See the bottom of the page for further sources of reference
Alpha vs. Benchmark Index
Alpha is another statistic in Modern Portfolio Theory (MPT) generated from a linear
regression of the fund's returns less the risk free rate against the market's returns less the
risk free rate. It measures the difference between the fund's actual returns and its expected
performance given its level of risk (as measured by beta).
Alpha is frequently used to measure manager or strategy performance. A positive alpha figure
indicates the fund has performed better than its beta would predict. In contrast, a negative
alpha indicates a fund has underperformed given the expectations established by the fund's beta.
Some investors see the alpha as a measurement of the value added or subtracted by a fund's
However, there are limitations to alpha statistic's ability to accurately depict a manager's added
or subtracted value. In some cases, a negative alpha can result from the expenses that are present in
the fund figures but are not present in the figures of the comparison index. Alpha is dependent on the
accuracy of beta: If the investor accepts beta as a conclusive definition of risk, a positive alpha
would be a conclusive indicator of good fund performance. Of course, the value of beta is dependent
on another statistic, known as R-squared.
For Alpha, the calculation is listed below.
Alpha = (Fund Return - Treasury) - ((Beta x (Benchmark - Treasury))
Benchmark = Total Return of Benchmark Index
Treasury = Return on 13-week Treasury Bill
Annualized Benchmark Return
This is the annualized return on the benchmark index (e.g. Standard and Poor's 500).
This is the annualized total return on an asset.
A total return can be annualized in the expression:
|Annual ret. = (Tot. Ret. + 1)^(365.25 ⁄ days) - 1|
The rate of trading activity in a fund's portfolio of investments,
equal to the lesser of purchases or sales, for a year, divided by average total assets.
Beta vs. Benchmark Index
Beta is another statistic in Modern Portfolio Theory (MPT) generated from a linear regression of the fund's returns
less the risk free rate against the market's returns less the risk free rate.
It measures the fund's sensitivity to market movements.
For example, a fund that has a beta of 1.10 means that for every return in the S&P 500 (or the chosen benchmark),
the fund's returns, on average, will be 1.10 * the benchmark return.
So if the S&P returns 10%, the fund will return 11%.
The reverse is true if the benchmark declines. If the benchmark returns -10%, the fund will return -11%.
Conversely, a beta of 0.85 indicates that the fund has performed 15% worse than the index in up markets
and 15% better in down markets. Therefore, by definition, the beta of the benchmark is 1.
A low beta does not mean that the fund has a low level of volatility, though; rather, a low beta means only
that the fund's market-related risk is low. A specialty fund that invests primarily in gold, for example,
will often have a low beta (and a low R-squared), relative to the S&P 500 index, as its performance is tied
more closely to the price of gold and gold-mining stocks than to the overall stock market. Thus, though the
specialty fund might fluctuate wildly because of rapid changes in gold prices, its beta relative to the
S&P may remain low.
The correlation coefficient is a measure of the strength of the linear relationship between two random variables, where the value 0 indicates independent variables, and 1 completely correlated variables. So, intuitively, this can be used to determine how the returns on a fund and returns on a benchmark are correlated. By convention, correlation is denoted by the greek letter ρ, and the coefficient used here is found by dividing the covariance of the two variables by the product of their standard deviations.
can be loosely defined as the largest drop from a peak to a bottom in a certain time period.
R-Squared vs. Benchmark Index
The R-Squared statistic is computationally the square of the correlation statistic (so, ρ2). Conceptually, it represents the percentage of the fund's returns that are explained by the returns of the benchmark. An R-squared of 1 means that the fund's returns are completely explained by the returns of the index. Conversely, a low R-squared indicates that very few of the fund's returns are explained by the returns of benchmark index. For example, An R-Squared of 50% means that 50% of the fund's returns can be explained by the benchmark's returns. Therefore, R-squared can be used to judge the significance of the fund's beta or alpha statistics. Generally, a higher R-squared will indicate a more useful beta figure. If the R-squared is lower, then the beta is less relevant to the fund's performance.
The Sharpe ratio is a risk-adjusted measure developed by Nobel Laureate William Sharpe. It measures the return per unit of risk. In other words, it measures how efficiently the fund is performing relative to its level of risk - the higher the Sharpe ratio, the higher the return given its risk. The Sharpe Ratio is calculated as the ratio of return of the fund above the risk-free return to annualized standard deviation. Risk-free return is the average monthly return of the 10Y Note over the appropriate period.
|Sharpe Ratio = ( Annualized Return - Risk Free Return ) ⁄ Annualized Std. Dev.|
This ratio is computationally very similar to the Sharpe Ratio, but divides from the excess
return of the portfolio by the standard deviation of the negative returns. The Sortino Ratio therefore uses
downside standard deviation as the proxy for risk for investors, instead of using standard deviation of all
the fund's returns, as this number includes upside standard deviation. This in effect removes the negative
penalty that the Sharpe Ratio imposes on positive returns.
To help you intuitively use this ratio, imagine a hypothetical portfolio, Portfolio A, which never
experiences negative returns. However, Portfolio A has incredible standard deviation in its positive returns:
one day it returns 0.1% and another 1000%. The standard deviation of Portfolio A will therefore be very large.
When measured by Sharpe Ratio, Portfolio A will have a low ratio, because it is symmetric in its treatment of
upside and downside deviation. However, the Sortino Ratio of Portfolio A will be infinite! This is the case
because there is zero standard deviation in negative returns. The Sortino Ratio only considers downside standard
deviation as important.
Similarly, imagine Portfolio B, where there are only negative returns. In this case, the Sharpe Ratio and
the Sortino Ratio will be exactly the same.
Therefore, the higher the Sortino Ratio, the better the risk adjusted (as measured by downside standard deviation)
returns are for your portfolio.
Standard Deviation (Volatility)
This statistical measurement of dispersion about an average depicts how widely a model or simulation returns are varied over a certain period of time. When a fund has a high standard deviation, the predicted range of performance is wide, implying greater volatility.
Investors can use the standard deviation of historical performance to try to predict the range of returns that are most likely in the future. Since a model's returns are assumed to follow a normal distribution, then approximately 68% of the time the returns will fall within one standard deviation of the mean, and 95% of the time within two standard deviations. For example, for a fund with a mean annual return of 10% and a standard deviation of 2%, you would expect the return to be between 8% and 12% about 68%of the time, and between 6% and 14% about 95% of the time.
At Portfolio123, the standard deviation is computed using the three year trailing weekly returns, and since inception. The results are then annualized.
The total return on a fund is expressed as a percentage.
That is, it is calculated as a simple return in the formula:
|Tot. Ret. = ( Ending capital ⁄ Starting Capital ) - 1.|
|At Portfolio123, we calculate the total return on the fund since it's inception, and for the trailing day,
week, four weeks, thirteen weeks, twenty-six weeks, year and three years.|
Year to Date
This is the total return on an asset since the beginning of the financial year.
Notes on Portfolio123's calculations
We only calculate risk statistics for
portfolios and simulations with over 6 month's worth of data. On the "Risk" page we display the Modern Portfolio Theory and Volatility measurements
for that portfolio or simulation, for two time periods:
- from inception to end date
- for a three year period beginning three years before the end date,
given that the inception date for the fund is more than three years before the end date.
For an example of portfolio statistics in Microsoft's Excel, click here.
has an excellent guide to terminology in the form of a glossary.
Other references worth checking are RiskGlossary
If you have any other questions regarding statistics, we suggest you have a look at
Wikipedia's glossary of probability and statistics.