'Total return, when measuring performance, is the actual rate of return of an investment or a pool of investments over a given evaluation period. Total return includes interest, capital gains, dividends and distributions realized over a given period of time. Total return accounts for two categories of return: income including interest paid by fixed-income investments, distributions or dividends and capital appreciation, representing the change in the market price of an asset.'

Which means for our asset as example:- Looking at the total return of 30.5% in the last 5 years of ProShares Ultra Silver, we see it is relatively smaller, thus worse in comparison to the benchmark SPY (122%)
- Looking at total return, or increase in value in of 61.6% in the period of the last 3 years, we see it is relatively higher, thus better in comparison to SPY (61%).

'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:- The compounded annual growth rate (CAGR) over 5 years of ProShares Ultra Silver is 5.5%, which is lower, thus worse compared to the benchmark SPY (17.3%) in the same period.
- Looking at annual performance (CAGR) in of 17.4% in the period of the last 3 years, we see it is relatively greater, thus better in comparison to SPY (17.2%).

'In finance, volatility (symbol σ) is the degree of variation of a trading price series over time as measured by the standard deviation of logarithmic returns. Historic volatility measures a time series of past market prices. Implied volatility looks forward in time, being derived from the market price of a market-traded derivative (in particular, an option). Commonly, the higher the volatility, the riskier the security.'

Applying this definition to our asset in some examples:- Compared with the benchmark SPY (18.8%) in the period of the last 5 years, the historical 30 days volatility of 55.1% of ProShares Ultra Silver is greater, thus worse.
- Compared with SPY (22.5%) in the period of the last 3 years, the 30 days standard deviation of 63.7% is larger, thus worse.

'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:- Compared with the benchmark SPY (13.6%) in the period of the last 5 years, the downside risk of 39.4% of ProShares Ultra Silver is higher, thus worse.
- Looking at downside deviation in of 45.4% in the period of the last 3 years, we see it is relatively higher, thus worse in comparison to SPY (16.3%).

'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.'

Using this definition on our asset we see for example:- The ratio of return and volatility (Sharpe) over 5 years of ProShares Ultra Silver is 0.05, which is lower, thus worse compared to the benchmark SPY (0.79) in the same period.
- Compared with SPY (0.65) in the period of the last 3 years, the risk / return profile (Sharpe) of 0.23 is smaller, thus worse.

'The Sortino ratio measures the risk-adjusted return of an investment asset, portfolio, or strategy. It is a modification of the Sharpe ratio but penalizes only those returns falling below a user-specified target or required rate of return, while the Sharpe ratio penalizes both upside and downside volatility equally. Though both ratios measure an investment's risk-adjusted return, they do so in significantly different ways that will frequently lead to differing conclusions as to the true nature of the investment's return-generating efficiency. The Sortino ratio is used as a way to compare the risk-adjusted performance of programs with differing risk and return profiles. In general, risk-adjusted returns seek to normalize the risk across programs and then see which has the higher return unit per risk.'

Which means for our asset as example:- The excess return divided by the downside deviation over 5 years of ProShares Ultra Silver is 0.08, which is lower, thus worse compared to the benchmark SPY (1.09) in the same period.
- Looking at downside risk / excess return profile in of 0.33 in the period of the last 3 years, we see it is relatively smaller, thus worse in comparison to SPY (0.9).

'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 Ulcer Index of 43 of ProShares Ultra Silver is larger, thus worse.
- Looking at Ulcer Ratio in of 29 in the period of the last 3 years, we see it is relatively larger, thus worse in comparison to SPY (6.83 ).

'Maximum drawdown measures the loss in any losing period during a fund’s investment record. It is defined as the percent retrenchment from a fund’s peak value to the fund’s valley value. The drawdown is in effect from the time the fund’s retrenchment begins until a new fund high is reached. The maximum drawdown encompasses both the period from the fund’s peak to the fund’s valley (length), and the time from the fund’s valley to a new fund high (recovery). It measures the largest percentage drawdown that has occurred in any fund’s data record.'

Which means for our asset as example:- Looking at the maximum drop from peak to valley of -76.3 days in the last 5 years of ProShares Ultra Silver, we see it is relatively lower, thus worse in comparison to the benchmark SPY (-33.7 days)
- During the last 3 years, the maximum drop from peak to valley is -66.1 days, which is lower, 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). Many assume Max DD Duration is the length of time between new highs during which the Max DD (magnitude) occurred. But that isn’t always the case. The Max DD duration is the longest time between peaks, period. So it could be the time when the program also had its biggest peak to valley loss (and usually is, because the program needs a long time to recover from the largest loss), but it doesn’t have to be'

Applying this definition to our asset in some examples:- Looking at the maximum time in days below previous high water mark of 1007 days in the last 5 years of ProShares Ultra Silver, we see it is relatively higher, thus worse in comparison to the benchmark SPY (139 days)
- Looking at maximum days under water in of 301 days in the period of the last 3 years, we see it is relatively higher, thus worse in comparison to SPY (139 days).

'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 below previous high over 5 years of ProShares Ultra Silver is 427 days, which is larger, thus worse compared to the benchmark SPY (33 days) in the same period.
- Compared with SPY (34 days) in the period of the last 3 years, the average days below previous high of 124 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 Silver 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.