Supplement to 9 4 Introduction This handout parallels the extra material that was covered in the lecture on 9 4 Specifically you will learn how to use differential equations to analyze loans and plan for your retirement Continuous Growth Rates vs Discrete Growth Rates What is a continuous growth rate dy ry where r represents the dt rate that y is growing Because we are using derivatives this must be a continuous growth rate That is it is the rate expressed as a percent that the population is growing at a particular instance This concept is hard to get your brain around at first but after working through the example below it will make sense Recall that our model of constant continuous growth is Imagine that A and B are each opening a new bank and are trying to attract new customers Due to certain banking regulations they are only allowed to offer a return of 6 on savings account Trying to drum up business both approach a potential customer who is trying to decide where to deposit his 1 000 A If you put your 1 000 in my bank and come back in one year you will have 1 060 in your account B You will make more money by going with me Instead of giving you 6 at the end of the year I will give you 3 six months from now and 3 after the next six months Notice that you have extra money after the first six months and you will earn interest on your interest during the next six months This is how it will work After six months you will have 1 030 in your account 1 000 1 03 After the next six months you will have 1 060 90 1 030 1 03 In other words after one year you will have 1 000 1 03 1 03 1 060 90 A I can beat that I will pay you 1 5 interest every 3 months That is still 6 but now your interest earns interest and the interest on your interest will earn interest and that interest will earn interest too Here is how your account will grow After 3 months 1 000 1 015 1 015 00 After 6 months 1 000 1 015 1 015 1 030 23 After 9 months 1 000 1 015 1 045 68 3 1 000 1 015 1 061 36 4 After 1 year B Oh yeah I will compound the interest every day At the end of the year this fine 12 gentleman will have 1 000 1 06 12 1 061 68 You can imagine A and B offering to compound interest daily hourly and even by the second Eventually one will decide to take the limit of this process Mathematically the depositor will have lim 1 000 1 06 n n n 1061 84 in his account at the end of the year Using techniques you learned in Calculus 1 you know that lim 1 000 1 06 n n n 1000e 06 This is what is meant by continuously compounded interest This matches up with our original equation dy ry The solution as you know is dt y t Ce r t How do you compute a continuous growth rate Suppose you are trying to decide which investment is better 6 compounded annually or 5 85 compounded continuously The best way to compare is to convert the 6 annual compounding to a continuous rate Here is how to do this Imagine that you are investing 1 000 at 6 compounded annually At the end of the year you will have 1 060 If you had an interest rate of r compounded continuously and still made 1 060 by the end of the year r would satisfy the equation 1060 1000e r With a bit of algebra shows you see that r ln 1060 1000 0582 5 82 So the 5 85 compounded continuously is a better deal This technique will also work with other compounding periods Let s convert 6 compounded quarterly to a continuously compounded rate Calculate what 1 000 will grow to and find the equivalent continuous rate 1 000 1 064 1 061 36 4 1 061 36 1 000e r t r ln 1061 36 1000 0596 5 96 Planning Your Retirement Suppose that on the day of your retirement you have 1 000 000 in an account that earns 5 compounded continuously You also decide that you will withdraw 60 000 per year for your expenses How long will your money last Assume that the money is withdrawn continuously Solution Let x t the amount of money you have in the bank in year t The interest that the bank pays you increases the amount of money you have and will be a positive contribution to the rate of change of your balance Your withdrawals will cause your balance to decrease and will add a negative term to the rate your balance is changing dx Specifically the initial value problem is 05 x 60 000 x 0 1000 000 You want dt to find the value of t such that x t 0 Here is how you work it ou dx dx dx 05 x 1200 000 05 dt 05 dt dt x 1200 000 x 1200 000 ln x 1200 000 05t C x t Ce 05t 1200 000 Using the initial condition x 0 1000 000 we see that C 200 000 the amount of money left in the account at time t is x t 200 000e 05t 1200 000 If 0 200 000e 05t 1200 000 e 05t 1200000 200000 t 35 84 The money will last slightly less than 36 years Buying a House Imagine you are thinking about taking out a 30 year fixed rate mortgage You want to borrow 500 000 and can get an interest rate of 6 What would your monthly payment be Solution To use the tools of this section we will make some simplifying assumptions First quoted mortgage rates are compounded monthly We could get the continuous rate It turns out that this does not affect the answer significantly so we will stick with 06 which is easy to work with Secondly mortgage payments are made monthly Our model assumes that payments are made continuously Again this will not affect the final answer by too much Since the 6 is the yearly rate let Y be the yearly payment You will divide Y by 12 to get the monthly payment As in the previous problem let x t be the amount of money you owe on the loan at year t and determine what factors increase the balance of the loan and which ones decrease the balance Specifically the interest will add a term of 06x to the rate of change of your balance and your yearly payment Y will bring the amount of money you owe down Here is the equation dx 06 x Y dt Since the initial balance is 500 000 we have x 0 500 000 Since the loan will be paid off in 30 years we have …
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