THE MEASUREMENT OF INTEREST
CONTENTS • COMPOUND INTEREST • PRESENT VALUE • THE EFFECTIVE RATE OF DISCOUNT • NOMINAL RATE OF INTEREST
• NOMINAL RATE OF DISCOUNT
COMPOUND INTEREST •
• • •
The word “compoundâ€? refers to the process of interest being reinvested to earn additional interest. Consider the investment of 1 which accumulates to 1 + đ?‘– at the end of the first period. This balance of 1 + đ?‘– can be considered as principal at the beginning of the second period. The balance at the end of the second period is 1 + đ?‘– + đ?‘– 1 + đ?‘– = (1 + đ?‘–)2 . Continuing this process indefinitely, we obtain
� � = (1 + �)� for t = 1,2,3,‌..
Effective rate of interest for nth period
đ?‘–đ?‘› =
đ?‘Ž đ?‘› −đ?‘Ž(đ?‘›âˆ’1) đ?‘Ž(đ?‘›âˆ’1)
=
(1+đ?‘–)đ?‘› −(1+đ?‘–)đ?‘›âˆ’1 (1+đ?‘–)đ?‘›âˆ’1
đ?‘–
=
1+đ?‘– −1 1
=
COMPOUND INTEREST
EXAMPLE 1 Find the accumulated value of $10 000 at the end of 8 years and invested at 8.5% per annum. Assuming compound interest throughout. ANSWER đ?‘Ą = 8đ?‘– = 0.085
đ?‘˜ = 10 000
đ??´ đ?‘Ą = đ?‘˜. đ?‘Ž(đ?‘Ą) đ??´ đ?‘Ą = đ?‘˜. (1 + đ?‘–)đ?‘Ą = 10 000. 1 + 0.085 = 10 000. 1.085 8 = 19206.04338 = 19206.04
8
COMPOUND INTEREST EXAMPLE 2 It is known that $300 invested for two years will earn $150 in interest. Find the accumulated value of $1500 invested at the same rate of compound interest for 3.5 years. 1 2
ANSWER
450 −1 300 đ?‘– = 0.2247 đ?‘– = 0.22
đ?‘–= đ?‘Ą = 2đ??źđ?‘› = 150
đ?‘˜ = 300
đ??´ đ?‘Ą = đ?‘˜ + đ??źđ?‘› = 300 + 150 = 450 đ??´ đ?‘Ą = đ?‘˜. (1 + đ?‘–)đ?‘Ą 450 = 300. 1 + đ?‘– 450 300 450 300
1 2
= 1+đ?‘–
= 1+đ?‘–
2
As the rate of compound interest are same:
đ?‘Ą = 3.5đ?‘– = 0.22 2
đ?‘˜ = 1500
đ??´ đ?‘Ą = đ?‘˜. (1 + đ?‘–)đ?‘Ą = 1500. 1 + 0.22 = 1500 1.22 3.5 = 3008.5045 = 3008.50
3.5
PRESENT VALUE • • • •
An investment of 1 will accumulate to 1+ i at the end of one period. It is often necessary to determine how much a person must invest initially so that balance will be 1 at the end of one period. The answer is (1+i)-1, since this amount will accumulate to 1 at the end of one period. We now define a new symbol v, such that
v= • • •
1 1+đ?‘–
The term v is often called a discount factor, since it “discount� the value of an investment at the end of a period to its value at the beginning of the period. The term “present value� refers to payment to be made in the future. There are different type of formula for simple interest and compound interest. Simple interest
a-1 (t) =
1 (1+đ?‘–đ?‘Ą)
Compound interest
a-1 (t) =
1 (1+đ?‘–)đ?‘Ą
PRESENT VALUE EXAMPLE 1 Find the amount which must be invested at 9% per annum in order to accumulate $1000 at the end of three years: i. Assuming simple interest ii. By using simple interest formula ANSWER i.
A(t) = ka-1(t) =
1000 1+(0.09)(3)
=
1000 1.27
= $787.40
ii.
A(t) = ka-1(t) = 1000v3 =
1000 (1.09)3
= $772.18
PRESENT VALUE
EXAMPLE 2 How much Ahmad must invest if he wants $10,000 at the end of 5 years with the effective rate of interest 10% . ANSWER A(t) = kđ?‘Žâˆ’1 (t) = 10000Ă— =
1 (1+0.1)5
10000 (1+0.1)5
10000 0.6209213231 = $6,209.213
=
THE EFFECTIVE RATE OF DISCOUNT •
• •
The effective rate of discount d is the ratio of the amount of interest (sometimes called “amount of discount� or just “discount�) earned during the period to the amount invested at the end of the period. Interest – paid at the end of the period on the balance at the beginning of the period. Discount – paid at the beginning of the period on the balance at the end of the period.
đ?‘‘
đ?‘›=
đ??´ đ?‘› −đ??´(đ?‘›âˆ’1) đ??źđ?‘› = đ??´đ?‘› đ??´(đ?‘›)
•
From the basic definition of i as the ratio of the amount of interest (discount) to the principal, we obtain đ?‘– đ?‘‘ đ?‘‘ = , đ?‘–= 1+đ?‘– 1−đ?‘‘ đ?‘‘ = đ?‘–đ?‘Ł, đ?‘‘ = 1 − đ?‘Ł đ?‘Žđ?‘›đ?‘‘ đ?‘– − đ?‘‘ = đ?‘–đ?‘‘
•
Simple discount:
•
Compound discount: đ?‘Ž −1 đ?‘Ą = đ?‘Ł đ?‘Ą = 1 − đ?‘‘ đ?‘Ą
đ?‘Žâˆ’1 đ?‘Ą = 1 − đ?‘‘đ?‘Ą
THE EFFECTIVE RATE OF DISCOUNT EXAMPLE 1 If i and d are equivalent rates of simple interest and simple discount over t periods, show that:
đ?‘– − đ?‘‘ = đ?‘–đ?‘‘đ?‘Ą 1 + đ?‘–đ?‘Ą =
1 (1−đ?‘‘đ?‘Ą)
1 + đ?‘–đ?‘Ą 1 − đ?‘‘đ?‘Ą = 1 1 − đ?‘–đ?‘‘đ?‘Ą 2 − đ?‘‘đ?‘Ą + đ?‘–đ?‘Ą = 1 đ?‘–đ?‘Ą − đ?‘‘đ?‘Ą − đ?‘–đ?‘‘đ?‘Ą 2 = 0 đ?‘Ą đ?‘– − đ?‘‘ − đ?‘–đ?‘‘đ?‘Ą = 0 đ?‘– − đ?‘‘ − đ?‘–đ?‘‘đ?‘Ą = 0 đ?‘– − đ?‘‘ = đ?‘–đ?‘‘đ?‘Ą = đ?‘ â„Žđ?‘œđ?‘¤đ?‘›
THE EFFECTIVE RATE OF DISCOUNT EXAMPLE 2: Find the amount which must be invested at 8% per annum in order to accumulate $2000 at the end of five years: a) Assuming simple discount instead of simple interest. b) Assuming compound discount.
a) Simple discount = đ?‘˜(1 − đ?‘‘đ?‘Ą) = 2000(1 – 0.08(5)) = $ 1200 b) Compound discount =đ?‘˜ 1−đ?‘‘ đ?‘Ą = 2000 1 − 0.08 5 = $ 1318.16
NOMINAL RATE OF INTEREST •
In many instances where payment are made for a period less than a year, the period interest rate is states as a nominal rates, which is the interest rate per period multiplied by the number of period per year.
•
In the general case for which there are m payment periods per year, we denote the nominal rate by đ?‘– (đ?‘š) .
•
The periodic interest rate is
đ?‘– (đ?‘š) đ?‘š
and the effective rate is
� (�) � = 1+ � •
đ?‘š
−1
This has consequences that đ?‘– (đ?‘š) đ?‘–+1= 1+ đ?‘š
đ?‘š
NOMINAL RATE OF DISCOUNT •
The nominal discount rate convertible monthly is denoted by đ?‘‘ (đ?‘š) . 1−đ?‘‘ = 1 −
đ?‘‘(đ?‘š) đ?‘š
đ?‘š
•
This equation can be remembered by noting that the left side represent v and the right side represent the v for m-thly period raised to the m-th power.
•
It is possible to relate the nominal rate of interest and nominal rate of discount. The relationship is đ?‘– (đ?‘š) 1+ đ?‘š
đ?‘š
đ?‘‘ (đ?‘?) = 1− đ?‘?
−đ?‘?
NOMINAL RATE OF INTEREST EXAMPLE 1: Find the accumulated value of $1000 invested for three years at 6% per annum convertible semiannually đ?‘š = 2,
đ?‘– (đ?‘š) 1000 1 + đ?‘š
đ?‘Ą = 3,
đ?‘šđ?‘Ą
đ?‘– (2) = 0.06
0.06 = 1000 1 + 2 = 1000(1.03)6 = $1194.05
2Ă—3
NOMINAL RATE OF DISCOUNT EXAMPLE 2: Find the present value of $800 to be paid at the end of five years at 12% per annum payable in advance and convertible quarterly. đ?‘š = 4, đ?‘‘ (4) 800 1 − 4
đ?‘Ą = 5, đ?‘šđ?‘Ą
đ?‘‘ (4) = 0.12
0.12 = 800 1 − 4 = 800(0.97)20 = $435.04
4Ă—5