Exponents, Radicals and Logarithms

Transcription

1 Mathematics of Fiace Expoets, Radicals ad Logaithms Defiitio 1. x = x x x fo a positive itege. Defiitio 2. x = 1 x Defiitio 3. x is the umbe whose th powe is x. Defiitio 4. x 1/ = x Defiitio 5. x m/ = x m Defiitio 6. log b x is the powe b must be aised to i ode to obtai x. Cosequeces: log b b x ) = x, b log b x = x. Commo Logs Base 10 Natual Logs Base e, deoted by l

2 Popeties of Logaithms 1. logαβ) = log α+log β The log of a poduct is the sum of the logs. 2. logα/β) = log α log β The log of a quotiet is the diffeece of the logs. 3. logα β ) = β log α The log of a umbe aised to a powe is the powe times the log. Logaithms ae useful i solvig expoetial equatios. The key is to isolate the tem with the vaiable i the expoet ad the take logaithims of both sides. Example: Solve x = 93. Solutio: 3 5x = 75, l3 5x ) = l 75, 5x l 3 = l 75 l 75, x = 5 l 3.

3 Simple Iteest Notatio: P = picipal o peset value = aual iteest ate I = amout of iteest t = time geeally i yeas) F = balace o futue value afte t yeas With simple iteest, the amout of iteest i oe yea is the poduct of the picipal ad the iteest ate. Moe geeally, the amout of simple iteest eaed i a peiod of time is equal to the poduct of the piciple, the iteest ate ad the amout of time.

4 I =

5 Compoud Iteest Iteest is computed peiodically, added to the balace, ad futue iteest is computed based o the updated balace. If iteest is compouded times pe yea, each time the amout of iteest compouded will equal 1 times the amout that would be compouded fo a etie yea. I othe wods, the amout of iteest will be 1 P + 1 P = P 1 + ). P ad the ew balace will be Effectively, the old balace is multiplied by 1 + to get the ew balace. I t yeas, the oigial balace will be multiplied by 1+ t times, oce fo evey iteest peiod. Effectively, it will be multiplied by 1 + ) t, so the futue balace will be F = P 1 + ) t.

6 Compoud Iteest Fomula F = P 1 + ) t We may look at this as a equatio ivolvig five diffeet vaiables, F, P,,, t, ay of which may be foud if the values of the othes ae kow. At diffeet times, we may use this fomula to fid the futue value, the oigial balace, the amout of time it will take fo the balace to each a cetai amout o the aual iteest ate.

7 Cotiuous Iteest If iteest is compouded vey fequetly, which coespods to beig vey lage, the balace oe obtais does ot chage vey much. Oe may see this by takig the Compoud Iteest Fomula ad maipulatig it as follows: F = P 1 + ) t F = P F = P ) t / [ )] t / We may wite this as F = P [ N ) N ] t, whee N =. ) N Whe is vey lage, so is N, ad 1 + N 1 gets vey close to the mathematical costat

8 e, which is a iatioal umbe appoximately We thus fid F will be close to P e t. The fomula F = P e t is kow as the Cotiuous Iteest Fomula.

9 Aual Pecetage Yield APY) o Effective Aual Yield Defiitio 7 Aual Pecetage Yield). The aual pecetage yield is the aual iteest ate which would be eeded to obtai the same futue balace i oe yea if iteest was compouded aually. Suppose the aual iteest ate is. The balace afte a yea will be F = P 1 + ) 1 = P 1 + ). This must equal P 1 + AP Y ), the balace the accout would have if the aual ate was equal to AP Y ad compouded oce pe yea. It follows that we must have: P 1 + ) = P 1 + AP Y ) 1 + ) = 1 + AP Y AP Y = 1 + ) 1

10 If iteest is compouded cotiuously, the coespodig fomula is AP Y = e 1.

11 Systematic Savigs Systematic Savigs Plas: A deposit of amout D is made at the ed of each of iteest peiods each yea fo a peiod of t yeas. To fid the balace afte t yeas, we may teat each deposit as if it wee placed i a sepaate accout. Afte t yeas... the fist deposit would have gow to D 1 + ) t 1 the secod deposit would have gow to D 1 + ) t 2 the thid deposit would have gow to D 1 + ) t 3

12 ... the fial deposit would ot have collected iteest yet ad would have gow to D. The total balace afte the t yeas of all the deposits would be F = D+D 1 + ) +D 1 + ) D 1 + t 1. ) Factoig out the commo facto of D, we may wite F = D ) ) ) ) t 1. The sum i paetheses is i the fom 1+a+a a k 1, whee a = 1+ ad k = t

13 Thee is a fomula fo such sums, which ae called geometic seies. Fom the tedious but elatively outie calculatio 1 a) 1 + a + a a k 1) = 1 a k, which may also be looked at as a factoizatio fomula, oe obtais the fomula 1 + a + a a k 1 = 1 ak 1 a. Usig this fomula i the fomula fo the balace, we obtai F = D ) t ) F = D 1 + ) t 1

14 This ca be used to fid the futue balace as well as to fid the size of the peiodic deposits eeded to obtai a give futue balace. To fid the peiodic deposits eeded, we may use the fomula as is o solve it fo D as follows: D 1 + ) ) t 1 = F D = F 1 + ) t. 1

15 Amotized Loas I a typical loa, the boowe is let a amout P ad makes peiodic epaymets R to educe the balace owed util the balace is educed to 0. This may be aalyzed diectly i a mae simila to the aalysis of peiodic savigs. We may cut the pocess shot by viewig the epaymets as peiodic deposits made by the boowe desiged so that they would gow, at the ed of the epaymet peiod, to a amout equal to what the oigial loa amout would gow to if it wee ivested at a iteest ate equal to the ate chaged. Usig the fomula obtaied fo systematic savigs, the peiodic paymets would gow to a balace F = R 1 + ) t 1.

16 A loa i the amout P would gow to a futue value P 1 + ) t. We must theefoe have P 1 + ) t = R 1 + ) t 1. P = ) t R 1 + ) t 1 P = R ) t This could be used to detemie the maximum loa someoe could affod if the maximum size of the mothly paymet they could make was R.

17 Oe may take the same fomula ad solve as follows fo R to obtai the mothly paymet ecessay o a loa of a cetai size. P = R ) t R = 1 P 1 + ) t R = 1 P 1 + ) t Note the umeato P is a moth s woth of iteest o the oigial balace. Obviously, the mothly paymet must be geate tha that sice othewise the outstadig balace would keep iceasig.

18 Defiitio 8 Amotizatio Schedule). A amotizatio schedule is a list of paymets to be made o a loa which beaks dow each paymet ito picipal ad iteest. A amotizatio schedule ca be set up faily easily usig a speadsheet.