The essence of estimating the value of a profit-making enterprise is that the current value of the profit that will be received in the forecast period is determined. The sum received tomorrow is worth less than the sum received today. This is due to the fact that, firstly, money brings income over time; and secondly, - inflationary processes devalue the soum. To determine the current value of tomorrow's sum, it is necessary to carry out appropriate calculations.

The six functions of money associated with the use of compound interest are discussed below, which the assessor should be aware of and constantly use in the practice of valuation.

Let us briefly characterize the main concepts encountered in this chapter.

Cash amounts. When evaluating the value of an enterprise that brings net income, it is important to determine the amounts of money that will be invested in it and received from these investments in the course of the operation of the enterprise. Determining the size of these amounts of money allows us to conclude whether these investments will provide a positive rate of return at which cash inflows exceed their outflows to cover future costs.

Time. The most precious thing in this world is time - it cannot be returned. The capital invested in the business earns interest over time, which, in turn, is used to earn even more interest. Time is measured in periods or intervals that make up a day, month, quarter, year, and so on.

Risk. Investment risk is understood as the uncertainty in obtaining net income from invested investments.

Income rate. The rate of net income from investments is the percentage of net income to invested capital. The rate of return involves an estimate of the amount of expected net income and the time of their receipt. The rate of return on investment is often referred to as the rate of return. Of the various options for investment projects, the one for which the rate of return is the highest is selected (if the experts are guided by economic criteria). If the rates of return of the two projects are the same, the project with the lower risk is selected. To select an investment option, the rates of return and risks corresponding to these options are compared. Only after analyzing these comparisons can we conclude on the choice of investment option.

net income. Net income is defined as the sum of net profit received after paying taxes and other obligatory payments and depreciation.

Annuity (usual) - a series of equal payments, the first of which is made in one period, starting from now, that is, the payment is made at the end of the periods under consideration.

Compound interest. Compound (cumulative) interest means that the interest received, put on the deposit along with the initial investment, becomes part of the principal amount. In the next period of time, along with the initial deposit, he himself brings interest. Simple interest does not imply income from interest. Special tables of six functions of the monetary unit (Appendix 1) help expert appraisers to make calculations using compound interest. The tables consist of six columns, which contain the values ​​obtained from the six functions of the monetary unit.

The first function is the accumulation of the amount of the monetary unit. The second function is the accumulation of a monetary unit over a period. The third function is the reimbursement fund factor. The fourth function is the current value of the monetary unit. The fifth function is the current value of the annuity. The sixth function is the contribution to the depreciation of the monetary unit. The calculation procedure and the use of the six functions of the monetary unit are discussed next.

5.1. The first function of compound interest

(future value of the currency - column 1)

When calculating the rate of return on investments, as the main criterion for choosing an investment project, the effect of compound interest is used, that is, calculation and accounting for the invested interest.

Cash in the examples in this tutorial is measured primarily in dollars. This allows not to take into account inflationary processes in the economy and simplify the calculations.

It is assumed that $100 is deposited in a special account and generates an annual income that accumulates. In the first year, $100 will earn $10 as interest (10% of $100 = $10). At the end of the year, the balance on the special account will be (PO dollars QOO dollars + 10 dollars = 110 dollars). If further the entire amount of 110 dollars is on deposit during the second year, then by the end of the second year the interest on it will be already 11 dollars (10% of BUT dollars = 11 dollars). If the entire balance remains on deposit, then by the end of the fifth year, the balance will be $161.05. At a simple interest of 10%, the annual income will be $10. $10 = $150). The difference from different forms of deposit was $11.05.

Due to the fact that compound interest functions are often used in calculating cash flows and in estimating the value of enterprises, it is necessary to get acquainted with special tables of six functions of the monetary unit, containing pre-calculated elements (separate multipliers) of compound interest. The calculation of compound interest in a special table is carried out according to the following formula:

Where: S t- deposit amount after periods, if $1 is invested;

1 - one dollar; i- periodic interest rate; t- number of periods.

If an investor knows from the table how much one dollar will cost in 10 years with an annual accumulation of 10%, then he will know how much the amount invested by him will cost by the end of 10 years, for example, $ 5,000. For this, the cost of $ 1 to the end of the 10-year period, taken in the special compound interest table (column 1), is multiplied by $5,000 (2.594-5,000 = $12,970).

Accumulation of funds can occur more frequently than a year: daily, monthly, quarterly or every six months. With more frequent accumulation of funds, the effective interest rate decreases. The calculation is made according to the main formula with its certain adjustment, the number of years ( i), during which accumulation occurs, is multiplied by the frequency of accumulation during the year (if accumulation is carried out once a quarter, then by 4, if once a month, then by 12), and the nominal annual interest rate is divided by the frequency of accumulation "

5.2. The second function of compound interest

(current value of the currency- column 4)

The current value of a monetary unit (reversion value, V) is the reciprocal of the accumulated amount of the unit:

The present value of a currency is the current value of one dollar that will be received in the future.

The present value ratio of a currency is used to estimate the present value of a known (or predicted) one-time cash inflow, given a given percentage (including a discount rate).

Tomorrow's monetary unit costs less than it costs today, and how much depends, firstly, on the time gap between the outflow and receipt of funds, and secondly, on the value of the required interest rate (discount rate).

If the discount rate is 10%, then the $100 we receive in a year has a present value of $90.91. To check, we will perform the reverse procedure. If an investor today has $90.91 and can receive 10% within a year, then the income received from interest will be $9.09. In this case, in a year the balance will increase to $100 (90.91+ 9.09=100)

The connection between the calculations carried out and the valuation of enterprises is as follows. Suppose an investor needs to determine how much he needs to pay today for the enterprise being evaluated in order to receive an income of 10% per annum from it, and in two years to sell it, for example, for $ 10 million. If the investor is going to receive 10% on the invested capital, then the amount he can offer for the venture today is $8.264 million.

The frequent use in practical calculations of the coefficient of the current cost of a unit led to the development of special tables with which you can quickly find the desired coefficient of the current cost of a unit (column-4)

In the case of discounting more frequently than one year, the nominal (annual rate) of the discount is divided by the frequency of intervals, and the number of periods in a year is multiplied by the number of years. The number of periods in a year is assumed to be either 4 or 12 if the interval is a quarter or a month, respectively.

5.3. The third function of compound interest

(current value of a cash single annuity - column 5)

This function of money reveals the present value of an ordinary annuity, that is, the present value of a series of equal payments.

This situation can arise if the owner leases the assets of the enterprise and wants to receive an annual rent of $100,000 over the next 4 years. At a 10% discount rate, the present value of the first lease payment of $100,000 in a year is $90.91 thousand ($100,000 - $0.9091=90.91 thousand), payment - 82.64 thousand dollars (100 thousand dollars - 0.8264 = 82.64 thousand dollars), the third lease payment - 75.13 thousand dollars, the fourth - 63.30 thousand dollars. Thus, the present value of the $100K lease payments over the next 4 years at a 10% discount rate is $316.98K. The latter amount is the fair current equivalent of $100K annual income over the next 4 years from the lease of the enterprise.

For the practical use of an ordinary annuity, special tables have been developed. The phenomenon of ordinary annuity is also called the Inwood factor after the American scientist William Inwood (1771-1843), who discovered this phenomenon.

The Inwood factor (a) is calculated using the following formula:

The present value of an annuity (a i) can be calculated as the sum of the present values ​​of $1 over a given period of time:

To build a table of ordinary annuity, it is necessary to add up the data on the current cost of a unit for the corresponding number of years.

If periodic payments are received more than once a year, the nominal (annual) interest rate must be divided by the number of periods in a year. The total number of periods is equal to the number of years times the number of periods in a year.

If the owner agrees with the tenant that he (the tenant) will make equal advance payments according to the following scheme: the first payment immediately after signing the contract, and subsequent, equal payments after a certain period, then such payments are called advance annuity.

With an advance annuity, the first payment is not discounted, since it is paid immediately, subsequent receipts are discounted: the second payment is discounted using the factor of the present value of the unit for the first interval, which can be taken from special compound interest tables (column-5). To convert an ordinary annuity into an advance one, it is necessary to add one to the factor of an ordinary annuity shortened by one period. When adding a unit, the first receipt is taken into account, which is carried out immediately after the signing of the contract. Thus, when reducing the cash flow for one period, the present value of the remaining payments is taken into account.

Example. The rent for the use of the company's property is $100,000 and is paid under the contract for 4 years at the beginning of each year. The current value of the advance annuity at a discount rate of 10% is 348.68 thousand dollars, and is distributed as follows: the present value of the first payment is 100 thousand dollars, the second - 90.91 thousand dollars, the third - 82.64 thousand. dollars, the fourth - 75.13 thousand dollars.

Income from the ownership of an enterprise can be received: 1) in the form of a cash flow from rental payments for the leased property of the enterprise or from profits; 2) in the form of a one-time proceeds from the sale of the company's assets. Two different compound interest factors are used to value these types of income: for cash flow, the annuity's present value factor is used; for a one-time income from a sale, the present value factor of the unit.

Example. For 25 years, at the end of each year, the enterprise brings to the owner a profit equal to 65 thousand dollars. The owner decided to sell the enterprise for 500 thousand dollars. The discount rate is 12%. To assess the income from the profit of the enterprise fro a special table of compound interest (column-5) determine the current value of the annuity. It amounts to 7.8431 at a discount rate of 12% and a duration of 25 years. It will amount to $509,804.

To estimate the current value from the sale of the enterprise in 25 years, we use the factor of the current unit cost (column-4). It is equal to 0.0588. Multiplying the received income from the sale of the enterprise (500 thousand dollars) by the factor of the present value of the unit (0.0588), we get the present value of the income from the sale of the enterprise (29.411 thousand dollars). Then the total current value of the company's assets is estimated at 539.215 thousand dollars. This example uses two compound interest factors: the present value of a unit and the present value of a regular annuity.

A situation is possible when the income from the sale of an enterprise can be more or less than 500 thousand dollars, that is, there is uncertainty. This uncertainty can be taken into account by using a discount rate of 15% instead of 12% for income from sales, for example. In this case, the estimated current value of the company's assets will be:

$65,000 x 7.8431 = $509,802

$500,000 x 0.0304 = $15,200

$525,002

5.4. The fourth function of compound interest

(contribution for the depreciation of the monetary unit- column-6)

A currency depreciation deposit is a regular periodic payment to repay an interest-bearing loan. This is the reciprocal of the present value of the annuity.

Amortization in this case is the repayment (reimbursement, liquidation) of debt over a certain period of time. The loan amortization contribution is mathematically defined as the ratio of one payment to the original principal amount of the loan. The unit depreciation contribution is equal to the obligatory periodic payment on the loan, which includes interest and the repayment of a part of the principal amount. This allows you to repay the loan and interest on it within a specified period.

As shown above, $1 due at the end of each year for 4 years has a present value of 3.1698 at 10% per annum. The first dollar will cost $0.90909, the second $0.8264, the third $0.7513, the fourth $0.6830. The four-year total is $3.1698 (0.90909 + 0, 8264 + 0.7513 + +0.6830 » 3.1698). This is the present value of the annuity.

The amount of depreciation for depreciation of a unit is equal to the reciprocal of the present value of the annuity, that is, the depreciation fee of $1 is the reciprocal of $3.1698. With a loan of $3.1698 at 10% per annum, the annual payment to repay it over 4 years is 1 dollar

The mathematical ratio of one payment to the initial annual amount of the loan, that is, the loan amortization contribution, is

This value shows the size of the periodic payment to repay the debt on the loan 3.1698 dollars. Thus, in order to fully repay the debt - its initial amount and accrued on the balance of 10% per annum for each dollar of the loan at the end of each year for 4 years - $0.315477 must be paid.

The higher the interest rate and/or the shorter the amortization period, the higher the mandatory periodic installment must be. Conversely, the lower the interest rate and/or the longer the loan repayment period, the lower the percentage of the regular installment.

Each unit depreciation installment includes interest and a repayment of a portion of the original principal amount of the loan. The ratio of these components changes with each payment.

The practical use of the unit depreciation contribution factor led to the development of special tables that contain the value of this factor per dollar of credit or $100, etc. When compiling tables, a formula is used that is inverse to the formula for the present value of an annuity:

Where: RMT - unit depreciation contribution factor; i - periodic interest rate; t - number of periods; a is the present value of the annuity.

If the conditions for issuing loans provide for monthly or quarterly repayment for positions, then the nominal annual interest rate is divided by the frequency of interest calculation (by 12 or 4, respectively), and in order to determine the total number of periods, the number of periods during the year is multiplied by the total number years.

As mentioned above, over time, the amount of interest paid decreases, as the balance decreases (the percentage accrued on the balance), and the amount of the main payment increases.

5.5. The fifth function of compound interest

(accumulation of the monetary unit for the period - column 2)

The unit accumulation factor answers the question of what the value of a series of equal contributions deposited at the end of each of the periodic intervals will be at the end of the specified period. If we invest $1 over three years, then at a rate of 10% per annum, the dollar deposited at the end of the first year will bear interest for the next two years; a dollar deposited at the end of the second year will bear interest for the next one year; a dollar deposited at the end of the third year will earn no interest at all.

Example. The entrepreneur wants to save up a certain amount to buy a new machine. The machine costs $4,641.

He puts aside one dollar each year (at the end of the year), which brings a 10% annual return. By the end of the fourth year, he has accumulated the necessary amount ($4,641) and buys a machine.

The calculation of special tables of accumulation of a unit for the period S(ti i) is carried out according to the following formula:

The calculation results are placed in column 2 of the special compound interest table.

5.6. The sixth function of compound interest

(reimbursement fund factor - column 3)

The compensation fund factor indicates the amount that must be deposited at the end of each period (periodic deposit) so that after a given number of periods the account balance will be $1. This takes into account the interest received on deposits.

Example. To receive one dollar in four years at zero interest, you must deposit 25 cents at the end of each year. If the interest rate is 10%, then at the end of each year, only 21.5471 cents must be deposited. The difference between $1 and four deposits (4-21.5471 = 86.1884 cents), equal to 13.8116 cents (100 cents-861884 cents), is the interest earned on the deposits.

Example. Suppose an entrepreneur needs to save $4,641 over four years to buy a machine tool. How much money does he need to set aside every year at 10% per annum in order to buy a $4,641 machine in four years?

Answer: the annual contribution should be $1 (0.215471 4.641=$1).

In the special compound interest table (see Appendix 1), the compensation fund factor is in column 3.

The Reimbursement Fund factor indicates the amount that must be deposited in each period so that after a given number of periods, the balance reaches one dollar. This value is the reciprocal of the unit accumulation factor for the period (column 2).

The compensation fund factor is equal to a portion of the $1 depreciation contribution, which in turn consists of two terms: the first is the interest rate, the second is the compensation fund factor or the return on the invested amount.

Annex 1

Compound interest tables - six functions

monetary unit

Interest accrual - annual

Year	Future unit value	Accumulation of a unit per period	Reimbursement Fund Factor	Current unit cost	Present value of a single annuity	Unit depreciation contribution
1	1,06000	1,00000	1,00000	0,94340	0,94340	1,06000
2	1,12360	2,06000	0,48544	0,89000	1,83339	0,54544
3	1,19102	3,18360	0,31411	0,83962	2,67301	0,37411
4	1,26248	4,37462	0,22859	0,79209	3,46511	0,28859
5	1,33823	5,63709	0,17740	0,74726	4,21236	0,23740
6	1,41852	6,97532	0,14336	0,70496	4,91732	0,20336
7	1,50363	8,39384	0,11914	0,66506	5,58238	0,17914
8	1,59385	9,89747	0,10104	0,62741	6,20979	0,16104
9	1,68948	11,49132	0,08702	0,59190	6,80169	0,14702
10	1,79085	13,18079	0,07587	0,55839	7,36009	0,13587
11	1,89830	14,97164	0,06679	0,52679	7,88687	0,12679
12	2,01220	16,86994	0,05928	0,49697	8,38384	0,11928
13	2,13293	18,88214	0,05296	0,46884	8,85268	0,11296
14	2,26090	21,01507	0,04758	0,44230	9,29498	0,10758
15	2,39656	23,27597	0,04296	0,41727	9,71225	0,10296
16	2,54035	25,67253	0,03895	0,39365	10,10590	0,09895
»7	2,69277	28,21288	0,03544	0,37136	10,47726	0,09544
18	2,85434	30,90565	0,03236	0,35034	10,82760	0,09236
19	3,02560	33,75999	0,02962	0,33051	11,15812	0,08962
20	3,20714	36,78559	0,02718	0,31180	11,46992	0,08718
21	3,39956	39,99273	0,02500	0,29416	11,76408	0,08500
22	3,60354	43,39229	0,02305	0,27751	12,04158	0,08305
23	3,81975	46,99583	0,02128	0,26180	12,30338	0,08128
24	4,04893	50,81558	0,01968	0,24698	12,55036	0,07968
25	4,29187	54,86451	0,01823	0,23300	12,78336	0,07823
26	4,54933	59,15638	0,01690	0,21981	13,00317	0,07690
27	4,82235	63,70576	0,01570	0,20737	13,21053	0,07570
28	5,11169	68,52811	0,01459	0,19563	13,40616	0,07459
29	5,41839	73,63980	0,01358	0,18456	13,59072	0,07358
30	5,74349	79,05818	0,01265	0,17411	13,76483	0,07265
31	6,08810	84,80168	0,01179	0,16425	13,92909	0,07179
32	6,45339	90,88978	0,01100	0,15496	14,08404	0,07100
33	6,84059	97,34316	0,01027	0,14619	14,23023	0,07027
34	7,25102	104,18375	0,00960	0,13791	14,36814	0,06960
35	7,68609	111,43478	0,00897	0,13011	14,49825	0,06897
36	8,14725	119,12087	0,00839	0,12274	14,62099	0,06839
37	8,63609	127,26812	0,00786	0,11579	14,73678	0,06786
38	9,15425	135,90421	0,00736	0,10924	14,84602	0,06736
39	9,70351	145,05846	0,00689	0,10306	14,94907	0,06689
40	10,28572	154,76197	0,00646	0,09722	15,04630	0,06646

6 MONEY FUNCTIONS. COMPOUND INTEREST FORMULA

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

TERMS AND DEFINITIONS

Accumulation is the process of converting the present value of money to its future value, provided that the invested amount is kept in the account for a certain time, bringing periodically accrued interest.

Discounting is the process of converting cash receipts from investments to their present value.

Annuity payments (PMT)- this is a series of equal payments (receipts) separated from each other by the same period of time. Allocate If payments are made at the end of each period, then the annuity is ordinary, if at the beginning - advance.

current value(PV)(eng. Present value) - the original amount of debt or an estimate of the current value of the amount 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 the effectiveness of investments (rate of return), characterizing the rate of growth in value over the period.

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

Accumulation frequency per year (k) - frequency of interest accrual affects the amount of accumulation. The more interest is compounded, the greater the accumulated amount.

FORMULA NOTATION

FV is the future value of the monetary unit;

PV is the current value of the monetary unit;

PMT - equal periodic payments;

i - income rate or interest rate;

n is the number of accumulation periods, in years;

k is the frequency of savings per year.

6 CURRENCY FUNCTIONS

Compound interest formula - 1 function

The future value of the currency ( FV) - the accumulated amount of the monetary unit. The accumulated amount of the monetary unit shows how much the monetary unit invested today will be after a certain period of time at a certain discount rate (yield).

Interest accrual once a year:FV = PV* [(1+ i) n] or FV = PV *

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

Compound interest formula - 2 function

The current value of the currency (P V) or present value of reversion (resale) shows how much you need to have today in order to get 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 to a monetary unit today, which we expect to receive in the future after a certain period of time.

Interest accrual 1 time per year: PV = FV * or PV = FV *

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

Compound interest formula - 3 function

Present value of the annuity shows how much money today is equivalent to a series of equal payments in the future, equal to one monetary unit, for a certain number of periods at a certain discount rate.

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

Ordinary annuity:

Interest accrual once a year:

Interest accrual more than once a year:

Advance annuity:

Compound interest formula - 4 function

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

Accumulation - it is the process of converting the present value of money to its future value, provided that the invested amount will be in the account for a certain time, bringing periodically accrued interest.

Discounting - the process of converting cash receipts from investments to their present value.

1 function. Determine the future value of the monetary unit (the accumulated amount of monetary units)

FV - the future value of the monetary unit,

PV is the current value of the currency,

i - income rate,

n is the 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 put 10 thousand rubles into the account at 10% per annum.

2 function. Current value of the currency (current resale value)

Task. How much you need to invest today in an investment project in order to get 8 thousand rubles by the end of 5 years. The rate of return is 10%.

3 function. Determining the current value of an annuity.

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

Allocate ordinary and advance annuity. 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 a regular annuity is:

PMT - Equal Periodic Payments.

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

4 function. Accumulation of a monetary unit for 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 the account, bringing 12% per annum, by the end of the 5th year, if 10 thousand rubles are annually deposited into the account.

5 function. Contribution to the depreciation of the monetary unit.

This function is the reciprocal of the present value of a regular annuity.

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

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

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. Such a loan is called self-absorbing loan.

6 function. Considers the placement fund factor and is the inverse function of the accumulation of a unit over a period.

The following formula is used to determine the payment amount:

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

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transcript

1 The six functions of compound interest are not that hard! Volnova Vera Alexandrovna certified ROO real estate appraiser TEGoVA appraiser

2 Theory BASIC CONCEPTS PV current value (present value) FV - future value (future value) PMT - payment, installment, payment (payment) n - number of periods (year) i - interest rate for the period (annual) k number. accruals for the period (per year) Annuity - a series of even equal payments Self-absorbing loan repayment is made in equal installments for the entire loan term and includes a part of the debt and accrued interest With payments once a period and a rate for the period (i) (n) With annual payments and annual rate (k=1) (i = i) (n = n) With monthly payments and annual rate (k=12) (i = i/k) (n = nk) 2

3 Theory SCHEME OF SIX FUNCTIONS 3

4 Theory WHY THERE ARE SIX FUNCTIONS? 4

5 Theory BASIC FORMULA 1. Future value of a unit (compound interest; how much will cost what we have today) FV = PV (1 + i) n 4. Current cost of a unit (discounting; how much is today what we get in the future) function , reciprocal of the first one Annual or monthly interest calculation 5

6 Theory BASIC FORMULA 2. Future value of an annuity (accumulation of a unit for a period; accumulation of a unit for n periods) (how much we will receive in the future if we invest 1 in each period) 2.1. (usual) if payments at the end of each year (i = i) (n = n) 2.2. (upfront) if payments at the beginning of each year (i = i) (n = n+1) (-1) Annual or monthly interest 6

7 Reimbursement fund factor (how much to pay to get 1) Theory BASIC FORMULA 3. Reimbursement fund factor (periodic contribution to fund accumulation; how much to pay in each period to accumulate a known amount) inverse second function 5. Annuity present value (current value of a single annuity; how much is a series of future payments in each period today) 5.1. (usual) if payments at the end of each period (i = i) (n = n) 5.2. (upfront) if payments at the beginning of each period (i = i) (n = n-1) (+1) Annual or monthly interest 7

8 Theory BASIC FORMULA 6. Unit depreciation fee (periodic installment to repay the loan; what is the amount of payments in each period to repay the amount taken) inverse fifth function rate and monthly payments (n = nk) (i = i/k) 8

9 Theory HOW TO REMEMBER BASIC FORMULA 9

10 Theory TEST QUESTIONS 1. To compare the value of two cash flows that differ in size, period of existence and interest rate, it is necessary to calculate: A. total present value. B. total future value. 2. If the terms of accumulation are given by the annual interest rate, the period expressed in years and the frequency of accrual of interest more frequent than once a year, it is necessary to correct: A. the number of periods of accumulation. B. rate of return. V. both options. 3. The statement that the function "Periodic contribution to the accumulation of the fund" and "Periodic contribution to the repayment of the loan" are inversely related: A. is true. B. is incorrect. 10

11 Table 6 of compound interest functions ANNUAL CALCULATIONS % 11

12 Table 6 of compound interest functions MONTHLY ACCRUED % 12

13 Table 6 Compound Interest Functions ANNUAL ACCRUES % MONTHLY ACCRUES % Column 1. Future Unit Value Shows the growth of 1 denominator deposited when interest is accrued. Interest is calculated on the amount of the initial deposit and the previously received interest. Column 4. Current Unit Value Shows the present value of 1 de, which should be received at a time in the future. This factor is the inverse of the value in column 1. Column 2. Accumulation of one per period Shows the growth of the savings account, into which 1 deu is deposited at the end of each period. Money on deposit during the period brings interest. 13

14 Table 6 Compound Interest Functions Each periodic amount is paid at the end of each period. This factor is the inverse of the value in column 2. Column 5. Present Value of a Single (Regular) Annuity Shows the present value of a uniform stream of income. The first entry in this flow occurs at the end of the first period; subsequent receipts at the end of each subsequent period. Column 6. Unit Amortization Contribution Shows the equal periodic payment required to fully amortize a loan that pays interest. This factor is the inverse of the value in column 5. The depreciation contribution 1 is sometimes called the mortgage constant. 14

15 Table 6 of compound interest functions ALGORITHM FOR USING TABLES Select the table of annual or monthly accumulation. 2. Find a page with the appropriate interest rate. 3. Find the column corresponding to the determined factor. 4. Find the number of years on the left or the number of periods on the right. 5. The intersection of a column and a row (periods) gives a factor. 6. Multiply the factor by the appropriate principal or deposit. Annual: 6% to 30% from 1 year to 40 years Monthly: 8% to 15% from 1 month up to 360 months (30 years old) 15

16 EXAMPLE OF USING TABLES 1. To what amount will the contribution of 1 de. for 5 years at 10% per annum, with annual interest.? 2. To what amount will the contribution of 1 de. for 5 years at 10% per annum, with monthly interest? Table 6 of compound interest functions 16

17 Table 6 of compound interest functions EXAMPLE OF USING TABLES (solution) 1. To what amount will the contribution of 1 de. for 5 years at 10% per annum, with annual interest? FV-? PV=1; i = 10%; n = 5 years; k =1 According to tab. (column 1, annual): future value of the unit at 10% -5 years = 1.61 1*f = 1* 1.61 = 1.61 deu. 2. To what amount will the contribution of 1 de. for 5 years at 10% per annum, with monthly interest? FV-? PV=1; i = 10%; n = 5 years; k =12 (n*k = 5*12 = 60) (column 1 monthly): future value of unit at 10% -5 years = 1.6453 1*f = 1* 1.65 = 1.65 deu. 17

18 EXAMPLE OF USING TABLES 3. How much can you accumulate if you save 1 de. at the beginning of the period. for 4 years at 10% per annum, with annual interest? FV-? RMT = 1; i = 10%; n = 4 years; k \u003d 1 Table of 6 functions of compound interest According to tab. (column 2, annual): future value of unit at 10% -4+1 years = 6.1 1*f = 1* (6.1-1) = 5.1 deu. 18

19 Theory TEST QUESTIONS 1. If the cash flow occurs at different intervals, compound interest tables should be used: A. expedient. B. inappropriate. 2. The use of compound interest tables requires adjustment if the cash flow occurs: A. at the end of the period. B. at the beginning of the period. 3. To determine the current value of the amount known in the future, it is necessary: ​​A. The factor “Current unit cost” determined from the table is divided by the amount known in the future. B. The factor “Current unit cost” determined from the table is multiplied by the amount known in the future. B. Divide the amount known in the future by the factor “Current Unit Value” determined from the table. 19

20 Typical tasks Group Income approach 6 functions of the monetary unit Defined values ​​1. First function The future value of the unit (the accumulated amount of the unit; the accumulation of the unit for the period; the future value of the known amount) 1. the amount accumulated over the period 2. to what amount the contribution will grow 3. value of the object 4. what is the accumulated amount to be returned 4. Fourth function Present value of the unit (present value of the future known amount) 1. cost of the object, the purchase of which will cost X 2. how much to put in order to accumulate X 3. what is the price paid today , will allow you to receive income X% 2. Second function Future value of an annuity (accumulation of a unit for a period; accumulation of a unit for n periods; future value of a series of payments) 1. the amount accumulated through periodic payments (deposits) 2. the marginal value of the object when depositing period 3. the amount accumulated by the owner after n years from the lease of the object 20

21 Typical tasks Group Income approach 6 functions of the monetary unit Defined values ​​3. Third function Compensation fund factor (value of payment at a known future value) 1. how much you need to save in order to save up for the purchase of an object 2. how much you need to save in order to replace the element in n years 3. what amount to receive from the tenant in order to save up for the object 5. Fifth function Current value of a single annuity (accumulation of the amount over n periods; current value of a known series of payments) 1. the right to receive rental income from the object 2. how much the object cost in installments, if the annual contribution is known 3. what amount to put in order to receive annually opr. payment 6. The sixth function Unit depreciation contribution (amount of necessary payments that will pay for the return on investment and interest; the amount of payment to repay a known current amount) withdraw from the account if you know how much was due 21

22 Typical tasks Group Income approach 6 functions of a monetary unit Defined values ​​Tasks for two functions 1. How much to contribute annually in order to accumulate funds, the amount of which is known today 2. Will there be enough funds for an object, the price of which is known today, if certain payments are made 3. What is the price of a property that brings the same annual income, which will then be sold 4. How much to sell this property now, if the annual income from it is known 5. What is the current value of the rental stream 22

23 First function 1. What amount will be accumulated in 4 years if the rate of return is 12% per annum, and ruble is initially deferred? 2. You deposited 100 units of money in the Bank for 5 years at an annual interest rate of 10%. How much money will you withdraw from your account after 5 years? 3. The apartment was sold for 400 de, the money brings 15% of the annual income. What is the marginal value of real estate that can be bought in 10 years? 4. A loan of 150 million rubles was received. for a period of 2 years, at 15% per annum; interest accrual occurs quarterly. Determine the accumulated amount to be returned. 23

24 First function 1. What amount will be accumulated in 4 years, if the rate of return is 12% per annum, and ruble is initially deferred? Calculation formula: FV = PV (1+i) n FV -? PV = i = 12% n = 4 k =1 FV = * (1+0.12) 4 = *1.12 4 = *1.574 = rub. According to the table: the future value of the unit (1 number) at 12% - 4 years \u003d 1, * f \u003d * 1.574 \u003d rub. 24

25 First Function 2. You have deposited 100 units of currency in the Bank for 5 years at an annual interest rate of 10%. How much money will you withdraw from your account after 5 years? Calculation formula: FV = PV (1+i) n FV -? PV = 100 i = 10% n = 5 k =1 FV = 100*(1+0.1) 5 = 100*1.1 5 = 161 de or: According to tab. (1 count) future value of a unit at 10% -5 years = 1, *f = 100* 1.61 = 161 de 25

26 First function 3. The apartment is sold for 400 deu, the money brings 15% of the annual income. What is the marginal value of real estate that can be bought in 10 years? Calculation formula: FV = PV (1+i) n FV -? PV = 400 i = 15% n = 10 k =1 FV = 400*(1+0.15) 10 = 400*1.15 10 = 400*4.046 = 1618.4 under 15% -10 years = 4, *f = 400* 4.04556 = 1,618.22 de 26

27 First function 4. Received a loan of 150 million rubles. for a period of 2 years, at 15% per annum; interest accrual occurs quarterly. Determine the accumulated amount to be returned. Calculation formula: FV = PV (1+i/k) n*k FV -? PV = 150 i = 15% n = 2 k = 4 i/k = 0.15/4 = 0.0375 n*k = 2*4 = 8 FV = 150*(1+0.0375) 8 = 150* 1, \u003d 150 * 1.342 \u003d 201.3 million rubles. 27

28 Fourth function 1. Calculate the cost of an apartment, for the purchase of which in 5 years you will need 500 deu, provided that the money brings an income of 15% per annum. 2. What amount must be deposited for 3 years at 10% per annum in order to receive de? 3. The investor plans that in 4 years the value of the object will be 2000 deu. What price must be paid today if the rate of return in this market is 11%? 4. What is the present value of the money received at the end of the third year at 10% per annum with monthly interest? 28

29 The fourth function 1. Calculate the cost of an apartment, for the purchase of which in 5 years you will need 500 deu, provided that the money brings an income of 15% per annum. Calculation formula: PV -? FV = 500 i = 15% n = 5 k = 1 PV= 500 * 1/(1+0.15) 5 = 500* 1/1.15 5 = 500*1/2.011 = 500*0.497 = 248.5 de or: According to tab: current unit value at 15% -5 years = 4, *f = 500* 0.497 = 248.5 de 29

30 Fourth function 2. What amount should be put for 3 years at 10% per annum to get de? Calculation formula: PV -? FV = 1000 i = 10% n = 3 k = 1 PV= * 1/(1+0.1) 3 = 1000* 1/1.1 3 = 1000* 1/1.331 = 1000 * 0.751 = 751de or : According to tab: current unit cost under 10% -3 years = 0, *f = 1000* 0.751 = 751 de 30

31 Fourth function 3. The investor plans that in 4 years the value of the object will be 2000 deu. What price must be paid for an object today if the rate of return in this market is 11%? Calculation formula: PV -? FV = 2000 i = 11% n = 4 k = 1 PV = * 1/(1+0.11) 4 = 2000* 1/1.11 4 = 2000* 1/1.518 = *0.659 = 1318de or : According to tab: current value of a unit at 11% -4 years = 0, *f = 2,000* 0.659 = de 31

32 Fourth function 4. What is the present value of de. received at the end of the third year at 10% per annum with monthly interest? Calculation formula: PV = FV PV -? FV = 1000 i = 10% n = 3 k = 12 i/k = 0.10/12 = 0.00834 n*k = 3*12 = 36 PV = * 1/(1+0.00834) 36 = 1 000* 1/1, = 1,000* 1/1.349 = *0.742 = 742de or: According to the tab: current unit cost at 10% -3 years (monthly) = 0, *f = 1,000* 0.741 = 742 de 32

33 Second Function 1. In order to earn your pension, you decide to save 100 ye to the bank at the end of the year. How much money will you withdraw from the account after 5 years if the bank accrues 10% annually? 2. What is the marginal value of real estate that can be bought in 10 years if you set aside 400 deu annually. at 15% per annum? 3. The owner leases the property, receiving at the end of each year 1000 ye. The profitability of similar objects is 12%. How much will the owner accumulate after 4 years? 4. Determine the future value of regular monthly payments of 10 thousand de. for 4 years at a rate of 12% and monthly accumulation. 33

34 Second function 1. In order to earn your pension, you decide to save 100 ye to the bank at the end of the year. How much money will you withdraw from the account after 5 years if the bank accrues 10% annually? Calculation formula: FV -? RMT = 100 i = 10% n = 5 k = 1 FV = 100* (1.1 5-1)/0.10 = 100*(1.61-1)/0.10 = 100*6.1 = 610 ye. or: According to the table: the future value of an annuity at 10% -5 years = 6, * f = 100 * 6.10 = 610 ye. 34

35 The second function 2. What is the marginal value of real estate that can be bought in 10 years, if annually set aside 400 deu. at 15% per annum? Calculation formula: FV -? RMT = 400 i = 15% n = 10 k = 1 FV = 400*(1.)/0.15 = 400*(4.046-1)/0.15 = 400*20.307 = 8122.8 de. or: According to the tab: the future value of an annuity at 15% -10 years = 20, * f = 400 * 20.304 = 8 122.2 de. 35

36 Second function 3. The owner rents out the property, receiving at the end of each year 1000 ye. The profitability of similar objects is 12%. How much will the owner accumulate after 4 years? Calculation formula: FV -? RMT 1000 i \u003d 12% n \u003d 4 k \u003d 1 FV \u003d 1000 * (1.12 4-1) / 0.12 \u003d 1000 * (1.574-1) / 0.12 \u003d 1000 * 4.78 \u003d 4 780 or: According to the tab: the future value of an annuity at 12% - 4 years = 4, * f = 1000 * 4.779 = 4779 ye 36

37 Second function 4. Determine the future value of regular monthly payments of 10 thousand de. for 4 years at a rate of 12% and monthly accumulation. Calculation formula: FV -? RMT = 10 i = 12% n = 4 k = 12 i/k = 0.12/12 = 0.01 n*k = 4*12 = 48 FV = 10*(1,)/0.01 = 10* (1.612-1) / 0.01 \u003d 10 * 0.612 / 0.01 \u003d 10 * 61.2 \u003d 612 thousand de. or: According to the tab: the future value of the annuity at 12% - 4 years = 61.222 10 * f = 10 * 61.222 = 612.2 thousand de 37

38 Third function 1. Calculate the annual installment at 15% per annum for the purchase of an apartment in 10 years for 500 deu. 2. What is the same amount that must be set aside each year in a fund that brings 10% of annual income in order to replace the roof in the amount of 150 thousand rubles in 10 years? 3. You borrowed 1 million ye. for 5 years at 10% per annum, every year you pay only %. How much do you have to deposit at the end of each year to save up a million? 4. You want to buy a country house. The estimated cost of the future purchase is 70 thousand ye. How much should be deposited monthly in the bank at 10% per annum from wages (at the end of the month) so that after 3 years this dream will come true? 38

39 Third function 1. Calculate the annual installment at 15% per annum for the purchase of an apartment in 10 years for 500 deu. Calculation formula: RMT -? FV \u003d 500 i \u003d 15% n \u003d 10 k \u003d 1 RMT \u003d 500 * (0.15 / 1.) \u003d 500 * (0.15 / 3.045) = 500 * 0.049 \u003d 24.5 de. or: According to tab: compensation fund factor at 15% - 10 years = 0, *f = 500* 0.049 = 24.5 de. 39

40 Third function 2. What is the same amount that must be set aside annually in a fund that brings 10% of annual income in order to replace the roof in the amount of 150 thousand rubles in 10 years? Calculation formula: RMT -? FV = 150 i = 10% n = 10 k = 1 RMT = 150 * (0.10 / 1.1 10-1) = 150 * (0.10 / 1.593) = 150 * 0.0628 = rub. or: According to the tab: compensation fund factor under 10% - 10 years = 0, * f = 150 * 0.0628 = rub. 40

41 The third function 3. What amount is desirable to receive from the tenant in order to save up for an object that in 5 years will cost 1 million cu, with a deposit rate of 10% per annum? Calculation formula: RMT -? FV = 1 i = 10% n = 5 k = 1 RMT = 1 * (0.10 / 1.10 5-1) = 1 * (0.10 / 0.610) = 1 * 0.164 = ye. or: According to the tab: compensation fund factor at 10% - 5 years = 0.164 1 * f = * 0.164 = ye. 41

42 Third function 4. You want to buy a country house. The estimated cost of the future purchase is 70 thousand de. How much should be deposited monthly in the bank at 10% per annum from wages (at the end of the month) so that after 3 years this dream will come true? Calculation formula: RMT -? FV = 70 i = 10% n = 3 k = 12 i/k = 0.10/12 = 0.0083 n*k =3*12 = 36 RMT = 70 * 0.0083/(1+0.0083) 36-1 \u003d 70 * 0.0083 / 1, \u003d \u003d 70 * 0.0083 / 0.347 \u003d 70 * 0.0239 \u003d 1.673 thousand de. or: According to the tab: compensation fund factor at 10% - 3 years (monthly) = 0, *f = 70* 0.0239 = 1.673 thousand deu. 42

43 Fifth function 1. You have the right to receive from real estate for 5 years every year at the end of the year 1 million rubles. net income in the form of rental income. How much is this right worth today, assuming that the rate of return (discount rate) is 10%? 2. How much did the apartment cost, bought in installments for 10 years at 13% per annum, if the annual installment is 1000 deu? 3. What amount should be deposited at present in a bank that accrues 8% per annum, so that then, within 5 years at the end of the year, withdraw 25 thousand rubles each? 4. Determine the amount of the loan, if it is known that its repayment is paid monthly at 3 thousand de for 4 years at a rate of 10% per annum. 43

44 Fifth function 1. You have the right to receive from real estate for 5 years every year at the end of the year 1 million rubles. net income in the form of rental income. How much is this right worth today, assuming that the rate of return (discount rate) is 10%? Calculation formula: РV -? RMT = 1 i = 10% n = 5 k = 1 PV = 1 * (1-1 / 1.10 5) / 0.10 = 1 * (1-1 / 1.61) / 0.10 = 1 * (1-0.62) / 0.10 \u003d 1 * (0.38 / 0.10) \u003d 1 * 3.8 \u003d 3.8 million rubles. or: According to the table: the current value of a single annuity at 10% - 5 years = 3.79 1 * f = 1 * 3.79 = 3.79 million rubles. 44

45 Fifth function 2. How much did the apartment cost, bought in installments for 10 years at 13% per annum, if the annual fee is 1000 deu? Calculation formula: РV -? RMT = 1000 i = 13% n = 10 k = 1 PV = 1000 * (1-1 / 1.13 10) / 0.13 = 1000 * (1-0.294) / 0.13 = 1000 * (0.706 / 0 .13) = 1000*5.43 = de. or: According to tab: current value of a single annuity at 13% - 10 years = 5, * f = 1000 * 5.426 = de. 45

46 Fifth function 3. What amount should be deposited at the present time in a bank that accrues 8% per annum, so that then, within 5 years at the end of the year, withdraw 25 thousand rubles each? Calculation formula: РV -? RMT = 25 i = 8% n = 5 k = 1 PV = 25 * (1-1 / 1.08 5) / 0.08 = 25 * (1-0.681) / 0.08 = 25 * (0.319 / 0 .08) \u003d 25 * 3.988 \u003d 99.7 thousand rubles. or: According to the table: the current value of a single annuity at 8% - 5 years = 3.99 25 * f = 25 * 3.99 = 99.75 thousand rubles. 46

47 Fifth function 4. Determine the amount of the loan, if it is known that its repayment is paid monthly at 3 thousand de for 4 years at a rate of 10% per annum. Calculation formula: РV -? RMT = 3 i = 10% n = 4 k = 12 i/k = 0.10/12 = 0.0083 n*k =4*12 = 48 PV = 3 * 1-(1/1,)/0, 0083 = 3*1-(1/1.48)/0.08 = 3* (1-0.672/0.0083) = 3* 0.328/0.0083 = 3* 39.518 = 118.554 thousand de. or: According to tab (5th column): current value of a single annuity at 10% - 4 years (monthly) = 39.428 3 * f = 3 * 39.428 = 118.284 thousand de. 47

48 The sixth function 1. Calculate the annual installment to pay for an apartment purchased in installments for 500 deu for 10 years at 15% per annum taken for 20 years? 3. What amount can be withdrawn annually for 5 years from an account to which 7% per annum is accrued, if the initial deposit is 850 thousand rubles, provided that the amounts withdrawn are equal? 4. What should be the monthly payments on a self-absorbing loan of 20 thousand deu, provided for 5 years at a nominal annual rate of 10%? paid at 3 thousand de for 4 years at a rate of 10% per annum. 48

49 The sixth function 1. Calculate the annual installment to pay for an apartment purchased in installments for 500 de for 10 years at 15% per annum Calculation formula: RMT -? PV \u003d 500 i \u003d 15% n \u003d 10 k \u003d 1 RMT \u003d 500 * 0.15 / 1- (1 / 1.15 10) \u003d 500 * 0.15 / 1-0.247 \u003d 500 * 0.15 / 0.753 \u003d 500 * 0.199 \u003d 99.5 de. or: According to tab: unit depreciation fee at 15% - 10 years = 0, *f = 500* 0.199 = 99.5 de. 49

50 The sixth function 2. What amount must be paid annually to repay a loan taken to buy an apartment worth 30 thousand cu. at 10% per annum, taken for 20 years? Calculation formula: RMT -? PV \u003d 30 i \u003d 10% n \u003d 20 k \u003d 1 RMT \u003d 30 * 0.10 / 1- (1 / 1.1 20) \u003d 30 * 0.10 / (1-0.148) \u003d 30 * 0.10 / 0.852 \u003d 30 * 0.117 \u003d 3.51 thousand cu. or: According to tab: payment for depreciation of the unit at 10% - 20 years = 0.0, *f = 30* 0.117 = 3.51 thousand cu. 50

51 The sixth function 3. What amount can be withdrawn annually for 5 years from an account to which 7% per annum is accrued, if the initial deposit is 850 thousand rubles, provided that the withdrawn amounts are equal? Calculation formula: RMT -? PV \u003d 850 i \u003d 7% n \u003d 5 k \u003d 1 RMT \u003d 850 * 0.07 / 1- (1 / 1.07 5) \u003d 850 * 0.07 / 1-0.713 \u003d 850 * 0.07 / 0.287 \u003d 850 * 0.243 \u003d 206.55 thousand rubles. or: According to the tab: contribution for depreciation of the unit at 7% - 5 years = 0.0, *f = 850* 0.243 = 206.55 thousand rubles. 51

52 The sixth function 4. What should be the monthly payments on a self-absorbing loan of 20 thousand deu, provided for 5 years at a nominal annual rate of 10%? Calculation formula: RMT -? РV = 20 i = 10% n = 5 k = 12 i/k = 0.10/12 = 0.0083 n*k =5*12 = 60 RMT = 20* 0.0083/ 1-(1/1, ) \u003d 20 * 0.0083 / 1-1 / 1.642 \u003d 20 * 0.0083 / 1-0.609 \u003d 20 * 0.0083 / 0.391 \u003d 20 * 0.021 \u003d 0.42 thousand. or: According to the table (column 6): contribution for depreciation of the unit at 10% - 5 years (monthly) = 0, * f = 20 * 0.021 = 0.42 thousand de. 52

53 Two functions 1. The owners of the condominium plan to change the roof covering in 10 years. Today it costs Rs. It is expected that this operation will rise in price by 12% per year (at compound interest). At the end of each year, how much should they pay into the 10% account in order to have enough money to replace the roof by the time indicated? 2. The couple plan to make a long tour in 5 years. At the moment, such a tour would cost $1. The cost of travel annually rises in price by 10% (at compound interest). Will the spouses have enough funds for the planned tour if they deposit 1,920de at the end of each year into an account that brings 12% per annum? 3. The owner of a parking lot expects to receive an annual rental income of 60 thousand deu within 6 years. At the end of year 6, the car park will be resold for 1,000 deu. Discount rate from income 15%, from resale 12%. Calculate the current value of the object. 4. Rented real estate for 3 years brings at the end of each year 10 thousand de. Over the next 2 years, the annual income will be 12 thousand de. Expected annual return of 15%. After 5 years, it is assumed that the property will be sold for 200 thousand de. For what sum it is expedient to sell this object now? 53

54 Two functions 1. The owners of the condominium plan to change the roof covering in 10 years. Today it costs Rs. It is expected that this operation will rise in price by 12% per year (at compound interest). At the end of each year, how much should they pay into the 10% account in order to have enough money to replace the roof by the time indicated? Calculation algorithm 1. Determine the future cost of coverage (the current value is known) 2. Determine the payment (the future value is known) 54

55 Two functions 1. Task 1 action: The future value of the unit (1f) FV = * (1 + 0.12) 10 = * 1.12 10 = * 3.106 = rub. Action 2: Compensation fund factor (3f) RMT = * (0.10 / (1.1 10-1) = * 0.10 / (2.59-1) = * 0.10 / 1.59 = * 0.063 = RUB Or: According to table 1 item: future unit st.

56 Two Functions 2. The couple plans to make a long tour in 5 years. At the moment, such a tour would cost $1. The cost of travel annually rises in price by 10% (at compound interest). Will the spouses have enough funds for the planned tour if they deposit 1,920de at the end of each year into an account that brings 12% per annum? Calculation algorithm 1. Determine the future cost of the cruise (the current value is known) The future value of the unit 2. Determine the future value of payments (the payment is known) The future value of the annuity 3. Compare the future and accumulated amounts 56

57 Two functions 2. Task 1 action Future value of the unit (1f) FV = * (1 + 0.10) 5 = *1.1 5 = * 1.61 = de 2 action Future value of payments (2f) FV = 1 920 * (1.12 5-1)/0.12 = 1920*(1.762-1)/0.12 = 1920*0.762/0.12 = 1920*6.35 = de. 3 action Required de. Accumulated funds are not enough 57

58 Two Functions 3. The owner of a car park expects to receive an annual rental income of 60 thousand deu for 6 years. At the end of year 6, the car park will be resold for 1,000 deu. Discount rate from income 15%, from resale 12%. Calculate the current value of the object. Calculation algorithm 1. Determine the current value of payments (payment is known) Current value of payments 2. Determine the current value of the sale (future is known) Present value of the future unit 3. Sum the current values ​​58

59 Two functions 3. Task 1 action Present value of payments (5f) PV = 60* (1-1/1.15 6)/0.15 = 60*(1-1/2.313)/0.15 = 60*( 1-0.432) / 0.15 \u003d 60 * 0.568 / 0.1 \u003d 60 * 3.786 \u003d 227.16 thousand de. 2 action The current value of the future unit (4f) PV \u003d 1350 * (1 / 1.12 6) \u003d 1350 * 1 / 1.97 \u003d 1350 * 0.507 \u003d 685.8 thousand de. 3 action Sum of current values ​​227.8 = 912.96 thousand de 59

60 Two functions 4. Rented real estate for 3 years brings at the end of each year 10 thousand deu. Over the next 2 years, the annual income will be 12 thousand de. Expected annual return of 15%. After 5 years, it is assumed that the property will be sold for 200 thousand de. For what sum it is expedient to sell this object now? Calculation algorithm 1. Generate income streams for periods PMTn 2. Determine the number of period n 3. Determine the discount rate (general rate of return) i 4. Calculate the discount factor Kd 5. Calculate the present value for each period PVn and sum 6. Calculate the present value of the sale object (reversion) PV P 7. Calculate the current market value of the object by summing the income stream and the cost of the reversion. 60

61 Two Functions 4. Task The market value of the property is CU135,050 thousand. 61

62 Two functions 5. The annual rental payment for the first 2 years is 100 thousand rubles, then it is reduced by 30 thousand rubles. and persists for 2 years, after which it increases by 50 thousand rubles. and will continue for 2 more years. Discount rate i = 15%, payments are received at the end of each year. What is the present value of the rental stream? Calculation algorithm 1. Generate income streams by periods (PMT) 2. Determine the number of the period (n) 3. Determine the discount factor (discount factor) (Kdn) 4. Calculate the present value of income of each period (PVn) as a product: PVn * Kdn 5 Calculate the present value of the lease payments by summing the result over the periods (PVn * Kdn) 62

63 SUCCESS IN PASSING THE QUALIFICATION EXAM IN THE DIRECTION OF REAL ESTATE APPRAISAL! +7 (383)

Appendix 2. Tables of six functions of compound interest. The tables of six functions proposed in this section can be used to solve a wide range of problems involving calculations.

Financial Mathematics Profit and profitability (yield) As a result of investments, the invested amount increases and an income is generated that is conveniently measured in% ... Task. The firm has purchased a promissory note

Ministry of Education and Science of the Russian Federation Federal State Budgetary Educational Institution of Higher Professional Education "Tomsk State Architectural and Construction

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION FEDERAL STATE BUDGET EDUCATIONAL INSTITUTION OF HIGHER VOCATIONAL EDUCATION "ST.

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So, to determine the value of income-generating property, it is necessary to determine the current value of the money that will be received at some time in the future.

It is known, and in conditions of inflation much more obvious, that money changes its value over time. The main operations that make it possible to compare different-time money are the operations of accumulation (building up) and discounting.

Accumulation is the process of converting the present value of money to its future value, provided that the invested amount is held in the account for a certain time, bringing periodically accrued interest.

Discounting is the process of converting cash receipts from investments to their present value.

In valuation, these financial calculations are based on a complex process whereby each subsequent accrual of the interest rate is made on both the principal amount and the unpaid interest accrued in previous periods.

In total, six functions of the monetary unit based on compound interest are considered. To simplify the calculations, tables of six functions have been developed for known rates of return and the accumulation period (I and n), in addition, you can use a financial calculator to calculate the desired value.

1 function: The future value of the currency (the accumulated amount of the currency), (fvf , i , n).

If accruals are made more often than once a year, then the formula is converted to the following:

k- the frequency of savings per year.

This function is used when the current value of money is known and it is necessary to determine the future value of the monetary unit at a known rate of return at the end of a certain period (n).

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Rule of 72x

For an approximate determination of the capital doubling period (in years), it is necessary to divide 72 by the integer value of the annual rate before the capital move. The rule is valid for rates from 3 to 18%.

A typical example for the future value of a monetary unit is a task.

Determine how much will be accumulated in the account by the end of the 3rd

year, if today you put into an account that brings 10% per annum, 10,000

FV=10000[(1+0.1) 3 ]=13310.

2 function : Current unit value (current reversion (resale) value), (pvf , i , n).

The present value of a unit is the reciprocal of the future value.

If interest is calculated more frequently than once a year, then

An example of a task is the following: How much should be invested today in order to get 8000 in the account by the end of the 5th year, if the annual rate of return is 10%.

3 function : The current value of the annuity (pvaf , i , n).

An annuity is a series of equal payments (receipts) separated 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 the beginning - advance.

The formula for the present value of a regular annuity is:

PMT - Equal Periodic Payments. If the frequency of accruals exceeds 1 time per year, then

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

Typical example:

The dacha lease agreement is made for 1 year. Payments are made monthly for 1000 rubles. Determine the present value of the lease payments at a 12% discount rate if (a) payments are made at the end of the month; b) payments are made at the beginning of each month.

4 function : Accumulation of the monetary unit for the period (fvfa , i , n).

As a result of using this function, the future value of a series of equal periodic payments (receipts) is determined.

Payments can also be made at the beginning and at the end of the period.

The formula for an ordinary annuity is:

Typical example:

Determine the amount that will be accumulated in the account, bringing 12% per annum, by the end of the 5th year, if annually set aside 10,000 rubles a) at the end of each year; b) at the beginning of each year.

5 function : Monetary depreciation contribution (iaof , i , n) The function is the reciprocal of the present value of an ordinary annuity. The monetary unit depreciation contribution is used to determine the amount of an annuity payment to repay a loan issued for a certain period at a given loan rate.

Amortization is the process defined by this function, including the interest on the loan and the payment of the principal amount of the debt.

For payments made more frequently than once a year, the following formula is used:

The following task can serve as an example: Determine what 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.

6 function : Recovery fund factor (sff , i , n)

This function is inverse to the function of accumulating a unit over a period. The compensation fund factor shows the annuity payment that must be deposited at a given percentage at the end of each period in order to receive the required amount after a given number of periods.

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

For payments (receipts) made more often than once a year:

An example would be a task.

Determine what payments should be in order to have 100,000 rubles in an account that brings 12% per annum by the end of the 5th year. Payments are made at the end of each year.

The annuity payment defined by this function includes the principal payment without interest payments.