3/9/12 Section 3.6/3.2Obj: SWBAT solve real world applications using exponential growth and decay functions.

Bell Ringer: Get Compound Interest Packet Part 2

HW Requests: pg 296 # 1-16 (3’s) 15, 32, 33 pg 342 #1-9 odds

Homework: pg 342 11-26 (3’s), 47, 50, 53, 55Announcements: Tuesday -Quiz Log and Exponential

Equations, Exponential Growth and Decay

Function for compound interest, compounding continuously:

P = the principal amountr = the interest ratet = the number of years

Problem

One thousand dollars is invested at 5% interest compounded continuously.

a. Give the formula for A(t), the compounded amount after t years.

b. How much will be in the account after 6 years?

c. How long is required to double the initial investment?

ProblemOne thousand dollars is invested at 5% interest compounded

continuously.

a. Give the formula for A(t), the compounded amount after t years.

b. How much will be in the account after 6 years?

c. How long is required to double the initial investment?

ProblemOne thousand dollars is invested at 5% interest compounded

continuously.

a. Give the formula for A(t), the compounded amount after t years.

b. How much will be in the account after 6 years?

= $1349.86

c. How long is required to double the initial investment?

ProblemOne thousand dollars is invested at 5% interest compounded

continuously.

a. Give the formula for A(t), the compounded amount after t years.

b. How much will be in the account after 6 years?

= $1349.86

c. How long is required to double the initial investment? 2000 = 1000 2 = ln(2) = ln() ln(2) = 0.05t ln(e) ln(2) = 0.05t ln(2) = t .05 t = 13.86 years

(0.5)(12)

(0.5)(12)

(0.5)(12)

Calculator http://iris.nyit.edu/appack/pdf/FINC201_4.pdf

An annuity is a sequence of equal periodic payments. We will be working with ordinary annuities. The deposits are made at the end of each period and at the same time the interest is posted or added to the account.

We can calculate the total of our periodic payments and the interest accrued. This value is called the Future Value of the annuity. The net amount of money returned from the annuity is its future value.

We make payments today and get the money some time in the future. (IRA’s, 401 k, mutual funds)

Future Value Equation:

wherer = the annual interest rate Ik = number of times interest is compounded per year C/Y

n = number of equal periodic payments P/Y (number of years * compounding period)R = payment amount PMTi = interest rate ()

Future Value Equation:

wherer = the annual interest ratek = number of times interest is compounded per year

n = number of equal periodic payments (number of years * compounding periods (k))R = payment amount i = interest rate ()

Geno contributes $50 per month into Morgan Park Mutual Fund that earns 7.26 APR. What is the value of Geno’s investment after heretires in 25 years?

R = $50, k = 12, r = 0.0726, n = 25*12, i = 0.0726/12

Geno contributes $50 per month into Morgan Park Mutual Fund that earns 7.26 annual interest rate. What is the value of Geno’s investment after he retires in 25 years?

R = $50, k = 12, r = 0.0726, n = 25*12, I = 0.0726/12

FV= $42211.46

How much money will he need to invest to make .5 million dollars?FV = .5 million, R = ?

Geno contributes $50 per month into Morgan Park Mutual Fund that earns 7.26 annual interest rate. What is the value of Geno’s investment after he retires in 25 years?

R = $50, k = 12, r = 0.0726, n = 25*12, I = 0.0726/12

FV= $42211.46 Geno will have $42, 211,46 for his retirement.

How much money will he need to invest to make .5 million dollars?FV = .5 million, R = ? R = $592.26 Geno needs to pay $592.26 monthly to have .5 milliondollars for retirement.

Loans and MortgagesWe want to get a loan or mortgage. We want to get the money today (present) and pay it back over time. The net amount of money put into an annuity is its present value. This value is called the Present Value of the Annuity. (Annual Percentage Rate – APR- the annual interestRate charged on consumer loans.)

We can calculate our monthly payments needed to pay back the loan using the Present Value Equation:

Pwhere r = the annual interest rate

k = number of times interest is compounded per year n = number of equal periodic payments

(number of years * compounding period)R = payment amount i = interest rate ()

Future Value Equation:

wherer = the annual interest ratek = number of times interest is compounded per year

n = number of equal periodic payments (number of years * compounding periods (k))R = payment amount i = interest rate ()

Genae buys a black C Class Mercedes Benz Coupe for $40,000. She gets financing and puts $5000 down for the car. The APR is 6%. What will be her monthly payments if she finances the car for 60 months?

Future Value Equation:

wherer = the annual interest ratek = number of times interest is compounded per year

n = number of equal periodic payments (number of years * compounding periods (k))R = payment amount i = interest rate ()

Genae buys a C Class Mercedes Benz for $40,000. She gets financing.and puts $5000 down for the car. The APR is 6%. What will be hermonthly payments if she finances the car for 60 months?

PV = 35000, k = 12, r = 0.06, n =5*12 =60, i = 0.06/12

P

Genae buys a C Class Mercedes Benz for $40,000. She gets financingand puts $5000 down for the car. The APR is 6%. What will be hermonthly payments if she finances the car for 60 months?

PV = 35000, k = 12, r = 0.06, n =5*12 =60, i = 0.06/12 35000

R= $676.65

Genae’s monthly bill will be $676.65 for 5 years to pay for her car.

How much interest will Genae pay in total?

How much interest will Genae pay?

First how much is she paying total? $676.65 * 60 = $40,598.88

Interest Paid: $40,598.88- $35000 = $5598.88

Compound InterestCompound interest is calculated each period on the original principal and all interest accumulated during past periods. Although the interest may be stated as a yearly rate, the compounding periods can be yearly, semiannually, quarterly, or even continuously.

Simple InterestSimple interest is calculated on the original principal only. Accumulated interest from prior periods is not used in calculations for the following periods. Simple interest is normally used for a single period of less than a year, such as 30 or 60 days. where:

Simple Interest I = P*r*n

p = principal (original amount borrowed or loaned)i = interest rate for one periodn = number of periods

Compound InterestCompound interest is calculated each period on the original principal and all interest accumulated during past periods. Although the interest may be stated as a yearly rate, the compounding periods can be yearly, semiannually, quarterly, or even continuously.

Type of Interest Principal Plus Interest Earned

Simple 46,000.00

Compounded Yearly 299,599.22

Compounded Quarterly 347,109.87

The power of compounding can have an astonishing effect on the accumulation of wealth. This table shows the results of making a one-time investment of $10,000 for 30 years using 12% simple interest, and 12% interest compounded yearly and quarterly.

10) The Fresh and Green Company has a savings plan for employees. If an employee makes an initial deposit of $1000, the company pays 8% interest compounded quarterly. If an employee withdraws the money after five years, how much is in the account?     11) Using the information in the question above, find the interest earned if the money is withdrawn after 35 years.  

12) Determine the amount of interest earned if $500 is invested at an interest rate of 4.25% compounded quarterly for 12 years. 

10) The Fresh and Green Company has a savings plan for employees. If an employee makes an initial deposit of $1000, the company pays 8% interest compounded quarterly. If an employee withdraws the money after five years, how much is in the account? A = $1485.95   11) Using the information in the question above, find the interest earned.Simple InterestCompound Interest  12) Determine the amount of interest earned if $500 is invested at an interest rate of 4.25% compounded quarterly for 12 years. 

10) The Fresh and Green Company has a savings plan for employees. If an employee makes an initial deposit of $1000, the company pays 8% interest compounded quarterly. If an employee withdraws the money after five years, how much is in the account? A = $1485.95

11) Using the information in the question above, find the interest earned.Simple Interest I = prt I = 1000*.08* 5 = $400Compound Interest = $1485.95 - $1000 = $485.95

12) Determine the amount of interest earned if $500 is invested at an interest rate of 4.25% compounded quarterly for 12 years.Simple Interest Compound Interest  

10) The Fresh and Green Company has a savings plan for employees. If an employee makes an initial deposit of $1000, the company pays 8% interest compounded quarterly. If an employee withdraws the money after five years, how much is in the account? A = $1485.95

11) Using the information in the question above, find the interest earned.Simple Interest I = prt I = 1000*.08* 5 = $400Compound Interest = $1485.95 - $1000 = $485.95

12) Determine the amount of interest earned if $500 is invested at an interest rate of 4.25% compounded quarterly for 12 years.Simple Interest $255 Compound Interest $830.41 

Future Value Equation: http://www.bugatti.com/en/veyron-16.4.html

wherer = the annual interest ratek = number of times interest is compounded per year

n = number of equal periodic payments (number of years * compounding periods (k))R = payment amount i = interest rate ()

Genae and Lucky want a Bugatti Veyron sports car. This car costs$1,700,000. Genae and Lucky are going to use the money Genae waspaying for the Mercedes to save up for the Bugatti. This means Genae contributes $680 per month @ 7.26 interest rate into the Genae/Lucky Bugatti fund. How long before Genae and Lucky will get their car?

Future Value Equation: http://www.bugatti.com/en/veyron-16.4.html

wherer = the annual interest ratek = number of times interest is compounded per year

n = number of equal periodic payments (number of years * compounding periods (k))R = payment amount i = interest rate ()

Genae and Lucky want a Buggatti Veyron sports car. This car costs$1,700,000. Genae and Lucky are going to use the money Genae waspaying for the Mercedes to save up for the Bugatti. This means Genae contributes $680 per month @ 7.26 interest rate into the Genae/Lucky Bugatti fund. How long before Genae and Lucky will get their car?

R = $680, k = 12, r = 0.0726, n = ?, i = 0.0726/12

Future Value Equation: http://www.bugatti.com/en/veyron-16.4.html

wherer = the annual interest ratek = number of times interest is compounded per year

n = number of equal periodic payments (number of years * compounding periods (k))R = payment amount i = interest rate ()

Genae and Lucky want a Buggatti Veyron sports car. This car costs$1,700,000. Genae and Lucky are going to use the money Genae waspaying for the Mercedes to save up for the Bugatti. This means Genae contributes $680 per month @ 7.26 interest rate into the Genae/Lucky Bugatti fund. How long before Genae and Lucky will get their car?

R = $680, k = 12, r = 0.0726, n = ?, i = 0.0726/12