## Unit 4 Practice

Complete these exercises and problems and then check your work.

### Exercises

##### QUESTION 1

Explain why $1 received today is worth more than $1 received one year from today.

##### QUESTION 2

What do we mean when we refer to an annuity? How is an annuity different from an annuity due?

##### QUESTION 3

What is the relationship between present value and future value?

##### QUESTION 4

How do we determine the appropriate discount rate to use when finding present value?

##### QUESTION 5

Why is compounding on a monthly basis better for us than compounding on an annual basis?

##### PROBLEM 1

Determine the answer to each of the following questions.

1a. Find the Future Value of $2500 invested today at 11% for 10 years.

1b. Find the Future Value of $2500 invested today at 11% for 30 years.

1c. Find the Present Value of $6000 received 10 years from today if the discount rate is 5%.

1d. Find the Present Value of $6000 received 10 years from today if the discount rate is 10%.

1e. Find the Future Value of $3000 per year (at the end of each year) invested at 6% for 30 years.

1f. Find the Future Value of $3000 per year (at the end of each year) invested at 12% for 30 years.

1g. Find the Present Value of $4000 per year (at the end of each year) if the discount rate is 15% for 20 years.

1h. Find the Present Value of $4000 per year (at the end of each year) if the discount rate is 15% for 40 years.

##### PROBLEM 2

Find the interest rates implied by each of the following:

2a. You borrow $1500 today and promise to repay the loan by making a single payment of $2114.00 in 5 years.

2b. You invest $500 today and receive a promise of receiving back $193.50 for each of the next 4 years.

##### PROBLEM 3

If $2000 is invested today at a 12% nominal interest rate, how much will it be worth in 15 years if interest is compounded

3a. Annually

3b. Quarterly

3c. Monthly

3d. Daily (365-days per year)

##### PROBLEM 4

How long will it take your money to triple given the following interest rates?

4a. 5%

4b. 10%

4c. 15%

##### PROBLEM 5

After graduating from college you make it big – all because of your success in business finance. You decide to endow a scholarship for needy finance students that will provide $5000 per year indefinitely, beginning 1 year from now. How much must be deposited today to fund the scholarship under the following conditions.

5a. The interest rate is 10%

5b. The interest rate is 10% and the first payment is made 6 years from today instead of 1 year from today.

##### PROBLEM 6

Find the present value of the following cash flow stream if the discount rate is 12%:

Years 1-10 $4000 per year

Years 11-15 $6000 per year

Years 16-20 $8000 per year

##### PROBLEM 7

Find the value of the following cash flow stream at the end of year 30 if the rate of return is 8.75%:

Years 1-5 $3000 per year

Year 6 $7500

Years 7-15 $9000 per year

Years 16-30 $12,000 per year

##### PROBLEM 8

Find the effective annual rate of interest for a nominal rate of 9% compounded

8a. Annually

8b. Quarterly

8c. Monthly

8d. Daily (365 days per year)

##### PROBLEM 9

Your firm has a retirement plan that matches all contributions on a one-to-two basis. That is, if you contribute $3000 per year, the company will add $1500 to make it $4500. The firm guarantees a 9% return on your investment. Alternatively, you can "do-it-yourself" and you think you can earn 12% on your money by doing it this way. The first contribution will be made 1 year from today. At that time, and every year thereafter, you will put $3000 into the retirement account. If you want to retire in 25 years, which way are you better off?

##### PROBLEM 10

Jen is planning for retirement. She plans to work for 32 more years. She currently has $15,000 saved and, for the next 15 years, she can save $6,000 at the end of each year. Fifteen years from now, she wants to buy a weekend vacation home that she estimates will require her to withdraw $100,000. How much will she have to save in years 16 through 32 so that she has exactly $750,000 saved when she retires? Assume she can earn 9% throughout the 32-year period.

##### PROBLEM 11

You are a recent college graduate and want to start saving for retirement. You plan to save $2000 per year for the next 15 years. After that you will stop contributing and just allow your savings to accumulate for another 20 years. Your twin brother would rather wait awhile before he starts saving. He is not going to put away anything for the next ten years, then he will start making contributions at the end of each year for the final 25 years. You both anticipate earning a 9.5% rate of return on your investments. How much must your brother put away at the end of each year to have the same amount of money for retirement as you?

##### PROBLEM 12

You are considering purchasing a new home. The house you are looking at costs $120,000 and you plan to make a 10% down payment. You checked with a bank and they have two mortgage loan options for you. The first is a 15-year mortgage at 6.25%. The second is a 30-year mortgage at 6.50%.

12a. What are your monthly payments for each loan?

12b. What is the total you will pay over the life of the loan for each loan?

12c. After one year you get a job transfer and have to sell the house. What is the payoff value of your remaining loan balance (hint: find PV of remaining payments)?

12d. Over the first year, how much did you pay in principal and how much did you pay in interest?

Source: Kevin Bracker, Fang Lin, and Jennifer Pursley, https://businessfinanceessentials.pressbooks.com/chapter/chapter-three-time-value-of-money-2/

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 License.