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1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "Bullet I
tem" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 
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1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 29 "Cvi\350en\355 z finan\350
n\355 matematiky" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Relativn\355
 hodnota a m\355ra" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "i-t\341 komo
dita, i=1,2,3 se obchoduje v okam\236iku t=1,2,3,4,5 v kurzu " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "restart;\nkappa:=(i,t)->abs(
2*i^3+(5*i^2)*sin(i*t/6))+1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "pl
ot(\{kappa(1,t),kappa(2,t),kappa(3,t)\},t=0..6, color=[GREEN,NAVY,RED]
,title=`kurzy komodit v z\341vislosti na \350ase`);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 5 "Kurzy" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "Dig
its:=3;\nj:='j';\nfor i from 0 to 5 do\nprint(i,evalf(kappa('j',i)) $'
j'=1..3);\n#print(i,(kappa(i,j)) $j=1..3);\nod;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Rozd
\355ly kurz\371 (p\370\355r\372stek kurzu na jednotku mno\236stv\355 k
omodity)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "for i from 0 to 5. do
\nxxx:=evalf(kappa(j,i)):yyy:=evalf(kappa(j,i+1)):\nzzz:=([xxx,yyy] $j
=1..3);\nzzzz:=evalf(kappa(j,i+1)-kappa(j,i));\nprint(i,evalf(kappa(j,
i+1)-kappa(j,i)) \n$j=1..3);\nod:\nevalf([5, -3.70+5.*sin(1), -19.9+20
.*sin(2), -26.9+45.*sin(3)]);\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 330 "Od okam\236iku 0 po okam\236
ik 2 m\341 nejv\354t\232\355 p\370\355r\372stek kurzu za jednotku \350
asu, kter\341 po dan\351m okam\236iku n\341sleduje, komodita 3. tato i
nformace ale nen\355 podstatn\341 pro rozhodov\341n\355 o investic\355
ch. N\341s nezaj\355m\341, kolik vyd\354l\341me na jednotce mno\236stv
\355 komodity, ale na jednotce mno\236stv\355 kapit\341lu, kter\354 do
 komodity investujeme. Proto si spo\350\355t\341me:" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 78 "Relativn\355 p\370\355r\371stky kurz\371 (p\370
\355r\372stek kurzu na jednotku investovan\351ho kapit\341lu)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "for i from 0.1 to 5.1 do\ni:=i-.1;
\nprint(i,evalf(kappa(j,i+1.)-kappa(j,i))/(kappa(j,i)) $j=1..3);\ni:=i
+.1\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "for i from 1 t
o 3 do\nplot(\{kappa(i,t),kappa(i,t+1)-kappa(i,t),(kappa(i,t+1)-kappa(
i,t))/kappa(i,t)\},t=0..6, color=[GREEN,NAVY,RED]);\nod;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 346 "\n\nplot(\{kappa(1,t+1)-kappa(1,t)
,kappa(2,t+1)-kappa(2,t),kappa(3,t+1)-kappa(3,t)\},t=0..6, color=[GREE
N,NAVY,RED],title=`p\370\355r\371stky kurz\371 za dobu 1`);\n\nplot(\{
(kappa(1,t+1)-kappa(1,t))/kappa(1,t),\n(kappa(2,t+1)-kappa(2,t))/kappa
(2,t),\n(kappa(3,t+1)-kappa(3,t))/kappa(1,t)\},t=0..6, color=[GREEN,NA
VY,RED],title=`relativn\355 p\370\355r\371stky kurz\371 za dobu 1`);\n
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Derivace kurzu podle \350asu \+
je " }{XPPEDIT 18 0 "limit((kappa(i,t)-kappa(i,t+delta(t)))/delta(t),d
elta(t)=0)" "6#-%&limitG6$*&,&-%&kappaG6$%\"iG%\"tG\"\"\"-F)6$F+,&F,F-
-%&deltaG6#F,F-!\"\"F--F26#F,F4/-F26#F,\"\"!" }{TEXT -1 40 " ur\350uje
 okam\236itou absolutn\355 zm\354nu kurzu." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 139 "\nplot(\{diff(kappa(1,t),t),diff(kappa(2,t),t),diff(
kappa(3,t),t)\},t=0..6, color=[GREEN,NAVY,RED],title=`derivace kurz
\371 komodit podle \350asu`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 62 "diff(Kappa(i,t)/Kappa(i,t0),t)=diff(Kappa(i,t),t)/Kappa(i,t0);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Relativn\355 okam\236it\341 zm
\354na  kurzu je " }{XPPEDIT 18 0 "limit((kappa(i,t)-kappa(i,t+delta(t
)))/(kappa(i,t)*delta(t)),delta(t)=0)=diff(kappa(i,t),t)/kappa(i,t)" "
6#/-%&limitG6$*&,&-%&kappaG6$%\"iG%\"tG\"\"\"-F*6$F,,&F-F.-%&deltaG6#F
-F.!\"\"F.*&-F*6$F,F-F.-F36#F-F.F5/-F36#F-\"\"!*&-%%diffG6$-F*6$F,F-F-
F.-F*6$F,F-F5" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 193 "plot(\n\{diff(kappa(1,t),t)/kappa(1,t)\n,diff(kappa(2,t),t)/kap
pa(2,t)\n,diff(kappa(3,t),t)/kappa(3,t)\n\},t=0..6, color=[GREEN,NAVY,
RED],title=`relativn\355 hodnoty derivace kurz\371 komodit podle \350a
su`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=12;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Kdybychom tedy mohli obchodovat ko
modity v ka\236d\351m okam\236iku, bylo by pro n\341s nejv\354hodn\354
j\232\355 nakoupit " }{TEXT 256 5 "t\370et\355" }{TEXT -1 28 " a podr
\236et ji a\236 do okam\236iku" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 
19 1 "T1:=fsolve(diff(kappa(2,t),t)/kappa(2,t)=diff(kappa(3,t),t)/kapp
a(3,t));" "6#>%#T1G-%'fsolveG6#/*&-%%diffG6$-%&kappaG6$\"\"#%\"tGF1\"
\"\"-F.6$F0F1!\"\"*&-F+6$-F.6$\"\"$F1F1F2-F.6$F;F1F5" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 25 "Pak je prodat a nakoupit " }{TEXT 257 6 "druhou
" }{TEXT -1 37 " komoditu a tu podr\236et a\236 do okam\236iku" }}}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "T2:=fsolve(diff(kappa(1,t),t)/
kappa(1,t)=diff(kappa(2,t),t)/kappa(2,t));\n" "6#>%#T2G-%'fsolveG6#/*&
-%%diffG6$-%&kappaG6$\"\"\"%\"tGF1F0-F.6$F0F1!\"\"*&-F+6$-F.6$\"\"#F1F
1F0-F.6$F:F1F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "a pak koupiprvn
t " }{TEXT 258 6 "prvn\355 " }{TEXT -1 69 "komoditu. Porovn\341me v
\354nos s obchodov\341n\355 v diskr\351tn\355m a spojit\351m \350ase:
" }{MPLTEXT 1 0 1 " " }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 11 "Obchodov
\341n\355" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 279 " M\341te k dispozici
 100 korun a p\370esnou znalost v\232ech kurz\371 p\370edem.  Jakou ne
jv\354t\232\355 \350\341stku m\371\236ete vyobchodovat (p\370edpokl
\341dejme, \236e komodity jsou ide\341ln\354 d\354liteln\351, \236e m
\371\236ete koupit jakoukoliv \350\341st jednotky komodity a \236e za
\350\355n\341te obchodovat v okam\236iku t=0 a kon\350\355te v okam
\236iku t=4)?" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 48 "Obchodov\341n
\355 v diskr\351tn\355ch okam\236ic\355ch t=1,2,. . ." }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 132 "PocetTitulu:=3;\nKapital[0]:=100;\nVynos
nost:=(i,t)->kappa(i,t+1)/kappa(i,t);\nHodnota:=t->Sum(kappa(i,t)*Poce
t[i],i=1..PocetTitulu);\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
46 "Max:=t->max(Vynosnost(i,t) $i=1..PocetTitulu);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 123 "Vyber:=proc(t)\nglobal CisloTitulu;\nfor i
 from 1 to PocetTitulu do\nif Max(t)=Vynosnost(i,t)\nthen CisloTitulu:
=i;\nfi \nod\nend;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 380 "for \+
t from  0 to 4 do \ni:='i';\nVyber(t); \n  for i from 1 to PocetTitulu
 do\n   Pocet[i]:=0;\n  od:\nPocet[CisloTitulu]:=Kapital[t]/kappa(Cisl
oTitulu,t);\nKapital[t+1]:=Pocet[CisloTitulu]*kappa(CisloTitulu,t+1);
\nprint(`v \350ase `,t, ` kupuji `, evalf(Pocet[CisloTitulu]),\n ` jed
notek komodity \350\355slo `,   CisloTitulu, `v \350ase`, t+1, `budu m
\355t kapit\341l velikosti`, evalf(Kapital[t+1]));\nod:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "V okam
\236ic\355ch 3 a 4  nemus\355me chodit na trh, sta\350\355 st\341le dr
\236et komoditu \350\355slo 1." }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 
29 "Obcodov\341n\355 v libovoln\351m \350ase:" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 161 "Do okam\236iku t=5 m\371\236eme vyobchodovat o n\354co
 v\354t\232\355 kapit\341l, pokud uskute\350n\355me obchody v okam\236
ic\355ch 0, T1 a T2 nam\355sto obchod\371 v okam\236ic\355ch 0,1,2. Je
ho velikost je:" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "Vynosnost:=(i,t1,t
2)->kappa(i,t2)/kappa(i,t1);\n\nevalf((100*Vynosnost(3,0,T1)*Vynosnost
(2,T1,T2))*Vynosnost(1,T2,5));" "6#C$>%*VynosnostGf*6%%\"iG%#t1G%#t2G7
\"6$%)operatorG%&arrowG6\"*&-%&kappaG6$F(F*\"\"\"-F26$F(F)!\"\"F/F/F/-
%&evalfG6#**\"$+\"F4-F%6%\"\"$\"\"!%#T1GF4-F%6%\"\"#FA%#T2GF4-F%6%F4FE
\"\"&F4" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 "Slo
\236en\354 \372rok (Coumpound interest)" }}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 13 "Obecn\354 vzorec" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 147 "restart;\nPhi[1]:=(x0,xi,t)->x0*(1+xi)^(floor(t));\n\nPhi[2]:=(
x0,xi,t)->x0*(1+xi)^(floor(t))*(1+xi*(t-floor(t)));\n\nPhi[3]:=(x0,xi,
t)->x0*(1+xi)^(t);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Dosad\355
me n\354jak\351 hodnoty: " }{XPPEDIT 18 0 "x0=10,xi=1.9" "6$/%#x0G\"#5
/%#xiG-%&FloatG6$\"#>!\"\"" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "Arg:
=(10,1.9,t);\nplot(\{\nPhi[1](Arg),\nPhi[2](Arg),\nPhi[3](Arg)\},\nt=0
..5,\nthickness=3,\ncolor=[coral,gold,maroon],discont=true);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "Rozd\355l mezi hodnotami spojit
\351ho a sm\355\232en\351ho uro\350en\355, tedy rozd\355l mezi hodnota
mi exponenc\355\341ln\355 funkce a hodnotami na jej\355ch se\350n\341c
h." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot(Phi[2](Arg)-Phi[3](Arg),
t=0..5);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Ur\350\355me okam\236
ik, kdy je rozd\355l roven polovin\354 p\371vodn\355ho vkladu" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "T:=solve(Phi[2](Arg)-Phi[3](
Arg)=5,t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "evalf(subs(t=
T,Phi[2](Arg)));\nevalf(subs(t=T,Phi[3](Arg)));" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 8 "P\370\355klady" }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 
171 "1.  1. m\341me  na kont\354 10. zlat\354ch  1. 3. ulo \236\355me \+
dal\232\355ch 10 zlat\354ch a 1. 9. je\232t\354 20 zlat\354ch. Jak\354
 bude stav \372\350tu p\370i ro\350n\355 \372rokov\351 m\355\370e  0.0
5   1. 1. n\341sleduj\355c\355ho roku?" }}{SECT 1 {PARA 5 "" 0 "" 
{TEXT -1 6 "\374e\232en\355" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
401 "PocetDniVMesici[1]:=31:\nPocetDniVMesici[2]:=28:\nPocetDniVMesici
[3]:=31:\nPocetDniVMesici[4]:=30:\nPocetDniVMesici[5]:=31:\nPocetDniVM
esici[6]:=30:\nPocetDniVMesici[7]:=31:\nPocetDniVMesici[8]:=31:\nPocet
DniVMesici[9]:=30:\nPocetDniVMesici[10]:=31:\nPocetDniVMesici[11]:=30:
\nPocetDniVMesici[12]:=31:\nPocetDniDoKonceRoku:=(d,m)->\nPocetDniVMes
ici[m]-d+sum(PocetDniVMesici[i],i=m+1..12);\nPocetDniDoKonceRoku(1,1);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "zeta := (1+5/100)^(1/3
65)-1;  evalf(10*(1+zeta)^PocetDniDoKonceRoku(21,1)+10*(1+zeta)^PocetD
niDoKonceRoku(16,3)+20*(1+zeta)^PocetDniDoKonceRoku(7,9));" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "PocetDniDoKonceRoku(21,1);\n
PocetDniDoKonceRoku(16,3);\nPocetDniDoKonceRoku(7,9);" }}}}}{PARA 4 "
" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 232 "Za jak dl
ouhou dobu bude nomin\341ln\355 stav va\232eho \372\350tu 120 checht
\341k\371, kdy\236 v na n\354j \350ase 0 ulo\236\355te 90 chcecht\341k
\371 a \372rokov\341 m\355ra je 0.05 po dobu, kdy je nomiin\341ln\355 \+
stav \372\350ztu men\232\355 ne\236 100 chcet\341k\371 a 0.07 po dobu,
 kdy je v\354t\232\355, ne\236 100?" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 
-1 6 "\374e\232en\355" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Nejprve v
ypo\350\355t\341me, za jak dlouhou dobu bude na na\232em \372\350tu 10
0checht\341k\371:" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "restart;
  T[1]:=solve(90*1.05^t=100,t);" "6#C$%(restartG>&%\"TG6#\"\"\"-%&solv
eG6$/*&\"#!*F))-%&FloatG6$\"$0\"!\"#%\"tGF)\"$+\"F6" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 69 "Potom vypo\350\355t\341me, za jak dlouhou dobu \+
stav \372\350tu naroste ze 100 na 120" }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "T[2]:=solve(100*1.07^t=120,t);" "6#>&%\"TG6#\"\"#-%&sol
veG6$/*&\"$+\"\"\"\")-%&FloatG6$\"$2\"!\"#%\"tGF.\"$?\"F5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 84 "Nakonec ob\354 doby se\350teme. Stav \372
\350tu bude m\355t nomin\341ln\355 hodnotu 100 chcet\341k\371 za dobu \+
" }{XPPEDIT 19 1 "Tt:=T[1]+T[2];" "6#>%#TtG,&&%\"TG6#\"\"\"F)&F'6#\"\"
#F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "P\370itom jednotka \350as
u je t\341\236,  v jak\351 je vyj\341d\370ena m\355ra \372roku. Pokud \+
se nap\370\355klad \372roky p\370ipisuj\355 pouze v okam\236ic\355ch \+
" }{XPPEDIT 18 0 "t=1/365,2/365,3/365" "6%/%\"tG*&\"\"\"F&\"$l$!\"\"*&
\"\"#F&F'F(*&\"\"$F&F'F(" }{TEXT -1 129 ",... m\236eme \370\355ci, \+
\236e nomin\341ln\355 stav \372\350tu nikdy 120 nebude. Ale prvn\355 o
kam\236ik, kdy m\371\236eme na \372\350tu s \350\341stkou 120chect\341
k\371 po\350\355tat je:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 
"T[1]:=ceil(solve(T[1]=x*1/365,x));\nT[2] := solve((90*1.05^(T[1]/365)
)*1.07^t = 120,t);\n#(90*1.07^(T[1]/365))*1.07^T[2];\nT[2]:=ceil(solve
(T[2]=x*1/365,x));\nTt:=T[1]+T[2];\n\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 21 "t\354 okam\236ik, tedy doba" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "evalf(Tt/365);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
37 "V tu chv\355li ov\232em budeme na \372\350t\354 m\355t:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "(90*1.05^(T[1]/365))*1.07^(T
[2]/365);" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 65 "P\370i jak\351 \+
\372rokov\351 m\355\370e za jenotku \350asu je stav obou \372\350t\371
 po dob\354  " }{XPPEDIT 18 0 "T:=2;" "6#>%\"TG\"\"#" }{TEXT -1 38 " s
tejn\354, je-li na prvn\355m  na po\350\341tku  " }{XPPEDIT 18 0 "x1:=
1234;" "6#>%#x1G\"%M7" }{TEXT -1 54 " zlat\354ch a  jednoduch\351 \372
ro\350en\355 a na druh\351m na po\350\341tku " }{XPPEDIT 18 0 "x2:=123
0; " "6#>%#x2G\"%I7" }{TEXT -1 28 " zalt\354ch a slo\236en\351m \372r
\350en\355 ?" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "P\370i jak\351 \+
\372rokov\351 m\355\370e za jenotku \350asu je stejn\354 v\354hodn\351
 spo\370it " }{XPPEDIT 19 1 "x[1]:=1234;" "6#>&%\"xG6#\"\"\"\"%M7" }
{TEXT -1 25 " jednoduch\354m \372rokem jako " }{XPPEDIT 19 1 "x[2]:=12
30; " "6#>&%\"xG6#\"\"#\"%I7" }{TEXT -1 25 " slo\236en\354m \372rokem \+
po dobu " }{XPPEDIT 19 1 "T:=2;" "6#>%\"TG\"\"#" }{TEXT -1 1 "." }}}
{PARA 4 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 6 "
\374e\232en\355" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "rovnice:=
x[1]*(1+xi*T)=x[2]*(1+xi)^T;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "res
eni:=solve(\{rovnice,xi>0\},xi);\nallvalues(reseni);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 30 "zeta:=op(2,op(evalf(reseni)));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 149 "plot(x[2]*(1+xi)^T-x[1]*(1+xi*T),xi=0..z
eta+0.1,title=`Rozd\355l v sou\350. hodn. jedn. a slo\236 spo\370. v z
\341v. na vel. \372rokov\351 m\355ry`,titlefont=[HELVETICA,7]);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 44 "Ukl\341d\341n\355 \350\341stek v ekvidistantn\355ch okam
\236ic\355ch" }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 185 "P\370edpokl\341d
ejme rok o 360 dnech s 12 m\354s\355ci po 30 dnech. Ukl\341d\341te na \+
\372\350et 100 k\350. \351\350et se \372ro\350\355 a m\355ra \372roku \+
je 0.05 p. a.. kolik bude na u\350t\354 na konci roku kdy\236 \nUlo
\236\355te jen jednou a to" }}{PARA 15 "" 0 "" {TEXT -1 0 "" }}{PARA 
16 "" 0 "" {TEXT -1 15 "na za\350\341tku roku" }}{PARA 16 "" 0 "" 
{TEXT -1 15 "v polovin\354 roku" }}{PARA 15 "" 0 "" {TEXT -1 13 "dvakr
\341t a to " }}{PARA 16 "" 0 "" {TEXT -1 33 "na za\350\341tku roku a v
 polovin\354 roku" }}{PARA 16 "" 0 "" {TEXT -1 31 "v polovin\354 roku \+
a na konci roku" }}{PARA 15 "" 0 "" {TEXT -1 18 "dvan\341ctkr\341t a a
 to" }}{PARA 16 "" 0 "" {TEXT -1 25 "na za\350\341tku ka\236d\351ho m
\354s\355ce" }}{PARA 16 "" 0 "" {TEXT -1 24 "na konci ka\236d\351ho m
\354s\355ce." }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 81 "Sou\350asnou  hodnotu \372\350tu (present value
) ozna\350\355me PV. \254as budeme po\350\355tat ve dnech." }}}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "restart;\nPV:=sum(Z(t)*(1+xi)^((360-t
)/360), t=1..360):" "6#C$%(restartG>%#PVG-%$sumG6$*&-%\"ZG6#%\"tG\"\"
\"),&F/F/%#xiGF/*&,&\"$g$F/F.!\"\"F/F5F6F//F.;F/F5" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 48 "PVdem:=Sum(Z(t)*(1+xi)^((360-t)/360), t=1..360):" }
}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "xi:=.05;" "6#>%#xiG-%&FloatG6$\"\"&!
\"#" }{TEXT -1 21 ", t je \350as v roc\355ch, " }{XPPEDIT 18 0 "z(t);
" "6#-%\"zG6#%\"tG" }{TEXT -1 28 " je velikost \372lo\236ky v \350ase \+
t" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "z[1]:=t->piecewise(t=
1,100,t<>0,0);\nz[1](t);\nz[2]:=t->piecewise(t=360/2,100,t<>0,0);\nz[2
](t);\nz[3]:=t->piecewise(t=1,100,t=360/2,100,0);\nz[3](t);\nz[4]:=t->
piecewise(t=360/2,100,t=360,100,0);\nz[4](t);\n\n" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 60 "for i from 1 to 4 do\nZ:=t->z[i](t);\nPresen
tValue[i]:=PV;\nod;" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "PresentV
alue[3]=PresentValue[2]+PresentValue[1]" "6#/&%-PresentValueG6#\"\"$,&
&F%6#\"\"#\"\"\"&F%6#F,F," }{TEXT -1 2 ", " }{XPPEDIT 18 0 "PresentVal
ue[4]=Presentvalue[2]+100" "6#/&%-PresentValueG6#\"\"%,&&%-Presentvalu
eG6#\"\"#\"\"\"\"$+\"F-" }{TEXT -1 40 ", proto\236e posledn\355 \372lo
\236ka se u\236 ne\372ro\350\355." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 102 "\nz[5]:=t->piecewise(type((t-1)/(30),integer),100,0);\nz[5](31)
;\npointplot(\{seq([n,z[5](n)],n=0..95)\});\n" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 64 "# z[6]:=t->piecewise(type((t)/(30),integer),100,0);\n
# z[6](31);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Z:=t->z[5]
(t);\nPresentValue[5]:=PV;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "V
e skute\350nosti zde s\350\355t\341me \370adu" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "#restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "
i:='i';Xi:=xi;Doba:='Doba':\nxi:='xi';\nxxx:=Sum(100*(1+xi)^(Doba(i)/3
60),i=1..12)=sum(100*(1+xi)^(Doba(i)/360),i=1..12);" }}{PARA 0 "" 0 "
" {TEXT -1 3 "kde" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "Doba:=i-
>360-(i-1)*30-1;" "6#>%%DobaGf*6#%\"iG7\"6$%)operatorG%&arrowG6\",(\"$
g$\"\"\"*&,&F'F/F/!\"\"F/\"#IF/F2F/F2F,F,F," }{TEXT -1 45 "je doba od \+
ulo\236en\355 i-t\351 \372lo\236ky do konce roku." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "xi:=Xi;\nxxx;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 42 "jedn\341 se o geometrickou \370adu s kvo\355cientem" }}}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "q:=(100*(1+xi)^(Doba(i+1)/360)
)/(100*(1+xi)^(Doba(i)/360));" "6#>%\"qG*(\"$+\"\"\"\"),&F'F'%#xiGF'*&
-%%DobaG6#,&%\"iGF'F'F'F'\"$g$!\"\"F'*&F&F'),&F'F'F*F'*&-F-6#F0F'F1F2F
'F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {XPPEDIT 19 1 "q:=simplify(q);\na1:=subs(i=1,100*(1+xi)^(Doba(i)/36
0));" "6#C$>%\"qG-%)simplifyG6#F%>%#a1G-%%subsG6$/%\"iG\"\"\"*&\"$+\"F
0),&F0F0%#xiGF0*&-%%DobaG6#F/F0\"$g$!\"\"F0" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 40 "a tu m\371\236eme se\350\355st podle obecn\351ho vzorce
" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "PresentValue[5]:=a1*(1-q^
12)/(1-q);" "6#>&%-PresentValueG6#\"\"&*(%#a1G\"\"\",&F*F**$%\"qG\"#7!
\"\"F*,&F*F*F-F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 181 "P\370\355p
ad\354, \236e spo\370\355me na konci m\354s\355ce se li\232\355 od p
\370edchoz\355ho pouze funkc\355 Doba a sou\350asnou hodnotou prvn\355
 \372lo\236ky. Kvocient  je stejn\354jako v p\370edchoz\355m p\370\355
pad\354 a po\350et \350len\371 \370ady tak\351:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 124 "Doba := i->360-(i-1)*30-30;\na1:=subs(i=1,100
*(1+xi)^(Doba(i)/360));\nPresentValue[6]:=sum(100*(1+xi)^(Doba(i)/360)
,i=1..12);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "\n" }{TEXT 
259 9 "a plat\355:\n" }{XPPEDIT 19 1 "(PresentValue[5]/PresentValue[6]
=(1+xi)^(29/360));\n" "6#/*&&%-PresentValueG6#\"\"&\"\"\"&F&6#\"\"'!\"
\"),&F)F)%#xiGF)*&\"#HF)\"$g$F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 16 "Spo\370en\355 (saving)" }}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 55 "Spo\370en\355 p\370i slo\236en\351m \372ro\350en\355 a ek
vidistantn\355ch \372lo\236k\341ch" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 13 "Obecn\354 vzorec" }}{PARA 0 "" 0 "" {TEXT -1 93 "P\370edpokl
\341dejme, \236e ukl\341d\341me v ekvidistantn\355ch okam\236ic\355ch,
 kter\351 jsou od sebe v \350ase vzd\341leny " }{XPPEDIT 18 0 "tau" "6
#%$tauG" }{TEXT -1 37 ", \236e prvn\355 \372lo\236ka se d\354je v okam
\236iku " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 19 ", dru
h\341 v okam\236iku " }{XPPEDIT 18 0 "epsilon+tau;" "6#,&%(epsilonG\"
\"\"%$tauGF%" }{TEXT -1 20 ", t\370et\355 v okam\236iku  " }{XPPEDIT 
18 0 "epsilon+2*tau;" "6#,&%(epsilonG\"\"\"*&\"\"#F%%$tauGF%F%" }
{TEXT -1 23 " a posledn\355 v okam\236iku " }{XPPEDIT 18 0 "epsilon+T;
" "6#,&%(epsilonG\"\"\"%\"TGF%" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 32 "#restart;\nCelaCast:=x->floor(x);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 16 "Ukl\341d\341me v \350ase " }{XPPEDIT 18 
0 "n*tau+epsilon,n = 0 .. T;" "6$,&*&%\"nG\"\"\"%$tauGF&F&%(epsilonGF&
/F%;\"\"!%\"TG" }{TEXT -1 24 "\nKolik je \372lo\236ek v \350ase " }
{XPPEDIT 18 0 "t < T+epsilon;" "6#2%\"tG,&%\"TG\"\"\"%(epsilonGF'" }
{TEXT -1 1 "?" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "pocetA:=CelaCast((t-
epsilon)/tau)+1:" "6#>%'pocetAG,&-%)CelaCastG6#*&,&%\"tG\"\"\"%(epsilo
nG!\"\"F,%$tauGF.F,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "a v \+
\350ase  " }{XPPEDIT 18 0 "t > T+epsilon;" "6#2,&%\"TG\"\"\"%(epsilonG
F&%\"tG" }{TEXT -1 2 "? " }{XPPEDIT 19 1 "pocetB:=T/tau+1;" "6#>%'poce
tBG,&*&%\"TG\"\"\"%$tauG!\"\"F(F(F(" }{TEXT -1 20 ", co\236 je cel\351
 \350\355slo." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 32 "Tedy obecn\354, ukl\341d\341me-li v \350ase " }
{XPPEDIT 18 0 "n*tau+epsilon,n = 0 .. T;" "6$,&*&%\"nG\"\"\"%$tauGF&F&
%(epsilonGF&/F%;\"\"!%\"TG" }{TEXT -1 13 "\nm\341me v \350ase " }
{XPPEDIT 18 0 "t" "6#%\"tG" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "pocet:=
min(pocetA,pocetB); #pocet:=K;" "6#>%&pocetG-%$minG6$%'pocetAG%'pocetB
G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "\372lo\236ek. Ve kter\351m o
kam\236iku se ukl\341d\341 " }{XPPEDIT 18 0 "i" "6#%\"iG" }{TEXT -1 
13 "-t\341 polo\236kla?" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "okamzik:=i
->i*tau+epsilon;" "6#>%(okamzikGf*6#%\"iG7\"6$%)operatorG%&arrowG6\",&
*&F'\"\"\"%$tauGF/F/%(epsilonGF/F,F,F," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 26 "Jak dlouho se do okam\236iku " }{XPPEDIT 18 0 "t" "6#%\"t
G" }{TEXT -1 7 " \372ro\350\355 " }{XPPEDIT 18 0 "i" "6#%\"iG" }{TEXT 
-1 17 "-t\341 \372lo\236ka je-li " }{XPPEDIT 19 1 "okamzik(i)<=t;" "6#
1-%(okamzikG6#%\"iG%\"tG" }{TEXT -1 1 "?" }}{PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "doba:=i->t-okamzik(i): doba(i);" "6#C$>%%dobaGf*6#%\"iG
7\"6$%)operatorG%&arrowG6\",&%\"tG\"\"\"-%(okamzikG6#F(!\"\"F-F-F--F%6
#F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Jak\341 je hodnota  " }
{XPPEDIT 18 0 "i" "6#%\"iG" }{TEXT -1 18 "-t\351 \372lo\236ky v \350as
e " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 18 " p\370\355 \372rokov
\351 m\355\370e " }{XPPEDIT 18 0 "xi" "6#%#xiG" }{TEXT -1 9 ",  je-li \+
" }{XPPEDIT 19 1 "okamzik(i)<=t;" "6#1-%(okamzikG6#%\"iG%\"tG" }{TEXT 
-1 57 " a pokud byla v okam\236iku ukl\341d\341n\355 jej\355 nomin\341
ln\355 hodnota " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 1 "?" }}{PARA 
0 "> " 0 "" {XPPEDIT 19 1 "hodnota:=i->z*(1+xi)^doba(i): hodnota(i);" 
"6#C$>%(hodnotaGf*6#%\"iG7\"6$%)operatorG%&arrowG6\"*&%\"zG\"\"\"),&F0
F0%#xiGF0-%%dobaG6#F(F0F-F-F--F%6#F(" }{TEXT -1 23 " je tedy za p\370e
dpokladu" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Chceme-li se\350\355
st sou\350asn\351 hodnoty v\232ech \372lo\236ek proveden\354ch v \350a
se p\370edch\341zej\355c\355m nobo rovn\351m  okam\236iku " }{XPPEDIT 
18 0 "t" "6#%\"tG" }{TEXT -1 16 " Dost\341v\341me \370adu " }{XPPEDIT 
18 0 "Sum(hodnota(i),i=0..pocet)" "6#-%$SumG6$-%(hodnotaG6#%\"iG/F);\"
\"!%&pocetG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Pod\355l i-t\351ho
 a i+1-n\355ho \350lenu  t\351to \370ady  nez\341vis\355 na i, a m\371
\236eme jej ozna\350it kvocient, nebo\234 \n" }}{PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "kvocient:=hodnota(i+1)/hodnota(i): kvocient:=simplify(k
vocient);\n\n" "6#C$>%)kvocientG*&-%(hodnotaG6#,&%\"iG\"\"\"F,F,F,-F(6
#F+!\"\">F%-%)simplifyG6#F%" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "A tedy jedn\341
 se o geometrickou \370adu, s t\355mto kvocientem." }}}{EXCHG {PARA 
16 "" 0 "" {TEXT -1 15 "po\350et \350len\371 je\n" }}{PARA 16 "> " 0 "
" {XPPEDIT 19 1 "PocetClenu=pocet;" "6#/%+PocetClenuG%&pocetG" }}}
{EXCHG {PARA 16 "" 0 "" {TEXT -1 14 "nult\354 \350len je\n" }}{PARA 
16 "> " 0 "" {XPPEDIT 19 1 "NultyClen:=hodnota(0);" "6#>%*NultyClenG-%
(hodnotaG6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "tedy sou\350e
t je:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "Vzorecek:=Sum(hodnota(i),i
=0..pocet-1)=simplify(sum(hodnota(i),i=0..pocet-1));" }}{PARA 0 "" 0 "
" {TEXT -1 34 "Pou\236ijeme jej k definic\355m funkc\355:\n" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 96 "Vzorecek:=simplify(op(2,Vzorecek)):\n#Vzo
recek:=subs(tau=1/k,Vzorecek);\n#limit(subs(k=K,\"),K=k);\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "SoucetGeometrickeRady:=Sum(a
[0]*q^i,i=0..n)=simplify(sum(a[0]*q^i,i=0..n));\n" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 75 "Vzorecek2:=subs(a[0]=NultyClen,q=kvocient,n=
pocet-1,SoucetGeometrickeRady):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "
Vzorecek2:=op(2,Vzorecek2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
179 "psi2:=(Tau,Epsilon,Xi,Z,TT,cas)->subs(tau=Tau,epsilon=Epsilon,xi=
Xi,z=Z,T=TT,t=cas,Vzorecek2);\nprint(`----------`);\nsimplify(psi2(tau
,epsilon,xi,z,T,t)-psi(tau,epsilon,xi,z,T,t));" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 67 "Prvn\355 funkce ud\341v\341 hodnotu naspo\370en\351 \+
\350\341stky, ze zadan\354ch paramatr\371:" }}{PARA 14 "" 0 "" {TEXT 
-1 67 "Vz\341jemn\341 zd\341kenost \372lo\236ek v \350ase (pro ka\236d
\351 dv\354 sousedn\355 je stejn\341)" }}{PARA 14 "" 0 "" {TEXT -1 34 
"\254as v n\354m\236 je ulo\236ena prvn\355 \372lo\236ka" }}{PARA 14 "
" 0 "" {TEXT -1 25 "\351rokov\341 m\355ra (konstantn\355)" }}{PARA 14 
"" 0 "" {TEXT -1 41 "Hodnota jednotliv\354ch \372l\371o\236ek (konstan
tn\355)" }}{PARA 14 "" 0 "" {TEXT -1 33 "Doba po kterou se \372lo\236k
y ukl\341daj\355" }}{PARA 14 "" 0 "" {TEXT -1 49 "okam\236ik ve kter
\351m n\341s zaj\355m\341 nomin\341ln\355 stav \372\350tu." }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 92 "psi:=(Tau,Epsilon,Xi,Z,TT,cas)->subs(tau=Ta
u,epsilon=Epsilon,xi=Xi,z=Z,T=TT,t=cas,Vzorecek);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 59 "Pokud n\341s zaj\355m\341 stav \372\350tu v dob
\354, kdy je\232t\354 spo\370\355me, je i " }{XPPEDIT 18 0 "trunc(t-ep
silon/tau) <= T+epsilon+1;" "6#1-%&truncG6#,&%\"tG\"\"\"*&%(epsilonGF)
%$tauG!\"\"F-,(%\"TGF)F+F)F)F)" }{TEXT -1 13 " a p[otom je " }
{XPPEDIT 18 0 "min(trunc((t-epsilon)/tau),T/tau)=trunc((t-epsilon)/tau
)" "6#/-%$minG6$-%&truncG6#*&,&%\"tG\"\"\"%(epsilonG!\"\"F-%$tauGF/*&%
\"TGF-F0F/-F(6#*&,&F,F-F.F/F-F0F/" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 326 "Vzorecek_:=subs(min(trunc((t-epsilon)/tau),T/tau)=
trunc((t-epsilon)/tau),Vzorecek):\nVzorecek_:=subs(min((T/tau,trunc((t
-epsilon+trunc(abs((t-epsilon)/tau))*tau+tau)/tau)-trunc(abs((t-epsilo
n)/tau))))=\ntrunc(T/tau,trunc((t-epsilon+trunc(abs((t-epsilon)/tau))*
tau+tau)/tau)-trunc(abs((t-epsilon)/tau)))\n,Vzorecek_):\nVzorecek_;\n
\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 9 "A funkce " }{XPPEDIT 18 0 "psi " "6#%$psiG" }{TEXT -1 55 "
 ud\341v\341 hodnotu naspo\370en\351 \350\341stky, ze zadan\354ch para
matr\371:" }}{PARA 14 "" 0 "" {TEXT -1 67 "Vz\341jemn\341 zd\341kenost
 \372lo\236ek v \350ase (pro ka\236d\351 dv\354 sousedn\355 je stejn
\341)" }}{PARA 14 "" 0 "" {TEXT -1 34 "\254as v n\354m\236 je ulo\236e
na prvn\355 \372lo\236ka" }}{PARA 14 "" 0 "" {TEXT -1 25 "\351rokov
\341 m\355ra (konstantn\355)" }}{PARA 14 "" 0 "" {TEXT -1 40 "Hodnota \+
jednotliv\354ch \372lo\236ek (konstantn\355)" }}{PARA 14 "" 0 "" 
{TEXT -1 49 "okam\236ik ve kter\351m n\341s zaj\355m\341 nomin\341ln
\355 stav \372\350tu." }}{PARA 0 "" 0 "" {TEXT -1 9 "Pokud je " }
{XPPEDIT 18 0 "trunc((t-epsilon)/tau) <= T/tau;" "6#1-%&truncG6#*&,&%
\"tG\"\"\"%(epsilonG!\"\"F*%$tauGF,*&%\"TGF*F-F," }{TEXT -1 4 ", je" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "psi_:=(Tau,Epsilon,Xi,Z,T)->subs(t
au=Tau,epsilon=Epsilon,xi=Xi,z=Z,N=TT,t=T,Vzorecek_);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 12 "Je-li nav\355c " }{XPPEDIT 18 0 "t-epsilo
n)/tau" "6#*&,&%\"tG\"\"\"%(epsilonG!\"\"F&%$tauGF(" }{TEXT -1 105 " c
el\351 \350\355slo, tj. pokud se pt\341me na stav \372\350tu jen v oka
m\236iku, kdy ukl\341d\341me, bude m\355t p\370edchoz\355 funkce tvar
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "VzorecekC:=subs(trunc((t-epsilo
n)/tau)=(t-epsilon)/tau,Vzorecek_);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 86 "Phi:=(Tau,Epsilon,Xi,Z,T)->subs(tau=Tau,epsilon=Epsil
on,xi=Xi,z=Z,N=TT,t=T,VzorecekC);" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT 
-1 7 "P\370\355klad" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Bu\324 " }
{XPPEDIT 18 0 "tau=1/3, epsilon=1/6, xi=3/4, z=5, T=2 a nebo T=1.4, t=
0..2.27" "6#%#%?G" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "p2:=plot(psi2(1/3,1/6,3/4,5,2-1/3,t
),t=0..2.75,discont=false):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
123 "p3:=plot(psi2(1/3,1/6,3/4,5,2-1/3,1.4)*(1+3/4)^(t-1.4),t=1.4..2.7
5#,title=`y\371statek se \372ro\350\355 d\341l. i kdy\236 u\236 #neukl
\341d\341me`\n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "l := [
[n/3+1/6,(n+1)*5] $n=0..5]:\np1:=plot(l, x=0..2.5, style=point,symbol=
circle#,title=`body zn\341zor\345uj\355 sou\350et nomin\341ln\355ch ho
dnot ulo\236en\354ch \350\341stek`\n):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "with(plots):\ndisplay(p1,p2,p3);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 54 "`plotsetup/devices`[jpeg]:=[jpeg,`plot.jpg`,
[],[],``]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "#plotsetup(jpeg, plotoutput=`sporeni.jpg`);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 49 "#plotsetup(plotoptions=`height=1200,width
=1200`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "#eval(plotsetup):" }}}
{EXCHG {PARA 12 "> " 1 "" {MPLTEXT 1 0 34 " evalf(psi2(1/3,1/6,1/3,5,2
,0.1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "ll := [trunc(10
0*(n/3+1/6))/100 $n=0..5];\nplot(evalf(subs(tau=1/3,epsilon=1/6,T=2-1/
3,t=t,pocet)),t=0..4,y=0..7,xtickmarks=ll, title=`graf z\341vislosti p
o\350tu ulo\236en\354ch \350\341stek na \350ase`);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 48 "evalf(subs(tau=1/3,epsilon=1/6,T=2,t=.1,poc
et));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "evalf(subs(T=2,tau
=1,t=10/4,epsilon=1/2,pocet));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "P\370\355klad" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 187 "P\370edpokl\341dejme, \236e ukl
\341d\341me 10. den ka\236d\351ho m\354s\355ce 250K\350. Za\350neme v \+
b\370eznu a sko\350\355me v kv\354tnu o t\370i roky pozd\354ji. Jak
\354 bude stav \372\350tu na konci onoho kv\354tna, je-li \372rokov
\341 m\355ra st\341le 2/50?" }}}{EXCHG {PARA 20 "" 0 "" {TEXT -1 0 "" 
}}{PARA 14 "" 0 "" {TEXT -1 45 "Od 1. b\370ezna do 31. kv\354tna ulo
\236\355me 15  \372lo\236ek" }}{PARA 14 "" 0 "" {TEXT -1 57 "vzd\341le
nost sousedn\355ch dvou v \350ase  je jeden m\354s\355c, to je " }
{XPPEDIT 18 0 "1/12" "6#*&\"\"\"F$\"#7!\"\"" }{TEXT -1 54 " roku (n
\354kdy je to 30 dn\355, n\354kdy 31 dn\355, n\354kdy 28 dn\355)" }}}
{SECT 1 {PARA 20 "" 0 "" {TEXT -1 6 "\374e\232en\355" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 25 "Prvn\355 (p\370ibli\236n\351) \370e\232en\355:
" }}{PARA 16 "" 0 "" {TEXT -1 43 "po\350\355t\341me v\232echny m\354s
\355ce t\370icetidenn\355 a tdy " }{XPPEDIT 18 0 "epsilon=1/3" "6#/%(e
psilonG*&\"\"\"F&\"\"$!\"\"" }}{PARA 16 "" 0 "" {TEXT -1 19 "m\354s
\355c\371 je 12 za rok" }}{PARA 16 "" 0 "" {TEXT -1 29 "za po\350\341t
ek vezmeme 1. b\370ezen." }}{PARA 16 "" 0 "" {TEXT -1 35 "za jednotku \+
\350asu bereme rok a tedy " }{XPPEDIT 18 0 "t=3+3/12" "6#/%\"tG,&\"\"$
\"\"\"*&F&F'\"#7!\"\"F'" }}{PARA 0 "" 0 "" {TEXT -1 44 "pak jde o mezi
lh\371tn\355 spo\370en\355 a podle vzorce:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "evalf(psi(1/12,(1/12)/3,2/50,250,3+3/12,3+3/12));" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 25 "Druh\351 (p\370ibli\236n\351) \370e\232en\355:" }}{PARA 16 "" 
0 "" {TEXT -1 35 "za po\350\341tek vezmeme 10. 3.   Pak je " }
{XPPEDIT 18 0 "epsilon=0" "6#/%(epsilonG\"\"!" }{TEXT -1 3 " a " }
{XPPEDIT 18 0 "t=12+1/12-(1/12)/3" "6#/%\"tG,(\"#7\"\"\"*&F'F'F&!\"\"F
'*(F'F'F&F)\"\"$F)F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 47 "evalf(psi(1/12,0,2/50,250,3+3/12,3+3/12-1/36));
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "T\370et\355 (p\370ibli\236n
\351) \370e\232en\355:" }}{PARA 0 "" 0 "" {TEXT -1 64 "Spo\350\355t
\341me efektivn\355 \372rok za m\354s\355c.\nza jednotku \350asu berem
e m\354s\355c" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "xi[mesic]:=(1+2/50
)^(1/12)-1;\nevalf(xi[mesic]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 46 "evalf(psi(1,1/3,xi[mesic],250,3*12+3,3*12+3));" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 149 "Uveden\351 \372vahy m\372\236eme zobecni
t a pak v\236dy vysta\350\355me se vzorcem . . .\ntedy bez epsilon, ze
jm\351na tedy nen\355 rozd\355l mezi p\370edlh\371tn\355m a polh\371tn
\355m spo\370en\355. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 221 "P\370es
n\351 \370e\232en\355 z\341le\236\355 na tom, je-li rok p\370estupn
\354, p\370edpokl\341dejme, \236e nen\355, nebo nen\355 a po jakou dob
u z\371st\341v\341j\355 \372roky konstantn\355, p\370edpokl\341dejme, \+
\236e po jeden den. Pak doby, po kterou jsou \372lo\236ky \372ro\350en
y jsou posupn\354: . . ." }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 59 "Spo\370en\355 p\370i jednoduch
\351m  \372ro\350en\355 a ekvidistantn\355ch \372lo\236k\341ch" }}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Obecn\354 vzorec" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 27 "#restart; \nCelaCast:=floor;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 93 "P\370edpokl\341dejme, \236e ukl\341d\341m
e v ekvidistantn\355ch okam\236ic\355ch, kter\351 jsou od sebe v \350a
se vzd\341leny " }{XPPEDIT 18 0 "tau" "6#%$tauG" }{TEXT -1 37 ", \236e
 prvn\355 \372lo\236ka se d\354je v okam\236iku " }{XPPEDIT 18 0 "epsi
lon;" "6#%(epsilonG" }{TEXT -1 19 " a posledn\355 v \350ase " }
{XPPEDIT 18 0 "`T+epsilon`;" "6#%*T+epsilonG" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "V \350as
e " }{XPPEDIT 18 0 "t < T+epsilon;" "6#2%\"tG,&%\"TG\"\"\"%(epsilonGF'
" }{TEXT -1 6 " m\341me " }{XPPEDIT 19 1 "pocetA:=CelaCast((t-epsilon)
/tau)+1:" "6#>%'pocetAG,&-%)CelaCastG6#*&,&%\"tG\"\"\"%(epsilonG!\"\"F
,%$tauGF.F,F,F," }{TEXT -1 11 " a v \350ase  " }{XPPEDIT 18 0 "t > T+e
psilon;" "6#2,&%\"TG\"\"\"%(epsilonGF&%\"tG" }{TEXT -1 6 " m\341me " }
{XPPEDIT 19 1 "pocetB:=T/tau+1:" "6#>%'pocetBG,&*&%\"TG\"\"\"%$tauG!\"
\"F(F(F(" }{TEXT -1 21 " \372lo\236ek, tedy v \350ase " }{XPPEDIT 18 
0 "t>0" "6#2\"\"!%\"tG" }{TEXT -1 7 " m\341me  " }{XPPEDIT 19 1 "pocet
:=min(pocetA,pocetB):" "6#>%&pocetG-%$minG6$%'pocetAG%'pocetBG" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "
\372lo\236ek, i-t\341 \372lo\236ka se ukl\341d\341 v okam\236iku " }
{XPPEDIT 19 1 "okamzik:=i->i*tau+epsilon;" "6#>%(okamzikGf*6#%\"iG7\"6
$%)operatorG%&arrowG6\",&*&F'\"\"\"%$tauGF/F/%(epsilonGF/F,F,F," }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "A do okam\236iku " }{XPPEDIT 18 0 
"t" "6#%\"tG" }{TEXT -1 18 " se \372ro\350\355 po dobu " }{XPPEDIT 19 
1 "doba:=i->t-okamzik(i): doba(i);" "6#C$>%%dobaGf*6#%\"iG7\"6$%)opera
torG%&arrowG6\",&%\"tG\"\"\"-%(okamzikG6#F(!\"\"F-F-F--F%6#F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Jej\355 hodnota v okam\236iku " }
{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 4 " je " }{XPPEDIT 19 1 "hodnota
:=i->z*(1+xi*doba(i)): hodnota(i);" "6#C$>%(hodnotaGf*6#%\"iG7\"6$%)op
eratorG%&arrowG6\"*&%\"zG\"\"\",&F0F0*&%#xiGF0-%%dobaG6#F(F0F0F0F-F-F-
-F%6#F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "M\341me se\350\355st s
ou\350asn\351 hodnoty (jej\355ch p\371vodn\355 hodnoty a \372roky) v
\232ech \372lo\236ek a\236 do okam\236iku " }{XPPEDIT 18 0 "t" "6#%\"t
G" }{TEXT -1 17 ". Rozd\355l hodnoty " }{XPPEDIT 18 0 "i" "6#%\"iG" }
{TEXT -1 6 "-t\351 a " }{XPPEDIT 18 0 "i+1" "6#,&%\"iG\"\"\"F%F%" }
{TEXT -1 13 "-n\355 \372lo\236ky je" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
45 "diference:=simplify(hodnota(i+1)-hodnota(i));" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 19 "A tedy nez\341vis\355 na " }{XPPEDIT 18 0 "t" "6#%
\"tG" }{TEXT -1 50 " a tedy \370ada je aritmetick\341. Nult\354 \350le
n m\341 v \350ase " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 9 " hodnotu
 " }{XPPEDIT 19 1 "hodnota(0);" "6#-%(hodnotaG6#\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 28 "a posledn\355 \372lo\236ka m\341 v \350as
e " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 13 " u\236  hodnotu " }
{XPPEDIT 19 1 "hodnota(pocet-1);" "6#-%(hodnotaG6#,&%&pocetG\"\"\"F(!
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "tedy sou\350et je:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Vzorecek:=Sum(hodnota(i),i=0..pocet
-1)\n=simplify(sum(hodnota(i),i=0..pocet-1));" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "Vzorecek:=op(2,Vzorecek);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Vzorece
k2:=simplify(1/2*(hodnota(0)+hodnota(pocet-1))*pocet);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "subs(min(T/tau,-trunc((-t+epsilon)/
tau))=K,\n2*simplify(xxx/sum(hodnota(i),i=0..pocet)));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "simplify(subs(min(trunc((t-epsilon)
/tau),T/tau)=K, Vzorecek-Vzorecek2));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 380 "W:=(Tau,Epsilon,Xi,Z,TT,cas)->\n[evalf(subs(tau=Tau,
epsilon=Epsilon,xi=Xi,z=Z,T=TT,t=cas,Vzorecek)),evalf(subs(tau=Tau,eps
ilon=Epsilon,xi=Xi,z=Z,T=TT,t=cas,Vzorecek2)),\nevalf(subs(tau=Tau,eps
ilon=Epsilon,xi=Xi,z=Z,T=TT,t=cas,hodnota(0))),\nevalf(subs(tau=Tau,ep
silon=Epsilon,xi=Xi,z=Z,T=TT,t=cas,hodnota(pocet-1))),\nevalf(subs(tau
=Tau,epsilon=Epsilon,xi=Xi,z=Z,T=TT,t=cas,pocet))\n];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "W(1,0,1,5,10,1);\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "psi:=(Tau,Epsilon,Xi,Z,TT,cas)->su
bs(tau=Tau,epsilon=Epsilon,xi=Xi,z=Z,T=TT,t=cas,op(2,Vzorecek));\npsi2
:=(Tau,Epsilon,Xi,Z,TT,cas)->subs(tau=Tau,epsilon=Epsilon,xi=Xi,z=Z,T=
TT,t=cas,Vzorecek2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "e
valf(psi(1,0,1,5,2,1));\nevalf(psi2(1,0,1,5,2,1));" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 284 "Na za\350\341tku roku \+
za\350nete spo\370it, ukl\341d\341te na za\350\341tku ka\236d\351ho m
\354s\355ce 114,7 oobol\371, a na konci ka\236d\351ho roku plat\355te \+
da\345 1/15 z \372rok\371. \351rokov\341 m\355ra je 1/50 p. m. a da
\345 z \372rok\371 1/15 se po\350\355t\341 ro\350n\354.  Kolik naspo
\370\355te za 11 let? (jak\354 bude stav \372\350tu na konci 11 roku, \+
kdy u\236 nic neukl\341d\341te)." }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 
6 "\374e\232en\355" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "restar
t;\nz:=114.7;xi:=1/50;delta:=1/15;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 6 "Za rok" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "xxx := 12.*z+(sum(z*(
1+xi)^t,t = 1 .. 12)-12*z)*(1-delta);" "6#>%$xxxG,&*&-%&FloatG6$\"#7\"
\"!\"\"\"%\"zGF,F,*&,&-%$sumG6$*&F-F,),&F,F,%#xiGF,%\"tGF,/F7;F,F*F,*&
F*F,F-F,!\"\"F,,&F,F,%&deltaGF;F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 9 "Za 11 let" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "\nsum(xxx*(1+xi*(1
-delta))^(12*t),t=0..10);" "6#-%$sumG6$*&%$xxxG\"\"\"),&F(F(*&%#xiGF(,
&F(F(%&deltaG!\"\"F(F(*&\"#7F(%\"tGF(F(/F2;\"\"!\"#5" }}}}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 45 "Vn\355t\370
n\355 m\355ra v\354nosu (internal rate of return)" }}{SECT 1 {PARA 0 "
" 0 "" {TEXT -1 285 "Dan si koupil auto na leasing, m\341 zaplatit  zb
yl\354ch130 000 K\350 z ceny 36 m\354s\355\350n\355mi spl\341tkami po \+
4592,5K\350.  Peripetie \236ivota jej donutili spl\341cet leasing z \+
\372spor. Na jak\354 \372rok by musel m\355t pen\355ze ulo\236eny, aby
 jej toto spl\341cen\355 vy\232lo levn\354ji, ne\236 kdyby hned zaplat
il celou \350\341stku 130 000?" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 6 "
\374e\232en\355" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "\374e\232\355me rovnici 36 s ne
zn\341mou " }{XPPEDIT 18 0 "xi" "6#%#xiG" }{TEXT -1 2 ". " }{XPPEDIT 
18 0 "xi" "6#%#xiG" }{TEXT -1 20 " je v tomto p\370\355pad\354 " }
{TEXT 260 8 "m\354s\355\350n\355 " }{TEXT -1 13 "\372rokov\341 m\355ra
." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "rce:=sum(4592.5*(1+xi)^i
,i=0..35)=130000*(1+xi)^36;" "6#>%$rceG/-%$sumG6$*&-%&FloatG6$\"&Df%!
\"\"\"\"\"),&F/F/%#xiGF/%\"iGF//F3;\"\"!\"#N*&\"'++8F/*$,&F/F/F2F/\"#O
F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Tuton rovnici mus\355me \+
\370e\232it numericky. Maple m\341 pvestav\354nou funkci " }{HYPERLNK 
17 "fsolve:" 2 "fsolve" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 20 "xxx:=fsolve(rce,xi);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 104 "Dostali jsme dv\354 \370e\232n\355, n\341s zaj\355m
\341 pouze to nez\341porn\351. M\354s\355\350n\355 \372rokovou m\355ru
 mus\355me p\370epo\350\355tat na ro\350n\355:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 72 "`Spl\341cen\355 se vyplat\355 p\370i ro\350n
\355 \372rokov\351 m\355\370e v\354t\232\355 ne\236`, (1+xxx[2])^12-1;
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Tuto rovnici m\371\236eme p
\370epsat pomoc\355 vzorce pro sou\350et geometrick\351 \370ady do tva
ru:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "rce2:=4592.5*(1-(1+x
i)^36)/(1-(1+xi))=rhs(rce);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "a \+
\370e\232it ji p\370\355mo. Rozkladem na sou\350in z\355sk\341me p\370
ehled o ko\370enech:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "rce
3:=factor((lhs(rce2)-rhs(rce2)))=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 44 "Jedin\354 kladn\354 ko\370en je m\354s\355\350n\355  \372rokov
\341 m\355ra" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(\{rce
3,xi>0\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "stejn\341 jakou jsm
e dostali p\370i p\370edchoz\355m postupu." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Tato \372ro
kov\341 m\355ra je vnit\370n\355 m\355rou v\354nosu  toku pen\354z (" 
}{TEXT 261 9 "Cash flow" }{TEXT -1 44 "), -130 000, 4592.5, 4592.5, . \+
. . , 4592.5." }}}}{PARA 0 "" 0 "" {TEXT -1 120 "Nyn\355 p\370edpokl
\341dejme, \236e by cht\354l Dan  rad\354ji zaplatit posledn\355ch 13 \+
spl\341tek najednou. Cht\354l by ov\232em zaplatit m\351n\354, ne\236 \+
" }{XPPEDIT 18 0 "13*4592.5" "6#*&\"#8\"\"\"-%&FloatG6$\"&Df%!\"\"F%" 
}{TEXT -1 129 " K\350, prot\236e pen\355ze zaplat\355 hned. P\370edsta
vuje si, \236e zaplatit hned na za\350\341tku 130000 je ekvivalentn
\355 tomu, zapaltit 36 spl\341tek po " }{XPPEDIT 18 0 "13*4592.5" "6#*
&\"#8\"\"\"-%&FloatG6$\"&Df%!\"\"F%" }{TEXT -1 409 " K\350. Co je v to
mto smzslu ekvivalentn\355 zapklacen\355 13 spl\341tek? Pokud by popt
\341vka po pen\354z\355ch leasingov\351 spole\350nosti p\370evy\232ova
la nab\355dku, tedy pokud by v\232echny pen\355ze, kter\351 dostane mo
hla ihned zase investovat za stejn\354ch podm\355nek (tedy p\370i stej
n\351m \372roku), m\371\236eme ot\341zku formulovat takto: p\370i jak \+
vysok\351 spl\341tce (jedinn\351 nam\355sto 13) by byl zisk leasingov
\351 spole\350nosti zachov\341n? (Manipula\350n\355 n\341klady zanedb
\341v\341me)" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 6 "\374e\232en\355" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart; " }{TEXT -1 110 "Zn
ovu vypo\350\355t\341me vnit\370n\355 m\355ru v\354nosu, tedy \372roko
vou m\355ru s jakou m\341 leasingov\341 spole\350nost zisk ze sv\354ch
 pen\354z:" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "rce:=sum(45925/
10*(1+xi)^(-i),i=1..36)=130000:" "6#>%$rceG/-%$sumG6$*(\"&Df%\"\"\"\"#
5!\"\"),&F+F+%#xiGF+,$%\"iGF-F+/F2;F+\"#O\"'++8" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 33 "Xi:=fsolve(rce,xi=0.013..0.014); " }{TEXT -1 
67 "s touto \372rokouvou m\355rou diskontujeme hodnotu 13 spl\341tek. \+
Dostaneme:" }}{PARA 0 "> " 0 "" {XPPEDIT 19 1 "xi:= Xi;\n`spravedliv
\240 cena je`,sum(45925/10*(1+xi)^(-i),i = 0 .. 13);" "6#C$>%#xiG%#XiG
6$%4spravedliv|[u~cena~jeG-%$sumG6$*(\"&Df%\"\"\"\"#5!\"\"),&F.F.F%F.,
$%\"iGF0F./F4;\"\"!\"#8" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Jednozna\350nost" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Uva\236ujme kapit\341lov\354 tok (
cash flow)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "CF:=[-1000,36
00,-4310,1716];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "PV:=sum(
CF[i]*(1+xi)^(-i),i=1..4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "solve(PV);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 41 "Spekula\350n\355 popt\341vka po p
en\354z\355ch (p\370\355klad)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 485 "
Jedne ze t\370\355 klasick\354ch motiv\371 popt\341vky po pen\354z\355
ch podle Johna Maynarda Keynese  je spekula\350n\355 motiv. Keynes se \+
zab\354val ot\341zkou, pro\350 sub\354kty dr\236\355 (popt\341vaj\355)
 v\354t\232\355 mno\236stv\355 pen\354z, ne\236  je objem pen\354z pop
t\341van\354  z transak\350n\355ho a opatrnostn\355ho motivu. Soudil, \+
\236e tyto pen\355ze jsou dr\236eny v d\371sledku nejistoty o pohybu b
udouc\355ch \372rokov\354ch sazeb a v d\371sledku vztahu mezi \372roko
vou sazbou a tr\236n\355mi cenami obligac\355. D\371sledkem r\371stu \+
\372rokov\354ch sazeb  jsou kapit\341lov\351 ztr\341ty z dr\236en\355 \+
obligac\355." }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 355 "Uva\236ujme  pe
rpetuitu, tj obligaci, kter\341 nen\355 nikdy dosp\354l\341 (zral\341)
 --- jej\355 jistina nebude nikdy splacena ale p\370in\341\232\355 sv
\351mu dr\236iteli pevnou ro\350n\355 kup\242novou  platbu. Nech\234 j
e investorem koupena tato obligace za  tr\236n\355 cenu 1000K\350, a n
ech\234 jsou v\354nosy z kup\242n\371 100K\350 ro\350n\354. Jakou hodn
otu m\341 tato obligace pokud b\354\236n\341 tr\236n\355 \372rokov\341
 sazba klesla na polovinu?" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 6 "\374
e\232en\355" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart; " }
{TEXT -1 213 "Dr\236en\355 obligace odpov\355d\341 v\354\350n\351mu d
\371chodu o pravideln\354ch v\354plat\341ch 100K\350 ro\350n\354 p\370
i ulo\236en\351m kapit\341lu 1000K\350. Ro\350n\355 \372rokov\341 m
\355ra, odpov\355daj\355c\355 ro\350n\355mu uroku 100K\350 z 1000K\350
 v okam\236iku, kdy investor obligaci koupil je" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 26 "xi[0]:=solve(1000*xi=100);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 36 "nynyn\355 je ale \372rokov\341 m\355ra polovi\350
n\355" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "xi[1]:=xi[0]/2;" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "a p\370i n\355 je hodnota obliga
ce" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Z:=solve(z*xi[1]=100)
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Tedy kapit\341lov\354 zisk j
e 1000K\350" }}}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 26 "Dva p\370\355k
lady --- opakov\341n\355" }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 41 "Jak
\341 je m\355ra inflace za t\370et\355 m\354s\355c, kdy\236" }}{PARA 
15 "" 0 "" {TEXT -1 79 "za rok je m\355ra inflace 0.05 a za ka\236d
\354 m\354s\355c byla stejn\341 jako ka\236d\354 jin\354 m\354s\355c.
" }}{PARA 15 "" 0 "" {TEXT -1 79 "za prvn\355 m\354s\355c byla 0.01, z
a druh\354 m\354s\355c byla 0.02 a za 1. \350tvrtlet\355 byla 0.06." }
}{PARA 15 "" 0 "" {TEXT -1 187 "Za rok je m\355ra inflace 0.1,\nza 4. \+
\350tvrtlet\355 je m\355ra inflace 0.02,\nza 3. \350tvrtlet\355 je m
\355ra inflace 0.03,\nza 2. \350tvrtlet\355 je m\355ra inflace 0.02,\n
za 1. m\354s\355c je 0.003,\nza druh\354 m\354s\355c je 0.004" }}}
{SECT 1 {PARA 0 "" 0 "" {TEXT -1 84 "Jak dlouho budete spl\341cet m
\354s\355\350n\355mi polh\372tn\355mi spl\341tkami dluh 10 000 spl\341
tkami 1000 " }}{PARA 15 "" 0 "" {TEXT -1 25 "p\370i \372rokov\351 m
\355\370e 0 p. m.," }}{PARA 15 "" 0 "" {TEXT -1 27 "p\370i \372rokov
\351 m\355\370e 0.05 p. m." }}{PARA 15 "" 0 "" {TEXT -1 26 "p\370i \+
\372rokov\351 m\355\370e 0.1 p. m." }}{PARA 0 "" 0 "" {TEXT -1 76 "Jak
\351 by musely b\354t spl\341tky, aby ve  druh\351m p\370\355pad\354 b
yla doba o m\354s\355c krat\232\355?" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "pom:=Sum(1000*(1+xi)^(t-i),i=1..t)=simplify(sum(1000*
(1+xi)^(t-i),i=1..t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "p
om:=op(2,pom)=10000*(1+xi)^t;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "solve(\{pom,xi=.1\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
28 "simplify(sum(a*q^i,i=1..n));" }}}{EXCHG {PARA 0 "> " 0 "" 
{XPPEDIT 19 1 "10*(1-1.1)^(20)/0.1;" "6#*(\"#5\"\"\"*$,&F%F%-%&FloatG6
$\"#6!\"\"F,\"#?F%-F)6$F%F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 
"" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 41 "Durace (duration) a konvexi
ta (convexity)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Bu\324 " }
{XPPEDIT 18 0 "PV(CF,xi)" "6#-%#PVG6$%#CFG%#xiG" }{TEXT -1 130 " sou
\350asn\341 hodnota z\341visl\341 na toku pen\354z a \372rokov\351 m
\355\370e. Zkoum\341me relativn\355 zm\354nu PV v z\341vislosti na zm
\354n\354 \372rokov\351 m\355ry, tedy v\354raz" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 68 "restart;\nwith(linalg):\nF:=(PV(CF,xi+Delta[xi
])-PV(CF,xi))/PV(CF,xi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 
"F:=unapply(F,Delta[xi]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "V
\354ray rozvineme do taylorovy \370ady" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "Taylor[f]:=taylor(F(Delta),Delta);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 5 "V\354raz" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "D
uration:=op(1,Taylor[f]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 191 "Je durace (v literatu\370e se tak
 naz\354vaj\355 i r\371zn\351 jin\351 modifikace tohoto v\354razu)\nDu
ration=discount mean term of the cash flow=elasticity of the net prese
nt value\nJe vyj\341d\370ena v jednotk\341ch \350asu." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 5 "V\354raz" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "Convexity:=2*op(3,Taylor[f]);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 100 "se naz\354v\341 konvexnost (konvexita?) konvexity o
f a cash flow. Je vyj\341d\370ena ve \350tverc\355ch jednotek \350asu.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Uva\236ujme p\354t projekt
\371:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "CF[1]:=(-1000,300
,500,200,100);\nCF[2]:=(-1000,47,47,47,1047);\nCF[3]:=(-851.18586,281.
0005,170.39716,300,200);\nCF[4]:=(-600,-500,-300,400,500,600);\nCF[5]:
=(1200,-400,-300,-200,-400);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "
projekty 1,2,3,4 jsou prvn\355ho druhu, projekt 5 je druh\351ho druhu.
 Projekt 2 p\370\350edstavuje kup\242nov\354 dluhopis. Sou\350asn\341 \+
hodnota z\341vis\355 na \372rokov\351 m\355\370e:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 79 "j:='j':i:='i':\nfor i from 1 to 5 do\nN[i]:=
vectdim(convert([CF[i]],vector));\nod;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "PV:=(i,xi)->sum(CF[i][j]*(1+xi)^(1-j),j=1..N[i]);" }}
}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 89 "for k from 1 to 5 do\nXi[k]:=fsolve(PV(k,xi)=0,xi=0.01..1);\nplo
t(PV(k,xi),xi=0..0.05);\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 28 "\nprint([Xi['i'] $'i'=1..5]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 46 "Tedy akceptujeme 1, 2., 3., 4., projekt je li " }{XPPEDIT 19 1 
"xiXi[3];" "6#2&%#XiG6#\"\"$%#xiG" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 37 "Porovn\341me nyn\355 pouze 1. a 3. projekt." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "plot(PV(3,xi)-PV(1,xi),xi=0
.01..0.03,title=`rozd\355l PV`);\nplot(PV(3,xi)-PV(1,xi),xi=0.0195..0.
0205,title=`rozd\355l PV`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 " j
e zanedbateln\354" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Rozdil
Duraci:=(diff(PV(3,xi),xi)/PV(3,xi)\n-\ndiff(PV(1,xi),xi)/PV(1,xi))\n;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(RozdilDuraci,xi=0.
01..0.03);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 106 "V bod\354 0.02 se p\370ibli\236n\354 rovnaj\355 s
ou\350asn\351 hodnoty i durace. Ale v okol\355 tohoto bodu je rozd\355
l konvexnost\355:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Rozdil
Konv:=(diff(PV(3,xi),xi,xi)/PV(3,xi)\n-\ndiff(PV(1,xi,xi),xi)/PV(1,xi)
)\n;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot(RozdilKonv,xi=
0..0.03);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "je v okol\355 bodu 0
.02 kladn\354, proto d\341me p\370ednost 3. projektu" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 38 "Depozita denominovan\341 v r\371zn\354ch \+
m\354n\341ch" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Probl\351m zm\354
ny kurs\371" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "Kurs  jen-min-piao
 (juan) je 36:37 k rupii je. O\350ek\341van\354 kurs v n\341sleduj\355
c\355m obdob\355 je 34:39. Jakou z t\354chto dvou m\354n je v\354hodn
\354j\232\355 podr\236et. vyj\341d\370ete tuto v\354hodu kvantitativn
\354." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "restart;\nE[0]:=36
/37;\nE[1]:=34/39;" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "E[t]" "6#
&%\"EG6#%\"tG" }{TEXT -1 82 " je hodnota rupie v jenech v \350ase t. H
odnota jenu v rupi\355ch je p\370evr\341cen\341 hodnota " }{XPPEDIT 
18 0 "E[t]" "6#&%\"EG6#%\"tG" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 
1 "if E[0]>E[1] then `kurs rupie kles\240` else `kurs rupie stoup\240`
 fi;" "6#@%2&%\"EG6#\"\"\"&F&6#\"\"!%1kurs~rupie~kles|[uG%2kurs~rupie~
stoup|[uG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "M\354\370me zisk v \+
t\351to m\355\370e: podr\236\355me-li rupie, nic nez\355sk\341me, ani \+
neztrat\355me. Podr\236\355me-li m\355sto rupi\355 jeny, bude n\341
\232 (relativn\355) zisk v rupi\355ch roven:" }}}{EXCHG {PARA 0 "> " 
0 "" {XPPEDIT 19 1 "x*E[0]/E[1]*`rupi\241`;" "6#**%\"xG\"\"\"&%\"EG6#
\"\"!F%&F'6#F%!\"\"%&rupi|\\uGF%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
21 "x rupi\355 investujeme, " }{XPPEDIT 18 0 "x*E[0]/E[1]" "6#*(%\"xG
\"\"\"&%\"EG6#\"\"!F%&F'6#F%!\"\"" }{TEXT -1 36 " rupi\355 z\355sk\341
me zp\341tky. M\355ra zisku je" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 
19 1 "E[0]/E[1]-1;" "6#,&*&&%\"EG6#\"\"!\"\"\"&F&6#F)!\"\"F)F)F," }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 244 "Pokud bychom ct\354li m\354\370it
 zisk v jenech, zisk bychom p\370e\232k\341lovali tak, \236e dr\236en
\355 jenu by pro n\341s znamenalo zisk 0 znamenalo by dr\236en\355 rup
i\355 ztr\341tu, kterou vy\350\355sl\355me takto: pokudinvestujeme do \+
rupii x jen\371, budeme m\355t v n\341sleduj\355c\355m obdob\355 u\236
 jen" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "x*E[0]^(-1)*E[1]*`jen
\205`;" "6#**%\"xG\"\"\")&%\"EG6#\"\"!,$F%!\"\"F%&F(6#F%F%%%jen|`sGF%
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "a m\355ra zisku by byla" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "E[0]^(-1)*E[1]-1;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 241 "Jak to, \236e n\341m nevy\232lo \+
\350\355slo s tout\351\236 absolutn\355 hodnotou, jako v p\370edchoz
\355m p\370\355pad\354? Proto\236e jsme zvolili jinou (v\354t\232\355)
 jednotku (p\370ij\355t o jen je v\354t\232\355 ztr\341tam ne\236 p
\370ij\355t o rupii). Vhodn\351 by tedy bylo pro podobn\351 v\354po
\350ty zvolit pevnou m\354nu, " }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 
116 " Kurz m\354n (1) jen-min-piao (2) ngultrum (bh\371t\341n) a (3) k
yata (Barma) je k D\341nsk\341 korun\354 v \350ase , t=1,2,3,4,5  rove
n " }{XPPEDIT 18 0 "kappa(i,t)=9/i+abs(sin(i*t*Pi/6))" "6#/-%&kappaG6$
%\"iG%\"tG,&*&\"\"*\"\"\"F'!\"\"F,-%$absG6#-%$sinG6#**F'F,F(F,%#PiGF,
\"\"'F-F," }{TEXT -1 88 " (tj. sou\350et absolutni hodnoty sinu \232es
tiny sou\350inu po\370adov\351ho \350\355sla m\354ny  \350asu a \350
\355sla " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 270 " s \350\355slem
 dev\354t d\354leno po\370adov\351 \350\355slo m\354ny  je cena v d
\341nsk\354ch korun\341ch).   M\341te k dispozici 100 d\341nsk\354ch k
orun a p\370esnou znalost v\232ech kurz\371 p\370edem. Sm\354\344ovat \+
m\371\236ete m\354ny v aktualn\355ch kurzech v \350asech i=1,2,3,4,5 b
ez poplatk\371. Jakou nejv\354t\232\355 \350\341stku m\371\236ete vyob
chodovat ?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "restart;\nkapp
a:=(i,t)->9/i+abs(sin(i*t*Pi/6));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
78 "plot(\{kappa(1,t),kappa(2,t),kappa(3,t)\},t=0..6, color=[navy,blue
,aquamarine]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "kappa(1,1
);" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 6 "\370e\232en\355" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "PocetTitulu:=3;\nKapital[1]:=100;
\nVynosnost:=(i,t)->kappa(i,t+1)/kappa(i,t);\nHodnota:=t->Sum(kappa(i)
*Pocet[i],i=1..PocetTitulu);\n\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "Max:=t->max(Vynosnost(i,t) $i=1..PocetTitulu);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 250 "Vyber:=proc(t)\nglobal Cisl
oTitulu;\ni:='i';\n#print(evalf(Vynosnost(i,t)) $i=1..PocetTitulu);\nf
or i from 1 to PocetTitulu do\nif Max(t)=Vynosnost(i,t)\nthen CisloTit
ulu:=i;\n#print(`nejv\354t\232\355 v\354nosnost m\341 titul`,i,`a to`,
evalf(Vynosnost(i,t)));\nfi \nod\nend;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 644 "for t from  1 to 6 do \ni:='i';\nVyber(t); MAX:=eval
f(Max(t));#print(MAX);\nif MAX>1 then\n  for i from 1 to PocetTitulu d
o\n   Pocet[i]:=0;\n  od:\nPocet[CisloTitulu]:=Kapital[t]/kappa(CisloT
itulu,t);\nKapital[t+1]:=Pocet[CisloTitulu]*kappa(CisloTitulu,t+1);\np
rint(`v \350ase `,t, ` kupuji `, evalf(Pocet[CisloTitulu]),\n ` jednot
ek m\354ny \350\355slo `,   CisloTitulu, `v \350ase`, t+1, `budu m\355
t kapit\341l velikosti`, evalf(Kapital[t+1]),`p\371vodn\355 m\354ny`);
\nelse\nKapital[t+1]:=Kapital[t];\nprint(MAX,evalf(Vynosnost(i,t)) $i=
1..PocetTitulu,`V \350ase`,t,`nech\341v\341m pen\355ze v p\371vodn\355
 m\354n\354, v \350ase`, t+1, `budu m\355t kapit\341l velikosti`, eval
f(Kapital[t+1]),`p\371vodn\355 m\354ny`)\nfi\nod:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 99 "Podm\355nka nekryt\351 \+
\372rokov\351 parity (uncovered interest parity condition), mezin\341r
odn\355 \372rokov\341 arbitr\341\236" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 99 "Uva\236ujme dv\354 m\354ny, t\370eba CZK a USD, jejich kurzy v \+
\350ase 0 a (skuite\350n\354) a v \350ase 1 (p\370edpokl\341dan\354) \+
" }{XPPEDIT 18 0 "E(0)" "6#-%\"EG6#\"\"!" }{TEXT -1 3 " a " }{XPPEDIT 
18 0 "E(1)" "6#-%\"EG6#\"\"\"" }{TEXT -1 64 " (tj. E(i) je cena dolaru
 v korun\341ch v \350ase i)dv\354 \372rokov\351 sazby " }{XPPEDIT 18 
0 "xi[CZK]" "6#&%#xiG6#%$CZKG" }{TEXT -1 3 " a " }{XPPEDIT 18 0 "xi[US
D]" "6#&%#xiG6#%$USDG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 102 "Pokud investujeme v \350ase 0 koruny (o objemu x CZK) do
 dolarov\354ch depozit, bude n\341\232 zisk v \350ase 1 roven" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {XPPEDIT 19 1 "Vynos[US]:=x*E(0)^(-1)*(1+xi[USD])*E(1)-x;" "6
#>&%&VynosG6#%#USG,&**%\"xG\"\"\")-%\"EG6#\"\"!,$F+!\"\"F+,&F+F+&%#xiG
6#%$USDGF+F+-F.6#F+F+F+F*F2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "P\370i jak\351 \372rokov\351 m\355
\370e " }{XPPEDIT 18 0 "xi[CZK]" "6#&%#xiG6#%$CZKG" }{TEXT -1 57 " by \+
tento v\354nos byl stejn\354, jako v\354nos z \254esk\354ch depozit?" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Vynos[CZ]:=x*xi[CZK];" }}
}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "rce:=Vynos[US]=Vynos[CZ];" "6
#>%$rceG/&%&VynosG6#%#USG&F'6#%#CZG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "xxx:=solve(rce,xi[CZK]);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Uprav\355me. \+
Ozna\350me o\350ek\341vanou m\355ru depreciace koruny proti dolaru Q:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Q:=(E(1)-E(0))/E(0);\nx
xx2:=sort(expand(xxx/Q),xi[USD]);\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 122 "PodmRovnovahy:=xi[CZK]=sort(\n(simplify((op(1,xxx2)-
xi[USD])*Q)+xi[USD]*Q)+\nsimplify(op(2,xxx2)*Q+op(3,xxx2)*Q)\n,xi[USD]
);\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 5 "V\354raz" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "op(2,op(2,PodmRov
novahy));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "je velmi mal\354 (po
kud je " }{XPPEDIT 18 0 "xi[USD]" "6#&%#xiG6#%$USDG" }{TEXT -1 167 " i
 m\355ra o\350ek\341van\351 depreciace mal\341, nebo\234 jej jejich so
u\350inem). M\371\236eeme shrnout: v rovnov\341\236n\351m stavu je roz
d\355l \372rokov\354ch m\354r roven  z\372ro\350en\351  o\350ek\341van
e m\355\370e depreciace:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
115 "lhs(PodmRovnovahy)=op(1,rhs(PodmRovnovahy))+\nfactor(\nsimplify(o
p(2,rhs(PodmRovnovahy))+op(3,rhs(PodmRovnovahy)))\n);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 33 "Citlivost sporeni na urokovu miru" }}{SECT 1 {PARA 4 "" 
0 "" {TEXT -1 13 "Present value" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 39 "Z:=simplify(sum(z*(1+xi)^(-t),t=0..T));" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 208 "F:=xi-> z*(-(1/(1+xi))^T+1+xi)/xi;\nxxx:=norm
al((F(xi+delta)-F(xi))/F(xi));\nyyy:=simplify(taylor(xxx,delta=0,2));
\nzzz:=expand(simplify(numer(op(1,yyy))/(1+xi)^T)/\nsimplify(denom(op(
1,yyy))/(1+xi)^T));\ni:='i';\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 27 "G:=unapply(xxx,delta,xi,T);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "G(0.01,0.1,144);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 136 "plot(G(delta,0.03,144),delta=0..0.02);\nplot(G(0.01,
xi,144),xi=0..0.5);\nplot(G(0.1*xi,xi,144),xi=0..0.75);\nplot(G(0.01,0
.05,T),T=0..150);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 12 "Future value" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Cit
livost nasporene castky" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Z:=simpl
ify(sum(z*(1+xi)^t,t=0..T));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 29 "F:=xi->z*((1+xi)^(T+1)-1)/xi;" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 41 "xxx:=normal((F(xi+delta)-F(xi))/F(xi));\n\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "#readlib(coeftayl):\nDiff(F,
delta)=simplify(diff(xxx,delta));\nyyy:=simplify(taylor(xxx,delta=0,2)
);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "expand(op(1,yyy));
\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "zzz:=expand(simplify(
numer(op(1,yyy))/(1+xi)^T)/\nsimplify(denom(op(1,yyy))/(1+xi)^T));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "i:='i';\nxxxx:=sum(simplif
y(op(i,zzz)),i=1..nops(zzz));\nsubs(q=1+xi,\nq^T*simplify(subs(1+xi=q,
\nop(1,xxxx)+op(2,xxxx))/q^T)\n)+sum(op(i,xxxx),i=3..nops(xxxx));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "G:=unapply(xxx,xi,T);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "[(G(j/100,i)) $i=1..3] $j=1.
.4;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "subs(delta=0.01,(G
(0.5,144)));\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "plot(G(0.
05,144),delta=0..0.03);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
134 "plot(subs(delta=0.001,G(xi,144)),xi=0..0.01);\nplot(subs(delta=0.
001,G(xi,144)),xi=0..1.0);\nplot(subs(delta=0.01,G(xi,144)),xi=0..1.0)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot(subs(delta=0.01,G
(0.03,T)),T=1..210);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "nor
mal((taylor(xxx,delta,2)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{MARK "10" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }