This page details all investing calculations and math equations in FastTrack products and popularly used in risk adjusted investing. FastTrack products apply all popular investment calculations to FastTrack’s high-quality dividend-adjusted database.
Alpha is used to measure a mutual fund, ETF, stock or portfolio’s ability to ‘beat’ the market. i.e. Alpha is the measure of the excess return of an investment compared to a benchmark.
Alpha is used to assess the performance of a portfolio manager or strategy.
Alpha is shown as a single percentage and can be positive or negative.
A positive alpha indicates a better return relative to how the benchmark performed, a lower alpha means the investment performed worse than it should, given the return of its benchmark.
Essentially, there are two components of a security’s return. First, the return attributed to the general market. When market as a whole goes up, how much does the security rise (see Beta).
The second component is an increase in the price of the security not explained by the market’s movement. This is important because it measures the skill of a portfolio manager in identifying undervalued securities or market trends that can generate higher returns than the benchmark.
Investors often look for managers or strategies that have historically delivered positive alpha, as it suggests that they may be able to continue to generate excess returns in the future.
Alpha = Annualized security return - risk free rate - beta * (annualized benchmark return - risk free rate) * 100
!! risk free rate = basis ticker in FT Cloud or FastTrack
!!! beta is calculated against benchmark in FT Cloud or FastTrack
Annualized return is a measure of the average rate of return on an investment over a period of time, expressed as an annual percentage rate (APR). It is used to compare the performance of different investments or investment strategies over time.
Annualized return is important because it allows investors to compare the performance of investments with different holding periods and/or to evaluate the performance of a portfolio over a longer time horizon.
( (TotalReturn + 1 ) ^ ( 252.25 / MarketDays ) ) - 1
- Total Return = (endprice – startprice) / startprice
- MarketDays = A count of market days over which the total return was achieved
- 252.25 = The number of market days in a year.
This calculation is known in the investment industry as a Compound Annual Growth Rate (CAGR). There is another calculation called Average Annual Growth Rate (AAGR) which takes the return each year and produces a simple average of the annual return. This calculation does not account for the effect of compounding and we think this is a highly misleading number. AAGR IS NOT provided or used by FT Cloud or FT4web.
Beta is a measure of the volatility or systematic risk of an investment compared to the market as a whole. It is used to evaluate the risk of an investment and to determine its expected return
eta is typically represented as a number, with a beta of 1 indicating that the investment’s price will move in line with the market. A beta greater than 1 indicates that the investment is more volatile than the market, while a beta less than 1 indicates that it is less volatile.
Beta = (Correlation of Issue and Basis * Issue’s standard deviation) / Basis standard deviation
The calculation used to compute FastTrack’s Beta was published in “Guide to Portfolio Management”, James L. Farrell, Jr., McGraw-Hill – 1983, pages 41-43.
When Beta is positive, the security’s price tends to move in the same direction as the market. The magnitude of Beta indicates how much the security moves with the market. If a security’s Beta is greater than 1, that means that when the market index goes up 1%, we expect the security will go up by more than 1%. Or, if the market goes down by 1%, we expect the stock to go down by more than 1%.
Negative betas signify a negative correlation. When the market goes up, a stock with a negative beta would be expected to go down.
There are a number of different ways to compute Beta so do not expect FastTrack’s method to compute exactly the Beta you might see in some publications.
Note: for Beta to have ANY validity, the issue’s and the basis MUST be closely correlated (usually a Correlation of 0.90 or higher).
Correlation is a statistical measure that describes the degree to which two securities or assets move in relation to each other. It is used to evaluate the relationship between two investments and to determine the diversification benefits of including them in a portfolio.
Correlation is important because it allows investors to evaluate the diversification benefits of combining different investments in a portfolio. Investments that have low or negative correlations can help to reduce the overall risk of a portfolio by spreading risk across different asset classes or industries.
Correlation is typically represented as a number between -1 and +1, with a correlation of +1 indicating a perfect positive correlation, a correlation of -1 indicating a perfect negative correlation, and a correlation of 0 indicating no correlation.
A positive correlation means that when the value of one investment goes up, the value of the other investment also tends to go up, and vice versa. For example, two technology stocks might have a positive correlation because they tend to move together in response to market trends and news.
A negative correlation means that when the value of one investment goes up, the value of the other investment tends to go down, and vice versa. For example, a bond fund and a stock fund might have a negative correlation because investors tend to move money out of stocks and into bonds during times of economic uncertainty.
Correlation is calculated over a time period, so the calculations below are done in a loop over the time period.
- L= The number of days in the period for which correlation is computed
- Prices = an array of periodic “independent” returns
- Cor = an array of periodic “dependent” periodic returns
- I = Starting day; Continue for all days; Each repetition I = I + L
- Y = the length in days of the period for which correlation is calculated – Start Day + 1
- XC = [ ( Cor [ I + N ] – Cor [ I ] ) / Cor [ I ] ]
- XP = [ ( Prices [ I + N ] – Prices [ I ] ) / Prices [ I ] ]
- Sum Cor = XC + Sum Cor
- Sum Prices = XP + Sum Prices
- Sum Cor Squared = Sum Cor Squared + XC * XC
- Sum Prices Squared = Sum Prices Squared + XP * XP
- Sum Cor & Prices = (Sum Cor & Prices) + XC * XP
- Sum Cor Squared = Sum Cor Squared – (Sum Cor * Sum Cor ) / Y
- Sum Prices Squared = Sum Prices Squared – (Sum Prices * Sum Prices) / Y
- Sum Product = Sum Cor & Prices – (Sum Cor * Sum Prices) / Y
- Correlation Coefficient = Sum Product / Square Root of (Sum Prices Squared
* Sum Cor Squared)
FastTrack’s correlation is the relation of the basis to the relevant fund, ETF, stock, or equity curve. It is based on the daily change in prices, not the closing prices.
The correlation length is set using the “correlation length” number box at the lower right of the login page.
Dividend adjusting is a method used to adjust the historical prices of a security or index to reflect the impact of dividends, distributions, capital gains, etc paid out to shareholders.
When a security pays a dividend, its price typically drops by the amount of the dividend on the ex-dividend date, which is the first day that the stock trades without the dividend. For example, if a company pays a dividend of $1 per share and the stock is trading at $50 per share, the stock price is likely to drop to $49 per share on the ex-dividend date.
Dividend adjusting is used to remove the impact of these price drops from historical stock prices, in order to provide a more accurate representation of the performance of the security or index over time. This allows investors to compare the performance of the stock or index with and without the impact of dividends.
Visit the following link to see all details on FastTrack dividend adjustments, including a dividend adjustment worksheet.
Downside Deviation is a measure of the volatility of an investment or portfolio based on the downward price movements or negative returns below a certain threshold, typically the risk-free rate or the expected rate of return. It measures the dispersion of these negative returns from the average or expected return of the investment or portfolio, providing a measure of downside risk.
Downside Deviation provides a more accurate measure of downside risk compared to standard deviation, as it only considers negative deviations from the threshold, which are the movements that investors are more concerned about. It helps investors to evaluate the downside risk of an investment or portfolio and to compare the downside risks of different investments or portfolios.
Investors can use Downside Deviation to build more efficient portfolios by identifying investments with lower downside risks or by balancing the downside risks with the expected returns. It can help investors to make more informed investment decisions by providing a measure of the potential downside risk of an investment or portfolio.
Downside Deviation is calculated by taking the square root of the average of the squared differences between the actual returns and the threshold or expected return, only considering negative deviations.
The formula for Downside Deviation is as follows:
Downside Deviation = Square Root [ (1/N) * Σ[max(0, Ri – Threshold)^2] ]
Where: Ri = the return of the investment or portfolio on day i Threshold = the minimum acceptable return, typically the risk-free rate or expected return N = the number of trading days in the period
52 Week High and Low
The 52 week high and low refer to the highest and lowest prices at which a particular stock has traded over the previous 52 weeks, or one year.
The 52 week high and low are commonly used as a reference point for investors and analysts when evaluating the performance of a stock. This information can be useful for determining the stock’s potential for growth or decline, as well as for setting buy and sell targets.
If an investment is trading near its 52 week high, it may indicate it is currently in high demand and that investors are willing to pay a premium for it. Conversely, if a stock is trading near its 52 week low, it may indicate that the stock is currently undervalued and may present a buying opportunity.
To calculate the 52 week high and low, you need to look at the highest and lowest prices at which the investment has traded over the previous 52 weeks.
For example, if you are looking at a stock that is currently trading at $50 per share, and its 52 week high is $75 and its 52 week low is $25, this means that the stock has traded as high as $75 and as low as $25 over the past year.
FT Alpha is a proprietary calculation developed by Investors FastTrack to identify high quality securites to add to a portfolio. It blends risk, return, and correlation into a single metric.
FastTrack’s FT Alpha will identify securites that will increase the return of your portfolio, reduce the risk, and increase the diversification
FT Alpha is a single number, either positive or negative, calculated over a time span.
A number above 0 indicates a security would increase the return, reduce the risk, and has low correlation to the benchmark.
A number below 0 indicates the security would not increase return, reduce risk, and/or is highly correlated to the benchmark.
A low ranking FT Alpha could indicate the security has a poor risk return profile, therefore will not increase the risk return of the benchmark. Or, it could also indicate a high correlation. For example, if you own SPY, adding more SPY to your portfolio will not have a positive impact on risk or return.
We do not disclose the mathematics for this indicator. FTAlpha is the sole proprietary formula within FastTrack.
A moving average is calculated by taking the average price of an asset over a specific time period, with the resulting line representing the smoothed-out trend of the asset’s price movement. The smoothed line helps filter out temporary, and possibly random, short term fluctuations in prices and makes for better trend and direction decisions.
FastTrack products use exponential moving averages by default. Exponential moving average give more weight to recent prices and are more responsive to new data.
Moving averages are often used by investors to identify trend reversals and potential entry and exit points for a stock or other financial instrument. For example, if the current price of a stock crosses above its 50-day moving average, it could be interpreted as a buy signal, indicating that the stock is on an upward trend. Conversely, if the current price of a stock crosses below its 50-day moving average, it could be interpreted as a sell signal, indicating that the stock is on a downward trend.
Moving averages can also be used to identify support and resistance levels, which are key levels at which the price of a stock is likely to experience increased buying or selling pressure. For example, if the 50-day moving average of a stock is currently at $100, it could be seen as a support level for the stock, indicating that buyers are likely to step in if the stock’s price drops to that level.
Moving averages are used to identify trends and potential buy or sell signals in a stock or other financial instrument.
To calculate a moving average:
- Calculate a smoothing factor (SFactor) derived from a parameter value (typically a number of days) . This factor is a constant throughout the averaging process.
SFactor = 2 / (Parameter + 1)
- Each day’s moving average value (MA) is computed as follows,
MA = ( Price * SFactor ) + ( prior MA * (1 – SFactor) )
On the first day, MA is set to the first day’s Price. The MA is carried forward to the next day to be used as the prior MA in the formula.
Maximum drawdown (MDD) is a measure of the largest decline in the value of an investment over a specified period of time, expressed as a percentage.
It is used to evaluate the risk of an investment and to assess the potential losses that an investor may face in a worst-case scenario.
Maximum drawdown is important because it provides an indication of the level of risk associated with an investment. A higher maximum drawdown means that the investment is more volatile and carries a greater risk of significant losses. Investors should be aware of the maximum drawdown associated with an investment and should consider it when evaluating the risk of that investment.
Maximum drawdown can also be used to compare the risk of different investments or investment strategies. For example, two investments with similar returns may have very different maximum drawdowns, indicating that one may be more risky than the other.
In general, Max Drawdown will be the highest for issues whose standard deviation is the highest.
The maximum drawdown is calculated by identifying the peak value of an investment and then calculating the largest percentage decline in value from that peak to the trough that follows. For example, if an investment reaches a peak value of $1,000 and then declines to a low of $700 before recovering, the maximum drawdown would be 30% ($300 divided by $1,000).
R-squared (also known as the coefficient of determination) is a statistical measure that represents the percentage of a fund or portfolio’s movements that can be explained by the movements of a benchmark index, typically the S&P 500 or a similar index. R-squared is used to evaluate the degree to which an investment or portfolio’s performance is related to the performance of a benchmark index.
R-squared is a value between 0 and 1, with 1 representing a perfect correlation between the investment or portfolio and the benchmark index. An R-squared of 0 indicates that there is no correlation between the investment or portfolio and the benchmark index.
Investors can use R-squared to evaluate the degree to which an investment or portfolio is influenced by the performance of a benchmark index. A high R-squared value indicates that the investment or portfolio’s movements are closely related to the movements of the benchmark index, while a low R-squared value indicates that the investment or portfolio’s movements are not related to the movements of the benchmark index.
R-squared is related to correlation in that they both measure the degree to which two variables are related to each other. But, importantly R-squared is a measure of how much of an investment or portfolio’s performance can be explained by the performance of a benchmark index, while correlation measures the strength and direction of the relationship between the investment or portfolio and the benchmark index
The Sortino Ratio is a risk-adjusted performance measure that is used to evaluate the return of an investment relative to its downside risk. Unlike other risk-adjusted measures, such as the Sharpe ratio, which considers both upside and downside volatility, the Sortino Ratio only considers downside volatility.
Investors use the Sortino Ratio to evaluate the performance of an investment or portfolio relative to its downside risk. It helps investors to identify investments that have generated higher returns per unit of downside risk, and to compare the risk-adjusted performance of different investments.
The Sortino Ratio is useful for investors who are more concerned about downside risk than upside volatility, as it only considers downside risk.
The Sortino Ratio provides a measure of the return per unit of downside risk. A higher Sortino Ratio indicates that the investment has generated a higher return per unit of downside risk, while a lower Sortino Ratio indicates the opposite.
The Sortino Ratio is calculated by dividing the excess return of an investment over a minimum acceptable return (MAR) by the downside deviation of the investment. The MAR is typically set to be the risk-free rate or the investor’s required rate of return.
The formula for the Sortino Ratio is as follows:
Sortino Ratio = (R – MAR) / Downside Deviation
Where: R = the average return of the investment MAR = the minimum acceptable return Downside Deviation = the standard deviation of the negative returns below the MAR
Total return is the gain or loss of a security/equity curve for a particular time period.
Total return is a measure of the overall performance of an investment, taking into account both price changes and income generated from the investment, such as dividends or interest. It represents the total profit or loss on an investment over a specified period of time, expressed as a percentage of the initial investment.
Total return is important because it provides a more accurate representation of the performance of an investment than price changes alone. It takes into account any income generated by the investment, which can significantly impact the overall return. For example, a stock that has a price increase of 10% but also pays a dividend of 2% would have a total return of 12%.
Total return is often used to compare the performance of different investments or investment strategies. Investors can compare the total return of a particular stock or bond with the return of a broader market index, such as the S&P 500 or the Dow Jones Industrial Average.
Total return can also be used to evaluate the performance of a portfolio over time, taking into account all of the investments in the portfolio and any income generated by those investments. This can help investors to make more informed decisions about their asset allocation and to adjust their investment strategies as needed to meet their financial goals.
All statistics and returns in FastTrack products are based on dividend adjsuted data, therefore incorporating the true total return of the securities and portfolios.
( Price on last day- price on the first day) / price on the first day
The Sharpe ratio is a measure of risk-adjusted return that takes into account the volatility of an investment. It is used to evaluate the performance of an investment or portfolio by comparing the returns earned above a risk-free rate, such as Treasury bills, with the volatility of those returns.
The classical Sharpe ratio is computed by using IRX-X (US T-Bill 13-Week) as the low risk basis.
FastTrack calculates the Sharpe Ratio against the security/index/Risk Basis of YOUR choice. Using IRX-X is a mediocre choice when trying to select the best fund/ETF.
A higher Sharpe ratio indicates that an investment or portfolio has generated higher returns for the level of risk taken, while a lower Sharpe ratio indicates that an investment or portfolio has generated lower returns relative to the amount of risk taken.
Using FastTrack Sharpe with a highly correlated risk/return basis works better than using uniformly IRX-X for all issues. For example, using SP-DA – S&P 500 Total Return Index as the basis for computing Sharpe on equity funds produces far superior, more relevant risk-adjusted return calculations.
The Sharpe ratio is important because it allows investors to compare the risk-adjusted returns of different investments or portfolios. It helps investors to evaluate whether the returns generated by an investment or portfolio are sufficient to compensate for the level of risk taken.
The Sharpe ratio is calculated by subtracting the risk-free rate from the expected return of the investment or portfolio, and then dividing by the standard deviation of the returns. The formula for the Sharpe ratio is as follows:
Sharpe Ratio = (Issue's Annualized Return - Low Risk Basis Annualized Return) / (Issue's Monthly Standard Deviation * SQR(12))
Similar securities are financial instruments that share similar characteristics, such as investment objectives, risk profile, asset class, and/or sector.
Investors often use similar securities as a benchmark to evaluate the performance of a particular security or portfolio.
Similar securities are used as a benchmark to compare the performance of a security or portfolio to other securities that have similar characteristics. This helps investors evaluate whether the security is outperforming or underperforming relative to similar securities.
For example, a mutual fund that invests primarily in technology stocks may be compared to other mutual funds that have a similar investment objective and asset class. By comparing the performance of the fund to other similar funds, investors can determine whether the fund is generating above-average returns or underperforming relative to its peers.
To identify similar securities, investors can use FastTrack’s screening tools to filter securities based on various criteria, such as asset class, sector, investment objective, and risk profile. Once the criteria are set, FastTrack will generate a list of securities that match the specified criteria, which can then be used to benchmark the performance of a particular security or portfolio.
Standard deviation is a statistical measure of the degree to which an investment’s returns vary from its average return over a specific period of time. It is used to evaluate the volatility and risk associated with an investment.
Standard deviation is important because it provides a measure of the risk associated with an investment, which is a key consideration for investors. An investment with a higher standard deviation is generally considered to be riskier than an investment with a lower standard deviation.
Investors can use standard deviation to evaluate the volatility of an investment relative to a benchmark or to compare the volatility of different investments. It can help investors to make more informed investment decisions by providing a measure of the risk associated with a particular investment.
A higher standard deviation indicates that an investment’s returns have been more volatile, meaning that the investment has experienced a wider range of returns over the specific period of time.
A lower standard deviation indicates that an investment’s returns have been less volatile, meaning that the investment has experienced a narrower range of returns over the specific period of time.
Standard deviations in FastTrack and FT Cloud are the monthly standard deviations of the daily percentage returns.
FastTrack calculates returns on a daily basis, then adjusted to a monthly basis by multiplying by the square root of 21.25 (the average number of market days in a month.)
SD = SqRt of Daily Price Change Variance * SqRt of 21.25
FastTrack’s algorithm is generally described as “The Difference between the Mean and the Square.” This is the most commonly used method used in evaluating investment data. FastTrack’s Standard Deviation extremely accurate since it is calculated on a daily basis. This means that if the period you evaluate contains 252 market days (approx. 12 months), then there are 251 daily change values included in FT’s SD calculation.
Some other data vendors may use monthly samples to compute an Annual SD. Using this method, a 12 month SD would only uses 12 percent changes (one per month) in the calculation.
While this SD will be fairly close to FastTrack, but FastTrack uses more samples and is more accurate.
FastTrack’s monthly Standard Deviation can be converted to Morningstar’s annual SD by multiplying FastTrack’s monthly Standard Deviation value by the 3.4 (the square root of 12). Starting v220.127.116.11 (released 11/2021), FT Cloud now includes this converted values under the “SD Ann” spreadsheet column.
Treynor ratio is a measure of risk-adjusted return that takes into account the level of systematic risk, or beta, associated with an investment or portfolio. It is used to evaluate the performance of an investment by comparing the returns earned above the risk-free rate, such as Treasury bills, with the level of systematic risk taken on by the investment.
The Treynor ratio is named after Jack Treynor
A higher Treynor Ratio indicates a better risk/ reward profile.
A higher Treynor ratio indicates that an investment or portfolio has generated higher returns for the level of systematic risk taken on, while a lower Treynor ratio indicates that an investment or portfolio has generated lower returns relative to the amount of systematic risk taken on.
The Treynor ratio is important because it allows investors to evaluate the performance of an investment or portfolio relative to the level of systematic risk taken on. It can help investors to determine whether an investment or portfolio has generated sufficient returns to compensate for the level of systematic risk taken on.
Treynor Ratio = (Annualized Total Return - Risk Free Return) / (Beta)
- Risk Free Return is calculated using the Basis ticker in FT Cloud
- Beta is calculated against the Benchmark ticker in FT Coud
Ulcer Index is a measure of the depth and duration of drawdowns in an investment or portfolio. It is used to evaluate the risk of an investment or portfolio by quantifying the level of pain or discomfort that an investor might feel during periods of poor performance.
The Ulcer Index provides a measure of the level of discomfort associated with an investment or portfolio during periods of poor performance. A higher Ulcer Index indicates that an investment or portfolio has experienced deeper and longer drawdowns, while a lower Ulcer Index indicates that an investment or portfolio has experienced less severe drawdowns.
Investors can use the Ulcer Index to evaluate the risk associated with an investment or portfolio and to compare the risk of different investments or portfolios. It can help investors to make more informed investment decisions by providing a measure of the potential pain or discomfort that they may experience during periods of poor performance.
The Ulcer Index is calculated by determining the square root of the average of the sum of the squared percentage drawdowns over a specific period of time. The formula for the Ulcer Index is as follows:
- Every day, determine the % amount ‘R’ that a mutual fund is below it’s highest previous value.
- Calculate a running total of R-squared.
- Then divide this product by N, the total number of days in the period and take the square root of the quotient to obtain UI.
- The lower the Ulcer Index the easier an investment will be to live with and the less troubling it will be on the down days.
Ulcer Index = SquareRoot((the sum of all R² values) / N)
Ulcer Performance Index
Ulcer Performance Index (UPI) is a risk-adjusted measure of investment performance that takes into account both the return and the risk associated with an investment or portfolio. It is used to evaluate the performance of an investment or portfolio by quantifying the level of pain or discomfort that an investor might feel during periods of poor performance.
The UPI provides a measure of the risk-adjusted return of an investment or portfolio, taking into account both the return and the risk associated with the investment or portfolio. A higher UPI indicates that an investment or portfolio has generated a higher risk-adjusted return, while a lower UPI indicates that an investment or portfolio has generated a lower risk-adjusted return.
The Ulcer Performance Index is a very good measure of the risk-adjusted return of an investment. It measures how well an investment outperforms a low-risk basis compared with the number of ulcers it gives you. The higher the value, the better the investment.
Investors can use the UPI to evaluate the performance of an investment or portfolio relative to the level of risk taken on. It can help investors to determine whether an investment or portfolio has generated sufficient returns to compensate for the level of risk taken on.
The UPI is calculated by dividing the annualized return of an investment or portfolio by its Ulcer Index, which is a measure of the depth and duration of drawdowns in the investment or portfolio. The formula for the UPI is as follows:
Ulcer Performance Index = (Annualized Return(Issue) - Annualized Return(Low Risk Base)) / Ulcer Index
Yield is a measure of the income generated by the underlying securities held in the fund over the prior 12 months, expressed as a percentage of the fund’s net asset value (NAV).
The yield is based on the income dividends or interest payments received by the fund’s holdings, not long and short term capital gains.
Yield is the ratio of income distributions a security paid in the past 12 months divided by the current share price of the security.
Investors use yield to evaluate the income generated by an ETF or mutual fund and to compare the income generated by different funds. Yield can help investors to determine whether a fund is generating sufficient income to meet their financial goals.
The distribution yield can fluctuate over time, depending on changes in the fund’s income and the fund’s NAV.
The distribution yield of a fund can vary depending on the fund’s investment strategy. For example, a high-yield bond fund like VWEHX may have a higher distribution yield than a growth stock fund such as FAGCX.
Yield = Sum of all Income Dividends for past 12 months / Current Security Price
Total Yield is the ratio of all the distributions (income, short term, and long term) a security paid in the past 12 months divided by the current share price of the security.
Large and frequent short and long term distributions can increase the tax consequenses of an investment in a taxable account.
Yield and Total Yield are different. The Yield will show you the percentage of the share price an investor received as an income dividends. Total Yield includes yield from all distributions: income PLUS long and short capital gains.
Total Yield = Sum of ALL Dividends for past 12 months / Current Security Price