# Rate of Return: Everything you need to know about calculating investment returns

Investors often start out thinking that calculating rate of return is quite simple, but as they begin to calculate returns in more and more scenarios, they begin to see some cracks in how simple they thought it would be. In reality, rate of return should be simple, but compounding, cash flows and more can complicate the accuracy of the measurement. In this guide on rate of return, we will walk through all the ways you can calculate returns on your investments and show how and why some investors get tripped up in various situations.

## A quick rate of return example to get us started

If an investment’s value is the same in the beginning as it is at the end of two periods, the total investment return should be 0%, right? No change means 0% return. But consider this scenario:

- The beginning investment value equals $100 and it falls to $50 by the end of this first period. That’s a 50% loss.
- By the end of the second period, the investment value recovers its loss and equals $100. That’s a 100% gain.
- To find the return for the two periods: take the average return, add the two returns (-50% and +100%) then divide the result by 2, the number of returns. The final result equals +25%. Right?

But the return should be 0% if the beginning equals the end, shouldn’t it? How can the average return for the two periods equal +25% if the investment value is the same at the end of the second period as it was at the beginning of the first?

This example already shows some of the pitfalls with over-simplified ways of calculating rate of return.

Calculating investment returns should be simple. But compounding and the intended purpose of the investment return can complicate the calculation a little. Dividends, interest, and additional principal transactions can also complicate calculations. It’ll all get straightened out here with easy-to-follow examples and succinct explanations that make it easy to decide the correct rate of return calculation.

## A few tips on using this guide

- You can use this as a reference; you don’t have to read all the way through.
- By its nature, investment returns are a quantitative thing. If you use the examples, tables, or spreadsheets to learn the methods, a calculator or spreadsheet will help.
- The method illustrations described here intentionally omit transaction fees, periodic fees, taxes, and other expenses that bear on investment returns. If they had been included, they would have unnecessarily complicated the illustration. The section titled “Fees, Inflation, and Tax” explains how to incorporate these into the calculations.

## Table of Contents

- Single-period arithmetic return
- Multiple-period returns
- Natural logarithmic returns
- Arithmetic vs. geometric vs log returns
- Other rate of return topics
- Fees, inflation and tax

## Rate of Return: Single-Period Arithmetic Return

A single-period return is a return for one interval like a year or month. This is useful for making comparisons of one investment or portfolio across different single periods or multiple investments for the same single period. The single-period arithmetic return is identical to what is also known as the “holding period” return..

### Single period returns with no dividends, interest or principal cash flows

Here’s the equation for a single-period return with no dividend, interest, or principal cash flows:

E is the ending value, and B is the beginning value. Here’s an example:

What return did Amazon (AMZN) earn in 2019? and in 2018?

**2019 Amazon (AMZN) single-period rate of return**

- Beginning price per share: $1,501.97
- Ending price per share: $1,846.89
- 2019 Return: ($1846.89 – $1501.97) / $1501.97 = 22.96%

**2018 Amazon (AMZN) single-period rate of return**

- Beginning price per share: $1,169.47
- Ending price per share: $1,501.97
- 2019 Return: ($1501.97 – $1169.47) / $1169.47 = 28.43%

We can say AMZN returned 22.96% in 2019 and 28.43% in 2018. We can also say AMZN’s 2019 return declined by 5.47 percentage points from its 2018 return. This decline is a change in returns. If we compared two investments’ returns, say AMZN and MSFT, we would report differences in returns the same way.

We wouldn’t say the AMZN decline in the return equaled 5.47% because the decline went from 28.43% to 22.96%. Rather, the change would equal a 19.23% decline. That is: (.2296 – .2843) / .2813 = 19.23%.

### Single period returns with dividends or interest, no principal cash flows

If positive cash flows occur in a single period – dividends or interest – we add them in the numerator. Here we illustrate dividend cash flows, which are handled the same as interest cash flows:

Again, “E” is the ending price, and “B” is the beginning price. “D” is dividends or interest. Stock issuers pay owners cash dividends on a per-share basis. Issuers of debt instruments like governments and corporations pay bondholders cash to compensate them for the risk they assume lending issuers money.

AMZN didn’t pay dividends, but Microsoft (MSFT) did. Here’s an example with dividend payments for a single one-year period:

**2019 Microsoft (MSFT) single-period rate of return**

- Beginning price per share: $101.57
- Dividends per share during year: $1.89
- Ending price per share: $157.70
- 2019 Return: ($157.70 + $1.89 – $101.57) / $101.57 = 57.12%

“Periods” don’t have to be one year. We could ask the same question – the MSFT return – for the two-calendar-year period starting in 2018:

**2018-2019 Microsoft (MSFT) single-period rate of return**

- Beginning price per share: $85.50
- Dividends per share, 2018: $1.72
- Dividends per share, 2019: $1.89
- Ending price per share: $157.70
- 2019 Return: ($157.70 + $1.72 + $1.89 – $85.50) / $85.50 = 88.67%

It’s tempting to take the two-year return of 88.67% and divide it by two to express the average annual return for the two years. Under certain circumstances, that is the right way to calculate the average annual return. But it’s often wrong. Hold off on that until we annualize multiple-year returns a little later.

These same concepts apply to corporate bonds as well.

Corporate bond prices are usually reported as a percent of par where par is the amount of principal the issuer will repay upon maturity. For example, a bond with par equal to $1,000 might be priced at 95. That price quote is equivalent to $950. Let’s look at some quick examples.

**2019 Newell Brands 4.00% 12/1/2024 (Par = $1,000) single-period rate of return**

- Beginning price per bond: 98.516
- Interest per bond: $40
- Ending price per bond: 102.640
- 2019 Return: ($1026.40 + $40 – $985.16) / $985.16 = 8.25%

**2019 Express Scripts Holding Co. 3.50% 6/15/2024 (Par = $1,000) single-period rate of return**

- Beginning price per bond: 97.112
- Interest per bond: $35
- Ending price per bond: 103.922
- 2019 Return: ($1039.22 + $35 – $971.12) / $971.12 = 10.62%

We can say the Express Scripts bond return exceeded the Newell Brands bond return by 237 basis points.

Next, we illustrate dividends and the purchase of additional shares of stock.

### Single period returns with dividends or interest and negative principal cash flows (purchase)

The purchase of shares creates a negative cash flow because cash flows out of the account to make the purchase. Let’s jump right into an example to demonstrate.

Suppose you owned 100 shares of MSFT at the beginning of 2019 and in July you bought 10 additional shares. MSFT paid per-share dividends of $0.46 in February, $0.46 in May, $0.46 in August, and $0.51 in November. What return did your MSFT position yield in 2019?

Because you bought the 10 additional shares in July, they benefitted only from the last two dividend payments of $0.46 and $0.51. Also, the stock price rose to $138.40 when you purchased the shares in July.

One way to handle this single-period rate of return is to separately calculate returns for the 100- and 10-share lots and assign weights to the results. So let’s do just that.

**100-Share lot**

- Beginning share price: $101.57
- Beginning total purchase amount: $10,157.00
- Dividends per share, over 4 dividend payments: $1.89
- Total dividend payments received: $189.00

**10-Share lot**

- Price per share paid in July: $138.40
- Total purchased in July: $1,384.00
- Dividends per share, over 2 dividend payments: $.97
- Total dividend payments received: $9.70

**Ending prices**

- Ending price per share: $157.70
- Ending total for 100 shares: $15,770.00
- Ending total for 10 shares: $1,577.00

**Calculating the return**

- Return on 100-share lot: ($15770 + $189 – $10157) / $10157 = 57.12%
- Return on 10-share lot: ($1384 + $9.70 – $1384) / $1384 = 14.65%

The 10 shares purchased in July produced a lower return than the 100 shares owned at the beginning of the year because the beginning price – the purchase price – for the 10 shares was greater than the price at the beginning of the year and because it earned less in dividends.

Next, we take the weighted average return of multiple lots using this equation:

“Lot” is the group of shares priced or purchased at the same time at the same price. “*w*” is the weight expressed as a percentage of total shares for a certain lot. In the example, the weight of the 100-share lot equals 100 ÷ 110 = 0.9091. The other ten shares make up the difference with 0.0909 so the total of the two lot percentages equals 100%.

“r” is the return for a certain lot. “*n*” is any number; the number of possible lots is unlimited and has no bearing on the weighted average return as long as all lots are treated the same by the equation. The absence of an operand between “*w*” and “R” means “*w*” and “R” are multiplied by each other. You could put a multiplication sign in there, but the equation is already cluttered. The ellipsis means any number of *w*/R combinations could be included.

Let’s use this equation to calculate the weighted average return in our example.

- Weighted average return = (.9091 x .5712) + (.0909 x .1465) = 53.26%

The share-weighted returns could also have been weighted by the number of periods each lot was held. This is true for examples appearing later too. The method multiplies the weights by number of periods. When reporting returns, be sure to disclose weighting factors (i.e., shares, periods).

### Single period returns with dividends or interest and positive principal cash flows (sale)

The last example was about the purchase of shares. That transaction creates a negative cash flow. This example is about the sale of shares. This transaction creates a positive cash flow because cash comes into the account. In this example we’ll start with 100 shares, and sell off 10 shares mid-year. We will separate the shares into lots in a similar fashion: 90 shares representing the shares held throughout the period, and 10 shares representing the shares we owned, but then sold mid-year.

Here are the inputs and calculation for a sale using the same formula:

**90-Share lot**

- Beginning share price: $101.57
- Beginning total purchase amount: $9,141.30
- Dividends per share, over 4 dividend payments: $1.89
- Total dividend payments received: $170.10

- Ending price per share: $157.70
- Total ending price: $14,193.00
- Weight for 90 of 100 shares: 0.9

**10-Share lot**

- Beginning share price: $101.57
- Beginning total purchase amount: $1,015.70
- Dividends per share, over 2 dividend payments: $.92
- Total dividend payments received: $9.20

- Sale price per share: $138.40
- Total sale: $1,384.00
- Weight for 10 of 100 shares: 0.1

**Calculating the return**

- Return on 90-share lot: ($14193 + $170.10 – $9141.30) / $9141.30 = 57.12%
- Return on 10-share lot: ($1384 + $9.20 – $1015.70) / $1015.70 = 37.17%
- Weighted average return = (.9 x .5712) + (.1 x .3717) = 55.13%

The 10-share lot return drags down the return because it is sold at a price lower than the year-end price (plus fewer dividends were collected on the 10-share lot).

## Rate of Return: Multiple-Period Returns

Single-period return calculations do not consider the timing of dividend, interest, or principal cash flows falling between the beginning and ending cash flows. Ignoring the timing of these cash flows is a useful, but simplifying assumption.

Yet money does have time value. Time value bears less on calculation outcomes for most short-term single periods and is usually more important for multiple-period rate of return calculations.

### Two concepts form the foundation of time value

- The future value of one dollar is always less than the present value of one dollar. Put differently, would you prefer receiving a payment of one dollar today or one dollar a year from now? You’d prefer payment today for at least three reasons:
- Inflation will probably have eroded the dollar’s purchasing power one year from now.
- It’s possible whoever promised to pay the dollar a year from now will fail to satisfy their promise.
- By accepting payment today, you can invest the dollar and earn interest you would not have earned if you had accepted payment one year from now.

- Investment values compound because of interest, dividends, and changes in the value of appreciable assets. For example, a one-year, $100 investment in an interest-bearing account paying 5% is worth $105 at the end of the year. The entire ending balance – both the $100 principal and the $5 interest – is the basis for interest in the second year. If the account pays 5% the second year, then the account earns $5.25 the second year and its ending value equals $110.25.

Because of these time value ideas, we usually treat multiple-period returns differently than single-period returns. We start with the less common case of multiple-period returns in which we are only concerned with a *representative* return. Subsequent examples address multiple-period returns and time value.

### Multiple-Period Arithmetic Returns

Sometimes you only want to know the *representative* return on an investment. **You aren’t concerned with compounding**, and you do not state what return an investment would have produced over multiple periods.

A similar objective to calculating the *representative *return with the same solution is that you want to know what one dollar, for example, would have been worth at the end of one period. At the end of the period, you sell the investment, then reinvest at the beginning of the next period, only to sell again at the end of that period. This process could repeat indefinitely.

When you calculate the average of these periodic returns to achieve these two objectives, the result is the mean multiple-period arithmetic return.

For example, you might be interested in the mean annual return of a stock index to get an idea of the return to expect in each period.

Here’s the equation for multiple-period arithmetic returns:

“R” is the period return; calculate R using Equation 1. The subscripts 1 and 2 returns reference the periods. The ellipsis between the plus signs means there could be any number of periods, and they’re all added together the same way R_{1 }and R_{2} were. “n” is the number of the last period in the sequence; it could be any number. Here’s an example:

What is the mean annual price return (excluding dividends) for the S&P 500 Index for the 10-year periods ending in 2019?

Using equation 4, we can determine the S&P 500 index annual price return over this period:

- S&P 500 Index Annual Price Return = (.1278 + 0.0 + .1341 + .2960 + .1139 + (-.0073) + .0954 + .1942 + (-.0624) + .2888) / 10 = .118

The arithmetic mean return for these 10 years equals 11.8%. The 11.8% value is *representative* of the periodic returns. But this representative value does not say anything about what an investment replicating exactly the Index would have been worth after ten years if the annual return equaled 11.8%.

We used this multiple-period arithmetic return method to calculate the mean annual return example at the beginning of this article. The example went like this: A $100 investment lost 50% by the end of the first period; it was worth $50. By the end of the second period, the investment recovered to its original $100 price, which is a 100% increase:

- Multiple-period mean arithmetic return = (-.5 + 1.0) / 2 = .25

Even though the beginning and ending investment values were the same, the average return equaled +25%. If we wanted merely a *representative* periodic return, +25% is correct. **However, it’s more common to perform multiple-period calculations when we want to know what an investment value would have been worth if it were invested for multiple periods. For these answers, we need methods that consider compounding**.

### Multiple-Period Geometric Returns

If you had an investment that exactly replicated the price-only S&P 500 Index for the ten years ending in 2019, what return would you have earned? The correct method uses geometric, not arithmetic returns.

The equation for the geometric mean return can be expressed like this:

Let’s briefly return to our initial example of losing 50% of a $100 investment in the first period, then the 100% increase in the second period which brought the ending value equal to the beginning value. As we showed above using the multiple-period mean arithmetic return, it gave us an average return of 25% which is a bit nonsensical since the investment started and ended at $100. What happens when we use the geometric mean return formula? Let’s plug in the numbers and find out:

- Multiple-period geometric mean return = [(1 + (-.5)) x (1 + 1)]^(1/2) – 1
- = (.5 x 2)^(1/2) – 1
- = 1^(1/2) – 1
- = 1 – 1
- = 0%

Using the geometric mean return formula, it tells us that the return was 0%. Since we started with $100 and ended with $100 after two periods, this would make more sense than the 25% return the arithmetic formula gave us.

To further demonstrate how this works and the differences you can encounter with taking the geometric mean as compared to the arithmetic mean, let’s circle back to the previous example of the S&P 500 returns over the ten year period of 2010-2019. As we showed before, the simple arithmetic average return was 11.8%. We mentioned it was representative of the return, but not accurate in communicating what an actual investment would be worth had it been invested in the S&P 500 over those ten years. Let’s use the geometric mean return.

- Let’s first calculate the product of returns+1 which is the main component of our formula. (.1278 + 1) x (0.0 + 1) x (.1341 + 1) x (.2960 + 1) x (.1139 + 1) x ((-.0073) + 1) x (.0954 + 1) x (.1942 + 1) x ((-.0624) + 1) x (.2888 + 1) = 1.1278 x 1 x 1.1341 x 1.2960 x 1.1139 x .9927 x 1.0954 x 1.1942 x .9376 x 1.2888 = 2.8974
- Now we can plug this in as follows (2.8974)^(1/10) – 1 = 1.1122 – 1 = .1122 = 11.22%

For additional clarity, here is a table showing the same data as the formula calculation we just outlined:

This calculation tells us that our annual return using the geometric mean would have been 11.22% (as compared to the 11.8% as determined from the arithmetic mean formula). The 11.22% is more accurate. It reflects the compounding that takes place. The compounding rate of return, or the geometric mean return, is always going to be lower than the arithmetic mean because it factors in the compounding that is taking place between each period. *Note that we aren’t factoring in dividends in this example.*

**Future Value and Present Value**

The proof of the utility of the geometric mean return lies in a useful planning tool. This is not just an academic proof. Here are equations for calculating future and present values:

“FV” means future value, “PV” means present value (today’s value), “r” means geometric return, and “n” means the number of periods.

Here’s the proof. In the case of the S&P 500 Index above, what is the future value of $1,115.10 at the end of 10 years if the annual geometric return equals 11.22%, the value we calculated in the table above? The $1,115.10 is the starting value, the value at the end of 2009.

- FV = 1,115.10(1 + 0.1122)^10 = 1,115.10(2.8973) = $3,230.78

This result equals the emboldened value in the 2019 row in the table above. If we had used the arithmetic mean return, 11.80%, we would have gotten a different answer, $3,403.39. That’s the wrong answer because you can’t use an arithmetic mean return when compounding matters.

By the way, we can calculate the present value if we know the future value:

- PV = 3,230.78 / (1 + .1122)^10 = 1,115.10

This result matches the emboldened starting value in the table, the value at the end of 2009.

If you want a fast solution instead of setting up a table and you already know FV and PV, you can directly solve for the geometric return with this equation:

“r” is the geometric return, “n” is the number of periods, “FV” is the future value, and “PV” is the present value. Note that this is not a standard square root calculation since the “n” modifies it. For example, if n is 3, it is the cube root of the number under the radical. In our example n is 10:

**More About Calculating Geometric Mean Returns and Equations 6 & 8 – Averages**

That 11.22% calculated above – it’s also called the compound annual growth rate (CAGR). It’s an average return, and all averages have good and bad properties:

**Good**: It’s useful for summarizing many observations. That’s what you do when you calculate investment returns. You take a lot of numbers and summarize them with one number.

**Good**: Two of the three kinds of averages – the mean and median – identify the middle of a lot of numbers. A mode – the third kind of average – identifies the most frequent observation, so it doesn’t usually help to find the middle. The nice thing about knowing the middle is that the middle is the single best number to represent the other numbers from which it was calculated.

**Good**: It’s useful for planning over long investment horizons. For example, if you plan to start spending your retirement savings in 30 years, it’s reasonable to plan your retirement savings using average historical returns.

Here’s are examples of how to do this with Equations 6 and 8:

Suppose the average return, which includes stocks, bonds, and other investments, equals 6%. What will the value of your savings, $100,000, be worth in 30 years?

- FV = 100,000(1 + .06)^30 = $574,349.

Here’s another way to look at it. You have $100,000 and you wonder what annual rate of return is required to grow the value to $1,000,000 in 30 years.

**Bad**: An average is not *sufficient *to define all the numbers that contribute to the average, and some people forget about that. For example, the social security administration publishes the “Actuarial Life Table.” One of the statistics it reports is the “average remaining number of years” expected before death for each yearly age from birth to 119. In 2017, a 35-year-old female had an average life expectancy of 47.20 years. That is, she’ll live to age 82.20 if she’s average.

But to calculate the “average,” the method counted many older and younger deaths. This is the idea of dispersion around the average and it’s what presents the problem. Once she reaches age 82, the probability she will die at that age is only about 5.4% (if the actuarial estimates don’t change). So even though the average is about 82 years, few people die at the average death age.

The same is true of investment averages. The average annual S&P 500 index return including dividends is between 10% and 11%. But through 2019, we haven’t seen a return in that range since 2004, and it only happened that one time in the last 50 years. The average geometric return implies that the return is the same for each period, which never happens.

### Multi-period geometric returns with reinvested dividends or Interest, no principal cash flows

Suppose you bought 100 shares of Boeing (BA) in early March 2020. Besides the trouble the corporation had already endured surrounding the tragic Boeing 373 Max crashes, the markets started recognizing the effect the coronavirus pandemic might have on civilization in general and on the aviation industry. You wanted to calculate your Boeing position return as of mid-March including the dividend the company paid on the 6^{th}.

The method of calculating daily geometric returns for the Boeing stock is the same as we did calculating the annual geometric returns for the S&P 500 Index above. But the dividend adds a wrinkle. We can assume the dividend payment gets reinvested in more Boeing shares or we can assume the dividend payment gets dropped in a cash account. The difference in this example is small because our interest lies only in a short period to keep it simple. But a longer time frame would make the difference between dividend reinvestment and dividend cash significant.

Let’s first assume the dividend gets reinvested in additional shares. The trick to simplifying dividend reinvestment lies in accounting for the share count. The simplest illustration uses a table:

The daily return during this period equals -5.9673%. We can double-check the result using either Equation 6 or 7. Here’s the double-check using Equation 6:

- FV = 28,062(1 + (-.059673))^8 = 28,062(.611265) = $17,153.33

Otherwise, we can use Equation 8 to find the geometric mean daily return:

This outcome – a daily return equal to -5.9673% – is good to know. But it’s customary to talk in terms of annual returns. We can annualize either the arithmetic or geometric mean daily return with this equation:

“r” is the geometric or arithmetic return, and “n” is the number of periods that would total one year. In Finance, we figure there are 252 trading days in one year; you probably wouldn’t be faulted for using 365 days, but 252 days is the convention. Of course, if the returns applied to months or quarters, n would equal 12 or 4, respectively.

In this example, the geometric mean return is the average return applicable to just one day; it’s the daily average for the 8 days. Therefore, 252 is the correct value for n to annualize the daily return:

- Annualized Geometric Return = (1 + (-.059673))^252 – 1 = -100%

If we had taken the single-period mean for the entire 8-day period – not the daily mean for the 8 daily periods – then the exponent would have been 252 ÷ 8 = 31.5.

This -100% outcome means if the daily mean loss of 5.9673% persists every day for a full 252-day year, the investment will have lost all its value. As it turns out, it would have taken a lot less than a full year for the stock price to go bust.

Another way to use the annualized return is to take just one day’s return from the example and annualize it. Let’s take the 03/06/2020 return equal to 1.5420%:

- Annualized Geometric Return = (1 + .015420)^252 – 1 = 4,628.89%

Both examples illustrate how annualizing daily returns can lead to practically meaningless results. The equation *assumes* the return for the sampled period will persist for an entire year. That rarely happens. But there are situations in which annualizing multi-period returns can be more practical. The most useful case is probably when you have a large sample of annual returns, say, 20 years, and estimate the return for, say, a one year period.

### Multi-period geometric returns with dividends or interest and principal cash flows

The last two sections illustrated multi-period geometric returns with dividends; this section adds principal cash flows.

A simple approach here can be to divide shares into lots, similar to what we did in some of our earlier arithmetic return calculations. Other methods which we’ll look at next include time-weighted return, internal rate of return and the Modified Dietz Method.

For now, let’s look at the simple approach of diving shares into lots. Then we’ll compare it against what we get with a time-weighted return approach.

Here are the details of our example:

- An investor buys 100 shares of the popular Invesco QQQ Trust (QQQ) exchange-traded fund at the 2014 year-end closing price of $103.25;
- The investor sells 10 shares at the 2015 year-end closing price of 111.86 and holds the remaining 90 shares until the end of 2019;
- Lastly, the investor buys 20 shares at the closing price of $132.50 on the first day of trading in April 2017 after the first 2017 dividend payment in March and holds the 20 shares until the end of 2019.

**Dividing shares into lots**

- There are 3 lots.
- Lot 1 is the 10 shares bought at the 2014 year-end closing price and were sold at the 2015 year-end closing price.

- Lot 2 is the 90 shares bought at the 2014 year-end closing price and were held until the end of 2019.

- Lot 3 is the 20 shares bought at the closing price on the first trading day in April 2017 and held until the end of 2019.

- Because Lot 3 shares were purchased after the first dividend in 2017, the investment benefited only from the remaining three dividend payments that year.
- Because the Lot 3 shares were purchased after the first quarter, the 9-month return was annualized to make it comparable to the other annual returns in 2018 and 2019.
- Because the QQQ stock price had risen in the first three months of 2017 and because Lot 3 only earned 3 dividend payments that year, Lot 3’s 2017 return was less than Lot 2’s 2017 return.
- The QQQ shares paid dividends quarterly and the dividends were held in a non-interest-bearing account until the first trading day of the next year when they were reinvested. Fractional shares are permitted. This is different than earlier illustrations merely to show a different way of treating dividends in return calculation.
- The sale of 10 shares in Lot 1 generated proceeds of $1,129.64. The profit on the 10 shares contributed to the total in the 2015 return.

Using this method, we get a weighted geometric mean return of 16.61%.

### Time-weighted rate of return

The time-weighted rate of return (TWRR) is designed to measure the performance of an investment or a portfolio over an entire time period invested while removing the effect of principal cash flows. The time-weighted rate of return is commonly used by brokerages to communicate portfolio performance to its users and clients. Similarly, investment advisors will communicate the time-weighted rate of return to clients. By eliminating the impact of capital flows in and out of an investment, the time-weighted rate of return attempts to communicate the actual performance of the investments themselves.

In our previous example, we broke out the example into share lots and then calculated a weighted geometric mean return. Time-weighted rate of return takes a different approach. When using the TWRR formula, we will break out the example by sub-periods as defined by capital flows. We will get the rate of return for each period, then we will take the geometric mean of the periods.

Here are the steps:

- Establish the holding period return (HPR) for each sub-period.
- Take each HPR and add 1
- Multiple all the (1 + HPR) terms together
- Subtract 1 from the product to get the time-weighted rate of return

Here is the formula:

Let’s go back to our previous example and use the TWRR formula.

**Holding period 1**

We know from the example that the investor buys 100 shares of the popular Invesco QQQ Trust (QQQ) exchange-traded fund at the 2014 year-end closing price of $103.25. A year later, the investors sells 10 of these shares at $111.86 each. These are the two end-points of the first sub-period.

Inputs:

- Beginning Value: 100 shares x $103.25 = $10,325.
- Dividends received: $1.104 per share. Total of $110.40.
- Ending Value: 100 shares x $111.86 = $11,186.

Let’s calculate the return for holding period 1 (you’ll recognize we’re using equation 2 here).

- HPR1 = (11186 – 10325 + 110.40) / 10325 = 971.4 / 10325 = 9.41%

**Holding period 2**

Holding period 2 will begin right where holding period 1 left off, and it will start off with the new balance after the cash flow occurred. The beginning value is determined by calculating the value after the sale of the 10 shares the investor made at the end of HPR1 (beginning of 2016).

We need to go back to the example to determine the end point of HPR2. We know that the investor buys 20 shares at the closing price of $132.50 on the first day of trading in April 2017 after the first 2017 dividend payment in March. This will be our end point.

Inputs:

- Beginning Value: 90 shares x $111.86 = $10,067.40.
- Dividends received 2016: $1.254 per share. Total of $112.86.
- Dividends received 2017: $0.274 per share. Total of $24.66
- Ending Value: 90 shares x $132.50 = $11,925.

Let’s calculate the return for holding period 2.

- HPR2 = (11925 – 10067.40 + 112.86 + 24.66) / 10067.40 = 1995.12 / 10067.40 = 19.82%

**Holding period 3**

Holding period 3 will go from right after those 20 shares were purchased in April 2017 to the end of 2019 and the end of our example.

Inputs:

- Beginning Value: 110 shares x $132.50 = $14,575.40.
- Dividends received 2017: $1.026 per share. Total of $112.86.
- Dividends received 2018: $1.406 per share. Total of $154.66.
- Dividends received 2019: $1.582 per share. Total of $174.02.
- Ending Value: 110 shares x $212.61 = $23,387.10.

Let’s calculate the return for holding period 3.

- HPR3 = (23387.10 – 14575.40 + 112.86 + 154.66 + 174.02) / 14575.40 = 9253.24 / 14575.40 = 63.49%

**Putting it together**

Now that we have the individual returns for each holding period, we can chain them together to get the time-weighted rate of return.

- TWRR = [(1 + HPR1) x (1 + HPR2) x (1 + HPR3)] – 1
- TWRR = (1.0941 x 1.1982 x 1.6349) – 1
- TWRR = 2.14327316864 – 1
- TWRR = 1.1432 = 114.32%

We can get the annualized TWRR with the following formula:

Using this formula and noting that n = 5 since our total time period is 5 years, we can calculate the annualized TWRR as follows:

- Annualized TWRR = (1 + TWRR)^(1/n) – 1
- Annualized TWRR = (1 + 1.1432)^(1/5) – 1
- Annualized TWRR = 2.1432^.2 – 1
- Annualized TWRR = 1.1647 – 1 = 16.47%

When we compare the annualized TWRR to the share-weighted approach from before, we can see that they are fairly close. 16.47% vs 16.61%.

## Natural Logarithmic Returns

Investors sometimes encounter natural logarithmic or log returns. They can be useful for multi-period returns, but they’re also used for single-period returns. Arithmetic returns are almost always superior to log returns for single periods. You can use log returns without spending a lot of time learning about logs because they’re easy to use.

The single-period log-return equation can be expressed like this:

“LN” is the natural log, “E” is the ending price, and “B” is the beginning price.

Let’s start with the example from the beginning of this article to compare the multi-period arithmetic, geometric, and log returns: An investment’s beginning price equals $100, it loses 50%, then it gains 100% so that it recovers its beginning price of $100.

In the “Average Return” row we calculate three kinds of average returns:

- The arithmetic return equals the sum of the periodic returns divided by the number of periodic returns. The result is the disturbing +25% gain. This is Equation 4.

- The geometric return works by adding 1 to each of the periodic arithmetic returns, taking the product of those results, and raising the product to the reciprocal of the number of periods. We subtract 1 from that result. This is Equation 5.

- The log return uses Equation 10. It sums the two periodic log returns and divides the result by the number of log returns. This is Equation 4 again. How easy is that?!

You can annualize log returns by simply adding each sub-period return.

For a more realistic example, we use one of the most widely held exchange-traded funds to illustrate log returns, SPDR S&P 500 Trust (SPY). In this example, the period is one month.

You can annualize log returns by simply adding each sub-period return.

For a more realistic example, we use one of the most widely held exchange-traded funds to illustrate log returns, SPDR S&P 500 Trust (SPY). In this example, the period is one month.

- Beginning price per share, October 2020 = $334.98
- Ending price per share, October 2020 = $326.54
- Log Return = LN(326.54/334.98) = -2.55%

Compare the log return to the arithmetic return:

- Arithmetic Return = (326.54-334.98)/334.98 = -2.52%

The results are pretty close. That changes when returns get larger. Here is the comparison of 2019 log and arithmetic returns for SPY, excluding dividends:

- Beginning price per share, 2020 = $249.92
- Ending price per share, 2020 = $321.86
- Log Return = LN(321.86/249.92) = 25.30%

Compare to the arithmetic return:

- Arithmetic Return = (321.86-249.92)/249.92 = 33.11%

A log return practically always assumes a value less than the arithmetic return when the two returns apply to the same investment for the same period. An explanation for the reason for this relationship is in a comparison of arithmetic, geometric, and log returns below.

In this next example, we illustrate why single-period log returns can be more useful than arithmetic returns and how log returns go hand-in-hand with geometric mean returns. The example uses the stock returns of Moderna (MRNA), one of a handful of companies projecting the release of a COVID vaccine in late 2020. You can imagine that Moderna’s stock price would have risen in anticipation of its vaccine.

These are the main things to notice or conclude about results in the table:

- The arithmetic mean return represents the return an investor could expect in each month. It’s not the return an investor would earn if they invested at the December 2019 closing price and held the investment until November.

- The cumulative arithmetic return is the beginning return divided into the result of the ending price minus the beginning price. This is Equation 1. It takes the 11 multiple periods and treats them like one period. This roughly 400% gain means the stock price increased by a factor of 5 from its starting price of $19.56. Whereas the arithmetic mean return of the 11 periods gives only a
*representative*value, this now-single period (cumulative) return is a correct expression of the investment return because there is no compounding within the single period.

- The average geometric return is the return an investor would have earned if they had purchased the stock at $19.56 in December and held it until November. We can prove this using equation 8:

- The cumulative geometric return equals the cumulative arithmetic return.

- Each log return uses Equation 12.

- Notice that each periodic log return is smaller than its corresponding periodic arithmetic return. As returns get bigger, so does the difference between arithmetic and log returns.

- The cumulative log return is simply the sum of the periodic log returns. The ease in calculating cumulative returns is one reason some investors prefer log returns.

- The cumulative log return is much smaller than the cumulative arithmetic and geometric returns. We can check this outcome by converting the cumulative log return to an arithmetic return using Equation 12:

The “e” constant always equals approximately 2.78182845904. This equation works for any log return, single period or cumulative.

- 2.7818^1.6075 – 1 = 399.03%

It worked! The cumulative log return of 160.75% matches Table 7 arithmetic return by applying Equation 14. It works for individual log returns like the May 2020 29.06% log return:

- 2.7818^.2906 – 1 = 33.72%

### Arithmetic vs. Geometric vs. Log Returns

Compounding periods explain why geometric returns are less than arithmetic returns and why log returns are less than geometric returns. The shorter the compounding period, the smaller the return:

- Arithmetic returns effectively use the period covered as the single compounding period; there’s no compounding within the period. But geometric returns compound based on the intervals according to which the data are grouped. For example, geometric return compounding periods in Table 7 are months, so the geometric return compounds monthly.

- Log returns compound continuously. That whole idea of continuous compounding isn’t intuitive, but continuous compounding “periods” are much shorter than any other compounding period.

Here’s an intuitive explanation: For a daily compounding period, the return would naturally be less than for a monthly compounding period because you’d expect the return to be greater for a longer period. Arithmetic returns don’t have compounding periods whether the returns apply to a single period or multiple periods. They just act like you put your money in at the beginning of each period and took it out at the end of each period. Geometric returns compound based on the compounding periods you specify. That’s why the arithmetic and geometric returns are equal for single periods. But if you have multiple periods, geometric return values are less than arithmetic returns. Log returns compound continuously. You can’t get shorter compounding periods than that, so log returns are less than the others.

Here are a few warnings about these different kinds of returns:

- Don’t mix them. For example, you can’t compare or do any math with the returns of one investment expressed in log returns with one expressed in geometric returns. There are ways to convert returns to one of the other forms, but you must do that before you make comparisons or do any math with them. Also, you can’t calculate, say, a “Sharpe Ratio” with log or geometric returns in the numerator and a standard deviation in the denominator. The standard deviation is ultimately an arithmetic mean of the dispersion of individual observations around the mean, so that would be a mixing violation.

- With log returns, you can add contiguous observations in a time series to arrive at the cumulative return – all of the tables above include contiguous observations in a time series. They’re just ordered periodic (e.g., daily, monthly) returns without gaps. Also, if you divide the cumulative log return by the number of periods, you can arrive at the periodic average. However, with log returns, you cannot add them across investments for the same period to get a cumulative or per-investment return. Adding log returns compounds the things you’re adding. It wouldn’t make sense to compound across investments for the same period.

- Log and geometric returns assume the investment returns distribute in normal form. In almost all cases, that’s a false assumption. Investment returns don’t fit in a “bell curve.” Too many investment returns fall close to the mean – those are periods when the markets are “in irons”; returns don’t go up or down much. Also, too many investment returns fall far from the mean on both sides. This assumption matters because log and geometric returns can give a false reading.

## Other Rate of Return Topics

### The Modified Dietz Method

The Modified Dietz Method (MDM) streamlines the “lot” method of handling principal cash flows and does a better job assigning time value to the cash flows. The MDM often comes up in evaluations of the performance of money managers, but it can equally apply to an individual’s investment return. Its recommended use is usually limited to historical returns, not projected returns. It works by applying to the principal cash flows the rate of return applicable to the other cash flows. The method can be expressed like this:

“B” is the beginning market value or beginning principal of the investment, “E” is the ending market value of the investment, and “P” is the net principal cash flows where incoming principal is a positive cash flow and outgoing principal is a negative cash flow. “w” is different from its use above. In the MDM case, “w” is time expressed as a percentage of the period elapsed in which the principal cash flow occurred.

Here’s an example of how to calculate the one-year investment return using the Modified Dietz Method. Suppose an investor’s market value at the beginning of the period equals $1,000 and they make a new principal contribution of $100 at the beginning of the 10^{th} month. Three-fourths of the year would have passed by the time of this new principal contribution, so “w” equals 0.75. At the end of the year the market value of the investment equals $1,155:

For multi-period returns, you can apply this method for each period then calculate the most suitable investment return, arithmetic, geometric, or log.

### Internal Rate of Return (IRR)

The internal rate of return, also known as the “money-weighted return” or “discounted cash flow return,” is a multi-period return that takes into account time value (c.f., Equations 6 and 7 for calculating future and present values) and compounding like geometric and log mean returns:

- Its main use is to find a rate of return for expected cash flows associated with business investments, though a general investment example is provided here.
- “Net present value” (NPV) is linked to IRR.

First, we’ll illustrate NPV because the process is similar to other time value return calculations (above) and it helps to illustrate IRR.

NPV is the discounted value of projected cash flows associated with an investment. It’s called “net” present value because investments typically include cash flows that are both positive (cash generated by the project) and negative (cash used to fund the project). Then the process adds these positive and negative discounted cash flows to arrive at NPV. The “discount” used in NPV problems is the business’s minimum rate of return on its investments. Sometimes this minimum return is called the business’s “cost of capital, “hurdle rate” or “required” rate of return. The idea is the business will reject investments if their expected returns are less than the required rate of return. The NPV can be expressed like this when the period durations are equal:

“CF_{0}” means the cash flow that occurs at time zero. The discount doesn’t apply to time-zero cash flows because they are present values. You could apply Equation 7 to the first cash flow, but the exponent in the denominator would equal zero rendering the denominator equal to one and the ratio equal to the value of the numerator. “FV” means the future value, which is the expected value of the future cash flow. The “CF” subscript with sub-subscripts 1 and 2 mean the cash flows at time 1 and 2. These are the cash flows at the end of periods 1 and 2. “r” is the required return, and exponents 1 and 2 in the denominator take 1 plus the required return to the power of the period number. The ellipsis means an unlimited of periodic cash flows can be included in the calculation. “n” in the numerator and denominator represents the period number for the last cash flow.

Suppose a business owner wants to know whether an investment will satisfy the 14% required rate of return. It projects these cash flows:

Values in the “Cash Flows” column are the projected cash flows, and values in the “Present Values” column are the results of applying Equation 7 to each projected cash flow to get its discounted value using the required rate of return. The NPV value in the bottom row is the sum of the present values.

The NPV is negative. The negative NPV means the project fails to earn the required return. That is, this project’s investment return is less than the minimum of 14%. IRR calculates the project return assuming the NPV equals zero. In that way, the IRR is the *breakeven* return on a potentially acceptable investment. It is the minimum return an investment would produce if it is potentially acceptable.

One way to think about IRR uses Equation 14. If you set NPV equal to zero with known projected cash flows and solve for r, the answer is IRR. Unfortunately, the IRR calculation amounts to a trial-and-error process. That’s a tedious and time-consuming task, so electronic calculators and spreadsheets make the calculation more tolerable.

Some investors might use NPV and IRR to compare competing investments. But there are significant caveats with IRR: In more realistic, complicated scenarios, IRR can give more than one answer. Things can also get a little complicated when the periods have unequal intervals or if the return varies in different periods.

When we’ve calculated multi-period returns in other parts of this article, we did it with periodic returns. It doesn’t work that way with IRR. We calculate IRR with positive and negative principal transactions only. Also, unlike other principal transactions illustrated in this article, the IRR method assigns a negative sign to investment purchases and a positive sign to investment sales; sales are analogous to business investment proceeds. So the IRR perspective is from the business, not from the investment portfolio. At the end of the periods, IRR assumes the investor cashes in the investment.

Suppose an investor wants to know whether a project will satisfy their 10% required rate of return. They project these cash flows:

In this example, the investor purchases shares at time 0 and at the ends of the first and second periods. At the end of the third period, the investor sells $3,000 worth of shares. At the end of the fourth period, the investor liquidates the position. The NPV is positive; that fact is sufficient to inform the investor that the IRR exceeds the 10% required rate of return. The IRR calculation reveals a 17.46% return.

### Annual Percentage Rate (APR) and Effective Annual Rate (EAR)

Investors frequently encounter APR in lending transactions with banks; to a lesser degree they encounter EAR. These types of return are associated with annuities – that is, payments that occur in equal amounts at equal intervals. The difference between APR and EAR concerns compounding.

APR annualizes the returns of cash flows that occur in periods shorter than one year like monthly mortgage payments or semiannual bond coupon payments. However, APR annualizes assuming simple interest – interest without compounding:

“APR” means annual percentage rate, “Per-period rate” means the rate applicable to a period less than one year, and “Periods per year” means the number of payment intervals that occur in a year. Suppose a corporate bond pays $15 interest every six months year and the principal or par equals $1,000:

- APR = (30 / 2 x .5 x 2) = 3.00%

This equation is equivalent to the single-period arithmetic return and assuming the interest would not be reinvested. That is, the value of the interest at the end of the two-periods would equal its nominal amount, and the time value would be irrelevant.

Alternatively, if the interest payments were reinvested and earned the rate applicable to the investment, we would use the effective annual rate. This rate compounds the interest. The method recovers the nominal per-period rate by dividing APR by the number of periods and then compound it based on the number of periods in the year:

“EAR” is the effective annual rate, “APR” is the annual percentage rate, and “n” is the number of periods in the year. Using the semi-annual bond interest payment example from above:

- EAR = (1 + (.03/2))^2 – 1 = 3.0225%

If everything else is held the same, a greater number of periods yields a larger EAR value.

### Excess Return

The excess return issue offers a more precise way to measure returns and for comparisons of returns across investments in a Sharpe ratio or other risk-adjusted return. For most individual investment return calculations, it’s not critical. But investors come across it occasionally.

The idea is rooted in the Capital Asset Pricing Model (CAPM), the dominant perspective on the relationship between risk and return. Part of the value of the CAPM is its claim that investment returns are positively related to investment risk. Excess return enters the discussion because part of an investment’s return is attributable to risk-free returns. This idea claims that short-term U.S. Treasury issues – Treasury bills – impose no risk. The idea is that the Treasury cannot default on its debt, and the maturities are so short – the usual proxy is a 90-day bill, though others are suitable, too – that changes in interest rates usually have no meaningful effect on price.

The calculation subtracts the risk-free rate from the realized return. The Federal Reserve Bank of St. Louis offers a convenient source for Treasury bill rates and other interest rates. Be careful with this because the rates are quoted in annualized terms. For example, a recent rate was reported as 0.16%, but the bill’s maturity isn’t even a year. You can’t divide the one-year rate by 4 to get the 90-day rate because of compounding. The equation for reducing an annual rate to a daily rate can be expressed as:

“r_{annual}”is the annual rate, 252 is the number of days in a year. This can be modified for different periods. For example, the quarterly rate replaces 252 with 4.

Suppose the annual risk-free rate equals 0.16% and you have an investment’s quarterly return equals 20%:

- Daily Rate = (1 + 0.0016)^1/4 – 1 = .04%

The excess return equals 19.96%.

Okay, you’re thinking you could have divided the annual rate by 4 and gotten the same answer. You’re right if we round to the basis point in this example. If we carry out the answer a few more decimal points, it’s 0.03998%. And you’re right, the difference here doesn’t matter because the risk-free rate is so small. That might change someday.

## Fees, inflation & tax

Here is a practical and powerful principle related to investment expense: Investment expense is money that otherwise would have been invested.

### One-time expense

This principle is important because investment expense compounds right along with the investment principal. Therefore, when somebody says an investment expense is a certain dollar amount, it’s usually an understatement. Suppose a mutual fund charges a 5.75% front-end load, the pre-expense investment values equals $1,000, and the fund value grows 5% per year. That is, the fund charges a fee upon purchase of the investment. Here’s how you analyze the expense and treat it in the investment return calculation:

Table 8 uses Equation 6 in the “Expense Time Value” column to calculate the future value of the expense for each year.

The period for each calculation equals 1. When the exponent is 1, then you can ignore it. The value at the end of Year 1 is calculated this way:

- FV = 50(1 + .05)^1 = 52.50

Table 8’s purpose is to show a) that the nominal value of the front-end load is less than the true future value, b) the expense is deducted from the value at the time it is incurred, and c) the effect of investment expense on investment value.

### Periodic Expense

Periodic expense is the kind of expense deducted repeatedly and at certain intervals during the life of an investment. The most common example is the fees mutual funds, variable annuities, and other packaged investments charge. They’re reported in the prospectus – the document prospective investors are supposed to read and understand before investing – in percentage terms. The simplest way to treat periodic expense is to deduct the percentage from the interest or growth rate. Again, the method uses Equation 6. Suppose an investment company charges a 1% periodic fee on a $1,000 initial investment and the investment grows at 5% per year before expense. At the end of the first year, the investment value will equal:

- FV = 1000(1 + (.05 – .01))^1 = 1,040

This equation is repeated for each period using the ending value from the previous period to serve as PV. For example, in the second year of this $1,000 initial investment and no change in the growth or expense rate, the calculation would be:

- FV = 1040(1 + (.05 – .01))^1 = 1,081.60

### Inflation

Recent inflation rates have been low. But they’re not insignificant for projecting investment values because they compound.For example, if the Bureau of Labor Statistics – the best source for inflation data – reports 2% inflation for one year, then you might say the price of a $100 item would cost $102 at the end of that year. If the inflation rate the next year equals 2%, then after two years, the $100 item would cost $104.04. These results use Equation 6, too:

- FV = 100(1 + .02)^2 = 104.04

Inflation should be incorporated in multiple-year investment projections because the compounding effect significantly diminishes investment value. For example, if annual inflation equals 2% and you’re accustomed to, say, a $50,000 per-year income, 20 years from now it’ll take almost a 50% increase in your income to maintain your lifestyle:

- FV = 50,000(1 + .02)^20 = 74,297.37

A simple way to incorporate inflation uses the same method used to incorporate periodic expense. It can come off the growth rate each period. Suppose you expect your investments to grow 6% per year, you expect an annual periodic expense equal to 1%, and you expect 3% annual inflation. The future value of a $1,000 investment after 20 years equals $1,485.95:

- FV = 1000(1 + (.06 – .01 – .03))^20 = 1,485.95

### Tax

The way to handle income tax is usually to assign the expense in the period it is incurred. However, in many cases, investors might prefer to deduct taxes from all taxable cash flows in the periods they occur. Some taxpayers make quarterly estimated payments, others pay with each income payment, and some pay only once a year. For planning, the investor can either reduce the growth rate (as with investment expense and inflation) or deduct a dollar amount. A tax adviser is the best source for the correct period to realize tax expense. Here are some investment examples:

- For investors using tax-deferred retirement plans (e.g., 401k, Traditional IRA), the tax (and potential fees) can be deducted from withdrawals.
- For Roth plans, investors can recognize the expense for the tax year in which they contribute.
- For taxable investments, the timing can be based on when the taxpayer makes tax payments.