SCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT BUSINESS MATHEMATICS / MATHEMATICAL ANALYSIS

Transcription

1 SCHOOL OF BUSINESS, ECONOMICS AND MANAGEMENT BUSINESS MATHEMATICS / MATHEMATICAL ANALYSIS Unit Five Moses Mwale moses.mwale@ictar.ac.zm

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3 BBA 120 Business Mathematics Contents Unit 5: Mathematics of Finance Simple and Compound Interest Calculations Simple interest... 3 Example... 3 Practice Exercise Compound Interest... 4 Example... 5 Example When interest is compounded several times per year... 6 Example Annual percentage rate (APR) or Effective rate... 8 Example... 8 Practice Problems... 9

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5 BBA 120 Business Mathematics Unit 5: Mathematics of Finance 5.0 Simple and Compound Interest Calculations Simple interest Example Today, businesses and individuals are faced with a bewildering array of loan facilities and investment opportunities. In this section we explain how these financial calculations are carried out to enable an informed choice to be made between the various possibilities available. We begin by considering what happens when a single lump sum is invested and show how to calculate the amount accumulated over a period of time. Simple interest is a fixed percentage of the principal, P 0, that is paid to an investor each year, irrespective of the number of years the principal has been left on deposit; that is, money invested at simple interest will increase in value by the same amount each year. So, if the investor is paid a fixed annual amount, i% of P 0, then the amount of simple interest, I, received over t years is given by the formula I = P 0 i t Therefore, the total value after t years, P t is the principal plus interest and is given by P t = P 0 + (P 0 i t) P t = P 0 (1 + it) When the total value (future value), the interest rate and time are known, the principal (present value) may be calculated by rewriting formula as P 0 (1 + it) = P t P 0 = P t (1 + it) An investment of $3,000 is made at an annual simple interest rate of 5%. How much additional money must be invested at an annual simple interest rate of 9% so that the total annual interest earned is 7.5% of the total investment? Solution I P i t 3

6 4 Unit 5: Mathematics of Finance Practice Exercise first (3,000)(0.05) = 150 3, additional 0.09 x x total (3,000 + x)(0.075) 3,000 + x First I fill in the P, i, and t columns with the given values. Then I multiply across the rows (from the right to the left) in order to fill in the I column. Then add down the I column to get the equation x = (3,000 + x)(0.075). To find the solution, I would solve for the value of x. 1. How long does it take a principal of $25,000 at a simple interest rate of 5% to become $30,000? 2. $45,000 is deposited into a savings account. After one year, 4 months and 20 days it totals $52,500. Calculate the simple interest rate for this account. 3. Determine the simple interest rate applied to a principal over 20 years if the total interest paid equals the borrowed principal. 4. How long does it take a principal payment to triple at a simple interst rate of 6%? 5. Find the total amount of simple interest that is paid over a perod of five years on a principal of $ 30,000 at a simple interest rate of 6%. 6. Calculate the total worth of an investment after six months with a principal of $10,000 at a simple interest rate of 3.5% Compound Interest Under compound interest, at the end of each interest period, the interest earned for that period is added to the principal (the invested amount) so that it too earns interest over the next interest period. The basic formula for the value (or compound amount) of an investment after n interest periods' under compound interest is as follows: P t = P 0 (1 + i) t

7 BBA 120 Business Mathematics The compound amount is also called the accumulated amount, and the difference between the compound amount and the original principal, P t P 0, is called the compound interest. Example Solution Example Solution Find the value, in 4 years time, of $ invested at 5% interest compounded annually. In this problem, P 0 = , i = 5% and t = 4, so the formula P t = P 0 (1 + i) t gives P t = 10000( ) 4 = 10000(1.05) 4 = The compound interest formula derived above involves four variables, r, t, P t and P 0. Provided that we know any three of these, we can use the formula to determine the remaining variable. This is illustrated in the following example. A principal of $ is invested at 12% interest compounded annually. After how many years will the investment first exceed $ ? We want to save a total of $ starting with an initial investment of $ The problem is to determine the number of years required for this on the assumption that the interest is fixed at 12% throughout this time. The formula for compound interest is P t = P 0 (1 + i) t We are given that 5

8 6 Unit 5: Mathematics of Finance P 0 = , P t = , i = 12 so we need to solve the equation = ( ) t for n. One way of doing this would just be to keep on guessing values of n until we find the one that works. However, a more mathematical approach is to use logarithms, because we are being asked to solve an equation in which the unknown occurs as a power. Following the method described in Section 2.3, we first divide both sides by to get 10 = (1.12) n Taking logarithms of both sides gives log(10) = log(1.12) n and if you apply rule 3 of logarithms you get Hence log(10) = n log(1.12) n = = log(10) log(1.12) = 20.3 Now we know that n must be a whole number because interest is only added on at the end of each year. We assume that the first interest payment occurs exactly 12 months after the initial investment and every 12 months thereafter. The answer, 20.3, tells us that after only 20 years the amount is less than $ , so we need to wait until 21 years have elapsed before it exceeds this amount When interest is compounded several times per year So far, it has been assumed that compound interest is compounded once a year. In reality, interest may be compounded several times per year, for example it may be compounded daily, weekly, monthly, quarterly, semiannually or continuously. Each time period is known as a conversion period or interest period. The number of conversion periods per year is denoted by the symbol, m; the interest rate applied at each conversion is i m. For example, an investment compounded twelve times per year will have twelve conversion periods; therefore if a five-year investment was compounded twelve times annually, then the investment would have sixty conversion periods; that is,

9 BBA 120 Business Mathematics Example n = m t where n = total number of conversion periods m = conversion periods per year t = number of years The value of the investment at the end of n conversion periods is P t = P 0 (1 + i n m ) = P 0 (1 + i m t m ) 5000 is invested for three years at 8% per annum compounded semiannually. Solution a) Calculate the total value of the investment. b) Compare the return on the investment when interest is compounded annually to that when compounded semi-annually. c) Calculate the total value of the investment when compounded, (i) monthly, (ii) daily. Assume all months consist of 365/12 days. a) P 0 = 5000, i = 0.08, t = 3 and m = 2. Using Compound interest formula, the total value after three years with n = m x t = 6 conversion periods is calculated as P 3 = 5000 ( ) (2)(3) = 5000( ) 6 = b) The total value of 5000 after three years compounded annually at 8% interest is P t = 5000( ) 3 = When the same investment is compounded semi-annually, the total value is , a gain of over the value when compounded annually. 7

10 8 Unit 5: Mathematics of Finance c) The value of the investment at the end of three years for monthly and daily compounding is calculated as i) Monthly compounding P 3 = P 0 (1 + i m ) mt = 5000 ( ) (12)(3) = 5000(1.2702) = ii) Daily compounding P 3 = P 0 (1 + i m ) mt = 5000 ( ) (365)(3) = 5000(1.2712) = Annual percentage rate (APR) or Effective rate Example Interest rates are usually cited as nominal rates of interest expressed as per annum figures. However, as compounding may occur several times during the year with the nominal rate, the amount owed or accumulated will be different from that calculated by compounding once a year. So, a standard measure is needed to compare the amount earned (or owed) at quoted nominal rates of interest when compounding is carried out several times per year. This standard measure is called the annual percentage rate (APR) or effective annual rate. If interest is compounded once a year at the APR rate, the investment would yield exactly the same return, P t, at the end of t years, as it would if interest were compounded m times per year at the nominal rate. The formula for the annual percentage rate (APR) is given as follows: APR = (1 + i m ) m 1 1. What effective rate is equivalent to a nominal rate of 6% compounded (a) semi-annually and (b) quarterly? Solution a) APR = ( )2 1 = (1.03) 2 1 = = 6.09% b) APR = ( )4 1 = (1.015) 4 = = 6.14% This example illustrates that, for a given nominal rate i, the effective rate increases as the number of interest periods per year (n) increases.

11 BBA 120 Business Mathematics 2. If an investor has a choice of investing money at 6% compounded daily or Solution 6 1 % quarterly, which is the better choice? 8 We determine the equivalent effective rate of interest for each nominal rate and then compare our results. The respective effective rates of interest are And APR = ( ) % APR = ( ) % 4 Since the second choice gives the higher effective rate, it is the better choice (in spite of the fact that daily compounding may be psychologically more appealing). Practice Problems 1. A bank offers a return of 7% interest compounded annually. Find the future value of a principal of $4500 after 6 years. What is the overall percentage rise over this period? 2. Find the future value of $ in 2 years time if compounded quarterly at 8% interest. 3. Midwest Bank offers a return of 5% compounded annually for each and every year. The rival BFB offers a return of 3% for the first year and 7% in the second and subsequent years (both compounded annually). Which bank would you choose to invest in if you decided to invest a principal for (a) 2 years; (b) 3 years? 4. The value of an asset, currently priced at $ , is expected to increase by 20% a year. (a) Find its value in 10 years time. (b) After how many years will it be worth $1 million? 5. How long will it take for a sum of money to double if it is invested at 5% interest compounded annually? 6. Find the future value of $100 compounded continuously at an annual rate of 6% for 12 years. 7. How long will it take for a sum of money to triple in value if invested at an annual rate of 3% compounded continuously? 8. If a piece of machinery depreciates continuously at an annual rate of 4%, how many years will it take for the value of the machinery to halve? 9. A department store has its own credit card facilities, for which it charges interest at a rate of 2% each month. Explain briefly why this 9

12 10 is not the same as an annual rate of 24%. What is the annual percentage rate? 10. Determine the APR if the nominal rate is 7% compounded continuously. 11. Current annual consumption of energy is 78 billion units and this is expected to rise at a fixed rate of 5.8% each year. The capacity of the industry to supply energy is currently 104 billion units. (a) Assuming that the supply remains steady, after how many years will demand exceed supply? (b) What constant rate of growth of energy production would be needed to satisfy demand for the next 50 years?