# Interest Rates: Loans, Credit Cards, and Annuties. Interest Rates: Loans, Credit Cards, and Annuties 1/43

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1 Interest Rates: Loans, Credit Cards, and Annuties Interest Rates: Loans, Credit Cards, and Annuties 1/43

2 Last Time Last time we discussed compound interest and saw that money can grow very large given enough time, or a high enough interest rate. We ll see how this is relevant for discussing loans. Home loans, which often run for 30 years, are over a long enough period of time that rates of interest are very significant. We ll see why. Interest Rates: Loans, Credit Cards, and Annuties 2/43

3 Loans If you borrow money, how is the monthly payment determined? What does it even mean to say you get a car loan at an annual interest rate of 6%? Interest Rates: Loans, Credit Cards, and Annuties 3/43

4 Say you get a \$20,000 car loan at 6% per year for 5 years. From the loan company s point of view, here are two scenarios: 1 The company invests the \$20,000 at 6% per year, compounded monthly. It would have \$26,977 after 5 years. 2 The company gives you the loan, and then invests each payment you make at 6% per year, compounded monthly. To say you have a 6% loan for 5 years means the loan company would have the same amount of money in each of the two scenarios. Interest Rates: Loans, Credit Cards, and Annuties 4/43

5 Interest Rates: Loans, Credit Cards, and Annuties 5/43

6 If the company invests each payment P at an annual interest rate r, then the last payment does not generate interest, so is worth exactly P. To simplify writing we ll abbreviate r/12 by q. This is the monthly interest rate. The second to last payment generates 1 month interest, so is worth P (1 + q) at the end of the loan. The third to last payment generates 2 months interest, so is worth P (1 + q) 2 at the end of the loan. And so on. If you have n payments, then the first payment generates n 1 months interest, so is worth P (1 + q) n 1 at the end. If L is the loan amount, if the company invested the money instead of giving it to you, after n months it would have L (1 + q) n. Interest Rates: Loans, Credit Cards, and Annuties 6/43

7 So, the payment satisfies the equation L (1 + q) n = P + P (1 + q) + + P (1 + q) n 1 = P (1 + (1 + q) + + (1 + q) n 1) since both sides represent how much money the loan company would have at the end of your loan in the two different scenarios we mentioned above. Fortunately, expressions like the one on the right occur often, and people have found formulas to simplify them. Interest Rates: Loans, Credit Cards, and Annuties 7/43

8 For example, suppose we consider the expression Then s = n 1 s + 3 n = n n = ( n 1) Rearranging gives 3 n 1 = 2s, and so s = 3n 1 2 = 1 + 3s Interest Rates: Loans, Credit Cards, and Annuties 8/43

9 More generally, if a is any number (other than 1), then 1 + a + a a n 1 = an 1 a 1 Applying this to the loan situation, with a = 1 + q gives us ( 1 + (1 + q) + + (1 + q) n 1 ) = (1 + q)n 1 q Thus, our loan formula simplifies as L (1 + q) n = P (1 + (1 + q) + + (1 + q) n 1) ( (1 + q) n ) 1 = P q Interest Rates: Loans, Credit Cards, and Annuties 9/43

10 Solving for P gives ( (1 + q) L (1 + q) n n ) 1 = P q P = Lq (1 + q)n (1 + q) n 1 or, if we divide the top and bottom of the fraction by (1 + q) n, P = Lq 1 (1 + q) n Interest Rates: Loans, Credit Cards, and Annuties 10/43

11 Loan Formula To summarize, if we borrow L at an interest rate of r per year, and make n payments, then the monthly payment P is P = Lr ( 12 1 ( 1 + r ) ) n 12 Using the Interest Calculator spreadsheet or an online financial calculator is a good way to do these calculations. Last time we saw the Moneychimp.com calculator that can do these sort of calculations without having to use the formulas. Interest Rates: Loans, Credit Cards, and Annuties 11/43

12 Clicker Questions Q What is the monthly payment for a \$20,000 car loan at an annual interest rate of 6% for 5 years? The loan formula is P = Lr ( 12 1 ( 1 + r ) ) n 12 but it is a lot easier to do in the MoneyChimp calculator. Interest Rates: Loans, Credit Cards, and Annuties 12/43

13 Answer A The monthly payment (not including insurance, taxes, license, etc.) would be \$ Q What if the loan was at 8%? A The monthly payment would now be \$ Q A 2014 Mercedes-Benz SLS AMG convertible starts at \$208,000. If you borrowed \$200,000 at 6% for 5 years to buy one, what would your monthly payment be? A It would be 10 times the first answer, or \$3, Interest Rates: Loans, Credit Cards, and Annuties 13/43

14 If you bought the Mercedes, your total payments for the loan would be \$3, = \$231, This means you d pay about \$32,000 in interest in order to borrow \$200,000 for 5 years at 6%. Interest Rates: Loans, Credit Cards, and Annuties 14/43

15 Home Loans With home loans typically for 30 years, interest is much more an issue than for car loans. Getting a 15 year loan can save a huge amount of money! Interest Rates: Loans, Credit Cards, and Annuties 15/43

16 Clicker Question Let s suppose you borrow \$150,000 to buy a house. Let s first compare doing this now to the early 1980s, when interest rates were much higher. How much more money in interest do you think you d pay with a 15% loan versus a 4% loan over the lifetime of the loan? A \$4,000 more B \$40,000 more C \$400,000 more D \$4,000,000 more E \$40,000,000 more Interest Rates: Loans, Credit Cards, and Annuties 16/43

17 Answer C You d pay about \$400,000 more in interest. Let s see why. We ll use the Interest Calculator spreadsheet to do this. The results do not include real estate taxes and insurance, which can be a few hundred dollars a month. Rate Monthly Payment Total Payments Interest Paid 4% \$ \$257,804 \$107,804 6% \$ \$323,757 \$173,757 10% \$1,316,36 \$473,889 \$323,889 15% \$1, \$682,800 \$532,800 Interest Rates: Loans, Credit Cards, and Annuties 17/43

18 Mortgage rates of 15% were common in in the 1980s. By 1990 rates were down around 10%. Currently the lowest rates are at or under 4%. If you get a 15 year home loan, generally you ll get a better interest rate. Let s compare a 30 year loan at 10% to a 15 year loan at 9%. We ll continue to consider a \$150,000 loan. Again, we ll use the Interest Calculator spreadsheet for this. The results are Rate Monthly Payment Total Payments 10% for 30 years \$1,316,36 \$473,889 9% for 15 years \$1, \$273,852 Interest Rates: Loans, Credit Cards, and Annuties 18/43

19 Let s do the same but for a 30 year loan at 6% and a 15 year loan at 4.5%. The results are Rate Monthly Payment Total Payments 6% for 30 years \$ \$323, % for 15 years \$1, \$206,548 Interest Rates: Loans, Credit Cards, and Annuties 19/43

20 Rate Monthly Payment Total Payments 10% for 30 years \$1,316,36 \$473,889 9% for 15 years \$1, \$273,852 6% for 30 years \$ \$323, % for 15 years \$1, \$206,548 In the first example, going from a 30 year loan to a 15 year loan would save about \$200,000 over the life of the loan. In the second example the savings isn t as much, but it is still about \$115,000. While affording the higher monthly payment may not always be possible, if you can do it you ll save a lot of money in the long run. Interest Rates: Loans, Credit Cards, and Annuties 20/43

21 Refinancing While this isn t as relevant now since interest rates are so low, at times interest rates drop a few points after you buy a house. If you borrow \$150,000 for 30 years at 7%, and you have the opportunity to refinance with a 15 year loan at 4%, is it a good idea? At least, what are the monthly payments? Rate Monthly Payment Total Payments 7% for 30 years \$ \$359,263 4% for 15 years \$ \$199,716 Refinancing would increase your monthly payment by about \$115, but you d save about \$160,000 over the life of the loan. Interest Rates: Loans, Credit Cards, and Annuties 21/43

22 The calculation above represents what would happen if you refinance at the very beginning of your loan. For a little more realistic example, suppose you had a 7% loan and paid on it for 5 years, at which time you could refinance for 4% for a 15 year loan. To do this calculation we d need to know how much you still owe. The Loan Schedule tab of the Interest Calculator spreadsheet does this kind of calculation. After 5 years you d still owe \$141,000. Rate Monthly Payment Total Payments 7% for 25 more years \$ \$299,386 4% for 15 years \$ \$187,733 Interest Rates: Loans, Credit Cards, and Annuties 22/43

23 To get the amount we d pay with the 7% loan, we take the \$359,263 we d pay over 30 years, and multiply by 25/30 to see how much we d pay in 25 years. Or, to come close to this amount, we could see how much we d pay by borrowing \$141,000 for 25 years at 7%. Since there are up-front costs in refinancing, typically a few thousand dollars, whether this is a good idea depends on how long you ll keep the house and how far into your mortgage you are. If you plan to keep your house for at least 5 years, refinancing when you can get a lower interest rate is often a good deal. Interest Rates: Loans, Credit Cards, and Annuties 23/43

24 Credit Cards Credit cards work the same as loans.the main difference is the high interest rate most charge. Interest rates up to 20% per year have been common. If you pay off your credit card in full each month, then you don t get charged interest. Using a credit card this way amounts to treating it as a debit card from your checking account. What happens if you don t pay it off in full? More particularly, what if you pay the minimum payment each month? Interest Rates: Loans, Credit Cards, and Annuties 24/43

25 The minimum payment on a credit card bill is typically the larger of a fixed amount and a certain percentage of your balance. Suppose your minimum payment is the larger of \$20 or 1.5% of your balance. Let s suppose the credit card company charges you 15% interest on unpaid balances. Let s also suppose you have a \$10,000 credit limit, and you max out your credit card. Your next statement then shows a \$10,000 balance. Interest Rates: Loans, Credit Cards, and Annuties 25/43

26 Clicker Question Q What is your minimum payment on a \$10,000 balance, when the credit card company requires you to pay at least the larger of \$20 or 1.5% of your balance? A Since 1.5% of \$10,000 is your minimum payment is \$150. \$10, = \$150 Interest Rates: Loans, Credit Cards, and Annuties 26/43

27 Let s now think what will happen if you make the minimum payment each month on your credit card. Your credit card company will charge you 15% = 1.25% interest on an unpaid balance each month. 12 You pay your minimum payment of \$150 the first month. The next statement you ll have a starting balance of \$9,850 since you paid \$150 of your \$10,000 balance. The company will then charge you 1.25% of that in interest. Interest Rates: Loans, Credit Cards, and Annuties 27/43

28 Clicker Questions Q If your balance is \$9,850 and you pay 1.25% interest, how much interest do you owe that month? A You owe in interest charges that month. \$9, = \$123 Q What will be your next monthly payment, if you pay the minimum payment? A Your new balance is \$9,850 + \$123 = \$9,973, and you will have to pay 1.5% of that for your minimum payment. So, your minimum payment is \$9, = \$ Interest Rates: Loans, Credit Cards, and Annuties 28/43

29 We can continue seeing what happens when we pay the minimum payment each month. The result for the first year will be the following table. Interest Rates: Loans, Credit Cards, and Annuties 29/43

30 After 12 months of payments you ll have paid around \$1,774 with \$1,336 of that in interest. You will still owe around \$9,700. One problem with this, besides the large amount of interest you pay, is that you won t be able to use your credit card very much, because your balance is close to your credit limit. To continue, we could keep going month by month. However, with some algebra we can be more efficient. This will help to see what will happen after several years. Interest Rates: Loans, Credit Cards, and Annuties 30/43

31 Suppose the original balance in some month (after the first) is B. We then get charged B interest. The new balance is then B + B = B Our minimum payment is 1.5% of this, or B Our next month s balance is then the previous month s new balance minus the minimum payment. That is, the next month s original balance is new balance payment = = B B = B = B Interest Rates: Loans, Credit Cards, and Annuties 31/43

32 Using the same reasoning, the original balance two month s later is (B ) = B (0.9973) 2 In general, n months later, the original balance would be B (0.9973) n Interest Rates: Loans, Credit Cards, and Annuties 32/43

33 This helps to figure out interest and payments years at a time. The results are summarized in the spreadsheet Credit Card Calculations.xlsx. These calculations are the reasons credit card companies are happy for people to pay the minimum payment. Interest Rates: Loans, Credit Cards, and Annuties 33/43

34 Annuities An annuity is a repeating payment, typically of a fixed amount, over a period of time. An annuity is like a loan in reverse; rather than paying a loan company, a bank or investment company pays you the monthly payment. There are two typical calculations for annuities. 1 Paying into an annuity 2 Collecting from an annuity Interest Rates: Loans, Credit Cards, and Annuties 34/43

35 Paying into an annuity means investing or saving money in order to receive an annuity in the future. Collecting from annuity then happens once you have invested enough money to receive an annuity. The MoneyChimp website isn t so easy to use for annuities. We ll use a bankrate.com website for collecting from annuity calculations. The Interest Calculator spreadsheet is a convenient way to do all these calculations. You can use the compound interest calculator on MoneyChimp.com to do paying into an annuity calculation with some trial and error. Interest Rates: Loans, Credit Cards, and Annuties 35/43

36 Paying Into an Annuity Q Suppose you need to have \$100,000 saved 20 years from now. If you can invest at 6% per year, how much do you need to put away each month? A We d have to invest \$216 each month to end up with \$100,000 in 20 years if we received 6% on our money. Q What if instead you want to end up with \$500,000 in 20 years? How much do you have to invest? A You want to end up with 5 times more money, so you need to invest 5 times more each month, or \$1,082. Interest Rates: Loans, Credit Cards, and Annuties 36/43

37 We saw that if you want to end up with \$500,000 by saving for 20 years at a 6% annual return, you need to invest \$1,082 each month. Q How much do you need to invest to end up with \$500,000 if you save for 30 years? A You d need to invest \$498. Saving for a longer time means you have to save far less money each month. Interest Rates: Loans, Credit Cards, and Annuties 37/43

38 Collecting From an Annuity Q Suppose you ve saved \$250,000 that you put into an annuity paying 5% per year, and you wish to collect from it for 20 years. How much monthly income will you receive? A You ll receive \$1,643. If you do this sort of calculation 30 years before you plan to retire, you need to realize the amount you ll receive will be in future dollars and will sound better than the same amount today. Q What if you want to collect for 30 years? A You will collect \$1,336 each month for 30 years. Interest Rates: Loans, Credit Cards, and Annuties 38/43

39 Suppose you ve invested \$250,000 into an annuity paying 6% and want to be paid from it indefinitely. How much will you receive? Collecting indefinitely is treated as collected forever. This is called a perpetual annuity. Surely you d get almost nothing if they let you collect forever. Let s look at some data, where we take a larger and larger number of years to collect from the annuity. Interest Rates: Loans, Credit Cards, and Annuties 39/43

40 If you invest \$250,000 in an annuity paying 5% per year, this table shows your monthly income depending on how long you collect. Number of Years Monthly to Collect Income 30 \$1, \$1, \$1, \$1, \$1, \$1, \$1, \$1,042 In fact, you can collect a reasonable amount indefinitely. Interest Rates: Loans, Credit Cards, and Annuties 40/43

41 Collecting From a Perpetual Annuity If you have L invested in an annuity paying an annual rate of r, then you ll collect L r 12 each month from a perpetual annuity. For example, in the previous example, L = \$250, 000 and r = So, the monthly return would be \$250, = \$1, This is an easier calculation and doesn t require you to estimate how long you ll collect. Interest Rates: Loans, Credit Cards, and Annuties 41/43

42 One way to see that collecting from a perpetual annuity is a reasonable thing in terms of getting a decent amount of money is to realize that what you are doing is simply withdrawing the interest earned each period. By withdrawing the interest the principal stays untouched. Doing this allows you to earn income each month but keep the principal unchanged. This is a reasonable retirement income method, as long as you have enough principal to survive on the interest and any pension/social security you have. Interest Rates: Loans, Credit Cards, and Annuties 42/43

43 Next Week and Assignment 9 Next time we will discuss issues of statistics. Assignment 9 is on the website. It is due next Thursday. Interest Rates: Loans, Credit Cards, and Annuties 43/43

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