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13. Discounting with the Euler number exponential function in economic models of vessel or the property complex of marine company’s real assets

The material of the given section material contains the analysis of approximation accuracy of the discounting factor determined by arithmetic summation (1+i) and estimations of present value for investing into development of the ocean wealth, by the discount financial functions, and it also contains the analysis of methodological advantages of formulas of the discount financial functions based on application of exponential functions of the Euler number e = 2.71828283..., that, is the bases of the natural logarithm.
           It is considered that when investing in industrial activity for the ocean resources development is more effectively and accordingly more risky economically than investing in the financial deposit in respect of which the discounting by the exponential functions of the factor calculated by the arithmetic summation (1+i) is accepted.
           In applied tasks to determine the total risk depending on its separate components the expert methods of the hypothec–investment group in particular, the following methods of the discounting rate calculation are usually recommended: the cumulative estimation, the models of capital assets estimation, the models of the average weighted cost of the capital etc., and also methods of the arbitration pricing or the extraction which though are not formally based on risks summation, but are alternative ways of calculation of the index of the discounting rate which is total of the risks.
           At the same time for present values estimation is required not only correct calculation of the total risk depending on separate components, but also the development of the adequate discounting method when the properties of summable risk components which correspond to actions in sequence of innovative investing in real assets of the marine companies’ complexes of the assets are taken into account.
           The analysis of applicability of discounting by the Euler number exponential function and advantages of this method, for present value estimation of the marine company’s real assets with the account of the formal–logic properties of risks probablistic concept.
           The financial functions based on discounting by exponential of the factor calculated by arithmetic summation (1+i) based on monetary rules of interest accruing on the financial deposit determination (or the mortgage) are the approximate ways of present value and financial indexes determination and are rather exact at the limited risks i correlated, on the other hand, with investments efficiency.
           The arithmetic summation of the discounting rate to a one in case of factor of discounting (1+i) determination is based on the general assumption that the risk is the one–factorial index corresponding to the discounting rate i, and it is rather true if financial investment strategy means a single action – the financial deposit (or the mortgage).
           If innovative strategy in the marine industrial activity on investing financial assets in real assets is an actions sequence, the general risk insufficiently precisely corresponds to the estimation by the one–factorial index and each action in strategic sequence influences strategy total risk value.
           It should be reasonable to research the methodological correctness of risks summation from the mathematical point and discounting by the factor (1+i).
           The commercial risk in estimation can be correlated with the discounting index i or 1-(1-i). When determining the total risk as a probablistic characteristic of the sequence of casual events it is necessary to take into account that the probability of events sequence corresponds to the product of probability of each event by the probability of previous events or to the multiplication of events probabilities in sequence.

 If the probability of the positive result for a single n–th action in the general sequence of innovative strategy actions is equal (1-in), where i n – is a risk, that is a probability of negative result for a single n–th action, then the probability of the successful innovative strategy consisting of n of actions is equal: (1-i1)×(1-i2)× ... ×(1-in), and the total risk of innovative strategy makes:
i=1-(1-i)=1-(1-i1)×(1-i2)×...×(1-in)=1-(1-i1- i2-...- in+ i1×i2+...+i1×in+i1×in-...- i1×i2×...×in

(13.1)

           The difference of the risks total i (13.1) from the arithmetic sum applied usually when the discounting rate is determined by cumulative estimation is visible

i=1-(1-i)=1-(1-i1-i2-...-in,)

(13.2)

the difference makes

1.

(13.3)

And the reminder (13.3), that is a difference between the total risk value determined by the complete–probablistic scheme (13.1) and by the cumulative estimation (13.2), apparently, can be neglected only provided that risks separately and in total are insignificant (assuming that i ≤0,15÷0,20).
           To determine the discounting rate, from the methodological point, the formula (13.1) i=1-(1-i)=1-(1-i1)×(1-i2)×...×(1-in), is more correct than the formula of cumulative estimation by risks arithmetic summation (13.2) i1+i2+...+in (which in total can appear to be more than 100 %, that does not mean that the real total risk i is more than 100 %).
           If in order to determine the total risk of innovative strategy we break it into actions and as a methodological technique to break the strategy into such actions to which correspond equal parts of risk in=i/n with the selection of macroeconomic definition for each of n equal parts of risk, then the total commercial risk is approximately calculated by the method of cumulative estimation of the equal parts total as: 13 or 13  . 
           Then according to the complete–probablistic scheme (13.1) taking into account the assumption about strategy division into equally risky separate actions the value of the total commercial risk can be estimated as:

13
or i=1-(1-i/n)n.

The estimation of the limiting value of the total commercial risk in the given definition 1-(1-i/n)n is of interest with a great number n of risk divisions into equal parts by used methodical technique.

 The limiting value of commercial risk at its division into the great number of the components 1 is equal:
13

As, on the other hand, the total commercial risk corresponds (is approximately equal) to the discounting rate, then i ≈1-1/ei, or, after derivation, the factor of discounting based on the exponential functions of the Euler number application is determined as ei≈1+i.
           The discounting rate as an indicator of the commercial risks total methodologically more precisely corresponds to the factor in exponential function of the Euler number ei or exp(i), than the member in the sum of discounting factor (1+i), as the factor of discounting as exponential function of the Euler number ei grows out in case of summation of risks in frameworks of the complete–probablistic estimation scheme with multifactorial characteristics of the general risk (13.3) as a form of use of the discounting rate of innovative strategy commercial risk unlike the factor of the discounting calculated by arithmetic summation (1+i) which is more suitable for determining the present values with one–factorial risks of the financial investment in the monetary deposit (or the loan).
           With rather small total investment risks i≤0,15÷0,20 of discounting factor the value of the discounting factor determined on the basis of the Euler number exponential function ei and on the basis of arithmetic summation (1+i) are approximately equal and in applied tasks lead to a rather close present value. In case of increase in investment risks i there is a difference of discounting factors ei and (1+i), tab. 13.1, fig. 13 .

Table 13.1

The dependences of the factors determined by the arithmetic summation (1+i) and the Euler number exponential function ei on discounting rate i

i
5 %
10 %
15%
20%
25%
30%
35%
40%
45%
50%
(1+i)
105 %
110 %
115%
120%
125%
130%
135%
140%
145%
150%
ei
105 %
111 %
116%
122%
128%
135%
142%
149%
157%
165%

It should be noted that the discounting factor as exponential function of the Euler number 1 is the form of the remainder account (13.3) that is a difference 1 between the total risk values determined by the complete–probablistic scheme (13.1) and by the cumulative estimation (13.2).
           The factor of discounting as an exponential function of the Euler number 1corresponds to present value estimation of innovative strategy with the multifactorial risks of consecutive actions if the total risk is rather great, and the strategy is as effective as in case of investing in real assets of the marine companies’ complexes of the assets using which the investor gets an access to marine wealth.
           Though with smaller risks and rates, as in case of financial investing in the monetary deposit, the application of the factor of discounting as the exponential function of the Euler number 1 is quite reasonable, as the factor of discounting determined by summation (1+i) with small risks correlated with index of the discounting rate 1, is rather close numerical approximation of discounting factor as the exponential function of the Euler number 1.
           Perhaps, the choice of discounting method from the two considered options with small risks depends basically on the expert’s ability to determine exponential function value and on formal–logic understanding properties of risks probablistic concept that provides the confidence of choice of present values of real assets as a part of the marine company’s property complex.
           It should be added that discounting by the Euler number exponential function 1 is also determined on the basis of present value potential equation integration (Chapter 4, the formula 4.1) when developing the form of the statistical regression for generalization of the marketing researches data concerning value of the assets, which pricing quite precisely can be laid in the property complex evaluation scheme with value of the asset proportional to the company complex of assets total value, for example of the marine commercial vessels and such assets, which value essentially depends on years.
           With the account the recommendation concerning preference of discounting by the Euler number exponential function the method of the discounted cash flow capitalization for vessel value estimation (or the isolated asset) as a part of the marine company’s property complex becomes:
           – For the continuously accumulating sum of the discounted cash flow and with the set duration of the forecasting period T it is determined as:

13.4

(13.4)

instead of discounting by the formula (11.1);
           – For the interval method (12.1) of capitalizations for the option of present values of cash flow determination as the sum of present values of N–th categories «adhered» to the planned schedule

13.5.

(13.5)

For designation of operations of discounting the term «money function» is not used in tab. 13.2 and the term «discount financial functions» is applied as the «functions of money» concept has own meaning in social and economic terminology (functions of: turnover, accumulation, savings, treasures etc.) other than mathematical transformations over present value indexes.
           As the addition to the recommendations concerning discounting by the Euler number exponential function it is possible to transform the received above formula (12.4) and recommended for working out of the financial assets in real assets of the marine company planned investment schedule.
           With the requirement of continuity for the formula (13.6) it is possible to «adhere» the cash flow set of values (1, 1, …, 1) to the planned investment schedule (1, 1, …, 1), having determined the duration of stages by stability intervals of the net operating income cash flow from 1 to 1 using transformation by the formula 13, that is, having evaluated the intervals of stability duration from 1 to 1 of cash flow before accumulation of the finances sufficient for investing in real assets of the marine company:

Tn+1-Tn=ln(NOIn+1/NOIn)/i.

(13.6)

Table 13.2

Six standard discount financial functions corresponding to the discounting factors determined by summation 1+i and by the Euler number exponential function ei  or exp(i)

The factor of discounting defined by summation 1+i

The discounting by the Euler number exponential function ei or exp(i)

If the rate i is defined based on methods of hypotec–investment group (cumulative estimation, etc.) it is recommended to correct the value of the discounting rate by operation i≈ei-1 for use in case of discounting by the factor (1+i)

If the rate i is determined according to market research by transformation of data on the basis of discounting by factor (1+i) it is recommended to correct the value of the rate by operation i≈ln(1+i) for use in case of discounting by the Euler number exponential function ei

1. The future value FV (complex percent), that is the present single investment, t – present time (or a number of the periods of interest accrual)

FV=(1+i)t

FV=i×t

2. The future value of annuity FVA (the future at the end of the period to Ò value of single cash flow of the uniform periodicity)

13

13

3. The factor of fund of compensation PMT/FVA or 1 (the constant value of the periodic payment in accumulation of the single sum before the end of the period to Ò)

13

13

4. The present value PV of unit (discounting) that is present value of future single investment after time t

PV=1/(1+i)t=(1+i)-t

PV=1/ei×t=e-i×t

5. The present value of single annuity PVA (the single cash flow uniform of periodicity to Ò)

13

13

6. The payment for unit amortization PMT/PVA (repayment of the single loan by equal payments uniform of periodicity to Ò)

13

13

The note. The functions in pairs 1 and 4, 2 and 3, 5 and 6 are reversible in relation to arithmetic dividing.
           After transformation it is visible, that the duration of the net operating income stability interval from Tn  to Tn+1 correlates with the duration of economic payback period term TR≈1 /i, that is Tn+1-Tn=TR×ln(NOIn+1/NOIn).
It is curiously to note some formal similarity, on the one hand, the formulas of the discounting factor defined by summation conformity and the discounting factor determined by the Euler number exponential function, that is  ei≈1+i, and on the other hand, the exponential form of a complex number 1, where i – denotes the square root of minus one 1 – imaginary number.
The external similarity of formulas can push to a guess about interrelation with mathematics of complex numbers which opens the number of possibilities of the phenomena and the processes in applied economics modeling.
           The value and efficiency of investments economic estimation should be developed on the basis of the strict mathematical logic. If the preconditions for value and efficiency definition were not formulated from the mathematical point of view their generalization for complex marine industrial objects or the property complex should be speculative that is erroneous.
           Concerning the interrelation with mathematics there is an interesting example that Nobel Prizes in economics are awarded usually for the contribution to science for the outstanding mathematical bases. At the same time this Prize is not awarded in the area of mathematics without economic or other applications (natural–science, etc.) and also actually is not awarded in economics, when there is no mathematical formalization.
The necessary knowledge of mathematics for the expert in the applied economics covers number of the mathematical analysis application: lines and sequences, linear differential equations, probability theory, mathematical statistics.

           Test questions

1.Discounting rate, the discounting factor.
            2. Arithmetic summation of risks for determination of the discounting rate by the formula of cumulative estimation.
            3. Summation with the account probabilistic characteristic of each risk with the multiplication by the reminder after subtraction of other risks (the formula derivation).
            4. Factor of discounting by the Euler number exponential function.
            5. Recommendations of discounting by the Euler number exponential function.
            6. Formal–logic sense of commercial risk index.
            7. Method of cash flow capitalization in case of use of discounting by the Euler number exponential function for the interval form (the formula at the set time interval of discounting to reversion).
            8. Form of cash flow capitalization in case of using of the discounting by the Euler number exponential function for intervals of cash flow stability.
            9. Definition of investment stages duration with the account continuity of a cash flow by the transformation based on discounting by the Euler number exponential function.
            10. Discount financial functions when using arithmetic discounting and the Euler number exponential function (the formula derivation): the future value, the future value of annuity, the factor of fund of compensation, the present value of a unit (discounting), the present value of single annuity, the payment for unit amortization.
            11. Factor of discounting determined by summation, based on operation of interest accrual for the mortgage or the deposit.
            12. Substantiation of the form of the discounting by the Euler number exponential function factor depending on summation rules of probablistic risks independent components.

 

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