'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.'

Applying this definition to our asset in some examples:- Looking at the total return, or increase in value of 63.8% in the last 5 years of ProShares Ultra Gold, we see it is relatively lower, thus worse in comparison to the benchmark SPY (94.9%)
- Looking at total return in of 42.2% in the period of the last 3 years, we see it is relatively greater, thus better in comparison to SPY (22.5%).

'The compound annual growth rate isn't a true return rate, but rather a representational figure. It is essentially a number that describes the rate at which an investment would have grown if it had grown the same rate every year and the profits were reinvested at the end of each year. In reality, this sort of performance is unlikely. However, CAGR can be used to smooth returns so that they may be more easily understood when compared to alternative investments.'

Applying this definition to our asset in some examples:- The annual performance (CAGR) over 5 years of ProShares Ultra Gold is 10.4%, which is smaller, thus worse compared to the benchmark SPY (14.3%) in the same period.
- Compared with SPY (7%) in the period of the last 3 years, the compounded annual growth rate (CAGR) of 12.5% is higher, thus better.

'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.'

Which means for our asset as example:- Looking at the volatility of 31.3% in the last 5 years of ProShares Ultra Gold, we see it is relatively greater, thus worse in comparison to the benchmark SPY (20.9%)
- Compared with SPY (17.5%) in the period of the last 3 years, the 30 days standard deviation of 28.6% is higher, 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.'

Applying this definition to our asset in some examples:- The downside risk over 5 years of ProShares Ultra Gold is 22.1%, which is larger, thus worse compared to the benchmark SPY (15%) in the same period.
- During the last 3 years, the downside volatility is 19.7%, which is larger, thus worse than the value of 12.3% 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:- Looking at the risk / return profile (Sharpe) of 0.25 in the last 5 years of ProShares Ultra Gold, we see it is relatively lower, thus worse in comparison to the benchmark SPY (0.56)
- During the last 3 years, the ratio of return and volatility (Sharpe) is 0.35, which is larger, thus better than the value of 0.26 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.'

Applying this definition to our asset in some examples:- The downside risk / excess return profile over 5 years of ProShares Ultra Gold is 0.36, which is lower, thus worse compared to the benchmark SPY (0.79) in the same period.
- Compared with SPY (0.37) in the period of the last 3 years, the downside risk / excess return profile of 0.51 is higher, thus better.

'The ulcer index is a stock market risk measure or technical analysis indicator devised by Peter Martin in 1987, and published by him and Byron McCann in their 1989 book The Investors Guide to Fidelity Funds. It's designed as a measure of volatility, but only volatility in the downward direction, i.e. the amount of drawdown or retracement occurring over a period. Other volatility measures like standard deviation treat up and down movement equally, but a trader doesn't mind upward movement, it's the downside that causes stress and stomach ulcers that the index's name suggests.'

Using this definition on our asset we see for example:- The Downside risk index over 5 years of ProShares Ultra Gold is 25 , which is higher, thus worse compared to the benchmark SPY (9.32 ) in the same period.
- Compared with SPY (10 ) in the period of the last 3 years, the Ulcer Index of 20 is higher, thus worse.

'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.'

Applying this definition to our asset in some examples:- The maximum DrawDown over 5 years of ProShares Ultra Gold is -46.2 days, which is lower, thus worse compared to the benchmark SPY (-33.7 days) in the same period.
- During the last 3 years, the maximum reduction from previous high is -40.2 days, which is smaller, thus worse than the value of -24.5 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'

Using this definition on our asset we see for example:- Compared with the benchmark SPY (488 days) in the period of the last 5 years, the maximum days under water of 950 days of ProShares Ultra Gold is higher, thus worse.
- Compared with SPY (488 days) in the period of the last 3 years, the maximum time in days below previous high water mark of 518 days is greater, thus worse.

'The Average Drawdown Duration is an extension of the Maximum Drawdown. However, this metric does not explain the drawdown in dollars or percentages, rather in days, weeks, or months. 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:- Looking at the average days under water of 377 days in the last 5 years of ProShares Ultra Gold, we see it is relatively greater, thus worse in comparison to the benchmark SPY (123 days)
- During the last 3 years, the average days under water is 191 days, which is higher, thus worse than the value of 179 days from the benchmark.

Historical returns have been extended using synthetic data.
[Show Details]

- Note that yearly returns do not equal the sum of monthly returns due to compounding.
- Performance results of ProShares Ultra Gold are hypothetical and do not account for slippage, fees or taxes.