 ## What is Compound Annual Return (CAR) and how is it calculated?

The following text originally appeared as an article in the Dec 2003 issue of BetterInvesting

One of the most often asked questions by Club Accounting users is, "How is the compound annual and total return shown on the Valuation Statement and Members Status Report calculated?" Club Accounting uses what is called an "Internal Rate of Return" (IRR) calculation to calculate these returns. This type of calculation takes into account both the time when money was invested and the amount of money that was invested. It is a complex calculation that is best not attempted manually but rather left to the computer.

First, a general explanation of what is taken into account when calculating the Compound Annual Return:

All cash flows into and out of an investment are considered when making these calculations. Cash flows for a security would include dividends, and returns of capital as well as the amount to purchase the security. For members, cash flows include payments, fees and partial withdrawals. Timing of the cash flows is also a factor in the calculations. These calculations use annual, not daily, compounding. Annual Compounding makes the return figures easier to compare with market index return figures.

Due to the way these calculations are done, it is possible to be confused at times. Below are some times when the return calculations may be confusing.

When a company is acquired through a spinoff or merger, Club Accounting uses the value of the company on the day the club acquired the stock (the spinoff or merger date) as the starting point in the return calculations. The adjusted cost basis is not used as the starting value. It is therefore possible for your cost basis to be less than the current value of the stock but for Club Accounting to show a negative return for the stock.

This occurs when the new stock loses value compared to the spinoff/merger date but is still above the cost basis in the stock.

When a stock makes a large move in price, either higher or lower, within a short time, the return calculations may look wrong. The problem is not the accuracy of the calculations but the usefulness of Compound Annual Return calculations when an investment has been held a short period. In this case, you can think of the CAR as the return you will receive if the rate of gain received in the short period continues for a full year.

Following is a more detailed explanation of Compound Annual Return, followed by the formula we use in our Club Accounting system:

CAR is the rate of return most of us are familiar with when we talk about interest paid on a bank savings account, certificate of deposit or a home mortgage. It is the yearly rate of return that is earned or paid and that usually is compounded daily, monthly or yearly. The more often the compounding is done (interest earned on interest paid) the larger the amount will be for the same rate of CAR. Club Accounting uses daily compounding. However, for simplicity we will use yearly compounding in all of our examples.

Total Return is the return earned over a specific period of time. For example, if one invests \$100 at 6% CAR for six months you would have \$103 or a 3% total return for the period. If the money was invested for 2 years at the same CAR you would have \$106 at the end of the first year and \$106 x 1.06 = \$112.36 at the end of the second year for a total return of 12.36%.

As you see, if you know the CAR and the period of time involved you can calculate the total return. Notice that the CAR and Total Return are the same at the one-year point. During the first year the CAR is always greater than the Total Return. After one year the Total Return is always greater than the CAR. It is important to understand the difference between these two returns because during the first year they can look very strange and yet be correct.

For instance, suppose you bought a stock and one week later it had gained 2%. The CAR is the rate you would earn if this same gain continued for a year. A 2% gain in one week equates to about a 180% CAR. On the other hand, the Total Return is only 2%. By using an Internal Rate of Return calculation, which takes into account both the time and amounts of money invested, we can calculate an accurate CAR. Here is an explanation of how it is done:

If you go to your local bank and open a savings account, the banker tells you that they will pay you a certain annual percent return on your money, say 6% compounded daily. You agree and deposit a sum in the account. From time to time you may deposit more money or make withdrawals from the account. At any time, you can ask a bank teller what your balance is and they can tell you. They can do so because they have recorded each of the transactions you have made in the account and they know they are paying you at a fixed 6 percent interest rate. Knowing these two factors allows them to calculate the third (your present balance) by adding the proper amount of interest to your balance each day.

The problem is different when you invest in an investment in which the return varies, such as an investment club where the value of the security or members' units change whenever the club is valued. However, like the bank, the Club Accounting program does know two of the three factors needed to calculate the third one.

In this case, the program knows all of the transactions that have been entered into the club and it knows the current balance in the account on the valuation date. What it doesn't know is the compound annual interest rate that would have to be paid on all these transactions to make them total to this current balance. It can calculate this interest rate but must do so by using trial and error to find an interest rate that when applied to the daily account balance will result in the current balance. The CAR rate calculated in this way is called an Internal Rate of Return (IRR). Once the CAR is found, a total return can easily be calculated for any period of time.

A simple example will illustrate the effect of time and dollar amount on the CAR calculation.

Suppose you open a bank account on January 1st and the bank pays you 6% CAR. You deposit \$100 to open the account. Then you do nothing with the account until December the 1st when you deposit \$200 more. On December 31st you run a valuation statement. The first \$100 would have made 6% a year for one year, and grown to \$106. The second \$200 would have made about \$1.00 in interest during the month of December. Therefore, the current value would be \$307 and the amount invested or cost is \$300. By using the cost and value to calculate the return one would get (7.00/300) x 100 = 2.33%. However, we know that we actually earned 6.0% CAR for the year not 2.33%. The difference is the result of having most of the money invested for only one month. The IRR calculates the true interest rate you earned as 6.0%. The formulas for this calculation are included at the bottom of the page.

As you see, one can't just use the cost of a stock, or the amount paid in by a member along with the current value to calculate the CAR. Every security or member transaction involving money flowing into or out of the club must be considered separately both in time and amount.

Consequently, almost all security and member transactions have an impact on the returns that are calculated. Club Accounting uses the IRR method to calculate the CAR and Total Return for several different combinations of individual and groups of securities, cash, and member transactions. These are shown on the Valuation Statement and Members Status Report.

Mathematical Formulas

The following description is for those readers who have a mathematical background and who want an exact definition of the way total return for each investment period is calculated.

Definition of Variables

Cf(i) Amount of the i-th cash flow transaction during the period. If the transaction by the investor puts money into the investment (member payments or fees, or security purchase), the amount is negative. If the transaction by the investor receives money from the investment (member withdrawals, security sales, cash dividends), the amount is positive. The four cash flows used for the four return calculations are described in the bulleted items in previous section.

N Number of cash flows during the investment period.

V Value of the investment at the end of the investment period.

d(i) Date the i-th cash flow was posted.

D Date of the end of the investment period.

r Estimated annual rate of return.

v Computed value of the investment at the end of the investment period.

The estimated rate of return (r) is varied in the following equation to make the computed value (v) equal to the actual value (V). An iterative procedure is used. The value of r that makes the two value figures equal (v=V) is the actual annual rate of return.

Formulas

v=Sum(cf(i) * (1 + r)^((D - d(i)) / 365.25))

where i = 1 to n and

D - d(i) is the number of days between D and d(i)

Finally, the total return during the investment period is:

Total Return = (1 + r)^((D - d(n)) / 365.25) -1