Preliminary Data Preparation

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both offers the lowest trading and financing costs simultaneously. Hence, the paper seeks a compromise which is found on the online trading platform Interactive Brokers. The paper does not assume that the trades in this paper would be implemented using this broker, but rather use the information as a point of departure to gain some bearings in terms of what an investor could expect to be charged in the real world.

The trading cost of almost all the instruments in the dataset is 0.1% of transaction size. A select few of the instruments offer a slightly lower rate. US ETFs are traded at a fixed price of USD 0.005 per share and USD 1.00 per order (Interactive Brokers, 2019b). For simplicity, the paper will use 0.1% of transaction size as the broker commission in the analysis. As will be explained in Section 4.2, a sensitivity analysis is conducted where the transaction costs are both higher and lower than 0.1% providing deeper insights into the effects of transaction costs.

The annualized financing cost quoted on InteractiveBrokers.com is given by the risk-free rate plus a premium of 2.5% (Interactive Brokers, 2019a). To arrive at the monthly rate this number is then divided by 12. This will be the rate that the paper uses as its benchmark financing cost in the forthcoming analysis. It must be noted that cheaper rates are obtainable if the trader is able to qualify for a PRO membership, where the rate decrease depending on investment size. As with the transaction costs, a sensitivity analysis using both a higher and lower financing cost will be conducted

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𝑟 = 𝑅 − 𝑟𝑓 (3.2)

Different perspectives exist concerning what the excess return means. One perspective is that the investor borrows at the risk-free rate and invests in an asset, meaning that the risk-free rate is an actual cost that is incurred. A second perspective is that the risk-free rate is the rate at which an investor could safely house his wealth and is therefore not an actual cost but rather an opportunity cost. This paper follows the second perspective.

3.2.2 Estimating Ex-ante Volatility

When calculating the weights in the LTSMOM and UTSMOM strategies, the standard deviation of each asset is the central component. To this end, ex ante volatilities must be estimated. This paper will follow the methodology of Moskowitz et al. (2012), using an exponentially weighted lagged squared daily returns model. The ex-ante variance is calculated using the formula:

𝜎 = 261 (1 − 𝛿)𝛿 (𝑟 − 𝑟̅ ) (3.3)

The standard deviation is then calculated by simply taking the square root of the variance:

𝜎 = 𝜎 (3.4)

In Equation 3.3 t is time measures in months, whereas i is measured in days. The scalar 261 annualizes the daily variance. Moskowitz et al. (2012) vary 𝛿 so that the centre of mass is 60 days. For simplicity this paper follows the methodology of Babu, Levine, Ooi, Pedersen, and Stamelos (2019) and sets the input 𝛿 to 0.98.

3.2.3 Modelling Spread Transaction Costs

Lesmond et al. (2004) and Korajczyk and Sadka (2004) acquire bid-ask spread data from the NYSE TAQ database, Frazzini et al. (2015) use unique live trading data on bid and ask quotes. Unfortunately, this paper has access to neither of these sources of data. Moreover, the availability and reliability of the data from the BLOOMBERG and Capital IQ (COMPUSTAT) databases for historical bid-ask data is underwhelming. Therefore bid-ask spreads must be estimated. While many methods are available to estimate transaction costs, each method possesses both strengths and weaknesses. When modelling the transaction costs used in the analysis, careful consideration has been taken to select the appropriate technique. One method which has been considered is derived by Roll (1984) and is captured by the expression: 𝑆𝑝𝑟𝑒𝑎𝑑 = 2√−𝑐𝑜𝑣. However, an underlying premise for this method is that markets are efficient an autocovariances are therefore negative. Harris (1990) finds that many of the autocovariances are nonnegative resulting in undefined results. Given the lack of reliable bid-ask quotes and the

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flaws of the Roll estimate, the paper uses a different approach to estimate the spread. The paper will use the method prescribed by Corwin and Schultz (2012), drawing on daily high and low prices. The approach rests on the idea that the high-low price ratio of an asset consists of its true variance and the bid-ask spread. The authors argue that the variance component changes in proportion with time, whereas the bid-ask component does not. Hence, by deriving one equation which is a function of the high-low ratios over two consecutive days and another that is a function of the ratio from a single two-day period, it is possible to solve for the spread and variance components individually. By path of several derivational steps, which this paper will overt, the following spread estimate is constructed:

𝑆 =2(𝑒 − 1)

1 + 𝑒 (3.5)

where,

𝛼 =

2𝛽 − 𝛽

3 − 2√2 − 𝛾 3 − 2√2 ,

𝛾 = ln 𝐻_,

𝐿_, ,

𝛽 = ln 𝐻

𝐿

Here 𝐻_, is the high price over the two days 𝑡 and 𝑡 + 1, and 𝐿_, is the low price over the same two days.

𝐻 is the high price on day 𝑡 + 𝑗 and 𝐿 is the low price on the same day. Since the spread is the cost for a round-trip, that is the cost of buying and selling an asset, the spread estimate must be halved, to account for a one-way cost. Moreover, this paper will take the arithmetic mean of all daily half-spread estimates in a given month for each asset and use these as the bid-ask transaction costs resulting in the following formula:

𝐵𝐴 =1 𝐷

𝑆

2 (3.6)

Where 𝐵𝐴 is the average bid-ask half spread in month t on asset s. D represents the total number of observations in the given month and will vary slightly from month to month. This estimate will be used to calculate transaction costs arising from the bid-ask spread. The reason for using an average spread estimate instead of the precise daily

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spread estimate on the execution day is simply to account for the fact that on a single day a trade cost may be unusually high or low than what is representative for the period of time. For instance, if the bid-ask spread is unusually high for some reason on a specific day, the trader would in a real-life situation perhaps wait a day before executing the trade. If the trader is in a period where the spread is high, this will still be represented in the results, contrarily is the spread on that day reveals itself as an outlier the trading cost will be reduced. It could be argued that this approach aligns better with reality than simply using the potentially extreme value on a specific day, since many traders use limits on the sale and purchase of assets.