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Консорциум экономических исследований и образования

Серия «Научные доклады»

Стимулирование

инвестиционных проектов

с

помощью механизма амортизации

abcd

В.И. Аркин

А.Д. Сластников

С.В. Аркина

Научный доклад № 02/05 Проект (№ 01-080) реализован при поддержке Консорциума экономических исследований и образования Мнение авторов может не совпадать с точкой зрения Консорциума Доклад публикуется в рамках направления Предприятия и рынки товаров В.И. Аркин, А.Д. Сластников, С.В. Аркина 2002 Классификация JEL: C61, D81, E22, H3, H7 АРКИН В.И., СЛАСТНИКОВ А.Д., АРКИНА С.В. Стимулирование инвестиционных проектов с помощью механизма амортизации. — М.: EERC, 2002. — 89 c.

Построена модель поведения инвестора в реальном секторе российской экономики с учетом факторов риска и неопределенности. Модель учитывает такие элементы рос сийской налоговой системы, как налог на прибыль предприятий, НДС, единый соци альный налог, налог на имущество предприятий, механизмы амортизации и налого вых каникул. Получено оптимальное правило выбора момента инвестирования. Об наружены новые эффекты совместного влияния механизмов ускоренной амортизации и налоговых каникул на поведение инвестора. Найдена политика амор тизации, максимизирующая налоговые поступления в региональный бюджет. Прове ден сравнительный анализ старой и новой систем налогообложения прибыли. Изуче ны эффекты, возникающие при замене налога на имущества налогом на недвижи мость. Исследованы возможности компенсации процесса риска с помощью уменьшения ставки налога на прибыль и ускоренной амортизации. Установлено нали чие зон риска, которые не могут быть скомпенсированы.

Ключевые слова. Россия, налоговая система, инвестиционный проект, неопределенность и риск, амортизация, налоговые льготы.

Благодарности. Авторы выражают благодарность Майклу Алексееву и Ричарду Эриксону за полезные замечания и дискуссии. Авторы также благодарны РФФИ (проект 02-06-80262) и РГНФ (проект 01-02-00415) за частичную поддержку работы.

Вадим Иосифович Аркин Александр Дмитриевич Сластников Светлана Вадимовна Аркина Центральный экономико-математический институт РАН, лаборатория Стохастических моделей экономики 117418, Москва, Нахимовский проспект, Тел.: (095) 332 42 14, (095) 332 42 Факс: (095) 718 96 E-mail: arkin@cemi.rssi.ru, slast@cemi.rssi.ru Формат 60*90/16. Объем 5.75 п.л. Тираж 500 экз.

Отпечатано в ОАО "Экос".

117209, Москва, ул. Зюзинская, 6, корп. 2. Тел. (095) 332 35 36.

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.,. MacKie-Mason (1990)., (, ).

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Mt = Pt + s St.

Zt = Xt Ct = t St Dt Mt 5, (3.1) t = Xt Yt 6.

,.

i , f i 7 i. r p .

va  (), va va.

f r s ,, s s.

f r pi , f pi pi.

r, t, va t + i (t St Dt Mt ) + Pt + s St.

f f va t + i (t St Dt Mt ) + (s + pi )St, f f (3.2)  va t + i (t St Dt Mt ) + Pt + (s + pi )St.

r r r r (3.3) (  ).

Xt Yt 4 5,..

. 6 7 8  20 t Xt Yt St Mt i (Xt Ct ) = (1i )(t St Mt )+i Dt. (3.4) 3.2.,,.

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, 10 ( 1 30 ). :

( ),,.. (, 0 1);

( ), ( ).

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at, bt 0, at dt = bt dt = 1.

0 3. (, ),.. ().

0 t L, at = (3.6) t L, 0, L, = 1/L.

()., at = et, (3.7) 0.

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,,, ( ), (NPV) :

E (V I ) e max, (3.11).

( (3.2) (3.3)),.,, f T = E [va t +i (t St Dt Mt )+s St ]e(t ) dt F, f f f (3.12) ,, 24 + T = E r [va t +i (t St Dt Mt )+Pt +s St ]e(t ) dt r 0 r [va t +i (t St Dt Mt )+Pt +s St ]e(t ) dt F, (3.13) r r r + + f s = s + pi, s = s + pi.

f f r r r B,,,,..

+ B = E [va t + s St ]e(t ) dt + f i Zt e(t ) dt i Zt e(t ) dt F E t e(t ) dt F, (3.14) + + (3.1).

Zt 4., ().,.

4.1.

It t It = I0 + Is (1 ds + 1 dws ), t 0, (4.1) t 0) , 1 1  (1 0), (wt, I0 .

4. t, t t t = + s (2 ds + 2 dws ), t, (4.2) F -,(wt, t 0) , 2 2  (2 0).

wt wt 1 r,.. E(wt wt ) = rt t 0.

,,, =,,..  ,. (t, t 0) .

,, ( (3.10)),.

,, ((t, It ), t 0).,,, - Ft t,.. Ft = (s, Is ;

0 s t).

, (t, t 0) t t = 0 + s (2 ds + 2 dws ), t 0, (4.3) 0.

, ( ).,, Real Options Theory (Dixit and Pindyck, 26 1994;

Trigeorgis, 1996). (.,,., 1999) (,, ).,, ( dt) ;

(dwt ) i ( )..

(t, t, 0) ( ) Nt t = (1 j ), j= 0 j 1, Nt   (,, ), [, t].

j, j = 1, 2,... (wt, t 0). .,  Nt = Nt, (Ns, s 0).

,,..

 Ej,,.

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,,, St t, St = µt, µ .,,. -, 2,,, ().

(3.1) Xt Ct = t St Dt Pt s St = t ( µ) Dt Pt, µ = (1 + s )., µ,, µ, (1 + s ) 1.

µ 4.2..

F E.

E t, t. j Nt (wt ), Nt n P{Nt E t = E (1 j ) = = n}E (1 j ) j=1 j= n n (t )n (t ) (1 q)n = e(t ), = e n!

n q = Ej, = q.

28 (4.4) :

e2 (t ), E t = I e1 (t ), E It = E Pt p I [1 at (1 )bt ]e1 (t ), = I [at + (1 )bt ]e1 (t ), E Dt = :

s bs es ds, = + 1, At = as e ds, Bt = (4.5) t t t s (et es )bs ds.

At = (e e )as ds, Bt = (4.6) t t (3.9), :

1 at et dt = et dt ds = bt et dt = as A0, B0, s 0 0 s at et dt = et dt ds et dt ds = [e A ], as as bt et dt = [e B ], + p E Pt e(+)(t ) dt = I [1 (A0 + A ) (1 )(B0 + B )], p E Pt e(+)(t ) dt = I [ A + (1 )B ], + + E Dt e(+)(t ) dt = I [(A0 A ) + (1 )(B0 B )], E Dt e(+)(t ) dt = I [A + (1 )B ].

+ 4. + {(1 i )[(1 µ)E t E Pt ] + i E Dt }e(+)(t ) dt V = {(1 i )[(1 µ)E t E Pt ] + i E Dt }e(+)(t ) dt + + (1µ)(1i ) = +I [(i A0 +i A )+(1)(i B0 +i B )] + p I {1i [(1i )A0 +i A ](1)[(1i )B0 +i B ]} + (1 µ)(1 i ) p (4.7) = + I H1 H2, + 2 + r i = i + i e(+2 ), i = i i, H1 = (i A0 + i A ) + (1 )(i B0 + i B ), H2 = 1 i [(1 i )A0 + i A ] (1 )[(1 i )B0 + i B ].

(4.7), t ( ).

(3.12) (3.13), (3.14):

f f f va + i (1 µ) + s µ p p f f T = i I + 2 1 A0 + (1 )B, va + (i + i e(2 ) )(1 µ) + s µ r r p r H0, T = I H 1 I ( 2 ) p B = va + i (1 µ) + s µ (1 K0 ) i K0, = (i A0 + i A0 ) + (1 )(i B0 + i B ), 0 00 H1 0 30 = 1 i [(1 i )A0 + i A0 ] (1 )[(1 i )B0 + i B ], 0 0 0 0 H2 0 0 K0 = A0 + (1 )B0, A0, B0, A0, B (4.5)(4.6) 0 0.

,., (1µ)(1i ) p (1i ) p (1i ) V = +I i + K, (4.8) +2 +1 + K = A0 + (1 )B0.

,,,, (, ) at et dt). ( 6.

4.3.,,.

( 1) + (2 1 ) ( 1 ) = 0, (4.9) 2 = 1 2r1 2 +2  .

2, 1 2. 0, 1 2 1 2 1 1 2( 1 ) = + +.

2 2 2 2.

4. 1. (4.1), t  (4.3).

, 0 :

12 2 2 1 1, max(1, 2 ), + 1.

2 = min{t 0 : t p It }, (4.10) p + 2 p =, H1, H 1 H1 + H2 · · + 1 (1 µ)(1 i ) i (4.7).

,.

. = 0 i p + 2 p = 1 + + (1 K) · ·, 1 i + 1 1µ K (4.8).

, p.

(4.10).

(4.7) V,,, V /I p p H2.

1 H1 + H2 + H 1 + 1 +, = 0,, 0 p I0.

,.

32 (.. (3.11)) f N, T f = E T e, T r = E T e r , B = B . E.

4 7,.

2. (4.1), t  (4.3).

12, 0, 2 2 1 1 max(1, 2 ).

2 :

0 p 1) N = I0 ;

1 H1 + H2 · I0 p + 1 f f f 0 va + i (1 µ) + s µ p f 2) T f = I0 p i I0 p 2 p, + 1 K va + (i + i e(2 ) )(1 µ) + s µ r r 3) T r = I0 p I0 p p H0 ;

H1 + 1 p (1K0 )i K0 (p )1 ;

4) B = va +i (1µ)+s µ(2 ) I0 p 12 5) E = (2 2 1 + 1 )1 log ;

2 2 p 1, K0 = A0 + (1 )B0.

( 1 2. 7.) 4. 4.4..

,.

,.

(, )..

p. 1, p ( ), ( ).

,,,,.,.,,, ( )  11.

p,.

(.. p = 0).

+ 2 p = (1 H1 ), p = p ·, (4.11) (1 µ)(1 i ) i = i +i e(+2 ), H1 = i [A0 +(1)B0 ]+i [A +(1)B ], A0, A, B0, B (4.5).

,,, -- (. Mintz, 1990) 34 p H1 (1H1 ) i p = pi e(+1 ) [a +(1)b ] = p + · 1i (1H1 )p i (+2 )e(+2 ) 1i = pi e(+) c e1 (1 µ)p e2, c = a + (1 )b.

p, c e1 (1 µ)p e, () I, = I c e1 (1 µ) e (,, !).

., (3.5) (4.4) I c e1 = E D + | F ) ( ) +, (4.3) e2 = E + | F ) ( ) +.

p,.,..

0,,,.,.,, : ( p ),...

4..

, ( )., = E0 E  ( ).

4.1.

2 = 2%, 2 = 0.1, 1 = 1 = 0 = 10% ( ).

35% ( ) 3% ( ).

4. = 0.9 = 0. 1 -3.2 -1. 2 -4.6 -1. 3 -5.1 -1. 4 -5.1 -0. 6 -4.2 0.,,,., 10%, ( = 0.9).

p 12.

(4.11),. p = p (A), A = i A0 + i A .

A.

36 6 - 0 1/ 0. 1. A, p (A)., (i = 0) ( ) A. A : (. (3.6)) (. (3.7)) .1.

, A : 0, 1 + / = e/ ;

 0, =.

+, 0,,.. 0 = 0 = = 0.

, p.  ( ), p,,. ( ), ( p ),...

5.,.,.

A ( i ). A, i + i e(+) [ ( + )] = 0.

.

5.

,.

,,,,,,, ( 3)., ( ),. (, ),,.

,,.,,.

38 5.1..

( NPV ).

, ( ) p t /It.

p = p (D), D,, 1, = (D)., D, (. 2 4.3):

va + (i + i e(2 ) )(1 µ) + s µ r 0 r 0 T r (D) = I0 p I0 p p H0, (5.1) H1 + 1 H1 = H1 (D), H2 = H2 (D).

0 0 0,,,., ( ), (5.1)., T r (D) D:

T r (D) max. (5.2) DD D (, ),.,, 5. NPV.

( ),..

Zt ( (3.1)). Arkin and Slastnikov (2000).

5.2. ( ),.,,, (at, t 0). (bt, t 0).

D ( ) D. D = (at, t 0) at e(1 )t dt, A = A(D) =.

min A(D) = A, max A(D) = A.

DD DD, ",, (C) a, A a A D D, A(D) = a.

min / max, inf / sup,, D 40, {a : a = A(D), D D} [A, A].

,,,,,.. = 0 = 0. (5.1) :

T r (D) = I0 (1 u) [q r (1 u) h1 u + h2 ], (5.3) I0 p p (1i ) u=u(D)=(i +)K, =, K=K(D)=A(D)+(1)B, r + i (1 µ) + s µ r r q = va r bt e(1 )t dt, ·, B= (1 µ)(1 i ) r + r i r i r r p (1 i ) h1 = i, r =.

, h2 = i + i +, (5.2) ( (5.3)),, g(u) max, (5.4) uu u g(u) = (1 u) [q r (1 u) h1 u + h2 ], u = (i + )[A + (1 )B], u = (i + )[ A + (1 )B], g(u), :

g (u) = (1 u)1 [q r (1 u) h1 u + h2 ] (1 u) (q r + h1 ) = (1 u)1 G(u), (5.5) r r G(u) = [q (1 u) h1 u + h2 ] (1 u) (q + h1 ) = ( 1)q r (1 u) h1 u (1 u)h1 + h (5.6) = (Q h1 )(1 u) h1 u + h2, r + i (1 µ) + s µ r r = va Q = q r.

(1 µ)(1 i ) G(u), G (u)=Q(1)h1 0.

(5.4). u (5.4).

5. G(u) 0, G(u) 0 u u, (5.5) g(u) u, u = u,, D, A(D ) = A. (5.6), G(u) 0, [Q (Q + h1 )u + h2 ] (1 u)h1. (5.7), G(u) 0 u u, [Q (Q + h1 ) + h2 ] (1 u)h1.

u (5.8) g(u) u,, u = u.

(5.7) (5.8), g(u) u, G(u ) = 0,..

Q h1 + h u =. (5.9) Q + h1 h,.

3. (C). D = (a, t 0) t, a e(1 )t dt :

A= t (5.7) A, u + A= (5.7) (5.8) i + (1 )B /, (5.8) A, u (5.9).

, ( 3.2).

(. (3.6)) A = ASL () = 1 e(1 )/.

, ASL ().

42,,.. 0., (C).

A = ASL (), A = ASL ().

1. :

(5.7), (5.7) (5.8), =, (5.8) u + ASL () = (1 )B, u i + (5.9).

(. (3.7)) A = ADB () =.

1 + ADB ().

, (C). A = ADB (), A = ADB ().

2. (5.7), (5.7) (5.8) ( 1 )A/( A), =, (5.8) u + (1 )B, u (5.9).

A = i + 5. 5.3. ( ).

,,.

,, ( ) Zt.

,,..

,,, () :

t E(Zt | F ) 0 (..) (5.10) Zt (3.1) (4.3)(4.4),, (5.10) : t (1 µ) e2 t I e1 t {at + (1 )bt (5.7) (..), (5.8) + p [1at (1)bt ] at, bt (3.9).

1,, (5.10) : t 2 (2 1 )t (1 u) · e at + (1 )bt 1 1 i (5.11) + p [1at (1)bt ], u (5.3).

(5.11)..

44 (3.7) a, b .

a b at = a ea t, bt = b eb t, K = + (1 ), a +1 b + (5.11) : t 2 (2 1 )t (p +a )ea t +(1)(p +b )eb t.

(1u) · e 1 1i (5.12), a b ( ), 2 + b 1 (5.12) t 0, :

(1 u) · p + (a b ). (5.13) 1 1 i, (5.13) a 0, 0 (1 u) · = p + (a b ). (5.14) 1 1 i (5.10) = min(, 0 ), 2 3, 0 (5.14).

(3.6) a , b . at = min(a t, 1), bt = min(b t, 1), a 1e(1 )/a +(1)b 1 e(1 )/b K=. (5.15), 2 1. (5.11) t,  t., 5. (5.11) t 0, t = 0,..

(1 u) · p + a + (1 )b. (5.16) 1 1 i,, (5.16) a 0, 0 (1 u) · = p + a + (1 )b. (5.17) 1 1 i, (5.10) = min(, 0 ), 1 3, (5.17).

, (5.10),. 0,.

5.4. -.

(.. 1 = 1 = 0). " 2 1% 2%, 2 0.1 0.15, 10% 20% ( ).

(,, ).

,,.. = = 0, = =.

( 1 2), 0 0  (5.10),.. (5.17) (5.14).

46, ( 16.5%15 ), (  2%) (13% ).

5.1, 0 0 (5.10) 2. 2 = 0.12, 10%, = 0.9, 5% (, 3.5% ), ( ) 30%.

5. 2 0 0.15 0. -1% 0.47 0. 0.14 0. 0% 0.24 0. 0.14 0. 1% 0.14 0. 0.08 0. 2% 0.13 0., (5.10),..

.

,, (5.10).., (5.10).

5.. 2 = 1%, 2 = 0..

5. 0 0.14 0. 0.9 0.14 0. 0.15 0. 0.8 0.17 0. 0.17 0. 0.7 0.24 0. 0.20 0. 0.6 0.47 0. 1 2 3.

5. 0 (2 = 1%) 0.16 0. 10% 0.21 0. 0.23 0. 15% 0.49 0. 0.30 0. 20% 1.08 2. (2 = 0%) 0.14 0. 10% 0.15 0. 0.22 0. 15% 0.32 0. 0.29 0. 20% 0.65 1. (2 = 1%) 0.09 0. 10% 0.15 0. 0.21 0. 15% 0.22 0. 0.29 0. 20% 0.44 0. 5.3 2. 2 = 0.15, = 0.9.

48,,.

,, .

5.5. ?

,,,,, 4.4.

,, (NPV). .,.  0 = 20%,.

,, 5.3,., NPV. T r ()16 (5.3). T f () NPV N () :

5. T f () = I0 (1 u) [q f (1 u) h3 u h4 ], (5.18) I0 p I0 (1 u)+1, N () = (5.19) 1 I0 p f f f va + i (1 µ) + s µ q f =, u = u() (5.3), · (1 µ)(1 i ) f f i p i p.

h3 = 1, h4 = + i i + 1 i +, :

T r ( ) 1 u0 q r (1 u ) h1 u + h r E = =, (5.20) T r (0 ) 1 u q r (1 u0 ) h1 u0 + h T f ( ) 1 u0 q f (1 u ) h3 u h f E = =, (5.21) T f (0 ) 1 u q f (1 u0 ) h3 u0 h N ( ) 1 u N E = =, (5.22) N (0 ) 1 u u, u0  0.

 20,. T r (20 ) 1 u0 q r (1 u2 ) h1 u2 + h r E2 = =, (5.23) T r (0 ) 1 u2 q r (1 u0 ) h1 u0 + h T f (20 ) 1 u0 q f (1 u2 ) h3 u2 h f E2 = =, (5.24) T f (0 ) 1 u2 q f (1 u0 ) h3 u0 h N (20 ) 1 u N E2 = =, (5.25) N (0 ) 1 u u2  20.

50,.

, :

( 7.5%), ( 20%) (35% ).,,,,.  2001.

1 3 (.. = 0, = ).

,.

,,  ( ) (, ). , (, = 0,, = ).,,, - ( ).

.

, = 10%, 3%.

5.4 (5.20)(5.25) ( = 0.9), (2 = 2%) 2 ( )  ( µ).

5.5 ( 2 = 0).

5. 5. f f r r N N µ E E2 E E2 E E (2 = 0.05) 0.2 1.01 0.97 0.91 1.12 0.90 1. 0.35 1.01 1.01 1.12 1.13 1.13 1. 0.5 1.11 1.06 1.32 1.13 1.34 1. 0.65 1.25 1.11 1.33 1.14 1.34 1. (2 = 0.25) 0.2 1.15 0.95 0.80 1.03 0.76 1. 0.35 1.06 0.97 0.82 1.04 0.80 1. 0.5 1.00 0.99 0.96 1.04 0.96 1. 0.65 1.05 1.02 1.09 1.04 1.09 1. 5. f f r r N N µ E E2 E E2 E E (2 = 0.15) 0.2 1.03 0.95 0.85 1.10 0.83 1. 0.35 1.00.99 1.00 1.10 1.00 1. 0.5 1.06 1.03 1.25 1.10 1.26 1. 0.65 1.18 1.08 1.25 1.11 1.26 1. (2 = 0.25) 0.2 1.11 0.94 0.78 1.05 0.74 1. 0.35 1.04 0.97 0.85 1.05 0.83 1. 0.5 1.00 1.00 1.04 1.05 1.04 1. 0.65 1.08 1.03 1.13 1.06 1.13 1. 52, 2, (.. t /It p ). 5.5, (2 = 0.15).

( = 0.5).

, (2 = 0.25).

5. f f r r N N µ E E2 E E2 E E (2 = 2%) 0.2 1.06 0.99 0.88 1.02 0.86 1. 0.35 1.02 0.99 0.89 1.02 0.87 1. 0.5 1.00 1.00 1.04 1.02 1.05 1. 0.65 1.03 1.01 1.04 1.02 1.05 1. (2 = 0) 0.2 1.04 0.99 0.86 1.02 0.83 1. 0.35 1.00 1.00 0.94 1.03 0.93 1. 0.5 1.01 1.01 1.06 1.03 1.06 1. 0.65 1.04 1.02 1.06 1.03 1.06 1. ( ).

,   (, 0.2).,, . 5., 1  2%. NPV.

( ).  ( ) µ . 10  15% ( ),. ( 10  25%),. 10  15%. µ ( 0.3  0.4) .

, 10%. µ ( 0.5),.,. 30  40% ( ),.

.,.

, ( ), : 1) ;

2)  ( );

3).,. ( ).

54 6. 6.1..

, , 31 2001.,, :

1)  35% ( : 11% , 19%  5%  );

2), ;

3) ( ) ( ).

1 2002.,, 24% (7.5% , 14.5%  2%  ).

,,.

1. p,.

2. T, ().

3. N, ().

4. B.

6. :

 va = 20%;

 p = 2%;

 s = 35%;

 pi = 13%;

 i = 35% ( ) i = 24% ( )17.

1 = 0, 1 = 0 ( ), 2 -1% 2%, 2 0.05  0.5, = 10%.

,,.. = 0. 3% 20%.

= 3 ka = 2.

:

p T Nnew Bnew Rp = new, RT = new, RN = RB = p Told Nold Bold old.

µ.   (,..), µ   ( ).

6.1 ( = 0.9, µ = 0.2).

,, 56 6. 2 2 = 0.25 2 = 0. -1% 0.85 0. Rp 1.12 1. RT 1.30 1. RN 0.86 0. RB 0% 0.84 0. Rp 1.09 1. RT 1.29 1. RN 0.86 0. RB 1% 0.83 0. Rp 1.07 0. RT 1.28 1. RN 0.85 0. RB 2% 0.83 0. Rp 1.04 0. RT 1.26 1. RN 0.85 0. RB 6.2  ( = 0.2, µ = 0.7).

, .

.,,.

6. 6. 2 2 = 0.25 2 = 0. -1% 0.96 0. Rp 1.07 1. RT 1.09 1. RN 1.00 1. RB 0% 0.95 0. Rp 1.07 1. RT 1.11 1. RN 1.00 1. RB 1% 0.94 0. Rp 1.08 1. RT 1.12 1. RN 0.99 1. RB 2% 0.93 0. Rp 1.07 1. RT 1.13 1. RN 0.99 0. RB ( 0.9). ( 15%  20%, ). ( ). ( 0.25) 10%  20%,,, ( 5% µ 0.2 1%  2% 58 µ 0.5). NPV,, 30% ( ) 5%  7% ( )., : 10%  15% ( ) 5% ( ).

( 0.2). µ., ( ). ( ). ( ).,,, NPV : 10%  15% ( ) 3% ( ).

6.2. 1997. (, ).

, N 110- 20. 06.1997  ,, 2000..

(, ).

18,.

(,,,,, ), 6., -..

,. 95%, 81%  52%.,, 10% 70%. 40  80% 19.

, ( ).

, -,.

, -.

,, Pt Pt = p (1 )It, p, (1 )It. (1 µ)(1 i ) p (1 ) V = + I H1 H2, + 2 +  http://www.valnet.ru, 60 i H1 (4.7), H2 = 1 i + i 1 e(+1 ).

= min{t 0 : t p It }, p (1 ) + 2 p = ( 1 H1 + H2 · · + 1 (1 µ)(1 i ) (4.10)).

( ) ( ).,, ( = 0). :

i p 1+ (1 K) + (1 ) p 1 i + Rp = =, i p p 1+ (1 K) + (1 K) 1 i + K = A0 + (1 )B0, A0, B0 (4.5).

, p p (1 ) (1 K)., + 1 +, (..

Rp 1), :

p + 1K · = 1 B0 + (1 A0 ).

p + 1 1,,, p + p + = B0 + · 1 B0 + · A0.

p + 1 p + ( . (3.6), 6. 3%)., ( = 0), = 10%,, (p = p = 2%), ( ) 1 = 1%21.

6. = 15% = 25% 0% 0.60 0. 5% 0.42 0. 10% 0.31 0. 15% 0.24 0.,,  . 60%  70% ( ), ( ).,   ( ),.

6.3..

 1 2.

, (,..).

62,,,.

.

..

,,.,,, :,,  ( ).

, N 22.

,. :

, (, ).

M,, M, N (, M ) N (0, M0 ). (6.1) :

.,, ( ).

.

2,, i ( (6.1)), :

1 i 1 i (1 i K ) (1 i K0 ), (6.2) + 2 N (·), NPV 6. i  , (+1 )t bt e(+1 )t dt.

K = at e dt + (1 ) 0 (6.2), 1 i 1 i 1+. (6.3) 0 K )11/ K )11/ (1 i (1 i (6.3) i ( ) 1 i = 0, (6.3) ( i ), 1 i 0, 1+ 1, (1 i K0 )11/ 2 0 11/ 0 = 1 + i.

0 (1 i K0 ) 1 i,  0, ( NPV ).

, i K0,, ( ) 0 0 0 = ( 2 )i /(1 i ). i = 35% ( ), 0 0.54( 2 ).

, (a ). t a e(+1 )t dt+(1) bt e(+1 )t dt ( K = t 0 ).

, (a ) t (6.1), :

(1 i K ) (1 i K0 ), ( + 2 ) ( 2 ) (a0 ) ( K0 )  t , 64 11/ 1 i K 1+. (6.4) 2 1 i K K + (1 )B0, bt e(+1 )t dt, (6.4) B0 =, 11/ 1 i K 1, 1+, 2 1 i [ + (1 )B0 ] 11/ 1 i K 1 = ( 2 ) 1.

0 + 0 (1 )(1 B )] 1 i i,,  1,.

0 1., 11/ 0 (1 i )11/ 1 i, 0 1 i + i (1 )(1 B0 )] 1 0.,,,,,.

, -, 0 0.02  0.04, 1 (  ).

,,.,.

6.4., ( 258). - ( 20 ) (),  () ( 259).

6.  ( ).

. (, ), ( )  23.

,.,,,, () at et dt.

A=.

,  ,. 259., 20%,.

,.,  .

,,.

, L, q ( ),. Cummins et al. (1996) 66 1 eL ASL =. (6.5) L, T RT = eT = q, T = ln q 1.

T L RT t t t ADB = e e dt + e dt.

LT 0 T = k/L, 1/L, k q 1 qeyx + eyx ex, ADB = (6.6) k+x x(1 y) x = L, y = (ln q 1 )/k.

,  k ASL = ADB. (6.7) (6.5)(6.7), k0 q x. 0.1 x 2, L  ( - ).

, q = 0.2, k 2.,, ASL ADB, (6.8) ASL /ADB 0.99 ( ) 0.85 ( )., (6.8) k ln q 1 ( T L)., ( ).

6., q, . q = 0.3,..

30%. 6., (6.7), ( ).

6. = 10% = 20% 1 1.48 1. 2 1.47 1. 3 1.46 1. 4 1.45 1. 5 1.44 1. 6 1.42 1. 7 1.39 1., 30- ,, 1.4  1.5, ( ).

, 20-. 2,, 1%  2% ( ) 15%  20% ( ). 30-,.  68, (  ) 1.4  1.5,.

6.5. :

, ( !).,,,,.

(3.10),.

,  , ().

, ., (.. ).

, ( 274),.

:, -, ( 10 );

, -,, 30% ( 283).

,.

,,.  ,,,..

,,,  .

:

.

6. 1..

, ( = 0),.

t (t ) Zt = Qt Dt, Qt ,, Dt  24.

, ( t + t0 ), t + t0.,.,, :

t + t 0, i (Zt lt ), + t0 t + t1, Tt = t + t i Zt, lt,, " +t ( (Zt ) dt)., t1, +t1 +t lt dt = Zt dt.

+t0, (0 1), lt = Zt. t 24, 25 (,,,, ...., 2000) 70 +t0 +t Zt dt + Zt dt = 0. (6.9) +t,   +t (t ) Qt e(t ) dt V = (Qt Tt )e dt = +t1 (t ) [(1i )Qt +i Dt ]e(t ) dt.

+ [Qt i (1)Zt ]e dt+ +t0 + t,   ( (3.10)), +t [Qt (1 i )Qt i Dt ]e(t ) dt V V = +t [Qt i (1 )Zt (1 i )Qt i Dt ]e(t ) dt + +t +t0 +t Zt e(t ) dt + Zt e(t ) dt. (6.10) = i +t Zt, +t0 +t0 +t1 +t (t ) t0 (t ) t Zt e dt e Zt dt, Zt e dt e Zt dt.

+t0 +t (6.10) +t0 +t V V i et0 Zt dt + Zt dt = +t (6.9).

6.,,   (, ),  .

, t (6.9),,..

+t1 +t (t ) Zt e(t ) dt, lt e dt = +t0, (6.10), V V.,, t1., t t1, (6.9).

2..

, (.. It I), t 2 =, (.. p = 0).

Qt = (1 µ)t ., ( (at, t 0)).

:

+t0 t0 t (t ) t at et dt Zt e dt = Q +t e dt I 0 1µ 1 et0 I(A0 At0 ), = + t1 t1 t (t ) t at et dt Zt e dt = +t e dt I +t0 t0 t 1µ et0 et1 I(At0 At1 ), = 72 as es ds. (6.10) =, At = t 1µ 1(1)et0 et1 I A0 (1)At0 At1, V V = (1 µ)(1 i ),, V = +i IA0 (. (4.8)),    :

V V p (1 µ)E A R =, V p (1 µ) + i A p = /I, E=1(1)et0 et1, A=A0 (1)At0 At1.

, 1, i 1 i A0 R = E ( )A (6.11) i A0 1 i (6.11) t0 t1,., t t0 = max{t 0 : Qt+ Dt }, (6.12) (,,, Q D0, ), t1 g1 (t) = g2 (t), (6.13) t0 t t t g1 (t)=(1) Ds ds+ Ds ds, g2 (t)=(1) Q +s ds+ Q +s ds.

0 0 0 t0 t1 (6.12) (6.13).. 7. 1% 2%, 10% 15% (  ), i = 24%, = 30% ( ).,  (  5-10 ) R   5%.

,.

7.

,., (3.11).

.,, 26.

7.1. (t = (t,..., t ), t 0) 1 m Rm, :

m i k 0 = x0, (7.1) dt = ai (t )dt + bki (t )dwt, (t 0), i = 1,..., m, k= a = (a1,..., am ), bk = (bk1,..., bkm )  Rm, (wt, t 0), k = 1,..., m  k, x .

 "(, 2002) (, 2002) 74 Ee g( ) max, (7.2) g : Rm R1, ( M).

(.,,, 1969;

Oksendal, 1998), ( ).

(7.2)  .

L (7.1), C 2 (Rm ), m m m Lf (x) = ai (x)fxi + bki (x)bkj (x)fxi xj.

2 i,j= i=1 k=,, L,.. m k=1 bki (x)bkj (x) x = 0.

F (x)  (7.2), G = {x Rm : g(x) F (x)}   .

F (x) 0 = x, LF (x) = x G   F (x) = g(x) G., G.

, G F (x) g(x) ( )27.,,  , (.,,, 1969).

,.

 ,..  ,, (1969), Oksendal (1998). 28,, 7. 7.2. (7.2).,,   (.,,, 1969;

Oksendal, 1998).  .

G   Rm G G.

G = G (x) = min{t 0 : t G}  / G t, (7.1) 0 = x. M(G) = {G, G G}  G.

, u(x) = EeG g(G ) (7.3) Lu(x) = u(x), x G, g(a), x a, x D, a D. (7.4) u(x) (,, Karatzas and Shreve, 1991;

Krylov, 1996;

Oksendal, 1998) (7.1) x0 G G (7.3)-(7.4) uG (x0 ), G.

, (7.2) M(G) uG (x0 ) max. (7.5) GG,., Dixit and Pindyck (1994), McDonald and Siegel (1986), Trigeorgis (1996), 76, G (7.5), M(G) : (G) = G.

G ,, (G) (7.2) M. t g.

,,.,, (7.3)-(7.4) (7.5).

7.3.

,.

t = (t, t ), t 0, :

1 1 dt = t (1 dt + 1 dwt ), 0 = x 1, (7.6) 2 2 dt = t (2 dt + 2 dwt ), 0 = x 2, wt wt :

1 Ewt wt = rt, (|r| 1).

, (7.1), 1 2 1 1 r2.

wt = wt, wt = (wt rwt )/, wt wt 1 (Ewt wt = 0),,. t :

1 1 dt = t (1 dt + 1 dwt ), (7.7) 2 2 1 dt = t [2 dt + 2 (rdwt + 1 r2 dwt )], wt wt .

1 7. R+, p, Gp = {(x1, x2 ) R+ : x2 px1 }, p 0.

= (t, t ), (7.6) x = (x1, x2 ) R+, p (x) = min{t 0 :

t Gp } = min{t 0 : t pt }  2 / Gp.

Fp (x) = Ex ep (x) g(p (x) ), x R+ ( x, t x).

, x Gp, p (x) = 0,, Fp (x) = g(x).

/, g : R+ R1 q, g(x) = q g(x) x R+ 0.

2 = 1 2r1 2 + 2  " 2 (7.6), 0.

4. g(x) q q1 (q 0). i = i + (i = 1, 2), 2i :

12 2 2 1 1, (7.8) 2 1 q.

(7.9) :

g(1, p)p xq x, x2 px 1 Fp (x1, x2 ) =, x2 px g(x1, x2 ),  12 q1 ( 1) + ( 2 ) ( 1 q) = 0.


(7.10) 2.

78 (7.5) Fp (x0 ) max. (7.11) p Fp 4 (7.11) (7.2) M(G0 ), G0 = {Gp, p 0}.

h(p) = g(1, p)p (0 p ).

5. 4, p h(p)29. :

1) (7.11) p = p ( x0 );

2) (7.2) M(G0 ) = min{t 0 : t p t }, 2 3) (7.2) M(G0 ), (x1, x2 ) (7.6) h(p )xq x, x2 p x 1 (x1, x2 ) =. (7.12) x2 p x g(x1, x2 ),, Gp (7.2) M.

6. 4, g C 2 (R+ ), p h(p),, p p :

h (p) 0, (7.13) pgx2 x2 (1, p) ( 1)gx2 (1, p) 0. (7.14) = min{t 0 : t p t } 2 (7.2) M, (7.2), (x1, x2 ) (7.6) (7.12).

, (.,,,, 1987;

Oksendal, 1998).

.. h(p ) h(p) p = p 7. g(x1, x2 ) = x2 x1..

. g(x1, x2 ) = x2 x1, (7.8) max(1, 2 ). (7.2) M = min{t 0 : t p t }, 2 p = /( 1),  ( 1) + (2 1 ) ( 1 ) = 0.

( ), -, McDonald and Siegel (1986).

,,, (Hu and Oksendal, 1998).

, p 0, (7.11) h (p ) = 0, p gx2 (1, p ) = g(1, p )   (7.12):

x2 (x1, p x1 0) = gx2 (x1, p x1 ).

,   5.

7.4.

4.. (7.10), p (x) (..) x R+ p 0.

. (7.6) :

t x 2 1 = x exp{( 2 1 )t + 2 wt 1 wt }, (7.15) t i = i 1 i (i = 1, 2).

80 (.

Karatzas and Shreve, 1991) i..

lim sup |wt |/ 2t log log t = 1 (i = 1, 2).

t, (7.15), 2..

lim sup t /t = t, p (x) = min{t 0 : t /t p} x R+ 21 p 0 (7.8).

4.,, Fp (x) q.

,, p (x) t /t p, (7.15), p (x) x = (x1, x2 ).

, t g :

= Ex ep (x) g(p (x) ) = Ex ep (x) g(p (x) ) Fp (x) = Ex ep (x) g(p (x) ) = q Fp (x),.. Fp (x) q.

Fp (x) (7.3)-(7.4). Fp (x) x Fp (x1, x2 ) = xq f (y), y =, f (y) = Fp (1, y).

x :

y yx1 =, y x2 =, x1 x y Fx1 = qxq1 f (y) + xq f (y) = xq1 [qf (y) yf (y)], 1 1 x Fx2 = xq1 f (y), Fx2 x2 = xq2 f (y), y Fx1 x2 = (q1)xq2 f (y)+xq1 f (y) =xq2 [(q1)f (y)yf (y)], 1 1 x 7. y y Fx1 x1 = (q 1)xq2 [qf (y) yf (y)] + xq1 qf (y) + f (y) 1 x1 x y ] = xq2 {(q 1)[qf (y) yf (y)] yf (y) x y[(q1)f (y)yf (y)] } =xq2 {(q1)[qf (y)2yf (y)]+y 2 f (y)}.

(7.7), t :

12 LF (x) = 1 x1 Fx1 +2 x2 Fx2 + 1 x2 Fx1 x1 +r1 2 x1 x2 Fx1 x2 + 2 x2 Fx2 x2.

1 2 (7.16) (7.3) (7.16) L f (y) = 1 [qf (y)yf (y)]+2 yf (y)+ 1 (q1)[qf (y)2yf (y)]+y 2 f (y) + r1 2 y[(q 1)f (y) yf (y)] + 2 y 2 f (y), 12 q1 y f (y) 2 + yf (y)[ 2 ] f (y)( 1 q) = 0. (7.17) 2 f (y) (7.17) f (y) = Cy, C . (7.10).

(7.9), (7.10) :

1 2., (7.17) 0 y p :

f (y) = C1 y 1 + C2 y 2, 1 0, 2 0,,, Fp (x1, x2 )=C1 xq1 x1 +C2 xq2 x2, 0x2 px1, x1 0. (7.18) 1 2 1 (7.4) x2 0 x2 px Fp (x1, x2 ) g(x1, 0), (7.18) C2 = 0. C1 {x2 = px1 },, Fp (x1, px1 ) = C1 x1 p1 = g(x1, px1 ) = xq g(1, p), q.. C1 = g(1, p)p..

82 5. x = (x1, x2 ) R+, Fp (x) Fp (x) p 0.

p g:

x2 x2 x2 x g(x) = xq g 1, xq x h(p )xq x.

=h 1 1 2 1 x1 x1 x1 x p p. 4 :

x2 p x1 Fp (x) = g(x) = Fp (x);

px1 x2 p x1 Fp (x) = g(x) h(p )xq x = Fp (x);

1 x2 px1 Fp (x) = h(p)xq x h(p )xq x = Fp (x).

1 2 1, Fp (x) Fp (x) p p. Fp (x) Fp (x) p p., (7.11) p = p. M(G0 ) 5.

M  , (.,, 1987;

Oksendal, 1998)..

P x  t ( ) 0 = x, Ex  P x.

(Oksendal, 1998)., : R+ R1, :

m 1) C 1 (R+ ), C 2 (R+ \ D);

m m D={x R+ : (x)g(x)}, D  D, m 2) D Ex D (t ) dt = 0 x R+ ;

m 3) (x) g(x) x R+ ;

m 4) L = x D;

5) L x R+ \ D (D  D);

m 6) D = inf{t 0 : t D}.. ( P x ) x R+ ;

m / {g( )e, D } ( 7) P x ) x D.

= D (7.2), (x) .

7. 6. (x1, x2 ), (7.12).

x = (x1, x2 ) R+, x1 = 0 p(x) = x2 /x1.

h(p ) h(p) p = p, x2 p x x h(p )xq x h(p)xq (x1, x2 ) = 1 2 x x2 x2 x xq g 1, = = g(x1, x2 ) x1 x1 x ( g).

, (x)g(x) xR+, D={xR+ :

2 (x) g(x)} {x2 p x1 } = {(x1, x2 ) : 0 p(x) p }.

D = inf{t 0 : t D} = inf{t 0 : t p t }..

2 / x R+.

, (7.9) 7)., D p,, 2 )( )q e h(p )(p ) ( )q e =g(1, p )( )q e.

1 ( )e =h(p :

Ex [( )e ]k g k (1, p )xkq Ex exp{[ + q(1 1 ) + q1 w ]k} = g k (1, p )xkq Ex exp{[ 1 q q 2 1 (k1)]k +kq1 w 2 1 k 2 q 2 1 } g k (1, p )xkq Ex exp{kq1 w k 2 q 2 1 }, 2 1 2 k 1, 1 q 2 q 2 1 (k 1) 0 ( (7.9)). Mt = exp{q1 wt 1 q 2 1 t} 1 (.,, Karatzas and Shreve, 1991), EM =EM0 =1.

sup Ex [( )e ]k g k (1, p )xkq, G {( )e, G }.

4) x2 p x1.

84 x2 p x1,.. pp,, (x1, x2 )=g(x1, x2 )=xq g(1, p).

, (7.17), 12 q1 Lgg=xq p gx2 x2 (1, p) 2 +pgx2 (1, p)( 2 )g(1, p)( 1 q).

2 (7.13), pgx2 (1, p) g(1, p). (7.9) :

1 q1 2 xq (Lgg) p2 gx2 x2 (1, p) 2 +pgx2 (1, p) 2 ( 1 q) 2 2 = p 2 [pgx2 x2 (1, p) ( 1)gx2 (1, p)] (, (7.10) (7.14))., 5).

,,, D = (7.2) M.

6, 2 1.

1 6 (4.7).

2. 1)4) 4 ( g(x) = x2 x1 ).

5) g(x) = 1 ( ). 1 :

Ee =, (7.19) p I 12 ( 1) + (2 1 + 1 r1 2 ) = 0.

(7.20) (7.19), p I E e = log. (7.21) p I0 8. (7.20), 1 1 12 = 2 +2 1 +1 r1 2 = 2 +2 1 + 1.

2 2 (7.21):

p I 0 12 E e = 2 +2 1 + 1 2 log. (7.22) p I0 2 2 (7.20) 0 0., (7.22) 0, 4).

.

8.

.

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.,.,.(1999), No 99/ (, ).

..,.. (2000),, 7, 302-304.

..,.. (2001), # WP/2001/107 (, ).

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