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# Introductory maths analysis chapter 05 official

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• ### Introductory maths analysis chapter 05 official

1. 1. INTRODUCTORY MATHEMATICALINTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences ©2007 Pearson Education Asia Chapter 5Chapter 5 Mathematics of FinanceMathematics of Finance
2. 2. ©2007 Pearson Education Asia INTRODUCTORY MATHEMATICAL ANALYSIS 0. Review of Algebra 1. Applications and More Algebra 2. Functions and Graphs 3. Lines, Parabolas, and Systems 4. Exponential and Logarithmic Functions 5. Mathematics of Finance 6. Matrix Algebra 7. Linear Programming 8. Introduction to Probability and Statistics
3. 3. ©2007 Pearson Education Asia 9. Additional Topics in Probability 10. Limits and Continuity 11. Differentiation 12. Additional Differentiation Topics 13. Curve Sketching 14. Integration 15. Methods and Applications of Integration 16. Continuous Random Variables 17. Multivariable Calculus INTRODUCTORY MATHEMATICAL ANALYSIS
4. 4. ©2007 Pearson Education Asia • To solve interest problems which require logarithms. • To solve problems involving the time value of money. • To solve problems with interest is compounded continuously. • To introduce the notions of ordinary annuities and annuities due. • To learn how to amortize a loan and set up an amortization schedule. Chapter 5: Mathematics of Finance Chapter ObjectivesChapter Objectives
5. 5. ©2007 Pearson Education Asia Compound Interest Present Value Interest Compounded Continuously Annuities Amortization of Loans 5.1) 5.2) 5.3) 5.4) Chapter 5: Mathematics of Finance Chapter OutlineChapter Outline 5.5)
6. 6. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.1 Compound Interest5.1 Compound Interest Example 1 – Compound Interest • Compound amount S at the end of n interest periods at the periodic rate of r is as ( )n rPS += 1 Suppose that \$500 amounted to \$588.38 in a savings account after three years. If interest was compounded semiannually, find the nominal rate of interest, compounded semiannually, that was earned by the money.
7. 7. ©2007 Pearson Education Asia Solution: There are 2 × 3 = 6 interest periods. The semiannual rate was 2.75%, so the nominal rate was 5.5 % compounded semiannually. Chapter 5: Mathematics of Finance 5.1 Compound Interest Example 1 – Compound Interest ( ) ( ) 0275.01 500 38.588 500 38.588 1 500 38.588 1 38.5881500 6 6 6 6 ≈−= =+ =+ =+ r r r r
8. 8. ©2007 Pearson Education Asia How long will it take for \$600 to amount to \$900 at an annual rate of 6% compounded quarterly? Solution: The periodic rate is r = 0.06/4 = 0.015. It will take . Chapter 5: Mathematics of Finance 5.1 Compound Interest Example 3 – Compound Interest ( ) ( ) ( ) 233.27 015.1ln 5.1ln 5.1ln015.1ln 5.1ln015.1ln 5.1015.1 015.1600900 ≈= = = = = n n n n n months9years,68083.6 2 1 4 233.27 =≈
9. 9. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.1 Compound Interest Example 5 – Effective Rate Effective Rate • The effective rate re for a year is given by 11 −      += n e n r r To what amount will \$12,000 accumulate in 15 years if it is invested at an effective rate of 5%? Solution: ( ) 14.947,24\$05.1000,12 15 ≈=S
10. 10. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.1 Compound Interest Example 7 – Comparing Interest Rates If an investor has a choice of investing money at 6% compounded daily or % compounded quarterly, which is the better choice? Solution: Respective effective rates of interest are The 2nd choice gives a higher effective rate. %27.61 4 06125.0 1 and%18.61 365 06.0 1 4 365 ≈−      += ≈−      += e e r r 8 1 6
11. 11. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.2 Present Value5.2 Present Value Example 1 – Present Value • P that must be invested at r for n interest periods so that the present value, S is given by Find the present value of \$1000 due after three years if the interest rate is 9% compounded monthly. Solution: For interest rate, . Principle value is . ( ) n rSP − += 1 ( ) 15.764\$0075.11000 36 ≈= − P 0075.012/09.0 ==r
12. 12. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.2 Present Value Example 3 – Equation of Value A debt of \$3000 due six years from now is instead to be paid off by three payments: • \$500 now, • \$1500 in three years, and • a final payment at the end of five years. What would this payment be if an interest rate of 6% compounded annually is assumed?
13. 13. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.2 Present Value Solution: The equation of value is ( ) ( ) 27.1257\$ 02.160002.11000 208 ≈ += −− x
14. 14. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.2 Present Value Example 5 – Net Present Value You can invest \$20,000 in a business that guarantees you cash flows at the end of years 2, 3, and 5 as indicated in the table. Assume an interest rate of 7% compounded annually and find the net present value of the cash flows. Net Present Value ( ) investmentInitial-valuespresentofSumNPVValuePresentNet = Year Cash Flow 2 \$10,000 3 8000 5 6000
15. 15. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.2 Present Value Example 5 – Net Present Value Solution: ( ) ( ) ( ) 31.457\$ 000,2007.1600007.1800007.1000,10NPV 532 −≈ −++= −−−
16. 16. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.3 Interest Compounded Continuously5.3 Interest Compounded Continuously Example 1 – Compound Amount Compound Amount under Continuous Interest • The compound amount S is defined as If \$100 is invested at an annual rate of 5% compounded continuously, find the compound amount at the end of a. 1 year. b. 5 years. kt k r PS       += 1 ( )( ) 13.105\$100 105.0 ≈== ePeS rt ( )( ) 40.128\$100100 25.0505.0 ≈== eeS
17. 17. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.3 Interest Compounded Continuously Effective Rate under Continuous Interest • Effective rate with annual r compounded continuously is .1−= r e er Present Value under Continuous Interest • Present value P at the end of t years at an annual r compounded continuously is .rt SeP − =
18. 18. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.3 Interest Compounded Continuously Example 3 – Trust Fund A trust fund is being set up by a single payment so that at the end of 20 years there will be \$25,000 in the fund. If interest compounded continuously at an annual rate of 7%, how much money should be paid into the fund initially? Solution: We want the present value of \$25,000 due in 20 years. ( )( ) 6165\$000,25 000,25 4.1 2007.0 ≈= == − −− e eSeP rt
19. 19. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.4 Annuities5.4 Annuities Example 1 – Geometric Sequences Sequences and Geometric Series • A geometric sequence with first term a and common ratio r is defined as a. The geometric sequence with a = 3, common ratio 1/2 , and n = 5 is 0where,...,,,, 132 ≠− aarararara n 432 2 1 3, 2 1 3, 2 1 3, 2 1 3,3                        
20. 20. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.4 Annuities Example 1 – Geometric Sequences b. Geometric sequence with a = 1, r = 0.1, and n = 4. c. Geometric sequence with a = Pe−kI , r = e−kI , n = d. 001.0,01.0,1.0,1 dkIkIkI PePePe −−− ,...,, 2 Sum of Geometric Series • The sum of a geometric series of n terms, with first term a, is given by ( ) 1rfor 1 11 0 ≠ − − == ∑ − = r ra ars nn i i
21. 21. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.4 Annuities Example 3 – Sum of Geometric Series Find the sum of the geometric series: Solution: For a = 1, r = 1/2, and n = 7 62 2 1 ... 2 1 2 1 1       ++      ++ ( ) 64 127 2 1 1 2 1 11 1 1 2 1 128 127 7 == −               − = − − = r ra s n Present Value of an Annuity • The present value of an annuity (A) is the sum of the present values of all the payments. ( ) ( ) ( ) n rRrRrRA −−− ++++++= 1...11 21
22. 22. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.4 Annuities Example 5 – Present Value of Annuity Find the present value of an annuity of \$100 per month for years at an interest rate of 6% compounded monthly. Solution: For R = 100, r = 0.06/12 = 0.005, n = ( )(12) = 42 From Appendix B, . Hence, 005.042 __100aA = 798300.37 005.042 __ =a ( ) 83.3779\$798300.37100 =≈A 2 1 3 2 1 3
23. 23. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.4 Annuities Example 7 – Periodic Payment of Annuity If \$10,000 is used to purchase an annuity consisting of equal payments at the end of each year for the next four years and the interest rate is 6% compounded annually, find the amount of each payment. Solution: For A= \$10,000, n = 4, r = 0.06, 91.2885\$ 465106.3 000,10000,10 000,10 06.04 06.04 ____ __ =≈== = aa A R Ra rn
24. 24. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.4 Annuities Example 9 – Amount of Annuity Amount of an Annuity • The amount S of ordinary annuity of R for n periods at r per period is Find S consisting of payments of \$50 at the end of every 3 months for 3 years at 6% compounded quarterly. Also, find the compound interest. Solution: For R=50, n=4(3)=12, r=0.06/4=0.015, ( ) r r RS n 11 −+ ⋅= ( ) 06.652\$041211.135050 015.012 __ ≈≈=S ( ) 06.52\$501206.652InterestCompund =−=
25. 25. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.4 Annuities Example 11 – Sinking Fund A sinking fund is a fund into which periodic payments are made in order to satisfy a future obligation. A machine costing \$7000 is replaced at the end of 8 years, at which time it will have a salvage value of \$700. A sinking fund is set up. The amount in the fund at the end of 8 years is to be the difference between the replacement cost and the salvage value. If equal payments are placed in the fund at the end of each quarter and the fund earns 8% compounded quarterly, what should each payment be?
26. 26. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.4 Annuities Example 11 – Sinking Fund Solution: Amount needed after 8 years = 7000 − 700 = \$6300. For n = 4(8) = 32, r = 0.08/4 = 0.02, and S = 6300, the periodic payment R of an annuity is 45.142\$ 6300 6300 02.032 02.032 ________ ____ ≈== = ss S R Rs rn
27. 27. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.5 Amortization of Loans5.5 Amortization of Loans Amortization Formulas
28. 28. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.5 Amortization of Loans Example 1 – Amortizing a Loan A person amortizes a loan of \$170,000 by obtaining a 20-year mortgage at 7.5% compounded monthly. Find a.monthly payment, b.total interest charges, and c.principal remaining after five years.
29. 29. ©2007 Pearson Education Asia Chapter 5: Mathematics of Finance 5.5 Amortization of Loans Example 1 – Amortizing a Loan Solution: a. Monthly payment: b. Total interest charge: c. Principal value: ( ) 51.1369\$ 00625.01 00625.0 000,170 240 ≈        − = − R ( ) 40.682,158\$000,17051.1369240 =− ( ) 74.733,147\$ 00625.0 00625.01 51.1369 180 ≈        − −