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Unformatted text preview: EC3070 FINANCIAL DERIVATIVES COMPOUND INTEREST Investments Imagine that a sum of y = 100 is invested at an annual rate of interest of r = 5% per annum. After one year has elapsed, the sum will have grown to y (1+ r ) = 105; and the opportunity will arise for deciding how to dispose of the funds. There are two options which might be considered. On the one hand, one might decide leave 100 permanently on account and to treat the interest payment of r = 5 as a small annual income or annuity. Alternatively, one might decide leave all of the money on account, and to watch it grow steadily through successive years. In that case, one would be interested in knowing what the investment would amount to after a definite period of years; and, for this purpose, one should need to understand the principle of geometric growth. Geometric Growth A process of geometric growth is defined by the equation (1) y t = y (1 + r ) t , wherein y stands for the value of the process at time 0 and y t stands for the value at time t . Here, t , which denotes time, takes only integer values. There are two ways in which the value of y t can be calculated from those of y and r . The first method uses logarithms: (2) ln( y t ) = t ln(1 + r ) + ln( y ) . The value of y t is found by applying antilogarithms to the value obtained from the RHS of the equation. The second method of computing y t is by iteration. Imagine that t = 3. Then (3) y 3 = (1 + r ) 3 y = n (1 + r ) (1 + r ) { (1 + r ) y } o . This equation can be decomposed into three stages: (4) y 1 = (1 + r ) y , y 2 = (1 + r ) y 1 , y 3 = (1 + r ) y 2 . The generic form of these equations is (5) y t = (1 + r ) y t 1 ; c D.S.G. Pollock: stephen pollock@sigmapi.unet.com COMPOUND INTEREST and one can imagine pursuing the iteration though any number of stages. In fact, if we were writing a computer program for the purpose of finding values of y t , then we should use the iterative scheme on the grounds that it is the...
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 Spring '12
 D.S.G.Pollock

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