Compound Interest. S.Y.Tan. Compound Interest

Transcription

1 Compoud Iterest S.Y.Ta Compoud Iterest

2 The yield of simple iterest is costat all throughout the ivestmet or loa term. =2000 r = 0% = 0. t = year =? I =? = 2000 (+ (0.)()) = 3200 I = - = = 200 I =200 = = 0 yr = = 0 6 moths yr = = 0 3 moths 6 moths 9 moths yr Note that the iterest yield at a certai cut off date or time iterval is costat all throughout the ivestmet or loa term. S.Y.Ta Compoud Iterest

3 Whe iterest yield or eared is added to the pricipal at regular time iterval ad the sum becomes the ew pricipal the iterest is said to be compouded or coverted. At compoud iterest, iterest eared at a certai cut-off date is automatically reivested to ear more iterest. Whe iterest is beig coverted or compouded oce or more tha oce a year, the time betwee successive coversios of iterest is called a coversio or iterest period or simply period. The umber of coversio or iterest periods i a year is called the frequecy of coversio (m). S.Y.Ta Compoud Iterest

4 So iterest may be compouded or coverted (i) Aually (oce a year / every year) m = (years) (ii) Semi-aually (twice a year / every 6 moths) m = ( semi-aual periods) 0 6 moths yr /2 yrs 2 yrs (iii) Quarterly (4 times a year / every 3 moths) m = (quarters) 0 3 moths 6 moths 9 moths yr S.Y.Ta Compoud Iterest

5 (iv) Mothly (2 times year / every moth) m = (moths) 0 year (v) Every 4 moths (3 times year ) m = (periods) 0 4 moths 8 moths year (vi) Every 2 moths (6 times a year) m = (periods) 0 2 mths 4 mths 6 mths 8 mths 0 mths year S.Y.Ta Compoud Iterest

6 The stated aual rate of iterest (coverted m times a year) is called the omial rate j. The rate of iterest per period is i = j/m ad the total umber of coversio period is = t m. The fial amout uder compoud iterest is called the compoud amout (). The differece betwee compoud amout ad origial pricipal is called the compoud iterest. id the compoud amout if is ivested at omial rate j coverted m times a year for a term of t years. S.Y.Ta Compoud Iterest

7 Let 0 = (origial pricipal) i = j/m = t m I k = iterest eared at the ed of the kth period k = ew pricipal at the ed of the kth period = k- + I k = I = 0 i = i I I 2 I 3 I 4 I = 0 + I = + i = ( +i ) I 2 = i = (+i) i 2 = + I 2 = (+i)+(+i)i = (+i )(+i)=(+i) 2 I 3 = 2 i = (+i) 2 i = 2 + I 3 = (+i) 2 +(+i) 2 i = (+i ) 2 (+i)=(+i) 3 I 4 = 3 i = (+i) 3 i 4 = 3 + I 4 = (+i) 3 +(+i) 3 i = (+i ) 3 (+i)=(+i) 4 - = (+i) - ad I = - i = (+i) - i = - (periods) = = - + I = (+i) - + (+i) - i = (+i) - (+i) = (+i) S.Y.Ta = ( + i) t years Compoud Iterest

8 =2000 j = 0% = 0.0 (compouded quarterly) m = 4 i =j/m = 0.0/4 = t = year = (4) = 4 = 2000 ( ) 4 = (compoud amout) I = - = = (compoud iterest) = = (quarters) 0 3 moths 6 moths 9 moths year ial amout uder simple iterest = = ial amout uder compoud iterest (quarters) 0 3 moths 6 moths 9 moths year I =2000(0.025) = 300 = + I = = = + I 2 = = I 2 =2300(0.025) = I 3 = (0.025) = = 2 + I 3 = = I 4 = (0.025) = = 4 = 3 + I 4 = = S.Y.Ta Compoud Iterest

9 ormula for the compoud amout : ( i) Accumulatio factor - origial pricipal j - rate of iterest per year (omial rate) m - frequecy of coversio i t - iterest rate per priod i - term i years j m - total umber of coversio periods tm S.Y.Ta Compoud Iterest

10 Table of the requecy of Coversio Nomial Rate Coverted requecy of Coversio (m) Aually Semi-aually 2 Quarterly 4 Mothly 2 Every 4 moths 3 Every 2 moths 6 Compoud Iterest

11 = ( + i) accumulatio factor Compoud amout is the accumulated value of pricipal at the ed of periods. To accumulate meas to fid. Ex 2 Accumulate 80,000 for 7 years at 5% compouded every 4 moths. =80,000 m = 3 j = 5% = 0.5 i = 0.5/3 =0.05 t = 7 years = 7(3)= 2 =? = ( + i) = ( +0.05) 2 = 222,877.0 S.Y.Ta Compoud Iterest

12 Ex3. id the compoud amout ad iterest at the ed of 6 years if 80,00 is ivested at 2 3 compouded a) semi-aually b) mothly. a) =80,000 t = 6 yr j = m = 2 i I = (6)(2)= ( i) 80000( 600) 64, , i b) m = = (6)(2) = 72 I ( i) 80000( ) , , % S.Y.Ta Compoud Iterest

13 discout factor = ( + i) - = ( + i) reset value of a amout due i periods is the value (pricipal) which is ivested ow at a give omial rate j. To discout meas to fid its preset value at periods before is due. ( i) Discout factor S.Y.Ta Compoud Iterest

14 Ex A ma eeds 500,000 i 3 years to start a small busiess. How much moey should he place i a accout ow that gives 4.02% compouded semi-aually so he ca start the busiess by the? =500,000 m = 2 j = 4.02% = i = /2 =0.020 t = 3 years = 3(2)= 6 =? = ( + i) - = ( ) - 6 = 443,724.6 S.Y.Ta Compoud Iterest

15 Ex 2 I purchasig a uit of I-phoe 6S, Has makes a dow paymet of 5000 ad agrees to pay 50,000 5 moths later. id the cash value of the I-phoe if moey is worth 9% compouded mothly. Cash value (CV) = Dow paymet (D) + reset Value () =50,000 m = 2 j = 9% = 0.09 i = 0.09/2 = t = 5 moths = 5/2 years =(5/2)(2)=5 D = 5000 =? CV =? = ( + i) - = ( ) - 5 = 44, CV = D + CV = CV = 49, S.Y.Ta Compoud Iterest

16 Ex 3 O her 8 th birthday, Liza receives 20,000 as gift from her parets. If she ivests this moey i a bak that gives 3% iterest coverted every 2 moths, how much moey will she have o her 25 th birthday? How much iterest will she ear? = t = 7 years m = 6 = 7(6) = 42 i =0.03/6 =0.005 As: = (+0.005) 42 =24, I = Ex 4 The buyer of a car pays 50,000 dow paymet ad the balace of 500,000 to be paid two years later. What is the cash price of the car if moey is worth 2% compouded aually? D = 50,000 = 500,000 m = t = 2 yrs = 2 j = 0.2 i = 0.2 As: = (+0.2) - 2 = 398, CV = C = 50, , = 548, S.Y.Ta Compoud Iterest

17 Ex 5 What is the maturity value of a 75,000- peso, three-year ivestmet earig 5% compouded mothly? 75,000 m 2 j 0.05 i 2 3(2) , ,0.42 Do this if i is ot a termiatig decimal S.Y.Ta Compoud Iterest

18 Ex 6 id the compoud amout after 5 years ad 9 moths if the pricipal is 50,000 ad the rate is 7% compouded quarterly. 50,000 t yrs m 4 4 (4) j 0.07 i ,000( 0 ) , S.Y.Ta Compoud Iterest

19 idig Iterest Rate (Compoud Iterest) j? i j m j i(m) ( i) ( i) ( i) i i Nomial rate j i(m) S.Y.Ta Compoud Iterest

20 Ex At what omial rate compouded quarterly will 30,000 amout to 45,000 i 3 years? 30,000 45,000 t 3yrs m 4 3(4) 2 j i(4) i? ( i) ,000 30,000( i) i i j i(4) % S.Y.Ta Compoud Iterest

21 Ex 2 Alla borrows 35,000 ad agrees to pay 42,000 for a debt i year ad 3 moths from ow. At what rate compouded mothly is he payig iterest? 35,000 t j i(2) 42,000 yrs m 2 (2) i? ( i) ,000 35,000( i) i i j i(2) % S.Y.Ta Compoud Iterest

22 Ex 3 If Bobby get 56,47.27 at the ed of 4 years ad 6 moths for ivestig 25,000 ow. At what rate compouded semi-aually is he earig iterest? m j 25,000 2 i(2) i 9 2 (2)? 56, t yrs ( i) , ,000( i) i i j i(2) % S.Y.Ta Compoud Iterest

23 Ex 4 O Jue 30, 200, Cyril ivested 30,000 i a bak that pays iterest coverted quarterly. If she wats her moey to be 4 times as large o Dec 30, 206, at what rate should her moey ear iterest? 6 / 30 /0 6 / 30 /6 30,000 4(30,000) 3 O.D. 6 / 30 /0 M.D. 2 / 30 / yrs (4) 26 m j 4 i(4) i? 4(30000) 26 4 ( i) (30,000) 30,000( i) j to i i i(4) 6 / 30 /6 to 2 / 30 /6 6 6 yrs moths t % S.Y.Ta Compoud Iterest

24 roperties of Logarithm or Laws of Logarithm ) logb(mn) logb M logb N l(mn) l M l N M M log M log N l l M l N 2) logb N b b N 3) log b N r r log b N l N r r l N 4) logb b l e 5) logb 0 l 0 6) log b b x x l e x x 3 2 log (8) 3 3 EX. start with log2 2 7) b log b x x e l x x EX. start with 8 log 2 log S.Y.Ta Compoud Iterest

25 Ex How log will it take 50,000 to accumulate to 58,000 at 2% coverted every 2 moths? 50,000 58,000 m 6 j 0.2 i t 6? 58,000 50,000( log t log(.02) 6 ) log log 0.02 log(.02) yrs S.Y.Ta Compoud Iterest

26 Ex 2 O March 5, 203, a ma ivested 50,000 i a bak that gives 5% iterest compouded every 4 moths. If he decided to withdraw his moey whe it accumulated to 60,000, whe did he make his withdrawal? 50,000 60,000 m 3 j 0.5 i 3 t 3? possible date? ,000 50,000( ) log log 0.05 log(.05) log log(.05) t yrs yr & 3 moths Jue 5, 204 S.Y.Ta Compoud Iterest

27 Ex 3 If 80,000 is ivested at the rate of 6 ½% compouded aually, whe will it ear iterest of 5,000? 3 80,000 I 5,000 95,000 m j % i t? 95,000 80,000( ) log log log(.065) log log(.065) t 2.73 yrs S.Y.Ta Compoud Iterest

28 Ex 4 O April 5, 20, Justi borrowed.4m. He agreed to pay the pricipal ad the iterest at 8% compouded semi-aually o Oct. 5, 206. How much will he pay the?,400,000 m 2 j 0.08 i O.D. 4 /5 / M.D. 7 /5 /6 5 years& 6 moths t 5 5 (2) ?.08,400,000( 0 ) 2 2,55, S.Y.Ta Compoud Iterest

29 CONTINUOUS COMOUNDING Iterest may be coverted very frequetly like weekly, daily or hourly. Let us observe the value of 000 after year at omial rate of 5% at differet frequecies of coversio m. 000 j 0.05 t year m = i icrease aually semi-aually / quarterly / mothly / weekly / daily / hourly / requet compoudig will oly icrease iterest eared very slightly. Thus whe iterest is beig compouded very frequetly we say it is beig compouded cotiuously. S.Y.Ta Compoud Iterest

30 Whe iterest is beig compouded cotiuously, j t we use e as accumulatio factor istead of ( i). That is, Ad cosequetly, So 000 e 0.05() j e e t j t m = i icrease daily / hourly / CONTINUOUSLY S.Y.Ta Compoud Iterest

31 Ex How much should be ivested ow i order to have 50,000 i 3 ¼ years if it is ivested at 6 2/3 % compouded cotiuously? 50,000 t yrs j 6 % 3 % ,000 e 40, Ex 2 How much is the accumulated value of 93,450 after 5 years if it ears 2.25% compouded cotiuously? 93, 450 t 5 yrs cotiuously j (0.0225)(5) 93, 450 e 04, , S.Y.Ta Compoud Iterest