This page details all math equations in FastTrack products. FastTrack products apply all popular investment calculations to FastTrack’s high-quality dividend-adjusted database.

## Alpha

Alpha is used to measure a mutual fund, ETF, stock or portfolio’s ability to ‘beat’ the market.

Excess returns of an investment, relative to the returns of a benchmark will be the investment’s alpha. This 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 component is a security’s alpha.

Alpha = Annualized security return - risk free rate - beta * (annualized benchmark return - risk free rate) * 100

!! risk free rate = basis ticker in FT Cloud

!!! beta is calculated against benchmark in FT Cloud

## Annual Return

The calculation is performed in the following equation:

( (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

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.

Beta = (Correlation of Issue and Basis * Issue’s standard deviation) / Basis standard deviation

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).

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.

## Correlation

FT Cloud’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.

**Advanced Details: **

- 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

*Periodic Calculation *

- 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

*Final Calculation *

- 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)

## Dividend Adjustment

Visit the following link to see all details on FastTrack dividend adjustments, including a dividend adjustment worksheet.

http://www.fasttrack.net/newhelp/DistributionWorkSheet.asp

## FTAlpha

We do not disclose the mathematics for this indicator. FTAlpha is the sole proprietary formula within FastTrack.

FTAlpha combines correlation, risk, and return. The computations for the spreadsheet FTAlpha column compare an issue to the Low Risk Basis for the correlation portion of the computation.

## Moving Average

Moving averages are used to smooth out data by constantly updating the average price. This smooths out temporary, and possibly random, short term fluctuations in prices and makes for better trend and direction decisions.

FastTrack, FT Cloud, and our APIs use exponential moving averages by default. Exponential moving average give more weight to recent prices and are more responsive to new data.

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.

## Max Drawdown

Maximum drawdown is the percentage loss from a ticker’s peak value to its lowest value, over a specified time period.

In general, Max Drawdown will be the highest for issues whose standard deviation is the highest.

## Return

The gain or loss of a security/equity curve for a particular time period.

( Price on last day- price on the first day) / price on the first day

## Sharpe Ratio

The Sharpe Ratio measures the risk-adjusted return of an asset by calculating the average return earned in excess of the risk-free rate per unit of volatility.

Sharpe Ratio = (Issue's Annualized Return - Low Risk Basis Annualized Return) / (Issue's Monthly Standard Deviation * SQR(12))

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 on the Low Risk Basis of YOUR choice. Using IRX-X is a mediocre choice when trying to select the best fund/ETF.

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.

## Standard Deviation

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 v4.0.2.11 (released 11/2021), FT Cloud now includes this converted values under the “SD Ann” spreadsheet column.*

## Treynor Ratio

The Treynor Ratio measures how much an investment returned given the amount of risk the investment assumed. A higher Treynor Ratio indicates a better risk/ reward profile.

The Treynor ratio is named after Jack Treynor

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

To calculate UI:

- 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

The Ulcer Performance Index is obtained as follows:

- Subtract the Annual Return of a Low Risk Base Mutual Fund from the Annual Return of an issue.
- Divide the result by Ulcer Index to get the UPI.

Ulcer Performance Index = (Annualized Return(Issue) - Annualized Return(Low Risk Base)) / Ulcer Index

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.

## Yield

Yield = Sum of all Income Dividends for past 12 months / Current Security Price

Yield is the ratio of income distributions a security paid in the past 12 months divided by the current share price of the security.

## Total Yield

Total Yield = Sum of ALL Dividends for past 12 months / Current Security Price

Total Yield is the ratio of all the distributions a security paid in the past 12 months divided by the current share price of the security.

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.