serious obstacle to benefiting from this dynamic risk control technique. In this paper we propose a modified volatility weighting strategy that allows one to reduce dramatically the amount of trading costs. The empirical evidence shows that the advantages of the modified volatility weighting strategy persist even in the presence of high transaction costs.
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Numerous empirical studies demonstrate the superiority of dynamic strategies with volatility weighting over time mechanism. These strategies control the portfolio risk over time by adjusting the risk exposure according to updated volatility forecasts. Yet, in order to reap all benefits promised by volatility weighting over time, the composition of the active portfolio must be revised rather frequently. Transaction costs represent a serious obstacle to benefiting from this dynamic risk control technique. In this paper we propose a modified volatility weighting strategy that allows one to reduce dramatically the amount of trading costs. The empirical evidence shows that the advantages of the modified volatility weighting strategy persist even in the presence of high transaction costs.
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Market Timing With a Robust Moving Average
by Valeriy Zakamulin, working paper
In this paper we entertain a method of finding the most robust moving average weighting scheme to use for the purpose of timing the market. Robustness of a weighting scheme is defined its ability to generate sustainable performance under all possible market scenarios regardless of the size of the averaging window. The method is illustrated using the long-run historical data on the Standard and Poor's Composite stock price index. We find the most robust moving average weighting scheme, demonstrates its advantages, and discuss its practical implementation.
The current version of the paper on the SSRN
The current version of the paper on the SSRN
Optimal Dynamic Portfolio Risk Management
by Valeriy Zakamulin, published in the Journal of Portfolio Management
Numerous econometric studies report that financial asset volatilities and correlations are time-varying and predictable. Over the recent decade, this knowledge has stimulated an increasing interest in various dynamic portfolio risk control techniques. The two basic types of risk control techniques are: risk control across assets and risk control over time. At present, the two types of risk control techniques are not implemented simultaneously. Surprisingly little has been done from a theoretical perspective in terms of studying the optimal dynamic portfolio risk management. In this paper we fill this gap in the literature by formulating and solving the multi-period portfolio choice problem of an investor with mean-variance preferences. In terms of dynamic portfolio risk control, our solution shows that it is optimal to control portfolio risk both across assets and over time simultaneously. Using several datasets and performing out-of-sample simulations, we demonstrate the superiority of dynamic portfolio risk control both across assets and over time. Specifically, we show that portfolios with risk control only across assets outperform the equally-weighted portfolios and that portfolios with risk control both across assets and over time outperform portfolios with risk control across assets only.
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The Real-Life Performance of Market Timing with Moving Average and Time-Series Momentum Rules
by Valeriy Zakamulin, published in the Journal of Asset Management
In this paper, we revisit the myths regarding the superior performance of market timing strategies based on moving average and time-series momentum rules. These active timing strategies are very appealing to investors because of their extraordinary simplicity and because they promise substantial advantages over their passive counterparts (see, for example, the paper by M. Faber (2007) "A Quantitative Approach to Tactical Asset Allocation" published in the Journal of Wealth Management). However, the "too good to be true" reported performance of these market timing rules raises a legitimate concern as to whether this performance is realistic and whether investors can expect that future performance will be the same as the documented historical performance. We argue that the reported performance of market timing strategies usually contains a considerable data-mining bias and ignores important market frictions. To address these issues, we perform out-of-sample tests of these two timing models in which we account for realistic transaction costs. Our findings reveal that the performance of market timing strategies is highly overstated, to say the least.
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Email me if you want a copy of this paper
Dynamic Asset Allocation Strategies Based on Unexpected Volatility
by Valeriy Zakamulin, published in the Journal of Alternative Investments
In this paper we document that at the aggregate stock market level the unexpected volatility is negatively related to expected future returns and positively related to future volatility. We demonstrate how the predictive ability of unexpected volatility can be utilized in dynamic asset allocation strategies that deliver a substantial improvement in risk-adjusted performance as compared to traditional buy-and-hold strategies. In addition, we demonstrate that active strategies based on unexpected volatility outperform the popular active strategy with volatility target mechanism and have the edge over the widely reputed market timing strategy with 10-month simple moving average rule.
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Email me if you want a copy of this paper
Secular Mean Reversion and Long-Run Predictability of the Stock Market
by Valeriy Zakamulin, published in the Bulletin of Economic Research
The empirical financial literature reports evidence of mean reversion in stock prices and the absence of out-of-sample return predictability over horizons shorter than 10 years. Anecdotal evidence suggests the presence of mean reversion in stock prices and return predictability over horizons longer than 10 years, but thus far, there is no empirical evidence confirming such anecdotal evidence. The goal of this paper is to fill this gap in the literature. Specifically, using 141 years of data, this paper begins by performing formal tests of the random walk hypothesis in the prices of the real S\&P Composite Index over increasing time horizons of up to 40 years. Although our results cannot support the conventional wisdom that the stock market is safer for long-term investors, our findings speak in favor of the mean reversion hypothesis. In particular, we find statistically significant in-sample evidence that past 15-17 year returns are able to predict the future 15-17 year returns. This finding is robust to the choice of data source, deflator, and test statistic. The paper continues by investigating the out-of-sample performance of long-horizon return forecasting based on the mean-reverting model. These latter tests demonstrate that the forecast accuracy provided by the mean-reverting model is statistically significantly better than the forecast accuracy provided by the naive historical-mean model. Moreover, we show that the predictive ability of the mean-reverting model is economically significant and translates into substantial performance gains.
The current version of the paper on the SSRN
The current version of the paper on the SSRN
A Test of Covariance Matrix Forecasting Methods
by Valeriy Zakamulin, published in the Journal of Portfolio Management
Providing a more accurate covariance matrix forecast can substantially improve the performance of optimized portfolios. Using out-of-sample tests, in this paper, we evaluate alternative covariance matrix forecasting methods by looking at (1) their forecast accuracy, (2) their ability to track the volatility of the minimum-variance portfolio, and (3) their ability to keep the volatility of the minimum-variance portfolio at a target level. We find large differences between the methods. Our results suggest that shrinkage of the sample covariance matrix improves neither the forecast accuracy nor the performance of minimum-variance portfolios. In contrast, switching from the sample covariance matrix forecast to a multivariate GARCH forecast reduces forecasting error and portfolio tracking error by at least half. Our findings also reveal that the exponentially weighted covariance matrix forecast performs only slightly worse than the multivariate GARCH forecast.
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