PRINCIPLES OF MONEY AND TIME RELATIONSHIPS INTEREST
– the amount of the money paid for the use of borrowed capital.
Simple Interest – the interest paid that is directly proportional to the length of time the amount or principal is borrowed. Principal – the amount of money borrowed and on which interest is charged. Rate of Interest – the amount earned by one unit of principal during a unit of time. I = Pin Where: I – total interest earned by the principal, P – amount of the principal I - rate of interest expressed in decimal form, n – number of interest periods F = P + I = P (1 + in) Where: F - total amount to be repaid . Ordinary simple interest – interest – computed on the basis of one banker’s year, or 1 banker’s year = 12 months, each consisting of 30 days = 360 days. Exact Compound Interest – Interest – based on the exact number of days, 365 for an ordinary year and 366 days for a leap year. Ordinary simple interest = Pi (d/360) Exact simple interest = Pi (d/365) – For ordinary year = Pi (d/360) – For leap year Compound Interest – the interest earned by the principal if not paid at the end of each interest period (considered as added to the principal that will also earn interest for the succeeding periods). F = P (1 + I) n, where the factor (1 + n) n = (F/P, i%, n) or Single Payment Compound Amount Factor (SPCAF). Derivation Interest Period 1 2 n
Principal at start of period P P (1 + I) P (1 + I) exp(n-1)
Interest earned during period Pi P (1 + I) I P (1 + I) exp (n-1) I
Compound amount at end of the period
P (1+I) exp (n-1) + P (1+I) exp (n-1) = P (1+I) exp n = F
Continuous Compounding F = P (1 + r/m) mn Where: r – nominal annual interest rate, m – number of interest period each year i = r/m – interest rate per interest period, m – number of interest periods in n years.
Nominal Rate of Interest – the rate of interest in compound interest that specifies the rate of interest and the number of interest periods per year. Effective Rate of Interest – the actual rate of interest on the principal for one year. Effective rate of interest(ERI) = (1+I) n -1 Present Value ( P) P) – the value of the compound amount F at present, or the present value of the amount F, or the amount which when invested now will become F after n periods. P = F (1+I) –n = F/(1+I) n where the factor (1+I) –n is the “Single Payment Present Worth Factor” or SPPWF = (P/F, i%, n). Discount (d) – the difference between what is worth in the future and its present worth. Rate of discount – the discount on one unit of principal per unit time. d = 1 – 1/(1+I) = 1 – (P/F, i%, 1) or d = i/(1+i) = (P/F, i%, 1)I Equivalent rate of interest corresponding to the rate of interest I, I = d/(1-d) = d/(P/F, i%, 1) Problems: 1. Determine Determine the exact exact and ordinary ordinary simple simple interests interests on P1200 for for the period from from January January 16 to November November 26, 1992, 1992, if the rate of interest is 24%. 2. A man borrows borrows P10,000 P10,000 from from a loan firm. firm. The rate of simple simple interest interest is 15%, 15%, but the interest interest to be be deducted from from the loan at at the time the money is borrowed. At the end of one year he has to pay back P10,000. What is the actual rate of interest? 3. A man borrows borrows P6400 P6400 from a loan loan association. association. In repaying repaying this debt debt he has to pay pay P400 at the end end of every 3 months months on the principal and a simple interest of 16% on the principal outstanding at that time. Determine the total amount he has paid after paying all his debt. 4. A man possesses possesses a promissory promissory note, note, due 3 years hence, hence, whose whose maturity maturity value is P6,700.4 P6,700.48. 8. If the rate of of interest interest is 10% compounded semi-annually, what is the value of this note now? 5. If you are investing investing your your money which which is better: better: 12% compou compounded nded monthly monthly or 12.5% compou compounded nded annually? annually? 6. An advertisem advertisement ent of an investment investment firm firm states states that if you invest invest P500 P500 in their firm firm today you you will get P1000 P1000 at the end of of 4 ½ years. What nominal rate is implied if interest is compounded quarterly?
References: Eng’g Economy by M. Arreola, Eng’g Economy by E. Paul De Garmo, Eng’g Economics by C. S. Park
