## Understanding Returns

This article will help you define and distinguish realized returns from unrealized returns. By the end of this section, you will be able to calculate an investment's dollar return and percentage return. You will also be able to describe how to use historical and average returns to predict future performance.

### Calculating and Understanding Average Returns

Average returns are commonly found using average ROI, CAGR, or IRR.

#### LEARNING OBJECTIVE]

• Differentiate between the different methods for calculating the average return of an investment

#### KEY POINTS

• Average return on investment (ROI) is the arithmetic average of the total cash returns divided by the initial investment. It is useful for quick calculations and specific securities (such as bonds purchased at par), but does not account for compounding returns.
• Compound annual growth rate (CAGR) is derived from the future value formula with compounding interest. It accounts for compounding returns.
• Internal rate of return (IRR) is the discount rate at which the NPV equals 0. It is used because it allows for easy comparison between investment options and is easy to understand.
• For all three methods, the higher the average rate of return, the more attractive the investment is.

#### TERM

• compounding returns

Returns earned on previous returns. Akin to compounding interest.

The average return of an investment can be calculated a number of ways. The three main methods are

1. Return on investment (ROI)
2. Compound annual growth rate (CAGR)
3. Internal rate of return (IRR).

For all three methods, the higher the rate, the more desirable the investment.

#### ROI

To calculate the total ROI of an investment, simply divide the total dollar returns of the investment by the initial value. The average ROI is the arithmetic average: divide the total ROI by the number of periods. If the purchase of a stock led to an ROI of 15% over 5 years, the average ROI is 3% per year. This is a simple way to calculate the average return. It is useful for certain securities such as bonds. If the only source of return on a bond is the coupon payments, then this is an accurate method. A bond purchased at par that pays a 5% coupon per year, will have a return of 25% over 5 years. However, this is a very special case. Average ROI generally does not calculate the actual average rate of return, because it does not incorporate compounding returns. A stock that appreciates by 3% per year would not actually be worth 15% more over 5 years, because the gains compound.

#### CAGR

CAGR stands for compound annual growth rate. It is calculated by the following formula where Vf is the future value, Vi is the initial value, and t is the number of years:

CAGR, unlike average ROI, does consider compounding returns. CAGR is derived from the compounding interest formula, $FV=PV(1+i)^t$, where $PV$ is the initial value, $FV$ is the future value, $i$ is the interest rate, and $t$ is the number of periods. The CAGR formula is what results when solving for $i$: the interest rate becomes CAGR, $FV$ becomes $V_f$, $PV$ becomes $V_i$, and the number of periods is generally assumed to be in years.

CAGR is very useful for finding the rate of return that the investment would have to earn every year for the life of the investment to turn the initial value into the future value over the given time frame.

#### IRR

The internal rate of return (IRR) is another commonly used method for calculating the average return . IRR is the discount rate at which the net present value (NPV) is equal to 0. Using IRR allows for easy comparison between investment options. It is also known as the effective interest rate.

Internal Rate of Return: The IRR is calculated by finding the discount rate at which the NPV of the investment equals 0.