Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Math 102 class: unit 1 finance
Typology: Assignments
1 / 80
This page cannot be seen from the preview
Don't miss anything!
This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 4.0 International License 4 th^ Edition (Fall 2019)
As authors, we don’t know the battles you face every day. We don’t know everything about your past that has led you to this moment, to this school, and to this class. But you are here now and others are not. This is a chance for you to learn a lot of new things that will help you in your life. With money, you could learn some things that will help you plan for your financial future and be better prepared. But it can be scary. All we ask is that you step forward and give yourself a chance. It’s not easy to step forward; being in your comfort zone of safety is easier. We believe in you and hope that you believe in yourself too. Historical Note: In 1879, about 15 years after the civil war, the US moved to a gold standard, which meant that you could exchange paper money for actual gold (or silver). The civil war produced problems with currency as both the North and South created their own. After re-unifying, the country moved to gold as a basis for currency to make things easier. The first major crisis was the Great Depression, causing the public to hoard gold as the way to build and maintain wealth. Great Britain moved away from the gold standard in 1931, and in 1933, President Roosevelt started the movement away from the gold standard. However, it took until 1971 for the US to completely abandon the gold standard. How does money work? Picture this, you have a nice car and someone wants to borrow it for a while. Because it is your car, would you let them borrow it for free? Or would you charge them a little money to cover the fact that you aren’t able to use the car when you want to. This idea of renting a car is similar to the idea of interest. Interest is like renting money for a while when you need more than you currently have. But how is the rental cost determined and what would influence the amount of rent that should be charged? Would you charge someone more rent for keeping the car longer or shorter? Would you charge someone more rent if the car was more expensive or if it was cheaper? And what rate would you charge – is it the same all the time or could the rates change from time to time? Would you charge more when there aren’t as many cars to rent or maybe charge less when everyone is renting a car? Just like “surge” pricing with Uber! All of these concepts come into play when renting money too. Now the value of our currency fluctuates with inflation and deflation. And along with these fluctuations go the interest rates charged to borrow money. The interest charged (like the rent for a car) is measured by three factors: the amount you owe ( principal ), the length of time to repay ( term ), and the cost for borrowing ( interest rate as a %). The difference between borrowing and lending is who earns the interest! So renting money is just like renting a car: if you rent a fancier car that is more expensive, then it will cost you more (principal); if you rent it longer, it will cost you more (term); and if you’re trying to rent a car when there aren’t many because the demand is high, then it will also cost you more (interest rate). All three things come together to produce the amount of rent called interest, and we need all three things in order to form the basis for an interest charge. In order to help us with the finance section, a few basic review questions are required as well as links to some ideas you may already understand! NOTE: Currently, the Federal Reserve adjusts interest rates up or down to control inflation and economic growth. If the “Fed” raises rates, loans will typically cost more but investments may earn more interest. If the “Fed” lowers rates, loans will typically cost less but investments may earn less interest. For a great infographic about the US budget, check out: https://tinyurl.com/USBudget
Solving an equation is finding a value that when substituted for the variable, makes the statement true. Example 1: Solve the following equations for the missing variable. A) 5, 000 = P 0. B) 4, 000 250 P =
Solution: When solving equations, the key is to “undo” the operation. If you are multiplying the variable, to solve, we’ll need to undo multiplying by dividing. A) Multiplying the variable by 0.25 is what we see in 5, 000 = P 0.25, so we need to divide by 0.25 to find the value. 5, 000 0.25 = P P =20, 000. You can check your answer by putting the 20,000 in for P and multiplying: 20, 000 0.25 =5, 000. We got it! B) Dividing the variable by 50 is what we see in4, 000 250 P = , so we need to multiply by 250 to find the value. 4, 000 250 = P P =1, 000, 000. You can check your answer by putting the 1,000, in for P and dividing: 1, 000, 000^ ^250 =4, 000. We got it! C) Subtracting 60 from the variable is what we see in 4, 250 = P − 60 , so we need to add 60 to find the value. 4, 250 + 60 = P P =4, 310. You can check your answer like the other solutions. D) Adding 230 to the variable is what we see in 230 + P =4, 250, so we need to subtract 230 to find the value. 4, 250 − 230 = P P =4, 020. You can check your answer like the other solutions. EXPLORE! (1) Solve the following for the missing variable. Check your answers! A) 700 + I =2, 940 Addition is used, so subtract to solve: I = 2, 940 – 700 = 2, B) 9, 700^ = P^ −^450 Subtraction is used, so add to solve: P = 9,700 + 450 = 10,
Division is used, so multiply to solve: P = 5 , 400 × 90 = 486 , 000 D) P^ ^20 =5, 000 Multiplication is used, so divide to solve: P = 5 , 000 ÷ 20 = 250
Percentages are a quick way to represent a rate of return. It’s helpful to recall 5% means 5 per 100, which is the same as 1005 or 0.05. Moving from percent to decimal is quick and moving back is just as easy. EXPLORE! (1.5) Complete the table to show how to convert from percent to decimal and back. Percent Decimal Percent Decimal A) 25% 0. 25 B) 61% 0. C) 4.2% 0.042 D) 3.25% 0. E) 0.3% 0.003 F) 143% 1.
When we see some answers, the results are completely unreasonable. For example, if you paid interest of $400,000 on a loan of $10,000, that would be an unreasonable amount of interest! Likewise, if you were paying back a loan of $1,000 and your payments were $10 per month for a year, that would be unreasonable… because you wouldn’t even pay back the loan amount! Example 3 : Determine the information requested. A) Is a $125 monthly payment reasonable for a 3 - year loan of $4,000? Explain. B) Is a $42 monthly payment reasonable for a 4 - year loan of $5,000? Explain. C) Is a $6,342 monthly payment reasonable for a 2-year loan of $10,000? Explain. D) What can you say about a loan of $819,000 where the monthly payments are $2,275 per month for 30 years? Is it reasonable? Solutions: A) With 3 years being 36 months, you would pay 125 36 =4, 500. On a loan of $4,000, this would be interest of $500 which is reasonable. B) With 4 years being 48 months, you would pay 42 48 =2, 016. This is an unreasonable payment because the loan was $5,000 so you didn’t even pay off the loan! C) With 2 years being 24 months, you would pay 6, 342 24 =152, 208. On a loan of $10,000, this is way too much to pay back and is unreasonable. D) With 30 years being 30 12 = 360 months, you would pay back 2, 275 360 =819, 000. If this was a real loan, it means you would be able to pay it off exactly, so there would be no interest. It would be unreasonable from a bank, so maybe this would be reasonable if it was a loan from a very nice family member! EXPLORE! ( 3 ) Solve the following for the requested information. Be able to explain your answers! A) Is a $200 monthly payment reasonable for a 4-year loan of $9,000? Explain. 4 years is 48 months, so 48 × 20 0 = 9 ,6 0 0. On a loan of $ 9 , 0 00, the interest is the amount we overpaid: 9 ,6 00 – 9 , 00 0 = $ 600. This is reasonable. B) Is a $55 monthly payment reasonable for a 3-year loan of $2,600? Explain. 3 years is 36 months, so 36 × 55 = 1 , 98 0. On a loan of $ 2 , 6 00, this payment is unreasonable because we didn’t even pay the loan off! C) Is a $400 monthly payment reasonable for a 2-year loan of $500. 2 years is 24 months, so 24 × 40 0 = 9 , 60 0. On a loan of $500, the interest is the amount we overpaid: 9 , 600 – 500 = $ 9 , 1 00. This is unreasonable because the amount of interest is extremely high!
There are three main sources of credit scores in the United States and they are called credit bureaus: Equifax, Experian, and TransUnion. All consumers can request a free credit report once per year and can then dispute any inaccuracies. Your credit score is extremely important for obtaining loans, credit, and even low insurance rates. The credit score links to you as a person and shows lenders at a glance whether you are good with credit. Poor Fair Good Very Good Excellent Credit Score 300 – 579 580 – 669 670 – 739 740 – 799 800 – 850 Here’s what it means:
When you think about saving for retirement, often we have to put money into investments where the interest rate is not always the same. Think of an investment where you earn 5% the first year, 6% the second year, and 4% the third year. It will be very challenging to compute information when the rate changes, so in this book, we’ll use a long-term average as the interest rate. There are several retirement accounts that may be mentioned in the text, so this is a place where we can show some of the similarities and differences.
Stock Investing to own part of a publicly traded company. Usually, these are very risky. Usually no – pay capital gains on growth. No. You sell later for a price offered by another. Bond Like investing in business or government debt, interest is paid. Usually these are low risk. Usually no – could have tax free interest for some types. No. You sell when you want or when the term is over. Mutual Fund Investing in a group of stocks or bonds which “diversifies” or breaks up the risk. Usually no – pay capital gains on growth. No. You sell later for a price offered by another. IRA Retirement account set up for an individual (outside an employer). Tax deferred: Contributions are not taxed, but payouts will be taxed in the future. Yes – you must start taking money out when you turn 70 ½ yrs old. SEP IRA Retirement account for someone who is self-employed typically. Tax deferred. See above, but this account offers higher limits than a typical IRA. Yes – you must start taking money out when you turn 70 ½ yrs old. 401(k) Retirement account that you invest in through a private employer, usually mutual funds. Tax deferred. See above. Employers often match your contributions = free $$ Yes – you must start taking money out when you turn 70 ½ yrs old. 403(b) or 457(b) Retirement account that you invest in through a public employer, usually mutual funds. Tax deferred. See above. Employers rarely match your contributions = free $$ Yes – you must start taking money out when you turn 70 ½ yrs old. Roth A special type of the 3 accounts above that offers different benefits related to taxes. Tax free growth. Tax paid on contributions, but payouts (with interest) are not taxed! No – you can just let the money grow. 529 account A type of savings account typically for education purposes like a college education. Contributions are taxed, but if the money is used for education, no tax on payouts. Not for most types of accounts. NOTE: The government will most often tax your money once for all these accounts, and only in rare situations will any interest be untaxed (see Roth or 529). For that reason, the annual contributions for the Roth plans are much lower each year than the 403(b) or 401(k) options.
Objectives:
_1. Understand basic finance terms related to simple interest.
Investment Called principal or present value. Interest adds to the investment so that the balance increases over time. The end result or future value that includes the interest should be higher than the present value. Loan Called principal or loan amount. As payments are made, interest is paid to the bank or lender and some portion of the payment will reduce the balance. With either a loan or investment, the principal amount will not change. Because it is always the same, any interest calculated based on the principal amount will be much easier to deal with. We call this type of interest simple interest because the interest rate is applied to the principal amount only. There is a formula associated with this type of growth called the simple interest formula : I = P r t. From the preview section, the relationship between interest ($) and interest rate (%) should be clear as one measures money and the other a percentage. In this formula, the capital letters relate to monetary value (dollars, euro, yen, etc) and the lowercase letters relate to other quantities. I represents the amount of simple interest (earned or paid), P represents the amount of principal (invested or borrowed), r is the interest rate (as a decimal), and t is the time. When dealing with the interest rate and the time, it is very important to see that the interest rate has a time factor associate with it. We will see interest rates of 5% annual simple interest as well as 4% monthly simple interest. The label on the type of interest needs to match with the time amount. Monthly interest needs time, t , to be measured in months, while annual interest needs time to be measured in years. Example 1 : Find the interest earned when investing… A) $5,000 at 6% annual simple interest for 7 years. B) $5,000 at 6% monthly simple interest for 7 yrs. C) $8,000 at 5% annual simple interest for 3 mths. D) $500 at 4% annual simple interest for 45 days. Solution: The interest earned is… A) I = P r t = 5, 000 0.06 7 =2,100.The interest earned is $2,100. B) The interest rate is monthly, so we need to measure time in months. 7 years is 84 months (each year is 12 months), so I = P r t = 5, 000 0.06 84 =25, 200.The interest earned is $25,200. C) The interest rate is annual, so we need to measure time in years. 3 months is 123 of a year, so 3 I = P r t = 8,000 0.05 12 =100. The interest earned is $100. D) The interest rate is annual, so we need to measure time in years. 45 days is 36545 of a year, so 45 I = P r t = 500 0.04 365 2.47. The interest earned is $2.47, rounded to the nearest penny.
EXPLORE! (2) Solve these simple interest problems for the missing piece. A) ** Find the annual simple interest rate for a loan of 3 years with $500 interest if $6,500 is invested. Since this is annual interest, make sure it matches with the time. It does, so we’re good to go: 500 500 I = P r t 500 = 6,500 r 3 (^) 6,500 = r 3 r = (^) 6,500 3 0.02564... ; the interest rate is about 2.564% (we round normally). B) Find the annual simple interest rate for a loan of 60 days requiring $50 interest on principal of $650. Since this is annual interest, make sure it matches with the time. Turn days into a fraction of a year and then compute: I = P r t 50 = 650 r 36560 65050 = r 36560 r = 65050 36560 0.4679487... ; the interest rate is about 46.795% (we round normally). C) Find the annual simple interest rate for a loan of 60 months with $2,350 interest on principal of $32,000. Since this is annual interest, make sure it matches with the time. Turn months into a fraction of a year and then compute:I = P r t 2,350 = 32, 000 r 1260 (^) 32,0002,350 = r 6012 2,350 (^60) r = (^) 32,000 12 =0.0146875 ; the interest rate is about 1.469% (we round normally). Example 3 : We have solved for I and r , so let’s try a few for P or t. An investment guarantees interest of $600 on a simple annual interest rate of 10% over 2 years. How much do you need to invest? Solution: We are missing something, but can use the formula to plug in everything else. I = P r t 600 = P 0.10 2. To solve for P , we will divide both sides by 2 and then divide by 0.10. 600 600 600 = P 0.10 2 2 = P 0.10 P = 2 0.10 =3,000^. We need to invest $3,000^ in the account. EXPLORE! (3) Solve these simple interest problems for the missing piece. A) ** Find the principal needed to earn $54 in interest on 6% annual simple interest over 3 months. Turn months into a fraction of a year and then compute: 3 54 3 54 3 I = P r t 54 = P 0.06 12 (^) 0.06 = r 12 r = (^) 0.06 12 =3,600 ; the principal is $3,600. B) Find the principal needed to earn $2 9 in interest on 5% annual simple interest over 3 years. Time matches, so we’re good to go: I = P r t 29 = P 0.05 3 (^) 0.05^29 = r 3 29 r = (^) 0.05 3 =193.3333... ; the principal is $193.3 3. C) Find the time needed to earn $ 6 0 in interest on a $500 investment at 4% annual simple interest. We can solve for years so the time matches and we find it takes 3 years: I = P r t 60 = 500 0.04 t t = 60 500 0.04 = 3_._ ROUNDING NOTE REMINDER: When we calculate principal or payments needed for an investment, we need to make sure we pay off the loan! In Explore (3) part (B), the calculator shows $193.3333… If we round to $193.33 and check our work, the interest earned is $28.9995, which rounds to $29. If the result didn’t get us to $29, we would have to round up to $193.34. With principal and payment calculations, your instructor may ask you to always round up to the nearest penny to avoid the extra calculations. In the book, we will round these up to be consistent. Circle what your instructor will be using: Round Up to Nearest Penny on Payment or Principal Only Standard Rounding on All Calculations
When considering simple interest to save money, remember that simple interest is earned on the principal only. We’ll use a spreadsheet to calculate dollar amounts and match them below. As an example, we’ll invest $1,000 into an account paying 12% annual interest, so each month the interest is 1%. Month Interest Balance Principal 0 1,000 1,00 0 1 10 1, 2 10 1, 3 10 1, 4 10 1, 5 10 1, As you can see, the principal amount never changes, so the interest stays at $10 per month for the length of the loan. The balance grows each month by $10, creating a sequence you may have seen in Math 28 – an Arithmetic Sequence! If we were to graph our money over time, it would form a straight line and the slope of that line would be the rate of change ($10/mo). We could create a new formula that would provide us with the future value of a simple interest investment, instead of just giving us the interest. The future value would include the principal and the interest, so the formula would be: FV^ =^ P + I = P + P r t = P (^1 + r t ). Because it is money, FV will be in capital letters. Future value simple interest formula : FV = P ( 1 + r t ). In the formula FV = P ( 1 + r t ), there is an ending amount and a starting amount. The starting amount can be called the principal or the present value. The present value is the amount of money invested now to generate a specific future value. Example 4 : What is the future value of $750 at 4% annual simple interest for 13 months? Solution: Put in all the information into FV^ =^ P (^1 + r t ). 13 750 1 0.04 782. 12 FV = (^) + (^) =
Calculator note: When putting in the information, you can put it all in with the parenthesis in one step, but if you prefer to go in steps, then compute what is inside the parenthesis first. Hit = and then multiply by the $750. EXPLORE! (4) Determine the following: A) ** Determine the future value of $8,000 at 3.2% annual simple interest for 5 years. Time matches, so we’re good to go: FV = P (^) ( 1 + r t (^) ) = 8, 000 1( + 0.032 (^5) ) =9,280 ; a principal amount of $8,000 will have a future value of $9,280. B) Determine the future value of $600 at 4.25% monthly simple interest for 10 months. Time matches (both monthly), so we’re good to go: (^) FV = P ( 1 + r t ) = 600 1( + 0.0425 10 )= 855 ; a principal amount of $600 will have a future value of $ 855.
EXPLORE! (6) Find the monthly payment and total interest paid for the following add-on interest loans. A) ** The principal amount is $800 at 12% annual simple interest for 2 years. First compute the interest; time matches so we are good to go: I = P r t = 800 0.12 2 = 192_. We will pay $192 in interest over the 2 years along with the $800 for a grand total of $992. Next, compute the monthly payment. 2 years is 24 months, so the monthly payment would be_ 992 41.33333... 24 ^. Because this is a monthly payment, we^ need to round up^ to pay it off^ in time so the monthly payment would be $41.34. B) The principal amount is $38,000 at 5.25% annual simple interest for 8 years. First compute the interest; time matches so we are good to go: I = P r t = 38, 000 0.0525 8 =15,960_. We will pay $15,960 in interest over the 8 years along with the $38,000 for a grand total of $53,960. Next, compute the monthly payment. 8 years is 96 months, so the monthly payment would be_ 53, 562.0833... 96 ^. Because this is a monthly payment, we^ need to round up^ to pay it off in time so the monthly payment would be $562.09. C) The principal amount is $18,000 at 5.25% monthly simple interest for 2 years. Turn years into a number of months first and then compute interest: 2 years = 24 months. I = P r t = 18, 000 0.0525 24 =22,680_. We will pay $22,680 in interest over the 2 years along with the $18,000 for a grand total of $40,680. Next, compute the monthly payment. 8 years is 96 months, so the monthly payment would be_ 40, 680 =1, 24 .The monthly payment would be $1,695. Yowsers!
Credit cards use the simple interest formula to calculate the finance charge each month. The finance charge (interest) is based on the average daily balance ( ADB – principal), the interest rate, and the number of days in the month. To help, it is good to remember the following about our calendar and the number of days in each.
Example 6 : Compute the finance charge for an average daily balance of $517 at 1 7 .99% annual simple interest during March. Solution: 31 517 0.1799 7. 365 I = P r t I = . The finance charge would be $7.90.
EXPLORE! (7) Determine the finance charge for the following credit cards. A) ** ADB is $1,100 at 19.99% annual simple interest over June. The finance charge over June (30 days) is: 30 1,100 0.1999 18.07315... 365 I = P r t = . We round normally so the finance charge would be $18.07. B) (https://www.valuepenguin.com/average-credit-card-debt, July 2019) The average debt for the poorest households in the United States was $10,700 at 1 5. 32 % annual simple interest. Determine the finance charge for a person in this situation over the month of October. The finance charge over October (31 days) is: 31 10, 700 0.1532 139.22312... 365 I = P r t = . We round normally so the finance charge would be $139.22. C) ADB is $152 at 1 1 .99% annual simple interest over January. First – ballpark this and decide if you think the finance charge will be less than $5, between $5 and $15, or more than $15? Explain your thinking. This is about 18% interest for the year, which is less than 2% in a month. 2% of $200 is $4, so this should be less than $5. Computing, the actual finance charge over January (31 days) is: 31 152 0.1199 1.5478597... 365 I = P r t = , which would round to $ 1. 55. Credit cards can be complicated tools for many people, and some have sworn them off entirely as the risk outweighs the reward. Used properly, they can help build credit and are very convenient. Some credit card concepts are: new balance, minimum payment amount, finance charge, and remaining balance. At the end of a billing cycle (usually a month), there is an amount still owed – this amount is your remaining balance. The credit card company computes your finance charge (and any other fees) and adds that to the remaining balance, creating the new balance. The new balance is used to compute the minimum payment amount which can vary between 1 % and 5% of the new balance. For these examples, we will use 4 % as the rate required for the minimum payment amount. Here is sample language from a credit card agreement: Minimum Due is calculated as 2% of the Statement Balance rounded down to the nearest $1. When
Example 7 : Petra has a credit card with an ADB of $684.28, with a remaining balance of $650.13. Her interest rate is 18.99% simple interest over November. Determine (A) her finance charge, (B) the new balance, and (C) the minimum payment amount. Solution: A) The finance charge is done as above: 30 684.28 0.1899 10. 365 I = P r t I = or $10.68. B) The new balance is $650.13 + $10.68 = $660.81. C) The minimum payment is 660.81^ ^ 0.04^ 26.4324which would be $26.43. (Some banks may round this to $30 – read your cardmember agreement to find out. We will just round normally.)
Formula Rolling Summary At the end of each section in this unit, we will place the formulas you’ve seen in the section with other formulas from previous sections. 1 .1: I = P r t and FV = P ( 1 + r t ) Simple Interest and Future Value Simple Interest Use for simple interest, add-on interest loans, and finding credit card finance charge (with average daily balance). SOLVE FOR: Any variable. EXPLORE! (9) Determine when the formulas are appropriate (without doing any computations) : A) When do you use I = P r t compared to FV = P ( 1 + r t )? We use I = P r t to compute the interest and FV = P (^) ( 1 + r t ) when we want to find the future value (the amount including both the principal and interest). B) How do you find the simple interest amount using FV = P ( 1 + r t )? To find the interest with only FV = P (^) ( 1 + r t ) , you would need to calculate the FV and then subtract the principal amount, P. FV = P + I, so FV – P = I. C) How could you find the future value with simple interest using I^ = P r t? If you want to find the future value with just I^ =^ P r t ^ , you would need to calculate the interest (I) and then add the principal amount, P. FV = P + I. D) With an add-on interest loan, to find the total amount you have to repay, could we just use FV = P ( 1 + r t )? Explain why or why not. YES! The future value of a simple interest loan is the loan amount added to the interest, which is exactly how we find the total amount to repay with an add-on interest loan. If you wanted to find the interest, it may be easier to find interest, then add it on (as the name suggests). However, if you’re just looking for the monthly payment, it may be quicker to find the future value and divide by the number of months. Finishing this section is a great time to complete the cover page and project section A.1 – find your job!
Objectives:
_1. Understand basic finance terms related to compound interest.
12
Since we often have time of 1 month or 1 quarter or 1 year, the numerator will be 1 in the fraction representing time. This technique of taking the rate and the time and combining them into one fraction is known as the periodic rate. The periodic rate takes an annual rate and computes what portion you’ll earn for a given time period. Example 1 : Find the periodic rate for these rate and time values. A) 6% annual interest, monthly B) 40% annual interest, quarterly C) 8% annual interest, semiannually D) 20% annual interest, annually Solutions:
The periodic rate is often simplified to the symbol i , and uses the following form: n r i = , where i is the periodic rate, r is the annual interest rate, and n is the number of compounding periods per year. This table may be helpful during the homework and these are the most common periods: Time Measurement # of Compounding Periods Time Measurement # of Compounding Periods Annually 1 Semi-Annually 2 Quarterly 4 Monthly 12 Semi-Monthly 24 Bi-weekly 26 Weekly 52 Daily 365
