Mathematics Of Finance
This document was uploaded by user and they confirmed that they have the permission to share it. If you are author or own the copyright of this book, please report to us by using this DMCA report form. Report DMCA
Overview
Download & View Mathematics Of Finance as PDF for free.
More details
- Words: 1,666
- Pages: 60
Chapter 5 Mathematics of Finance
Section 5.1 Simple Interest Section 5.2 Compound Interest Section 5.3 Annuities and Sinking Funds Section 5.4 Present Value of an Annuity and Amortization
Section 5.1
Simple Interest
Simple interest is most often used for __________of _____________ duration. The money borrowed in a loan is called the ___________. The number of dollars received by the borrower is the ________________________. In a simple interest loan, the ___________ and present value are the same. The ______________ is the fee for a simple interest loan and usually is expressed as a percent of the principal. Simple interest is paid on the principal _____________ and not paid on interest already earned.
Simple Interest The simple interest of a loan can be calculated using the following formula.
Example An individual borrows $300 for 6 months at 1% simple interest per month. How much interest is paid?
Another Example Jane borrowed $950 for 15 months. The interest was $83.13. Find the interest rate.
Future Value A loan made at simple interest requires that the borrower pay back the sum borrowed (principal) plus the interest. This total is called the future value, or amount and is equal to P + I.
Example Find the amount (future value) of a $2400 loan for 9 months at 11% interest.
Simple Discount The simple discount loan differs from the simple interest loan in that the interest is deducted from the principal and the borrower receives less than the principal.
This type of loan is referred to as a simple ____________ note. The interest deducted is the _________________. The amount received by the borrower is the ______________. The discount _______________is the percentage used. The amount repaid is the ___________________________.
Simple Discount Note D= PR = = =
Example Find the discount and the amount a borrower receives (proceeds) on a $1500 simple discount loan at 8% discount rate for 1.5 years.
Example A bank paid $987,410 for a 90-day $1 million treasury bill. What was the simple discount rate?
Example A bank wants to earn 7.5% simple discount interest on a 90-day $1 million treasury bill. How much should it bid?
Example How much should a bank bid on a 30-day $2 million treasury bill if the bank wants to earn 5.125% on its money.
HW 5.1 Pg 348-350 1-49 odd 51-65
Section 5.2
Compound Interest Suppose you deposit money into a savings account, the bank will typically pay you interest for the use of your money at a specific period of time, say every three months. The interest is usually credited to your savings account at each time period. At the next time period, the bank will pay interest on the new total, this is called _________________________. Amount of Annual Compound Interest When P dollars are invested at an annual interest rate r and the interest is compounded annually, the amount A at the end of t years is
__________________
Example Suppose $800 is invested at 6%, and it is compounded annually. What is the amount in the account at the end of 4 years?
Amount (Future Value) The general formula for finding the amount after a specified number of compound periods is
Example Suppose $800 is invested at 12% for 2 years. Find the amount at the end of 2 years if the interest is compounded (a) annually, (b) semiannually, and (c) quarterly.
Example Suppose $800 is invested at 12% for 2 years. Find the amount at the end of 2 years if the interest is compounded (a) annually
Example Suppose $800 is invested at 12% for 2 years. Find the amount at the end of 2 years if the interest is compounded (b) semiannually
Example Suppose $800 is invested at 12% for 2 years. Find the amount at the end of 2 years if the interest is compounded (c) quarterly.
Vocabulary •
Nominal Rate
Effective Rate
Effective Rate The effective rate of an annual interest rate r compounded m times per year is the simple interest rate that produces the same total value of investment per year as the compound interest.
Example The Mattson Brothers Investment Firm advertises Certificates of Deposit paying a 7.2% effective rate. Find the annual interest rate, compounded quarterly, that gives the effective rate. SOLUTION If we let i = quarterly rate, then
0.072 (1 i) 4 1 1.072 (1 i) 4 4 1.072 1 i 1.017533 1 i i 0.017533 The annual rate = 4(0.017533) = 0.070133 = 7.013% (rounded). The annual rate just found is also called the nominal rate.
HW 5.2 Pg 359-360 1-17 odd, 18-63 every 3rd
Section 5.3
Ordinary Annuity
An annuity refers to equal __________ paid at equal ________ intervals. The time between successive payments is called the ________________________________. The amount of each payment is the _________________________________. The interest on an annuity is _________________ interest.
An _______________________ is an annuity with periodic payments made at the end of each payment period.
Future Value (Amount) Payments are made at the end of each period.
where i = n= R= A=
Example How much money will you have when you retire if you save $20 each month from graduation until retirement? Let’s assume you start saving at age 22 until age 65, 43 years, and the interest rate averages 6.6% annual rate compounded monthly.
How much money will you have when you retire if you save $20 each month from graduation until retirement? Let’s assume you start saving at age 22 until age 65, 43 years, and the interest rate averages 6.6% annual rate compounded monthly.
Sinking Funds A sinking fund refers to a fund that is created when an amount of money will be __________ at some future date. For example, a family may need a new car in 3 years, or a company may expect to replace a piece of equipment in the future.
Example Darden Publishing Company plans to replace a piece of equipment at an expected cost of $65,000 in 10 years. The company establishes a sinking fund with annual payments. The fund draws 7% interest, compounded annually. What are the periodic payments?
Darden Publishing Company plans to replace a piece of equipment at an expected cost of $65,000 in 10 years. The company establishes a sinking fund with annual payments. The fund draws 7% interest, compounded annually. What are the periodic payments?
You figure you will need $30,000 to use as a down payment on a house you want to buy in 5 years. How much should you put in your bank account every month to accumulate $30,000 in 5 years if your bank account pays 5% annual interest compounded monthly?
A couple wants to start a college fund for their new born child. They figure they will need $120,000 in 18 years to pay for the child’s education in college. If they set up an annuity that pays 6.5% compounded quarterly, what is the amount of the quarterly payment they will need to achieve this goal?
HW 5.3 Pg 372-374 1-47 odd
Section 5.4
Present Value The present value of an annuity is the ______________ payment that yields the same total amount as that obtained through equal _____________ payments made over the same period of time.
Example Find the present value of an annuity with periodic payments of $2000, semiannually, for a period of 10 years at an interest rate of 6% compounded semiannually.
Equal Periodic Payments The amount needed to provide equal periodic payments can be found using the formula
or equivalently,
where P = amount needed in the fund R = amount of periodic payments i = periodic interest rate n = number of payments
Example Find the present value of an annuity (lump sum investment) that will pay $1000 per quarter for 4 years. The annual interest rate is 10%, compounded quarterly.
Two Ways to Save Money
Future Value of an Annuity
Present Value of an Annuity
Two Ways to Save Money
You decide to save for retirement by making regular payments of $50 to an annuity that pays 5% compounded monthly. How much money will you have in 20 years?
Find the lump sum payment that will yield the same amount after 20 years.
Present Value of an Annuity
The __________ required to pay out regular payments over time.
Determine the amount required to collect 60 monthly payments of $2000 at 5% compounded monthly.
Determine the amount required to collect 60 monthly payments of $2000 at 5% compounded monthly.
Amortization The amortization of a debt (___________________) requires no new formula because the amount borrowed is just the present value of an annuity.
George want to buy a car for $32,595. He decides to put nothing down on it a want to finance it for 5 years. The car company offers him an interest rate of 5.95%. What are George’s monthly payments and how much does the car actually cost him.
Example A student obtained a 24-month loan on a car. The monthly payments are $395.42 and are based on a 12% interest rate. What was the amount borrowed?
Balance of an Amortization The balance after n periods is the amount of compound interest minus the amount of an annuity. Mathematically we can find the balance using the formula n (1 i ) 1 n Balance P(1 i) R i
where P = the amount borrowed i = periodic interest rate n = number of time periods elapsed R = monthly payments Bal(# of Payments)
---- Be careful the solver menu must be filled in correctly
Example A family borrowed $60,000 to buy a house. The loan was for 30 years at 12% interest rate. The monthly payments were $617.17. What is the balance of their loan after 2 years?
HW 5.4 Pg 389-391 1-47 Odd
Section 5.1 Simple Interest Section 5.2 Compound Interest Section 5.3 Annuities and Sinking Funds Section 5.4 Present Value of an Annuity and Amortization
Section 5.1
Simple Interest
Simple interest is most often used for __________of _____________ duration. The money borrowed in a loan is called the ___________. The number of dollars received by the borrower is the ________________________. In a simple interest loan, the ___________ and present value are the same. The ______________ is the fee for a simple interest loan and usually is expressed as a percent of the principal. Simple interest is paid on the principal _____________ and not paid on interest already earned.
Simple Interest The simple interest of a loan can be calculated using the following formula.
Example An individual borrows $300 for 6 months at 1% simple interest per month. How much interest is paid?
Another Example Jane borrowed $950 for 15 months. The interest was $83.13. Find the interest rate.
Future Value A loan made at simple interest requires that the borrower pay back the sum borrowed (principal) plus the interest. This total is called the future value, or amount and is equal to P + I.
Example Find the amount (future value) of a $2400 loan for 9 months at 11% interest.
Simple Discount The simple discount loan differs from the simple interest loan in that the interest is deducted from the principal and the borrower receives less than the principal.
This type of loan is referred to as a simple ____________ note. The interest deducted is the _________________. The amount received by the borrower is the ______________. The discount _______________is the percentage used. The amount repaid is the ___________________________.
Simple Discount Note D= PR = = =
Example Find the discount and the amount a borrower receives (proceeds) on a $1500 simple discount loan at 8% discount rate for 1.5 years.
Example A bank paid $987,410 for a 90-day $1 million treasury bill. What was the simple discount rate?
Example A bank wants to earn 7.5% simple discount interest on a 90-day $1 million treasury bill. How much should it bid?
Example How much should a bank bid on a 30-day $2 million treasury bill if the bank wants to earn 5.125% on its money.
HW 5.1 Pg 348-350 1-49 odd 51-65
Section 5.2
Compound Interest Suppose you deposit money into a savings account, the bank will typically pay you interest for the use of your money at a specific period of time, say every three months. The interest is usually credited to your savings account at each time period. At the next time period, the bank will pay interest on the new total, this is called _________________________. Amount of Annual Compound Interest When P dollars are invested at an annual interest rate r and the interest is compounded annually, the amount A at the end of t years is
__________________
Example Suppose $800 is invested at 6%, and it is compounded annually. What is the amount in the account at the end of 4 years?
Amount (Future Value) The general formula for finding the amount after a specified number of compound periods is
Example Suppose $800 is invested at 12% for 2 years. Find the amount at the end of 2 years if the interest is compounded (a) annually, (b) semiannually, and (c) quarterly.
Example Suppose $800 is invested at 12% for 2 years. Find the amount at the end of 2 years if the interest is compounded (a) annually
Example Suppose $800 is invested at 12% for 2 years. Find the amount at the end of 2 years if the interest is compounded (b) semiannually
Example Suppose $800 is invested at 12% for 2 years. Find the amount at the end of 2 years if the interest is compounded (c) quarterly.
Vocabulary •
Nominal Rate
Effective Rate
Effective Rate The effective rate of an annual interest rate r compounded m times per year is the simple interest rate that produces the same total value of investment per year as the compound interest.
Example The Mattson Brothers Investment Firm advertises Certificates of Deposit paying a 7.2% effective rate. Find the annual interest rate, compounded quarterly, that gives the effective rate. SOLUTION If we let i = quarterly rate, then
0.072 (1 i) 4 1 1.072 (1 i) 4 4 1.072 1 i 1.017533 1 i i 0.017533 The annual rate = 4(0.017533) = 0.070133 = 7.013% (rounded). The annual rate just found is also called the nominal rate.
HW 5.2 Pg 359-360 1-17 odd, 18-63 every 3rd
Section 5.3
Ordinary Annuity
An annuity refers to equal __________ paid at equal ________ intervals. The time between successive payments is called the ________________________________. The amount of each payment is the _________________________________. The interest on an annuity is _________________ interest.
An _______________________ is an annuity with periodic payments made at the end of each payment period.
Future Value (Amount) Payments are made at the end of each period.
where i = n= R= A=
Example How much money will you have when you retire if you save $20 each month from graduation until retirement? Let’s assume you start saving at age 22 until age 65, 43 years, and the interest rate averages 6.6% annual rate compounded monthly.
How much money will you have when you retire if you save $20 each month from graduation until retirement? Let’s assume you start saving at age 22 until age 65, 43 years, and the interest rate averages 6.6% annual rate compounded monthly.
Sinking Funds A sinking fund refers to a fund that is created when an amount of money will be __________ at some future date. For example, a family may need a new car in 3 years, or a company may expect to replace a piece of equipment in the future.
Example Darden Publishing Company plans to replace a piece of equipment at an expected cost of $65,000 in 10 years. The company establishes a sinking fund with annual payments. The fund draws 7% interest, compounded annually. What are the periodic payments?
Darden Publishing Company plans to replace a piece of equipment at an expected cost of $65,000 in 10 years. The company establishes a sinking fund with annual payments. The fund draws 7% interest, compounded annually. What are the periodic payments?
You figure you will need $30,000 to use as a down payment on a house you want to buy in 5 years. How much should you put in your bank account every month to accumulate $30,000 in 5 years if your bank account pays 5% annual interest compounded monthly?
A couple wants to start a college fund for their new born child. They figure they will need $120,000 in 18 years to pay for the child’s education in college. If they set up an annuity that pays 6.5% compounded quarterly, what is the amount of the quarterly payment they will need to achieve this goal?
HW 5.3 Pg 372-374 1-47 odd
Section 5.4
Present Value The present value of an annuity is the ______________ payment that yields the same total amount as that obtained through equal _____________ payments made over the same period of time.
Example Find the present value of an annuity with periodic payments of $2000, semiannually, for a period of 10 years at an interest rate of 6% compounded semiannually.
Equal Periodic Payments The amount needed to provide equal periodic payments can be found using the formula
or equivalently,
where P = amount needed in the fund R = amount of periodic payments i = periodic interest rate n = number of payments
Example Find the present value of an annuity (lump sum investment) that will pay $1000 per quarter for 4 years. The annual interest rate is 10%, compounded quarterly.
Two Ways to Save Money
Future Value of an Annuity
Present Value of an Annuity
Two Ways to Save Money
You decide to save for retirement by making regular payments of $50 to an annuity that pays 5% compounded monthly. How much money will you have in 20 years?
Find the lump sum payment that will yield the same amount after 20 years.
Present Value of an Annuity
The __________ required to pay out regular payments over time.
Determine the amount required to collect 60 monthly payments of $2000 at 5% compounded monthly.
Determine the amount required to collect 60 monthly payments of $2000 at 5% compounded monthly.
Amortization The amortization of a debt (___________________) requires no new formula because the amount borrowed is just the present value of an annuity.
George want to buy a car for $32,595. He decides to put nothing down on it a want to finance it for 5 years. The car company offers him an interest rate of 5.95%. What are George’s monthly payments and how much does the car actually cost him.
Example A student obtained a 24-month loan on a car. The monthly payments are $395.42 and are based on a 12% interest rate. What was the amount borrowed?
Balance of an Amortization The balance after n periods is the amount of compound interest minus the amount of an annuity. Mathematically we can find the balance using the formula n (1 i ) 1 n Balance P(1 i) R i
where P = the amount borrowed i = periodic interest rate n = number of time periods elapsed R = monthly payments Bal(# of Payments)
---- Be careful the solver menu must be filled in correctly
Example A family borrowed $60,000 to buy a house. The loan was for 30 years at 12% interest rate. The monthly payments were $617.17. What is the balance of their loan after 2 years?
HW 5.4 Pg 389-391 1-47 Odd
Related Documents
Mathematics Of Finance
December 2019 6
Mathematics-of-finance-and-marketing-capital-management.docx
November 2019 11
Finance
November 2019 63
Mathematics
June 2020 27
Mathematics
June 2020 26
