Лекции по финансовой математики
TRANSCRIPT
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080105
2008
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.. : . .: , 2008. 98 . : ..-.., .. 080105 29.03.2005 . , 23.01.06, , . , , , . : , , , . .
- 17.03.2008 ., 5.
, 2008 .., 2008
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. - , . .
, , , , .. . , . , . ( ): (, ), , .. , . , . . : 1) ; 2) ; 3) ; 4) ; 5) ; 6) ; 7) ; 8) ; 9) ; 10) ; 11) . , , - , .
-
5
1.
1.1. , , , , .
- , . , , , , , , , , , .
, . , , , .
, , - : 1) ( , , . .); 2) ( , , . .); 3) ( ).
1.2.
, , , , , , , , , . :
1) , , , ;
2) ; 3)
, :
-
6
4) , ;
5) ; 6)
, ; 7) ; 8) .
, , .
1.3. , -
, : .
, , . , , , , , , , , . ., , : , .
, , .. , : , , , . .
, . , . .
. , . , .
. , (, , , ) - .
-
7
, , , , .
. , .
, . , , , . - . , , .
. , .
, , , - . , , , , . : , , .
. , .
1.4.
. : , , , . , . , (), (). , .
. , . () ,
-
8
( ) .
1.5.
-
. () ().
, , .
. , , . , .
() . , , . . ( ). . , , .
. , -, - rote of interest ( i ) , , . -, () - discount rote ( d ). , , .
PV , n
FV .
(1),
(2):
nPVPVFVi
= (1.1)
-
9
nFVPVFVd
= (1.2)
, , , , , , - , . , .. , (3) (4) :
ddi = 1 (1.3)
i
id += 1 (1.4) ,
, .
, , , . () .
. , . , .. , .
. , - .
. 100
. . 140 . . . : .2;..140;.100 nFVPV ===
%.202,020040
2100100140
nPVPVFVi ==
==
%.7,16167,0
2,012,0
1
iid =+=+=
-
10
1.6.
,
, . , . . 1.
. 1. .
, () . , (future value - FV ) (present value - PV ) , .
, .
PV , , ( ) FV .
(FV)
(PV)
(FV)
( ) (PV)
-
11
. .
2.
2.1. , , . . PV . () FV ; , .. - n ; - i . : .iPV , )1( iPViPVPV +=+ .
iPV : )21()1( iPViPViPV +=++ .
- )31()21( iPViPViPV +=++ .. , n-
: )1(])1(1[ niPViPVinPV +=++ . ,
PV iPV
, n : ( )inPVFV += 1 (2.1) (2.1) , , , ; . ( )in+1 . , ( ) . : PVFVD = . PVFV . . 100 000 3 4,5 % . . : .3;045,0.;100000 niPV === (5):
-
12
( ) .113500135,1100000)3045,01(1000001 inPVFV ==+=+= ( ) 13500
.
3 . , , , (n
-
13
. 365/365. , .
2) . , (Bankers Rule), , . 365/360. , .
3) . , , . . 360/360.
. . 1 . . 20 5 18 % . ? .
. 20 20 ( 1 1). 5 - 278. , 258 (278 20).
. 30 . 8 ( 20 20 ). 20 5 15 : (30-20)+5. : 830+15=240+15=255 .
: 1. (365/365):
+= 18,036525811000000FV = 88,1127232 .
2. (365/360)
+= 18,036025811000000FV .1129000 =
3. (360/360)
.112750018,036025511000000 FV =
+=
2.2
, , .
-
14
, 1n 1i ,
11 inPV . 2n
2i , 22 inPV . - m . , ..
, : =++++= mm inPVinPVinPVPVFV ...2211
+=++++= m mmmm inPVinininPV t
12211 1)...1( , (2.3)
ki kn , .,...,2,1 mk = ; . : 12% , 2,5%. .
, 400,0 . .
: .17,0;25,0;145,1;25,0;12,0;5,0;.400 332211 ======= inininPV
..6,455139,1400)043,0036,0060,01(400)17,025,0145,025,012,05,01(400)...1( 2211
inininPVFV mm
==+++==+++=++++=
2.3.
, ( ) . , .. .
, 1n 1i , )1( 11 inPV + .
2n 2i . 2n )1)(1( 2211 ininPV ++ ..
, : ),1(...)1()1( 2211 tt inininPVFV +++= (2.4.)
tnnn ,..., 21 ; tiii ,..., 21 - , .
-
15
. 500,0 . . 3 , 10% , 5% ? :
.360;2,0;15,0;1,0;30;30;30;.500 321321
YiiitttPV
========
inininPVFV
.896,531
2,036030115,0
3603011,0
360301500)1)(1)(1( 332211
==
+
+
+=+++=
2.4.
, , : PV , , , n i FV . . , .
, , FV , i . : () . (), .
, . (2.1) PV , :
niFVPV += 1 , (2.5)
PV - () FV ; n - ( ); i - .
in +11 ,
PV FV . . 2 . 12% .
-
16
, 1 500 000 ?
: FV=1500 000 ; n=2 ; i= 0,12
inFVPV 1209677
12,0211500000
1=+=+= ,
D 29032312096771500000 == .
, (2.2) n
-
17
).
, , , . . .
, .
, , , . , , i . d .
, :
nFVPVFVd
= ,
, dnFVD = ,
, : )1( dnFVdnFVFVPV == , :
)1( dnFVPV = (2.7) dn 1 . ,
=PV )1( dYtFV , (2.8)
t - ( ); Y - ( ).
360=Y . ,
, 01 dn , d
n 1 . , , 0 (
dn 1= ) ( n >
d1 ).
, . . 1 . . 17
.
-
18
23 20 %. ( ) . : FV=1000 000 ; d=0,2. , t: 17 - 321; 23 - 266. t=266-321=55;
, 360 , , Y=360.
, , :
.9694442,036055110000001 d
YtFVPV =
=
=
30556 (1000000- 969444). ,
. :
dYt
PVdn
PVFV
== 11 (13)
. , 18% 150 . 925000 . .
. , .
: PV=925 000 ; d=0,18; t=150 . .100000
18,03601501
92500
1d
Yt
PVFV =
=
=
3.
3.1.
- , , , . . , , .
-
19
, , PV i . (, ) ,
)1( iPViPVPV +=+ . i . , : 2)1()1()]1([)]1([)1( iPViiPViiPViPV +=++=+++ . , , :
3222 )1()1(])1([])1([)1( iPViiPViiPViPV +=++=+++ . ,
, PVa =0 , iq += 1 .
n - , n - FV , : =FV ( )niPV +1
, n - , , :
=FV ( )niPV +1 , (3.1) PV - , i - , n - . ( n ) :
( ) ( )[ ]111 +=+== nn iPVPViPVPVFVD ( )ni+1
. . . , 1000 000 , 5 15,5% ? : PV= 1000 000 ;i = 0,155; n=5.
.2,105546410000002,2055464.2,2055464)155,01(1000000)1( 5
DiPVFV n
===+=+=
3.2.
-
, , . , ,
-
20
, (margin) . , .
, .
knk
nn iiiPVFV )1...()1()1( 21 21 +++= , (3.2) kiii ,..., 21 - ; knnn ,..., 21 - ,
. . 1000000 . 5
12% 0,5% 0,75%- . .
:
.3;1275,00075,012,0;2;125,0005,012,0;5;1000000
22
11
nininPV
==+===+===
.1814072433341,1265625,11000000)1275,01()125,01(1000000)1()1( 3221 21
iiPVFV nn
===++=++=
814072 .
3.3. .
: .
. , n,
Ytn = . . 13 1000$,
6% . 1.
: 13 - 13; 1 - 244. t=244-13=231. PV=1000$; i=0,06.
$1038038,1100006,11000)6,01(1000)1( 642,0360231
===+=+= Yt
iPVFV .
, , , , , () .
-
21
( ) , . . :
),1()1( biiPVFV a ++= (3.3) a - ; b - .
, , .
. 3 . . 3 160 16,5% . .
1) : iPVFV n 5,507193)165,01(300000)1( 43836,3 =+=+= .
2) : .8,508659)165,043836,01()165,01(300000)1()1( 3 biiPVFV a =++=++=
, .
3.4.
,
, , ( in +1 ) ni)1( + . , 10% . , 365 . .
30 180 1 5 10 20 50
1,00822 1,04932 1,10 1,5 2,0 3,0 6,0
1,00786 1,04812 1,10 1,61051 2,59374 6,72750 117,39085
, 0 < n 1 : ni)1( + > in +1 . n = 1 .
, i . , ()
-
22
, , , .
3.5. m
, , , , , . . , , , , .
() i , n , m .
mi ,
: nmN = .
: N
miPVFV )1( += , (3.4)
i - ; N - , nmN = . . , 1 . .
15,5% , ? ; PV= 100 000 ; n=5 ; i= 0,155; m= 4.
miPVFV
mn
9,2139044155,011000001
54
=
+=
+=
.
3.6.
, , , %12=i .
, 4=m ,
mi .
mi 03,0
412,0 == ,
:
( ) 1255,103,011 4 =+=
+m
mi
, 12,55%. , 12,55%, - 12%.
-
23
, , , . , . , , , , m
mi .
i , m
mi ,
f :
,11 fmi m +=
+
: 11
+=m
mif (3.5)
. , 24% .
: i=0,24; m=12.
268,0102,111224,01 12
12
==
+=f .. 26,8%
3.7.
, PV
, FV , .
PV 3.1, :
,)1( ni
FVPV += (3.6)
ni)1(1+ - ,
3. m , 3.6
: mn
mi
FVPV
+=
1, (3.7)
-
24
PV - ( ) FV .
+== niFVPVFVD )1(11 .
m ,
+= mn
mi
FVD1
11 .
, , .
. 500 000 5 . , 12% . : FV = 500 000 ; n = 5 ; i = 0,12.
.7,283713
76234,1500000
)12,01(500000
)1( 5
iFVPV n ==+=+=
3.8.
(, , , ). ( , , ) , .
i. m
mi
11
+=m
mif
, m
mi
+1 . , .. m , , :
ii
im
m
iim
m
m
me
mi
mi
mi =
+=
+=
+
1lim1lim1lim
e - ( ), e 2,718. , i
PV ie ine n .
-
25
, , . .
, :
inePVFV = . (3.8) ,
. :
nePVFV = (3.9) , n
. . 10 000
8% . 3,5 .
: .5,3;08,0.;10000 nPV === .3,1323132313,11000010000 5,308,0 eePVFV n ====
(3.9),
: neFVPV = (3.10)
.
, 5 300 000 , 8%?
: .5;08,0.;300000 nPV ===
eeeFVPV n 201096670320,0300000300000300000 4,0508.0 =====
3.9. ()
. , , .
FV n d . ,
dFV . )1( dFVdFVFV =
-
26
,
)1( dFV , . , :
22 )1()21()1()1()1( dFVddFVdFVddFVdFV =+== .. n ,
( )ndFV 1 . ,
: ( )ndFVPV = 1 d - . (3.11)
( )nd1 - . : ])1(1[)1( nn dFVdFVFVPVFVD ===
, : ( )ndFVPV = 1
.1mn
mdFVPV
= (3.12) . 500 000 , 3
15% . ?
: .15,0;3.;500000 === dnFV
( ) dFVPV n 5,307006614125,0500000)15,01(5000001 3 ==== .
: .5,1929375,307062500000 D == . ,
.
: .4;15,0;3.;500000 ==== mdnFV .31606563213,0500000
415,015000001
43
mdFVPV
mn
==
=
=
, , , .
, :
) 0< n nd )1( ; ) n >1 )1( nd < nd )1( ; ) n =1 :
)1( nd = nd )1( ; , n
-
27
, , , , . n >1, .
3.10.
FV (3.13),
:
ndPVFV
)1( = , (3.13) d - ; n - .
:
nm
md
PVFV
=
1 , (3.14)
d - ; - ; n - . . 350000 2,5 .
12% . , :
) ; ) . : .2);1);12,0;5,2.;350000 ===== mmdnFV ) ;7,481793
726452,0350000
)12,01(350000
)1( 5,2
dPVFV n ====
)
md
PVFV nm 6,476901733904,0350000
212,01
350000
15,22 ==
=
= .
3.11.
, m , ..
nm ,
md , d - ,
.
-
28
f . , :
m
mdf
= 11 .
, m
mdf
= 11 (3.15) . . 500000 , 3 , 12% . .
: .12;15,0;3.;500000 ==== mdnFV PV 5,348206696413,0500000
1212,01500000
312
==
=
D 5,1517935,348206500000 == 1136,0
1212,011
12
=
=f , 11,36%. 4.
4.1.
, , , , , .. , . , . . , .
, , .. .
-
29
, .
, , .
4.2.
. ) . , 360 . , . , :
.1
11nd
ni =+ i d, 1.3 1.4. 1. , :
dtdi =
360360 (4.1)
itid +=
360360 (4.2)
i, ,
365 , d 360 , :
dtdi =
360365 (4.3)
itid +=
365360 (4.4)
. 100 . 20% . ? : .2,0;100 == it
.187,038572
2,01003652,0360
365360 ==+
=+=
itid
, 18,7%.
-
30
) . :
( )njin +=+ 11 i - ; j - ; n - . i j .
( )nji
n 11 +=, (4.5)
n nij +=+ 11 , 11 += n nij . (4.6)
. 20% 0,5 . . ; i =0,2; n = 0,5.
21,0121,11)1,01(12,05,0111 25,0 ==+=+=+= n nij 21%
4.3.
, , ( ) . , , , .. . , , . . , , 2 . 2 , 2,5 . 3 . 15% . : .15,0,3.;2500000;2.;2000000 2211 ===== inFVnFV :
.;33,15122873225,1
2000000)15,01(
200000021 PV ==+=
.58,1643790520875,1
2500000)15,01(
250000032 PV ==+=
-
31
, .
, , , , . , , , . .
400 . . 700 . . 1 1 ( ). : 600 . 1 , 1 . , 20% . .
: .
- 1 . 1 (1=400 . ) 1 212
(365-213 +60). 1 (2=700 . ) 1 59
(60-1). 1 (3=600 . ) 1 120
(365-305+60). :
.2,036512016002,0
3655917002,0
3652121400 4P+
+=
++
+
,45,63963,72247,446 4P+=+
4=529,65 . . . 200, 300 500 .
1,5 , 2 4 . 3,5 . , 18% .
: ( )
..31,112329,46054,38448,278)18,01(
500)18,01(30018,01200 5,3425,35,15,3
4
P
=++==+++++=
-
32
5.
5.1. .
. . . , , , , , .
PV , n FV .
: ) ,
nPVPVFVi
= , (5.1) ) ,
niPVFV )1( += : 1= n
PVFVi (5.2)
) , i, m :
(5.3)
2,5 . 2
, 3,5 . . .
: .2.;.5,3.;.5,2 nFVPV === 2,0
25,25,25,3 =
==
nPVPVFVi 20%.
183,015,25,31 === n
PVFVi 18,3%.
, 16% , . . : i=0,16; m=4.
.11
+=m
mif
-
33
%.99,161699,01416,0111
4
mif
m
=
+=
+=
. , , .
, , . , . , . , .
, , .
100 . 240 12% .
. 1. . .
: PV=100 . .; t = 240 ; Y = 360 . :
iYtPVFV .10812,0
36024011001 =
+=
+=
99 . . (100-1). .66667,0
360240 n ==
: 1394,010909,1199
1081 5,1360240
==== nPVFVi .
, 13,94%.
5.2.
. , . , ,
-
34
, .
) , 1n 1i , 2n 2i .. -m . :
=
=+++=m
kkm nnnnN
121 ...
,
, i . :
k
m
kk iniN +=+
=1
_11
Nin
i kk= _
(5.4) mddd ;...;; 21
: N
dnd
m
kkk
=
= 1 . (5.5)
i ( d )
, , .
.
, , , .
. 400 000 , .
ki
nk ,
kk in
1
0,12
0,75
0,09
2
0,11
2,0
0,22
3
0,08
1,25
0,1
-
35
4,0
0,41
.
: .08,0;25,1;11,0;0,2;12,0;75,0;400000 332211 ======= inininPV
: .0,425,10,275,0321 =++=++= nnnN
:
1025,00,441,0_ ===
Nin
i kk
10,25 % . :
.564000)1025,041(400000 FV =+= ) ,
. , , :
( ) ( ) ( ) mnknnN
iiii +++=
+ 1...111 21 21_
.
( ) ( ) ( ) 11...11 21 21_ +++= N nknn miiii , (5.6)
=
=+++=m
kkm nnnnN
121 ...
. ,
. .
8,5 % 0,5 % 2 0,75 % 3 . .
: .0925,00075,0085,0;3;09,0005,0085,0;2 2211 =+===+== inin
: .532 =+=N :
( ) ( ) 0915,010925,0109,015 32_ =++=i 9,15% .
-
36
5.3.
. , . , . , , . , .
. .
. 5%. ?
: 1+ 0,05=1,05
%.2,95952,005,11
. i cn == (S)
, FV ,..iFVS = (5.7)
FV - . . 100000 .
12% . , . (3 ) 30%. .
: ).3,01(3,1;3;12,0.;100000 +==== iniPV
: ( ) .136000)12,031(10000011 niPVFV =+=+=
: ( ) .14049340493,1100000)12,01(1000001 32 iPVFV n ==+=+=
: 77,0
3,111
. ===
cn ii
.
.10818077,0140493.;10472077,0136000
..22
..11
iFVSiFVS
======
-
37
. 1i , - 2i .
01549,011000001047201 311 === n PV
Si 01549,01 =i
02656,011000001081801 322 === n PV
Si 02656,02 =i ,
1,5% , - 2,7% .
5.4.
PV , . , ( ), , .. . , , : n
niS)1(
1+
+= . (5.8) :
niPVS
++= 1
1 (5.9)
PV - , ; i - ; - ; n - .
( )( )nnn iPViPVS +
+=
++=
11
11 ,
: , .
: 1). i > , S > PV , .. . .
, , , .
-
38
2). i = , S = PV , .. . , . . 3). i < , S < PV . .. , . . . , . . 6000 . 3 4,5% . 7%. . : .07,0;045,0;3.;6000 ==== inPV
.2,558993153,0600007,01045,016000
11
3
iPVSn
==
++=
++=
.. . . , , . , . , . . 7% ( ). 200 . ., 7 10% . .
..1815,13160578,1058333,1200
)07,01(
1,01271
200)1(
1
7 i
Yt
PVS n ==+
+=+
+=
, S,
i i - - . :
( ) ( )( )nn
n ii ++=+
111
,i : .1
+= ii (5.10)
-
39
. 20% , 15%. : i = 0,2; = 0,15.
i = ==+
15,105,0
15,0115,02,0 0,0435
- 4,35%, .. 4,35% , 13% ( =
15,11 0,87 87% 100-87=13%)
5.5.
.
+=
1ii .
, i> i . , . . , j , ).1()1(1 ++=+ ij ,
iij ++= . (5.11) . i ,
. , : += ij .
. 300000 2 . 8% . . , 10% . .
: .1,0;08,0;2.;300000 ==== inPV ;188,008,01,008,01,0 =++=++= iij
, , 18,8%.
:
( ) .2,423403188,01300000 2 FV =+= 123403,2 .
-
40
6.
6.1.
- , . , , , ..
. .
, , , ..
. , , . . . , . , , . , : , , .
6.2.
. . . , . , , , , -, . . - . , , , .
6.3.
.
R . , .
-
41
R )1( i+ , i - . fFV , :
( )( )
( )iiR
iiRFV
nn
f11
1111 +=+
+= ; (6.1)
( )iif
n
in11
.+=
. , . ( )inf , 4.
, .. . , , )1( i+ . :
( ) ),1(, if in + ,
( ) ( )iiiRFV
n
f ++= 111 (6.2) I . 4 50 . . 10%. : ; R = 50 . ., n = 4, i = 0,1
( ) =+=1,0
11,01504
fFV ..05,2321,04641,050 =
. 10 . . 5 , 20%. . : .2,0;5.;.10 === inFV pref R .
;2,12,0
12,110);1(1)1(5
=++= RiiiRFV
npre
f
..1198,193,8
10;93,8102;2,12,0
48832,110 RRR ====
-
42
6.4. m
, ,
. , m .
mi , i - .
n .
R m
mi
+1 . :
,11
11
+
+= m
mn
f
mi
mi
RFV (6.3)
R- . . 10 000
7 . 15% . , .
: R=10 000 .; m = 4 ; n =7 ; i = 0, 15.
.92,113663366392,11100001
415,01
1415,01
1000011
11
4
74
mi
mi
RFV m
mn
postf ==
+
+=
+
+=
6.5. - , , . R - ; n - (); i - ; p ; m .
pR
nm , mi .
-
43
pR
pm
mi
+1 . n. :
.11
11
+
+=
pm
mn
postf
mi
mi
pRFV (6.4)
: .111
11
+
+
+=
mi
mi
mi
pRFV
pm
mn
pref (6.5)
. 250 . . 3 . , , 10% . , . :
pR =250 . ., R =500 . .; i=0,1; p=2 ; m=4 ; n=3
.
..5,1699798,62500506,0344,0250
141,01
141,01
250
11
11
24
34
mi
mi
pRFV
pm
mn
f ===
+
+=
+
+=
. . 5 . 4 . . 18,5% . , 1). , .( 1== pm ). R = 4 ..; i = 18,5% ; n=5 .
( ) ( )185,0
3366,14185,0
1185,014115
=+=+=iiRFV
n
f =28,90 . .
2). , ( m = 4, p =1),
..669,291982,04701,14
14185,01
14185,01
4
11
11
4
54
mi
mi
pRFV
pm
mn
postf ==
+
+=
+
+=
-
44
. 3). , , . ( m = 1, p = 4);
( )( ) ..833,3004335,0
3366,141185,01
1185,0144
41
51
FV f ==++=
4). . (m = p = 4);
..786,3104625,04701,11
14185,01
14185,01
44
1
54
FV f ==
+
+=
5). , ( m = 12, p = 4).
..025,32046967,0504132,1
1054167,11054167,11
14185,01
112185,01
44
3
60
412
512
FV f ===
+
+=
6.6.
, . ni
Ri
Ri
Ri
R)1(
;...;)1(
;)1(
;1 32 ++++
i
R+1
i+1
1 . :
( )i
iR
i
ii
RPVn
n
postf
+=+
++=
111
11
1
11
1
1 (6.6)
i
i n)1(11 +
, . 5.
-
45
(1+i) , , :
i
iiRPVn
pref
)1(11
)1( +
+= (6.7) . 8 . . 2,2 . . , 10% . :
, . , 2,2 . (R=2,2 . .; i=0,1; n=4 ):
( ) ( ) ..974,61699,32,21,0
1,0111
2,2111 4
i
iRPVn
f ==+
=+
= - 8+ 6,974 = - 1,026 . .< 0. , , . 6.7.
m
m , mn . R. :
nmmm
mi
mi
mi
+
+
+ 1
1;...;1
1;1
12
-
46
, m
mi
R
+1
m
mi
+11 . : m
mn
m
mn
mf
mi
mi
R
mi
mi
mi
RPV
+
+=
+
+
+=
11
11
11
1
11
1
1. (6.8)
. 10000 . 15% . . : R=10 000 .; i=0,15; m=12; n=7.
.72,40546054672,410000
415,011
1415,01
1000011
11
4
74
mi
mi
RPV m
mn
f ==
+
+=
+
+=
6.8. -
m
, m n i .
mi . mn .
.
11
111:
11
11
1
+
+=
+
+
+=
+=
pm
mn
mn
pm
mn
nmf
f
mi
mi
pR
mi
mi
mi
pR
mi
FVPV (6.9)
. 2 100 . . , 12% . : =
pR 100 . .; p = 4; i = 0,12; n = 2; m =12.
.17,701079030301,0212434,0100000
11212,01
1212,011
100000
11
11
412
122
mi
mi
pRPV
pm
mn
f ==
+
+=
+
+=
-
47
.., 701 079 .
6.9.
, . , , , .
: ( )iR
iiRPV
n
n=+
=
111
lim (6.10)
. 10000 $ . 5%? : R = 10 000 $; i = 0,05.
- . $200000
05,010000 === i
RPV .
, 200000 $ 5% , 10 000 $.
- m :
.
11
1
11
11lim
+=
+
+=
pm
pm
mn
n
mip
R
mi
mi
pRPV (6.11)
. , 2 . 12% . :
.4;12;12,0.;2000 ==== mpipR
.4,201979009902,0
12000
1412,01
12000
11
1
124
mip
RPVpm ==
+=
+=
-
48
6.10.
, . , , . . . , , . , . . 5 , R =2000 . i =6% 8 . . : R1=2 000 .; i=0,06; n1=5;n2=8. 1). .
.21,893006,0
)06,01(11
)06,01(2000)1(11
)1()15
1 iiiRPV
npre
f =+
+=+
+= 2). . :
( ) ( ) .21,893006,0
06,0111
06,018
2 =+
+R 3). :2R
.68,1356
582381,621,8930
;21,8930582381,6
2
2
R
R
===
6.11.
, . , .
- , .
: 1) - ;
-
49
2) ; 3) , , . . - : - 5 1000 $. , - 8 800 $. 8%. : .8$;800;5$;1000 2211 nRnR ====
( ) $7,39929927,3100008,0
)08,01(11
1000111
)15
11
1 ==+
=+
=i
iRPVn
f -
.
( ) $28,45977466,580008,0
)08,01(11
800111
)28
22
2 ==+
=+
=i
iRPVn
f -
. 3) -: 3992,7+4597,28=8590,02=8589,98 $. -, . , 6 , :
;98,85896229,4;98,858908,008,111
3
6
3 ==
RR
: $14,18586229,4
98,85893 ==R .
6.12.
R , in, , p m . , fFV fPV . . ). fFV fPV , R . R .
-
50
. 100 . 5 . 20%. : .2,0;5;100000 === inFV f
;1000002,0
1)2,01(.;1000005
=+= RFV postf
( ) ( ) 4416,72,0
48832,12,0
12,0111 5 ==+=+ii n ,
1000004416,7 =R , : R = .438.134416,7
100000 = ) .
, . . 100 . . , 1 . ., 24 % . : 24,0;12.;12000;12.;1000 ===== imRp
pR .
: .11
11
+
+=
pm
mn
postf
mi
mi
pRFV
: .1
1224,01
11224,01
10001000001212
12
+
+=
n
n: ;02,0100102,1 12 =n
;302,1;1202,1 1212 =+= nn ;3ln02,1ln12 =n
.6,40198,012
0986,102,1ln12
3ln n ===
6.13.
, , , . .
, , ,
-
51
, , .
:
( ) ( ) ( )= +=+++++=n
kk
kn
npostf i
Ri
Ri
Ri
RPV1
221
11...
11 (6.12)
kR k ; n - .
. . 20,12,8,45,30 (. .). , 25% , .. . , ? : ..30;45;8;12;20;5 54321 RRRRRn ======
( ) ( )..039,56
25,130
25,145
25,18
25,112
25,120
)1()1(111 543255
44
33
221
i
Ri
Ri
Ri
Ri
RPV postf
==++++=+++++++++=
25,125,011 =+=+ i .
..049,7025,1039,56)1( iPVPV postfpre
f ==+=
7.
7.1.
- . , , , 2
5 . , , .
, , , , , . .
. .
-
52
. , .. .
, , , . , () , , , , . , , .. , .
, .. , , . , , , .. , , . .
7.2. .
. , . , , . , .
, , , : )1( inPVFV += . , m , :
mnFVR = .
. 8 . . 10 % . . . : .1,0;12;2.;8000 ==== imnPV , :
.9600)1,021(8000 FV =+= : .400
2129600 R ==
-
53
7.3. - 78.
, , . , , . 78.
, :
1 . : 1+2+3+4+5+6+7+8+9+10+11+12=78. 12/78 . . : 11/78 , ..
: kkN +=
21 , .
, : .1;...;2;1;
NNk
Nk
Nk
, , . .
- . , , , . , . , .. .
. 15000 2 20% . . . . : .2,0;12;2.;15000 ==== imnPV
.21000)202,01(15000 FV =+= D = 21 000 15 000 = 6 000 . - 12, , = 122 = 24.
.300242241
21 =+=+= kkN
-
54
: .87524
21000 mnFVR ===
78 24/300 (6 000 ). . 23/300 , 6 ..
: (.)
Nk
15000 24/300 480 395 14605 23/300 460 415 14190 22/300 440 435 13755 21/300 420 455 13300 20/300 400 475 12825 19/300 380 495 12330 18/300 360 515 11815 17/300 340 535 11280 16/300 320 555 10725 15/300 300 575 10150 14/300 280 595 9555 13/300 260 615 8940 12/300 240 635 8305 11/300 220 655 7650 10/300 200 675 6975 9/300 180 695 6280 8/300 160 715 5565 7/300 140 735 4830 6/300 120 755 4075 5/300 100 775 3300 4/300 80 795 2505 3/300 60 815 1690 2/300 40 835 855 1/300 20 855 1,000 6000 15000
7.4.
D n i .
n - ( )niD +1 .
-
55
, . . 20000 . 8 10% . . . : .1,0;8.;20000 === inD :
.8,4287114359,220000)1,01(20000)1( 8 iD n ==+=+
7.5.
, , . D n i . iD . , D .
iD , . n - iD + D - .
. 100000 . 3 15% . , iD , - iD + D . : .15,0;3.;100000 === inD 100000 . : .1500010000015,0 iD == : 1 - 15000 .; 2 15000 .; 3 15000+100000=115000 .
7.6.
D n i . n - , ..
nD .
, D , , . . iD . :
iDnDR +=1 ;
-
56
nD
nDD .
2-
+=nDDi
nDR2 ..
. 5000 $ 5 10% . , . : .1,0;5$;5000 === inD
: $.10005
5000 ==nD
50000,1=500$. , 1500$ (1000+500).
1000$ 4000$. , 40000,1=400$. , , 1400 $, . .
, :
7.7.
n i . R .
R n R . .
( )i
iRDn+= 1
11,
( )i
i n+ 111
- .
5000 4000 3000 2000 1000 0
0 1 2 3 4 5 1000 1000 1000 1000 1000 + + + + + 500 400 300 200 100 = = = = = 1500 1400 1300 1200 1100
-
57
:
( )niDiR
+=
111
.
. 5000 $ 5 10% . , . : .1,0;5$;5000 === inD
.13233779,0500
6209,01500
61051,111
500
1,111
1,05000
5
R ====
=
7.8.
m n. m n .
m mi . .
:
mi
mi
YFV
mn
f
11
+=
D - . : mn
miD
+1 . , :
mn
miD
+1 = mi
mi
Y
mn
11
+ .
.
11
1
+
+= mn
mn
mi
mi
miD
Y ;
. 10000 $ 3 12 . ) ( m = 4) ) ( m =12) .
-
58
: .12,0;3$;10000 === inD ) .4=m
.56,100414258,1
03,04258,110000
1412,01
412,0
412,0110000
12
43
Y ==
+
+=
) .12=m
$13,33214308,1
01,04308,110000
11212,01
1212,0
1212,0110000
36
123
==
+
+=
Y .
. 500 000 . 5 10% , . . : .1,0;4;5.;500000 ==== imnD
$.07,3207416386,1
025,06386,1500000
141,01
41,0
41,01500000
54
54
==
+
+=
Y
7.9.
. ,
, . , , , , .
D n i . n - ( )niD +1 .
R ig > . n - ( )
ggR
n 11 + , ( )niD +1 . :
-
59
( )niD +1 = ( )ggR
n 11 + . . 9000 $ 4% 10 . 8% . , . . : .08,0;04,0;10$;9000 ==== ginD , , : ( ) ( ) $.8,133214802,1900004,190001 10 ===+ niD
, , :
( ) .8,1332108,0
108,01 10 =+R R, :
$.6,9194866,14
8,13321 ==R
8.
8.1.
invest . , , . , . . , . , 2,5 . . . , . . . :
-
60
, . . . , , .
8.2.
NPV (Net Present Value) . , , . , , , . .. , : r = i + r, i ( ), rp . , (NPV >0), . (NPVmax). , . , NPV. NPV (I) . . r, () , , . , , IC n nPPP ,...,, 21 . :
-
61
= +=+++++n
kk
kn
n
rP
rP
rP
rP
12
21
)1()1(...
)1()1(,
:
( ) .11 ICrP
NPVn
kk
k += = , NPV>0, , , NPV=+++++++=+= = ICr
PNPVn
kk
k
( 930 ), , . , m ,;...;; 10 mICICIC NPV :
( ) ( ) =+= ++=m
jj
jn
mkk
k
rIC
rPNPV
01 11
. . . . 500 . . 3 . , 3 . 1. .; 1,5 . . 2 . . . , 10%. : 0- .
..0,2.;.5,1.;.1.;.3.;.5,0 43210 PICIC =====
( ) ( ) ( ) ( ) ( ) ( ) ( )
rIC
rPNPV
m
jj
jn
mkk
k
.0921,01,01
0,31,01
5,01,01
0,21,01
5,11,01
0,111 1043201
==+++++++=++= =+=
NPV= 92,1 . . >0, . ( ),
-
62
NPV . . 55 . . , 10 . 10 . 9,5%. . : .095,0;10.;.10.;.550 ==== inRIC 55 . 10 , 10 . . .
( ) ( ) ..7880,62095,0
095,0111
10111 10
i
iRPVn
f =+
=+
= NPV .88,70,55788,62 == .
NPV > 0, .
8.3.
, , .
, I n ,...,, 21 , :
ICr
PPI
n
kk
k :)1(1
= +=
:
( ) ( ) =+= ++=m
jj
jn
mkk
k
rIC
rPPI
01 1:
1.
. , .. . , , PI >1 , . PI
-
63
. 1000 . . 6 . 250; 300; 350; 400; 450; 500 . . . , 20%. : .12,0;500;450;400;350;300;250;1000 654321 ======== iPPPPPPIC
( ) ( ) ( ) ( ) ( ).1604,11000:4,1160
1000:2,01
5002,01
4502,01
4002,01
3502.01
3002,01
250:)1( 654321==
=
+++++++++++=+== ICr
PPI
n
kk
k
PI= 1,1604>1, .
8.4.
( ) , . , .
:
D=min n, : ( ) ICrPn
kk
k +=1 1
: ) , ; ) , (, 5 ). . , : -250; 100; 150; 160; 100 . 11%. : : (. )
-
- -
0 - 250 1,000 - 250,00 - 250,00 1 100 0,901 90,10 - 159,90 2 150 0,812. 121,80 - 38,10 3 160 0,731 116,96 78,86>0 4 100 0,659 65,90
-
64
, .
, D, : .326,0
96,11610,38 =
, D= 2, 326 2 119 . 8.5. IRR (international rate of return) -
, .
, . , - ..
, , () . .
, , , , .
, (NPV0).
.
, , .. , , .
IRR , .
, , IRR , .
, , IRR (cost of capital - CC).
IRR>, ;
-
65
IRR0 ( )1(rf = rNPVrNPV ;
2r - , NPV, .. { }0)(max)( 2
-
66
I II
1r =15%
)( 1rNPV 2r =20%
)( 2rNPV
0- 20,0 1,0 20,0 1,0 20,0 1- 6,0 0,8696 5,2176 0,8333 4,9998 2- 8,0 0,7561 6,0488 0,6944 5,5552 3- 14,0 0,6575 9,2050 0,5787 8,1018 +0,4714 1,3432 IRR :
%3,1630,11558146,14714,015
)1520()3432,1(4714,0
4714,015)()()(
)(12
21
11
=+=+=
=+=+= rrrNPVrNPVrNPVrIRR
. , 1r =16% 2r =17% , .. 16,3 (16, 17).
.
I II
1r =16%
)( 1rNPV 2r =17%
)( 2rNPV
0- 20,0 1,0 20,0 1,0 20,0 1- 6,0 0,8662 5,1972 0,8547 5,1282 2- 8,0 0,7432 5,9200 0,7305 5,8440 3- 14,0 0,6407 8,9698 0,6244 8,7416 +0,0870 0,2862
%23,1623,01613732,00870,016
)1617()2862,0(0870,0
0870,016)()()(
)(12
21
11
=+=+=
=+=+= rrrNPVrNPVrNPVrIRR
, IRR=16,23 , . 16,23%.
-
67
9.
9.1.
. . : ; . . , , . . .
. , . , , , . , , . , , , , . , , , . , , , .
, ( ) . ( ) . , (). . , () . , , . . .
-
68
, , . . ( ). .
, , () . , , . . , .
. : ( mP ), , , , ( tV ). , . , , , , . , , . , , , .. . , , , , .. ( tV ) :
( ) ( ) ( )= +=++++++++=n
kk
kn
nk
kt r
Pr
Pr
Pr
Pr
PV1
221
1)1(...
1...
11,
nk PPPP ;...;;...;; 21 , r , n . , , . .
-
69
: , , , , . , . , , - , , . . , , : prir += , i ( ), pr .
, :
) , :
nPVPVFVi
=,
) :
1= nPVFVi
9.2. . . . . . , . , , , . , ( ), . .
-
70
: , , . , , . (: , ) , , . , , , . () . )( mP ( M ) . , .
%100MPm
. 500 . 465 ., .
%93%100500465%100 ==
MPm .
9.3.
, , , nnt r
Pr
Pr
PV)1(
...)1()1( 2
21
+++++= :
,)1( nt r
CV +=
C , ; n , . . 1000 . 560,35 . , 14%. : . 1 . .5;14,0.;.1000 nrC ===
-
71
:
Vt 4,5195194,01000)14,01(1000
5 ==+= .
, , (519,4 .) , , (560,35 .). 2 . , .5.;35,560.;1000 nPVFV ===
%28,12,1228,0135,560
10005 i ==
(12,28%) (14%), .
9.4. .
AP =( k ), += k kkt rPV
)1(
.rAVt =
. , 1 . ., 16%. : .16,0.;1000 == rA
: .625016,0
1000 rAVt ===
, 6250 . .
9.5.
( )
-
72
, . , , :
n
n
n
n
kkt r
Mr
rAr
Mr
AV)1(
)1(11
)1()1(1 +++=+++= =
. , , 1000 ., 10% . , 15% . : .15,0;5.;1001,01000.;1000 ===== rnAM 100 ., :
.42,8322,49722,3354972,010003522,310015,1
100015,015,111
100 55
Vt =+=+=+
=
:
%24,83%1001000
42,832%100 ==MVt
: 15% (832,42 .) ( 100 .) (1000 .) .
9.6. .
, . , .. , , ( ). , ,
, , . . , , , , , ,
-
73
. , , .
, .
,
: += k kkt rPV
)1(, kP
k- ; r . , ,
.. . . ( )
. () . : , , , , . , : , , . , , .
, , .
, ,
DPk = , D , , : .
rDVt = ,
(, ).
9.7.
, , ..
+= k kkt rPV
)1(.
. : ; ; .
-
74
, .. ,
rDVt = D
, r . , .
. 6 . . . 35 . ., 15% ?
: .15,0.;6000 == rD
rDVt = ,
.4000015,0
6000 rDVt === , ,
35 . . .
, . .
(.. ) D . , g . )1( gD + , - 2)1( gD + , k- ( )kgD +1 .. += k kkt r
PV)1(
:
k
kkk
k
t rgD
rgDV
=
=
++=+
+=11 1
1)1()1(
D
++
rg
11 . , 1
11 : gr
rDgr
rD
rg
D+=+
+=++
)1(11
)1(
111
.
, gr
rDVt += )1( .
gr > . , . . 2,7 . . . 4%
-
75
. 20 . ., 14% .
: .14,0;04,0.;.7,2 === rgD , : ( ) ..08,28
04,014,004,017,2)1(
grgDVt =
+=+=
20 . ., . . , , . ) , , S, g . S- : .;...;; 21 sDDD :
.)1(
1)1()1(
...)1(1 2
21s
ss
st rgr
gDr
Dr
Dr
DV +++++++++=
. 1,2; 1,8; 2; 2,4 . . , 5% . , 14%. : .4;14,0;05,0$;4,2$;0,2$;8,1$;2,1 4321 ======= SrgDDDD :
( ) ( ) ( )( )
( ) .79,2114,011
05,014,005,014,2
14,014,2
14,012
14,018,1
14,012,1
4432 Vt =++++++++++=
, , 21,79 . ) g : . , . , g q ,
= +
+=1 )1(
)1(k
k
k
t rgDV :
( )( )
( )( ) s
s
k
sk
k
skk
k
s
s
kk
k
ot rqrqD
qgD
rqD
rgDV
)1(1)1(
)1()1(
11
11
10
11 ++++
+=++++
+= =
+==
-
76
0D , ; sD S - g S ; q . . 10$ . , 3% , 2% . , 10%. : D0 =10$; g = 0,03; q =0,02; S = 3.
$.927,10)03,01(10 33 =+=D ( )( )
$18,12884,10134,26731,0319,139)821,0877,0936,0(101,11
02,01,0)02,01(927,10
1,103,1
1,103,1
1,103,110
)1(1)1(
11
33
3
2
2
1
=+=+++==
++
++=+
++++=
=s
ss
kk
k
ot rqrqD
rgDV
9.8.
i , ,
mPDi =
D - ; mP - . . , 60 ., 1000 .
: %6,06,01000
60.,1000.60 ===== rrPD m . , :
0
01
Pn
PP
PDi
o
+=
0P ; 1P ; n ; D n ( ). . 5 . . 8 . . 300 ., 450 ., 600 . .
-
77
: .3.,8000.,5000 10 === nPP :
.4503
600450300 D =++= :
.29,05000
350008000
5000450 =
+=i
, 29% .
9.9. , . . -, . . FV . 1d 1t . ). 1t .
= 111 1 dYtFVPV
Y - , , Y = 360 2t 2d
= 222 1 dYtFVPV
) - , , .
1n :
1)1( 11ndFVPV =
2n 2d 2)1( 22
ndFVPV =
. :
)( 21112
nnPVPVPVi
=
-
78
: Ytn 11 = ; Y
tn 22 = ;
:
1211
2 = nnPVPVi
. 100 . 150 , - 15%. 30 12%. . : .12,0;15,0;12030150;150.;100000 2121 ====== ddttFV
: PV 93750)15,0
3601501(1000001 ==
:
PV 96000)12,03601201(1000002 ==
:
288,03093750
3602250
3603093750
9375096000 ==
=i 28,8%
. 200000 5 . - 10%. 8%. . :
.235,5;08,0;1,0,200000 2121 nnddFV ====== PV 118098)1,01(200000 51 == PV 169280)08,01(200000 22 ==
1275,011275,11118098169280
3 ===i 12,75%
10.
10. 1.
,
-
79
() . , ,
. , , . , . .
, , 100 . - 100 .
. . , , , .
10. 2.
,
. , PV . n .
PV ; FV - ;
0K 1 - , , , ./. ./. , 0K - , 1 - . , , : ,1
0kIPVKK
PVFV == .0
1
KKI k =
n , S :
p
k
IIPVS = . pI - n .
npI )1( += , - .
i :
.)1( np
k iPVIIPV +=
-
80
i , :
1= np
k
IIi .
, pk II = . pk II > , pk II < - .
. , 500 . , , 1,5 . 24, 15 ., - 24,20 . 34, 65 . 34, 75 . , 24,75 ., 24,85 . 36, 50 . 36, 60 . 7,5%. : ) , - ; ) : ) ; ) , . : 1146,1)075,01()1( 5,1 =+=+= npI .
, 1,5 11,46% ,
) .02273,120,2475,24
0
1 ===KKI k
IPVFV k .36,51102273,1500 === .05036,1
75,3450,36
0
1 ===KKI k
IPVFV k .18,52505036,1500 === , . ) :
.783,4581146,1
36,511 I
FVI
IPVSp
p
k
====
.825,4711146,1
18,525 I
FVI
IPVSp
p
k
==== , .
) .
-
81
0557,019443,011146,102273,11 5,1 ==== n
p
k
IIi .
. 5,57%.
.0388,0196120,011146,105036,11 5,1 ==== n
p
k
IIi
. 3,88%. ) , , - . : pk II = .
, ,1146,120,241 == I
.97,261 = , 26,97 . . , .
:
1146,175,341 = . .73,381 =
, , .
10.3.
,
. ( ), . : 1) ; 2) ; 3) 4) ) ; . 1 3 , .. .
: , . : . ( ), . ,
-
82
, . . : *
*PV ; PV ; FV ;
*FV ; 0 ; 1 ;
n ; i ; j
. ( * ,
).
*PV , PV i . , : 1) ;* 0KPVPV = 2) );1(*)1( 0 inKPVinPVFV +=+= 3)
1
0
1
0
1
)1(*)1(*
*KKinPV
KinKPV
KFVFV +=+== .
, :
.)1(**1
0
KKinPVFV +=
, :
.)1(**1
0
KKiPVFV n +=
, *,* PVFV = .. 1)1(1
0 =+KKin
1)1(1
0 =+KKi n
. , )1(01 inKK += , ,
niKK )1(01 += . )1(01 inK +> ,
.
-
83
, : .)1(01 niK +> . 5000 $
4 . 25,3. . 25,85. . 9%. . :
) ; ) ; ) ,
. : ) :
$.43,5040.85,253,25
12409,015000)1(**
1
0 =
+=+=KKinPVFV
) %.4,2024,0
1245000
500043,5040 ik ==
) .06,2612409,013,25)1(01 inKK =
+=+= .06,261 > , .
1 , , . .
, j . 1
:
;)1(11
0
KKinjn +=+
jnin +
+=11
01 .
, ,11
01 jnin +
+> .
. , 4%.
:
-
84
.72,25
3104,01
3109,01
3,251 =++
=
, 25,85. (, 25,72 . ), , .
, , :
$43,5040$67,50663104,015000)1(** >=
+=+= jnPVFV , .
PV , *PV j . , : 1) ;*
0KPVPV =
2) ;)1()1(**0K
jnPVjnPVFV +=+=
3) 0
1
0
11 )1(
)1(*KKjnPV
KjnKPVKFVFV +=+== .
, :
.)1(0
1
KKjnPVFV +=
:
.)1(0
1
KKjPVFV n +=
, ,PVFV = .. 1)1(0
1 =+KKjn
1)1(0
1 =+KKj n
.
jn
KK += 10
1 , , njKK
)1(0
1 += .
-
85
jn
KK +< 10
1 ,
.
, : njKK
)1(0
1 +< 1K ,
. .
1 :
;)1(10
1
KKjnin +=+
jnin +
+=11
01 .
jnin +
+