### What is a Rate of Return

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.

The formula for rate of return is:

This simple rate of return is sometimes called the return on investment, or ROI. Taking into account 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.

#### Rate of Return

### BREAKING DOWN Rate of Return

A rate of return can be applied to any investment vehicle, from real estate to bonds, stocks and fine art, 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.

### Calculating the Rate of Return

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: (335,000-250,000)/250,000 x 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: (187,500-250,000)/250,000 x 100 = -25%.

### The Differences Between 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, therefore, $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 Rate 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 and 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. 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 period length, and subtract one from the subsequent result.

This can be written as follows:

Therefore, if we use our original example of a home purchased and then sold for a profit six years later, we can compute the CAGR as: (335,000/250,000)^{(1/6)} - 1 x 100 = 5% per year. This figure is somewhat less than dividing the 34% RoR obtained above by six years (which would be 5.67% per year). This is because CAGR smooths returns so that they are the same for each year, and then compounds them.

### How to Discount Cash Flows and the Internal Rate of Return

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 an 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. 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: