'The total return on a portfolio of investments takes into account not only the capital appreciation on the portfolio, but also the income received on the portfolio. The income typically consists of interest, dividends, and securities lending fees. This contrasts with the price return, which takes into account only the capital gain on an investment.'

Which means for our asset as example:- Looking at the total return, or increase in value of 258.9% in the last 5 years of ProShares Ultra QQQ, we see it is relatively larger, thus better in comparison to the benchmark SPY (95.5%)
- During the last 3 years, the total return is 12.4%, which is lower, thus worse than the value of 25.3% from the benchmark.

'The compound annual growth rate (CAGR) is a useful measure of growth over multiple time periods. It can be thought of as the growth rate that gets you from the initial investment value to the ending investment value if you assume that the investment has been compounding over the time period.'

Which means for our asset as example:- Looking at the compounded annual growth rate (CAGR) of 29.2% in the last 5 years of ProShares Ultra QQQ, we see it is relatively higher, thus better in comparison to the benchmark SPY (14.4%)
- Compared with SPY (7.8%) in the period of the last 3 years, the compounded annual growth rate (CAGR) of 4% is lower, thus worse.

'Volatility is a statistical measure of the dispersion of returns for a given security or market index. Volatility can either be measured by using the standard deviation or variance between returns from that same security or market index. Commonly, the higher the volatility, the riskier the security. In the securities markets, volatility is often associated with big swings in either direction. For example, when the stock market rises and falls more than one percent over a sustained period of time, it is called a 'volatile' market.'

Using this definition on our asset we see for example:- The 30 days standard deviation over 5 years of ProShares Ultra QQQ is 51.1%, which is higher, thus worse compared to the benchmark SPY (20.9%) in the same period.
- Compared with SPY (17.5%) in the period of the last 3 years, the 30 days standard deviation of 47.2% is higher, thus worse.

'The downside volatility is similar to the volatility, or standard deviation, but only takes losing/negative periods into account.'

Applying this definition to our asset in some examples:- The downside volatility over 5 years of ProShares Ultra QQQ is 36.3%, which is higher, thus worse compared to the benchmark SPY (15%) in the same period.
- Looking at downside risk in of 33.4% in the period of the last 3 years, we see it is relatively higher, thus worse in comparison to SPY (12.3%).

'The Sharpe ratio (also known as the Sharpe index, the Sharpe measure, and the reward-to-variability ratio) is a way to examine the performance of an investment by adjusting for its risk. The ratio measures the excess return (or risk premium) per unit of deviation in an investment asset or a trading strategy, typically referred to as risk, named after William F. Sharpe.'

Applying this definition to our asset in some examples:- Looking at the ratio of return and volatility (Sharpe) of 0.52 in the last 5 years of ProShares Ultra QQQ, we see it is relatively lower, thus worse in comparison to the benchmark SPY (0.57)
- During the last 3 years, the ratio of return and volatility (Sharpe) is 0.03, which is lower, thus worse than the value of 0.3 from the benchmark.

'The Sortino ratio, a variation of the Sharpe ratio only factors in the downside, or negative volatility, rather than the total volatility used in calculating the Sharpe ratio. The theory behind the Sortino variation is that upside volatility is a plus for the investment, and it, therefore, should not be included in the risk calculation. Therefore, the Sortino ratio takes upside volatility out of the equation and uses only the downside standard deviation in its calculation instead of the total standard deviation that is used in calculating the Sharpe ratio.'

Which means for our asset as example:- Compared with the benchmark SPY (0.79) in the period of the last 5 years, the excess return divided by the downside deviation of 0.74 of ProShares Ultra QQQ is lower, thus worse.
- Compared with SPY (0.43) in the period of the last 3 years, the downside risk / excess return profile of 0.04 is lower, thus worse.

'Ulcer Index is a method for measuring investment risk that addresses the real concerns of investors, unlike the widely used standard deviation of return. UI is a measure of the depth and duration of drawdowns in prices from earlier highs. Using Ulcer Index instead of standard deviation can lead to very different conclusions about investment risk and risk-adjusted return, especially when evaluating strategies that seek to avoid major declines in portfolio value (market timing, dynamic asset allocation, hedge funds, etc.). The Ulcer Index was originally developed in 1987. Since then, it has been widely recognized and adopted by the investment community. According to Nelson Freeburg, editor of Formula Research, Ulcer Index is “perhaps the most fully realized statistical portrait of risk there is.'

Applying this definition to our asset in some examples:- Compared with the benchmark SPY (9.32 ) in the period of the last 5 years, the Ulcer Ratio of 29 of ProShares Ultra QQQ is higher, thus worse.
- Looking at Ulcer Ratio in of 36 in the period of the last 3 years, we see it is relatively greater, thus worse in comparison to SPY (10 ).

'Maximum drawdown is defined as the peak-to-trough decline of an investment during a specific period. It is usually quoted as a percentage of the peak value. The maximum drawdown can be calculated based on absolute returns, in order to identify strategies that suffer less during market downturns, such as low-volatility strategies. However, the maximum drawdown can also be calculated based on returns relative to a benchmark index, for identifying strategies that show steady outperformance over time.'

Using this definition on our asset we see for example:- Looking at the maximum reduction from previous high of -63.7 days in the last 5 years of ProShares Ultra QQQ, we see it is relatively lower, thus worse in comparison to the benchmark SPY (-33.7 days)
- Looking at maximum drop from peak to valley in of -63.7 days in the period of the last 3 years, we see it is relatively smaller, thus worse in comparison to SPY (-24.5 days).

'The Drawdown Duration is the length of any peak to peak period, or the time between new equity highs. The Max Drawdown Duration is the worst (the maximum/longest) amount of time an investment has seen between peaks (equity highs) in days.'

Using this definition on our asset we see for example:- The maximum days below previous high over 5 years of ProShares Ultra QQQ is 629 days, which is higher, thus worse compared to the benchmark SPY (488 days) in the same period.
- Compared with SPY (488 days) in the period of the last 3 years, the maximum days below previous high of 629 days is greater, thus worse.

'The Drawdown Duration is the length of any peak to peak period, or the time between new equity highs. The Avg Drawdown Duration is the average amount of time an investment has seen between peaks (equity highs), or in other terms the average of time under water of all drawdowns. So in contrast to the Maximum duration it does not measure only one drawdown event but calculates the average of all.'

Using this definition on our asset we see for example:- The average days under water over 5 years of ProShares Ultra QQQ is 186 days, which is larger, thus worse compared to the benchmark SPY (123 days) in the same period.
- Looking at average time in days below previous high water mark in of 277 days in the period of the last 3 years, we see it is relatively larger, thus worse in comparison to SPY (179 days).

Historical returns have been extended using synthetic data.
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- Note that yearly returns do not equal the sum of monthly returns due to compounding.
- Performance results of ProShares Ultra QQQ are hypothetical and do not account for slippage, fees or taxes.