Business Mathematics MTH367 Lecture 10 Chapter 8 Mathematics
![Business Mathematics MTH-367 Lecture 10 Business Mathematics MTH-367 Lecture 10](https://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-1.jpg)
Business Mathematics MTH-367 Lecture 10
![Chapter 8 Mathematics of Finance continued Chapter 8 Mathematics of Finance continued](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-2.jpg)
Chapter 8 Mathematics of Finance continued
![Objectives • Provide an understanding of the time value of money • Provide an Objectives • Provide an understanding of the time value of money • Provide an](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-3.jpg)
Objectives • Provide an understanding of the time value of money • Provide an understanding of the mathematics of interest computations for singly payment and annuity cash flow structures
![Review • • • Interest Simple interest Compound interest Graphical comparison Single payment and Review • • • Interest Simple interest Compound interest Graphical comparison Single payment and](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-4.jpg)
Review • • • Interest Simple interest Compound interest Graphical comparison Single payment and its computations
![Today’s Topics • Compound amount formula (Examples) • Computing present value • Further applications Today’s Topics • Compound amount formula (Examples) • Computing present value • Further applications](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-5.jpg)
Today’s Topics • Compound amount formula (Examples) • Computing present value • Further applications of compound amount formula • Effective interest rates • Annuities and their future values
![Compound amount formula • Compound amount formula •](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-6.jpg)
Compound amount formula •
![Examples 1) Suppose that $ 1000 is invested in a saving bank which earns Examples 1) Suppose that $ 1000 is invested in a saving bank which earns](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-7.jpg)
Examples 1) Suppose that $ 1000 is invested in a saving bank which earns interest at the rate of 8 % per year compounded annually. If all interest is left in the account, what will be the account balance after 10 years?
![Examples cont’d 2) A long term investment of $ 250000 has been made by Examples cont’d 2) A long term investment of $ 250000 has been made by](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-8.jpg)
Examples cont’d 2) A long term investment of $ 250000 has been made by a small company. The interest rate is 12 % per year, and interest is compounded quarterly. If all interest is reinvested at the same rate, what will the value of the investment be after 8 years? Solution: Note:
![Examples Here, the interest rate per quarter equals, The number of compounding periods over Examples Here, the interest rate per quarter equals, The number of compounding periods over](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-9.jpg)
Examples Here, the interest rate per quarter equals, The number of compounding periods over 8 -years period is
![Present value computation • Present value computation •](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-10.jpg)
Present value computation •
![Example • A person can invest money in saving account at the rate of Example • A person can invest money in saving account at the rate of](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-11.jpg)
Example • A person can invest money in saving account at the rate of 10 % per year compounded quarterly. • The person wishes to deposit a lump sum at the beginning of the year and have that sum grow to $ 20, 000 over the next 10 years. • How much money should be deposited?
![Example cont’d • How much amount of money should be invested at the rate Example cont’d • How much amount of money should be invested at the rate](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-12.jpg)
Example cont’d • How much amount of money should be invested at the rate of 10 % per year compounded quarterly, if the compound amount is $ 20000 after 10 years?
![Other applications of compound amount formula • Other applications of compound amount formula •](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-13.jpg)
Other applications of compound amount formula •
![Other applications of compound amount formula cont’d Example: A person wishes to invest $ Other applications of compound amount formula cont’d Example: A person wishes to invest $](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-14.jpg)
Other applications of compound amount formula cont’d Example: A person wishes to invest $ 10000 and wants the investment to grow to $ 20000 over the next 10 years. At what annual interest rate the required amount is obtained assuming annual compounding?
![](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-15.jpg)
![Effective Interest Rates • The stated annual interest rate is usually called nominal rate. Effective Interest Rates • The stated annual interest rate is usually called nominal rate.](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-16.jpg)
Effective Interest Rates • The stated annual interest rate is usually called nominal rate. • We know that when interest is compounded more frequently then interest earned is greater than earned when compounded annually. • When compounding is done more frequently than annually, then effective annual interest rates can be determined. • Two rates would be considered equivalent if both results in the same compound amount.
![Effective Interest Rates • Effective Interest Rates •](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-17.jpg)
Effective Interest Rates •
![](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-18.jpg)
![Example Nominal interest rate = 12 % / year, quarterly Example Nominal interest rate = 12 % / year, quarterly](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-19.jpg)
Example Nominal interest rate = 12 % / year, quarterly
![Annuities and their Future value • Annuities and their Future value •](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-20.jpg)
Annuities and their Future value •
![Example • Example •](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-21.jpg)
Example •
![Example cont’d • Example cont’d •](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-22.jpg)
Example cont’d •
![Formula • The procedure used in the above example is not practical when dealing Formula • The procedure used in the above example is not practical when dealing](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-23.jpg)
Formula • The procedure used in the above example is not practical when dealing with large number of payments.
![Formula cont’d Formula cont’d](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-24.jpg)
Formula cont’d
![](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-25.jpg)
![Examples A boy plans to deposit $ 50 in a savings account for the Examples A boy plans to deposit $ 50 in a savings account for the](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-26.jpg)
Examples A boy plans to deposit $ 50 in a savings account for the next 6 years. Interest is earned at the rate of 8% per year compounded quarterly. What should her account balance be 6 years from now? How much interest will he earn?
![](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-27.jpg)
![Summary • Compound amount formula (Examples) • Computing present value • Further applications of Summary • Compound amount formula (Examples) • Computing present value • Further applications of](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-28.jpg)
Summary • Compound amount formula (Examples) • Computing present value • Further applications of compound amount formula • Effective interest rates • Annuities and their future values by a mathematical formula
![Next Lecture • Determining the size of annuity • Annuities and their present value Next Lecture • Determining the size of annuity • Annuities and their present value](http://slidetodoc.com/presentation_image_h2/54dc42d4fa4ee983b7e326e7efd391b9/image-29.jpg)
Next Lecture • Determining the size of annuity • Annuities and their present value
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