5 Txoj Kev Siv Txoj Cai ntawm 72

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5 Txoj Kev Siv Txoj Cai ntawm 72
5 Txoj Kev Siv Txoj Cai ntawm 72

Video: 5 Txoj Kev Siv Txoj Cai ntawm 72

Video: 5 Txoj Kev Siv Txoj Cai ntawm 72
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Cov Txoj Cai 72 yog cov cuab yeej siv tau siv nyiaj txiag los kwv yees cov xyoo nws yuav siv ob npaug ntawm cov nyiaj los ntawm kev them paj, muab tus paj laum tshwj xeeb. Txoj cai kuj tseem tuaj yeem kwv yees tus paj laum txhua xyoo uas yuav tsum tau ua ob npaug ntawm cov nyiaj hauv cov xyoo tshwj xeeb. Txoj cai hais tias tus paj laum sib npaug los ntawm lub sijhawm yuav tsum tau ua ob npaug ntawm cov nyiaj yog kwv yees li sib npaug rau 72.

Txoj Cai ntawm 72 yog siv rau qhov xwm txheej ntawm kev nthuav dav, (raws li kev nyiam sib xyaw) lossis nthuav tawm "kev lwj," zoo li qhov poob ntawm kev yuav lub zog los ntawm kev nce nyiaj txiag.

Cov kauj ruam

Txoj Kev 1 ntawm 4: Kwv yees Lub Sijhawm "Doubling"

Siv Txoj Cai ntawm 72 Kauj Ruam 1
Siv Txoj Cai ntawm 72 Kauj Ruam 1

Kauj Ruam 1. Cia R x T = 72

R yog tus nqi ntawm kev loj hlob (tus paj laum txhua xyoo), thiab T yog lub sijhawm (xyoo) nws yuav siv nyiaj ntau npaum li ob npaug.

Siv Txoj Cai ntawm 72 Kauj Ruam 2
Siv Txoj Cai ntawm 72 Kauj Ruam 2

Kauj Ruam 2. Ntxig tus nqi rau R

Piv txwv li, nws siv sijhawm ntev npaum li cas los hloov $ 100 rau $ 200 ntawm qhov paj laum txhua xyoo ntawm 5%? Tso R = 5, peb tau txais 5 x T = 72.

Siv Txoj Cai ntawm 72 Kauj Ruam 3
Siv Txoj Cai ntawm 72 Kauj Ruam 3

Kauj Ruam 3. Kev daws qhov tsis paub sib txawv

Hauv qhov piv txwv no, faib ob sab ntawm qhov sib npaug saum toj no los ntawm R (uas yog, 5) kom tau T = 72 ÷ 5 = 14.4. Yog li nws yuav siv 14.4 xyoo rau $ 100 kom ob npaug ntawm tus paj laum ntawm 5% hauv ib xyoos. (Thawj qhov nyiaj tsis muaj teeb meem. Nws yuav siv sijhawm tib lub sijhawm los ua ob npaug tsis hais tus lej pib yog dab tsi.)

Siv Txoj Cai ntawm 72 Kauj Ruam 4
Siv Txoj Cai ntawm 72 Kauj Ruam 4

Kauj Ruam 4. Kawm cov piv txwv ntxiv no:

  • Nws siv sijhawm ntev npaum li cas los ua ob npaug cov nyiaj ntawm tus nqi 10% toj ib xyoo? 10 x T = 72. Faib ob sab ntawm qhov sib npaug ua 10, kom T = 7.2 xyoo.
  • Nws siv sijhawm ntev npaum li cas los hloov $ 100 rau $ 1600 ntawm tus nqi ntawm 7.2% toj xyoo? Paub tias 100 yuav tsum ob npaug plaub zaug kom txog 1600 ($ 100 → $ 200, $ 200 → $ 400, $ 400 → $ 800, $ 800 → $ 1600). Rau txhua qhov ob npaug, 7.2 x T = 72, yog li T = 10. Yog li, raws li txhua ob npaug siv sijhawm kaum xyoo, tag nrho lub sijhawm xav tau (hloov $ 100 rau $ 1, 600) yog 40 xyoo.

Txoj Kev 2 ntawm 4: Kwv yees Kev Loj Hlob

Siv Txoj Cai ntawm 72 Kauj Ruam 5
Siv Txoj Cai ntawm 72 Kauj Ruam 5

Kauj Ruam 1. Cia R x T = 72

R yog tus nqi ntawm kev loj hlob (tus paj laum), thiab T yog lub sijhawm (xyoo) nws yuav siv ob npaug ntawm cov nyiaj.

Siv Txoj Cai ntawm 72 Kauj Ruam 6
Siv Txoj Cai ntawm 72 Kauj Ruam 6

Kauj Ruam 2. Sau tus nqi ntawm T

Piv txwv, cia peb hais tias koj xav tau koj cov nyiaj ob npaug hauv kaum xyoo. Koj xav tau tus paj laum dab tsi thiaj li ua tau? Sau 10 rau T hauv kab zauv. Luas 10 x r = 72

Siv Txoj Cai ntawm 72 Kauj Ruam 7
Siv Txoj Cai ntawm 72 Kauj Ruam 7

Kauj Ruam 3. Kev daws rau R

Faib ob tog los ntawm 10 kom tau R = 72 ÷ 10 = 7.2. Yog li koj yuav xav tau tus paj laum txhua xyoo ntawm 7.2% txhawm rau txhawm rau txhawm rau koj cov nyiaj hauv kaum xyoo.

Txoj Kev 3 ntawm 4: Kwv yees kwv yees qhov "Decay" (Loss)

Siv Txoj Cai ntawm 72 Kauj Ruam 8
Siv Txoj Cai ntawm 72 Kauj Ruam 8

Kauj Ruam 1. Kwv yees lub sijhawm nws yuav siv kom poob ib nrab ntawm koj cov nyiaj (lossis nws lub zog yuav khoom tom qab kev nce nqi). Cia T = 72 ÷ R

Nov yog qhov sib npaug zoo ib yam li saum toj no, tsuas yog kho dua me ntsis. Tam sim no sau tus nqi rau R. Piv txwv li:

  • Ntev npaum li cas nws yuav siv $ 100 los txiav txim siab lub zog yuav khoom ntawm $ 50, muab tus nqi nce nqi ntawm 5% hauv ib xyoos?

    Cia 5 x T = 72, yog li T = 72 ÷ 5 = 14.4. Qhov ntawd yog pes tsawg xyoo nws yuav siv nyiaj kom poob ib nrab ntawm nws lub zog yuav hauv lub sijhawm 5% kev nce nqi. (Yog tias qhov kev nce nqi nce los ntawm ib xyoos rau ib xyoos, koj yuav tsum tau siv qhov nruab nrab kev nce nqi uas muaj nyob rau lub sijhawm puv sijhawm.)

Siv Txoj Cai ntawm 72 Kauj Ruam 9
Siv Txoj Cai ntawm 72 Kauj Ruam 9

Kauj Ruam 2. Kwv yees tus nqi ntawm kev lwj (R) dhau lub sijhawm muab:

R = 72 ÷ T. Sau tus nqi rau T, thiab daws rau R. Piv txwv li:

  • Yog tias kev yuav khoom ntawm $ 100 dhau los yog $ 50 hauv kaum xyoo, tus nqi nce nqi hauv lub sijhawm ntawd yog dab tsi?

    R x 10 = 72, qhov twg T = 10. Tom qab ntawd R = 72 ÷ 10 = 7.2%

Siv Txoj Cai ntawm 72 Kauj Ruam 10
Siv Txoj Cai ntawm 72 Kauj Ruam 10

Kauj Ruam 3. Tsis quav ntsej tej ntaub ntawv uas txawv txawv

Yog tias koj tuaj yeem tshawb pom qhov dav dav, tsis txhob txhawj xeeb txog cov lej ib ntus uas dhau mus deb heev. Tso lawv los ntawm kev txiav txim siab.

Daim Npav Sijhawm Doubling

Image
Image

Qauv Doubling Time Chart

Txoj Kev 4 ntawm 4: Derivation

Kauj Ruam 1. Nkag siab tias kev rho tawm ua haujlwm li cas rau kev sib xyaw ua ntu zus

  • Rau kev sib xyaw ua ntu zus, FV = PV (1 + r)^T, qhov twg FV = tus nqi yav tom ntej, PV = tus nqi tam sim no, r = kev loj hlob, T = sijhawm.
  • Yog tias nyiaj tau ob npaug, FV = 2*PV, yog li 2PV = PV (1 + r)^T, lossis 2 = (1 + r)^T, piv txwv tias tus nqi tam sim no tsis yog xoom.
  • Daws rau T los ntawm kev siv cov cav ntoo ntawm ob sab, thiab rov kho dua, kom tau T = ln (2) / ln (1 + r).
  • Taylor series rau ln (1 + r) ncig 0 yog r - r2/2 + r3/ 3 -… Rau qhov muaj nqis tsawg ntawm r, kev koom tes los ntawm cov lus muaj zog ntau dua yog qhov me me, thiab cov lus qhia kwv yees r, yog li t = ln (2) / r.
  • Nco tseg tias ln (2) ~ 0.693, yog li T ~ 0.693 / r (lossis T = 69.3 / R, qhia qhov paj laum raws li feem pua R los ntawm 0-100%), uas yog txoj cai ntawm 69.3. Lwm tus lej xws li 69, 70, thiab 72 tau siv rau kev suav yooj yim dua.

Kauj Ruam 2. Nkag siab tias kev rho tawm ua haujlwm li cas rau kev sib xyaw ua ntu zus

Rau kev sib xyaw ua ntu zus nrog ntau qhov sib xyaw hauv ib xyoos, tus nqi yav tom ntej yog muab los ntawm FV = PV (1 + r/n)^nT, qhov twg FV = tus nqi yav tom ntej, PV = tus nqi tam sim no, r = kev loj hlob, T = sijhawm, thiab n = tus naj npawb ntawm cov sijhawm sib xyaw hauv ib xyoos. Rau kev sib txuas tas mus li, n ua rau infinity. Siv lub ntsiab lus ntawm e = lim (1 + 1/n)^n raws li n mus rau qhov tsis muaj qhov kawg, qhov kev qhia ua FV = PV e^(rT).

  • Yog tias nyiaj tau ob npaug, FV = 2*PV, yog li 2PV = PV e^(rT), lossis 2 = e^(rT), piv txwv tias tus nqi tam sim no tsis yog xoom.
  • Kev daws rau T los ntawm kev siv cov cav ntoo ntawm ob sab, thiab rov kho dua, kom tau T = ln (2)/r = 69.3/R (qhov twg R = 100r los qhia txog kev loj hlob raws li feem pua). Nov yog txoj cai ntawm 69.3.
  • Rau kev sib txuas tas mus li, 69.3 (lossis kwv yees li 69) muab cov txiaj ntsig tau raug ntau dua, txij li ln (2) yog kwv yees li 69.3%, thiab R * T = ln (2), qhov twg R = kev loj hlob (lossis tawg), T = ob npaug (lossis ib nrab sijhawm), thiab ln (2) yog lub cav log ntawm 2. 70 kuj tseem tuaj yeem siv los ua qhov kwv yees txuas ntxiv lossis niaj hnub (uas ze rau txuas ntxiv mus) sib xyaw, kom yooj yim ntawm kev suav. Cov kev hloov pauv no hu ua txoj cai ntawm 69.3, txoj cai 69, los yog txoj cai 70.

    Ib qho kev hloov pauv zoo ib yam rau txoj cai ntawm 69.3 yog siv rau tus nqi siab nrog kev sib xyaw txhua hnub: T = (69.3 + R / 3) / R.

  • Cov Eckart-McHale txoj cai xaj thib ob, lossis E-M txoj cai, muab ntau qhov kev hloov kho rau Txoj Cai ntawm 69.3 lossis 70 (tab sis tsis yog 72), kom raug qhov tseeb rau cov paj laum ntau dua. Txhawm rau xam E-M kwv yees, muab Txoj Cai ntawm 69.3 (lossis 70) los ntawm 200/(200-R), piv txwv li, T = (69.3/R) * (200/(200-R)). Piv txwv, yog tias tus paj laum yog 18%, Txoj Cai ntawm 69.3 hais tias t = 3.85 xyoo. Txoj Cai E-M sib tw qhov no los ntawm 200/(200-18), muab sijhawm ob npaug ntawm 4.23 xyoo, uas zoo dua kwv yees lub sijhawm tiag tiag ob npaug 4.19 xyoo ntawm tus nqi no.

    Qhov kev txiav txim thib peb Padé kwv yees kwv yees kwv yees zoo dua, siv qhov ntsuas raug (600 + 4R) / (600 + R), piv txwv li, T = (69.3 / R) * ((600 + 4R) / (600 + R)). Yog tias tus paj laum yog 18%, qhov thib peb xaj Padé kwv yees muab T = 4.19 xyoo

  • Txhawm rau kwv yees sijhawm ob npaug rau tus nqi siab dua, kho 72 los ntawm kev ntxiv 1 rau txhua 3 feem pua ntau dua 8%. Ntawd yog, T = [72 + (R - 8%) / 3] / R. Piv txwv, yog tias tus paj laum yog 32%, lub sijhawm nws yuav siv ob npaug ntawm cov nyiaj tau muab yog T = [72 + (32 - 8) / 3] / 32 = 2.5 xyoos. Nco tseg tias 80 tau siv ntawm no tsis yog 72, uas yuav tau muab 2.25 xyoo rau lub sijhawm ob npaug.
  • Nov yog lub rooj muab tus naj npawb ntawm xyoo nws yuav siv ob npaug ntawm cov nyiaj tau muab los ntawm ntau cov paj laum, thiab sib piv qhov kwv yees nrog ntau txoj cai:

Xyoo

ua 72

ua 70

69.3

txoj cai

Tus nqi Tiag tiag Txoj Cai Txoj Cai Txoj cai ntawm E -M
0.25% 277.605 288.000 280.000 277.200 277.547
0.5% 138.976 144.000 140.000 138.600 138.947
1% 69.661 72.000 70.000 69.300 69.648
2% 35.003 36.000 35.000 34.650 35.000
3% 23.450 24.000 23.333 23.100 23.452
4% 17.673 18.000 17.500 17.325 17.679
5% 14.207 14.400 14.000 13.860 14.215
6% 11.896 12.000 11.667 11.550 11.907
7% 10.245 10.286 10.000 9.900 10.259
8% 9.006 9.000 8.750 8.663 9.023
9% 8.043 8.000 7.778 7.700 8.062
10% 7.273 7.200 7.000 6.930 7.295
11% 6.642 6.545 6.364 6.300 6.667
12% 6.116 6.000 5.833 5.775 6.144
15% 4.959 4.800 4.667 4.620 4.995
18% 4.188 4.000 3.889 3.850 4.231
20% 3.802 3.600 3.500 3.465 3.850
25% 3.106 2.880 2.800 2.772 3.168
30% 2.642 2.400 2.333 2.310 2.718
40% 2.060 1.800 1.750 1.733 2.166
50% 1.710 1.440 1.400 1.386 1.848
60% 1.475 1.200 1.167 1.155 1.650
70% 1.306 1.029 1.000 0.990 1.523

Yees duab - Los ntawm kev siv qhov kev pabcuam no, qee cov ntaub ntawv yuav raug muab qhia rau YouTube

Lub tswv yim

  • Cia Txoj Cai 72 ua haujlwm rau koj los ntawm pib txuag tam sim no.

    Ntawm qhov kev loj hlob ntawm 8% hauv ib xyoos (kwv yees kwv yees tus nqi rov qab los hauv khw muag khoom), koj yuav nce koj cov nyiaj hauv cuaj xyoo (72 ÷ 8 = 9), plaub npaug koj cov nyiaj hauv 18 xyoo, thiab muaj 16 zaug koj cov nyiaj hauv 36 xyoo.

  • Koj tuaj yeem siv Felix Corollary rau Txoj Cai ntawm 72 txhawm rau xam "qhov txiaj ntsig yav tom ntej" ntawm cov nyiaj ib xyoos (uas yog, qhov nyiaj them txhua xyoo lub ntsej muag tus nqi yuav nyob ntawm lub sijhawm yav tom ntej). Koj tuaj yeem nyeem txog kev txheeb xyuas ntawm ntau yam nyiaj txiag thiab nqis peev cov vev xaib.
  • Tus nqi ntawm 72 tau xaiv los ua tus lej yooj yim hauv qhov sib npaug saum toj no. 72 tau faib tau yooj yim los ntawm ob peb tus lej: 1, 2, 3, 4, 6, 8, 9, thiab 12. Nws muab qhov kwv yees zoo rau kev sib xyaw txhua xyoo ntawm tus nqi raug (los ntawm 6% txog 10%). Qhov kwv yees tsis muaj tseeb ntawm cov paj laum siab dua.

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