Rule of 72 - Wikipedia

In finance, the rule of 72, the rule of 70 and the rule of 69.3 are methods for estimating an investment's doubling time. The rule number (e.g., 72) is ... Ruleof72 FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Nottobeconfusedwith72-yearrule. Methodsofestimatingthedoublingtimeofaninvestment Infinance,theruleof72,theruleof70[1]andtheruleof69.3aremethodsforestimatinganinvestment'sdoublingtime.Therulenumber(e.g.,72)isdividedbytheinterestpercentageperperiod(usuallyyears)toobtaintheapproximatenumberofperiodsrequiredfordoubling.Althoughscientificcalculatorsandspreadsheetprogramshavefunctionstofindtheaccuratedoublingtime,therulesareusefulformentalcalculationsandwhenonlyabasiccalculatorisavailable.[2] Theserulesapplytoexponentialgrowthandarethereforeusedforcompoundinterestasopposedtosimpleinterestcalculations.Theycanalsobeusedfordecaytoobtainahalvingtime.Thechoiceofnumberismostlyamatterofpreference:69ismoreaccurateforcontinuouscompounding,while72workswellincommoninterestsituationsandismoreeasilydivisible. Thereareanumberofvariationstotherulesthatimproveaccuracy.Forperiodiccompounding,theexactdoublingtimeforaninterestrateofrpercentperperiodis t = ln ⁡ ( 2 ) ln ⁡ ( 1 + r / 100 ) ≈ 72 r {\displaystylet={\frac{\ln(2)}{\ln(1+r/100)}}\approx{\frac{72}{r}}} , wheretisthenumberofperiodsrequired.Theformulaabovecanbeusedformorethancalculatingthedoublingtime.Ifonewantstoknowthetriplingtime,forexample,replacetheconstant2inthenumeratorwith3.Asanotherexample,ifonewantstoknowthenumberofperiodsittakesfortheinitialvaluetoriseby50%,replacetheconstant2with1.5. Contents 1Usingtheruletoestimatecompoundingperiods 2Choiceofrule 3History 4Adjustmentsforhigheraccuracy 4.1E-Mrule 4.2Padéapproximant 5Derivation 5.1Periodiccompounding 5.2Continuouscompounding 6Seealso 7References 8Externallinks Usingtheruletoestimatecompoundingperiods[edit] Toestimatethenumberofperiodsrequiredtodoubleanoriginalinvestment,dividethemostconvenient"rule-quantity"bytheexpectedgrowthrate,expressedasapercentage. Forinstance,ifyouweretoinvest$100withcompoundinginterestatarateof9%perannum,theruleof72gives72/9=8yearsrequiredfortheinvestmenttobeworth$200;anexactcalculationgivesln(2)/ln(1+0.09)=8.0432years. Similarly,todeterminethetimeittakesforthevalueofmoneytohalveatagivenrate,dividetherulequantitybythatrate. Todeterminethetimeformoney'sbuyingpowertohalve,financiersdividetherule-quantitybytheinflationrate.Thusat3.5%inflationusingtheruleof70,itshouldtakeapproximately70/3.5=20yearsforthevalueofaunitofcurrencytohalve.[1] Toestimatetheimpactofadditionalfeesonfinancialpolicies(e.g.,mutualfundfeesandexpenses,loadingandexpensechargesonvariableuniversallifeinsuranceinvestmentportfolios),divide72bythefee.Forexample,iftheUniversalLifepolicychargesanannual3%feeoverandabovethecostoftheunderlyinginvestmentfund,thenthetotalaccountvaluewillbecutto50%in72/3=24years,andthento25%ofthevaluein48years,comparedtoholdingexactlythesameinvestmentoutsidethepolicy. Choiceofrule[edit] Thevalue72isaconvenientchoiceofnumerator,sinceithasmanysmalldivisors:1,2,3,4,6,8,9,and12.Itprovidesagoodapproximationforannualcompounding,andforcompoundingattypicalrates(from6%to10%).Theapproximationsarelessaccurateathigherinterestrates. Forcontinuouscompounding,69givesaccurateresultsforanyrate.Thisisbecauseln(2)isabout69.3%;seederivationbelow.Sincedailycompoundingiscloseenoughtocontinuouscompounding,formostpurposes69,69.3or70arebetterthan72fordailycompounding.Forlowerannualratesthanthoseabove,69.3wouldalsobemoreaccuratethan72.[3]Forhigherannualrates,78ismoreaccurate. Graphscomparingdoublingtimesandhalflivesofexponentialgrowths(boldlines)anddecay(faintlines),andtheir70/tand72/tapproximations.IntheSVGversion,hoveroveragraphtohighlightitanditscomplement. Rate ActualYears Rate×ActualYears Ruleof72 Ruleof70 Ruleof69.3 72adjusted E-Mrule 0.25% 277.605 69.401 288.000 280.000 277.200 277.667 277.547 0.5% 138.976 69.488 144.000 140.000 138.600 139.000 138.947 1% 69.661 69.661 72.000 70.000 69.300 69.667 69.648 2% 35.003 70.006 36.000 35.000 34.650 35.000 35.000 3% 23.450 70.349 24.000 23.333 23.100 23.444 23.452 4% 17.673 70.692 18.000 17.500 17.325 17.667 17.679 5% 14.207 71.033 14.400 14.000 13.860 14.200 14.215 6% 11.896 71.374 12.000 11.667 11.550 11.889 11.907 7% 10.245 71.713 10.286 10.000 9.900 10.238 10.259 8% 9.006 72.052 9.000 8.750 8.663 9.000 9.023 9% 8.043 72.389 8.000 7.778 7.700 8.037 8.062 10% 7.273 72.725 7.200 7.000 6.930 7.267 7.295 11% 6.642 73.061 6.545 6.364 6.300 6.636 6.667 12% 6.116 73.395 6.000 5.833 5.775 6.111 6.144 15% 4.959 74.392 4.800 4.667 4.620 4.956 4.995 18% 4.188 75.381 4.000 3.889 3.850 4.185 4.231 20% 3.802 76.036 3.600 3.500 3.465 3.800 3.850 25% 3.106 77.657 2.880 2.800 2.772 3.107 3.168 30% 2.642 79.258 2.400 2.333 2.310 2.644 2.718 40% 2.060 82.402 1.800 1.750 1.733 2.067 2.166 50% 1.710 85.476 1.440 1.400 1.386 1.720 1.848 60% 1.475 88.486 1.200 1.167 1.155 1.489 1.650 70% 1.306 91.439 1.029 1.000 0.990 1.324 1.523 Note:Themostaccuratevalueoneachrowisinitalics,andthemostaccurateofthesimplerrulesinbold. History[edit] AnearlyreferencetotheruleisintheSummadearithmetica(Venice,1494.Fol.181,n.44)ofLucaPacioli(1445–1514).Hepresentstheruleinadiscussionregardingtheestimationofthedoublingtimeofaninvestment,butdoesnotderiveorexplaintherule,anditisthusassumedthattherulepredatesPaciolibysometime. Avolersapereogniquantitàatantoper100l'anno,inquantiannisaràtornatadoppiatrautileecapitale,tieniperregola72,amente,ilqualesemprepartiraiperl'interesse,equellocheneviene,intantiannisaràraddoppiato.Esempio:Quandol'interesseèa6per100l'anno,dicochesiparta72per6;nevien12,ein12annisaràraddoppiatoilcapitale.(emphasisadded). Roughlytranslated: Inwantingtoknowofanycapital,atagivenyearlypercentage,inhowmanyyearsitwilldoubleaddingtheinteresttothecapital,keepasarule[thenumber]72inmind,whichyouwillalwaysdividebytheinterest,andwhatresults,inthatmanyyearsitwillbedoubled.Example:Whentheinterestis6percentperyear,Isaythatonedivides72by6;12results,andin12yearsthecapitalwillbedoubled. Adjustmentsforhigheraccuracy[edit] Forhigherrates,alargernumeratorwouldbebetter(e.g.,for20%,using76toget3.8yearswouldbeonlyabout0.002off,whereusing72toget3.6wouldbeabout0.2off).Thisisbecause,asabove,theruleof72isonlyanapproximationthatisaccurateforinterestratesfrom6%to10%. Foreverythreepercentagepointsawayfrom8%,thevalueof72couldbeadjustedby1: t ≈ 72 + ( r − 8 ) / 3 r {\displaystylet\approx{\frac{72+(r-8)/3}{r}}} or,forthesameresult: t ≈ 70 + ( r − 2 ) / 3 r {\displaystylet\approx{\frac{70+(r-2)/3}{r}}} Bothoftheseequationssimplifyto: t ≈ 208 3 r + 1 3 {\displaystylet\approx{\frac{208}{3r}}+{\frac{1}{3}}} Notethat 208 3 {\displaystyle{\frac{208}{3}}} isquitecloseto69.3. E-Mrule[edit] TheEckart–McHalesecond-orderrule(theE-Mrule)providesamultiplicativecorrectionfortheruleof69.3thatisveryaccurateforratesfrom0%to20%,whereastheruleisnormallyonlyaccurateatthelowestendofinterestrates,from0%toabout5%. TocomputetheE-Mapproximation,multiplytheruleof69.3resultby200/(200−r)asfollows: t ≈ 69.3 r × 200 200 − r {\displaystylet\approx{\frac{69.3}{r}}\times{\frac{200}{200-r}}} . Forexample,iftheinterestrateis18%,theruleof69.3givest=3.85years,whichtheE-Mrulemultipliesby 200 182 {\displaystyle{\frac{200}{182}}} (i.e.200/(200−18))togiveadoublingtimeof4.23years.Astheactualdoublingtimeatthisrateis4.19years,theE-Mrulethusgivesacloserapproximationthantheruleof72. Toobtainasimilarcorrectionfortheruleof70or72,oneofthenumeratorscanbesetandtheotheradjustedtokeeptheirproductapproximatelythesame.TheE-Mrulecouldthusbewrittenalsoas t ≈ 70 r × 198 200 − r {\displaystylet\approx{\frac{70}{r}}\times{\frac{198}{200-r}}} or t ≈ 72 r × 192 200 − r {\displaystylet\approx{\frac{72}{r}}\times{\frac{192}{200-r}}} Inthesevariants,themultiplicativecorrectionbecomes1respectivelyforr=2andr=8,thevaluesforwhichtherulesof70and72aremostaccurate. Padéapproximant[edit] Thethird-orderPadéapproximantgivesamoreaccurateansweroveranevenlargerrangeofr,butithasaslightlymorecomplicatedformula: t ≈ 69.3 r × 600 + 4 r 600 + r {\displaystylet\approx{\frac{69.3}{r}}\times{\frac{600+4r}{600+r}}} . Derivation[edit] Periodiccompounding[edit] Forperiodiccompounding,futurevalueisgivenby: F V = P V ⋅ ( 1 + r ) t {\displaystyleFV=PV\cdot(1+r)^{t}} where P V {\displaystylePV} isthepresentvalue, t {\displaystylet} isthenumberoftimeperiods,and r {\displaystyler} standsfortheinterestratepertimeperiod. Thefuturevalueisdoublethepresentvaluewhenthefollowingconditionismet: ( 1 + r ) t = 2 {\displaystyle(1+r)^{t}=2\,} Thisequationiseasilysolvedfor t {\displaystylet} : ln ⁡ ( ( 1 + r ) t ) = ln ⁡ 2 t ln ⁡ ( 1 + r ) = ln ⁡ 2 t = ln ⁡ 2 ln ⁡ ( 1 + r ) {\displaystyle{\begin{array}{ccc}\ln((1+r)^{t})&=&\ln2\\t\ln(1+r)&=&\ln2\\t&=&{\frac{\ln2}{\ln(1+r)}}\end{array}}} Asimplerearrangementshows: ln ⁡ 2 ln ⁡ ( 1 + r ) = ( ln ⁡ 2 r ) ( r ln ⁡ ( 1 + r ) ) {\displaystyle{\frac{\ln{2}}{\ln{(1+r)}}}={\bigg(}{\frac{\ln2}{r}}{\bigg)}{\bigg(}{\frac{r}{\ln(1+r)}}{\bigg)}} Ifrissmall,thenln(1+r)approximatelyequalsr(thisisthefirsttermintheTaylorseries).Thatis,thelatterfactorgrowsslowlywhen r {\displaystyler} isclosetozero. Callthislatterfactor f ( r ) = r ln ⁡ ( 1 + r ) {\displaystylef(r)={\frac{r}{\ln(1+r)}}} .Thefunction f ( r ) {\displaystylef(r)} isshowntobeaccurateintheapproximationof t {\displaystylet} forasmall,positiveinterestratewhen r = .08 {\displaystyler=.08} (seederivationbelow). f ( .08 ) ≈ 1.03949 {\displaystylef(.08)\approx1.03949} ,andwethereforeapproximatetime t {\displaystylet} as: t = ( ln ⁡ 2 r ) f ( .08 ) ≈ .72 r {\displaystylet={\bigg(}{\frac{\ln2}{r}}{\bigg)}f(.08)\approx{\frac{.72}{r}}} Writtenasapercentage: .72 r = 72 100 r {\displaystyle{\frac{.72}{r}}={\frac{72}{100r}}} Thisapproximationincreasesinaccuracyasthecompoundingofinterestbecomescontinuous(seederivationbelow). 100 r {\displaystyle100r} is r {\displaystyler} writtenasapercentage. Inordertoderivethemorepreciseadjustmentspresentedabove,itisnotedthat ln ⁡ ( 1 + r ) {\displaystyle\ln(1+r)\,} ismorecloselyapproximatedby r − r 2 2 {\displaystyler-{\frac{r^{2}}{2}}} (usingthesecondtermintheTaylorseries). 0.693 r − r 2 / 2 {\displaystyle{\frac{0.693}{r-r^{2}/2}}} canthenbefurthersimplifiedbyTaylorapproximations: 0.693 r − r 2 / 2 = 69.3 R − R 2 / 200 = 69.3 R 1 1 − R / 200 ≈ 69.3 ( 1 + R / 200 ) R = 69.3 R + 69.3 200 = 69.3 R + 0.3465 {\displaystyle{\begin{array}{ccc}{\frac{0.693}{r-r^{2}/2}}&=&{\frac{69.3}{R-R^{2}/200}}\\&&\\&=&{\frac{69.3}{R}}{\frac{1}{1-R/200}}\\&&\\&\approx&{\frac{69.3(1+R/200)}{R}}\\&&\\&=&{\frac{69.3}{R}}+{\frac{69.3}{200}}\\&&\\&=&{\frac{69.3}{R}}+0.3465\end{array}}} Replacingthe"R"inR/200onthethirdlinewith7.79gives72onthenumerator.Thisshowsthattheruleof72ismostaccurateforperiodicallycompoundedinterestsaround8%.Similarly,replacingthe"R"inR/200onthethirdlinewith2.02gives70onthenumerator,showingtheruleof70ismostaccurateforperiodicallycompoundedinterestsaround2%. Alternatively,theE-Mruleisobtainedifthesecond-orderTaylorapproximationisuseddirectly. Continuouscompounding[edit] Forcontinuouscompounding,thederivationissimplerandyieldsamoreaccuraterule: ( e r ) p = 2 e r p = 2 ln ⁡ e r p = ln ⁡ 2 r p = ln ⁡ 2 p = ln ⁡ 2 r p ≈ 0.693147 r {\displaystyle{\begin{array}{ccc}(e^{r})^{p}&=&2\\e^{rp}&=&2\\\lne^{rp}&=&\ln2\\rp&=&\ln2\\p&=&{\frac{\ln2}{r}}\\&&\\p&\approx&{\frac{0.693147}{r}}\end{array}}} Seealso[edit] Exponentialgrowth Timevalueofmoney Interest Discount Ruleof16 Ruleofthree(statistics) References[edit] ^abDonellaMeadows,ThinkinginSystems:APrimer,ChelseaGreenPublishing,2008,page33(box"Hintonreinforcingfeedbackloopsanddoublingtime"). ^Slavin,Steve(1989).AlltheMathYou'llEverNeed.JohnWiley&Sons.pp. 153–154.ISBN 0-471-50636-2. ^KalidAzadDemystifyingtheNaturalLogarithm(ln)fromBetterExplained Externallinks[edit] TheScalesOf70–extendstheruleof72beyondfixed-rategrowthtovariableratecompoundgrowthincludingpositiveandnegativerates. 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