College Math

©2018 of 120 2.8 Continuous Interest In order to make the offer lucrative, suppose the bank offers you an interest rate which is compounded infinite times; this is termed continuous compounding . It should be noted that as the number of compounding period increases, the compound amount also increases, but the proportion of increase falls continuously. Hence, P(1+r/n)^nt becomes closer and equivalent to Pe^rt. ‘e’ is an irrational number and bears a value of 2.718. Therefore, the formula for continuous compound interest is: = A = Compound amount P = Principal Sum r = Rate of interest t = Number of years For instance, what will be the compounded amount and interest earned when $3500 is invested at the rate of 6.25% compounded continuously for 7.5 years? Here, r = 0.0625, t = 7.5 and P =3500. Substituting the values in the formula, we get = 3500 ∗ 0.0625∗7.5 A = $5592.712 The interest would be 5592.712-3500 = $2092.712 2.9 Effective Interest Rate (Effective Annual Rate of Interest or EAR) The concept of effective interest rate is generally used when the compounding is not done annually but for different time periods, say semi-annually, quarterly, or monthly. In that case, the effective rate of interest calculated at the end of a year is different from the rate which is charged annually. For instance, suppose $100 is invested at 5% for one year compounded semi-annually, then the effective interest earned would be at the rate of 5.06%. This is because when the sum is invested for one year (with annual compounding), the interest becomes $5, but when it is compounded semi-annually, the interest earned would be $5.0625. In this case, 5.06% is termed as the effective interest while 5% is known as nominal interest rate. The effective rate of interest can be calculated by using the following formula: = �1 + � − 1 Achieve Page 38