'Total return is the amount of value an investor earns from a security over a specific period, typically one year, when all distributions are reinvested. Total return is expressed as a percentage of the amount invested. For example, a total return of 20% means the security increased by 20% of its original value due to a price increase, distribution of dividends (if a stock), coupons (if a bond) or capital gains (if a fund). Total return is a strong measure of an investment’s overall performance.'

Applying this definition to our asset in some examples:- Looking at the total return, or increase in value of 654.9% in the last 5 years of ProShares Ultra QQQ, we see it is relatively larger, thus better in comparison to the benchmark SPY (122.7%)
- Compared with SPY (65.3%) in the period of the last 3 years, the total return, or increase in value of 232.4% is greater, thus better.

'Compound annual growth rate (CAGR) is a business and investing specific term for the geometric progression ratio that provides a constant rate of return over the time period. CAGR is not an accounting term, but it is often used to describe some element of the business, for example revenue, units delivered, registered users, etc. CAGR dampens the effect of volatility of periodic returns that can render arithmetic means irrelevant. It is particularly useful to compare growth rates from various data sets of common domain such as revenue growth of companies in the same industry.'

Which means for our asset as example:- The compounded annual growth rate (CAGR) over 5 years of ProShares Ultra QQQ is 49.9%, which is higher, thus better compared to the benchmark SPY (17.4%) in the same period.
- Compared with SPY (18.2%) in the period of the last 3 years, the annual performance (CAGR) of 49.2% is larger, thus better.

'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:- Looking at the volatility of 44.2% in the last 5 years of ProShares Ultra QQQ, we see it is relatively higher, thus worse in comparison to the benchmark SPY (18.7%)
- During the last 3 years, the volatility is 52.3%, which is larger, thus worse than the value of 22.5% from the benchmark.

'Downside risk is the financial risk associated with losses. That is, it is the risk of the actual return being below the expected return, or the uncertainty about the magnitude of that difference. Risk measures typically quantify the downside risk, whereas the standard deviation (an example of a deviation risk measure) measures both the upside and downside risk. Specifically, downside risk in our definition is the semi-deviation, that is the standard deviation of all negative returns.'

Using this definition on our asset we see for example:- The downside volatility over 5 years of ProShares Ultra QQQ is 31.5%, which is higher, thus worse compared to the benchmark SPY (13.6%) in the same period.
- During the last 3 years, the downside deviation is 37.4%, which is higher, thus worse than the value of 16.3% from the benchmark.

'The Sharpe ratio was developed by Nobel laureate William F. Sharpe, and is used to help investors understand the return of an investment compared to its risk. The ratio is the average return earned in excess of the risk-free rate per unit of volatility or total risk. Subtracting the risk-free rate from the mean return allows an investor to better isolate the profits associated with risk-taking activities. One intuition of this calculation is that a portfolio engaging in 'zero risk' investments, such as the purchase of U.S. Treasury bills (for which the expected return is the risk-free rate), has a Sharpe ratio of exactly zero. Generally, the greater the value of the Sharpe ratio, the more attractive the risk-adjusted return.'

Applying this definition to our asset in some examples:- Compared with the benchmark SPY (0.8) in the period of the last 5 years, the ratio of return and volatility (Sharpe) of 1.07 of ProShares Ultra QQQ is higher, thus better.
- Looking at ratio of return and volatility (Sharpe) in of 0.89 in the period of the last 3 years, we see it is relatively greater, thus better in comparison to SPY (0.7).

'The Sortino ratio improves upon the Sharpe ratio by isolating downside volatility from total volatility by dividing excess return by the downside deviation. The Sortino ratio is a variation of the Sharpe ratio that differentiates harmful volatility from total overall volatility by using the asset's standard deviation of negative asset returns, called downside deviation. The Sortino ratio takes the asset's return and subtracts the risk-free rate, and then divides that amount by the asset's downside deviation. The ratio was named after Frank A. Sortino.'

Applying this definition to our asset in some examples:- Looking at the downside risk / excess return profile of 1.5 in the last 5 years of ProShares Ultra QQQ, we see it is relatively higher, thus better in comparison to the benchmark SPY (1.1)
- During the last 3 years, the ratio of annual return and downside deviation is 1.25, which is greater, thus better than the value of 0.96 from the benchmark.

'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.'

Using this definition on our asset we see for example:- Compared with the benchmark SPY (5.58 ) in the period of the last 5 years, the Downside risk index of 12 of ProShares Ultra QQQ is greater, thus worse.
- During the last 3 years, the Ulcer Index is 14 , which is greater, thus worse than the value of 6.83 from the benchmark.

'A maximum drawdown is the maximum loss from a peak to a trough of a portfolio, before a new peak is attained. Maximum Drawdown is an indicator of downside risk over a specified time period. It can be used both as a stand-alone measure or as an input into other metrics such as 'Return over Maximum Drawdown' and the Calmar Ratio. Maximum Drawdown is expressed in percentage terms.'

Applying this definition to our asset in some examples:- Compared with the benchmark SPY (-33.7 days) in the period of the last 5 years, the maximum reduction from previous high of -51.7 days of ProShares Ultra QQQ is smaller, thus worse.
- During the last 3 years, the maximum reduction from previous high is -51.7 days, which is smaller, thus worse than the value of -33.7 days from the benchmark.

'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.'

Applying this definition to our asset in some examples:- Looking at the maximum time in days below previous high water mark of 214 days in the last 5 years of ProShares Ultra QQQ, we see it is relatively larger, thus worse in comparison to the benchmark SPY (139 days)
- During the last 3 years, the maximum days under water is 214 days, which is larger, thus worse than the value of 139 days from the benchmark.

'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.'

Applying this definition to our asset in some examples:- Looking at the average days under water of 37 days in the last 5 years of ProShares Ultra QQQ, we see it is relatively higher, thus worse in comparison to the benchmark SPY (33 days)
- Compared with SPY (35 days) in the period of the last 3 years, the average time in days below previous high water mark of 52 days is larger, thus worse.

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, do not account for slippage, fees or taxes, and are based on backtesting, which has many inherent limitations, some of which are described in our Terms of Use.