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INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences
2007 Pearson Education Asia
Chapter 5 Chapter 5 Mathematics of Finance Mathematics of Finance
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
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
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
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)
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
nrPS 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.
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
0275.01500
38.588
500
38.5881
500
38.5881
38.5881500
6
6
6
6
r
r
r
r
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
233.27015.1ln
5.1ln
5.1ln015.1ln
5.1ln015.1ln
5.1015.1
015.1600900
n
n
n
n
n
months 9 years,68083.6 21
4233.27
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
rr
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
2007 Pearson Education Asia
Chapter 5: Mathematics of Finance
5.1 Compound Interest
Example 7 – Comparing Interest RatesIf 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.
%27.614
06125.01
and %18.61365
06.01
4
365
e
e
r
r
8
16
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 .
nrSP 1
15.764$0075.11000 36 P
0075.012/09.0 r
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, • $1500 in three years, and • a final payment at the end of five years.
What would this payment be if an interest rate of 6% compounded annually is assumed?
2007 Pearson Education Asia
Chapter 5: Mathematics of Finance
5.2 Present Value
Solution:
The equation of value is
27.1257$
02.160002.11000 208
x
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
investment Initial - values present of Sum NPV ValuePresent Net
Year Cash Flow
2 $10,000
3 8000
5 6000
2007 Pearson Education Asia
Chapter 5: Mathematics of Finance
5.2 Present Value
Example 5 – Net Present Value
Solution:
31.457$
000,2007.1600007.1800007.1000,10NPV 532
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 ofa. 1 year.
b. 5 years.
kt
k
rPS
1
13.105$100 105.0 ePeS rt
40.128$100100 25.0505.0 eeS
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 .rtSeP
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
000,254.1
2007.0
e
eSeP rt
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
0 where,...,,,, 132 aarararara n
432
2
13 ,
2
13 ,
2
13 ,
2
13 ,3
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
1r for
1
11
0
r
raars
nn
i
i
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
11
64
127
21
1
21
11
1
1
21
128127
7
r
ras
n
Present Value of an Annuity
• The present value of an annuity (A) is the sum of the present values of all the payments.
nrRrRrRA 1...11 21
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
798300.37005.042
__ a
83.3779$798300.37100 A
2
13
2
13
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
AR
Ra
rn
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
rRS
n 11
06.652$041211.135050015.012
__ S
06.52$501206.652 Interest Compund
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?
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
SR
Rs
rn
2007 Pearson Education Asia
Chapter 5: Mathematics of Finance
5.5 Amortization of Loans5.5 Amortization of LoansAmortization Formulas
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 $170,000 by obtaining a 20-year mortgage at 7.5% compounded monthly. Find
a.monthly payment,
b.total interest charges, and
c.principal remaining after five years.
2007 Pearson Education Asia
Chapter 5: Mathematics of Finance
5.5 Amortization of Loans
Example 1 – Amortizing a Loan
Solution:a. Monthly payment:
b. Total interest charge:
c. Principal value:
51.1369$
00625.01
00625.0000,170
240
R
40.682,158$000,17051.1369240
74.733,147$
00625.0
00625.0151.1369
180