QMT 3301 BUSINESS MATHEMATICS REV 00 CHAPTER 9 COMPOUND INTEREST QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS 9.1 Time Value of Money REV 00 Money has time value, that is a ringgit today is worth more than a ringgit tomorrow. Money has time value because of its investment opportunities. QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS 9.2 Compound Interest REV 00 In compounding, after the interest is calculated, it is then added to the principal and becomes an adjusted principal. Processes are repeated until the end of the loan or investment term. Normally used with long-term loan or investment, and the interest is calculated more than once during the loan or investment term. QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS REV 00 The interest earned is called compound interest, and the final sum at the end of the period of borrowing is called the compound amount. Therefore, compound interest is the difference between the original principal and the compound amount. QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS 9.3 Some Important Terms REV 00 Some of the common terms used in relation to compound interest are: 1. Original principal 2. Nominal interest rate 3. Interest period or conversion period 4. Frequency of conversions 5. Periodic interest rate 6. Number of interest periods in the investment period QMT 3301 BUSINESS MATHEMATICS

9.4 Compound Interest Formula REV 00 The method used in finding compound amount at the end of the nth period is as follows: S = P(1 + i)n Where: P = Principal / Present Value S = Future Value n = Number of Periods (number of years multiplied by number of times compounded per year) i = Interest rate per compound period QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS Example 1: REV 00 Find the future value of RM 1000 which was invested for a) 4 years at 4% compounded annually, b) 250 days at 10% compounded daily, and c) 2 years 3 months at 4% compounded quarterly. QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS REV 00 Solution: a) S = 1000(1 + 0.04)4 = RM 1169.86 b) S = 1000 1 + 0.1 250 = RM 1071.90 360 c) S = 1000 1 + 0.04 9 = RM 1093.69 4 QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS Example 2: REV 00 What is the nominal rate compounded monthly that will make RM 1000 become RM 2000 in five years? Solution: From S = P(1 + i)n, we get 2000 = 1000 1 + k 60 12 2 = 1 + k 60 k = 13.94% QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS Example 3: REV 00 How long does it take a sum of money to double itself at 14% compounded annually? Solution: Let the original principal = W Therefore sum after n years = 2W From S = P(1 + i)n, we get 2W = W(1 + 0.14)n 2 = (1 + 0.14)n lg 2 = n lg 1.14 n = 5.29 years QMT 3301 BUSINESS MATHEMATICS

9.5 Effective, Nominal and Equivalent Rates REV 00 Effective rate: Simple rate that will produce the same accumulated amount as the nominal rate that compounded each period after one year. Nominal rate: Stated annual interest rate at which interest is compounding more than once a year. Equivalent rate: Two different rates that yield the same value at the end of one year. QMT 3301 BUSINESS MATHEMATICS

9.6 Relationship Between Effective and Nominal Rates REV 00 9.6 Relationship Between Effective and Nominal Rates The relationship between the nominal rate and effective rate is derived as follows: Assume a sum RM P is invested for one year. Then the future value after one year: a) At r% effective = P(1 + r) b) At k% compounded m times a year = P(1 + k/m)m QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS Example 1: REV 00 Find the effective rate which is equivalent to 16% compounded semi-annually. Solution: QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS Example 2: REV 00 Find the nominal rate, compounded monthly which is equivalent to 9% effective rate. Solution: QMT 3301 BUSINESS MATHEMATICS

9.7 Relationship Between Two Nominal Rates REV 00 The relationship between two nominal rates is given as follows: (1 + k/m)m = (1 + K/M)M Where: k and K are two different annual rates with respectively two different frequencies of conversions, m and M. QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS Example : REV 00 Find k% compounded quarterly which is equivalent to 6% compounded monthly. Solution: QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS 9.8 Present Value REV 00 Present value or discounted value is the value which will yield the sum (S) after certain time and at a specific interest rate. We can find present value by transposing the formula as follows: S = P(1 + i)n P = S (1 + i)n OR P = S(1 + i)-n QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS REV 00 Example : A debt of RM 3000 will mature in three years’ time. Find a) The present value of this debt, b) The value of this debt at the end of the first year, Assuming money is worth 14% compounded semi annually. QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS REV 00 Solution: a) From P = S(1 + i)-n, we get P = 3000 1 + 0.14 -6 2 = RM 1999.03 b) P = S(1 + i)-n P = 3000 1 + 0.14 -4 = RM 2288.69 QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS 9.9 Equation of Value REV 00 An equation that expresses the equivalence of two sets of obligations at a focal date. In other words, it expresses the following: What is owed = What is owned at the focal date OR What is given = What is received at the focal date QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS REV 00 Example : A debt of RM 7000 matures at the end of the second year and another of RM 8000 at the end of six years. If the debtor wishes to pay his debts by making two equal payments at the end of the fourth year and the seventh year, what are these payments assuming money is worth 6% compounded semi-annually? QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS REV 00 Solution: 0 1 2 3 4 5 6 7 years 7000 X 8000 X Focal date QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS REV 00 Let the payment be RM X each. Formulating the equation of value at the focal date as shown, we get What is owed = What is owned X(1 + 3%)6 + X = 7000(1 + 3%)10 + 8000(1 + 3%)2 X = RM 8155.97 QMT 3301 BUSINESS MATHEMATICS

9.10 Continuous Compounding REV 00 We have been discussing compounding of interest on discrete time intervals (daily, monthly, etc). If compounding of interest is done on a continuous basis, then we will have a different picture of the future value as shown below: Discrete compounding Continuous compounding Future value Future value Time QMT 3301 BUSINESS MATHEMATICS Time

QMT 3301 BUSINESS MATHEMATICS REV 00 The future value of a sum of money compounded continuously is given by: S = Peit Where: S = Future value P = Original principal e = 2.718282… i = Continuous compounding rate t = Time in years QMT 3301 BUSINESS MATHEMATICS

QMT 3301 BUSINESS MATHEMATICS Example : REV 00 Find the accumulated value of RM 1000 for six months at 10% compounded continuously. Solution: From S = Peit, we get S = 1000[e10% x 0.5] = RM 1051.27 QMT 3301 BUSINESS MATHEMATICS