Chapter05_Mathematics of Finance_13e
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S P1 r
n
Example 1 – Compound Interest
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.
Chapter 5: Mathematics of Finance
Chapter Objectives
• To solve interest problems which require logarithms. • To solve problems involving the time value of money.
2
3 5
$10,000
8000 6000
Assume an interest rate of 7% compounded annually and find the net present value of the cash flows.
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance 5.2 Present Value
Example 5 – Net Present Value
Solution: Substracting the initial investment from the sum of the present values of the cash flows gives
NPV 10,0001.07 80001.07 60001.07 20,000
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance 5.2 Present Value
Net Present Value
Net Present Value NPV Sum of present values - Initial investment
5001 r 588.38 1 r 6 588.38 500
6
588.38 1 r 500
6
r 6
588.38 1 0.0275 500
The semiannual rate was 2.75%, so the nominal rate was 5.5 % compounded semiannually.
P S1 r
n
Example 1 – Present Value
Find the present value of $1000 due after three years if the interest rate is 9% compounded monthly. Solution: S=1000, r =0.09/12, n =3(12) For interest rate, r 0.09 / 12 0.0075. 36 Principle value is P 10001.0075 $764.15.
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance
5.2 Present Value
• P that must be invested at r for n interest periods so that the present value, S is given by
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance 5.1 Compound Interest
Example 2 – Compound Interest
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.
2011 Pearson Education, Inc.
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance 5.1 Compound Interest
Example 1 – Compound Interest
Solution: Let r be the semiannual rate. There are 2 × 3 = 6 interest periods.
INTRODUCTORY MATHEMATICAL ANALYSIS
For Business, Economics, and the Life and Social Sciences
Chapter 5 Mathematics of Finance
2011 Pearson Education, Inc.
900 6001.015
n
1.015n 1.5 n ln1.015 ln 1.5
n ln 1.015 Biblioteka Baiduln 1.5 ln 1.5 n 27.233 ln 1.015
It will take
27 .233 4
1 6.8083 6 years,9 2 months .
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance
5.1 Compound Interest
• Compound amount S at the end of n interest periods at the periodic rate of r is as
Example 2 – 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. Year Cash Flow
• To solve problems with interest is compounded continuously.
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance
Chapter Outline
5.1) Compound Interest 5.2) Present Value 5.3) Interest Compounded Continuously
2 3 5
$457.31
Since NPV <0, business venture is not profitable if one considers the time value of money. It would be better to invest the $20,000 in a bank paying 7%, since the venture is equivalent to investing only $20,000 -$457.31= $19,542.69
n
Example 1 – Compound Interest
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.
Chapter 5: Mathematics of Finance
Chapter Objectives
• To solve interest problems which require logarithms. • To solve problems involving the time value of money.
2
3 5
$10,000
8000 6000
Assume an interest rate of 7% compounded annually and find the net present value of the cash flows.
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance 5.2 Present Value
Example 5 – Net Present Value
Solution: Substracting the initial investment from the sum of the present values of the cash flows gives
NPV 10,0001.07 80001.07 60001.07 20,000
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance 5.2 Present Value
Net Present Value
Net Present Value NPV Sum of present values - Initial investment
5001 r 588.38 1 r 6 588.38 500
6
588.38 1 r 500
6
r 6
588.38 1 0.0275 500
The semiannual rate was 2.75%, so the nominal rate was 5.5 % compounded semiannually.
P S1 r
n
Example 1 – Present Value
Find the present value of $1000 due after three years if the interest rate is 9% compounded monthly. Solution: S=1000, r =0.09/12, n =3(12) For interest rate, r 0.09 / 12 0.0075. 36 Principle value is P 10001.0075 $764.15.
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance
5.2 Present Value
• P that must be invested at r for n interest periods so that the present value, S is given by
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance 5.1 Compound Interest
Example 2 – Compound Interest
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.
2011 Pearson Education, Inc.
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance 5.1 Compound Interest
Example 1 – Compound Interest
Solution: Let r be the semiannual rate. There are 2 × 3 = 6 interest periods.
INTRODUCTORY MATHEMATICAL ANALYSIS
For Business, Economics, and the Life and Social Sciences
Chapter 5 Mathematics of Finance
2011 Pearson Education, Inc.
900 6001.015
n
1.015n 1.5 n ln1.015 ln 1.5
n ln 1.015 Biblioteka Baiduln 1.5 ln 1.5 n 27.233 ln 1.015
It will take
27 .233 4
1 6.8083 6 years,9 2 months .
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance
5.1 Compound Interest
• Compound amount S at the end of n interest periods at the periodic rate of r is as
Example 2 – 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. Year Cash Flow
• To solve problems with interest is compounded continuously.
2011 Pearson Education, Inc.
Chapter 5: Mathematics of Finance
Chapter Outline
5.1) Compound Interest 5.2) Present Value 5.3) Interest Compounded Continuously
2 3 5
$457.31
Since NPV <0, business venture is not profitable if one considers the time value of money. It would be better to invest the $20,000 in a bank paying 7%, since the venture is equivalent to investing only $20,000 -$457.31= $19,542.69