Interest: the compensation that a borrower of capital pays to a lender of capital for its use. It can be viewed as a form of rent that the borrower pays to the lender to compensate for the loss of use of capital by the lender. Principal: the initial amount of money invested. Accumulated Value: the total amount received after a period of time, t Interest = Accumulated Value Principal Period: The unit in which time is measured. Unless otherwise stated, assume the period to be one year. Accumulation function, a(t): the accumulated value at time t ³ 0 of an original investment of 1. Properties: a) a(0) = 1 b) For positive interest rates, a(t) is a non-decreasing function. c) If effective interest accrues continuously, a(t) is continuous. Otherwise, a(t) will have discontinuities. Amount Function, A(t): the accumulated value at time t ³ 0 of an original investment of k. Properties: a) A(t) = k a(t) b) A(0) = k c) For positive interest rates, A(t) is a non-decreasing function. d) If effective interest accrues continuously, A(t) is continuous. Otherwise, A(t) will have discontinuities. Define I n = amount of interest earned during the n th period. It's given by: I n = A(n) A(n 1) Examples of Amount functions: 1
Effective rate of interest, i: the amount of money that 1 unit invested at the beginning of a period will earn during the period, where interest is paid at the end of the period. i = a(1) a(0), or a(1) = 1 + i Observations about the definition a) effective is used for rates of interest in which interest is paid once per measurement period. b) Effective rate of interest is often expressed as a percentage, e.g. i= 8% is equivalent to 0.08 earned per unit of principal. c) The amount of principal remains constant throughout the period. d) The interest is paid at the end of the period. 2
Example 1. An investment of $10,000 is made into a fund at time t = 0. The fund develops the following balances over the next 4 years: t A(t) 0 10,000 1 10,600 2 11,130 3 11,575 4 12,153 a) Find the effective rate of interest for each of the four years. b) If $5000 is invested at time t = 2, under the same interest environment, find the accumulated value of the $5000 at time t = 4. 3
Example 2. It is known that a(t) is of the form ae 0.09t + b. If $250 invested at time 0 accumulates to $257.10` at time 5, find the accumulated value at time 12 of $250 invested at time 3. 4
Simple Interest With simple interest, the amount of interest earned during each period is constant. a(t) = 1 + it for t = 1, 2, 3,... A constant rate of simple interest does not imply a constant effective rate of interest. Amount function A(t) for Simple Interest Example 3. Find the accumulated value of $3500 invested for 10 years if the simple interest rate is 4% per annum. 5
Compound Interest: interest is automatically reinvested to earn additional interest. a(t) = (1 + i) t for t = 1, 2, 3,... A constant rate of simple interest implies a constant effective rate of interest and that the two are equal. Note: Unless stated otherwise, we will assume that interest is accrued over functional periods according to a(t) = 1 + it for simple interest, and a(t) = (1 + i) t for compound interest. Example 4. Find the accumulated value of $100 at the end of 3 years and 8 months invested at 7% per annum. Example 5. Rework example 4 assuming simple interest during the final fractional period. Example 7. Which of the two accumulated values is higher one in example 4 or example 5. Explain. 6
Present Value: We have seen that an investment of 1 will accumulate to 1 + i at the end of one period. 1 + i is known as the accumulation factor. Now consider: How much should be invested initially, so that the balance will be 1 at the end of one period? Let that initial amount be x. v is called the discount factor, since it discounts the value of an investment at the end of the period to its value at the beginning of the period. How much should be invested initially such that the balance is 1 at the end of t periods? The discount factor is denoted by a 1 (t). We get, for t ³ 0, For simple interest : a 1 (t) = For compound interest : a 1 (t) = 7
Example 8. If an investment of $1000 will grow to $6000 after 20 years, find the sum of present values of two payments of $9000 each, which will occur at the end of 30 and 50 years, assuming the same interest rate. 8