MBD 1 Proj 2 - The Mathematical Association of America...

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Title Compound Interest Histograms Probability Distributions Random Sampling Simulation Wholesale Ordering Example © 2009 by The Arizona Board of Regents for The University of Arizona. All rights reserved. C I Video introduction by Prof. Christopher Lamoureux Department of Finance University of Arizona. Stock Option Pricing Project 2 Part 1: Probability & Simulation The M athematical A ssociation of A merica Published and Distributed by Mathematics for Business Decisions Part 1 Release 2, 2009 For Office 2007

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Compound Interest, Discrete Compounding (material continues) 1 . DISCRETE COMPOUNDING Suppose that money left on deposit earns interest at an annual rate r . This may be given as either a decimal, such as 0.0625, or as a percentage, such as 6.25%. Interest paid for an interval other than one year has the rate adjusted for time. For example, if the annual rate is 4.8% and interest is paid every month, then the monthly rate is 0.048/12 = 0.004, or 0.4%. The simple interest on P dollars after a time of t years, at an annual rate r , is P r t . Since the original money is still there, the value of the P dollars is P (1 + r t ). Interest is normally paid at regular intervals, while the money is on deposit. This is called compounding . Example 1 . We deposited \$3,200 on April 1,1998 and left it in the bank for one and one-half years at an annual rate of 5.45% , compounded quarterly (every three months). On July 1, 1998, interest made the deposit worth \$3,200(1 + 0.0545/4)  \$3,243.50. On October 1, 1998, further interest made the deposit worth \$3,200(1 + 0.0545/4)(1 + 0.0545/4) = \$3,200(1 + 0.0545/4) 2  \$3,287.79. Compound Interest. Discrete Compounding I T C
t n n r P F + = 1 Interest, Discrete There are four more interest payments, that made the deposit worth \$3,200(1 + 0.0545/4) 3  \$3,332.59, \$3,200(1 + 0.0545/4) 4  \$3,378.00, \$3,200(1 + 0.0545/4) 5 \$3,424.02, and \$3,200(1 + 0.0545/4) 6  \$3,470.67 on January 1, 1999; April 1, 1999; July 1, 1999; and October 1, 1999; respectively. Note that, after 3/2 years, at 4 periods per year, the value of the money was \$3,200(1 + 0.0545/4) 4(3/2) . Compound Interest. Discrete Compounding: page 2 (material continues) The more often interest is compounded during a given period, the more rapidly the investment grows. The sheet Frequency in the Excel file Comp. Int.xlsx shows the effect of frequency of compounding on the \$3,200 that was discussed in Example 1 . Open that sheet and study its contents now. Comp. Int.xlsx \$ I T P dollars invested at an annual rate r , compounded n times per year, has a value of F dollars after t years. C

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1 1 - + = n n r y \$ Interest, Discrete Compound Interest. Discrete Compounding: page 3 The spreadsheet shows that there is almost no gain when interest is compounded on more than a daily basis. How much more is the initial money worth, rounded to whole cents, if interest is compounded hourly, rather than daily?
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