LECTURE 10

Financial Mathematics : Further concepts

Continuous compounding

If a sum of money, P, were to be compounded continuously for n years at r percent, the sum will grow to:

A = Pe ^{i t}

Example 1:

RM5000 is to be compounded continuously for 6 years, at an interest rate of 10%. Find the amount at the end of this period.

Exercise 1

RM6000 is to be compounded continuously for 7 years, at an interest rate of 9%. Find the amount at the end of this period.

Concepts: PV, FV, n, i, R (PMT)

PV – present value – amount you receive/pay today (P)

FV – future value – amount you receive/pay in the future (F)

i – interest rate (%).

n – number of compounding periods

R – periodic payments received/paid (PMT)

Example 2:

Identify PV (P), FV (F), n, i, R (PMT) in the following (including the value you need to find). Note: you are not required to calculate the relevant values – only identify PV, FV, etc

a.) You deposit RM3000 in a back that pays 8% interest compounded annually for 2 years.

How much will you receive at the end of 2 years?

b.) You pay RM 100 monthly into an account that will provide RM20,000 at the

end of 6 years. Find the interest rate (assume monthly compounding).

c.) You have taken a housing loan for RM120,000. You will pay RM1200

monthly for the next 25 years..

Find the interest rate (assume monthly compounding).

d.) You want to take a 5-year car loan for RM50,000. The interest rate is 8%,

compounded monthly. How much do you need to pay monthly?

e.) A student’s parents want to deposit a sum of money now that will provide

RM10,000 every six months for the next four years. If the money earns interest of 4%, compounded semiannually, how much should they deposit today?

f.) An employee planning to retire in 25 years deposits a certain amount of money

at the end of each year so he can get a lump sum of RM500,000 when he retires. Given an interest rate of 6% compounded annually, how much does he need to pay each year?

Annuities

An annuity is a series of periodic payments. Examples include regular deposits on savings accounts, monthly payments for car loans, housing loans, insurance or periodic payments to a person from a retirement fund. We will assume that an annuity involves a series of equal payments.

Example 3a:

If you deposit $7500 on the first day of every year into an account which pays 6% annually, compounded annually, what is your total balance after 3 years?

Answer:

S = P (1 + i) n

After 3 years,

Yr 1 deposit accumulates to

S = 7500 (1 + 0.06) 3 = 8932.62

Yr 2 deposit accumulates to

S = 7500 (1 + 0.06) 2 = 8427

Yr 1 deposit accumulates to

S = 7500 (1 + 0.06) 1 = 7950

TOTAL = 8932.62 + 8427 + 7950 = 25,309.62

FINANCIAL CALCULATOR SOLUTION:

2nd F, CA (to clear previous memory)

BGN,

3, n

6, i

7500, +/-, PMT

COMP FV

Note: We are using –ve nos. for money paid out and positive nos. for money received.

These computations explain the background of annuity calculations. We can reduce the tediousness of these calculations by using the following formulae:

Future Value, F = R [(1+i)ⁿ -1]/i

Present Value, P = R [1 -(1+i) ^-
n ]/i

These formulae assume that the annuities are paid/received at the end of the period

Example 3b:

You deposit $1000 each year for four years in a savings account that earns 6% interest per year compounded annually. What is the sum in your account at the end of 4 years? How much interest was earned?

R = 1000; i = 0.06, n = 4, FV = ?

Future Value, F = R [(1+i)ⁿ -1]/i = 1000 [(1+0.06) ⁴ – 1]/0.06 = 4374.62

Interest = 4374.62 – (1000 X 4) = 374.62

Financial calculator:

2ndF, CA (clear memory)

1000, PMT

6,i

4,n

COMP FV

Answer: R = 1000; i = 0.06, n = 4, FV = 4374.62

Interest = 4374.62 – (1000 X 4) = 374.62

Example 3c:

How much must you invest today to receive RM20,000 per year for the next 4 years, at an interest rate of 8% per year compounded annually?

Answer:

R = 20,000; i = 0.08; n = 4; PV = ?

Present Value, P = R [1 -(1+i) ^{- n} ]/i

= 20,000 [1 - (1.08) ^-4 ]/0.08

= RM 66242.54

Exercise 3a:

You want to deposit a sum of money today that will pay you RM25,000 each year, for the next 10 years. Assuming an interest rate of 9% per year compounded annually, how much must you deposit today?

Exercise 3b:

An employee planning to retire in 25 years deposits a certain amount of money

at the end of each year so he can get a lump sum of RM500,000 when he retires. Given an interest rate of 6%, how much does he pay each year?

Amortization

When you take a loan and make constant periodic repayments, your repayment contains partly payment of interest and partly the repayment of principal (the initial amount). We say that the debt is amortized. The amount of interest and principal repayments can be computed with an amortization schedule.

Example 4:

A debt of RM5000 is amortized with equal semi-annual payments over the next three years. Given an interest rate of 5% compounded semi-annually,

a.) Find the value of each payment

b.) How much of this payment consists of interest, and how much consists of

repayment of principal?

a.) P = R [1 -(1+i) ^{- n} ]/i;

5000 = R [1 – (1 + (0.05/2)) ^-6 ]/(0.05/2)

5000 = R (5.508)

R = 907.75 *

b.) Amortization schedule:

Payment No.	Outstanding Debt (O)	Interest Paid (I)	Payment Amount PMT *	Principal repaid PMT - I
	A = O - D	B = A X i	C = PMT	D = C - B
1	5,000	5,000 X 0.5/2 = 125	907.75	907.75 - 125 =782.75
2	5,000 – 782.74 = 4217.25	4217.25 X 0.05/2 = 105.43	907.75	907.75 – 105.43 = 802.32
3	4217.25 – 802.32 = 3414.93	3414.93 X 0.05/2 = 85.37	907.75	907.75 – 85.37 = 822.38
4
5
6

Final balance: (0.01)

Independent reading: BUD, Chapter 8; FRA Ch. 23