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# Class Note for MATH 1313 at UH

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This 10 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 40 views.

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Date Created: 02/06/15

Math 1313 Section 51 Section 51 Simple Interest Future Value Present Value and Effective Rate Simple Interest is interest that is compounded on the original principal only P principal r interest rate per year to decimal t time in years Accumulated Amount is the sum of the principal and interest after t years Formula A P l A P Prt A P1rt Example 1 Find the simple interest on a 1000 investment made for 3 years at an interest rate of 5 per year What is the accumulated amount Example 2 Find the simple interest rate at which 1000 will grow to 1050 in 9 months Compounded Interest is earned interest that is periodically added to the principal and there after earns interest at the same rate Future Value with compound interest formula APliquot wherei1andn mt m A the Future Value or the accumulated amount at the end ofn conversion periods A conversion period refers to the interval of time between successive interest calculations P present value or principal r the interest rate per year m the number of compounding periods per year t time years Math 1313 Section 51 Example 3 Find the accumulated amount after 5 years if 1700 is invested at 625 per year compounded a quarterly b semiannually Present Value Recall A P1 1quot and P Present Value Why would we want to find P Well in certain instances an investor may wish to determine how much money he should invest now at a fixed rate ofinterest so that heshe will realize a certain sum ofmoney at some future date So solving the Future Value formula for P we obtain the Present Value with compound interest formula AP1i 2 p 111 PA1z39 Example 4 Kim and Ken find that they will need 15500 to build an addition to their home in 4 years How much should they invest now at 325 per year compounded quarterly to have the desired funds in 4 years Example 5 A newborn child receives a 5000 gift towards a college education from her grandparents How much will the 5000 be worth in 17 years if it is invested at 9 per year compounded quarterly Math 1313 Section 51 Example 6 An Individual Retirement Account IRA has 20000 in it and the owner decides not to add any more money to the account other than interest earned at 8 per year compounded monthly How much will be in the account 35 years from now when the owner reaches retirement age Example 7 Kim invested a sum of money 4 years ago in a savings account that has since paid interest at the rate of 65 per year compounded monthly Her investment is now worth 1944031 How much did she originally invest Example 8 Kate s son will be leaving to an outof state private university this year Twenty years ago she set up an account for him to help pay for his college tuition She decides now is the right time to pay out the total amount earned in this account and close the account The amount she receives is 25 67890 Her son would like to know how much his mother had invested in this account 20 years ago at the rate of 7 per year compounded monthly Math 1313 Section 51 Effective Rate Effective Rate of Interest Formula r m l eff 1 1 Where reff is the Effective rate of interest 139 is the nominal interest rate per year m is the number of conversion periods per year The effective rate formula shows that money invested at V4 simple interest earns the same amount ofinterest in one year as money invested at r per year compounded In times a year Example 9 Find the effective rate corresponding to the nominal rate of 9 per year compounded quarterly per year Section 52 7 Annuities 1 Section 52 Annuities An ordinary annuity is a sequence of equal periodic payments made at the end of each payment period Examples of annuities 1 Regular deposits into a savings account 2 Monthly home mortgage payments 3 Payments into a retirement account We will study annuities that are subject to the following conditions 1 The terms are given by fixed time intervals 2 The periodic payments are equal in size 3 The payments are made at the end of the payment periods 4 The payment periods coincide with the interest conversion periods The Future Value of an Annuity is sum of all payments made and interest earned on an account Future Value of an Annuity The future value 5 of an annuity ofn payments ofR dollars each paid at the end of each investment period into an account that earns interest at the rate ofi per period is s l S future value R payment r I m r annual interest rate m conversion periods per year n mt t time in years Example 1 Find the future value of the ordinary annuity of 1500 per semiannual period for 8 years at 9 per year compounded semiannually Section 52 7 Annuities 2 Present Value of an Annuity Why would we want to know the present value of an annuity Well in some cases we would like to know how much money we should put away now at the present time so that at a later time we could withdraw a certain amount of money for a certain length of time In other cases we may wish to know how much something cost us originally Present Value of an Annuity The present value P of an annuity of n payments ofR dollars each paid at the end of each investment period into an account that earns interest at the rate ofi per period is 139 Example 2 Find the present value of the ordinary annuity 0f 3000per quarterly period for 6 years at 11 per year compounded quarterly Example 3 Carrie opened an IRA on January 31 1990 with a contribution of 2000 She plans to make a contribution of 2000 thereafter on January 31 of each year until her retirement in the year 2009 20 payments If the account earns interest at the rate of 8 per year compounded yearly how much will Carrie have in her account when she retires Section 52 7 Annuities 3 Example 4 Parents ofa college student wish to set up an account that will pay 350 per month to the student for 4 years How much should they deposit now at 9 per year compounded monthly so that their child will have the desired amount of money each month for 4 years Example 5 Christopher s monthly net pay is 339400 He decides to make monthly deposits of 10 of his monthly net pay into an account that pays 95 per year compounded monthly How much will he have in the account after 2 years assuming his net pay remains the same for the next two years Example 6 Tim pays 320 per month for 4 years for a car making no down payment If the loan borrowed costs 6 per year compounded monthly what was the original cost of the car How much interest will be paid Section 52 7 Annuities 4 Example 7 Tom and Ierri paid 10000 down toward a new house They also have a 30year mortgage for which they pay 1100 per month Ifinterest is 635 per year compounded monthly What did the house that they purchased originally cost Example 8 Betsy estimates that she spends 150 per month on cigarettes She decides that she will quit and invest the monthly amount spent on cigarettes in an account earning 225 per year compounded monthly At the end of 5 years she will donate the interest earned on her account to her favorite charity How much money can the charity expect to get in 10 years Section 53 7 Amortizations and Sinking Funds 1 Section 53 Amortization and Sinking Funds Amortization You borrowed money from a bank and want to know how much should your monthly payments be so that the loan will be amortized paid off at the end of the term 0fthe loan Amortization Formula The periodic payment R on a loan of P dollars to be amortized over n periods with interest charged at the rate 0f139 per period is Pi R 1 1i n Sinking Fund A sinking fund is a fund is setup for a speci c purpose at some future date Sinking Fund Formula The periodic payment R required to accumulate a sum ofS dollars over n periods with interest charged at the rate 0f139 per period is Si R 1 iquot 1 Example 1 Kelly wishes to buy a car that costs 32998 The car dealer tells her that they can finance the car at 625 per year compounded monthly for 5 years She decides to secure the loan from the dealer How much will her monthly payments be Example 2 A person would like to have 200000 in an account for retirement 15 years from now How much should be deposited quarterly in an account paying 6 per year compounded quarterly to obtain this amount Section 53 7 Amortizations and Sinking Funds 2 Example 3 A sailboat costs 16000 You pay 15 down and secure a loan for the remaining balance How much are your monthly payments if 18 per year compounded monthly is charged over a period of6 years Example 4 Christina plans to go to Disney World in two summers and Wishes to have 7000 by then How much money should she deposit monthly for the next 2 years in an account paying 325 per year compounded monthly to achieve this goal Example 5 Business partners Bill and Bob buy an apartment house for 1250000 by making a down payment of 125000 and financing the rest with semiannual payments over the next 10 years The interest rate on the debt is 8 per year compounded semiannually How much is their semiannually payment

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