Six functions of compound interest can be used in real estate valuations. The accumulated amount of the unit allows us to answer the question: “How much can the property be sold for based on its current market value and the expected growth of the latter at compound interest?" The accumulation of one per period shows how regular deposits will grow at compound interest. The recovery fund factor shows how much must be deposited periodically in order to accumulate $1 after a certain number of periods at compound interest. It shows what should be the annual rate required to recoup investments in a given asset.
The present value of a unit shows the present value of a sum of money to be received in a lump sum in the future, such as from the expected sale of land. The annuity factor shows the value of the flow Money, such as income received from rental properties or payments for mortgage loan. The unit amortization contribution factor determines the size of the periodic payment required to amortize the loan, including interest and principal payments.
Each of the six functions is based on compound interest, which means that the entire principal amount held in the deposit account must earn interest, including interest remaining in the account from previous periods. Moreover, interest is paid only on the funds in the deposit account, and not on interest or principal withdrawn from it.
The six compound interest functions can be used to solve almost all arithmetic problems associated with valuing income-producing real estate properties.
Money has a time value, i.e. A ruble received today is worth more than a ruble received tomorrow. And not only because inflation can reduce its purchasing power, but also because a ruble invested today will bring concrete profit tomorrow. Time value of money - important aspect when making decisions in financial practice in general and when evaluating investments in particular.
Calculation based on compound (cumulative) interest means that the interest accrued on the initial amount is added to it, and interest is accrued in subsequent periods on the already accrued amount. The process of capital accumulation in this case occurs with acceleration. It is described by a geometric progression. The mechanism for increasing the initial amount (capital) using compound interest is called capitalization. In financial and economic terms, capitalization is defined as rate of return on invested capital. When assessing real estate and investments, this term takes on a slightly different meaning.
There are annual capitalization (the interest payment is calculated and added to the previously increased amount at the end of the year), semi-annual, quarterly, monthly and daily. There is also the concept of continuous compounding, which in its meaning is very close to daily compounding.
The calculation of the accrued amount at compound interest is carried out using the formula:
cash payment rent debt
where S is the accumulated amount;
P - the initial amount on which interest is calculated;
i - rate compound interest, expressed as a decimal fraction;
n is the number of years during which interest accrues.
The value is called the compound interest multiplier. It shows how much one monetary unit will increase when interest increases on it at rate i for n years.
However, in most cases, it is not the quarterly or monthly rate that is indicated, but the annual rate, which is called the nominal rate. In addition, the number of periods (t) of interest accrual per year is indicated. Then the formula is used to calculate the accrued amount:
where i is the nominal annual interest rate;
t - the number of interest calculation periods per year;
n - number of years;
tp - the number of interest periods for the entire term of the contract.
Using formulas (3.1) and (3.2), we carried out a discrete increase in interest, i.e. interest was accrued annually, quarterly or monthly. Continuous compounding means that interest is compounded over the shortest possible period of time. Although it is understood that this period will be infinitely short, the most accurate approximation of continuous compounding is daily compounding. In this case, formula (3.2) can be used to determine the accumulated amount. So, with an annual rate of 10% and a year length of 360 days (a similar year length is accepted in banking calculations in a number of countries) with daily interest accrual.
The term “discounting” is used very widely in financial practice. It can be understood as a method of finding the value P at a certain point in time, provided that in the future, when interest is calculated on it, it could amount to the accrued amount S. The value P, found by discounting the accrued value S, is called the modern, current or reduced value. With the help of discounting, the time factor is taken into account in financial calculations. The current value is the reciprocal of the accumulated value, i.e. Discounting and the discount rate are opposite to the concepts of “accumulation” and “interest rate”. For example, if in a year you should receive your own bank deposit 1100 rubles, and the bank accrued at the rate of 10% per annum, then the current value of your deposit is 1 thousand rubles.
Since the current value is the reciprocal of the accumulated amount, it is determined by the formula:
where is the discount factor. It shows the current value of one monetary unit, which should be received in the future.
When interest is calculated th times a year, the current value is calculated using the formula:
where is the discount factor.
When considering a modern value, it is necessary to pay attention to two of its properties. One of them is that the interest rate at which discounting is carried out and the modern value are inversely related, i.e. the higher the interest rate, the lower the current value, other equal conditions.
Also, the current value and payment term are inversely related. As the payment term (n) increases, the current value will become less and less. The limit of values of the modern value (P) with the payment term (p) tending to infinity will be:
With very long payment terms, its current value will be extremely insignificant. So, for example, if someone decides to bequeath to his descendants to receive an amount of 50 million rubles in 100 years, then for this he just needs to put 22.72 thousand rubles at 8% per annum.
As the value t (the number of interest calculation periods) increases, the discount factor decreases, and therefore the current value P decreases.
Meanwhile, payment for concluded transactions may include either a one-time payment or a series of payments distributed over time. Pay rent, payments for purchased property in installments, investing funds in various programs, etc. in most cases provide for payments to be made at certain intervals, i.e. a flow of payments is formed.
A series of consecutive fixed payments made at regular intervals is called financial rent, or annuity.
According to the moment of payments of annuity members, the latter are divided into ordinary (post-numerando), in which payments are made at the end of the corresponding periods (year, half-year, etc.), and pre-numerando, in which payments are made at the beginning of these periods. There are also annuities that provide for the receipt of payments in the middle of the period.
General indicators of annuity are: the accumulated amount and the modern (current, reduced) value.
The accrued amount is the sum of all members of the payment stream with interest accrued on them at the end of the term, i.e. on the date of the last payment. The accrued amount shows how much the capital will represent when contributed at regular intervals throughout the entire annuity term along with accrued interest.
The current value of the payment flow is the sum of all its members, reduced (discounted) by the interest rate at a certain point in time, coinciding with the beginning of the payment flow or preceding it.
The value is the annuity growth coefficient, which is also called the coefficient of accumulation of a monetary unit for the period.
It was previously stated that some annuities are realized immediately after the contract is concluded, i.e. The first payment is made immediately and subsequent payments are made at regular intervals. Such annuities (prenumerandos) are also called advance or entitlement annuities. The sum of the members of such an annuity is calculated by the formula:
That is, the sum of the annuity terms prenumerando is greater than the accumulated sum of the annuity postnumerando by a factor, therefore the accumulated sum of the annuity prenumerando is equal to:
where S is the accumulated postnumerando sum.
In cases where payments are made in the middle of periods, the accrued amount is calculated using the formula:
where S 0 is the accrued amount of payments paid at the end of each period (post-numerando annuity).
The modern value of the annuity (also called the current or reduced value) is the sum of all terms of the annuity, discounted at the time of reduction at the selected discount rate. For rent with terms equal to R, the modern value is calculated using the formula:
where A is the rent reduction coefficient, showing how many rent payments (R) are contained in the modern value;
i is the annual interest rate at which discounting is carried out;
n is the period of rental payments.
This figure is also called the present value of an ordinary annuity, or the present value of future payments. Rent reduction coefficients are tabulated.
Costs associated with debt repayment, i.e. repaying the amount of the debt itself (debt amortization), and paying interest on it, are called debt service expenses.
Exist various ways debt repayment. The parties to the transaction stipulate them when concluding the contract. In accordance with the terms of the contract, a debt repayment plan is drawn up.
One of the most important elements of the plan is determining the number of payments during the year, i.e. clarification of the number of so-called urgent payments and their magnitude.
Urgent payments are considered as funds intended to repay both the principal debt and current interest payments on it. In this case, the funds used to repay (amortize) the principal debt may be equal or vary according to some laws, and interest may be paid separately.
Debt can be repaid by annuities, i.e. payments made at regular intervals and containing both payment of principal and interest on it. The amount of the annuity can be constant, or it can change in arithmetic or geometric progression.
Below we will consider the case when the plan is drawn up in such a way that the loan is repaid at the end of each billing period in equal urgent payments, including payment of the principal amount of the debt and interest on it and allowing the loan to be fully repaid within deadline. Each urgent payment (Y) will be the sum of two quantities: the annual cost of repaying the principal debt (R) and the interest payment on it (I), i.e.
The urgent annual payment is calculated using the formula:
where i is the interest rate;
n - loan term;
D is the amount of debt.
The value is called the debt repayment ratio, or the contribution to the depreciation of the monetary unit. It can also be thought of as the inverse of the present value of the annuity, i.e. .
In practice, it may be necessary to know the amount of the outstanding principal balance for any period. This value is calculated by the formula:
where k is the number of the billing period in which the last urgent payment was made.
Buying real estate in most cases involves obtaining a loan. In this regard, it is necessary to know in advance how much will need to be deposited in each payment period in order to ensure the repayment of the principal amount of the debt (excluding interest payments) on time.
To solve this problem we use the formula:
where R 1 is the cost of repaying the principal debt in the first payment period;
D is the amount of the principal debt;
n - loan term;
i - interest rate.
The value is called the compensation fund factor. It shows how much will need to be deposited at the end of each payment period so that the principal loan amount will be fully repaid within a specified number of periods.
To calculate the amount used to repay the principal debt in any period, it is necessary to multiply the compensation fund factor and the compound interest multiplier for a given period, i.e.
where k is the number of periods for which the principal debt was repaid.
We examined the functions of compound interest using the basic formula that describes the accumulated amount of a unit. All considered formulas (factors) are derived from the main formula. Each of them provides that the money in the deposit account earns interest only as long as it remains in that account. Each of the formulas takes into account the effect of compound interest, i.e. that interest which, when received, is converted into the principal amount.
All of the above formulas are summarized in a table, which makes financial calculations somewhat easier. The table has the name: “Tables of compound interest. 6 functions of compound interest. The quantities included in the table are in a certain relationship with each other. Below in the table. this connection is given.
To determine the value of an investment project or property, it is necessary to determine the current 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 do you need to invest in an investment project today in order to receive 8 thousand rubles by the end of the 5th year? 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 the current value of lease 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 installment to repay a loan with a known number of installments 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.
TOPIC 3.^
Mathematical foundations of assessment activities
This topic examines the mathematical foundations of valuation activities, which include six functions of a monetary unit.
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3.1. Six functions of currency
To determine the value of income-producing property, it is necessary to determine the present value of money that will be received some time in the future.
It is known that in conditions of inflation it is much more obvious that 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 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.
In valuation, these financial calculations are based on a complex process in which each subsequent calculation of the interest rate is carried out both on the principal amount and on the accrued interest. previous periods unpaid interest.
In total, 6 functions of the monetary unit are considered (see Table 5), based on compound interest. To simplify calculations, tables of functions have been developed for known rates of income and the accumulation period (I and n); in addition, a financial calculator is used to calculate the required value.
^ Table 5
Table structure of the six functions of money
| Function of money | Future unit value | Unit accumulation per period | Indemnity Fund Factor | Current unit cost | Present value of the annuity | Unit depreciation contribution |
| Formula | | | | | |
|
| Asked by: | PV, i, n | PMT,i,n | FV,i,n | FV,i,n | PMT,i,n | PV, i, n |
| Define | F.V. | F.V. | PMT | PV | PV | PMT |
| Type of tasks to be solved | Future value of a current sum of money | Cost of payments at the end of the period | Loan principal repayment rate | The present value of a sum of money that will be received in the future | Current value cash payments | Regular periodic loan payment, including interest and loan repayment |
1 function:
The future value of a monetary unit (the accumulated amount of a monetary unit).
Where, FV is the future value of the monetary unit;
PV – current value of a monetary unit;
I – rate of income;
N – number of accumulation periods, in years.
If accruals are made more often than once a year, then the formula is converted to the following:
Where, k– frequency of accumulations per year.
This function is used when the current value of money is known and it is necessary to determine the future value of a monetary unit at a known rate of income at the end of a certain period (n).
Rule of 72 : To approximately determine the period of doubling of capital (in years), it is necessary to divide 72 by the integer value of the annual rate of return on capital. The rule applies to rates from 3 to 18%.
A typical example of determining the future value of a monetary unit is the following problem.
Determine what amount will be accumulated in the account by the end of the 3rd year, if today you put 10,000 rubles into an account that brings 10% per annum.
Solution: FV=10000[(1+0.1) 3 ]=13310
2 function:
Current unit value (current resale reversion value).
If interest is calculated more often than once a year, then
An example of a formula would be the following problem:
How much should you invest today in order to get 8000 in your account by the end of the 5th year, if the annual rate of return is 10%.
Solution: 
3 function:
The current value of the 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 the beginning, it is an advance annuity.
The formula for the present value of an ordinary annuity is:
Where, PMT – equal periodic payments.
If the frequency of accruals exceeds 1 time per year, then
Formula for the present value of an advance annuity:
Typical example:
The rental agreement for the dacha is for 1 year. Payments are made monthly in the amount of 1000 rubles. Determine the current value of 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 a monetary unit over a period. 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 end of the period.
Ordinary annuity formula:
Advance accrual (or advance annuity):
Typical example:
Determine the amount that will be accumulated in an account yielding 12% per annum by the end of the 5th year, if 10,000 rubles are annually deposited into the account a) at the end of each year; b) at the beginning of each year. Solution:
5 function:
Contribution for depreciation of a monetary unit. The function is the reciprocal of the present value of an ordinary annuity.
The contribution to the depreciation of a monetary unit is used to determine the amount of the annuity payment to repay a loan issued for a certain period at a given loan rate.
Depreciation- This is the process defined by this function, including interest on the loan and payment of the principal amount.
1
2
For payments made more often than once a year, the second formula is used
An annuity (by definition) can be either a receipt (incoming cash flow) or a payment (outgoing cash flow) to the investor. Therefore, this function can be used if it is necessary to calculate the amount of an equal installment to repay a loan with a known number of installments and a given interest rate. This kind of loan is called "self-amortizing loan" .
An example would be the following 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. Solution:
The borrower will pay the lender for 7 years:
24036 * 7 = 168,252 rubles
6 function:
The compensation fund factor. This function is the inverse of the function of accumulating a unit over a period. The compensation fund factor shows annuity payment, which 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 the following task:
Determine what payments should be in order to have 100,000 rubles in an account earning 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 payment of the principal amount without payment of interest.
^ TOPIC 4.
Preparing information during the assessment process
This topic examines all the information that an appraiser may need in the process of preparing a report on the assessment of an object of value. Dividing information into external and internal allows the student to better understand this topic.
Information used in the assessment process must meet the requirements:
reliability;
accuracy;
complexity.
There are different orders for organizing information: chronological, journalistic, logical.
Chronological order provides for a consistent transition from the past to the future (or from the future to the past). For example, in an appraisal report, the description of the production process begins with the history of the company.
At journalistic order The material is arranged from most important to least important. Thus, when analyzing financial information, as a rule, it does not make sense to describe all retrospective information; attention is focused on the most important proportions or ratios.
At logical order information is distributed from general to specific or from specific to general. For example, before moving on to the analysis of the company being assessed, a review of the macroeconomic situation is carried out to determine the investment climate in the country.
Business valuation is based on an analysis of the value of the enterprise as an investment product, that is, taking into account past costs, current state and future potential. To implement such integrated approach it is necessary to collect and analyze large quantity information that can be classified as follows:
external information characterizes the operating conditions of the enterprise in the region, industry and economy as a whole;
inside information reflects the activities of the enterprise being valued.
The analysis of all information blocks is based on the following sequence: 
Normal operation business is possible with the optimal combination of sales volume, profit received and financial resources to ensure planned growth, which is largely determined external factors functioning of the enterprise. The latter include macroeconomic and industry factors: inflation rate, economic development countries, competitive conditions in the industry, etc.
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4.1. External information
The block of external information, as noted earlier, covers the operating conditions of the enterprise in the industry and economy.
The amount and nature of external information varies depending on the purpose of the assessment. When preparing a report, it must be shown that information base, collected and studied by the appraiser, is necessary and sufficient for the final conclusion about the value of the enterprise. If the review of information is extended and not focused on the object being assessed, it should be considered inappropriate.
Macroeconomic indicators contain information about how changes in the macroeconomic situation are reflected or will affect the activities of the enterprise. These indicators characterize the investment climate in the country. Depending on the objectives of the assessment, the macroeconomic overview may be highlighted as a separate section of the assessment report or considered in the overall context of the report.
Macroeconomic risk factors form a systematic risk that arises from external events affecting the market economy and cannot be eliminated by diversification within the national economy.
Risk – degree of certainty characterizing the achievability of expected results in the future.
Diversification - risk reduction by portfolio investments(purchase of a wide range of securities).
In most cases, risk is perceived as the possibility of loss. Any possible deviation up or down from the predicted value is a reflection of risk. Risk factor analysis is subjective: appraisers who are confident in the future growth of a company determine its current value higher compared to an analyst who makes a pessimistic forecast. In other words, the wider the spread of expected future returns around the “best” estimate, the riskier the investment.
The current value of a company whose activities are related to high risk, will be lower than the current value of a similar company operating in less risky conditions.
The investor's understanding of the risk factor can be depicted graphically (see Fig. 6)
The higher the investor's assessment of the level of risk, the higher the rate of return he expects. Globally, most valuation assignments involve the analysis of closely held companies whose owners do not diversify their holdings to the same extent as the owners of publicly traded companies. Therefore, when evaluating closed companies, the appraiser, along with the analysis of systematic (macroeconomic) risk, must take into account unsystematic risk factors. The latter include industry risks and the risks of investing in a specific company.
Rice. 6. Relationship between expected risk and rate of return
Main macroeconomic risk factors:
inflation rate;
the pace of economic development of the country;
changes in interest rates;
change exchange rate;
level of political stability.
In accordance with these factors, the following risks are identified.
Inflation risk – This is the risk of unpredictable changes in price growth rates. The investor seeks to receive income that covers inflationary price changes. High or unpredictable inflation may undermine expected results production activities; inflation ensures the redistribution of income in the economy and increases business risk, resulting in an underestimation of the real value of the enterprise’s property.
Government programs;
Internet (sites “RosBusinessConsulting”, “Expert”, “Recep.ru”, “Finmarket”).
Main sources of information:
Government programs;
analytical reviews of news agencies;
periodical economic press;
Internet.
Main sources of information:
news agencies;
periodical economic press;
Internet.
When forecasting sales volumes for the next year, the appraiser can make calculations in rubles, taking into account projected inflation expectations, or recalculate the projected values at the dollar exchange rate, for which inflation expectations are lower. It is impossible not to take into account inflation expectations for any type of currency.
Main sources of information:
Government programs;
news agencies;
periodical economic press;
Internet.
Main sources of information:
Data from analytical reviews conducted by the agencies “EURO-MONEY”, “Moody’S”, “Standard & Poors”, “Valuation Center For Central & Eastern Europe”, “Dun & Bradstreet”;
Russian analytical rating and information agencies;
Legislation of the Russian Federation.
The level of country risk is measured based on:
quantitative assessment methods (statistical data);
qualitative assessment methods ( expert assessment);
econometric assessment methods (risk forecast based on identified trends in the study of statistical data);
combined assessment methods.
1. economic data (25%);
2. political risk (25%);
3. debt indicators (10%);
4. unpaid or time-restructured debts (10%);
6. access to banking finance (5%);
7. access to short-term finance (5%);
8. access to capital markets (5%);
9. Forfeiting discount (5%).
Political risk is assessed based on expert opinions on a scale from 0 to 10 (high risk).
External information. in addition to macroeconomic information, it includes industry information: the state and development prospects of the industry in which the enterprise being assessed operates. The content of this block is determined by the degree of availability of industry data. It must reflect the competitive conditions in the industry; sales markets and possible options use of manufactured products; factors influencing the potential volume of manufactured products and the dynamics of changes in demand for them. The operating conditions of an enterprise in the industry can have a serious impact on the final value of the cost.
Main industry risk factors:
regulatory framework;
sales markets;
conditions of competition.
Regulatory framework.
It is determined taking into account the presence of restrictions on entry into the industry, competition conditions and pricing.
Main sources of information:
legislation of the Russian Federation (legal databases “Garant”, “Consultant-plus”, etc.);
industry newsletters;
To analyze the product sales strategy chosen by an enterprise, you can use, for example, the Ansoff matrix, which involves four strategies:
Penetrating an already established market with the same product as competitors.
Market development through the creation of new market segments.
Development of fundamentally new products or modernization of existing ones
Diversification of manufactured products to develop new markets.
Demand represents the quantity of goods and services that will be purchased at a certain price during a certain period.
Demand for goods in quantitative terms is measured in inverse relation to price, all other things being equal. Market price ultimately established through the interaction of supply and demand.
In the process of collecting information, relationships with suppliers are also important, taking into account the legal certainty of contracts and their reliability.
The purpose of collecting this information is to determine the potential of the domestic (if necessary, and foreign) market for the sale of goods: sales volume at current prices, a retrospective for the last 2 - 5 years for the enterprise under evaluation, sales volume at current prices for competitors, forecasts for the expansion of sales markets in Russia and beyond.
Main sources of information::
data from the State Committee on Statistics of the Russian Federation;
data from the marketing department of the company being evaluated;
periodical economic press;
Internet (sites “KG Capital”, “Business List”, “Finmarket”);
personal contacts.
Conditions of competition. IN market economy The most typical markets are imperfect competition, in which the mechanism of free competitive pricing has serious limitations.
The assessment of the competitiveness of an enterprise is carried out taking into account the type of market, therefore, the presence of restrictions on the entry of competitors producing substitute goods into the industry. The analysis should be supplemented with information on the volume of production of a competing product in natural and in value terms, characteristics of competitors' products (volume, quality of service, prices, distribution channels, advertising), the share of products sold in the total volume of domestic production, as well as a list of the main Russian importers of this product.
Main sources of information:
State Committee on Statistics of the Russian Federation;
data from the marketing department of the company being evaluated;
dealer companies;
customs administration;
industry information publications;
business plan.
Particular attention should be paid to collecting accounting and pricing information on competing enterprises. It is required for profitable and comparative approaches to business valuation. The purpose of the analysis is to determine the place of the company being assessed in the industry depending on the most important financial indicators and calculating multipliers.
Main sources of information:
databases of information and analytical agencies (AK&M, RA Expert, etc.);
Russian-language Internet sites:
FCSM website - electronic information disclosure form for issuers of securities;
website SCREEN NAUFOR - NAUFOR comprehensive information disclosure system (provides freely accessible profiles of issuing enterprises, as well as quotes of ordinary and preferred shares);
RA Expert website;
RTS (Russian trading system);
MICEX (Moscow Interbank Currency Exchange);
MFB (Moscow stock Exchange);
SPEB (St. Petersburg Currency Exchange);
FB "SP" (stock exchange "St. Petersburg");
EFB (Ekaterinburg Stock Exchange);
NKS (over-the-counter market - National Quotation System), etc.;
English-language sites and resources:
B) Bloomberg et al.
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4.2. Inside information
Internal information characterizes the activities of the enterprise being valued. If the reader of the report is not familiar with the enterprise, he should obtain the most complete and accurate information in order to understand the characteristics of the enterprise being assessed.
The information block usually includes:
retrospective data on the company's history;
description of the enterprise's marketing strategy (competitive conditions);
production capacity;
information about working and managerial personnel;
internal financial information (balance sheet data, financial statements) financial results and cash flow for 3-5 years);
other information.
If the enterprise has developed a business plan, then the section devoted to the description of the enterprise provides basic information about the enterprise: types of activities, characteristics of the industry, factors influencing the activities of the enterprise, main indicators of the current financial condition enterprises, etc. In addition, the business plan must contain the following data: organizational and legal form; size authorized capital; information about the owners of the largest shares of the authorized capital, controlling blocks of shares; belonging of the enterprise to concerns, associations, holdings.
history of the company . The report describes the production process for each type of product manufactured and begins the description with the history of the company.
Marketing strategy of the enterprise
. The marketing strategy of an enterprise is determined by external factors, as well as the phase of the life cycle of the goods produced and the availability of production capacity (Fig. 8).
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Rice. 8. Product life cycle phases
Phases I – II - development and introduction of goods to the market; Phase III - growth in sales volumes of goods. An increase in product sales volumes ensures that the enterprise overcomes the break-even point. Semi-fixed costs are fixed, and revenue covers growing variable costs; Phase IV - market saturation with manufactured products, marginal returns decrease; Phase V - reduction in sales volumes, the need to develop a further strategy: modernization of manufactured products or development of new ones
When analyzing the marketing strategy of an enterprise, the appraiser must compare the following information:
sales volumes for the past (retrospective), current and forecast periods;
cost price products sold;
prices of goods and services, their dynamics;
projected change in demand volumes;
production capacity.
Production capacity . The volume of output is determined, on the one hand, by the demand for it; on the other hand, the availability of production capacity for its production. Therefore, the appraiser, especially when making forecasts, takes into account data on the availability of production capacity at the enterprise and future capital investments.
Example. Analyzing the sales markets, the appraiser came to the conclusion that, taking into account the development of the market of the CIS countries, it is possible to double the volume of products sold, which will amount to:
2003 - 200 million units;
2004 - 250 million units.
However, the production capacity of the enterprise, taking into account future capital investments, will allow the following volumes to be carried out:
2003 – 180 million units.
2004 – 200 million units.
As a result, the forecast for volumes of products sold will be adjusted to production capacity.
Workers and management personnel
. This factor of production has a significant impact on the value of the enterprise. In closely held companies, employees' labor may be partially compensated by company shares (employee profit-sharing program), and employees of the enterprise may be considered as co-owners of the enterprise, owning a certain block of shares.
The enterprise manager can be a “key figure”, providing effective management and business development. This fact must be taken into account during the assessment process, for example, when calculating discount rates, because in the event of the sale of an enterprise, its plans for future activities may change.
The level is also important wages at the enterprise in comparison with industry average data. A deviation upward or downward is considered by the appraiser in order to identify the characteristics of the business being valued and can be adjusted when reporting is normalized.
Main sources of information:
business plan;
interview with the head of the enterprise;
marketing department data;
retrospective financial statements.
Internal financial information . The purpose of the analysis of current and retrospective financial statements is to determine the real financial condition of the enterprise on the date of assessment, the actual value net profit, financial risk and market value of tangible and intangible assets.
Depending on the purpose of the assessment, the direction of analysis of the financial condition of the company changes. For example, if the value of a minority (non-controlling) stake in an enterprise is assessed, then a potential investor will be more interested in the forecast assessment of the company’s profitability and its ability to pay dividends.
Key financial reporting documents analyzed during the assessment process:
balance sheet;
income statement;
cash flow statement.
If an assessment of the company’s assets is required, it is necessary to request a transcript of the most important balance sheet accounts:
1. Non-current assets:
intangible assets;
fixed assets;
long-term financial investments.
Sample form of requested information
2. Working capital:
stocks;
accounts receivable;
short-term financial investments.
Sample form of requested information
3. Obligations:
accounts payable;
long-term debt.
Sample form of requested information
Sample form of requested information
The information request form may contain:
list of documents, analyzing which the appraiser collects necessary information;
a list of data filled out by responsible employees of the enterprise according to the form provided by the appraiser;
list of documents and data in accordance with the appraiser’s request.
Example. Here is a request for information that includes a list of only company documents:
Name and details of the enterprise:
Charter
Financial statements for the last 3 years (balance sheet, applications - forms 1-5), explanatory note to the annual balance.
Business plan.
Licenses for the types of activities carried out (copies of patents and licensing agreements, information on payment of fees).
Reports on revaluations of fixed assets carried out at the enterprise.
Depreciation sheet.
BTI passports for real estate objects.
Lease agreements.
Agreements with major debtors.
Loan agreements.
Agreements (contracts) for the supply of equipment.
The purpose of the appraiser’s work in collecting internal information is:
analysis of the company's history in order to identify future trends;
collecting information to forecast sales volumes, cash flows, arrived;
taking into account unsystematic risk factors specific to the business being assessed;
analysis of financial documentation;
interviews with managers and collection additional information, allowing you to make the assessment more realistic (in any company, especially a closed one, there is a set of important documents, as well as general information that can be obtained by the appraiser directly from the management of the enterprise).
L.O. Grigorieva
Investment management
Training module
Ulan-Ude
Publishing House VSTU
| introduction………………………………………………………………………………….………………………………… | ||
| Topic 1. Concept and classification of investments………………………………………..……. | ||
| 1.1. | The concept of investments and their classification……………………………………...……………………. | |
| 1.2. | Investment process and mechanism of the investment market……………………….…………. | |
| 1.3. | Six functions of compound interest……………………………………………………………….... | |
| Topic 2. Economic, legal and organizational foundations investment activity in the Russian Federation……………………..……………………….................... | ||
| 2.1 | Normative base investment activity in the Russian Federation……………………………………………………………… | |
| 2.2 | Methods government regulation investment activities………………………. | |
| Control questions………………………………………………………………………………………. | ||
| Tests…………………………………………………………………………………………………………. | ||
| Topic 3. Sources of financing investment activities…………. | ||
| 3.1 | Classification of sources of financing the investment activities of an enterprise...... | |
| 3.2 | Basic methods of financing investment activities……………………………………………………… | |
| 3.3 | Analysis of price and capital structure…………………………………………………………………………………. | |
| 3.4 | Methods for calculating investment needs………………………………………………………. | |
| Control questions………………………………………………………………………………………. | ||
| Tests…………………………………………………………………………………………………………. | ||
| Topic 4. Investment planning. Stages of drawing up a business plan……….. | ||
| 4.1 | Essence and classification investment projects…………………………………………… | |
| 4.2 | Life cycle investment project……………………………………………………….. | |
| 4.3 | Methodology for drawing up and structure of a business plan for an investment project………………………. | |
| Control questions………………………………………………………………………………………. | ||
| Tests…………………………………………………………………………………………………………. | ||
| Topic 5. Evaluating the effectiveness of an investment project…….………………….. | ||
| 5.1 | Main aspects of assessing the effectiveness of investment projects…………………………. | |
| 5.2 | Assessment of the financial viability of an investment project…………………………………… | |
| 5.3 | Grade economic efficiency investment projects……………………………… | |
| Control questions………………………………………………………………………………………. | ||
| Tests…………………………………………………………………………………………………………. | ||
| Tasks for practical exercises…………………………………………………………………………………………. | ||
| Topic 6. Risk management of an investment project………………………………. | ||
| 6.1 | Essence and classification of risks of an investment project…………………………………….. | |
| 6.2 | Risk management of an investment project………………………………………………………. | |
| 6.3 | Methods for assessing project risk…………………………………………………………………………………... | |
| 6.4 | Techniques for project risk management……………………………………………………………………………… | |
| Control questions……………………………………………………………………………………….. | ||
| Tests………………………………………………………………………………………………………….. | ||
| Topic 7. Assessment of investment qualities and efficiency financial investments ……………………………………………………………………………………………… | ||
| 7.1. | Calculation of profitability on transactions with securities………………………………………………………. | |
| 7.2 | Calculation of future capital in financial investments……………………………………………. | |
| 7.3 | Calculation of market value of securities…………………………………………………………... | |
| 7.4 | Peculiarities of investment evaluation in bill circulation…………………………………………………………. | |
| Control questions………………………………………………………………………………………. | ||
| Tests…………………………………………………………………………………………………………. | ||
| Problems for practical exercises………………………………………………………………………………………….. | ||
| Topic 8. Formation of an investment portfolio…………………………………… | ||
| 8.1 | Concept and types investment portfolios……………………………………………………… | |
| 8.2 | Portfolio return………………………………………………………………………………… | |
| 8.3 | Portfolio risk……………………………………………………………………………………… | |
| Control questions………………………………………………………………………………………. | ||
| Tests…………………………………………………………………………………………………………. | ||
| Problems for practical exercises……………………………………………………………………………………………… | ||
| Annex 1………………………………………………………………………………………………. | ||
| Appendix E2………………………………………………………………………………………………. | ||
| Appendix 3………………………………………………………………………………………………………… |
Topic 1. Investments. The essence of the investment process
Six functions of compound interest
The first function of compound interest is a factor in the future value of current (today's) capital.
| FV = PV*(1+i)n | (1.4) |
FV is the future value of current capital;
PV – current value of capital (present value);
i – interest rate;
n – number of periods.
In what cases is the compound interest formula used:
We have some amount of money. We want to put it in the bank at a certain percentage for a certain period (year, month, quarter). At the same time, we want to know: how much our money will be worth at the end of the deposit period.
Example. Let's say we have 1 rub. and we put it in the bank at the beginning of the year, at 10% per annum for 5 years. How much will this ruble cost? after 5 years?
FV = 1 rub.*(1+10%) 5 = 1.61 rub.
Example. You deposited money in the bank 1000 rubles. at 24% per annum for 1 year. Accumulation (i.e. accrual) occurs twice a year at a fixed annual rate. It is necessary to determine the periodic rate (i p), the future value of current capital (FV), the amount of return on capital (D) and the actual annual rate (i f).
Let us determine the periodic rate, in in this case– six-monthly: i p = i g /2 = 24% /2 =12%
Let's determine the future value of current capital: FV = 1000(1+0.12) 2 = 1254.4 rubles.
Let's determine the amount of income on capital: D = FV – PV = 1254.4 – 1000 = 254.4 rubles.
Let's determine the actual annual rate: i f = (FV–PV)/PV=(1254.4–1000)/1000=0.2544=25%
The actual rate includes compounded interest, so it is always higher than the nominal rate. In addition, the more interest periods there are in a year, the more significant this difference will be.
Example. How many years will it take for the capital to double if it is known that the annual nominal rate at which money was deposited in the bank is 12%?
The solution to this problem is based on the use of the so-called “rule of 72”. According to this rule, the number of years after which the invested amount will double is determined by the formula: 72 / nominal annual rate%
72 / 12% = 6 years.
The rule gives a satisfactory answer for rates ranging from 3 to 18%.
The second function of compound interest is to factor in the future value of an annuity.
It is intended to determine the future value of equal capital accumulations for a certain number of periods, i.e. when, for example, we will invest the same amount of money (RMT) for some time (1,2,3 years, etc.).
RMT ( payment) – one-time payment in period k. (the periods are the same).
A series of such payments is called annuity.
Distinguish ordinary And advance annuity.
The future value of an ordinary annuity (payments at the end of each period). Its future value is expressed in the formula:
Example. To save up for a car, you decide to put $1000 into the bank every year at 12% per annum for 5 years. What is the best way to save money (at the end or at the beginning of the year) in order to receive a larger amount in 5 years and how much money will be in your account after 5 years?
Let's first determine how much money we will receive in 5 years if we save at the end of each year:
Thus, it turns out that investing at the beginning of each year is much more profitable than at the end.
The third function of compound interest is the compensation fund factor.
Indemnity Fund Factor– this is the amount of payment that must be deposited (invested) in each period at a given rate annual interest in order to receive a certain (desired) amount in the account in the last period. Those. Let's say we want to get 1 million rubles in five years. To do this, you can put money in the bank. We know the amount of bank interest. The compensation fund factor (RFF) determines the amount of periodic equal payments that we will have to pay over these 5 years. That is, FFF is the same as RMT.
A distinction is made between the Regular Compensation Fund Factor and the Advance Compensation Fund Factor, depending on when (at the end or beginning of the period) payments are made.
Ordinary Compensation Fund Factor(payments at the end of each period):
The 2nd and 3rd functions of compound interest are interconnected through formulas. The 2nd function is the determination of FV, and the 3rd is the determination of PV.
Example. You borrowed money from a friend and in 5 years you must return $1000. To make it easier to pay off your debts, you decide to put money in the bank every year. Bank rate also equal to 15% per annum. What is the best way to deposit money: at the beginning of the year or at the end of the year? How much should you deposit in the bank to pay off this $1000 at the end of year 5?
1. Ordinary Compensation Fund Factor:
| FFOV = | _____15%___ | *1000$ | = 148 $ |
| (1+15%) 5 - 1 |
- Advance Compensation Fund Factor:
2. Advance Compensation Fund Factor:
| FFAW = | ________1,25%__________ | *10000$ | = 111,5 $ |
| (1+1,25%) 5*12+1 – (1+1,25%) |
It is more profitable for you to save $111.5 every month.
The fourth function of compound interest is the factor of the current value of future capital.
Present value of future capital is the present value of capital to be received in the future. The current value of future capital can be expressed mathematically as follows:
PV = FV /(1+i)n(1.9)
As you noticed, the 4th and 1st functions of compound interest are interconnected by one formula. The 1st function determines the future value of current capital.
Example. You decide to save $12,000. You will need this amount in 4 years. How much money should you put in the bank today at 10% per annum to get $12,000 in 4 years?
PV = $12,000 /(1+10%) 4 = $8,196
The fifth function of compound interest is the factor of the present value of an annuity.
The 5th function is intended to determine the current value (PV) of equal capital accumulations for a certain number of periods, i.e. when, for example, we invest the same amount of money (RMT) for some time (1,2,3 years, etc.) with a known rate of return ( i).
In this sense, the 5th function is somewhat similar to the 2nd compound interest function, with the only difference being that the 2nd determines FV.
A distinction is made between the present value factor of an Ordinary Annuity (payments at the end of each period) and an Advance Annuity (payments at the beginning of each period).
Present value of an ordinary annuity:
2. If payments are made at the beginning of each year:
Advance payment for depreciation(payments at the beginning of the period):
2. If payments are made at the beginning of the year:
| RMTn = | 15000$*12%_____ | = 3715$ |
| (1+12%) – (1+12%) – (5 – 1) |
Control questions
1. Describe the concept of investment, give options for their classification.
2. What are the main differences between investments and capital investments?
3. What is investment activity and what stages does it consist of?
4. What subjects of investment activity can be identified? Their differences and main characteristics?
5. Objects of investment activity, their differences and main characteristics.
6. Recipient as a subject of investment activity?
7. What is the structure of the investment market?
8. What is the structure of the investment market in Russia? List and describe its components.
1.1. Which of the following is not an investment in most cases?
a) acquisition foreign currency;
b) investments in bonds on the secondary market;
c) investments in certificates of deposit;
d) leasing financing;
e) investments in shares on the primary market.
1.2. The main investment goals are:
a) making a profit;
b) achieving social effect;
c) accumulation of capital
1.1. Direct investment involves:
a) attracting financial intermediaries to the implementation of investment projects;
b) use internal sources investment financing;
c) direct participation of the investor in the selection of investment objects and capital investment.
1.2. Which of the following economic entities is not a participant (executor) of investment activities?
a) investor;
b) performer;
c) designer;
d) contractor;
d) insurance company.
1.3. In what area does investment activity take place?
b) appeals;
d) intangible production.
1.4. Investment activities commercial banks in the field of real investment has the following forms:
a) investment lending;
b) investing in securities;
c) project financing;
d) equity participation.
1.7. Which of the following are the tangible elements of investment?
a) communications;
c) investments in human capital;
d) securities;
e) patents, licenses.
1.8. What underlies the division of investments into real, financial and investments in intangible assets?
a) investment objects;
b) reproductive forms;
c) stages of the investment process;
d) subjects of investment activities.
1.9. The concept of the investment multiplier was developed by:
a) R.F. Kahn;
b) Samuelson;
c) J.M. Keynes.
1.10. Investments in intangible assets are:
a) investments in brands, trademarks, copyrights, etc.;
b) costs for the acquisition of environmental management facilities;
c) investments in working capital enterprises.
Problems for practical exercises
Task 1.1.
Calculate the annual payment for an apartment worth 800 thousand rubles, purchased in installments for 10 years at 12%.
Problem 1.2.
Calculate the annual contribution at 12% for the purchase of an apartment worth 800 thousand rubles in 10 years.
Problem 1.3.
Calculate the contribution at 12% for the purchase of an apartment worth 800 thousand rubles in 10 years.
Problem 1.4.
The apartment was sold for 800 thousand rubles, money brings 12% of annual income. What is the marginal value of real estate that can be purchased in 10 years?
Problem 1.5.
What is the maximum cost of real estate that can be purchased in 10 years if you save 80 thousand rubles annually? at 12%?
Problem 1.6.
How much did it cost for an apartment bought in installments for 10 years at 12% per annum, if the annual payment is 80 thousand rubles?
Cash flow analysis should be carried out both in the short term and in the long term. The basis of long-term cash flow analysis is the understanding of time preference in the disposal of funds, or, in other words, the concept of the time value of money.
This concept is that money has a value that is determined by the time factor, that is, resources available today are worth more than the same resources received after a certain (significant) period of time.
The concept of cost of money affects a wide range of business decisions related to investing. Understanding this concept largely determines the effectiveness of decisions made.
Time preference in the disposal of funds is determined as follows. The current management of resources allows you to take actions that, over time, will lead to an increase in future income. Based on this, the value of funds is characterized by the opportunity to receive additional income. The greater the possible income, the higher the cost of funds. Thus, the cost of funds is determined by the lost opportunity to receive income in the case of the best option for their placement.
This provision is of great importance because the cost of funds is often mistakenly reduced to losses from inflation. Indeed, under the influence of the inflation factor purchasing power cash flow is decreasing. But it becomes fundamental to understand that even in the complete absence of inflation, funds have a value determined by the previously noted time preference and the possibility of receiving additional income from an earlier investment of funds.
The cost of cash or the cost of lost opportunities is not an abstraction, although it is not recorded in accounting. The quantitative expression of time preference in the use of funds is usually interest rates that reflect the rate of time preference in a given economic situation.
But if the interest rate reflects the greater value of the resources available now, then it follows that in order to determine the present value of funds expected to be received in the future, it is necessary to discount these amounts in accordance with the interest rate.
Let us note that the adopted Concept of Accounting in the Market Economy of Russia for the first time introduced the concept of discounted value into Russian accounting practice. According to the Concept, present value can be used to evaluate both assets and liabilities. Valuing assets at discounted value allows you to see the relationship between the costs associated with the creation (formation) of assets and the income arising in the future from their use.
The valuation of liabilities at present value represents the future payments associated with them reduced (recalculated) to the current moment.
Thus, definitions of the basic concepts of long-term financial analysis can be given.
Discounted (present) value - reduced to today the value of a payment or stream of payments to be made in the future.
Future value is the value expected to be obtained as a result of investing funds under certain conditions (interest rate, time period, interest accrual conditions, etc.) in the future.
Interest and discounting are the main techniques of long-term analysis. Their use is based on the understanding that, from an economic point of view, it makes no sense to directly (without reference to one time period) compare amounts of money received at different times. In this case, it does not matter at what point in time the monetary amounts will be reduced - the present or the future. However, since the need to compare cash flows arises for the purpose of making a specific management decision, for example, about investing money in order to generate income in the future, cash flows, as a rule, are reduced to the moment the decision was made (usually called time 0).
Bringing the future value of funds to the present time (moment 0) is usually called discounting. Economic sense The process of discounting cash flows consists of finding an amount equivalent to the future value of cash. Equivalence of future and discounted sums of money means that the investor should be indifferent whether he has a certain amount of cash today or after a certain period of time has the same amount, but increased by the amount of interest accrued over the period. It is in this case of temporary indifference that we can say that the discounted value of future flows has been found.
As we see, the following questions are fundamental: the actual amount of future monetary amounts; terms of their receipt; interest or discount rate (interest rate is used to determine the future value of monetary amounts, discount rate is used to find the present value of future amounts); a risk factor associated with receiving future amounts.
When determining the interest (discount) rate, the effect of compound interest must be taken into account. Compound interest assumes that the interest accrued over the period is not withdrawn, but added to the original amount. In the next period it brings in new income.
Thus, in order to determine the feasibility of making an investment, it is necessary to evaluate whether the present value of the amounts of money that will be received in the future exceeds the present value of the amounts of money that must be invested to obtain these incomes. The presence of an excess of the first amounts over the second is a criterion for how desirable the investment is.
A total of six functions of the monetary unit based on compound interest are considered. To simplify calculations, tables of six functions have been developed for known rates of income and the accumulation period (I and n); in addition, you can use a financial calculator to calculate the required value.
1 function: Future value of a monetary unit (accumulated amount of a monetary unit), (fvf, i, n).
If accruals are made more often than once a year, then the formula is converted to the following:
k– frequency of accumulations per year.
This function is used when the current value of money is known and it is necessary to determine the future value of a monetary unit at a known rate of income at the end of a certain period (n).
2 function : Current value of the unit (current value of reversion (resale)), (pvf, i, n).
The current value of a unit is the inverse of its future value.
If interest is calculated more often than once a year, then
An example of a problem is the following: How much should be invested today in order to get 8,000 in the account by the end of the 5th year, if the annual rate of return is 10%.
3 function : Present value of the annuity (pvaf, i, n).
An 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 the beginning, it is an advance annuity.
The formula for the present value of an ordinary annuity is:
PMT – equal periodic payments. If the frequency of accruals exceeds 1 time per year, then
Formula for the present value of an advance annuity:
4 function : Accumulation of a currency unit over a 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 end of the period.
Ordinary annuity formula:
5 function : Contribution depreciation of a monetary unit (iaof, i, n) .
The function is the inverse of the present value of an ordinary annuity. The contribution to the depreciation of a monetary unit is used to determine the amount of the annuity payment to repay a loan issued for a certain period at a given loan rate.
Amortization is a process defined by this function that includes interest on the loan and payment of the principal amount.
For payments made more often than once a year, the following formula is used:
6 function : Restitution fund factor (sff, i, n)
This function is the inverse of the function of accumulating a unit over a period. The recovery 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:
