Annual compounding, revisited

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Camilla Knight
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1 Sectio 1.: No-aual compouded iterest MATH 105: Cotemporary Mathematics Uiversity of Louisville August 2, 2017 Compoudig geeralized 2 / 15 Aual compoudig, revisited The idea behid aual compoudig is that ew iterest is computed ad added to the balace each year. For a xed-term multi-year deposit, this works, but what if we wat to withdraw our moey several moths ito a year? Oe thig we could do dieretly is to compute a smaller chuk of iterest more ofte.

2 Compoudig geeralized 3 / 15 Smaller iterest, more ofte A multiple-computatio study Suppose we wat to compute ad add i iterest quarterly o a $1000 balace with a aual iterest rate of 5%, ad wat to kow what the balace is after a full year. Recall that for aual compoudig we just did a simple iterest calculatio for each idividual year. Now we do a simple iterest calculatio for each quarter so t = 0.25): F 1 = = F 2 = F F so after a year the balace will be $ Note that this is more tha the omial 5% per year i the iterest rate! Compoudig geeralized / 15 Why compute iterest more frequetly? There are two cosequeces of the calcluatio we did i the last slide which are relevat: itermediary-stage values are ow kow; for istace, the balace halfway through the year was $ the actual iterest was higher tha if it were compouded aually. The rst eect is udeiably good; the secod maybe seems deceptive, but ca be addressed with proper iformatio.

3 Compoudig geeralized 5 / 15 Simplifyig our calculatios Same study, but with less butto-mashig How ca we simplify that calculatio of quarterly iterest o a $1000 balace with a aual iterest rate of 5% for a full year? Recall that the rst calculatio looked like this: F 1 = = which simplies to F 1 = ). We wat to apply that same multiplicative factor four times, so we might compute: F = ) Ad for more emphasis o the four quarters per year aspect, we may write it as: F = 1000 ) Compoudig geeralized 6 / 15 Applyig our simplicatio A extesio of the last questio Suppose, as before, we wat to compute ad add i iterest quarterly o a $1000 balace with a aual iterest rate of 5%, but ow we wat to kow what the balace is after 6 years. As previously, we see that every quarter's iterest applicatio is a multiplicatio by Six years measured i quarters is 6 = 2 quarters, so we wat to perform that multiplicatio twety-four times: F = ) for a al balace of $

4 Buildig a formula Compoudig geeralized 7 / 15 F = ) This calculatio makes use of the pricipal P = 1000, the aual iterest rate r = 0.05, ad the lifetime t = 6, but it also uses a ew quatity =, the umber of compoudig periods per year. Note that the expressio 0.05 is the periodic iterest rate, i.e., the proportio of the balace retured i iterest over a sigle compoudig period, while 6 is the lifetime measured i compoudig periods. This gives us the geeral formula: F = P 1 + r ) t Sometimes the periodic iterest rate is deoted by the letter i = r, ad the umber of compoudig periods by m = t. Example calculatios Compoudig geeralized 8 / 15 Why stop at quarters? I take out a $500 loa whose aual iterest rate of 18% is compouded mothly. How much would I eed to pay it o after 9 moths? After 2 years? I both scearios, P = 500, r = 0.18, ad = 12. I the rst sceario, sice the lifetime was give i moths, we could either establish t = 9 = 0.75 or, more straightforwardly, m = 9, so: 12 F = ) so I would have to pay back $ of which $71.69 is iterest). I the secod sceario, t = 2, givig: F = ) so I would have to pay back $71.75 of which $21.75 is iterest).

5 Compoudig geeralized 9 / 15 Variatios i compoudig periods I geeral, more frequet compoudig icreases the log-term balace, but ot by much! Hypothetical compariso Cosider a $500 loa with a 18% aual iterest rate. How would the balace dier over years usig dieret compoudig periods? $1000 $900 $800 $700 $600 $ Takig it to the limit Compoudig geeralized 10 / 15 Dimiishig returs How does a $500 loa with a 18% aual iterest rate for four years chage as we icrease the umber of compoudig periods? As the last slide idicated, the returs o icreasig compoudig frequecy decrease rapidly: ) ) ) ) ) ) If we compoud very ofte, this calculatio teds towards $

6 Compoudig geeralized 11 / 15 Compoudig cotiuously Whe is very large, the compoudig becomes cotiuous. There is a formula for what happes i this case too: As gets very large, P 1 + r ) t approaches Pe rt where e You wo't be expected to work out cotiuous-compoudig problems i this course, but kowig that there is a limitig behavior is useful! Uveilig the truth Aual percetage rates 12 / 15 Oe disadvatage of oaual compoudig is that it coceals the truth: 5% aual rate compouded mothly is't actually a 5% growth over a year! A useful measure is the aual percetage rate or aual percetage yield, which describes what percetage growth actually occurs yearly as a result of iterest. A APR example If I borrow $1000 at 7% aual iterest compouded mothly, what is the actual percetage growth after a year? After oe year, the future value is F = ) so the growth percetage is %.

7 Aual percetage rates 13 / 15 From the particular to the abstract Our calculatio i the last slide for the APR was ) Here 1000 was the pricipal, 0.07 the aual iterest rate, 12 the umber of compoudig periods per moth, so i the abstract the APR is P 1 + r ) P P = 1 + r ) 1 Note that the amout ad lifetime of the loa are ot ecessary to calculate a APR! Aual percetage rates 1 / 15 Oe iterest rate, may aual percetages Somethig as simple as a 5% aual iterest rate could mea may dieret thigs i dieret circumstaces: Compouded aually )1 1 = 5% APR. Compouded semiaually )2 1 = % APR. Compouded quarterly ) % APR. Compouded mothly ) % APR. Compouded weekly ) % APR. Compouded daily ) % APR. Compouded cotiuously e % APR.

8 All the formulas i oe place 15 / 15 All the formulas i oe place Aual compoudig = 1): F = P1 + r) t Periodic compoudig: F = P 1 + r ) t Cotiuous compoudig: F = P 1 + i) m where i = r ad m = t APR = 1 + r ) 1 F = Pe rt APR = e r 1

APPLICATION OF GEOMETRIC SEQUENCES AND SERIES: COMPOUND INTEREST AND ANNUITIES

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