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Compound InterestInterest earned by the principal is added to the principal and will also earn interest for the succeeding periods.Interest periodBeginning PrincipalInterest EarnedAmount at the end of the period1PPiP + Pi = P(1+i)2P(1+i)P(1+i)iP(1+i) + P(i+1)i = P(1+i)23P(1+i)2P(1+i)2iP(1+i)2 + P(1+i)2i = P(1+i)3……………………nP(1+i)n--1P(1+i)n-1i P(1+i)n—1 + P(1+i)n-1i = P(1+i)n2273131165735F = P(1+i)n00F = P(1+i)nWhere: F = total amount due after n periodsP = principali = rate of interest expressed in decimal formn = no. of interest periods(1+i)n -“single payment compound amount factor” -symbolized by (F/P,i%,n) -read as “F given P at i percent in n interest periods”Solving for P:2273417102124P = F(1+i)-n00P = F(1+i)-n(1+i)-n –“single payment present worth factor”-symbolized by (P/F, i%,n)-read as “P given F at i percent in n interest periods”Using the notations, we can write the formulas as:227341674947F = P(F/P, i%, n)00F = P(F/P, i%, n) 227457056515P = F(P/F, i%, n)00P = F(P/F, i%, n)Rates of interest:a. Nominal rate of interest (r)-The usual quoted rate of interest for compound interest problems-specifies the rate of interest and the number of interest periods per year221469592751i = r/m00i = r/mWhere: i = interest rate per periodr = nominal interest ratem = no. of compounding periods per yearex. 8% compounded quarterly -means 4 interest periods per year, the rate of interest per period being 8% / 4 = 2%b. Effective rate of interest-the actual rate of interest on the principal for one year-equal to the nominal rate of interest if it is compounded annually, but greater than the nominal rate if the number of interest periods per year exceeds one.Illustration:Imagine P 1.00 invested at nominal rate of 8% compounded quarterly, after one year…F = P(1+i)nWhere: P = 1, I = 0.08/4, n = 4F = (1)(1+.08/4)4 = P 1.0824After 1 year, actual interest is,I = F – 1 = 1.0824 – 1 = 0.0824Therefore the effective rate of interest is 8.24%General Formula for Effective Rate of Interest (E. R. I.):239925132636E. R. I. = F1 - 1 E. R. I. = (1+i)m - 100E. R. I. = F1 - 1 E. R. I. = (1+i)m - 1Ex. If the sum of P 12,000.00 is deposited in an account earning interest at the rate of 9% compounded quarterly, what will it become at the end of 8 years?Sol’n:F = P (1+i)nF = (12,000)(1+0.09/4)8x4 F = P 24, 457.24 Ex. A man possesses a promissory note, due 3 years hence, whose maturity value is P 6, 700.48. If the rate of interest is 10% compounded semi-annually, what is the value of this note now?Sol’n: P = ?; F= P6, 700.48; n = 3 x 2 = 6; i = 0.10/2 = 0.05P = F(1+i)-n = 6, 700.48(1+0.05)-6 P = P 5, 000.00 Exercise: If you are investing your money which is better:12 % compounded monthly or 12.5 % compounded annually? ................
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