## Accumulated Value

This function allows to determine the future value of an available amount of money, based on an assumed rate of return periodicity, period of accumulation, and interest accrual. The unit accumulated amount is a basic compound interest function that allows you to determine the future value for a given period, interest rate, and known amount in the future.

The future value accumulation factor is based on compound interest. Compound interest is a geometric relationship between the initial deposit, the interest rate, and the accumulation period. Thus, compound interest implies that interest is accrued not only on the main sum of the deposit, but also on the corresponding sum of the accrued interest. It is possible only in case of reinvestment of accrued interest saving it on the deposit. The simple interest technique assumes an arithmetic relationship between the amount deposited, the interest rate and the period in which the money is kept on deposit. Consequently, the simple interest accrual is assumed only once at the end of the deposit contract term.

Sometimes in calculations businessmen are faced with the task of determining the number of accrual periods, after which the originally deposited amount doubles. However, accumulation can occur not only once a year, but also more often, for example once a quarter, once a month. The more often interest is accrued, the greater the accumulated amount. The present value of a unit makes it possible to determine the present value of the amount, the amount of which is known in the future for a given interest rate period. It is a process completely inverse to compound interest, which is called discounting. An annuity is a cash flow in which all amounts are equal and occur at the same intervals. Therefore, an annuity is a cash flow represented by equal amounts. An annuity can be both payments and receipts. An annuity that occurs at the end of a period is called a regular annuity. When calculating, you can determine the present value of a contribution that provides a future flow of a given equal income for a known number of periods and interest rate.

To determine the present value of an advance annuity, you must trace the cash flow.

Since the first annuity coincides in time with the deposit of the principal contribution, it should not be discounted. All subsequent annuities are discounted as usual, but the discount period will always be one less. Consequently, the advance annuity factor corresponds to the regular annuity factor for the previous period, to which one is added. Thus, the advance annuity factor occurring 7 times at a discount rate of 12% is found as follows:

- First determine the advance annuity factor for period 7-1=6, discount rate 12%. For example its value is 4.1114.
- Then calculate the upfront annuity factor:
- 4,1114+1,0=5,1114

Assessment of the investment attractiveness of real estate is related to the possible differentiation of discount rates, depending on the level of risk of certain real estate operations. Accounting for these differences requires the analyst to apply the appropriate discount rates.

## Present Value

Present value (PV) is the return on investment to date. In simple terms, present value is the amount of money that an investor will receive in a future period, converted into today’s value. In addition to the concept of “present value,” there is such a term as “net present value” in economics. NPV is the total financial turnover from investments translated into value at the time of the analysis.

Net present value differs from PV in that the former takes into account the initial financial investment. That is, the amount that the company has invested to generate income is subtracted from the future value of the asset, reduced to its present value. The present value is calculated to determine the amount of cash that an investor will receive in the future in value as of the date of analysis (Belu, 2020). For example, today, the company has invested 1 million dollars in the development of the project. It is unknown how much the company will receive from the implementation of the program. However, this value can be calculated using the formula for calculating present value.

As a rule, in the future, the money has a very different price compared to today. And the peculiarity of the present value calculation is the fact that the analysis reflects the amount of income in today’s value. Calculation of present value is quite different from the calculation of future returns. In order to calculate present value, you must first find the future rate of return. It is from this amount that the analyst doing the analysis starts (Belu, 2020). The future value is then adjusted to today’s value by discounting. That is, when calculating the COP figure, it is first necessary to understand how much the investor wants to earn from the investment. Then it is possible to calculate by means of an arithmetic formula. The result of the analysis is the amount of money the company should have today to get the planned amount of income.

Here is an example of calculating present value based on the following conditions. The company plans to earn $1 million in income from depositing money for 5 years. The bank offers two options for calculating interest: compound (12% per annum) and simple (10% per annum). Before concluding the contract, it is necessary to choose the optimal method. To do this, the present value is calculated by taking compound and simple interest. First of all, it is important to determine the present value of the method of simple interest:

- $1 million / (1 + 0,1) 5 = $951465, 68.

Thus, in 5 years, to get income of $ 1 million at 10% per annum, you need to invest $ 951465, 68. Now find the present value of the method of compound interest. Suppose interest is accrued each month, there are no replenishments:

- $1 million / (1 + 0,12 / 12) 5 * 12 = $550375.73.

It turns out that investing 550375,73 dollars at a compound interest rate of 12% per annum, after 5 years, the company could have an income of $ 1 million. Thus, it is much more advantageous to put money on deposit at compound interest, because if individual receive the same income over a similar period of time to invest a lot less money. To calculate the net present value, person must subtract the calculated COP from the expected income:

- NPV by the simple interest method: $1 million. – $951465.68. = $48534.32;
- NPV by the compound interest method: $1 million. – $550375,73 = $449624,27.

## Internal Rate of Return

The rate of interest that ensures that all present values aspire to or are equal to zero is called the internal rate of return. It refers to the investor’s ability to return the investment that was made at the beginning, but no more. The investor needs to find out how to control and calculate the internal rate of return (Belu, 2020). A project is considered effective and possible for the adoption only if the IRR of the project is more.

In a detailed analysis, it becomes clear that this implies that the borrowed money will bring revenue only if the credit rate is less than the investment rate. The fact that the cost of capital will bring less income, the investment project assumed above. As an example, consider a bank that makes loans at 14% per annum. This loan is taken for a project, which in turn should bring 20% per annum income. If all the conditions are met, the project is considered profitable. However, if the calculations are wrong, or the internal rate of return is less than 14%, then the project is considered unprofitable, because the debt to the bank will be more than the income from the project.

The bank itself does exactly the same, namely, he attracts money from the public, say, at 10% per annum, and gives loans at 20% per annum. As long as the rate on deposits taken by the bank is less than the rate on loans issued by the bank, the bank will live on this difference. By calculating the IRR, you can find out the upper allowed level of the cost of borrowed capital to be invested. If the cost of capital is higher than the project’s internal rate of return, the project will make a loss. If the cost of capital for the company is lower than the IRR of the project, the company will, in a sense, operate like a bank – living on the difference between the interest rates of bank lending and the return on investment.

It is important to analyze an example, where an individual has $6,000,000 available. Right now he can make a term deposit in the bank for, say, three years. The amount is large, so you need the most reliable bank and a stable bank. The institution of choice is currently offering a rate for deposits over $2 million for three years of 9.0% per annum without capitalization and 10.29% per annum with monthly capitalization. Since interest will be withdrawn at the end of each year, it will be a deposit without interest capitalization and the rate will be 9 percent per annum. At the end of each year an amount equal to $6,000,000*0.09 = $540,000 could be withdrawn. At the end of the third year, the deposit can be closed, withdrawing the interest for the third year and the principal amount of $6 million. A deposit in a bank is also an investment project, because the initial investment is made first, and then the cash inflows from our project are collected. A bank deposit is a financial instrument, and the easiest way to invest available to the average person. Since it is an investment project, it can be calculated its internal rate of return. The internal rate of return on a bank deposit is equal to the interest rate on that deposit, which is 9%. Therefore, such an investment project would be profitable at any deposit rate. But taking a $6,000,000 from one bank and depositing that money in another bank at a profit will not work: the loan rate will always be knowingly higher than the investment rate.

## Reference

Belu, R. (2020). *Building electrical systems and distribution networks. An introduction*. CRC Press.