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Unformatted text preview: Revision for Mathematical Finance 1. For n >- a , log(1 + a/n ) = log( n + a )- log n = integraltext n + a n dx/x a (1 / ( n + a ) , 1 /n ) using the obvious bounds on the integrand and noting that the interval has signed length a . As na/ ( n + a ) a as n it follows that n log(1 + a/n ) a as n and hence by continuity of log that (1 + a/n ) n e a as n . There are various ways to show the convergence is actually monotone. 2. From the definition R j +1 = (1 + r ) R j- a or R j = (1 + r )- 1 ( a + R j +1 ). Check by induction back from j = n- 1 that R j = a n- 1 j (1 + r ) k- n and from this that R j = a n- 1 summationdisplay k = j (1 + r ) k- n = a 1 + r n- j- 1 summationdisplay k =0 (1 + r )- k = a r parenleftBig 1- (1 + r ) j- n parenrightBig Hence L = R = ( a/r ) ( 1- (1 + r )- n ) and thus a = rL/ ( 1- (1 + r )- n ) . It follows immediately that a > rL which is crucial or the interest added would exceed the payment and the loan would never be paid off.payment and the loan would never be paid off....
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This note was uploaded on 02/04/2011 for the course MATH 3301 taught by Professor Dri.m.macphee during the Spring '10 term at Durham.
- Spring '10