Unit of value and monetary unit. Fundamentals of the theory of the value of money over time. Standard Compound Interest Functions Future Unit Value

Value of money

Money is a commodity that has its own intrinsic value at the stage of the emergence and formation of market relations. Thanks to this, money played the role of a general value equivalent in the world of goods. Being in the form of paper money redeemable for gold, they are considered as signs of the value of a monetary commodity. Exchangeable paper money, which did not have its own internal value, represented in circulation the value of the weight fraction of gold, officially determined on the basis of a fixed state scale of prices. Modern cash must have relative value. As a result, they function in circulation as legal tender in that they are money declared by the state; their value is formed spontaneously under the influence of market forces.

Signs of the economic usefulness of money:

Having absolute liquidity, money can be exchanged for other goods;

Money is a convenient form of accumulating wealth, and storing it in this form requires a minimum of costs;

Money has unique property- ensuring connection between the present and the future.

The value of money is determined by its purchasing power, and the price of a particular monetary unit is its exchange rate.

The relative value of money as a medium of exchange is determined indirectly as its purchasing power; its value is compared with the cost of goods and services that can be bought with it. The dynamics of the value of money is determined by the dynamics of prices:

The value of money can be determined by one of the following indicators:

Based on the retail price index;

Based on the wholesale price index;

Through the GDP deflator (comparison of the nominal and real values ​​of GNP).

The relative value of money in the accumulation function, used in the form of financial assets (stocks, bonds, other securities), is determined by the interest rate, which is the fee for storing money in exactly one of the forms.

Functions of money

IN economic literature according to theory monetary relations The initial and central function in the system of monetary relations is the function of the measure of value, because it is this function that supplies the commodity mass with the necessary material to express its value. Cost, on the one hand, generates a measure function

value, on the other hand, manifests itself in the price of a product only on the basis of this function. A measure of value is a monetary unit "which is used to measure and compare the cost of goods and services. Based on the measure of value, a price is set, which is the monetary expression of the value of goods. The price depends, on the one hand, on the cost of goods, and on the other, on the value of the value itself monetary unit. The cost of goods may remain unchanged, but if the value of the monetary unit decreases, the prices of goods will rise, therefore, we are talking about the reverse proportional dependence prices and the value of the monetary unit. Money realizes its function as a measure of value through interaction with the price scale. The price scale is a purely technical function, that is, a counting function of money, reflecting the value of the commodity mass in monetary units. The scale of prices is associated with devaluation (the official decrease in the exchange rate of the ruble relative to another currency) and revaluation (increase in the exchange rate) of monetary units.

Money performs the function of circulation, therefore, it is a special commodity that can be exchanged for another commodity, and vice versa.

The amount of money needed in circulation (M) to perform the function of a medium of circulation is determined by the price of goods and services to be sold within a certain period of time:

Where r and- price of the i-th product; q i- quantity of the i-th product.

Each monetary unit is used more than once during the circulation process. Hence, the sum of the prices of goods must be divided by the value V - the average circulation number of each bill:

Consequently, the amount of money needed for circulation varies in direct proportion to the sum of the prices of goods and services sold and inversely proportional to the speed of circulation of money.

The features of the credit economy, that is, the realities of buying and selling goods on credit, with deferred payment, are reflected by the function of the means of payment. In this case, the means of circulation are not the money itself, but the obligations expressed in money. The following monetary payments are based on the use of the means of payment function:

Payments by bank transfer to enterprises, institutions, organizations for goods and services;

Salary;

Issuance and repayment of bank loans;

Settlements related to insurance, administrative and judicial obligations, etc.

Accumulation of value at the disposal of legal and individuals in the process of development of commodity production, the function of a means of accumulating money serves. The formation of savings accumulations causes certain expenses for their owners. During a period of inflation, cash turnover increases to 30 percent or more, and the function of a store of value is sharply reduced, because this leads to a loss from the depreciation of money. In accordance with this, the structure of money circulation changes, which is performed by various functions of money.

To determine the value of an investment project or property, it is necessary to determine the present value of money that will be received some time in the future. Under inflation, money changes its value over time. The main operations that make it possible to compare money at different times are the operations of accumulation (increase) and discounting.

Accumulation – This is the process of reducing the current value of money to its future value, provided that the invested amount will remain in the account for a certain time, earning periodically compounded interest.

Discounting – reduction process cash receipts from investments to their current value.

1 function. Let's determine the future value of a monetary unit (the accumulated amount of monetary units)

FV - future value of a monetary unit,

PV – current value of the monetary unit,

i – income rate,

n – number of accumulation periods in years.

Task. Determine what amount will be accumulated in the account by the end of 3 years, if today you deposit 10 thousand rubles into the account at 10% per annum.

2 function. Current value of a monetary unit (current resale reversion value)

Task. How much should you invest in today? investment project in order to receive 8 thousand rubles by the end of 5 years. Income rate is 10%.

3 function. Determining the current value of an annuity.

Annuity is a series of equal payments (receipts) spaced from each other by the same period of time.

There are ordinary and advance annuities. If payments are made at the end of each period, then the annuity is ordinary; if at first - advance.

The formula for the present value of an ordinary annuity is:

PMT – equal periodic payments.

Task. The rental agreement for the dacha is for 1 year. Payments are made monthly in the amount of 1 thousand rubles. Determine current value rental payments at a 12% discount rate. n = 12 (number of periods – months).

4 function. Accumulation of a monetary unit over a period. As a result of using this function, the future value of a series of equal periodic payments or receipts is determined.

Task. Determine the amount that will be accumulated in an account yielding 12% per annum by the end of the 5th year, if 10 thousand rubles are annually deposited into the account.

5 function. Contribution for depreciation of a monetary unit.

This function is the reciprocal of the present value of an ordinary annuity.

Depreciation is a process defined by this function and includes interest on the loan and payment of the principal amount.

Task. Determine what annual payments should be in order to repay a loan of 100,000 rubles issued at 15% per annum by the end of the 7th year.

An annuity can be either a receipt (incoming cash flow) or a payment (outgoing cash flow) to the investor. Therefore, this function can be used in the case of calculating the amount of an equal contribution to repay a loan with a known number of contributions and a given interest rate. This loan is called self-amortizing loan.

6 function. Considers the allocation fund factor and is the inverse of the unit accumulation function for the period.

To determine the amount of payment, the following formula is used:

Task. Determine what payments should be in order to have 100,000 rubles in the account at an annual rate of 12% by the end of 5 years.

Current value of a monetary unit – the second function of money. The idea is to give an estimate of the current value of the money that can be received at the end of a certain period at a given discount rate. Determined by the formulas:

a) when interest is calculated once a year:

b) when interest is calculated more often than once a year:

(7)

PV - present value, rub.;

FV – future value, rub.;

Current unit value factor;

k – number of accruals per year (period).

Problem 2 . Determine the current value of 5250 rubles, which will be received at the end of 6 years at a 12% rate. Accrual is quarterly.

Solution:

Answer: PV = 2609.09 rubles.

Accumulation of a monetary unit over a period - the third function of money. The economic meaning of this function is what amount will be accumulated in the account at a given rate, if one monetary unit is regularly deposited into the account over a certain period of time.

PMT – periodic equal payment.

1. Calculation of the future value of an ordinary annuity

a) when accrued at the end of each year:

(8)

b) for accruals made more often than once a year:

(9)

2. Calculation of the future value of the advance annuity (at the beginning of the year, month)

(10)

b) for payments made more often than once a year:

(11)

Task 3. Determine the amount that will be accumulated in an account yielding 12% per annum by the end of the 16th month, if you deposit 2,000 rubles into the account every month.

a) at the end of the month;

b) at the beginning of the month.

Solution:

a) formula (9)

b) formula (11)

Answer: a) FV = 34766.63 rubles.

b) FV = 34422.41 rub.

Compensation fund

Compensation fund – fourth function of money. This function shows how much needs to be deposited into the account regularly over a certain period of time in order for a given rate of income to have one monetary unit in the account by the end of this period.

a) for payments made once a year:

(12)

(13)

The compensation fund factor.

Problem 4 . Determine the amount of payments in order to have 20,000 rubles in an account earning 11% per annum by the end of 16 years. Payments are made:

1) annually k = 1,

2) monthly k = 12.

Solution:

2)

Answer: 1) PMT = 510.33 rub.

2) PMT = 38.47 rub.

Unit depreciation contribution

Depreciation contribution - the fifth function of money. Amortization in this case refers to the process of paying off debt over time. This function shows what annuity or equal payments should be to repay a loan of one monetary unit issued at a certain percentage for a certain period. The function is used to determine the mandatory periodic payments required to repay (repay) the loan within a specified period.

a) for payments made once a year:

(14)

b) for payments made more than once a year:

(15)

Problem 5 . A loan in the amount of 130,000 rubles was issued for 6 years at 15% per annum. Determine the amount of annuity payments. The loan is repaid monthly.

Solution:

Answer: PMT = 2748.85 rub.

Present value of the annuity

Present value of annuity – sixth function of money. The meaning is what, at a given discount rate, is the present value of a series of equal payments of one monetary unit over a certain period of time.

Annuity - a series of equal payments made over the same period of time, either regular or advance.

This function is the inverse of the depreciation function for the depreciation of a unit. Used to determine the present value of recurring payments received in the future over a specified period of time.

Calculating the present value of an ordinary annuity(payments are made at the end of the period).

a) for payments made once at the end of the year:

(16)

6 FUNCTIONS OF MONETARY UNIT. COMPOUND INTEREST FORMULAS

The theory of changes in the value of money is based on the assumption that money, being a specific product, over time change their value and, as a rule, depreciate. Changes in the value of money occur under the influence of a number of factors, the most important of which are inflation and the ability of money to generate income, provided they are wisely invested in alternative projects. The main operations that make it possible to compare money at different times are the operations of accumulation (increase) and discounting.

TERMS AND DEFINITIONS

Accumulation is the process of reducing the current value of money to its future value, provided that the invested amount is held in an account for a certain time, earning periodically compounded interest.

Discounting is the process of reducing cash flows from investments to their current value.

Annuity payments (PMT) is a series of equal payments (receipts) spaced from each other by the same period of time. Highlight If payments are made at the end of each period, then the annuity is ordinary; if at the beginning, it is an advance annuity.

Current value(PV)(English: Present value) - the original amount of debt or an estimate of the current value of a sum of money, the receipt of which is expected in the future, in terms of an earlier point in time.

Future Value (FV)(eng. Future value) - the amount of debt with accrued interest at the end of the term.

Rate of return or interest rate (i)(eng. Rate of interest) - is a relative indicator of investment efficiency (rate of return), characterizing the rate of increase in value over the period.

Debt repayment period (n)(eng. Number of periods) - the time interval after which the amount of debt and interest must be repaid. The term is measured by the number of billing periods, usually equal in length (for example, month, quarter, year), at the end of which interest is accrued regularly.

Frequency of savings per year (k) - frequency of interest calculation influences the amount of accumulation. The more often interest is calculated, the greater the accumulated amount.

NOTATION FOR FORMULAS

FV – future value of a monetary unit;

PV – current value of a monetary unit;

PMT – equal periodic payments;

i – income rate or interest rate;

n – number of accumulation periods, in years;

k – frequency of accumulations per year.

6 FUNCTIONS OF MONETARY UNIT

Compound interest formula - 1 function

Interest is calculated once a year:F.V. = PV* [(1+ i) n] or FV = PV *

Interest accrual more often than once a year: FV = PV * [(1+ i / k ) nk ]

Compound interest formula - function 2

Current value of a monetary unit (P V) or current value of reversion (resale) shows what amount you need to have today in order to receive an amount equal to a monetary unit after a certain period of time at a certain discount rate (yield), that is, what amount is equivalent today to the monetary unit that we expect to receive in the future after a certain period of time.

Interest is calculated once a year: PV = FV * or PV = FV *

Interest accrual more often than once a year: PV = FV *

Compound interest formula - 3rd function

Present value of the annuity shows what amount Money what is equivalent today is a series of equal payments in the future equal to one monetary unit over a certain number of periods at a certain discount rate.

Highlight ordinary and advance annuities. If payments are made at the end of each period, then the annuity is ordinary; if at the beginning, it is an advance annuity.

Ordinary annuity:

Interest is calculated once a year:

Interest accrual more often than once a year:

Advance annuity:

Compound interest formula - 4 function

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