Investing, for everything

How we Calculate Returns

The contents and data made available on performance charts, any stated index values, and other information is intended for illustrative and informational purposes only and is not intended to represent actual results that could be considered a recommendation of an investment or investment strategy a user could rely on to make an investment decision. The performance charts represent hypothetical results that are based on information over a defined period of time. The charts themselves attempt to follow a standardized and consistent methodology for performance reporting, which we describe below. While we believe the performance data is gathered from reliable sources, the information that generates charts and performance results, uses historical data that has not been audited and validated, and may contain errors in pricing or other conditions. Further, CITI EQUITY relies on third-party content providers for market data and information as the basis for the calculations it generates. However, CITI EQUITY cannot be held responsible for the accuracy and timeliness of the content provided. You are responsible to validate the information used to aid you in your research of investments and the decisions you make.

You understand that any investment decisions you make are based on your own individual needs and tolerance for risk and that the content that you gather from any performance charts are just one of many factors you should consider before making your investment decision. The presentation of index performance for a CITI EQUITY should not be regarded as an offer or a solicitation for the purchase of any CITI EQUITY, and you are responsible for all investment decisions you make. Past performance is not an indication of future results and the index performance for any CITI EQUITY is subject to fluctuation depending on shifting market conditions.

Calculating CITI EQUITY Performance

CITI EQUITYs are collections of stocks but are represented for purposes of performance as models, and as such, no actual money is invested in the underlying components of these models, no commissions or fees or other charges are reflected in CITI EQUITY’s performance calculations and no actual trades have been executed in the marketplace that could be expected to represent actual results. In calculating CITI EQUITY performance, we first assign each CITI EQUITY an index value of 1,000 at its origin date.

We then calculate the split-adjusted total returns (inclusive of cash and stock dividends) of each individual component (symbol) in the CITI EQUITY from the CITI EQUITY origin date. For example, if IBM were part of a particular CITI EQUITY, had a share price of $100 at the origin date, a current price of $110, had no stock splits, and had paid dividends of $10 during the intervening time, we would calculate its return as 20% (($110+$10)/$100-1). We repeat this calculation for each stock in the CITI EQUITY.

[The number of data points used in each return calculation depends on the length of the time period under consideration. For example, when calculating returns for a 5-year performance chart, we may use end-of-month trading data, or 60 data points for a 5-year period, while we may use end of day data for a 1-year performance chart or approximately 220 data points (trading days) for a 1-year period.]

Next, we multiply the return of each component in the CITI EQUITY by its respective weight within the CITI EQUITY. For example, if IBM represented 10% of the CITI EQUITY, we would calculate its weight-adjusted return as 2% (20% x 10%). We then sum the weight-adjusted returns of all of the components in the CITI EQUITY to generate the composite return for the CITI EQUITY.

The composite return for the components of the CITI EQUITY is then applied to the base index value of 1,000 to generate the current index value. Composite returns for each CITI EQUITY are calculated at the end of each trading day. During the trading day, an intraday change or delta factor using 20-minute delayed quotes is applied to the previous closing index value in order to calculate the current index’s value.

The weights of each component in a CITI EQUITY are established at the CITI EQUITY’s origin date, but those weights fluctuate as the price of each component’s stock rises and falls. On a periodic basis following the establishment of a CITI EQUITY (usually quarterly, following the origin date), we review the CITI EQUITY and may elect to update it to reflect changes in the companies, industries and trends relevant to the CITI EQUITY. In particular, we may remove or add companies based on any changes in their relevance to the idea on which the CITI EQUITY is based, as a result of our review process.

For monthly, quarterly, yearly, and year-to-date returns, the performance is shown as the difference between the ending index value and beginning index value of that period over the beginning index value.

Information about CITI EQUITY Returns

CITI EQUITY returns are based on stock prices from the end of the normal trading day and include any distribution and cash dividends. The stocks that make up a CITI EQUITY may change from time to time - certain stocks may be removed and replaced with new stocks added to a CITI EQUITY, or the weighting of the stock represented within the CITI EQUITY may change. These returns track those changes. However, commissions and fees that are incurred for actual transactions are not deducted from these hypothetical returns.

Actual customer returns will likely vary from the CITI EQUITY returns shown due conditions such as corporate actions, transaction fees and customization. The calculation of split-adjusted total returns involves the adjustment of historical prices for corporate actions such as cash dividends, splits, spin-offs etc. Since we do not adjust the original cost basis for cash dividends, the returns on owned CITI EQUITYs do not include cash dividends and will therefore underperform the stated CITI EQUITY returns. While the original cost basis is not adjusted for cash dividends, the end of day security prices are adjusted for cash dividends and therefore the chart of portfolio performance excluding the date of purchase represent split-adjusted total returns. In addition, actual transactions would include the expense of commissions and customers may change the stocks in their CITI EQUITY on their own. Customers may choose not to change their CITI EQUITY when the stocks in a CITI EQUITY are changed in the CITI EQUITY’s periodic rebalancing. There could be other corporate action events that may not be replicated in a CITI EQUITY until the rebalancing period.

Calculating Performance Returns

We calculate the returns of portfolios which are owned by customers using a time-weighted return methodology. Due to the fact that the amount of cash currently invested in a given portfolio can vary over time and cash gains can be realized with intraday prices, the return cannot be calculated as simply the amount of money made (or lost) divided by the amount of money currently in the portfolio.

To handle cash flowing into or out of a portfolio we break up the timeline of ownership of the portfolio into distinct segments each of which has a constant amount of money invested in the portfolio. To calculate the returns of a portfolio on a given day, we must start the calculation at the beginning of the history of the portfolio. We value the portfolio on every market day until reaching the end of the segment. As stated above a segment ends when a trade is placed which invests more in the portfolio or sells part of the portfolio.

Formulas

The returns from the start of the segment to a given day in a segment is calculated using the simple formula:

r = (Ending Market Value - Segment Starting Value - Cashflow Into Portfolio) / (Segment Starting Value)

where

Once we get to the next segment the computation within that segment is equally trivial, however computing the total returns up to a day beyond the first segment requires us to geometrically link the returns of the two segments. The geometric linking is accomplished by applying the following formula:

R = (1 + r_{last segment} ) * (1 + r_{this segment to desired date} ) - 1

where

By applying these two formulas we can draw a line which computes the return of the portfolio on any given day.

Example

Let’s walk through a simple example. Let’s assume there is a portfolio that owns a single stock with the symbol STOCK. The stock has the following price values over a given week.

Day of Week	Closing Price of ’STOCK’
Monday	$100
Tuesday	$125
Wednesday	$95
Thursday	$110
Friday	$115

The customer bought 10 shares the stock on Monday at a price of $100. S/he sold 5 shares of the stock on Wednesday slightly above the closing price at $101. We would now like to compute the returns of the portfolio on each day of this week. First we must break up the timeline of the portfolio into 3 segments. One segment for Monday to the close on Tuesday. One segment for the day we sold the stock on Wednesday. Finally, one segment that covers Thursday and Friday.

Day of Week	Customer Action	Formula	Daily Segment Return	Linking	Total Time Weighted Return Up To Day
Monday	Buy 10 @ $100	($100 * 10 - $100 * 10)/($100 * 10)	0%	N/A	0%
Tuesday	N/A	($125 * 10 - $100 * 10)/($100 * 10)	25%	N/A	25%
Wednesday	Sell 5 @ $101	($95 * 5 - $125 * 10✝ - $101 * 10)/($125 * 10✝)	-21.6%	(1 + 0.25) * (1 - 0.216) - 1	-2%
Thursday	N/A	($110 * 5 - $95 * 5)/($95 * 5)	15.8%	(1 - 0.02) * (1 + 0.158) - 1	13.5%
Friday	N/A	($115 * 5 - $95 * 5)/($95 * 5)	21.1%	(1 - 0.02) * (1 + 0.211) - 1	18.7%

^{✝ Note: We choose $125*10 on Wednesday as the segment starting values because we model all sells as happening at the end of day. Buys are effective for the start of the day, hence the $0 cash flow on Monday.}

Let’s look at this result and make sure it makes sense. As we can see from a trivial price returns calculation of the underlying we only expect to get 15% returns.

($115 - $100) / ($100) = 15%

However, this customer realized additional return by selling the stock intraday, above the closing price on Wednesday. By locking in realized gains at the higher price, the return for the overall portfolio is higher than the simple return of the underlying over the same period.

Simulated Performance Calculations vs. Backtesting

CITI EQUITY Investing endeavors to offer customers CITI EQUITYs corresponding to new and timely investing ideas. At the same time, investors may wish to see how those CITI EQUITYs could have potentially performed if they had owned them in the past, sometimes for time periods that pre-date the CITI EQUITY’s origin date. There are two methods by which we calculate those simulated returns, depending on the type of CITI EQUITY being examined.

Simulated performance refers to the total return of a CITI EQUITY hypothetically purchased at some point in the past through the current time based on 20-minute delayed quotes. This method assumes no updates or other changes to the components of the CITI EQUITY during the time period under examination. We simply calculate the returns that would have been generated if someone had held this same set of equities in the same share-weighting as established at the origin date for a period of time that extends to a point in time prior to the origin date.

Backtesting refers to the total return of a CITI EQUITY hypothetically purchased at some point in the past through the current time based on 20-minute delayed quotes that does assume and calculate the impact of periodic updates throughout the hypothetical time period. This method is primarily, though not necessarily exclusively, used to generate the performance returns of Model Based CITI EQUITYs. These CITI EQUITYs are constructed using mathematical rules based on financial metrics derived from the operating and/or trading results of the component stocks of the CITI EQUITY. For example, we might screen the universe of equities that are traded publicly in the United States for those that have had the highest amount of revenue over the past twelve months. We might then set a rule for a CITI EQUITY that the CITI EQUITY would select the 20 companies that topped that screen at the end of each financial quarter. In each subsequent quarter, we would update that CITI EQUITY to select whichever 20 companies over the previous twelve months had generated the highest amount of revenue. The list of companies in the CITI EQUITY might be completely different from one quarter to the next, because inclusion in the list was caused by the application of a numerical rule. Backtesting is a method that could then be used to calculate that CITI EQUITY’s hypothetical historical performance by applying the numerical rule retroactively on a quarterly basis over the time period being examined.

However, backtested results include inherent shortcomings that you should be aware of as you continue your research into an investment. These results do not represent actual returns that occurred in the marketplace, do not consider additional purchases and sales of stocks that can occur as an investor makes adjustments to their accounts, do not account for certain fees or corporate actions that can affect a stock, and cannot replicate the share prices you could have experienced had you personally entered buy and sell orders during market hours. The results of the performance calculations should not be considered an actual representation of results that a customer should expect to replicate, but is made available for informational purposes only.