LN1.pdf - 2019 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT8015 Actuarial Statistics LN1 The Measurement of Interest

LN1.pdf - 2019 THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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2019 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT8015 Actuarial Statistics LN1: The Measurement of Interest LN1 is based on the reference book ”The Theory of Interest” by S.G. Kellison. Interest may be regarded as a reward paid by the borrower for the use of an asset, referred to as capital, belonging to the lender. A. The Accumulation and Amount Functions 1. The initial amount of money invested is called the principal and the total amount received after a period of time is called the accumulated value . The difference between the accumulated value and the principal is the amount of interest . 2. Measurement Period — the unit in which time is measured. 3. An accumulation function a ( t ) which gives the accumulated value at time t 0 of an original investment of 1. It is clear that a (0) = 1. 4. An amount function A ( t ) gives the accumulated value at time t 0 of an original investment of k . Then, we have A ( t ) = k · a ( t ) and A (0) = k. 5. I n — the amount of interest earned during the n th period from the date of investment. i.e., I n = A ( n ) - A ( n - 1) for integral n 1 . Example 1. A ( t ) = t 2 + 2 t + 3. Find (a) a ( t ) and (b) I n . Soln: (a) Since A (0) = 3, a ( t ) = A ( t ) / 3 = 1 3 ( t 2 + 2 t + 3). (b) I n = A ( n ) - A ( n - 1) = ( n 2 + 2 n + 3) - (( n - 1) 2 + 2( n - 1) + 3) = 2 n + 1. 1
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B. The Effective Rate of Interest 1. The effective rate of interest is the amount of money that one unit invested at the beginning of a period will earn during the period, where interest is paid at the end of the period. 2. The use of the word “effective” is not intuitively clear. This term is used for rates of interest in which interest is paid once per measurement period. This will be contrasted with “nominal” rates of interest, in which interest is paid more frequently than once per measurement period. 3. Let i n be the effective rate of interest during the n th period from the date of investment. It is given by i n = A ( n ) - A ( n - 1) A ( n - 1) = I n A ( n - 1) = a ( n ) - a ( n - 1) a ( n - 1) for integral n 1 . Example 2. If A (4) = 1 , 000 and i n = 0 . 01 n , find A (7). Soln: A (7) = A (4)((1 + i 5 )(1 + i 6 )(1 + i 7 )) = 1 , 000(1 . 05)(1 . 06)(1 . 07) = 1 , 190 . 91 . C. Simple Interest 1. The accruing of interest according to a ( t ) = 1 + it for t 0 is called simple interest which possesses the following property a ( t + s ) = a ( t ) + a ( s ) - 1 for t 0 and s 0 . 2. Let i be the rate of simple interest and let i n be the effective rate of interest for the n th period. Then, we have i n = a ( n ) - a ( n - 1) a ( n - 1) = [1 + in ] - [1 + i ( n - 1)] 1 + i ( n - 1) = i 1 + i ( n - 1) for integral n 1, which is a decreasing function of n . Thus, a constant rate of simple interest implies a decreasing effective rate of interest. 2
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Example 3. Simple interest of i = 0 . 04 is being credited to a fund. In which period is this equivalent to an effective rate of 0.025? Soln: Under simple interest, the effective rate for the n th period is i n = a ( n ) - a ( n - 1) a ( n - 1) = i 1 + i ( n - 1) .
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