Financial Mathematics

Transcription

1 Financial Mathematics 1 Introduction In this section we will examine a number of techniques that relate to the world of financial mathematics. Calculations that revolve around interest calculations and differing types of interest; comparing interest rates from different institutions to discover the best deal; and also depreciation calculations that enable us to discover the future worth of an asset as it declines in value over a period of time. Required Knowledge In order to successfully work with the formulae and methods of this section we must ensure that we have a thorough understanding of the following principles: 1. Percentage calculations and conversion to decimals 2. Competence with exponents and rules governing their manipulation 3. Familiarity with the prinicples of logarithms 4. Familiarity with usage of calculators for calculating logarithms and customised exponents 1

2 2 Simple Interest When a person invests money in an account they generally expect to get some return on their investment. We call this return interest. There are two main types: The first is where the interest is calculated periodically and is then withdrawn. This is called Simple Interest, the second also computes interest periodically but the interest is reinvested each time. This is called Compound Interest. First, we will examine Simple Interest. If we invest e1000 for two years at 10% simple interest per annum then after the first year we have earned e100 in interest. Since this e100 is not reinvested we will earn the same amount the second year. Thus, our earnings from interest total e200 over the two years. This can be simplified as follows; Let: P = Principal (amount invested) r = periodic rate (as a decimal) t = no. of periods for which the principal is invested The interest earned is: I = P rt Examples: 1. How much interest is earned if e250 is invested in an account paying 5.75% simple interest annually after 6 years? 2. How much interest is earned if e300 is invested in an account paying 3% simple interest every six months after 5 years? 3. How much interest is earned if e1700 is invested in an account paying 7% simple interest annually, where interest is calculated monthly, after 10 years? The notion of the periodic interest rate is important. A bank will quote an annual interest rate but will frequently calculate interest on a monthly or quarterly basis. Therefore if the rate quoted is 7% and they calculate the interest monthly then the periodic interest rate is 7 12 %. 3 Compound Interest When a bank calculates interest however they do not withdraw the interest at the end of each period; instead they reinvest it. Consequently if we invest e1000 for two years at 10% interest per annum then after the first year we have earned e100 in interest. Since this e100 is now reinvested we will now earn more the second year. In fact 10% of e1100 is e110 and so our total interest is e210, which is more than what we earned in the 2

3 simple interest case. This will leave us with a total investment of e1210. These calculations can be summarized as follows; Let: P = Principal (amount invested) sometimes referred to as the present value r = periodic rate (as a decimal) t = no. of periods for which the principal is invested The amount in the account at the end of the investment, referred to as the future value (F), is: Examples: F = P (1 + r) t 1. How much interest is earned if e250 is invested in an account paying 5.75% compound interest annually after 6 years? 2. How much interest is earned if e300 is invested in an account paying 3% compound interest annually after 5 years? 3. How much interest is earned if e1700 is invested in an account paying 7% compound interest annually after 10 years? Some banks calculate interest every month and others do it every 3 months. Consequently when a bank says it offers 6% interest a year, the amount of interest you earn is dependent on how interest is calculated. Suppose the Bank A pays interest at 7% per annum, but calculate the interest once a year, and that Bank B pays interest at 6.9% but calculate the interest monthly. Which account will yield the most interest after one year? Compound Interest - Compounded multiple times per year 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, semi-annually 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: r m 3

4 For example an investment compounded twelve times per year will have 12 conversion periods; therefore if a 5 year investment was compounded twelve times annually, then the investment would have 60 conversion periods; that is, n = mt where m = total number of conversion periods per year and t = number of years. The compound interest formula is revised to: F = P (1 + r m )mt In practice, we typically calculate the r value and the mt value before applying the previous version of the formula, i.e. F = P (1 + i) t, swapping the newly calculated values for i m and t respectively. Note that we use i to distinguish between period rate and interest rate in this instance. Example 1 e5000 is invested for 3 years at 8% per annum. 1. Calculate the total value of the investment if the interest is compounded annually. 2. Calculate the total value of the investment if the interest is compounded semiannually. 3. Compare the return on the investment when interest is compounded annually to that when compounded semi-annually. 4. Calculate the total value of the investment when compounded (a) monthly (b) daily. Assume all months consist of days. Answer 1 1. F = 5000( )1 3 = e annually 2. F = 5000( )2 3 = e semi-annually 3. The more often it is compounded, the more return on the investment. 4. (a) F = 5000( )12 3 = e (b) F = 5000( ) = e Example 2 e500 is invested for three years at an interest rate of 8% per annum. Calculate the total value of savings at the end of the three years. If the interest is 8% per annum but compounded semi-annually what is the difference in the investment? 4

5 Example 3 Calculate the amount gained on an investment of e1000 over four years at an interest rate of 2.5%p.a. when it is compounded (i) annually (ii) semi-annually (iii) quarterly (iv) monthly. Example 4 How much should you invest to receive e6,000 in five years time if the interest is 7.5% p.a. and compounded semi-annually. Example 5 An investor receives e6,500 after a sum of money is invested for four years when interest is at 3.5% compounded quarterly. Find the sum of money. Example 6 Find the annual compound interest rate required for e10,000 to grow to e20,000 in six years if the interest is compounded monthly. Example 7 Find the annual compound interest rate required for e5,000 to grow to e9,000 in four years if the interest is compounded twice a year. Example 8* A bank pays 7.5% p.a. How long (in years) will it take for e10,000 to grow to e20,000 if the interest paid out quarterly? Example 9* Calculate the number of time periods it will take for a sum of e5,000 to grow to e20,000 when invested at 5.5% p.a. but compounded monthly? * Note: We have looked at many algebraic rules that are used to help solve and manipulate expressions. However, we have not yet discussed solving for an unknown power in exponential expressions. In the starred examples above, we need to find the value of t, the power of the exponent in the formula. To solve this, we need to know how to rephrase exponents so that the unknown can be isolate and calculated. The following log rule applies in such instances: if, x = y z then we can rewrite as, log 10 x = z log 10 y e.g. 100 = 10 n so, log = n log = n 1 thus, n = 2 5

6 4 Annual Percentage Rate - APR Interest rates are usually cited as nominal rates of interest expressed as per annum figures. However, as we saw compounding may occur several times during the year with nominal rate; the amount owed or accumulated will be different from that calculated by compounding once a year. A simple way of comparing different accounts is to use an effective rate of interest that accurately reflects the real interest rate of different banks. This is the rate of simple interest that will yield the same return as the compound interest account after one 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 a year. This standard measure is called annual percentage rate (APR). It is also called effective annual rate is also used. APR is expressed as a percentage. If we let: i= rate of simple interest r = periodic rate of compound interest t = no. of periods in one year, then P (1 + i) = P (1 + r) t 1 + i = (1 + r) t i = (1 + r) t 1 Example 1 Find the effective rate of interest if the quoted annual rate of interest is 8% and compounding is done (a) monthly (b) quarterly. From the workings above we conclude that the Annual Percentage Rate (APR) formula can be represented as: i = (1 + r m )m 1 where i stands for the effective rate of interest of the institution. Example 2 Suppose that you have some spare money to invest. Your local banks offer the following interest rates: Bank A: 10% per annum Bank B: 10% per annum, compounded semi-annually Bank C: 8% per annum, compounded quarterly 6

7 Bank D: 7% per annum, compounded monthly Bank E: 5% per annum, compounded daily Which bank would you choose and why? Example 3 (a) Find the APR on a loan corresponding to 6% compounded monthly. (b) Two banks in a local town quote the following nominal interest rates: Bank A: pays 6.6% interest on a savings account compounded monthly Bank B: pays 6.65% interest on a savings account compounded semi-annually Which bank pages its savers the most interest? 7