« » , " ! ! ,!, . . 6.12.
) , % ! , "# .
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$ 6.12. |
G%& ;+ % " % + ' %"%' +'+!# %7 *%$+"% |
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P |
6 % |
6 |
8 % |
' , ’ ’ |
7 % |
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4 % |
9 ! |
2 % |
;-# "$ #-# $ |
27 % |
+ ! ! ! ! " -8 ( . 3) ! .
4 " ! :
1. 0 ' (Net Present Value – NPV). 5 , ! -, " # "! # , ! ! " .
* " : (NPV) – # !" ! , ! !, # # -" " ! #! . N % ", ! , NPVr ! :
n |
CFk |
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NPV |
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IC, (6.11) |
(1 r) |
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k 1 |
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CF1, CF2, CFk ,....., CFn – # " n ; IC – ;
r – ;
n – ! " .
* " +, %! NPV > 0, * % % * ' "% ! & 7; # )% NPV < 0, * % % "; ! , $% &$% ; NPV = 0 – * % *$+ % &$% .
N % ! # m , ! :
n |
CF |
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ICj |
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NPV |
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(6.12) |
(1 r) |
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Ij – , j- .
$ , , , -# -r. ! :
n |
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CFk |
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NPV |
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IC, |
(6.13) |
n |
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k 1 |
(1 rk ) |
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k 1
n
(1 rk ) = (1+r1) (1+r2) … (1+rn).
k 1
G%&-!# '% * ! " .
9. & 30 . . - ! # " CF1 = 10 . ., CF2 = 16 . ., CF3 = 15 . . ) ! , %! 15 %.
4 - ! , . . 6.13.
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$ 6.13. |
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( . .) |
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G , t |
Q %F% |
@ % ' %6- |
@ % % |
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(15 %) |
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*% |
-%F% *% |
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1/ (1 + 0,15)t |
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0- |
-30 |
1,0 |
-30 |
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1- |
10,0 |
0,8696 |
8,696 |
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2- |
16,0 |
0,7561 |
12,097 |
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3- |
15,0 |
0,6575 |
9,862 |
* " : (NPV) + 0,655 ( . .)
# NPV = (+0,655) > 0, ,.
9. ) ! , , % ! ! : 15 %, 17 %, 16 %.
6 , !
6.10.
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NPV |
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10 |
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16 |
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15 |
30 0,156 . . |
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0,15)1 |
1,15 |
1,17 |
1,15 1,17 1,16 |
(1 |
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& " , NPV = (+0,156) > 0. |
8 |
, % ! - |
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. & " -NPV , !, % !# NPV " # .
6 , !, . . 6.14.
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$ 6.14. |
- "%! '%" ! NPV |
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G Q |
@ / |
1. 9 |
1. 4 ! |
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NPV " |
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2. 7 % ! |
2. & - |
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! ( ) NPV |
3. 4 " - |
3. ) " ! ! - |
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4. 4 " - |
4. 2 ! |
# ( !# ) |
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" NPV |
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5. + , , |
5. ) |
" ! NPV |
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NPV(1+P)=NPV(1)+NPV(P), % - |
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6. 9 ( - |
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7. 7 |
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2. 0 ( ) -(Profitability Index – PI) " ! . 8 , -NPV, PI – . )# :
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n CF |
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k 1 |
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PI |
NPV |
1 (6.14) |
PI |
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IC |
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N % !m , ! :
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9 ! , % PI !# . N % *% -1, ! !( ! NPV=0). N % *% # 1, ! ’ , % ! -( ! NPV ’ ).
7 NPV, ! . & ! '*" ! , % ! " -NPV, " # ! .
) ,’ , , . . 6.15.
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$ 6.15. |
- "%! ' %"+ %& ;+ + P |
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7 |
1. !# |
1. 7 # |
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2. 2 % NPV |
2. ' ! ! |
! - |
" ! |
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! NPV |
3. 4 |
3. ) " - |
!- |
NPV, |
( ) |
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4. 2 |
. # ! 6.13, - ! ! : 30,655/30 = 1,02.
8 " , PI>1, , , % "# 2 .
3. 0 # ' (Internal Rate of Return – IRR).
+ F # % ' "%; " % – ,! ## . 8 " , -
# ! (IRR) ! -! , ! - .
' IRR - ! :
n |
CFk |
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NPV |
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IC 0 , |
(6.16) |
(1 IRR) |
k |
k 1 |
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IRR – , NPV=0.
' " " :
n |
CF k |
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IC |
(6.17) |
(1 IRR ) |
k |
k 1 |
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N % ! m , ! :
n |
CF |
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IC j |
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(6.18) |
NPV |
k |
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k 1 |
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j 0 |
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6 , # !! -, ! " ! " , -NPV=0.
!# – IRR# % - , # ! . 9# ! ! IRR !# , " - ! r.
4 IRR Excel @ D," -" # . * ! ’ ,– . $ " ! .
1 @ D :
1)" -Excel. N % " ,! «0»;
2): f', « », &J"KH;
3) 4 Excel,;
4)4 ! -IRR, K! - " Excel.
9 ! ! , % - " . !! . 9 ! -r1 r2 , % (r1 r2) -NPV = f(r) «+» «-», . 4 - ! :
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IRR r |
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NPVr1 |
(r r ) , (6.19) |
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NPVr1 NPVr2 |
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1 |
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2 1 |
r1 – , NPV>0;
r2 – , NPV<0.
6 ! " (r1 r2),!# ! ! , " - ! ! 1 %.
. & # ! -20 . ., # - " CF1 = 6 . ., CF2 = 8 . ., CF3 = 14 . ., % -" 15–20 %.
NPV15 NPV20, # !6.16.
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$ 6.16. |
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( . .) |
G , t |
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% |
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r1 =15 % |
@ % % |
r2 =20 % |
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@ % % |
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1/(1 + |
*% |
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1/ (1 + |
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*% |
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0,15) t |
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0,2)t |
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0- |
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-20 |
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1,0 |
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1,0 |
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-20 |
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6,0 |
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0,870 |
5,22 |
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0,833 |
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4,999 |
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2- |
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8,0 |
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0,756 |
6,048 |
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0,694 |
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5,552 |
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3- |
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14,0 |
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0,658 |
9,212 |
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0,579 |
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8,106 |
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0,4714 |
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-1,3432 |
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NPV15 = + 0,48 |
NPV20 = – 1,343 |
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& NPV15 NPV20 IRR. |
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IRR 15 |
0,48 |
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(20 15) 16,31 % |
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0,48 ( 1,343) |
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6 , 16,3 % – " ," # .
+ # ! - , . . 6.17., ,
# . 4 ! %" %-+ F # % ' "%; " % (MIRR).
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$ 6.17. |
- "%! ' %"+ %& ;+ + RR |
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"%! |
1. K " ! - |
1. ( |
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2. & " ! !- |
2. S ! |
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( " ) ! |
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# |
3. 7 % - |
3. ) ! - |
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!# ! - |
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, % IRR ! |
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4. ) " ! |
4. 7 " - |
( , |
# ( |
! !# IRR, !# |
! ) |
) |
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5. 4 " ! ! - |
5. 2 |
" - |
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# ( !# ) |
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6. ) ! |
6. 9 % IRR - |
’ , |
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" " - |
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7. P !# , " NPV |
7. 8 , - |
« » |
" # - |
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! # ! |
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8. ) " ! ! |
8. 2 " |
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9. 7 !# - |
9. 7 " |
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% |
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10. & - |
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! ( . |
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Excel) |
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4. 0 # ' (Modifid Internal Rate of Return – MIRR).
+ # (MIRR), IRR, , ! -, ! #.
4 MIRR, IRR, " !:
, % #! , % ,IRR, IRR;
" ! ;
MIRR " (-) # ;
! NPV.
) , , % ! -
! , " - # ! MIRR.
) MIRR " :
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IC j |
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CF k (1 r ) n k |
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k 1 |
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(6.20) |
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* 6.20 MIRR: |
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(6.21) |
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MIRR n |
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IC j |
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9. & . 6.18 #, %12 %.
$ 6.18.
%" ( ) |
0 |
1 |
2 |
3 |
4 |
* (* ), |
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( . .) |
500 |
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120 |
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K # " (CF), |
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300 |
450 |
150 |
180 |
( . .) |
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9 ! # :
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500 |
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120 |
500 95,663 595,663 ( . .) |
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2 |
(1,12) |
1,12 |
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+ ! # |
" ! - |
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300·1,123+450·1,122+150·1,121+180·1,120 |
= |
421,478+564,48+168+180 = |
1333,958 ( . .) |
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6 , MIRR 4 |
1333 ,958 |
1 0,2233 , 22,33 % |
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595 ,663 |
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5. 0 (DPP).
@ % % * %" % +* % – ! ," # # - % ! , ! ! , -# .
4 ! ," , % -, # - ! .
5 , (**), " .
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IC n |
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IC j |
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DPP |
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CFk |
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k 1 (1 r)k |
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CFk – # " k- ; |
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r – ; |
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n – ! " |
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8Fk |
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n |
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m |
IC j |
D'' = min n, k 1 (1 r)k |
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j 0 (1 r) j |
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(6.23)
. & 6.19.
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$ 6.19. |
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* % N 1 |
* % N 2 |
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G% |
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@ % |
@ % % |
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@ % |
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@ % % |
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Q %F% |
' %6 & |
Q %F% |
' %6 & |
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-%F% *%- |
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-%F% *%- |
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*% % |
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*% % |
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% |
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10 % |
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12 % |
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0- |
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- 30 |
1 |
-30 |
- 28 |
1 |
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-28 |
1- |
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24 |
0,909 |
21,816 |
8 |
0,893 |
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7,144 |
2- |
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12 |
0,826 |
9,912 |
12 |
0,797 |
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9,564 |
3- |
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4 |
0,752 |
3,080 |
12 |
0,712 |
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8,544 |
4- |
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12 |
0,636 |
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7,632 |
9 1: , % ! - , # :
21,816 +9,912·' = 30;
x 30 21,816 0,83 , 10 (0,83·12)
9,912
"*% " %, DGG1 = 1 10 ' # .
9 2: ! ! , : 7,144+9,564+8,544+7,632·' = 28
x 28 (7,144 9,564 8,544) 0,36 , 5 (0,36·12) 7,632
"*% " %, DGG2 = 3 % 5 ' # .
6 , * T 1 # , #- # # ! # .
7 " ! , !# DPP, . . 6.20.
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$ 6.20. |
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- "%! ' %"+ %& ;+ + DPP |
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"%! |
1. |
9 |
1. 7 # , - |
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2. |
& - |
2. 7 ! ’ |
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( " ) |
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, !# !- |
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