### What Is a Rate of Return (RoR)?

A rate of return (RoR) is the net gain or loss on an investment over a specified time period, expressed as a percentage of the investment’s initial cost. Gains on investments are defined as income received plus any capital gains realized on the sale of the investment.

#### Rate of Return

### What Is the Formula for RoR?

$\text{Rate of return} = [\frac{(\text{Current value} - \text{Initial value})}{\text{Initial value}}]\times 100$

This simple rate of return is sometimes called the basic growth rate, or alternatively, return on investment, or ROI. If you also consider the effect of the time value of money and inflation, the real rate of return can also be defined as the net amount of discounted cash flows received on an investment after adjusting for inflation.

### What Does the RoR Tell You?

A rate of return can be applied to any investment vehicle, from real estate to bonds, stocks, and fine art. RoR works with any asset provided the asset is purchased at one point in time and produces cash flow at some point in the future. Investments are assessed based, in part, on past rates of return, which can be compared against assets of the same type to determine which investments are the most attractive.

- The rate of return is used to measure growth between two periods, rather than over several periods.
- The RoR can be used for many purposes, from evaluating investment growth to year-over-year changes in company revenues.
- The RoR calculation does not consider the effects of inflation.

### RoR vs. Stocks and Bonds

The rate of return calculations for stocks and bonds are slightly different. Assume an investor buys a stock for $60 a share, owns the stock for five years, and earns a total amount of $10 in dividends. If the investor sells the stock for $80, his per share gain is $80 - $60 = $20. In addition, he has earned $10 in dividend income for a total gain of $20 + $10 = $30. The rate of return for the stock is thus $30 gain per share, divided by the $60 cost per share, or 50%.

On the other hand, consider an investor that pays $1,000 for a $1,000 par value 5% coupon bond. The investment earns $50 in interest income per year. If the investor sells the bond for $1,100 premium value and earns $100 in total interest, the investor’s rate of return is the $100 gain on the sale plus $100 interest income divided by the $1,000 initial cost or 20%.

### Real vs. Nominal Rates of Return

The simple rate of return used in the first example above with buying a home is considered a nominal rate of return since it does not account for the effect of inflation over time. Inflation reduces the purchasing power of money, and so $335,000 six years from now is not the same as $335,000 today.

Likewise, $250,000 today is not worth the same as $250,000 six years from now. Discounting is one way to account for the time value of money. Once the effect of inflation is taken into account, we call that the real rate of return (or the inflation-adjusted rate of return).

### RoR vs. CAGR

A closely related concept to the simple rate of return is the compound annual growth rate, or CAGR. The CAGR is the mean annual rate of return of an investment over a specified period of time longer than one year, which means the calculation must factor in growth over multiple periods.

To calculate compound annual growth rate, we divide the value of an investment at the end of the period in question by its value at the beginning of that period, raise the result to the power of one divided by the number of holding periods, such as years, and subtract one from the subsequent result.

### Example of How to Use the RoR

The rate of return can be calculated for any investment, dealing with any kind of asset. Let's take the example of purchasing a home as a basic example for understanding how to calculate the RoR. Say that you buy a house for $250,000 (for simplicity let's assume you pay 100% cash).

Six years later, you decide to sell the house—maybe your family is growing and you need to move into a larger place. You are able to sell the house for $335,000, after deducting any realtor's fees and taxes. The simple rate of return on the purchase and sale of the house is as follows:

$\frac{(335,000-250,000)}{250,000} \times 100 = 34\%$

Now, what if, instead, you sold the house for less than you paid for it—say, for $187,500? The same equation can be used to calculate your loss, or the negative rate of return, on the transaction:

$\frac{(187,500 - 250,000)}{250,000} \times 100 = -25\%$

### Example of IRR and DCF

The next step in understanding RoR over time is to account for the time value of money (TVM), which the CAGR ignores. Discounted cash flows take the earnings on investment and discount each of the cash flows based on a discount rate. The discount rate represents a minimum rate of return acceptable to the investor, or an assumed rate of inflation. In addition to investors, businesses use discounted cash flows to assess the profitability of their investments.

Assume, for example, a company is considering the purchase of a new piece of equipment for $10,000, and the firm uses a discount rate of 5%. After a $10,000 cash outflow, the equipment is used in the operations of the business and increases cash inflows by $2,000 a year for five years. The business applies present value table factors to the $10,000 outflow and to the $2,000 inflow each year for five years.

The $2,000 inflow in year five would be discounted using the discount rate at 5% for five years. If the sum of all the adjusted cash inflows and outflows is greater than zero, the investment is profitable. A positive net cash inflow also means the rate of return is higher than the 5% discount rate.

The rate of return using discounted cash flows is also known as the internal rate of return, or IRR. The internal rate of return is a discount rate that makes the net present value (NPV) of all cash flows from a particular project or investment equal to zero. IRR calculations rely on the same formula as NPV does and utilize the time value of money (using interest rates). The formula for IRR is as follows:

$\begin{aligned} &IRR = NPV = \sum_{t = 1}^T \frac{C_t}{(1+ r)^t} - C_0 = 0 \\ &\textbf{where:}\\ &T=\text{total number of time periods}\\ &t = \text{time period}\\ &C_t = \text{net cash inflow-outflows during a single period }t \\ &C_0 = \text{baseline cash inflow-outflows}\\ &r = \text{discount rate}\\ \end{aligned}$