Will Rogers once quipped: “It is not the return on my investment that I am concerned about; it’s the return of my investment.”

In all seriousness though, calculating a rate of return; also known as “return on investment”, or ROI, is useful because it allows investors to compare various investments, regardless of the amount invested in each.  The formula is simple:

ROI = Gain / Cost

For example, an investor deposits $100,000 into an account, and it grows to $110,000:

• Cost = $100,000
• Gain = $10,000

Applying the formula:

• ROI = $10,000 / $100,000, or 10%

That sure seems basic.  But unfortunately, nothing in finance is simple. It turns out that there are different ways to calculate rates of return, depending upon what is being illustrated.  It is vital to understand what type of return you are seeing.  Is the type used appropriate to your situation?  Let’s evaluate various methods to calculate rate of return:

Cumulative return versus Annualized return

Our formula above was missing a key component: A period of time. The cumulative return is 10%, but without a time period, the 10% return is not a rate of return, any more than traveling ten miles is a rate of speed.  To establish a rate of return, a period of time must be specified. The conventional time- period used is one year.  A return that takes place over a one- year time period is known as an annual rate of return, and when a rate of return that takes place over a time frame other than one year is adjusted to a one year time frame, it is known as an annualized rate of return.

Annualized rates of return allow us to determine the rate at which an investment is growing, regardless of the amount invested.  This is a useful tool when comparing investment options.

A rate of return that takes place over a very short time frame may be inappropriate to annualize.  For example, an investment that grows by 10% over a one- month time frame is growing at an annualized rate of 120%!  Although the annualized rate is truly 120%, it is ridiculous to extrapolate the one month return as though it has bearing on a full year’s return.  It would be more descriptive to simply quote a cumulative 10% return over the one month’s time.

Simple Average Return versus Compounded Average Return

Simple average return is calculated by adding up annual returns and dividing by the number of years.  For example, over a three- year period, an investor earns:

Year 1	-40%
Year 2	5%
Year 3	35%

…and thereby obtains an average simple return of 0%.     (-40+5+35)/3 = 0

Compounded average return represents the cumulative effect of a series of gains and losses.  For example, if an investment claims to have produced a 10% compound return over the past three years, this means that at the end of its third year, the funds have grown to a size equal to what it would be if the funds on hand at the beginning of each year had earned exactly 10% by the end of each year.  Using our series of returns over the three- year period as listed above, the effects upon a $100,000 investment are as follows:

Year	Beginning Value	Percentage gain (loss)	Year end value
1	$100,000	-40%	$60,000
2	$60,000	5%	$63,000
3	$63,000	35%	$85,050

Without getting into the math, the resulting compounded average return is -5.25%, a more useful illustration of the $14,950 actual loss.  The average simple return of 0% is not descriptive.  Note that had the investment earned 0% in each of the three years, the average simple return would still have been 0%, but in that case there would have been no loss.

A more extreme example of the uselessness of average simple returns is the scenario in which an investment loses 50% of its value in year one.  The return required to erase the loss in year two is 100%.  The investment is worth no more than it was at the beginning, yet the average simple return is 25%!

Time Weighted versus Dollar Weighted Return 

The time weighted return is a compounded rate that excludes the effects of cash flows. Time-weighted returns are useful when comparing the performance of investment funds (such as mutual funds) over a set time period.  A typical time-weighted return example calculates the ending value after five years resulting from a $10,000 initial investment, with no additional investments or withdrawals.

Although a time-weighted return can be useful when comparing the returns of one fund to another, they shouldn’t be relied upon when calculating an investor’s personal return, especially if additional deposits or withdrawals have taken place.  For that, a dollar weighted return, also called an Internal Rate of Return (IRR) is appropriate.

Consider a scenario in which an investor deposited $10,000 into a fund exactly five years ago, and made an additional deposit of $50,000 at the beginning of year four.  Assume that the annual rates of return were:

Year 1	-10%
Year 2	2%
Year 3	-25%
Year 4	16%
Year 5	19%

The time- weighted return is a loss: -1%.  However, because of the large deposit in year four, the dollar weighted return (IRR) is a positive 18%.  This large difference results because there was relatively little money invested in the lackluster first three years, and the large $50,000 deposit was invested only during the last two high- return years.  Note that some reports will list time weighted returns, and these often differ from your personal IRR to a large extent.  Time weighted returns are an inappropriate method of communicating rates of return when additional deposits or withdrawals have taken place.

Published Return versus Investor Return

An investment fund will publish the time weighted return.  It is up to individual investors (or their advisors) to calculate the personal dollar weighted return.  However, Morningstar calculates an “Investor return”, which is really a dollar-weighted return, using the aggregate deposits and redemptions of all shareholders of the fund.

The investor return doesn’t have much use for investors as an indicator of their individual performance, because the beginning and ending date of the calculation is done on a monthly, quarterly, and annual basis, which will rarely match the actual dates that an individual investor was invested.  Secondly, the timing and amounts of aggregate deposits and withdrawals will be different than those of an individual investor.

Note that while the dollar weighted returns of individual investors will vary widely from the published return due to deposits and withdrawals, the investor return will not vary significantly unless investors are moving in or out of a fund in aggregate.  Sometimes, aggregate deposits or redemptions are simply a result of systematic cash flows, but often they are a reflection of investors engaged in market-timing.  Typically, investors deposit more into mutual funds after a run up in prices, and they redeem shares after a market drop.  This lowers the investor return.

The investor return is therefore useful in that it can be an indication of the extent to which shareholders in a given fund are market-timing. Market timing can be harmful to a mutual fund’s return.  All deposits to funds are made to cash, and all redemptions come from cash.  A manager of a fund awash with new cash who does not make purchases quickly suffers “cash drag”, which refers to the relative loss due to inferior long-run returns of cash.  A fund that experiences large redemptions after a market drop is forced to sell shares to meet those redemptions at a bad time.  This “hot money” makes it more difficult for fund managers and increases costly turnover. Market timing harms not only the individual engaged in it, but the fund itself.

Consider this comparison between two actual stock funds as of the first quarter of 2018, labeled here as fund A and fund B:

Fund A	1 year	3 year	5 year	10 year
Published return	15.54%	9.68%	11.93%	9.2%
Investor return	15.19%	9.99%	11.56%	10.97%
Fund B
Published return	7.77%	7.31%	8.59%	7.54%
Investor return	6.96%	6.61%	7.68%	6.44%

Both funds invest in the same universe of large cap US stocks.  Fund A has had superior performance, some of which may be attributable to the buy-and-hold behavior of its shareholders as evidenced by the Investor returns which are not generally lower than the published return.  Contrast that with fund B, where the investor returns are significantly lower than the published returns.  This indicates that in aggregate, shareholders are redeeming shares when they are low, and buying them when they are high.  This may be one factor in the lower published returns.

One qualifier here:  investor returns can vary from published returns even without “hot” money or market timing by fund investors.  For example, a fund might experience a very steady rate of deposits in a period of rising share prices, which will result in inferior investor returns.

Summary

When a rate of return is listed or quoted, inquire as to whether it is:

• Annualized or Cumulative
• Simple or Compounded
• Time Weighted or Dollar Weighted
• Published or Investor

Further, is the chosen method of calculating and communicating the rate of return appropriate to your situation?

Now that you understand the differences, you can avoid being confused by inappropriate methods of calculating rates of return.

For Clients of Financial Plan Only:

Financial Plan clients are exposed to various reports that illustrate rates of return.  The type of return varies depending upon the purpose:

TD Ameritrade View: The account overview prominently displays daily returns.  After selecting a security, returns for various time periods are shown.  These are all cumulative, compounded, time-weighted returns.  The “standardized” returns are adjusted for sales charges (commissions).  As Financial Plan does not charge commissions, the standardized returns will be identical to the total returns.  However, note that commission-loaded, legacy positions which were purchased at another firm will show a reduced standardized return due to the commission that was charged at the other firm.  None of the returns in the TD Ameritrade View are your personal returns.

Morningstar Portal Home Page: The home page of the Morningstar portal lists a “gross return” for the entire portfolio.  The time frame can be selected to show one year, three years, five years, ten years, or since inception. These returns are annualized, compounded, and dollar weighted, also known as Internal Rate of Return (IRR).  This is your personal return for your entire portfolio.

Morningstar Portal Account Page: When an account is selected, the gross return can be viewed for that account.  This return is also annualized, compounded, and dollar weighted (IRR). This is your personal return for that account.

Morningstar Portal Securities Page: When a mutual fund security is selected, rates of return for each year over the past ten years are listed.  Returns over 1 day, 1 week, 1 month, 3 month, YTD, 1 year, 3-year, 5-year, 10- year, 15-year, and since inception are also listed.  The returns for time frames shorter than one year are cumulative; while those longer than one year are annualized.  All of these returns are time-weighted, compounded, published returns; and not intended to reflect your personal return.

When the security selected is a stock or ETF, no return is illustrated.  A real time quote with the change in share price for the day is listed.  A bond will not have a securities page available.

Financial Plan Dashboard Reports:  Under Investments, select an account and an investment ticker symbol within that account, and then select performance, and then investor returns.  Here the investor return can be seen compared to the total (published) return, over 1, 3, 5, 10, and 15 years.   The returns are annualized and compounded.  The total return is time weighted, while the investor return is dollar weighted using aggregate shareholder cash flows. Neither are reflective of your personal performance.

Select reports/ report selection, fact summary, and select growth rates summary to view the assumed rate of return for each account used in your financial plan. We assume an annualized, compounded, time-weighted return on equity securities of 9%; close to the historical average of the CRSP database. On debt securities we assume a rate of 3%; also a close approximation of the historical average. The assumed rate of return on an account is derived by blending the rate of return on equity and debt.  For example, an account with 70% equity exposure would have an assumed rate of return of (.7*.09) + (.3*.03), or 7.2%.  This return is gross of costs.  In the financial plan assumptions, we subtract costs to arrive at a net return.  These returns are all hypothetical assumptions, used to make meaningful projections.  They are not intended to be reflective of your personal performance, nor are they promissory.

Advisor-Generated Performance Reports:  Performance reports generated by Financial Plan advisors may utilize any rate of return method, but typically these reports illustrate your personal performance, so they will use IRR:  compounded, annualized, dollar-weighted returns.  When the report is being used to compare one security to another or to a benchmark, then compounded, annualized, time-weighted returns will be used.

If you lack clarity when viewing any rate of return, don’t hesitate to ask your Financial Plan advisor for direction.