Financial Mathematics

Introduction

Interest can be defined in two ways.

1. Interest is money earned when money is invested.

Eg. You deposited RM 1000 in a bank for a year and you find that at the end of year 1, you have RM 1050 in your account. The additional amount of RM 50 is the interest you earned when you invested RM 1000.

2. Interest is charge incurred when a loan or credit is obtained.

Eg. You borrowed Rm 1000 from a bank for a year and you paid back RM 1080 at the end of that year. The additional amount of RM 80 is the charge or interest you need to pay when you borrow RM 1000.

Introduction • Principal - the initial amount of the debt, usually the price of

the item purchased.

• Interest Rate - the amount one will pay for the use of someone else's money. Usually expressed as a percentage so that this amount can be expressed for any period of time.

• Time- essentially the amount of time that will be taken to pay down (eliminate) the debt. Usually expressed in years, but best understood as the number of and interval of payments, i.e., 36 monthly payments.

Simple interest The interest calculated on the original principal for the entire period it

is borrowed or invested.

Simple interest is the product of the principal multiplied by the rate and time.

𝑠 = 𝑋 + 𝑛𝑟𝑋

where

• S = accrued amount at the end of n-th year

• X = principal amount

• r = interest rate per year

• n = number of years

Question 1

• How much will an investor have after five years if he invests RM 1000 at 10% simple interest per annum?

Question 2

• How much a simple interest would be earned on an investment of $1,200 at 15% per annum for 10 years?

𝑆 = 1000 + 5 0.1 1000 = 𝑅𝑀1500

𝑆𝐼 = 5 0.15 1200 = 𝑅𝑀900

Question 3

• Alicia invest RM 5000 in an investment fund for three years. At the end of the investment period, his investment will worth RM 6125. Find the simple interest rate that is offered.

Question 4

• How long does it take a sum of money to triple itself at a simple interest rate of 5% per annum?

𝑆𝐼 = 6125 − 5000 = 1125 1125 = 3 𝑟 5000

𝑟 = 0.075 = 7.5%

𝐿𝑒𝑡 𝑠𝑎𝑦, 𝑋 = 1000, 𝑆𝐼 = 3000 − 1000 = 2000

2000 = 𝑛 0.05 1000 n = 40𝑦𝑒𝑎𝑟𝑠

Exercise

1. Mr. Aryan deposited RM6000 in a bank and obtained RM120 simple interest after 5 years. Find the simple interest rate offered.

2. Ishak invests RM X in a bank. After 36 months, his investment will be worth RM5439. If the simple interest rate is 8.5% per annum, find the value of X.

3. 5 years ago, Jane invested RM 6660 in a bank at a simple interest rate of 7.2%. Find:

a) The amount in the account today

b) The number of years required if Jane wanted the amount in the account to become RM 9537.12

Compound Interest • Based on the principle, which changes from time to time.

• Interest that is earned is compounded or converted into principal and earns interest thereafter.

𝑠 = 𝑋(1 + 𝑟)𝑛

where

• S = accrued amount at the end of n-th year

• X = principal amount

• r = interest rate per year

• n = number of years

Question 5

• At what annual rate of compound interest will RM 2000 grow to RM 2721 after 4 years?

Question 6

• A sum of $ 800 is invested in a business which earns interest at a rate of 12% compounded annually for 3 years. Find the accumulated amount and the compound interest earned at the end of 3 years.

2721 = 2000(1 + 𝑟) 4

1.3605 = (1 + 𝑟) 4 𝑟 = 1.0800 − 1 = 0.08 = 8%

S = 800(1 + 0.12) 3 S = 829.15

CI = 829.15 − 800 = 29.15

Frequency Compounding

𝑆 = 𝑋(1 +𝑟

𝑘)𝑛×𝑘

• where

• S = accrued amount at the end of n-th year

• X = principal amount

• r = annual nominal rate

• n = number of years

• k = frequency of conversions

• For annually (once a year), k = 1.

• For semi-annually (twice a year), k =2 and subsequently.

Example: Suppose RM9000 is invested for 7 years at 12% compounded quarterly

• Original principal (X) = RM9000

• Annual nominal rate (𝑟), the interest rate for a year together with the frequency in which interest is calculated in a year. 𝑟 =12% compounded quarterly.

• Interest period is the length of time in which interest is calculated. Thus, interest period is 3 months.

• Frequency of conversions (𝑘), is the number of times interest is calculated in a year. 𝑘 = 4 times.

• Periodic interest rate (𝑖), is the interest rate for each interest period.

𝑖 =𝑟

𝑘=

12

4= 3%

• Number of interest periods in the investment period (t), is the number of times interest is calculated.

𝑡 = 𝑛 × 𝑘 = 7 × 4 = 28 𝑡𝑖𝑚𝑒𝑠.

Question 7

• Find the accumulated amount if a sum of $6000 is invested for two years at 15% per annum compounded

a) Annually

b) semi-annually

c) quarterly

d) monthly

(a) 𝑆 = 6000(1 +0.15

1)2×1 = 6181.35

(b) 𝑆 = 6000(1 +0.15

2)2×2 = 8012.81

(c) 𝑆 = 6000(1 +0.15

4)2×4 = 8054.82

(d) 𝑆 = 6000(1 +0.15

12)2×12 = 8084.11

Question 7

What is the nominal rate compounded monthly that will make RM1000 become RM2000 in 5 years?

2000 = 1000(1 +𝑟

12)5×12

2 = (1 +𝑟

12)60

260

= 1 +𝑟

12

𝑟 = 1.012 − 1 12 = 0.139 = 13.9%

Exercise

1. Find the future values for RM 30000 at 6% compounded semi-annually fro 5 years 6 months.

2. At the end of every year fro 3 years, RM 1000 is invested in an account that offers 8% compounded annually/ find the account amount at the end of the three years.

3. Find the interest earned for an investment if the accumulated amount at the end of 6 years is RM 2412.66 and the interest rate given is 8% compounded quarterly.

Problem A trader wishes to invest $ 60,000 in a business project. He has two choices: either to invest in Project A which earns compound interest at a rate of 9% per annum for 5 years, or to invest in Project B which earns interest at 8% per annum compounded monthly for 5 years. Which project the trader should choose?