Internal rate of return of an investment project: formula

An investment is a long-term investment of capital in something with the aim of obtaining an effect. This effect can be social and economic. The economic effect of an investment is called profit.

Interest rates required to calculate the feasibility of an investment

In financial mathematics, there are three types of interest rates that an investor uses when calculating the feasibility of his investments. The first rate is the investment project's internal rate of return (IRR). This index shows what percentage should be taken when calculating investment efficiency.

The second interest rate is the calculation interest itself. This is the rate that the investor calculates.

The third indicator is called "internal interest". It shows how much the investment has paid off as a percentage.

Difference between IRR of an investment project, internal and calculation interest

All of the above indicators may be equal or may differ. If you calculate the internal rate of return of an investment project, you can see that these three interest rates do not always have the same value.

The thing is that with a calculated interest, the depositor can receive both profit and loss in general and in comparison with an alternative way of using funds. The internal rate of return of an investment project shows the percentage at which the investor receives neither a loss nor a profit. If the net worth is above zero, it means that the percentage included in the calculation of the investment efficiency is below the rate of return. In the event that the net worth is below zero, the calculated interest exceeds the IRR of the investment project.

In these cases, it is necessary to calculate the internal interest, which shows how profitable the investment is.

The concept of the rate of return and the way to determine it

The key indicator for determining how effective an investment is is the internal rate of return of an investment project. This means that the amount of income received from the implementation of investment activities should be equal to the size of the investment. In this case, the flow of payments will be zero.

There are two ways to determine the rate of return. The first of them is to calculate the internal rate of return of an investment project, provided that the net value is 0. However, there are times when this indicator is higher or lower than zero. In this situation, it is necessary to "play" with the calculation percentage, increasing or decreasing its value.

It is necessary to find two calculation rates at which the net present value will have the minimum negative and minimum positive values. In this case, the payback ratio can be found as the arithimetic average of two calculated interest rates.

The role of present value in calculating the rate of return

The fair value plays a key role in determining the internal rate of return of an investment project. On the basis of the formula for its determination, the calculation of the internal rate of return of the investment project is carried out.

From the fair value method, it is known that the fair value is zero, which means that the invested capital is returned with an increase at the level of the calculated interest. When determining the internal interest rate, the interest rate is determined such that the present value of a series of payments will be zero. This means at the same time that the present value of receipts coincides with the present value of payments.

When you use an alternative accrued interest, you determine the one that results in a fair value of zero.

Net present value calculation

As you already know, the internal rate of return of an investment project is calculated using the net present value formula, which has the following form:

ChTS = CF t / (1 + VND) t, where

• CF - (flow of payments is the difference between receipts and expenses);
• IRR is the internal rate of return;
• t - period number.

Payback calculation

The formula for the internal rate of return of an investment project is derived from the formula that is used in the process of determining the net present value, and has the following form:

0 = CF / (1 + p) 1 ... + ... CF / (1 + VND) n, where

• CF is the difference between receipts and payments;
• IRR is the internal rate of return;
• n is the number of the period of the investment project.

Manual Calculation Issues

If the investment project is designed for a period of more than three years, the problem arises of calculating the internal rate of return using a simple calculator, since equations of the fourth degree arise to calculate the coefficient of a four-year project.

There are two ways to get out of this situation. First, you can use a financial calculator. The second way to solve the problem is much simpler. It consists in using the Excel program.

The program has a function for calculating the rate of return, which is called IRR. To determine the internal rate of return of investment projects in Excel, you need to select the SD function and put a range of cells with cash flow in the "Value" field.

Graphical calculation method

Investors were calculating the internal rate of return long before the first computers appeared. To do this, they used a graphical method.

On the ordinate axis it is necessary to display the difference between receipts and expenses for the project, and on the abscissa axis - the calculated percentage of the investment project. The type of graphs can be different depending on how the cash flow changes during the investment project. Ultimately, any project will cease to be profitable, and its schedule will cross the abscissa axis, on which the calculated percentage is displayed. The point at which the project schedule crosses the abscissa is the internal rate of return on investment.

An example of calculating the internal rate of return

You can analyze the method for determining the payback ratio of a deposit using the example of a bank deposit. Let's say its size is 6 million rubles. The term of the deposit will be three years.

The capitalization rate is 10 percent, and without capitalization it is 9 percent. Since the money earned will be withdrawn once a year, the rate without capitalization is applied, that is, 9 percent.

Thus, the payment is 6 million rubles, receipts - 6 million * 9% = 540 thousand rubles for the first two years. At the end of the third period, the amount of payments will be 6 million 540 thousand rubles. In this case, the GNI would be 9 percent.

Using 9% as the accrued interest, the NPV key figures are 0.

What influences the size of the rate of return?

The internal rate of return of an investment project depends on the size of payments and receipts, as well as on the duration of the project itself. Net present value and the rate of return are interrelated. The higher the coefficient, the lower the NPV will be, and vice versa.

However, there may be a situation where the relationship between NPV and the internal rate of return is difficult to keep track of. This happens when looking at several alternative financing options. For example, the first project may be more profitable at one rate of return, while the second project can generate more income at a different rate of return.

Internal interest

When calculating manually, it is accepted that the internal interest is determined by interpolating the nearby positive and negative current values. At the same time, it is desirable that the calculation percentages used differ by no more than 5%.

Example. What is the internal percentage of a series of payments?

1. Determine the calculated percentages that lead to negative and positive present values. The closer the current values ​​are to zero, the more accurate the result.
2. Determine the percentage using an approximate formula (linear interpolation).

The formula for calculating internal interest is as follows:

Vp = Kpm + Rkp * (ChTSm / Rchts), where

Bn - internal interest;

• Кпм - lower calculation percentage;
• Ркп - the difference between lower and higher calculation interest;
• ChTSm - net present value at a lower calculation percentage;
• Rchts - the absolute difference in current values.

From the data in the table, you can calculate the value of the internal percentage. The discounted flow of payments is calculated by multiplying the discount rate by the size of the flow of payments. The sum of the discounted cash flows is equal to the net present value. The internal percentage in this example is:

13 + 1 * (207 242 / (207 242 + 69 607)) = 13,75%

Interpretation of internal interest

A certain internal percentage can be interpreted:

1. If the internal interest is greater than the specified calculation interest p, then the investment is evaluated positively.
2. If the internal and calculated interest are equal, it means that the invested capital is returned with the necessary increase, but additional profit is not created.
3. If the internal interest is lower than p, then there is a loss of interest, because the invested capital, if used alternatively, would receive more accumulation.
4. If the internal interest is below 0, then there is a loss of capital, i.e. the invested capital is only partially returned from investment income. There is no accrual of interest on capital.

The advantage of internal interest is the fact that it does not depend on the volume of investment and thus is suitable for comparing investments with different investment volumes. This is a very big advantage over the present value method.