4.2 Methodology
4.2.4 Results and Performance
For both the one-factor and the three-factor model, we must check if the alphas and betas are significant. To do this, we use the associated p-value we obtain when performing the regressions to test for significance. We consider a p-value below 0.05 to be significant.
Next, we look at the average annual excess returns, annual standard deviation, and the Sharpe ratio. To calculate the average annual excess returns, we subtract the daily risk-free rate (three-month T-bill) from the daily log returns for the strategy and use the follow formula:
𝑒𝑟𝑎𝑣𝑒𝑟𝑎𝑔𝑒×252− 1, where the 𝑟𝑎𝑣𝑒𝑟𝑎𝑔𝑒 is the average excess returns over the period. For the standard deviation calculation, we calculate the sample standard deviation on the daily excess strategy returns for the period, and then multiply it by the square root of 252: 𝜎𝑠𝑎𝑚𝑝𝑙𝑒× √252.
Then, we calculate the Sharpe ratio to look at the relationship between standard deviation and returns. The Sharpe ratio measure the reward per standard deviation, or risk, and is calculated by dividing the expected excess returns by the standard deviation (Pedersen, 2015):
𝑆ℎ𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 =𝐸(𝑅 − 𝑟𝑓) 𝜎(𝑅 − 𝑟𝑓)
Another risk-reward measure is the Sortino ratio. Whereas the Sharpe ratio looks at both upside and downside risk, the Sortino ratio is only concerned about the downside risk. This estimate is interesting as the main concern for most investors is downside risk, and the Sortino ratio measures the reward per risk to the downside. To calculate the ratio, we first need to estimate the downside standard deviation. We do this by calculating the standard deviation of the returns where they are below some minimum acceptable return (MAR). For our analysis, we assume the MAR to be equal to the daily risk-free rate. The calculation is done be using the below formula, where we only calculate the standard deviation for the returns below the risk-free rate:
𝜎𝑑𝑜𝑤𝑛𝑠𝑖𝑑𝑒 = √𝐸((𝑀𝐴𝑅 − 𝑅)21{𝑅<𝑀𝐴𝑅})
When we have the downside standard deviation, we use the same annual excess returns as in the Sharpe ratio, and divide it by the downside standard deviation (Pedersen, 2015).
𝑆𝑜𝑟𝑡𝑖𝑛𝑜 𝑟𝑎𝑡𝑖𝑜 =𝐸(𝑅 − 𝑟𝑓) 𝜎𝑑𝑜𝑤𝑛𝑠𝑖𝑑𝑒
Next, we look at the drawdowns. This measure compares the highest value of the portfolio until that point in time with the current value of the portfolio. The drawdown describes the losses of the portfolio and is interesting to look at from a risk perspective. To calculate the drawdown, we first calculate the daily high water mark (HWM), which is equal to the highest portfolio value achieved up until that point in time. Then we subtract today’s price from the high water mark, and then divide it by the high water mark:
𝐷𝐷𝑡 =𝐻𝑊𝑀𝑡− 𝑃𝑡 𝐻𝑊𝑀𝑡
In our result section, we look at the maximum drawdown for the period, which displays the highest drop in the portfolio over the time period (Pedersen, 2015).
We also look at the implied leverage employed in each of the strategies. This is calculated by looking at price change in the strategy relative to the price change in the underlying. For our results, we look at the average daily leverage employed during the time period.
𝐿𝑒𝑣𝑒𝑟𝑎𝑔𝑒 𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑑𝑡= 𝑟𝑡𝑠𝑡𝑟𝑎𝑡𝑒𝑔𝑦 𝑟𝑡𝑢𝑛𝑑𝑒𝑟𝑙𝑦𝑖𝑛𝑔
Finally, we look at the skewness and excess kurtosis of the returns. Both estimates describe the distribution of the returns and is interesting at it provides information on how the strategies affect the distribution. In our calculation, we use normal returns as opposed to log returns when we make these estimates.
The skewness tells us if the returns skew right or left compared to a normal distribution. A positive skew means that the returns skew to the left and that we have fatter tails on the right.
While a negative skew means that the returns skew to the right, and that we have fatter tails to the left (Chen, 2021). This is interesting to know, as we get information on whether the fat tails are more to the upside or to the downside.
The kurtosis on the other hand describes the tails to both the right and to the left. We focus on excess kurtosis, which is obtained by subtracting three from the kurtosis estimate. A higher
kurtosis means that there are fatter tails and more extreme returns. For our analysis this is an interesting metric, as we wish to minimize extreme returns, at least to the downside.
4.2.4.2 Time Periods
In addition to looking at the full period 1996 through 2020, we will also look at the performance for some specific time periods. We have chosen the periods based on the characteristics of the returns. First, we will look at a period with few major drawdowns, then we look at three periods with substantial drawdowns. Looking at these specific periods will provide information on how the strategies perform under various market conditions.
The first additional time period we will analyze, is the period 2010 through 2017. This period is characterized by few and only minor drawdowns. During this period, it was mainly an uptrend in the index value, as can be seen in Figure 4.5 below. By analyzing this period, we will see how the strategies work when there are no large drawdowns.
Figure 4.5: The S&P 500 TR Performance and Drawdowns 2010-2017
Next, we look at the first of three periods with large drawdowns visualized in Figure 4.6.
During the period 2000 through 2003, the max drawdown was close to 50 percent. This period describes the aftermath of the dotcom bubble, where tech companies saw great valuation gains followed by large losses. This period represents a period with a prolonged downturn, as it took around two years for the index to hit its lows.
Figure 4.6: The S&P 500 TR Performance and Drawdowns 2000-2003
We also look at the period 2007 through 2009, pictured in Figure 4.7. This period shows how the S&P 500 TR performed under the global financial crisis. Here the maximum drawdown was above 50 percent, and the downturn was sharper than what we saw after the dotcom bubble.
That makes this period interesting, as we see how the strategies performed during a steeper and more immediate downturn.
Figure 4.7: The S&P 500 TR Performance and Drawdowns 2007-2009
Finally, we will look at a more recent down period in Figure 4.8. During 2020 we saw a very steep downturn in the S&P 500 TR early in the year, before the market recovered quickly thereafter. The performance during this period will provide information on how the strategies work with extremely steep downturns, followed by a speedy recovery.
Figure 4.8: The S&P 500 TR Performance and Drawdowns 2020