Modeluvannya ekonomiki - Vіtlіnskiy V.V.

4.3.3. Імовірнісна model впву чинників ризику

{Smolyak, SA Risk Accounting in Setting the Rate of Discount // Economics and Mathematical Methods. - 1992. - Vol. 28. - Vol. 5-6. - P. 794.}

Statement of tasks. The process is sustained by the income of the function of the object to be characterized by the intensity ( x ) ( x ). Todi dokhid, oderzhuvannyh for dos mali interval ( t , t + dt ), where x ( t ) dt . The energy intensity that is sustained in the income x (0) = X on the back of the function of the obtect (at the moment t = 0). On podalshu dinamiku chjogo otpazhnika vplyvayut dvі grupi chinnikiv.

  • Фізична зношуваність underlying засобів . Pripimo, scho zalizhno vіd vіk the basic засобів дохід зменшується лінійно:

X ( t ) = X - bt . (4.22)

Відомо, що об'єкт доцільно експлуатувати доти, вы дохід від ннього не від'ємний. Tomu in kitsy terminu ekspluotaatsii (at rik T ) moe bouti x ( T ) = 0. Zvidsy

Abo (4.23)

  • Riznogo kind vipadkovі chinniki, scho zumovlyuyut rizik vikoristania ob'ekta. Yakbi їх vplyvu not boolo vino vin buv neistotnim, then for vidomoi rates discount dokhid vіd functііonuvannya ob'ekta obkaslyulivayusya b for the formula:

(4.24)

Vvazhatimemo vidomimi (viokremimo) three tipip vipadkovichh chinnikiv, scho vplyavayut on dohіdnist 'ob'ekta: vypadkovi "zboi" in vibrobitnvі; Різкі зміни економічного середища ("catastrophe"); Випадкові коливання цін, податків та обсягів попиту.

Щоб adequately відобразити ці чинники в нормаі дисконту R , потріібі виконіти two варіанти розрахунку дохідності від роз -глядуваного об'єкта.

In the first, but the rate of discount R , zagadі vipadkovі chinniki vzagali not to take up respect, and the integral discounting of income is determined from (4.24).

In another, to spiroyaetsya on "bezryzikovu" norm Rj , tsі chinniki include without a middle type in the model of the vipadka process of zmini intensity of income.

Todi znachennya normi discount R , yak vrahovuyu chinniki riziku, dotsilno zrobiti such, shchob obidva varianti rozraunkiv nevertheless assessed the effectiveness of the function of the obekt.

Розгляньмо докладніше моделювання the income from урахуванням перелічених чинників ризику.

Vipadkovi "zboi" near the vibrobit. Do not forget that at the moment t the subject was characterized by the activity of intensity gained by the income x ( t ). Todi, in the course of an advancing small time interval, dt is forbidden to be "zbіy" for vibrobnitsy w іmіnіrnіstu w dt , abо об'єкт functііонутитиме "normally" зімірність 1 - w dt . Yaksho "zbiy" vidbuvsya, then the yogo usunennya potryben active hour t (vvazhatimemu tuu value small, or not unintentionally small), and takozh znobodlyatsya dodatkovi vitrati x, vzagali kauchuchi, vipadkovi.

Vvazhatimemo, scho після цього виробництво poverataetsya to his own alternate, tobto "zbіy" does not zmenshuyu odnutsionnuvannya ob'ekta.

"Catastrophe". The order with the buyer (vlaskom ob'ekta) that the very kind of disinfection and construction of the property. You can be so, so chtoos iz them rasrobit noviy efektivnyi sposib (tehnologii) vibrokittyva vidpіdnoї produktsii, y zv'yazku zim cim on product rizko vpade. Todi further away the function of the first obekta vzhe not zabezpecchuatime income, but for pidpriemtcya stane ekonomichnoy catastrophe. Analogical situation, can, and, if the situation is changed in the same way, the law does not fully respect the situation in the region.

Nekhi імовірність such a "catastrophe" in the interval ( t , t + dt ) dorivnyuє kdt, de k - intensity of "disasters", but do not lie in t . Otsinyuvati dovmovnist such situatsіy mozhlivo lishe ekspertno, z urahuuvannyam proizvodіv analizu naukovo-tehnichnogo progressu vidpovidnyi vibrochnichy galuzi (pidgaluzi) and prognosis of the economical and political situation.

Oskilki values k і t - mali, "catastrophe" in the end of the day in the form of "evil" in vvazhatimemo nemozlimivi.

Kolivannia tsіn podatkіv ta obyagіv popita . By the end of the function of the function of the goods on goods, yak vigotovlyaetsya, sirovinu, materiiali, complementary virobi, and takozh otsyagi popita that rate of payment mozhut zmіnuvatysya. Pid vlivom chich chinnikіv інтенсивність oderzhuvannogo income takozh vypadkova kolovatimetsya. Yakshcho pripustimo, scho at the time of otsinyuvannya ob'ekta buli correctly viznacheni rozmiri income, then kolivannya іntenivnosti x ( t ), zoumovleni gruzogo rozgljanutih schoyno chinnikiv, matimut nulove mathematicheskogo spodivannya, ale characterize the song with a spindle (dispersity). It is natural to adapt, but in the small interval of the vipadkovi kolivannya x ( t ), mute the dispersion, and the intensity x ( t ) should not be stored in such kolivans at the crosswind for an hour.

I will look at the pricelist, let me describe the model of the Vinar process in Vinera, toto the intensity of income in close proximity to the hour t i t + dt is satisfied with the sig- nificance

X ( t + dt ) = x ( t ) + s d w ( t ), (4.25)

De s - midweekwatches vidhilennya vipadkich kolivan intensity in income x ( t ) per oditnitsu hour (the middle square of such kolіvіn per hour dt дорівнютиме за цих умов s2 dt ; w ( t ) - звичайний вінерівський випадковий процес.

Mathematician spodivannya income in razi uroakhuvannya chinnikiv riziku.Vaznachimo through V ( x ) the value of the mathematical method of integrating the discounted income (for the rates of discount Rj ) vіd ekspluotaatsії ob'ekt to zakinchennya term ynogo funktsionovannya (Vipadkova value). Nehai in the rudimentary moment, the vibrokitsnut functioned "normal", and the intensity of the well-known income of the boule x . Obviously, V0 (0) = 0. It is natural to sight a vipadoc if x > 0. It is commendable, but for V ( x ) there is no meaning, such as, vague, moment of hour of brother for cobblestones. You can achieve the discount by the time t = 0.

Razglyanmo maliy interval (0, dt ). There are three situations here.

  • З імовірністю w dt will become збій у виробництві. On the basis of yogo nalidkіv, vithrati (Vipadkova value) should be known, the value of such x is discounted. Vibrobitzhno normalizuyutsya through vipadkivy prom for hours t (Vipadkova value), pislya chogo ob'ekt pereide to normal stanu, yakomu vidpodiada mathematicheskogo spodivannya Integral discounted income V ( x ). At the time, you can get up to the hour of devotion to "evil". To increase the number until the moment t = 0, then the value of V ( x ) should be multiplied by the discount discrete key .

For pripuschennya, scho hour lіkvіdatsії "zboyu" vipadkovyi і підлягає експоненціальному to the law роподілу із середнім значенням q, mathematica сподівання М disconnectable сафіцієнта mozhna podati v vyglyadі

(4.26)

Vvazhayuchi, sho dodatkovi vitrati in the process of usunnenya naslidkіv zboyu vidbuvayutsya rivnerno, and їh value for oditnitsyu hour to become z , mozhdemo mathematicheskogo spodivannya discounted vitrate, povyazanyh with one "zboem":

(4.27)

  • Упродовж інтервалу (0, dt ) часу з імовірністю kdt відбудеться економічна катастрофа. At such times, the vibrobitzt is hanged, and then the integrating discounting of the offensive function of the ob'ekt on a zero value zero.
  • At the interval (0, dt ), the number 1 - (w + k ) dt of the functional function is "normal". Todi for an hour dt before the establishment of x ( t ) dt , for which the intensity of yogic restraint is to be changed by the value of bdt for the races of the phizic old age of the main prehensions, and so on (4.25) - on s d w ( t ) in the vipadic kolivan cyn Abo) submissions. At the moment of the hour, the dt object is characterized by the intensity of the income x - bdt + s d w ( t ). Цьому відповідає Integral discount (up to the moment 0, not d ( t )) дохід, що дорівнює

Urakhovoychi dovmovnist kozhno z rozgljanutih situatsіy і ti stan, y yakomu perebuvayot ob'ekt pislja them, you can record visas for a mathematical disintegration of an integrative discounted income in the function of the obekt:

In the case of (4.25), (4.27), if we want to reduce the order to small values ​​to the order of the first order, we can:

(4.28)

It is permissible, however, to function smoothly, and for the sake of the x 0 її friend, the V '' ( x ) існує і є непереревною. Todi, having rooted the rest of the spinner in the Taylor series and having scooped, the value of d w ( t ) is equal to zero of the value of the mathematical dispersion and variance , Knowable:

Звідси маємо рівняння

(4.29)

De

D = Rj + k + (1 - q ) w. (4.30)

Lіnіyna funktsіya

V 0 ( x ) = ( x - C w) / (s - b / s2) (4.31)

Є one із розв'язків цього рівняння. Overwhelming consignments (4.29) note that the sum of V 0 ( x ) and the outflow of the unidirectional rivnyannya

(4.32)

Obviously, (4.32) there are two linearly independent decongestants - e l x i e m x . Here

. (4.33)

The values ​​of l, m are the values ​​of the characteristic characteristic rivnyannya.

Otzhe, zagalny rozv'yazok (4.29) moe viglyad:

V ( x ) = V 0 ( x ) + Ce l x + C 0 e m x .

(4.33) is bachimo, and m> 0> l. Otche, if C 0? 0, then the function V ( x ), xaxx x ® +, is exponential to + Abo eksponentsialno sladatime to - ?. Portivnyuchi with deterministic vipadkom, bachimo, scho function V ( x ) is not aging for x ( t ). The integer is bounded if C 0 = 0.

I call V (0) = 0 independently of C = - V 0 ( x ). Далі запишемо

V ( x ) = x / d - ( C w / d + b / d2) [1 - e l x ]. (4.34)

Zokrema, mathematician spodivannya integrally discounted income in the function of the objekt at the moment, if you become accustomed to a disproportionate deviation, then if you apply yoga to the power of the dowry, you can calculate for the formula (4.34) for the name, xo x = x , toto

.

Agreguvannya vplyvu vipadkovih chinnikiv in one show. It is commendable, if w = k = s = 0 i Rj = R , then formula (4.34) is rewritten by formula (4.24). Obviously, it is realistic for the income of a product to not be absorbed in a linear model (4.22).

Todi zruchnishe obmezhitisya prognozuvaniyam dinamiki srednnyogo znachennya income, aggruvshivshy in a single osyuyu nyayavnu іnformatsіy about vpplyv vypadkovy chinnikiv.

At rozvinutyh kraїnah vplyiv chinnikiv riziku i neiznachenstisti vrahovuyutsya, on suti, vstanovlennnyam vidpіdnії rates of discount (about shcho vzhe yshelsya earlier).

In order to calculate for the formulas (4.24) and (4.34) with the discount rates R i Rj, they were given the same results, be satisfied with the rivion

(4.35)

Can be learned:

With w / X = g, d (s T / C ) 2 = n , d T = a, RT = r. (4.36)

Spivvodnoshennya (4.35) can be filed in the following cases:

(4.37)

Otzhe, shcheob viznachiti nevіdomu norm discount, potrbno spogatku zgіdno z vihіdnoyu іnformatsієyu znajti? (Розв'язок рівняння (4.37)) і, нарешті, Calculate R :

(4.38)

Tse means, the norm of discount R z Urauuvanniam risiku vіdrіznyayatsya vіd? With a measurable criticism . Znachennya tsyogo koefitsynta, yak bachimo, to lay off the list of a, g and n .

It is possible to corrupt the proanalizovati into the skin of the skin of the kidneys.

Formulary such important imbibitions in the motivational model: the rate of discount R , the know-how in the manner that can not be taxed, the amount of money-free warehousing Rj, the activity of the charge, yaka vrachovuyu rizik (premi заa for the rizik) і є nezalezhnoju vіd Rj , ні у вигляді добутку цієї складової І якогось більшого від одниці коефіцієнта, який not lie down in the Rj і враховує ризик.

The cleavage of the stock of roses in the norms of discount to the hurricanes reduces the risk . For oditnitsu vimiryuvanya hour vizmemo one ric. Vvazhatimemo, scho "zboy" vibrobitvvy vinikayut at the middle one time on the river: , And the hour on the basis of the violation of the exponential law rose below the mean values ​​q = 0.04 (approximately two tyzhni). Pripustimo, sho vitrati, povyazhany z lіkvіdatsіyu nalіdkіv "zboyu", proporazatіyni vitratam odinitsyu hour, prichomu kozhen day "zboyu" is not deprived prisvodit vіsutnostіі vіdpіdіdnogo income, and i need dodatkovіkі vitrat, shоo become 50% of crocheted income Х. For the transfer of the transitions iz (4.26) and (4.27), we know:

Poyva novih technologii, dovorovazhenzhenya some zrobit rozglyaduvane vibrobnitsu zbitkim, you can vvazhat small. Yakshcho pripusti, scho tako "revolution" in the technology vidbuvayutsya in the middle three times for storichchya, then you can k = 0,03.
Запишемо співвідношення для параметрыів, які login to рівняння (4.37):

In the case of R & R, it is excluded from Rf і T t співвіdosenly .

Portivnyannaya sign R ta Rj pidtverdzhuyu visnovok about nonadivism and nonmultiplication vplyiv chinnikіv riziku on the rate of discount.