Chapter 5 Mathematics of Finance

INTRODUCTORY MATHEMATICAL ANALYSIS 0. Review of Algebra Applications and More Algebra Functions and Graphs Lines, Parabolas, and Systems Exponential and Logarithmic Functions Mathematics of Finance Matrix Algebra Linear Programming Introduction to Probability and Statistics

INTRODUCTORY MATHEMATICAL ANALYSIS Additional Topics in Probability Limits and Continuity Differentiation Additional Differentiation Topics Curve Sketching Integration Methods and Applications of Integration Continuous Random Variables Multivariable Calculus

Chapter 5: Mathematics of Finance Chapter Objectives 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 Outline Compound Interest 5.1) Present Value Chapter 5: Mathematics of Finance Chapter Outline Compound Interest Present Value Interest Compounded Continuously Annuities Amortization of Loans 5.1) 5.2) 5.3) 5.4) 5.5)

Chapter 5: Mathematics of Finance 5.1 Compound Interest Example 1 – Compound Interest Compound amount S at the end of n interest periods at the periodic rate of r is as 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.

There are 2 × 3 = 6 interest periods. Chapter 5: Mathematics of Finance 5.1 Compound Interest Example 1 – Compound Interest Solution: There are 2 × 3 = 6 interest periods. The semiannual rate was 2.75%, so the nominal rate was 5.5 % compounded semiannually.

The periodic rate is r = 0.06/4 = 0.015. Chapter 5: Mathematics of Finance 5.1 Compound Interest Example 3 – 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. It will take .

Example 5 – Effective Rate Effective Rate 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 To what amount will $12,000 accumulate in 15 years if it is invested at an effective rate of 5%? Solution:

Respective effective rates of interest are 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.

Chapter 5: Mathematics of Finance 5.2 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 .

a final payment at the end of five years. 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?

The equation of value is Chapter 5: Mathematics of Finance 5.2 Present Value Solution: The equation of value is

Example 5 – Net Present Value Chapter 5: Mathematics of Finance 5.2 Present Value Example 5 – Net Present Value 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. Year Cash Flow 2 $10,000 3 8000 5 6000

Solution: Chapter 5: Mathematics of Finance 5.2 Present Value Example 5 – Net Present Value Solution:

5.3 Interest Compounded Continuously Chapter 5: Mathematics of Finance 5.3 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.

Effective Rate under Continuous Interest Chapter 5: Mathematics of Finance 5.3 Interest Compounded Continuously Effective Rate under Continuous Interest Effective rate with annual r compounded continuously is . Present Value under Continuous Interest Present value P at the end of t years at an annual r compounded continuously is .

We want the present value of $25,000 due in 20 years. 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.

5.4 Annuities Sequences and Geometric Series Chapter 5: Mathematics of Finance 5.4 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

b. Geometric sequence with a = 1, r = 0.1, and n = 4. 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. Sum of Geometric Series The sum of a geometric series of n terms, with first term a, is given by

Find the sum of the geometric series: 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 Present Value of an Annuity The present value of an annuity (A) is the sum of the present values of all the payments.

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,

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,

The amount S of ordinary annuity of R for n periods at r per period is 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,

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?

Amount needed after 8 years = 7000 − 700 = $6300. 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

5.5 Amortization of Loans Amortization Formulas Chapter 5: Mathematics of Finance 5.5 Amortization of Loans Amortization Formulas

total interest charges, and principal remaining after five years. 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 monthly payment, total interest charges, and principal remaining after five years.

Solution: a. Monthly payment: b. Total interest charge: 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: