F2S: Mathematics of Finance Problem Set 1Name: ______

F2S: Mathematics of Finance—Problem Set 1Name: ______

Simple Interest. It costs to borrow money. The rent one pays for the use of money is called the interest. The amount of money that is being borrowed or loaned is called the principal or present value. Simple interest is paid only on the original amount borrowed. When the money is loaned out, the person who borrows the money generally pays a fixed rate of interest on the principal for the time period he keeps the money. Although the interest rate is often specified for a year, it may be specified for a week, a month, or a quarter, etc. The credit card companies often list their charges as monthly rates sometimes it is as high as 1.5% a month.

When the money is loaned or borrowed for a longer time period, the interest is paid (or charged) not only on the principal, but also on the past interest, and we say the interest is compounded.(Note: if there is no explicit mention of compounding value, use m = 1; i.e. annual compounding.)

See attached sheet for formulas.

Problem Set—Work on Separate Paper. Include all work to answer each question completely.

1. Jose deposited $2500 in an account that pays 6% simple interest. How much money will he have at the end of 3 years?
2. Darnel owes a total of $3060, which includes 12% interest for the three years he borrowed the money. How much did he originally borrow?(Assume simple interest)
3. If $3500 is invested at 9% compounded monthly, what will the future value be in four years?
4. How much should be invested in an account paying 9% compounded daily for it to accumulate to $5,000 in five years?
5. For comparison purposes, the government requires the bank to state their interest rate in terms of effective yield or effective interest rate.

For example, if one bank advertises its rate as 7.2% compounded monthly, and another bank advertises its rate as 7.5%, how are we to find out which is better? This requires us to use the effective rate formula on our sheet. Compare these two scenarios using $1000 as an initial investment in each account.

1. Think back to your work on exponential functions. If $3500 is invested at 9% compounded continuously, what will the future value be in four years?
2. If an amount is invested at 7% compounded continuously, what is the effective interest rate?(Note: consider using TABLE with increasingly large values of m.)
3. If an amount is invested at 7% compounded weekly, how long will it take our investment to double?
4. You borrow $4,500 for six months at a simple interest rate of 8%. How much is the interest?
5. Jamie just paid off a loan of $2,544, the principal and simple interest. If he took out the loan six months ago at 12% simple interest, what was the amount borrowed?
6. If $8,000 is invested at 9.2% compounded monthly, what will the final amount be in 4 years?
7. Lydia's aunt Rose left her $5,000. Lydia spent $1,000 on her wardrobe and deposited the rest in an account that pays 6.9% compounded daily. How much money will she have in 5 years?
8. Bank A pays 5% compounded daily, while Bank B pays 5.12% compounded monthly. Evaluate when $1000 is the initial investment in an account in each of the banks. Which bank pays more? Explain.
9. Jon's grandfather was planning to give him $12,000 in 10 years. Jon has convinced his grandfather to pay him $6,000 now, instead. If Jon invests this $6,000 at 7.5% compounded continuously, how much money will he have in 10 years?
10. What will be the price of a $20,000 car in 5 years if the inflation rate is 6%?

(Note: Assume m = 1 and use compound interest formula.)

1. An amount of $2000 is borrowed for 3 years. At the end of the three years, $2660 is paid back. What was the simple interest rate?
2. Shanti charged $800 on her charge card and did not make a payment for six months. If there is a monthly charge of 1.5%, how much does she owe?
3. If an investment earns 10% compounded continuously, in how many years will it triple?
4. Mr. and Mrs. Tran are expecting a baby girl in a few days. They want to put away money for her college education now. How much money should they deposit in an account paying 10.2% so they will have $100,000 in 18 years to pay for their daughter's educational expenses?
5. How long will it take $10,000 to grow to $15,000 if you receive 5.2% interest compounded quarterly? Solve this both graphically and with logs.
6. Mike is the beneficiary of a trust fund established for him 21 years ago. If the original amount placed in the trust was $10,000, how much will he receive if the money has earned interest at the rate of 8%/year compounded annually? Quarterly? Continuously?
7. Sarah invested some money into a mutual fund account 5 years ago. She received 6% interest. Her investment is now worth $22,289.22. How much did she invest originally?
8. Leonard’s current annual salary is $45,000. Ten years from now, how much will he need in order to retain his present purchasing power if the rate of inflation over that period is 3%/year compounded continuously?
9. Find the interest rate needed for an investment of $5,000 to grow at an amount of $8,000 in 5 years if the interest is compounded continuously.
10. How long will it take an investment of $2,000 to double if the investment earns interest at 4.2%/year compounded monthly? Solve by using logs and then graphically. Also, consider the Rule of 70 to estimate doubling time.

Reflections—Write your answers in complete sentences.Think about and work on your responses to these questions throughout this problem set. Final written reflections are due on Monday, Nov 10 in class.

1. What is the primary difference between simple and compound interest?
2. With compound interest, what is the effect of number of “compoundings” on accumulated amount? Which is more important, m or t? Explain with an example.
3. What types of problems are still giving you difficulties? Be specific.
4. Explain to someone who knows nothing about finances the meaning of effective rate.
5. If you receive $1,000 at graduation, save that money until you retire 50 years later, how much will you have? (You will have to make some assumption, which should be based on fact.) What could you do with that money?