2 Empirical Methodology
In this section, we introduce three method related to volatility management. The first one is target volatility strategy implemented by Dachraoui (2018), which is the main method that we use in the article. The next method is volatility-managed strategy of Moreira and Muir (2017). Last is a portion of Liu, Tang, and Zhou (2018). We only show the part which identifies look-ahead bias. In the beginning, we consider the method of Dachraoui (2018) and Moreira and Muir (2017). However, when we start to implement the model, we find that Dachraoui (2018) is more intuitive, more practical, and easier to implement. Therefore, we choose Dachraoui (2018) as our main method.
2.1 The Method of Dachraoui (2018)
Basically, target volatility strategy (TVS) is valid in the following conditions:
A. Volatility persistence
Famous scientist-mathematician Benoit Mandelbrot said, "large changes tend to be followed by large changes...and small changes tend to be followed by small changes." Volatility clustering is the tendency of large changes in prices of financial assets to cluster together, which results in the persistence of volatility.
The persistence in a period can help to predict the expectations of volatility from realized volatility.
B. Negative relationship between volatility and returns
This characteristic helps to predict the returns from realized volatility. We can easily conclude that the returns of remaining a low target volatility is better than holding index portfolio only.
From the conditions above, TVS is workable as long as the conducting period is
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short enough having volatility persistence and the underlying assets are selected suitable.
To confirm TVS is effective or not, we follow the method proposed by Dachraoui (2018). The investor applies TVS to a portfolio which is considered to be an index portfolio. Let L be the exposure level and there is no restriction for investors to determine the value of L. In other words, investors can borrow (or lend) in the
underlying portfolio. Therefore, the value of L can be less or greater than 1. When L is greater than 1 (L>1), investors will hold over 100% in the index portfolio. On the other hand, when L is less than 1 (L<1), investors will have a portion of risk-free assets, instead of holding index portfolio. The amount at time t of the exposure level Lt is determined by the constant volatility target σT divided by σt-1, which is estimated volatility according to the information available at time t.
The relationship is indicated as the following equation:
𝐿𝑡 = 𝜎𝑇 𝜎𝑡−1
If the estimated volatility is higher than the target volatilty, Lt becomes less than 1 and investors will invest more portion in risk-free assets. On the constrast, if the estimates volatility is lower than the target volatility, Lt becomes greater than 1. In this situation, because of low volatility, investors should increase the index portfolio and reduce cash investment to higher the volatility.
We take historical standard deviation of index return in whole data as our target volatility in this article. Although the estimated volatility can be implied or historical, we however use historical observations here because it is the most common way in practice. Depending on the windows or rebalanced period operating by the investors, we will have different σt-1 and so have different Lt , keeping the allocation between
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risk-free assets and index portfolio to the next rebalancing date. In addition, for the length of the window, it should not be too long. If investors use a long window of estimated volatility, it may be almost constant, and that leads to the exposure level also remains steady. On the other hand, using a short window may be more fluctuated so that we can manage our target volatility strategy by observing exposure level.
According to the above, at time t, the investors would have the returns until time t-1. The estimated volatility of historical data using rolling 60-business-day period
(the equivalent of twelve-week/ three-month trading period). We use daily data once in a week to compose the estimated volatility, that is, there are twelve historical values in our period. The standard deviation of the returns past twelve weeks is considered estimated volatility. Now, our TVS constructs as the equation as follow:
𝑅𝑡 = 𝐿𝑡𝑟𝑡+ (1 − 𝐿𝑡)𝑟𝑓
Where Rt denotes the levered portfolio, in other words, the returns that performed under the target volatility strategy, rf is the returns of risk-free assets at time t, rt
denotes the returns of index portfolio, and Lt is the exposure level, as we mention above.
2.2 The Method of Moreira and Muir (2017)
Same as the concept of Dachraoui (2018), Moreira and Muir (2017) also construct a method by scaling an excess return by the inverse of its conditional variance. In principle, the motivation of the strategy is from the perspective of a mean-variance investor, who owns an optimal allocation problem, that is, deciding how much to invest in a risky portfolio. There is a risk-return trade-off. Suppose there are a risky portfolio, with a random variable excess return rt over month t, and a risk-free rate rf. The expected excess return μt, and the return volatility σt. Assume that a
mean-‧
variance investor is now facing the decision—the weight ωt between risky portfolio and risk-free assets (1- ωt). As we only focus on the relationship with excess return, volatility, and the weight, we convert to a formula representation below:
𝜔𝑡 ∝ 𝜇𝑡 𝜎̂𝑡2
Knowing that volatility is highly variable, persistent, and difficult to predict returns, Moreira and Muir (2017) construct a volatility-managed portfolio by the inverse of the conditional variance. In other words, the variation of risk exposure is according to the change of variance. The managed portfolio is constructed as follow:
𝑟𝑡𝑀𝑀 = 𝐿 𝜎̂𝑡−12 𝑟𝑡
Where rt is the buy-and-hold portfolio excess return, L is a constant which controls the average exposure of the strategy, and 𝜎̂𝑡−12 is the realized return variance in month t-1, daily data into monthly. The appealing feature of this method is that it does not rely on any parameter estimation. Comparing to mean-variance approach, it does not need to estimate variances or expected returns. Then, Moreira and Muir (2017) did a time-series regression of the volatility-managed portfolio on the original factor:
𝑟𝑡𝑀𝑀 = 𝛼 + 𝛽𝑟𝑡+ 𝜖𝑡
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If the result of alpha is positive, it implies that the volatility-managed portfolio increase Sharpe ratios relative to the original factors. Hence, Moreira and Muir (2017) apply the analysis factor by factor, including excess market return, size factor, value factor, momentum factor, profitability factor and investment factor.
2.3 The Method of Liu, Tang, and Zhou (2018)
Liu, Tang, and Zhou (2018) first review three volatility-timing strategies in the article, the managed strategy of Moreira and Muir (2017), the volatility-targeting strategy of Barroso and Santa-Clara (2015), and the mean-variance portfolio allocation strategy under estimation risk of Kan and Zhou (2007). Then, they
challenge their strategy that there is a look-ahead bias.
We mainly focus on the strategy of Moreira and Muir (2017). From the above, we already know the definition and estimated method of L. For instance, using the returns of whole period from August 1926 to December 2017. Liu, Tang, and Zhou (2018) consider a look-ahead bias and hence construct two ways to estimate L. The first one is fix window approach, which use the initial ten years, August 1926 to July 1936, for L of the entire period. The second approach is ten-year rolling window, which allows L being variable every month. For example, the returns of August 1926 to July 1936 determine the L in August 1936, and the returns of September 1926 to August 1936 determine the L in September 1936.
Therefore, in our article, we also conduct the second approach in Liu, Tang, and Zhou (2018), using rolling window to predict target volatility to avoid look-ahead bias. In the next section, we take look-ahead bias into consideration and implement on robustness test.
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We begin by examining data on the iShares MSCI World ETF from 2013/01- 2019/02. The Morgan Stanley Capital International (MSCI) World Index tracks the performance of investments across 23 developed markets countries, such as United States, Canada, Europe, Asia-Pacific region and so on. The iShares MSCI World ETF is the only fund that directly tracks the MSCI World Index. The composition of the ETF is 62% of the fund which is invested in the United States, 8% in Japan, 5% in the United Kingdom, 3% in France, 3% in Canada, 3% Germany and etc. Its holdings include more than 1,190 securities across all of the major equity. As for the risk-free assets, we use iShares US Treasury Bond ETF, the same period as the index portfolio.
Because the United States account for a large proportion in MSCI World ETF, we decide to take “US” Treasury Bond ETF as our risk-free assets.
Both of the ETFs are daily data and are obtained from Thomson Reuters Eikon database. There is no missing value during 2013/01 to 2019/02. We adopt the data beginning from 2013 and delete the part from 2012/01-2012/12 because there are lots of missing values in the interval.
In Table 1, we show the descriptive statistics of US Treasury Bond ETF and MSCI World ETF from the full sample 2013/01 to 2019/02, including mean, variance, skewness, kurtosis, minimum, median, and maximum. As expected, the mean of US Treasury Bond ETF is significant lower than MSCI World ETF, with the negative mean -0.004 versus 0.073. The volatility is also lower in US Treasury Bond ETF, which is one quarter times comparing to MSCI World ETF. As for skewness and