Compound interest functions. Theory of the Time Value of Money

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Compound interest functions. Theory of the Time Value of Money
Compound interest functions. Theory of the Time Value of Money

Whether you plan to invest your capital in a friend's business or in your own life, you need to accurately calculate the money that you will receive in the future. To do this, there is a concept that financiers call "compound interest." Of course, there are a large number of online compound interest calculators. However, in order not to get into a puddle, it is better to understand the method of calculating this indicator yourself. In order to help you with this, this article was written.

Theory of the Time Value of Money

initial investment
initial investment

According to one of the many economic concepts, money tends to depreciate over time. Today's deposit, which costs, say, $1,000, will cease to cost the same amount in 5-6 years.

But the value of money is affected not only by the time period. There are three main factors that can affect the real value of money capital:

  • time;
  • inflation;
  • risk.

Given what investing in itself involvesmaking a profit in the future, it becomes necessary to calculate what it will be in a given period of time. After all, when an investor invests in a certain enterprise, he must feel the difference between what he has invested and what he will receive. For this, two basic concepts of contribution are introduced: the current and future value of money capital.

Current value of money

The invested present value of the money supply is the future financial receipts, which are adjusted to the current time period, taking into account the established interest rate. Establishing the current value of money is characterized by a process called "discounting". Reverse to accretion, it helps determine how much money you need to invest today to get $10,000 in 6 years.

This simple arithmetic operation is performed by multiplying future cash flows by a discount factor.

discount coefficient
discount coefficient

Where: α-discount factor; r - discount rate divided by 100%; t - serial number of the year for which the calculation is made.

Future value of capital

The future value of an investment unit is the amount that is obtained as a result of investing the n-th amount of money on today's date after a specified amount of time and a certain interest rate. This method of calculating future income is called "accumulation". It is a movement from the present to the future. When taking into account the stipulated rate of the year, the year occursgradual increase in initial investment. Thus, the first capital investments increase their value over time. When considering investment projects, the interest rate plays the role of a profitability ratio for operations.

The following formula is used to determine future earnings from investments invested today.

Future arrivals
Future arrivals

Where: Co - initial investment; r - interest rate; n - the agreed investment period.

It was the accumulation method that led to the emergence of compound interest.

What is compound interest?

interest rate
interest rate

Let's imagine that you have invested 200,000 rubles at 12% per annum. For the first year, your profit will be 24,000 rubles: 200,000 + 200,00012%=224,000 rubles. However, according to the agreement, you do not take this money, but they are transferred to the category of a deposit and already in the second year the interest is charged not on 200,000 rubles, but on 224,000 rubles, etc.

Such a scheme, in which interest is charged on the profit received in the previous period, is called compound interest or capitalization.

This method works for both deposits and loans, if you do not plan to return money to the bank in the first few years. Moreover, according to the agreement, interest is accrued either every month, or quarterly, or once a year.

Compound Interest Functions

When conducting a variety of financial calculations, you often have to resort to solving problems of creating a cash flow with the availablecharacteristics and their value. To simplify the calculations, to standardize them, they use the derived compound interest functions that display the dynamics of changes in the cost of capital investments over the allotted time period.

There are 6 such functions in total:

  • The amount of future savings, taking into account the compound interest rate.
  • Annuity future value or accumulation of a unit over a period.
  • The present value of the annuity.
  • Reimbursement fund factor.
  • Partial payment for unit depreciation.
  • Reversion factor or current unit cost.

The volume of future savings, taking into account the compound interest rate

This compound interest function was discussed above when we talked about the future cost of capital and accumulation. When determining future income, the following are taken as the basis: the initial investment, the rate on a complex loan and the period for which the investment is provided.

Annuity value in the future

Allows you to determine the amount of increase in the savings account, which involves regular deposits of the depositor, on which interest is charged in the specified period of time.

Calculated using the following formula:

FVA=M((1 + r)n - 1 / r, where: FVA - future price of money; M - the amount of the permanent payment; r - loan rate; n - time period.

Thus, if you pay 1,500 rubles every month for three years at a rate of 15%, then after all payments, your future value of constant paymentswill be equal to 67,673 rubles.

Regular equal contributions

The compensation fund factor shows the amount of the contribution that must be made on a regular basis in order to receive the planned amount using compound interest by the end of the set period.

For the calculation, you must use the formula:

M=FVAr / ((1 + r)n - 1).

Like all cash flow formulas, this one is easily derived from the previous one.

Return on investment
Return on investment

If you decide after 6 years to buy an apartment, the cost of which is, relatively speaking, $1,000,000, then at a fixed annual interest rate of 15%, you need to pay $8,645 to the bank every month.

Reversion factor

Receiving a profit
Receiving a profit

This compound interest function is the inverse of the first one. The calculation is made according to the following formula:

PV=FV / (1 + r) , where: PV - initial contribution; FV - future receipt; r - interest rate; n - number of years (months).

This function gives an idea of how much you need to invest today in order to get a guaranteed profit under given conditions (period and percentage).

For example, the current value of 20,000 rubles, expected to be received after 4 years at an annual rate of 15%, will be equal to 11,435 rubles.

The present value of a regular annuity

Demonstrates the cost of regular payouts to date. First arrivalsare expected at the end of the first year, month, quarter, and subsequent - at the end of each subsequent time interval.

The following formula is used for calculation:

PVA=M(1 - (1 + r)-n) / r.

A simple example where this technique is used can be a situation in which it is necessary to set the amount of a loan given for a certain period of time, given the interest rate and monthly payments to the bank.

Partial payment for unit depreciation

Demonstrates the amount of the equal periodic payment required to fully amortize an interest-bearing loan.

The formula looks like this:

M=PVAr / (1 - (1 + r)-n).

A good example would be to determine the amount of the installment that must be repaid to the bank within the allotted time period so that the loan is repaid on time, taking into account the repayment of the principal and interest payments.