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Exponential Functions in Business

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Exponential Functions in Business
Turgenbayeva Aiida
ID 20092726
Variant 2
Kazakhstan Institute of Management, Economics and Strategic Research

MSC1101 Mathematics for Business and Economics
Instructor: Dilyara Nartova
Section #2
Summer-I 2009
Abstract
This project reflects my knowledge and understanding of the interest rate, its types, formula and its evaluation in order to determine the most profitable type of investment scheme for National Bank wishing to increase its savings in developing country. My project will also illustrate the elements of money management in business, basically finding the deposit amount for investment to attain a given future value. Finally project will reflect the timing in money management, thus showing what time period needs to be in order to attain certain return on investments.

Exponential Functions in Business
Initial data
Principal – 2726* 10^6
Interest rate – 6%
Compound - 20092726
Interest Rate Types
Simple interest
[pic]
Simple interest is calculated on the original principal only. Accumulated interest from prior periods is not used in calculations for the following periods.
Compound interest
[pic]
Compounded interest calculation based on each period on the original principal and all interest accumulated during past periods. Even though the interest usually is stated as a yearly rate, there are also exist compounding periods such as annually, semiannually, quarterly and continuously.

Calculations I. The expected level of savings in 9 years if the money in bank is compounded: a) Annually [pic] S= 2726 (1+0, 06) ^9=4605, 5197* 10^6 Annually compounded interest is the interest that is paid or earned on the amount of interest accrued each year for as long as the investment exists. b) Semiannually [pic] S = 2726(1+0, 06/2) ^18= 4640, 8326* 10^6 Semiannually compounded interest means that the number of conversion periods per year is two. b) Quarterly [pic] S= 2726(1+0, 06/4) ^36 =4659, 1144 *10^6 Quarterly compounded interest rate means that four times a year there would be an "interest day", when investors balance will increase by one fourth of the going interest rate. c) Continuously [pic] S= 2726*e^0, 06*9= 2726* 2, 7182^0, 54=4677, 7587 * 10^6 Continuously compounded interest is the interest that is computed and added to the investors account on every instant. If we compare the amounts given by calculations, we can assume that the most profitable scheme for investors in National Bank is the continuously compounded interest rate. The least profitable is the annually compounded interest, which gives us the lowest possible amount of money. Let’s see the amounts calculated: Annually – 4605, 5197* 10^6 Semiannually – 4640, 8326* 10^6 Quarterly – 4659, 1144 *10^6 Continuously - 4677, 7587 * 10^6 As we see, the amount grows by each strategy; however the most profitable for investor would be continuously compounded interest in order to accumulate the large amount of money. II. The amount of principal to be deposited in order to accumulate after 9 years with compounded value of 20092726 if money is compounded annually is11, 8929. In order to calculate the amount above, the principal, we need to use the following formula: [pic] P= 20092726/ (1+0, 06) ^9 =11, 8929*10^6 Therefore, the amount of principal to be deposited to accumulate after 9 years compounded value of 20092726 if compounded annually is 11, 8029*10^6 III. In order to double its money, country needs to determine the interest rate that would double the amount in 9 years under annual compounding. The formula used to find the interest rate is : [pic] [pic] 8% - is the ideal interest rate that would double country’s money in 9 years under annual compounding. IV. What is the interest rate that would treble the money amount if compounded continuously for 9 years? The formula used to determine this is : [pic] The interest rate that would allow the amount of money to tremble is 12, 2% if compounded continuously. V. To determine years that would take to double the money at 12% interest rate under annual compounding, we need to use the following formula: [pic]

As result shows, we need 6 years and 1 month in order to double the money at 12% interest rate under annual compounding. Therefore, project reflected all the purposes that were aimed at the beginning. Project reflected the level of savings in different compounded periods and determined the continuously compounded interest rate as the most profitable for the investor and annually compounded interest as the least profitable. It also determined the amount of principal needed to be deposited, in order to accumulate after specific period of time under compounded annually. Moreover, project shows the interest rates that would be needed to allow the money be doubled in certain years under annual compounding and trebled if compounded continuously. Overall, project really shaped my understanding of interest rate, different schemes and money management elements very broadly.

References Barnett, R. A., Zieger, M. R., & Byleen, K. E. (2005). Compound Interest. Calculus for Business Economics, Life Sciences, and Social Sciences. (pp. 103-105). Upper Saddle River, New Jersey 07458: Pearson Education International. Tenth Edition. Compound Interest Formula (with Graph and Calculator Link). (n.d.). Moneychimp: learn Stock Investing, Index Funds, Valuation Models, and more. Retrieved June 4, 2010, from http://www.moneychimp.com/articles/finworks/fmfutval.htm Simple and Compound Interest Rates. (n.d.). Tutorials on Software Reuse and Time Value of Money Concepts. Retrieved June 3, 2010, from http://www.getobjects.com/Components/Finance/TVM/iy.html

References: Barnett, R. A., Zieger, M. R., & Byleen, K. E. (2005). Compound Interest. Calculus for Business Economics, Life Sciences, and Social Sciences. (pp. 103-105). Upper Saddle River, New Jersey 07458: Pearson Education International. Tenth Edition. Compound Interest Formula (with Graph and Calculator Link). (n.d.). Moneychimp: learn Stock Investing, Index Funds, Valuation Models, and more. Retrieved June 4, 2010, from http://www.moneychimp.com/articles/finworks/fmfutval.htm Simple and Compound Interest Rates. (n.d.). Tutorials on Software Reuse and Time Value of Money Concepts. Retrieved June 3, 2010, from http://www.getobjects.com/Components/Finance/TVM/iy.html

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