4-MATH158-MODULE-2-COMPOUND-INTEREST

COMPOUND INTEREST MATH 158 MODULE 2 Mapua University – Department of Mathematics

At the end of the lesson, the students are expected to  Define the Compound Interest;  Identify the conversion period ;  Determine the periodic rate;  Determine the total number of period of transactions Mapua University – Department of Mathematics

Compound Interest is an amount computed every conversion period whose principal amount includes the specified interest earned every end of the conversion date. When an interest is compounded more than once a year, the interest rate is nominal otherwise, it is considered as effective. Mapua University – Department of Mathematics

Compound interest ( I ) – an amount resulting from the periodic period addition of simple interest to the principal amount. Compound amount ( F ) – an accumulated amount composed of the principal amount and the compound interest. Compounding or conversion period ( m ) – a number of times in a year that the interest is being compounded. Mapua University – Department of Mathematics

Basis of one year conversion Annually m=1 Monthly m=12 Bimonthly m=6 Semi-annually m=2 Weekly m=52 Quarterly m=4 Daily m=360 Mapua University – Department of Mathematics

Total number of conversion periods ( n ) in a given number of years is given by: n =t ( m ) Periodic rate ( i ) is given by: ������ ������ = ������ Note: If the conversion period is not indicated in the problem then it is assumed to be compounded or converted annually. Mapua University – Department of Mathematics

1. Determine the amount and interest if Php7500 is invested at 12% compounded quarterly for 1 year. Show how the interest is compounded. Given: P = Php7500 r=12%= 0.12, i=12/4=3%=0.03 t= 1 year m= 4 Mapua University – Department of Mathematics

1st 2nd 3rd 4TH 3 months 3 months 3 months 3 months P = Php7500 P = Php7725 P = Php7956.75 P= Php8195.4525 I = Pi I = Pi = 7500(0.03) = 7725(0.03) I = Pi I = Pi= = 7956.75(0.03) 8195.4525(0.03) I = Php225 I = Php231.75 I = Php245.8636 I = Php238.7025 F=P+I F=P+I = 7500+225 = 7725+231.75 F=P+I F= P + I= = 7956.75+238.70 8195.45+245.863 F = Php7725 F = Php7956.75 6 F = Php8195.4525 F = Php8441.3161 F = Php8441.32 I = Php941.32 Mapua University – Department of Mathematics

1st m 2nd m 3rd m Nth m F=P(1+i)n P=P P = P(1+i) P = P(1+i)2 I=Pi I = Pi I = Pi I = P(1+i)i I = P(1+i)2i F=P+I F = P +I F = P +I = P +Pi = P(1+i) + P(1+i)i = P(1+i)2 + P(1+i)2i F=P(1+i) F = P(1+i)(1+i) = P(1+i)2(1+i) F=P(1+i)2 F=P(1+i)3 Mapua University – Department of Mathematics

Final Value ������ = ������ ������ + ������ ������ Present Value Interest Rate ������ = ������ ������ + ������ −������ Period of Transactions ������ ������ = ������ ������ − ������ ������ ������ ������ = ������������������(������������ ) ������������������(������ + ������) ������ = ������������������(������������ ) ������������������������(������ + ������) Mapua University – Department of Mathematics

2. Find the compound amount and interest on Php1,074, 500 for 8 years and 8 months at 16% compounded quarterly. Given: ������������������������: P = Php1,074,500 I, F m =4 t = 8 + 8 = 8 + 2 = 26 12 3 3 r = 16% = 0.16 Mapua University – Department of Mathematics

n =mt= 4(236)= 104 The compound amount 3 is Php4,185,000.51 and the interest is i=������������ = 16 = 4% = 0.04 Php3,110,500.51 4 F=P(1+i)n = 1, 074,500 1 + 0.04 (1034) F= Php4,185,000.51 I =F-P= 4185000.51-1074500 I=Php3,110,500.51 Mapua University – Department of Mathematics

3. Find the present value of Php55000 due 5 years if money is invested at 5.55% compounded monthly. Given: ������������������������: F = Php55000 P t = 5 years r = 5.55% m = 12 ������ = ������������ =(12)(5)=60 ������ 5.55 0.0555 ������ = ������ = 12 = 12 Mapua University – Department of Mathematics

������ = ������ 1 + ������ −������ = 55000 1 + 0.0555 (−60) 12 P = Php41698.82 The present value is Php41,698.82 Mapua University – Department of Mathematics

4. A piece of property is purchased on installment. The buyer makes a Php750, 000 down payment and owes a balance of Php1.5M to be paid in 9 years. Find the property’s cash value if money is worth 12% compounded monthly. Given: ������������������������: DP=Php750000 Cash Value F=Php1.5M t=9 years r=12% m=12 Mapua University – Department of Mathematics

n = mt = (12)(9)= 108 12% ������ = 12 = 1% = 0.01 Cash Value= Down Payment + Present Value CV = DP + F(1+i)-n = 750000 + 1,500,000( 1+ 0.01)(-108) CV = 750000+512.133.1526 CV =Php1,262,133.15 The Cash Value of the property is Php1,262,133.15 Mapua University – Department of Mathematics

5. How many payments will you pay for a loan of Php75000 at 15% compounded every four months if the interest paid is Php8400? Given: ������������������������: P =Php75000 n I= Php8400 r=15% m= 3 i=������������ = 15% = 5% = 0.05 3 Mapua University – Department of Mathematics

������ = log(������������) = log(7853040000) = 2.17 log(1 + ������) log(1 + 0.05) n = 2 payments There are two payments to pay for a loan of Php75000 Mapua University – Department of Mathematics

6. How long will it take Php160, 000 to mature t0 Php170, 000 if the interest is 12% compound monthly ? If the term starts today, Sept 16, 2020 when will it end? Given: ������������������������: P=Php160000 t F= Php170000 r = 12% m = 12 ������ 12% ������ = ������ = 12 = 1% = 0.01 Mapua University – Department of Mathematics

������ = log(������������) = log 170000 mlog(1 + 160000 ������) 12log(1 + .01) t= 0.51 year = 6 months and 3 days Therefore , it will take 6 months and 3 days to accumulate Php160000 to Php170000 Mapua University – Department of Mathematics

If the transaction start on Sept 16,2020 Sept 16, 2020 Actual time Oct 14 Nov 31 Dec 30 Jan, 2021 31 Feb 31 March 28 total 18 183days Mapua University – Department of Mathematics

If the transaction start on Sept 16,2020 Then using t = 183 days, and actual time The transaction will end on March 18,2021 Mapua University – Department of Mathematics

 Compound Interest is an amount computed every conversion period whose principal amount includes the specified interest earned every end of the conversion date.  Compound interest ( I ) – an amount resulting from the periodic period addition of simple interest to the principal amount.  Compound amount ( F ) – an accumulated amount composed of the principal amount and the compound interest. Mapua University – Department of Mathematics

 Compounding or conversion period ( m ) – a number of times in a year that the interest is being compounded Mapua University – Department of Mathematics

 Wiley Pathways Business Math, by Slavin, 1st ed.  Mathematics of Investment by William L. Hart, 5th ed.  Business Mathematics by Norma Lopez-Mariano, 2016 ed.  Investment Mathematics by Win Ballada, 2016 ed.  Mathematics of Investment Made Simple by Felina C. Young  Business Mathematics Comprehensive Approach by Altares et al, Mapua University – Department of Mathematics