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11. Features of vessel estimation or marine company’s property complexes of real assets based on income approach

The value MV of the vessel or any other isolated asset of the marine company is determined by capitalization of continuously accumulating discounted cash flow NOI sum by the formula

 , (11.1)

and the reversion Rv for revenue–generating vessel or other isolated asset can be determined, for example, as Rv≈NOIT / i, where NOI – the net operating income or the similar cash flow – for the property complex; i – the discounting rate; t – the vessel years; T – the life time of the commercial vessel or other real asset or the duration the predicted period before the reversion reception Rv at the end of the period of asset productive operation; tN –  the present time – discrete variable in summation of the discounted cash flow; OC – the working assets.
When determining the discounted present values of cash flow by summation of the sequence with the discrete variable of time with a range tN= t +1, T (as in the formula 11.1 and in the similar) or tN= 1, T, that is, with the account the indexes of cash flow in each summand at the end of the annual period, for applied tasks it is recommended to make the adjustment on accounting of the distributed payment within a year to which the summation of indexes in the middle of the annual period corresponds better than the summation of indexes at the end of the year.
The adjustment, when determining the present value present value of the discounted cash flow that accounts of the distributed payment within year, can be introduced by the multiplier (1+i)1/2 for the sum of sequence or by the multiplier ei/2 if the discounting by the Euler number exponential function is used. If, when defining the discounted flows present value, the summation is made in intervals tN= t,T-1 or tN= 0 ,T-1 it is necessary to apply the correction multiplier (1+i)-1/2 or e-i/2.
Despite formal similarity of the symbol NOI and some methodological similarity, the cash flow, of the property complex value as a part of the marine company when determined by the income approach, is calculated according to the concepts of business evaluation as the sum of parts (for the invested capital): net profit (without deductions for credit use), amortization (compensation deductions for the assets depreciation), increase in current asset and reduction of long–term obligations minus investments into the real assets.
The discounting rate i corresponds to the interest rate or with the macroeconomic index of risks and is evaluated by the applied methods in sphere of business evaluation, for example by the market extraction according to analogues, by the models: the cumulative estimation, the arbitration pricing, the average weighed value of the capital, the estimation of capital assets, etc.
For the formula (11.1) and for other formulas based on estimations of present values of cash flows it is possible to present the discounting rate as the sum of components i=i0+ i1+ i2+ i3+ ηMV+ ηINS, where i0 – the risks free rate evaluated by the available data of the issue centers; i1 – the adjustment with the account systematic (external) economic risk; i2 – the adjustment with the account non–systematic risk; i3 – the gain rate of consumer prices index (inflation index); ηMV – the rate of the property tax in real asset value estimation; ηINS – the rate of the insurance coverage depending on the estimated value.
The advice to determine the discounting rate i in the sum of the specified terms does not mean in this case the recommendation to determine the rate by summation, and the rate defined by any suitable approach from among used in theory of business and assets valuation shows that, by the economic sense, it is the sum of the components and generally of the commercial risks.
The planning period T recommended in valuation usually should not be less the investments payback period TR~ 1/i with the account the interest rate i or can be accepted equal to life time – depending on the subject assumptions.
It should be noted that the estimation method based on income approach from the point of the discounted cash flow capitalization to a greater extent corresponds to the value estimation of the marine company taken as a whole entity or economically self–supporting isolated property complex of real assets as part of the marine company.
The value MV of self–supporting objects of the similar type that is the property complexes evaluated based on income approach (11.1) includes also working assets, and the result is equal to total value of all assets as a part of the company’s property complex, – both the real assets and working assets including financial assets that is
.
When one estimate the marine company isolated real asset with gradually going down present value and the growing in value the economically self–supporting property complex for which the financial records are kept (or it can be kept) at an innovative investment stage or for the whole company based on income approach, despite the general formal similarity (11.1) there are available methodological distinctions according to the economic indexes used in calculation: the cash flow NOI, the discounting rate i and the reversion Rv.
As the accounting at the level of financial result of NOI estimation usually is not kept for the isolated asset of the marine company, in particular separate assets can be accounted only by expenses and be used for increase of the total property complex efficiency whereas other assets generate income and the record of expenses is not kept, then when determining the net operating income in order to estimate the asset in the sum with working assets based on income approach (by the formula 11.1 or on other) it is required to work out and use an asset economic model – the sequences of dependences, on the one hand, between characteristics of an asset X, expert indexes Y – parameters in evaluation and external data Z, and, on the other hand, the basic economic indexes: the annual income I and operating costs C, that concerning marine commercial vessels of various classes is considered in corresponding sections (article 14, 15).
The reversion Rv when the estimation is made by the income approach of the property complex value or the whole marine company can be determined based on methods of the property costs approach or the income approach on the basis of profitableness accounting for the requirement of the property complex profitability in the last forecasting years.
The vessel, in itself, without other property of the ship–owner used for commercial vessel productive operation for example, without taking into account of working assets for the advance payment and acquisition of fuel, supply, crew salary, vessel maintenance and other expenses, methodologically cannot be evaluated as a marine company’s property complex by the income approach.
Without the connected working assets the vessel can be a source of income only in case of sale, thus it is supposed, that the new owner will resupply working assets.
It should be taken into consideration that working assets OC are not completely a part of investments present value as they come back by their nominal value to the owner at the time of productive operation end TR.
The structure of investments includes, thus, the value of the working assets right–of–use during the period before commercial vessel or other asset of the marine company resale for the value equal the reversion Rv at the end of productive operation.
The part of working assets OC which is a part of investments present value with the account the presented explanatory, can be determined in case of limited life time by the next ways:
OC×i/(i+fm), OC×(1-fm / (i+fm)) or ,

where fm – the factor of compensation of the depreciated part of investments fund (see article 13, tab. 13.2), determined by the formulafm=i/((1+i)T-t-1.
It is necessary to point out, that with the great life time of an asset (vessel) or with the long enough forecasting period if risks in the sum i are insignificant the part of working assets OC which is a part of present value of investments is approximately equal to working assets completely.
Generalizing the stated material regarding the account of working assets OC when estimation is made based on income approach it should be noted that the present value of a cash flow (11.1) is equal to the sum of the vessel value (or other real asset of marine company) and current asset value or the value of the part of current asset – in case of the limited duration of forecasting or the limited life time. In this case when the estimation of present values of cash flow for the vessel evaluation or other asset have been done away the adjustment to working assets value should be subtracted.
On the other hand, the present value of a cash flow of the net operating income corresponds to the value of the whole marine company total property complex, and there is a task to estimate such an object and the adjustment to the value of working assets shouldn’t be subtracted from the present value of a cash flow for estimation of the object of the specified class.
The correcting reference of working assets to the value of the commercial vessel or other real asset of the marine company evaluated based on income approach is an supplement of the isolated object to the property complex, that is, a supplemented economic property of the object which value estimated based on income approach is equal to the present value of corresponding cash flow.
The  choice of discounted cash flow NOI capitalization method within the frameworks of the income approach sometimes is proved by the assumption that this method is more accurate in comparison with the direct capitalization method. In some cases it causes objections.
Unlike the direct capitalization, the increase in number of used parameters for the discounted cash flow accounted with the opinion on higher accuracy of this method can generate uncertainty of the specified parameters estimation – subjectivity of the period of planning estimation, apriority of the forecast cash flow variability during the planning period etc.
Apparently, more objective argument in favor of the discounted cash flow capitalization method can be a property of the cash flow predicted variability caused by the own properties of evaluated object, for example, an increase in a cash flow in case of reinvesting into the assets of growing innovative business, instead of properties of macroeconomic environment which can predetermine the unpredictable variability of cash flow, which is a risk factor and considered as the component of the discounting rate.
The preference area of the discounted cash flow capitalization method is limited by the objects possessing a property of cash flow predicted variability, for example, if there is a planned schedule of investment, and it is reasonable to apply direct capitalization method for other objects.
If there is no investment planned schedule even when there is a forecast of cash flow growth for the property complex or the whole marine company it is more logical to use direct capitalization method variants with the forecast of cash flow growth, for example the Gordon formula, than to summarize the discounted cash flow. When there is no planned schedule, the estimation of present value by cash flow direct capitalization is a clearer calculation and is free from superfluous illusions about higher accuracy of the sum estimation by the discount financial functions.
The predicted variability of a cash flow can be an integral property of the object of estimation when the object of estimation is an isolated business – the acting company’s property complex or the part of the property complex changing in a quantitative sense in process of growth, for example unidentifiable intangible asset – good will.
Usually for identified assets (real and intangible, including a vessel) which are separate of the marine company’s property complex variability of a cash flow has unpredictable property external in relation to the object, and variability is the characteristic of a macroeconomic environment and is taken into account in the sum of risks.
In a predicted sense the cash flow for separate assets as a part of the marine company’s property complex is invariable in the present prices and depends on functional and structural characteristics, in particular on the characteristics of commercial vessel which are her steady identification attributes.
Concerning the isolated real assets it is more correctly to consider that the variability of their cash flow having unpredictable property is the macroeconomic characteristic and is considered as a component in the risks total sum. Therefore, to estimate the value of the isolated real asset of the marine company’s property complex by the income approach including vessel value, owing to stability of an identification attribute it is preferable to estimate by the income approach on the basis of direct capitalization method.
Even if the vessel reconstruction or repair is predicted this circumstance can’t be considered an argument in favor of preference of application of a capitalization of the discounted cash flow method instead of direct capitalization.
It is more logical to determine a present value of expenses for reconstruction and take into account it as the adjustment to the vessel value estimated based on income approach on the basis of direct capitalization method (by the formula 11.4) in the assumption of reconstruction, along with estimation of servitudes and other adjustments.
In consecutive statement of the task to determine the economic indexes of marine commercial vessels the systematized approach with account of taxes and deductions is required. In spite of the fact that in Russia and in other states the rules of taxation can partially differ generally for the size of rates of taxes it is necessary to consider available similarity and universality of approaches.
For calculation of taxes in the states producing an overwhelming part of a cumulative industrial product, similar economic indexes are basic: income, profit, cost of assets, operating costs.
In view of universality of the taxation base in different states having generally close legal system, distinctions in taxation rules concern generally the rates of taxes, rather than the methods of taxes calculation, leaving aside separate distinctions in used terms and designations. For the specified reasons the conclusions concerning rules of taxes account for marine commercial vessels evaluation have a rather universal character for the various states.
In total amount of taxes and the deductions arriving in the budget the greatest part (more than 95 % of the general taxes and deductions receipts both federal and local) makes: the profit tax, the added value tax, the property tax, the deductions for the social insurance.
Formally it is taxes of the specified kinds that are subject for accounting in marine commercial vessels evaluations, as cumulative receipts of taxes of other kinds are insignificant by their value.
The sales tax, apparently, is not important in sphere of rather large investments into construction and productive operation of marine engineering and ocean technology, as mentioned investments are made almost exclusively by the clearing settlement.
When estimating the vessel or other real asset of marine company’s property complex the cash flow of net operating income NOI is equal to difference between the annual income I and operating costs C, and also the property tax and some payments, for example insurance premium.
At the same time the profit and the added value taxes are not subtracted when calculating net operating income NOI. And deductions for the social insurance are also included into the structure of operational expenses C to expenses for salary payment joins also.
The property tax is taken into account when the value estimated by the income approach by adding the rate ηMV of the property tax to the rate of net operating income capitalization R.
In the expert’s activity the exception of the profit tax and the tax for the added value from the results of estimation would be an infringement of rules and approaches as the results of estimation further usually used in contracts between the parties should include the profit tax and the added value tax unless otherwise stipulated.
Besides, while estimating of marine commercial vessel or other real asset of marine company usually it is impossible theoretically to distinguish the profit tax and the added value tax as the base of taxes of the specified kinds are indexes of the whole marine company financial accounting instead of separately taken asset for which the financial record usually is not kept and it is not reproduce–able to the full.
As the profit tax and the added value tax should be a part of commercial vessel or other real asset all economic indexes as basic indicators – the annual income, the operating costs, the investments – and derivative indicators – the profits, the net operating income, etc., defined by mathematical derivations over the basic indexes the possible exception of the profit tax and the tax for the added value of any economic indexes methodologically would be incorrect.
With account taken of the profit tax and the tax for the added value when estimating the vessel or other real asset of marine company consists in preservation of the profit tax and the tax for the added value as a part of economic indexes and not exclusion of these taxes from profit and added value (and from other economic indexes in estimating of an asset).
As rules of the account of taxes in process of value estimation have a rather universal characteristic in various states, the approach of taxes inclusion in total sum of the marine company asset value is done by means of:
– Determining of economic indexes and also value without withdrawal of the tax for the added value and the profit tax, that is including the profit tax and the added value tax;
– Taking into account, as a part of operating costs, those deductions which base of determination are items of operating costs;
– Taking into account the property tax by addition property tax rate ηMV for the capitalization rate when using the income approach;
– Neglecting in estimations of the basic economic indexes the numerous taxes which cumulative share in the total amount of tax revenues is little.
It should be noted, that the method of accounting of taxes as a part of a capitalized index in case of estimation of the commercial vessel or the property complex of the marine company based on income approach concerns the method of determination of the capitalization rate (or discounting).
If the rate is determined on the basis of market extraction according to analogues of estimated objects the way of the rate calculation should correspond to a way of its application and the accounting of taxes as a part of capitalized index – to a way of the accounting of taxes as a part of corresponding index in rate determination. At the same time the resultant value includes taxes in full, just as analogue value when determining the rate by the market extraction.
The estimation of profit minus deductions for real assets restoration, taking into account preconditions considered above, is determined by the formula

 p=NOI-fm×MV=I-C-fm×MV. (11.2)

The estimation based on methods of income approach assume that the equipped vessel (fuel charged, supplied etc.) is operated with all advance payments for operation costs which correspond to working assets OC.
At the unlimited or rather great life time of commercial vessel (or other asset of the marine company) the discounting rate i (the sum of risks) as an indicator of investments efficiency MV+OC is equal

 i=p/(MV+OC). (11.3)

The formula (11.3), on the one hand, is a result out of mathematical summation by the formula (11.1) in condition of constant net operating income NOI on years (in the present value on the day of estimation), and also at rather great life time T of an evaluated asset within a summation limit (or in case of the limited life time, but with the account that reversion Rv is defined too by summation by the formula 11.1 or direct capitalization).
From the other hand, the formula (11.3) serves as a definition of the basic economic law: the profit p is proportional to capital investments MV+OC. And the proportionality factor corresponds to the sum of commercial risks according to the discounting rate i.
Substituting the equation (11.2) for profit estimation p in the formula of the basic economic law (11.3), it is possible to derive

 MV+OC=(I-C)/R=NOI/R. (11.4)

The rate of net operating incomecapitalization R = i + fm is determined as a sum of the discounting rate i including rates of the property tax ηMV and insurance coverage ηINS, and factor fm of depreciated part of investments fund of compensation, fm=i/((1+i)T-t-1) (article 13, tab. 13), where Tt is the period which has remained before write–off operation with the account life time T and vessel (or other real asset of marine company) years t, that is residual life time.
Just as the commercial vessel or other real asset of marine company estimation based on income approach of present values of cash flow (11.1), the result of calculation by the formula (11.4) is not only the value, it is more exactly – the total sum of cost of investments, that is the sum of asset value and present value of working assets OC×i/R with the account the limited life time of the commercial vessel and difference due to this case between the capitalization rate R and the discounting rate i, i.e.

MV+OC×i/R=(I-C)/R.

Test questions

1. Evaluation of the discounted cash flow by capitalization (the formula derivation).
2. Elements of a cash flow.
3. Ways of reversion determination.
4. Predicted variability of the cash flow as the basis for application of discounted cash flow capitalization method.
5. Objects of estimation possessing predicted variability of the cash flow (the property complex of the marine company, unidentifiable intangible asset – good will, etc.).
6. Preference of the discounted cash flow capitalization of the marine company assets value possessing predicted variability of cash flow is determined by the income approach.
7. Unpredictable variability of cash flow for the isolated asset as macroeconomic risk factor.
8. Recommendations of direct capitalization method when the isolated assets of the marine company value characterized by stability of functional attributes is estimated by the income approach.
9. Assets characterized by unpredictable variability of a cash flow (the isolated real assets of the marine company).
10. Recommended period of planning when estimation is made on the basis of cash flow estimation.
11. Area of the effective application of methods on the basis of cash flows estimation.
12. Net operating income and profit.
13. Advantages of the method of direct capitalization of the net operating income for the isolated asset (vessel) estimation.
14. Accounting of taxes in expenses (operating costs).
15. Features of taxes when the net operating income and vessel value are estimated by the income approach.
16. Adjustment by the size of working assets value to the result of commercial vessel estimation by formulas of the income approach.

11.1. Methodological conformity of cost and income approaches in estimation of the marine company’s property complex

Despite the methodological distinctions of the estimation by income and cost approaches certain formal conformity is available.
As with the account a relation of reproduction cost and the market value of the new asset MV(t=0)=MV(0), RC×(1-D5)=MV(0), corresponding to estimation of the external obsolescence (economical) in equally influencing the same assets regardless of their internal depreciation (D1, D2, D3, D4 èëè D1-4), in this case defined by the absolute monetary magnitude collecting over the time, the reproduction cost RC of the marine company asset based can be estimated by the income approach as:

 , (11.5)

and the market value of not a new asset MV(t) , having age t is possible to determine based on income approach by the formula (11.1) or, on the other hand, based on cost approach MV(t)=RC×(1-D5)-D1-4, thus

, or
, than the internal depreciation   D1-4in the sum physical, and functional and for assets of the marine company mostly functional with rather great life time, corresponds to a difference between the depreciated reproduction cost with the subtraction of the external economic obsolescence RC×(1-D5)=MV(t=0,0) and the market value MV (t) of not a new asset or a difference between the market value of new and not a new asset of the marine company , that is

,

or after derrivations of the received formula

the accumulated internal depreciation D1-4of the marine company isolated real asset can be defined as the index methodologically corresponding to difference between the cash flow NOI of present value for period of use by an asset t (duration of depreciation, age of an asset) and the present value of asset use MV(t)×(1-1/(1+i)t)=MV(t)-MV(t)/(1+i)t  for the same period evaluated at the end of use period

 MV(t)=RC×(1-D5)-D1-4, and depreciation is (11.6)
It should be reminded, that the choice of the mathematical form of statistical regression (4.5–4.8, article 4.4) for value estimation of commercial vessel (or other asset of the marine company) based on comparable sales approach also takes into account estimation of asset ability to generate the income and depreciation of asset value in process of transformation of accumulated net operating income NOI into the financial form.
The derivation of the formula (11.6) pulls together the cost and income approaches to value estimation of the isolated real assets of the marine company

or

 . (11.7)
Using the direct capitalization of the invariable net operating income NOI in the present value, if the factor of fund of compensation fm of investments depreciated part as a component of the capitalization rate R=i+fm  of net operating income can be neglected with rather great life time of the asset, the formula (11.7) which is a combination of cost and income approaches, will be transformed as:

MV(t)≈(RC×(1-D5)-(1-1/(1+i)t)×NOI/i)/(2-1/(1+i)t).

The retrospective summation of the discounted cash flow in the subtracted part of the formulas (11.7) is a certain methodological difference from an estimation based on income approach: the cash flow is summarized for the last interval of period of vessel or other asset of the marine company use, whereas in estimation by the income approach the cash flow is summarized for the perspective forecasting period and is taken into account in an evaluation of present value with «plus» sign. Thus in order to forecast the cash flow to some extent the optimistical concept – the scenario can be used, as, for example, the scenario of the prospective highest and the best use is applied for value estimation of the real estate territory sites under the building.
Apparently, being for a vessel or other asset of the marine company a certain analogy of methods of estimation of depreciation on a basis of both the concepts of «life time» and the capitalization of the income loss, cash flow summation for the last period can be considered more objective estimation than cash flow capitalization expected in the future and even there is a possibility of keeping documentary records of cash flow in the past account.
The sum of the discounted cash flow evaluated for the last period methodologically corresponds to the scenario of the future continuity in relation to the past if the cash flow is predicted for the future continuously with estimation of retrospective flow.
One shouldn’t assume that depreciation of the real asset over the time determined by the formula (11.7) does not cover the accounting of physical depreciation factors and structural durability but only characterizes asset depreciation in process of its transformation into the financial form of accumulated net operating income NOI. By definition of net operating income NOI, the operating costs are subtracted from the annual income that is NOI = IC which contain also expenses for maintenance of the marine company asset in a suitable and efficient condition.
If to consider that the factor of the cost approach in the formula (11.7) corresponds the component containing reproduction cost RC×(1-D5), and the factor of the income approach – the component characterizing present value of net operating income NOI cash flow for a period of asset use within the age t the proportion of these factors varies over asset years in process of the net operating income accumulation.
In estimating by the formula (11.7) over asset years the cost approach factor influence decreases and influence the income approach increases.
For a new asset (for example, a vessel) substituting t = 0.0 in the formula (11.7) it is possible to accept resultant dependence MV(0)=RC×(1-D5) , that corresponds to the cost approach use.
And for not a new isolated real asset at considerable years t→∞ , assuming stability of net operating income NOI and other economic indexes in the present values and considering that value of a steady cash flow of the net operating income is equal
,
substituting t→∞ in the formula (11.7) and at MV(t→∞)≈NOI/i  based on income approach it is possible to receive a resultant dependence

MV(t→∞)≈(RC×(1-D5)-(1+i)1/2×NOI/i)/2
or MV(t→∞)≈(RC×(1-D5)/(2+(1+i)1/2) for the assumption that the asset of the marine company keeps ins profitability and structural durability.
The possibility of introducing to the property complex analogy of the depreciation concept applied in case of the isolated asset value estimation based on cost approach is also interesting. The depreciation corresponds to subtracted part of the formula (11.7). And investments for the property complex estimation by this formula as well as cash flow are recommended to consider by present value

 . (11.8)

Test questions

1. Determination of the net operating income capitalization rate with the account the factor of fund of depreciated investments into vessel compensation.
2. Attributes of formal conformity and analogy when using income and cost approaches.

11.2. Adjustment to the value of the marine company assets estimation by the income approach with the account of the right of dispose

The isolated real assets and the property complex of the marine company are usually evaluated by the income approach for cash flow of the net operating income present value or cash flow value of other methodologically close index, for example, profit. In order to determine the present value, the corresponding factor of the capitalization or the discounting is used, and necessary adjustments are introduced, and the index of profitableness applied in estimation reflects both functional characteristics of evaluated asset or the property complex and efficiency of the ocean resources.
As the value according to the income approach is the capitalization (accumulation), the capitalized indexes can serve both the net operating income (or profit) that corresponds to the standard income approach and the expense of the value accumulated in capitalization, in itself.
It is possible to point out a well-known economic analogy that obligations under the loan include both its return and repayment of percent (interest rate) that is obligations are a two–componential index.
The capitalized index from the system methodological concepts point of view is not only the net operating income (or profit) and includes also the value expense, for example, as a value derivative in time, in other words, as value potential that in details considered above (article 4.4).
The structure of a capitalized index in estimation based on income approach is relative and should include both components which influence the evaluation based on income approach in a certain relation. The relativity concept can be borrowed from physics: as it is impossible to distinguish in many respects the similar mechanical phenomena of inertia and gravitation they are defined as a single entity (whole).
At the same time, the expense of the value as a part of capitalized index also is considered by the factor of fund of investments compensation which are going down over the time in value. Using typical discount financial functions the factor of fund of compensation makes appreciable impact only on the value of not a new asset which years are proportional to life time.
The specified methodological contradiction in respect of underestimation of the value expense as a part of capitalized index can serve as the cause of observable discrepancy of dependence of its value estimated based on income approach on asset years.
This contradiction is noticeable for assets which present value decreases over years, for example, for commercial vessels, and the value estimated by capitalization of the net operating income (or profit) only varies over the time slightly (fig. 11.1). The discrepancies also arise when coordinating the value estimation results by the standard independent approaches.

According with the standard income approach the asset has the value if generates the net operating income (or profit). Without touching the details, according to the income approach based only on capitalization of the net operating income (or profit) the value of the asset which does not generates the net operating income is equal to zero, and the assets which generates in the equal net operating income have equal value. The specified consequences of the income approach based only on capitalization of the net operating income (or profit), come into contradictions with economic observations and cause doubts.
The standard approaches and, among them, the income approach to a certain extent are conditional methodological approximations and they should be corrected from time to time. Taking into account that the capitalized index is not reduced only to the net operating income (or profit) and is also connected with the value expense, the analysis and specification of applied recommendations of the income approach is required.
The capitalization of the net operating income (or profit) corresponds to estimation of possession of the right–of–use, thus the possession on the right of disposal is not evaluated. Inclusion of the value expense in the structure of capitalized index means the estimation of property right of disposal that is the good of possession of the right of disposal even if the asset use is not implimented.

The value estimation by capitalization of net operating income NOI looks like:
MV(t)+OC=NOI/R=NOI/(i+fm)=(NOI/i)/(1+1/((1+i)T-t-1)).
After mathematical substitutions and derivations the vessel value (or other asset of the marine company) is determined by the income approach, as
 MV(t)+OC=(NOI/i)×((1+i)T-t-1)/(1+i)T-t. (11.9)
With the account the working assets the estimation can be fulfilled on the basis of the potential equation of present value (4.3, article 4.4) integration by the formula

MV+OC=(MV(t=0,0)+OC)×(((1+i)T-t-1)/((1+i)T-1))i/ln(1+i),

which after replacement of symbol MV(t=0,0)=MV0 and in the assumption about small commercial risks, that is, the discounting rate i, with which i/ln(1+i)≈1 , becomes:
 MV(t)+OC=(MV0+OC)×((1+i)T-t-1)/((1+i)T-1). (11.10)
The formulas comparison shows formal partial similarity of estimation based on income approach (11.9) and integration of the present value potential equation (11.10).
After mathematical derivations the equation (11.10) becomes:

,

that for the option of discounting by the Euler number exponential function (1+i)t≈ei×t  corresponds to the formula:

 . (11.11)
For a rather new vessel (or other asset of the marine company), for which 0,0 ≤ (i×t)n <<1 , for example, at the stage of designing or construction the linearization of the discounting exponential function with the account fast increase of the denominator n! with number n of components of exponential function of ei×t decomposition in the Taylor range: ei×t= 1+i×t/1!+(i×t)2/2!+(i×t)3/3!+... , – corresponds to approximate mathematical formula ei×t≈1+i×t.
After mathematical substitutions the estimation of present value applying the results of integration of the linearized potential equation (11.10 and 11.11) is done by the formula
 MV(t)+OC≈(MV0+OC)/(1+t×i+t/(T-t)), or MV(t)+OC≈(MV0+OC)/(1+fMV), (11.12)
where fMV=t×i+t/(T-t) – the factor of present value potential, taking into account vessel depreciation (or other marine company asset) in process of the transformation to the financial form along with the net operating income accumulation.
If to continue analogy and conditionally to accept that the value of the new vessel (or other asset of the marine company) at the stage of finance investing corresponds to capitalization of net operating income MV0+OC=NOI/i, the estimation by integration of the linearized potential equation of present value (11.12) will become:
 MV(t)+OC≈NOI/RMV=NOI/(i+i2×t+i/(T-t))=NOI/(i+fNOI), (11.13)
where fNOI=i2×t+i/(T-t) – the factor of the net operating income NOI potential, evaluated with the account the factor of present value potential, as fNOI=i×fMV; RMV – the capitalization rate of the value and the net operating income potentials factors accounting RMV=i+fNOI.
Within the frameworks of the considered assumptions the rate of the net operating income capitalization, taking into account the factor of the net operating income potential defined by the factor of present value potential, is evaluated, as RMV=i+fNOI.
Without addressing the assumption of the equation linearization which is fair with rather big predicted life time, for example, for a new vessel (or other asset of the marine company), the definition of factor fMV of value potential according to the model of its expenditure along with accumulation of the net operating income looks like:
 fMV=(1+i)t+((1+i)t-1)/((1+i)(T-t)-1)-1, (11.14) or fMV=ei×t+(ei×t-1)/(ei×(T-t)-1)-1.
Outside a methodological assumption of correlation of initial value with present value of cash flow NOI/R, that is in more general statement, and in the assumption that initial value corresponds to value estimation by capitalization of the net operating income with the confidential probability α and with the confidential probability (1 - α ) is the alternative evaluation with account of present value potential, the vessel evaluation (or other marine company asset) balanced by the factor of probability α as dependence of value on years becomes:
 MV(t)+OC=α×NOI/(i+fm)+(1-α)×(MV0+OC)/(1+fMV, or (11.15) MV(t)+OC=α×NOI/(i+i/((1+i)T-t-1))+(1-α)×(MV0+OC)/(1+t×i+t/(T-t)).
The dependence (11.15) of the vessel value (or other real asset of marine company) on age in the sum with working assets by the relative value ( MV(t)+OC)/(MV0+OC)  is presented on the graph (fig. 11.2) and also reflects the value estimation by the capitalization of the net operating income with confidential probability α and the evaluation with the account the present value potential with confidential probability (1 - α).
In the assumption of rather great life time T (or residual life time Tt), when influence of the reversion Rvon the value determined by capitalization of the discounted cash flow of the net operating income by the formula (11.1) can be neglected, that is, if
and ,
in the assumption that the initial value corresponds to an evaluation by capitalization of the net operating income with the confidential probability α and is an alternative estimation with the confidential probability (1 - α), with account of the present value potential, the balanced evaluation of commercial vessel (or other asset of the marine company) as dependence of value on years looks like:
,
or in the linearized assumption that the factor of present value potential is defined asfMV=t×i+t/(T-t), the estimation balanced on confidential probability α by the capitalization of the discounted cash flow of the net operating income corresponds to the formula
 (11.16)

To estimate a new vessel (or other asset of the marine company), for example, at the stages of designing and construction (or purchase at the second hand market at value MV0  with t=0), the balanced value estimation by the formula (11.16) looks like: ,
with the account that MV(t=0)=MV0  and (1+t×i+t/(T-t))=(1+0×i+0/(T-0))=1
with t=0 after the mathematical derivations it is possible to receive the formula
,
or , that corresponds to value estimation based on income approach by the formula (11.1) with the account that we determine the new vessel value (or other asset of the marine company).
The same conclusion about the determination of a new vessel value (or other asset of the marine company) with rather great life time by the income approach, when fm≈0.0 , as about the result of generalization of estimation by capitalization of the net operating income with confidential probability α and the alternative evaluation with confidential probability (1 - α), with account on the present value potential it is possible to receive by the derivation of the formula (11.15): MV0+OC=NOI/i .
It is possible to find that the initial value MV0 estimation in the sum with working assets OC does not depend on the structure of a capitalized index that is on confidential probability balance α of the value estimation by capitalization of the discounted cash flow of the net operating income (the same can be seen at the graph, fig. 11.2).
The initial value MV0 estimation corresponds to the present value of a cash flow based on income approach  that, as an assumption, is used above for derivation of the formula (11.13). Though there is a methodological difference in the fact that dependence of the value in the sum with working assets on years by the formula (11.13) does not reflect the evaluation based on income approach, taking into account with confidential probability α by the formulas (11.15 and 11.16) but only an alternative estimation with confidential probability (1 - α), derived from the potential equation of present value solution.
With rather great life time T (or residual life time Tt) with which the reversion Rvinfluence for value of new vessel (or other asset of the marine company) can be neglected and whenfm≈0.0, the dependence of value on the age corresponding value estimation by capitalization of the net operating income with confidential probability α and the alternative evaluation with confidential probability (1 - α), with the account of present value potential it can be defined by the mathematical derivation of formulas (11.15 and 11.16) and substitution of MV0+OC=NOI/i êàê
 MV(t)+OC=(α+(1-α)/(1+fMV))×NOI/i, (11.17) or MV(t)+OC≈(α+(1-α)/(1+t×i+t/(T-t)))×NOI/i, (11.18) or
In further clarification of formulas (11.7 and 11.8) received on the basis of the methodological conformity of cost and income approaches (article 11.1) based on the value estimation by capitalization of the net operating income with confidential probability α and on the evaluation of potential of present value with confidential probability (1 - α), the value of the isolated real asset of the marine company can be estimated by the formula
 (11.19)

or, for a rather new asset of the marine company
,
and initial value minus external economic obsolescence corresponds to an estimation by the formula MV0=RC×(1-D5) .
With the account difference of investments for the years of summation tN , depreciated with the account the external obsolescence (economical)  , and the net operating income with expertly evaluated the confidential probability α, and also the alternative evaluation of present value potential with the confidential probability
(1 - α), that is

,

the marine company’s property complex value is equal:

 , (11.20)

or at the innovative stage of the marine company when financial assets are invested into acquisition of sets of real assets the marine company’s property complex value can be estimated, as:

.

Test questions

1. Capitalized index in case of the isolated real assets or the property complex evaluation based on income approach.
2. Two–componential structure of capitalized index in case of evaluation based on income approach.
3. Methodological contradiction in respect of underestimation of the value expenditure when it defined by capitalization of the net operating income.
4. Attributes of the contradiction of underestimation as a part of the capitalized index of the vessel value expense (or other real asset of the marine company).
5. Evaluation of property right of disposal by inclusion of the value expenditure into the structure of capitalized index.
6. Partial formal similarity of value estimation based on income approach and by the integration of the present value potential equation (to present formulas and to compare).
7. Estimation of present value with application of results of integration of the linearized potential equation.
8. Rate of the net operating income capitalization with the account expenditure of present value potential.
9. Formulas of evaluation by capitalization of the net operating income with confidential probability  and the alternative evaluation based on the account of present value potential with confidential probability (1 – ).
10. Methodological assumptions concerning reversion influence and the factor of fund of investments compensation for the value estimation with the great life time of commercial vessel (or other real asset of marine company).
11. Formulas of value estimation by capitalization of the discounted cash flow of the net operating income balanced by the confidential probability .
12. Independence of initial value estimation on the capitalized index structure.
13. Applied recommendations (the formula derivation) for definition of value dependence on years according to value estimation by capitalization of the net operating income with confidential probability  and evaluation with account of present value potential with confidential probability (1 – ).
14. Vessel value estimation or other isolated asset of the marine company for the methodological conformity of cost and income approaches based on capitalization of the net operating income with confidential probability  and based on evaluation of present value potential with confidential probability (1 – ).
15. Evaluation of the marine company’s property complex based on the methodological conformity of cost and income approaches with the account an annual difference of investments and the net operating income with expertly evaluated confidential probability ..

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