Compoud Iterest S.Y.Ta Compoud Iterest
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 = - = 3200-2000 = 200 I =200 =2000 3200 = 0 yr 600 600 =2000 3200 = 0 6 moths yr 300 300 300 300 =2000 3200 = 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
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
So iterest may be compouded or coverted (i) Aually (oce a year / every year) m = 0 2 3 4 (years) (ii) Semi-aually (twice a year / every 6 moths) m = 2 0 2 3 4 ( semi-aual periods) 0 6 moths yr /2 yrs 2 yrs (iii) Quarterly (4 times a year / every 3 moths) m = 4 0 2 3 4 (quarters) 0 3 moths 6 moths 9 moths yr S.Y.Ta Compoud Iterest
(iv) Mothly (2 times year / every moth) m = 2 0 3 6 9 2 (moths) 0 year (v) Every 4 moths (3 times year ) m = 3 0 2 3 (periods) 0 4 moths 8 moths year (vi) Every 2 moths (6 times a year) m = 6 0 2 3 4 5 6 (periods) 0 2 mths 4 mths 6 mths 8 mths 0 mths year S.Y.Ta Compoud Iterest
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
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 = 0 0 2 3 4 I = 0 i = i I I 2 I 3 I 4 I 2 3 4.. = 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.. - 3 = 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
=2000 j = 0% = 0.0 (compouded quarterly) m = 4 i =j/m = 0.0/4 = 0.025 t = year = (4) = 4 = 2000 (+ 0.025) 4 = 3245.75 (compoud amout) I = - = 3245.75-2000 = 245.75 (compoud iterest) =2000 300 300 300 300 3200 = 0 2 3 4 (quarters) 0 3 moths 6 moths 9 moths year ial amout uder simple iterest =2000 300 307.50 35.875 323.067875 3245.75 = ial amout uder compoud iterest 0 2 3 4 (quarters) 0 3 moths 6 moths 9 moths year I =2000(0.025) = 300 = + I = 2000 + 300 =2300 2 = + I 2 = 2300 + 307.50 =2607.50 I 2 =2300(0.025) = 307.50 I 3 =2607.50(0.025) = 35.875 3 = 2 + I 3 = 2607.50 + 35.875 =2922.6875 I 4 =2922.6875(0.025) = 323.067875 = 4 = 3 + I 4 =2922.6875 + 323.067875 = 3245.75469 S.Y.Ta Compoud Iterest
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
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
= ( + 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) = 80000 ( +0.05) 2 = 222,877.0 S.Y.Ta Compoud Iterest
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 = 37 300 m = 2 i I 37 300 2 37 600 = (6)(2)=2 37 2 ( i) 80000( 600) 64,039.40 64039.40 80000 84,039.40 i 37 300 b) m = 2 37 2 3600 = (6)(2) = 72 I ( i) 80000( 37 3600 ) 72 67042.74 80000 67,042.74 87,042.74 % S.Y.Ta Compoud Iterest
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
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% = 0.0402 i = 0.0402/2 =0.020 t = 3 years = 3(2)= 6 =? = ( + i) - = 500000 ( +0.020) - 6 = 443,724.6 S.Y.Ta Compoud Iterest
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 = 0.0075 t = 5 moths = 5/2 years =(5/2)(2)=5 D = 5000 =? CV =? = ( + i) - = 50000 ( +0.0075) - 5 = 44,698.63 CV = D + CV = 5000 + 44698.63 CV = 49,698.63 S.Y.Ta Compoud Iterest
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? = 20000 t = 7 years m = 6 = 7(6) = 42 i =0.03/6 =0.005 As: = 20000 (+0.005) 42 =24,660.65 I = 4660.65 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: = 500000 (+0.2) - 2 = 398,596.94 CV = C = 50,000 + 398,596.94 = 548,596.94 S.Y.Ta Compoud Iterest
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) 0.05 36 75,000 0.05 36 2 87,0.42 Do this if i is ot a termiatig decimal S.Y.Ta Compoud Iterest
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 5 5 3 23 2 4 4 yrs m 4 4 (4) 9 23 23 j 0.07 i 4 0.07 0.075 23 50,000( 0 ).07 4 223,554.22 S.Y.Ta Compoud Iterest
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
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? 45000 2 ( i) 2 2 45,000 30,000( i) 30000 45000 2 i i 0. 034366083 30000 j i(4) 0.37464332 0.375 3.75% S.Y.Ta Compoud Iterest
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) 3 5 5 2 4 4 4 i? 5 42000 5 ( i) 5 5 42,000 35,000( i) 35000 42000 5 35000 i i 0.0033758372 9 j i(2) 0.04050047 0.0405 4.05% S.Y.Ta Compoud Iterest
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)? 56,47.27 9 t 4 6 2 4 2 9 2 yrs 5647.27 9 ( i) 9 9 56,47.27 25,000( i) 25000 5647.27 9 25000 i i 0. 094764833 j i(2) 0.89529667 0.895 8.95% S.Y.Ta Compoud Iterest
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 /6 6 6 6 3 yrs (4) 26 m j 4 i(4) i? 4(30000) 26 4 ( i) 26 26 4(30,000) 30,000( i) 30000 4 26 j to i i 0. 054766076 i(4) 6 / 30 /6 to 2 / 30 /6 6 6 yrs moths 0.29064305 t 2 2 2 2 0.29 2.9% S.Y.Ta Compoud Iterest
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 3 2 8 log2 2 7) b log b x x e l x x EX. start with 8 log 2 log2 8 3 8 3 2 2 8 S.Y.Ta Compoud Iterest
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 6 0.2 0.02 t 6? 58,000 50,000( log 58000 50000 t log(.02) 6 ) log 58000 log 0.02 log(.02) 0.2 6 50000.24960889 7.494965335.25 yrs S.Y.Ta Compoud Iterest
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? 0.5 0.05 60,000 50,000( ) log 60000 log 0.05 log(.05) 0.5 3 50000 log 60000 50000 log(.05) 3.736850652 t 3.24566884.25 yrs yr & 3 moths Jue 5, 204 S.Y.Ta Compoud Iterest
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 % 3 2 200 i 200 3 0.065 t? 95,000 80,000( ) log 95000 log 0.065 log(.065) 3 200 80000 log 95000 80000 log(.065) 2.728873442 t 2.73 yrs S.Y.Ta Compoud Iterest
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 2 0.08 0.04 O.D. 4 /5 / M.D. 7 /5 /6 5 years& 6 moths t 5 5 (2) 6 2 2 2 2?.08,400,000( 0 ) 2 2,55,235.68 S.Y.Ta Compoud Iterest
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 0.05 050 2 semi-aually 2 0.05/2 050.625 0.625000 3 quarterly 4 0.05/4 050.945337 0.320337 4 mothly 2 0.05/2 05.6898 0.2656 5 weekly 52 0.05/52 05.245842 0.083944 6 daily 365 0.05/365 05.267496 0.02655 7 hourly 8760 0.05/8760 05.270946 0.003450 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
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 05.27096 m = i icrease daily 365 0.05/365 05.267496 hourly 8760 0.05/8760 05.270946 0.003450 CONTINUOUSLY 05.27096 0.00050 S.Y.Ta Compoud Iterest
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 3 3 2 20 4 4 yrs j 6 % 3 % 3 20 3 300 4 50,000 e 40,259.92 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 (0.0225)(5) 93, 450 e 04, 577.3024 04, 577.30 20 300 S.Y.Ta Compoud Iterest