How do all of those "don't pay until . ." plans work? The retail industry overflows with financing plans specifically designed to attract customers to purchase merchandise on credit. Most of these offers include terms such as "no money down" and "no payments for x months." You can find these plans at most furniture retailers such as The Brick and Leon's, electronic retailers such as Best Buy, along with many other establishments including jewelry stores and sleep centers.

But have you ever considered how this works on the business side? If every customer purchased merchandise under these no-money-required plans, how would the retailer stay in business? For example, perhaps The Brick’s customers purchase furniture in January 2014 that they do not have to pay for until January 2016. During these two years, The Brick does not get paid for the products sold; however, it has paid its suppliers for the merchandise. How can a retailer stay in business while it waits for all those postponed payments?

Consumers generally do not read the fine print on their contracts with these retailers. Few consumers realize that the retailers often sell these contracts (sometimes immediately) to finance companies they have partnered with. While the consumer sees no noticeable difference, behind the scenes the retailer receives cash today in exchange for the right to collect payment in the future when the contract becomes due. That way the finance company becomes responsible for collecting on the loan to the consumer.

This section introduces the mathematics behind the sale of promissory notes between companies. Recall from Section 8.4 that a *promissory note*, more commonly called a note, is a written debt instrument that details a promise made by a buyer to pay a specified amount to a seller at a predetermined and specified time. If the debt allows for interest to accumulate, then it is called an *interest-bearing promissory note*. If there is no allowance for interest, then it is called a *non-interest-bearing promissory note*. Interest-bearing notes are covered in this section and non-interest notes are discussed in the next section. When promissory notes extend more than one year, they involve compound interest instead of simple interest.

## Interest-Bearing Promissory Notes

When you participate in "don't pay until . ." promotions, you create a promissory note in which you promise to pay for your goods within the time interval stated. These promotions commonly carry 0% interest if paid before the stated deadline, so the notes are non-interest-bearing promissory notes. However, failure to pay the note before the deadline transforms the note into an interest-bearing note for which interest is retroactive to the date of sale, usually at a very high rate of interest such as 21%.

The mathematics of interest-bearing promissory notes deal primarily with the sale of long-term promissory notes between organizations. When the note is sold, the company buying it (usually a finance company) purchases the maturity value of the note and not the principal of the note. To the finance company, the transaction is an investment from which it intends to earn a profit through the difference between maturity value and purchase price. Thus, the finance company discounts the maturity value of the note using a discount rate that permits it to invest a smaller sum of money today to receive a larger sum of money in the future. The company selling the note is willing to take the smaller sum of money to cash in its accounts receivables and eliminate the risk of default on the debt.

### How It Works

Recall that in simple interest the sale of short-term promissory notes involved three steps. You use the same three step sequence for long-term compound interest promissory notes. On long-term promissory notes, a three-day grace period is not required, so the due date of the note is the same as the legal due date of the note.

**Step 1**: Draw a timeline, similar to the one on the next page, detailing the original promissory note and the sale of the note.

**Step 2**: Take the initial principal on the date of issue and determine the note's future value at the stated deadline using the stated rate of interest attached to the note. As most long-term promissory notes have a fixed rate of interest, this involves a future value calculation using Formula 9.3.

- Calculate the periodic interest rate using Formula 9.1, (i=dfrac{1 Y}{C Y}).
- Calculate the number of compounding periods between the issue date and due date using Formula 9.2, (N = CY × ext {Years}).
- Solve for the future value using Formula 9.3, (FV=PVleft(1+i ight)^N).

**Step 3**: Using the date of sale, discount the maturity value of the note using a new negotiated discount rate of interest to determine the proceeds of the sale. Most commonly the negotiated discount rate is a fixed rate and involves a present value calculation using Formula 9.3.

- Calculate the new periodic interest rate using Formula 9.1, (i=dfrac{1 Y}{C Y}).
- Calculate the number of compounding periods between the date of sale of the note and the due date using Formula 9.2, (N = CY × ext {Years}). Remember to use the (CY) for the discount rate, not the (CY) for the original interest rate.
- Solve for the present value using Formula 9.3, (FV=PVleft(1+i ight)^N), rearranging for (PV).

Assume that a three-year $5,000 promissory note with 9% compounded monthly interest is sold to a finance company 18 months before the due date at a discount rate of 16% compounded quarterly.

**Step 1**: The timeline to the right illustrates the situation.

**Step 2a**: The periodic interest rate on the note is (i) = 9%/12 = 0.75%.

**Step 2b**: The term is three years with monthly compounding, resulting in (N) = 12 × 3 = 36.

**Step 2c**: The maturity value of the note is (FV) = $5,000(1 + 0.0075)36 = $6,543.23.

**Step 3a**: Now sell the note. The periodic discount rate is (i) = 16%/4 = 4%.

**Step 3b**: The time before the due date is 1½ years at quarterly compounding. The number of compounding periods is N = 4 × 1½ = 6.

**Step 3c**: The proceeds of the sale of the note is ($6,543.23 = PV(1 + 0.04)^6), where PV= $5,171.21. The finance company purchases the note (invests in the note) for $5,171.21. Eighteen months later, when the note is paid, it receives $6,543.23.

### Important Notes

The assumption behind the three-step procedure for selling a long-term promissory note is that the process starts with the issuance of the note and ends with the proceeds of the sale. However, mathematically you may deal with any part of the transaction as an unknown. For example, perhaps the details of the original note are known, the finance company's offer on the date of sale is known, but the quarterly discounted rate used by the finance company needs to be calculated.

The best strategy in any of these scenarios is always to execute step 1 and create a timeline. Identify the known variables to visualize the process, then recognize any variable(s) remaining unknown. Keeping in mind how the selling of a promissory note works, you can adapt the three-step promissory note procedure using any of the techniques discussed in Chapter 9. Some examples of these adaptations include the following:

- The discounted rate is unknown. Execute steps 1 and 2 normally. In step 3, solve for i (then (IY)) instead of (PV).
- The original principal of the note is unknown. Execute step 1 normally. Work with step 3, but solve for (FV) instead of (PV). Then work with step 2 and solve for (PV) instead of (FV).
- The length of time by which the date of sale precedes the maturity date is unknown. In step 3, solve for (N) instead of (PV). As you can see, the three steps always stay intact. However, you may need to reverse steps 2 and 3 or calculate a different unknown variable.

### Things To Watch Out For

In working with compound interest long-term promissory notes, the most common mistakes relate to the maturity value and the two interest rates.

**Maturity Value**. Remember that the company purchasing the note is purchasing the maturity value of the note, not its principal on the issue date. Any promissory note situation always involves the maturity value of the promissory note on its due date.**Two Interest Rates**. The sale involves two interest rates: an interest rate tied to the note itself and an interest rate (the discount rate) used by the purchasing company to acquire the note. Do not confuse these two rates.

Example (PageIndex{1}): Proceeds on an Interest-Bearing Note

Jake's Fine Jewelers sold a diamond engagement ring to a customer for $4,479.95 and established a promissory note under one of its promotions on January 1, 2014. The note requires 6% compounded semi-annually interest and is due on January 1, 2017. On October 1, 2015, Jake's Fine Jewellers needed the money and sold the note to a finance company at a discount rate of 11% compounded quarterly. What are the proceeds of the sale?

**Solution**

Find the proceeds of the sale for Jake's Fine Jewellers on October 1, 2013. This is the present value of the note ((PV)) based on the maturity value and the discount rate.

**What You Already Know **

**Step 1**:

The issue date, maturity date, principal, interest rate on the note, date of sale, and the discount rate are known, as illustrated in the timeline.

**How You Will Get There **

**Step 2a**:

Working with the promissory note itself, calculate the periodic interest rate by applying Formula 9.1.

**Step 2b**:

Calculate the number of compound periods using Formula 9.2.

**Step 2c**:

Calculate the maturity value of the note using Formula 9.3.

**Step 3a**:

Working with the sale of the note, calculate the discount periodic interest rate by applying Formula 9.1.

**Step 3b**:

Calculate the number of compound periods elapsing between the sale and maturity. Use Formula 9.2.

**Perform**

**Step 2a**:

[IY=6 \%, CY=2, i=6 \% / 2=3 \% onumber ]

**Step 2b**:

Years = January 1, 2017 − January 1, 2014 = 3 Years, (N) = 2 × 3 = 6

**Step 2c**:

[egin{aligned} PV&=$ 4,479.95 FV&=$ 4,479.95(1+0.03)^{6}=$ 5,349.29 end{aligned} onumber ]

**Step 3a**:

[IY=11 \%, CY=4, i=11 \% / 4=2.75 \% onumber ]

**Step 3b**:

Years = January 1, 2017 − October 1, 2015 = 1¼ Years, (N) = 4 × 1¼ = 5

**Step 3c**:

[egin{aligned} $ 5,349.29 &=PV(1+0.0275)^{5} PV &=dfrac{$ 5,349.29}{1.0275^{5}} &=$ 4,670.75 end{aligned} onumber ]

**Calculator Instructions**

Part | N | I/Y | PV | PMT | FV | P/Y | C/Y |
---|---|---|---|---|---|---|---|

Maturity | 6 | 6 | -4479.95 | 0 | Answer: 5,349.294586 | 2 | 2 |

Sale | 5 | 11 | Answer: -4,670.753954 | (surd) | 5349.29 | 4 | 4 |

Jake's fine Jewelers made the sale for $4,479.95 on January 1, 2014. On October 1, 2015, it receives $4,670.75 in proceeds of the sale to the finance company. The finance company holds the note until maturity and receives $5,349.29 from the customer.

Example (PageIndex{2}): Finding an Unknown Discount Rate

A $6,825 two-year promissory note bearing interest of 12% compounded monthly is sold six months before maturity to a finance company for proceeds of $7,950.40. What semi-annually compounded discount rate was used by the finance company?

**Solution**

Calculate the semi-annually compounded negotiated discount rate ((IY)) used by the finance company when it purchased the note.

**What You Already Know **

**Step 1**:

The term, principal, promissory note interest rate, date of sale, and proceeds amount are known, as shown in the timeline.

**How You Will Get There **

**Step 2a**:

Working with the promissory note itself, calculate the periodic interest rate by applying Formula 9.1.

**Step 2b**:

Calculate the number of compound periods using Formula 9.2.

**Step 2c**:

Calculate the maturity value of the note using Formula 9.3.

**Step 3a**:

Note the (CY) on the discount rate.

**Step 3b**:

Calculate the number of compound periods elapsing between the sale and before maturity. Use Formula 9.2.

**Step 3c**:

Calculate the periodic discount rate used by the finance company by applying Formula 9.3 and rearranging for (i). Then substitute into Formula 9.1 and rearrange for (IY).

**Perform**

**Step 2a**:

[IY=12 \%, CY=12, i=12 \% / 12=1 \% onumber ]

**Step 2b**:

Years = 2, (N) = 12 × 2 = 24

**Step 2c**:

[PV=$ 6,825.00, FV=$ 6,825.00(1+0.01)^{24}=$ 8,665.94 onumber ]

**Step 3a**:

(CY) = 2

**Step 3b**:

Years = 6 months = ½ Year, (N) = 2 × ½ = 1

**Step 3c**:

Solve for (i):

[egin{aligned} $ 8,665.94&=$ 7,950.40(1+i)^1 1.090000&=1+i i&=0.090000 end{aligned} onumber ]

Solve for (IY: 0.090000 = IY div 2)

(IY) = 0.180001 or 18.0001% compounded semi-annually (most likely 18%; the difference is due to a rounding error)

**Calculator Instructions**

Part | N | I/Y | PV | PMT | FV | P/Y | C/Y |
---|---|---|---|---|---|---|---|

Maturity | 24 | 12 | -6825 | 0 | Answer: 8,665.938976 | 12 | 12 |

Discount Rate | 1 | Answer: 18.0001 | -7950.40 | (surd) | 8665.94 | 2 | 2 |

The sale of the promissory note is based on a maturity value of $8,665.94. The finance company used a discount rate of 18% compounded semi-annually to arrive at proceeds of $7,950.40.

## Non-interest-Bearing Promissory Notes

A non-interest-bearing promissory note involves either truly having 0% interest or else already including a flat fee or rate within the note’s face value. Therefore, the principal amount and maturity amount of the promissory note are the same.

### How It Works

A non-interest-bearing note simplifies the calculations involved with promissory notes. Instead of performing a future value calculation on the principal in step 2, your new step 2 involves equating the present value and maturity value to the same amount ((PV = FV)). You then proceed with step 3.

Assume a three-year $5,000 non-interest-bearing promissory note is sold to a finance company 18 months before the due date at a discount rate of 16% compounded quarterly.

**Step 1**: The timeline is illustrated here.

**Step 2**: The maturity value of the note three years from now is the same as the principal, or (FV) = $5,000.

**Step 3a**: Now sell the note. The number of compounding periods is (N) = 4 × 1½ = 6.

**Step 3c**: The proceeds of the sale of the note is ($5,000 = PV(1 + 0.04)^6), or PV = $3,951.57. The finance company purchases the note (invests in the note) for $3,951.57. Eighteen months later, when the note is paid, it receives $5,000.

Exercise (PageIndex{1}): Give It Some Thought

If a non-interest-bearing note is sold to another company at a discount rate, are the proceeds of the sale more than, equal to, or less than the note?

- Answer
Less than the note. The maturity value is discounted to the date of sale.

Example (PageIndex{3}): Selling a Noninterest-Bearing Long-Term Promissory Note

A five-year, noninterest-bearing promissory note for $8,000 was issued on June 23, 2011. The plan is to sell the note at a discounted rate of 4.5% compounded semi-annually on December 23, 2015. Calculate the expected proceeds on the note.

**Solution**

**What You Already Know **

**Step 1**:

The principal, non-interest on the note, issue date, due date, discount rate, and date of sale are known, as illustrated in the timeline.

**How You Will Get There **

**Step 2**:

Equate the (FV) of the note with the principal of the note.

**Step 3a**:

Working with the sale of the note, calculate the discount periodic interest rate by applying Formula 9.1.

**Step 3b**:

Calculate the number of compound periods elapsing between the sale and maturity. Use Formula 9.2.

**Step 3c**:

Calculate the proceeds of the sale by using Formula 9.3, rearranging for (PV).

**Perform**

**Step 2**:

(FV = PV) = $8,000

**Step 3a**:

[IY=4.5 \%, CY=2, i=4.5 \% / 2=2.25 \% onumber ]

**Step 3b**:

Years = June 23, 2017 − December 23, 2015 = 1½ Years, (N) = 2 × 1½ = 3

**Step 3c**:

[egin{aligned} $ 8,000 &=PV(1+0.0225)^{3} PV &=dfrac{$ 8,000}{1.0225^{3}} &=$ 7,483.42 end{aligned} onumber ]

**Calculator Instructions**

N | I/Y | PV | PMT | FV | P/Y | C/Y |
---|---|---|---|---|---|---|

6 | 4.5 | Answer: -7,483.418564 | 0 | 8000 | 2 | 2 |

The expected proceeds are $7,483.42 on December 23, 2015, with a maturity value of $8,000.00 on June 23, 2017.

## Business Math: A Step-by-Step Handbook

I have spoken with many math instructors and professors all across Canada. As educators, it is apparent that many of us are experiencing the same challenges with our students. These challenges include, but are not limited to:

Mathematics has a negative perception amongst most students and is viewed as the "dreaded" course to take

It is commonly accepted that mathematics is one of the few courses where students make the comment of "I'm not very good at it" and people just accept it as truth and

The declining ability of our students with regards to fundamental mathematical skills and problem-solving ability.

Compared to when we grew up, our students now face a very different Canada. Today’s society is extremely time-pressed and always on the go. Most students are not only attending college or university, but holding down full- or part-time jobs to pay their bills and tuition. There are more mature students returning from industry for upgrading and retraining who have families at home. There are increasing rates of cultural diversity and international students in the classroom. Many students do not go directly home at night and do their homework. Rather, they fit their schoolwork, assignments, and homework into their schedules where they can.

In addition to all of this, our students have grown up surrounded by technology at home, in school, and at work. It is the modern way of doing business. Microsoft Office is prevalent across most industries and companies, however LibreOffice is readily available and starting to compete with Microsoft Office. As technology continues to develop and prices become more affordable, we see more students with laptops, iPhones, and even iPads. Our students need us to embrace this technology which is a part of their life in order to help them become the leaders of tomorrow.

### What Do We Need To Do?

I set out to write this textbook to answer the question, “What can we, as educators, do to help?” We can’t change the way our students are or how they live. Nor can we change the skill sets they bring into our classroom. What we can do though is adapt our textbooks, resources, and the way we teach mathematics. After all, isn’t it our job to find teaching strategies that meet with the needs of our students?

You may ask, ‘what do students need in a textbook’? The answer requires us to listen to our students both in feedback and the questions that they pose. Term after term, year after year, do these questions sound familiar?

1. How do we approach the mathematical problem (I don’t know where to start)? 2. What are the steps needed to arrive at the answer (how do I get there)? 3. Why is this material so repetitive (particularly made with respect to annuities)?

4. Why do we use algebraic symbols that have absolutely no relevance to the variable they represent? 5. How and why does a formula work?

6. Is there a quick way to locate something in the book when I need it?

7. How does this material relate to me personally and my professional career (what's in it for me?)?

8. How does the modern world utilize technology to aid in math computations (does anyone do this by hand)? 9. Where are the most common errors so that I can try to avoid them?

10. Are there are any shortcuts or "trade secrets" that can help me understand the concepts better?

11. How do all the various mathematical concepts fit together when we only cover each piece one at a time? 12. Is there any way to judge whether I conceptually understand the material?

## More Information

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## Test Bank For Mathematics Of Finance 8th Editon by Robert Brown, Steve Kopp Professor

1. $10,000 was deposited into an investment account earning interest at a nominal rate of 9% compounded monthly for eight years. How much interest was earned in the fifth year?

A. $1468.72

B. $1342.76

C. $1303.11

D. $1270.42

2. You invest $1500 today and another $2000 18-months from today in a fund earning j 4 = 8% for the first 18 months followed by j 2 = 6% thereafter. How much do you have at the end of 4 years?

A. $4007.79

B. $4274.59

C. $4276.84

D. $4377.73

3. A loan of $15,000 is taken out with interest at j 4 = 8%. What is the total amount of interest due on the loan in the second year?

A. $1338.41

B. $1298.92

C. $1296.00

D. $1200.00

4. Jim deposits $5000 in an account earning j 12 = 9%. He leaves it there for 5 years. How much interest did he earn in the last year (that is, between years 4 and 5)?

A. $671.38

B. $644.13

C. $635.21

D. $565.82

5. You invest $5000 into a fund paying interest at j6 = 9%. How much is in the fund after 3 years?

A. $6350.00

B. $6475.15

C. $6511.30

D. $6536.70

6. What is the total amount of interest earned between the end of 18 months and the end of 36 months on an investment of $1000 if the interest rate is j 12 = 9%?

A. $308.65

B. $164.69

C. $154.94

D. $143.96

7. A long-term promissory note for $40,000 is taken out on June 15, 2010. The interest rate on the note is j 4 = 7% and the note is due on Dec 15, 2015. What is the maturity value of the note?

A. $58,597.50

B. $58,589.15

C. $58,398.79

D. $57,581.47

8. A long-term promissory note for $40,000 is taken out on June 15, 2010. The interest rate on the note is j 4 = 7% and the note is due on December 15, 2015. What is the maturity value of the note?

A. $58,589.15

B. $58,398.79

C. $58,597.50

D. $57,581.47

9. You deposit $10 000 today in a fund that earns interest at j1 = 4.5%. However, at the end of every year, an expense charge of 0.50% of the accumulated amount of the fund is withdrawn. How much do you have at the end of 3-years?

A. $11,241.34

B. $11,248.64

C. $11,354.60

D. $11,411.66

10. What simple interest rate, r , is equivalent to j 4 = 14% over 18 months?

A. 14.48%

B. 14.75%

C. 15.28%

D. 16.74%

11. You shop around for the best interest rate and have narrowed your choices to the following:

Bank I: j 1 = 12% Bank II: j 4 = 11.55% Bank III: j 12 = 11.30%

You wish to have $ S in 3 years. What is the present value of S ? Put the banks in order, from lowest to highest present value of S .

A. I < II < III

B. III < II < I

C. II < I < III

D. III < I < II

12. What simple interest rate, r , is equivalent to j 12 = 9% over a 2-year period?

13. You are investing a sum of money. Rank the following interest in order of which one will give you the most interest.

a. j 2 = 8.00% b. j 12 = 7.95% c. j 52 = 7.90%

A. c > b > a

B. b > c > a

C. a > b > c

D. b > a > c

14. What nominal interest rate compounded semi-annually is equivalent to j 52 = 13%?

A. 12.60%

B. 13.41%

C. 13.86%

D. 14.34%

15. Which of the following interest rates results in the most interest being charged on a loan?

A. j 1 = 15.0%

B. j 4= 14.2%

C. j 12 = 14.1%

D. j 52 = 14.0%

16. What simple interest rate, r , is equivalent to j 12 = 8% over 9 months?

17. What simple interest rate r is equivalent to j 4 = 8% if money is invested for 4 years?

18. What simple interest rate r is equivalent to j 12 = 9% if money is invested for 3.5 years?

19. What simple interest rate, r, is equivalent to j6 = 8% over 8 months?

20. Rank the following interest rates in the order in which they would give the highest to lowest amount of interest on an investment:

a. j 2 = 15.25% b. j 4 = 15.1% c. j 12 = 14.85%

21. Your portfolio of investments consists of a $10,000 loan due at the end of 5 years with interest at j 1 = 10%, and a $25,000 loan due at the end of 10 years with interest at j 12 = 6%. What is the present value of this portfolio at j 2 = 8%? (Answer to nearest dollar)

A. $18,165

B. $28,109

C. $31,639

D. $32,029

22. You buy a motor boat worth $13,400. You can pay cash, or chose one of two payment options:

Option 1: Pay $10,000 in one year and $5000 in two years Option 2: Pay $17,400 in 3 years

If the interest rate on both options is j 12 = 9%, which one should you take and how much cheaper is it compared to paying cash?

A. Best option is to pay cash

B. 1 $17.19

C. 1 $78.46

D. 2 $103.81

23. A woman plans to withdraw $1800 18-months from now and $2400 36-months from now. How much does she need to deposit today if the interest rate is j 12 = 12% for the first 2 years and j 12 = 6% thereafter?

A. $3182.25

B. $3285.18

C. $3510.38

D. $3765.40

24. A promissory note for $10,000, dated July 1, 2007, is due in four years with interest at j 2 = 8%. On October 1, 2008, it was sold to an investor who discounted the note at j 4 = 9%. What was the purchase price of the note on October 1, 2008?

A. $10,714.46

B. $10,955.53

C. $11,176.77

D. $11,975.31

25. A company has a loan that is due on December 30, 2009. At that time they are required to pay $20,000. What was the original amount of the loan if it was taken out on September 30, 2007 at j 4 = 12%?

A. $15,290.08

B. $15,328.33

C. $15,498.49

D. $15,516.81

26. You take out a loan of $25,000 from the ABC company. It is due in 5 years with interest at j 2 = 6%. After 18 months, ABC sells your loan to the XYZ company at a price that will earn XYZ a rate of return of j 4 = 5%. What price does XYZ pay?

A. $28,264.75

B. $28,234.61

C. $27,318.18

D. $26,934.58

27. A long term promissory notes for $20 000 is signed on April 15, 2010. It is due on October 15, 2013 at j2 = 6%. The maturity value of the note is $24 597.48. The note is sold on April 15, 2011 to a bank that discounts the note at j4 = 8%. What are the proceeds?

A. $21,218.00

B. $20,582.07

C. $20,292.30

D. $20,178.50

28. An individual borrowed $10,000 nine months ago and another $6,000 three months ago and he wishes to pay off this loan with a payment of $ X today. You are given that j 2 = 10% and that the practical method of crediting/charging interest is used. Determine $ X .

A. $16,918.90

B. $16,912.50

C. $16,907.46

D. $16,900.26

29. What is the accumulated value of $15,000 over 6 years and 5 months if j4 = 10% and the exact method of accumulating is used? (Answer to the nearest dollar)

A. $27,411

B. $27,981

C. $28,039

D. $28,271

30. You invest $50,000 today in a fund earning j 4 = 12%. How much have you accumulated 65 months later using the practical method?

A. $81,038

B. $93,945

C. $94,866

D. $94,875

31. What is the present value of $100,000 due exactly 4 years and 8 months from today if j 2 = 9% and the practical method of discounting is used?

A. $66,324.55

B. $66,310.35

C. $66,296.00

D. $65,358.66

32. You invest $20,000 today in a fund earning interest at j 1 = 4%. How much will you have in 5 years, 270 days if the exact method of accumulating is used?

A. $25,063.05

B. $25,059.46

C. $25,053.05

D. $25,049.36

33. You invest $50,000 today in a fund earning j 4 = 12%. How much have you accumulated 5 years and 5 months later using the exact method?

A. $93,935.73

B. $93,944.88

C. $94,865.84

D. $94,875.02

34. The maturity value of a promissory note due on Sept. 14, 2010 is $6200. What are the proceeds of the note using the practical (approximate) method on Jun. 2, 2007 if j 2 = 10%?

A. $4454.51

B. $4498.36

C. $4502.80

D. $4760.25

35. What is the accumulated value of $5000 at the end of 4 years, 11 months if j 4 = 9% using the exact method?

A. $7588.52

B. $7638.07

C. $7708.09

D. $7744.89

36. Using the exact method, what is the present value of $100,000 due in 19 months if j4 = 12%? (Answer to nearest dollar.)

A. $82,935

B. $82,927

C. $82,919

D. $82,901

37. What is the accumulated value of $5000 for 20 months at j 2 = 11% using the practical method?

A. $5978.85

B. $5976.93

C. $5957.00

D. $5949.88

38. Three payments of $10,000 are made at the end of 3, 9 and 15 months respectively. Calculate the total accumulated value of these payments at the end of two years using the exact method with j 6 = 6%.

A. $30,756.65

B. $32,334.08

C. $32,340.10

D. $34,869.82

39. A loan of $ A is taken out today. You are given that this loan is to be paid off with a payment of $20,000 in 3 years and 8 months. Determine A , if j 2 = 10% and the practical method of crediting/charging interest is used.

A. $13,988.01

B. $13,984.33

C. $13,980.62

D. $13,762.40

40. Mrs. Singh owes Mr. Valdy $55,513.78 in 14-months. Mr. Valdy agrees to let her repay the loan with a payment of $35,000 in 6-months, $10,000 in 8-months and $ X in 14-months. If money is worth j 4 = 6%, what is the value of X , using 14-months as the focal date along with the exact method of crediting/charging interest?

A. $8793.97

B. $9204.40

C. $9230.06

D. $9794.03

41. A long term promissory note is due on April 5, 2013. The maturity value of the note on that date is $4720.56. On June 7, 2010, the holder of the notes sells it to a bank who discounts the note at j 2 = 14%. Using the practical method of crediting/charging interest, calculate the proceeds of the sale.

A. $3217.60

B. $3218.30

C. $3219.24

D. $3221.52

42. A promissory note with a maturity value of $20,000 is sold to a bank 32-months before maturity. The bank discounts the note using j 2 =16%. What are the proceeds, if the practical method is used?

A. $13,275.57

B. $13,258.11

C. $13,266.92

D. $12,939.48

43. A non-interest bearing long term promissory note is due on February 1, 2011. The loan amount was $30,000. On October 15th, 2010 the note was sold to a bank that charges interest at j 12 = 12%. How much did the bank pay for this note? Assume the practical (or approximate) method is used for fractional time periods.

A. $28,962.10

B. $28,952.61

C. $28,829.41

D. $28,675.35

44. A non-interest bearing long term promissory note is due on February 1, 2011. The loan amount was $30,000. On October 15th, 2010 the note was sold to a bank that charges interest at j 12 = 12%. How much did the bank pay for this note? Assume the exact method is used for fractional time periods.

A. $28,959.25

B. $28,973.20

C. $28,977.85

D. $29,104.21

45. Using the practical method, what is the accumulated value of $20,000 over 7 years and 10 months if the investment earns interest at j 2 = 10%?

A. $42,898.52

B. $42,953.22

C. $42,964.52

D. $44,350.47

46. A loan of $A is taken out today. The loan is to be paid off with a payment of $20,000 in 44 months. If the interest rate on the loan is j2 = 10% and the practical method of crediting/charging interest is used, what is the value of A?

A. $13,988.01

B. $13,762.40

C. $13,984.33

D. $13,980.62

47. An individual borrowed $10 000 nine months ago and another $6000 three months ago and he wishes to pay off this loan with a payment of $ X today. You are given that j 2 = 10% and that the practical method of crediting/charging interest is used. Determine $ X .

A. $16 918.90

B. $16 912.50

C. $16 907.46

D. $16 900.26

48. Mrs. Singh owes Mr. Valdy $55 513.78 in 14-months. Mr. Valdy agrees to let her repay the loan with a payment of $35 000 in 6-months, $10 000 in 10-months and $X in 14-months. If money is worth j4 = 6%, what is the value of X, using 14-months as the focal date along with the exact method of crediting/charging interest?

A. $8894.58

B. $8895.72

C. $8896.37

D. $8897.69

49. A lump sum of $25,000 is due in 4-years and 5 months. What is the present value of this amount using the practical method if the interest rate is j2 = 8%?

A. $17,681.77

B. $17,679.86

C. $17,677.99

D. $17, 685.52

50. What is the present value of $100,000 due in 5-years and 10-months using the practical method if the interest rate is j 4 = 12%?

A. $49,685.31

B. $50,167.50

C. $50,172.39

D. $50,177.24

51. You deposit $1000 today in an account that pays interest at j 2 = 8% for the next 5 years and j 2 = 6% thereafter. How many complete interest periods will it take for you to accumulate at least double your original investment?

A. 11 periods

B. 18 periods

C. 21 periods

D. 24 periods

52. You invest $2130.22 today. Four and a half years later, you see that this investment has grown to $3316.08. What nominal rate of interest, j 12, have you been earning?

53. A $2000 loan is to be repaid with payments of $1200 in 1 year, $800 in 4 years, and $400 in n

years, assuming a nominal interest rate of 6% compounded annually. Determine n .

54. $15,000 is invested into an account that earns interest at j 4 = x%. There are no other deposits made into the account. At the end of 15 years, the accumulated value of the account is $45,000. Determine x .

55. A deposit of $100 is made into an account earning j 12 = 18%. Another $100 is deposited into a 2nd account earning j 2 = 10%. At what time, n (where n is in years), would the accumulated value of the first account be twice as much as the accumulated value in the 2nd account? (Answer in years)

56. How long does it take for a loan of $5000 to accumulate $1000 of interest if j 2 = 10%?

A. 1 year, 10 months, 13 days

B. 1 year, 10 months, 29 days

C. 3 years, 8 months, 26 days

D. 3 years, 9 months, 28 days

57. An investment doubles in 9 ½ years. What nominal rate of interest j 4 is being earned?

58. You invest $1000 today at j 12 = 6%. After 2 years, the interest rate changes to j 12 = 12%. How many years from today will it take the $1000 to grow to $10,000?

A. 21.28 years

B. 20.28 years

C. 19.28 years

D. 18.28 years

59. How long will it take for $750 to accumulate to $1000 if j 2 = 9%?

A. 6 years, 175 days

B. 6 years, 196 days

C. 3 years, 88 days

D. 3 years, 98 days

60. You wish to have $4000 in 3-years time. If you invest $3000 today, what nominal rate j 4 must you earn on your investment to reach your goal?

61. If money triples in 6 years, what rate of interest, j 2, is being earned?

62. $4000 is deposited into an account earning j 2 = 8% for the first 2 years and j 2 = 10% thereafter. How long will it take for it to grow to $9041.67?

A. 6 yrs, 9 months

B. 8 yrs, 9 months

C. 13 yrs, 6 months

D. 17 yrs, 6 months

63. If money triples in value in 8 years, what nominal rate of interest compounded semi-annually is being earned?

64. $25,000 was deposited into an investment account earning interest at a nominal rate of j 2 = x % for 10 years. You are given that the corresponding total amount of interest earned in the first four years is $9,012.22. Determine x .

65. A car insurance company charges you a premium of $1452 a year for your car insurance policy. You have two options. Option 1 is to pay the $1452 in cash today. Option 2 is to make three payments of $499 at the following times: today, 3-months from now and 6-months from now. What nominal rate of interest, j 4, are you being charged?

A. 11.73%

B. 12.12%

C. 12.53%

D. 13.06%

66. Which of the following rates would lead to the shortest length of time ( n , in years) needed to double an initial investment of $1000?

A. j 1 = 9.15%

B. j 2 = 8.90%

C. j 4 = 8.84%

D. j 12 = 8.77%

67. If money doubles at a certain rate of interest compounded monthly in 6 years, how long will it take for the same amount of money to triple in value?

A. 10.40 years

B. 9.51 years

C. 8.35 years

D. Cannot be determined

68. What is the nominal rate of interest convertible quarterly at which the discounted value (present value) of $15,000 due at the end of 186 months is $5000?

69. You are given that at a certain rate j 1, money will double itself in 12-years. At this same rate j 1, how many years will it take for $1500 to accumulate $700 of interest?

A. 8.2 years

B. 6.6 years

C. 5.6 years

D. 4.2 years

70. You deposit $1000 today in an account that pays interest at j2 = 8% for the next 5-years and j2 = 6% thereafter. How long in total (in years and days) will it take for you to at least double your original investment?

A. 10 years, 34 days

B. 10 years, 67 days

C. 10 years, 79 days

D. 10 years, 158 days

71. A car insurance company charges you a premium of $1452 a year for your car insurance policy. You have two options. Option 1 is to pay the $1452 in cash today. Option 2 is to make two payments of $749 at the following times: today and 6-months from today. What nominal rate of interest, j4, are you being charged?

A. 11.73%

B. 13.06%

C. 12.88%

D. 12.12%

72. Paul deposited $1000 in a savings account paying interest at j1 = 4.5%. The account has now grown to $1246.18. If he had been able to invest the same amount over twice as long in a fund paying interest at j1 = 5.5%, to what amount would his investment now have accumulated?

A. $1498.43

B. $1882.35

C. $1653.64

D. $1708.14

73. You borrows $10,000 today at j2 = 8%. You pay back $7000 at the end of 2-years and another $7000 at the end of n-years (from today). What is the value of n? (years, days)

A. 7 yrs, 30 days

B. 7 yrs, 60 days

C. 14 yrs, 30 days

D. 14 yrs, 60 days

74. Mr. Harry Leggs borrows $5000 today, due with interest at j 4 = 8% in one lump sum at the end of 2 years. Instead, Mr. Leggs wishes to pay $2000 six months from today and $ X in 18 months. If money is worth j 12 = 6%, what is X ?

A. $3252.09

B. $3346.29

C. $3465.95

D. $3562.23

75. A loan of $5000 is taken out today. It is due with interest at j 4 = 8% in 2 years. Instead, the borrower negotiates with the lender to pay $2500 in 1 year and $ X in 3 years. If the lender can reinvest any payment at j 2 = 5%, what is the value of X ?

A. $2493.59

B. $3038.93

C. $3395.34

D. $3581.68

76. A woman borrowed money and owes $3000 one-year from now and $3000 three-years from now. The loan is renegotiated so that the woman can instead pay $ X two-years from now and $4000 four-years from now which will fully pay back the loan. If the interest rate on the loan is j 2 = 8%, what is the value of X ?

A. $2599.25

B. $2588.42

C. $2580.78

D. $2306.39

77. A debt of $5700 is due, with interest at j 2 = 8%, in three years. It is agreed instead that the loan will be repaid with a payment of $ X in one year and $3000 in two years. If money is worth j 4 = 4%, what is the value of X ?

A. $3282.18

B. $3348.50

C. $3391.45

D. $3777.51

78. Payments of $1000 due in 6 months, $1500 due in 9 months, and $1200 due in 15 months are to be exchanged for a single payment $ X due in 12 months. What is X if j 4 = 6%?

A. $3770.73

B. $3734.99

C. $3809.54

D. $3927.05

79. You borrow $2000 today. The loan is due in 3 years, with interest at j 1 = 9%. It is agreed that you will instead pay $1000 one year from now and $ X two years from now. If money is worth j 2 = 6%, what is the value of X ?

A. $1190.12

B. $1286.20

C. $1380.48

D. $1529.16

80. A debt of $7000 is due with interest at j 2 = 8% at the end of 3 years. To repay this debt, a payment of $1500 is made at the end of 1 year, followed by a payment of $ X at the end of 2 years. If money is worth j 4 = 10%, what is the value of X so that the loan is fully paid off?

A. $4685.94

B. $4713.64

C. $5172.40

D. $6873.10

81. A student borrows $2,000 today and they agree to pay off the loan with one payment of $2,590.06 to be made at the end of 3 years time. It is then agreed that, instead of paying off the loan with one payment, the student can pay off the loan with a payment of $1000 one year from now and $ X two years from now. Given that j 2 = 6%, determine $ X .

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## Financing Options: Long-term Financing Flashcards Preview

**Long-term (Capital) Financing Defined:**

__ Long-term Financing:__ long-term, or capital, financing provided by funding which does not become due within one year.

- It's the primary source of funding for most firms
**The cost associated with each source used will determine firm's weighted average cost of capital (WACC).**

**Primary Forms of Long-term Financing:**

- Long-term notes
- Financial (capital) leases
- Bonds
- Preferred Stock
- Common Stock

**Long-term Notes:**

__ Long-term Notes:__ They result from acquiring cash through borrowing with repayment due in more than one year.

- Typically a promissory note is required
- Borrowings are commonly from one to ten yrs, but may be longer
- Repayment is usually in periodic installments
- Note may be secured (collateral) by a mortgage on property or real estate
- Promissory note often containes restrictive covenants.

**Common Restrictive Covenants - (to reduce risk)**

- Maintaining a certain working capital condition (e.g. a minimum working capital ratio)
- Restrictions on incurrence of additional debt without lender's approval.
- Specification of required frequency and nature of financial information provided to lender, perhaps audited FS.
- Restrictions on management changes without lender approval.

**Cost of Long-Term Notes: It will depend on:**

- General level of interest
- Creditworthiness of borrowing firm
- Nature and value of collateral, if any

**Interest rate is likely to be expressed as a function of a macroeconomic benchmark.**

- Commonly available to creditworthy firms
- Provides long-term financing, often w periodic repayment

**Disadvantages:**

- Poor credit rating results in higher interest rate, greater security requirements, and more restrictive covenants.
- Violation of restrictive covenants can trigger serious consequences, including technical default.

**Financial (Capital) Leases:**

__ Financial Leases:__ Leasing is a common way of acquiring use of certain assets. In some cases leasing may be less costly than buying.

**When leasing of assets is possible, the acquisition of assets should be evaluated under both purchase and lease options:**

- Is proposed project economically feasible if assets are purchased?
- Is proposed project economically feasible if assets are leased?

**Possible Outcomes:**

*Reject project*, if neither alternatives shows the project is feasible*Purchase assets*, if the purchase alternative is feasible and leasing alternative is not or if both are feasible, but purchase has higher return.*Lease assets,*if the leasing alternative is feasible and purchasing alternative is not or if both are feasible, but leasing has higher return.

__ Cost of Leasing:__ may be less than cost of buying because:

- Lessor has buying power or efficiencies that lessee does not have.
- Lessor has lower interest rate than the lessee
- Lessor has tax advantages the the lessee does not

**Lease Terms:**

**Net Lease:**Lessee assumes cost associated w ownership**(executory costs):**- Maintenance
- Taxes
- Insurance

**Advantages and Disadvantages of Financial Leases:**

- Limited immediate cash outlay
- Possible lower cost than purchasing
- Possible scheduling of payments to coincide with cash flows
- Debt (lease payments) is specific to amount needed.

**Disadvantages:**

- Not all assets available for leasing
- Lease terms may prove different than the period of asset usefulness
- Often chosen over buying for noneconomic reasons (e.g., convenience).

Which of the following long-term notes would best facilitate financial leverage for the borrowing firm?

Variable Rate Long-term Note Fixed Rate Long-term

Variable Rate Long-term Note: NO

Financial leverage derives from the use of debt with a fixed or determinable cost (rate of interest) for capital financing. Therefore, financial leverage would be possible with either fixed rate or variable rate debt (notes) however, **fixed rate debt would better facilitate financial leverage because the cost of the use of debt-financed capital would not change over the life of the financing. The cost of variable rate debt can change, thereby making the degree of leverage more uncertain over the life of the debt.**

What would be the **primary reason for a company to agree to a debt covenant** limiting the percentage of its long-term debt?

A. To cause the price of the company's stock to rise.

B. To lower the company's credit rating.

C. To reduce the risk of existing debt holders.

D. To reduce the interest rate on the debt being issued.

**D. To reduce the interest rate on the debt being issued.**

The primary reason for a company to agree to a debt covenant limiting the percentage of its long-term debt would be to **reduce the risk, and therefore the interest rate, on debt being issued**. Debt covenants place contractual limitations on activities of the borrower to help protect the lender. As such, they reduce the default risk associated with a debt issue and, therefore, reduce the interest rate on that debt.

**Bonds Defined and Features:**

__ Bonds:__ Long-term promissory notes wherein the borrower, in return for buyers'/lenders' funds, promises to pay the bondholders a fixed amount of interest each year and to repay the face value of the note at maturity.

**Bond Features:**

**Bond Indenture**= Bond contract**Par/Face Value**= Bond principal, commonly $1,000 per bond**Coupon rate (Stated rate)**= Annual rate of interest stated on the face of the bond.**Maturity**= Time at which issuer repays the bondholder principal and extinguishes debt.**Debenture Bonds**= Unsecured bonds, no specific assets are desginated as collateral.__Riskier and higher return and cost than secured bonds.__**Secured Bonds**= Have specific assets designated as collateral like:**Mortgage Bonds:**secured by real property like land or buildings

**Bond Selling Price and Value:**They depend on the relationship between the rate of interest the bonds pay (coupon or stated rate) and the rate of interest in the market for comparable risk when bond is issued.**Bond Selling Price and Value:****Coupon Rate > Market Effective Rate = Sells at***Premium***Coupon Rate < Market Effective Rate = Sells at***Discount*

**Coupon Rate = Market Effective Rate = Sells at***Par*- is detertmined as the PV of cash flows from the bonds:**Bond Selling Price or Fair Value****Periodic interest:**discounted ast PV of annuity at market effective rate.**Face Value:**discounted ast PV of single amount at market effective rate.

*Discount using the market rate of interest.**Sum of present values = selling price of bonds and reflects any premium or discount.*

**Market Rate of Interest and Market Price of Bonds:**The market price of bonds__Market Rate of Interest and Market Price of Bonds:__with changes in the market rate of interest:**changes inversely****Market rate of int goes down = Market price of bond goes up.**Assume $1,000 bonds outstanding that pay 4% - that rate doesn't change (coupon)

The market rate goes up to 5%

As a consequence, the value of 4% bonds goes down, no one will buy a 4% bond for $1,000 when they can get a better rate of interest (5%) on the new bonds. So your 4% bonds will sell in the market only if the price is such that they earn 5%.

What's the price? Market price of the $1,000 bond would be $800:

The bond would have to sell in the market for $800 in order for the buyer to earn 5% interest:

$1,000x.04 = $40 / .05 =

**$800**Bondholders face what is called

**"Market Interest Rate" Risk:**- The risk that market will go down due to interest rates going up.
__The longer the maturity of the bonds, the greater the risk of that happening (because of longer holding period) and the higher the required (stated) interest rate.__

**Describe the calculation of the Current Yield on a bond:****Current Yield of Bond:**The ratio of annual interest payments to the current market price of the bond. It is computed as:

**Annual interest payment/Current market price****Describe the Yield to Maturity for Bonds (also called the expected rate of return).****Yield to Maturity for Bonds:**The rate of return required by investors as implied by the current market price of the bonds determined as the

**discount rate that equates present value of cash flows from the bonds with the current price of the bonds.****Advantages and Disadvantages for Bonds:**- A source of large sums of capital
- Does not dilute ownership or earnings per share
- Interest payments are tax deductible.

**Disadvantages:**- Required periodic interest payments-default can result in bankruptcy
- Required principal repayment at maturity-default can result in bankruptcy
- May require security and/or have restrictive covenants.

Which of the following statements

**concerning debenture bonds and secured bonds is/are correct**?I. Debenture bonds are likely to have a greater par value than comparable secured bonds.

II. Debenture bonds are likely to be of longer duration than comparable secured bonds.

III. Debenture bonds are more likely to have a higher coupon rate than comparable secured bonds.

A. I only.

B. II only.

C. III only.

D. I, II, and III.**C. III only. Debenture bonds are more likely to have a higher coupon rate than comparable secured bonds.**Debenture bonds are unsecured bonds. Because they are unsecured, they are likely to have a higher coupon rate (interest rate) than comparable secured bonds.

Which of the following types of bonds is

**most likely to maintain a constant market value?**A. Zero-coupon.

B. Floating-rate.

C. Callable.

D. Convertible.**B. Floating-rate.****Floating-rate bonds**are most likely to maintain a constant market value. The rate of interest paid on floating-rate bonds (also called variable-rate bonds/debt) varies with the changes in some underlying benchmark, usually a market interest rate benchmark (e.g., LIBOR or the Fed Funds Rate). Because the interest rate changes with changes in the market rate of interest, they maintain a relatively stable (constant) market value.**Preferred Stock Defined:**ownership interest with preference claims (over common stock)**Preferred Stock Defined:****It has characteristics of both bonds and stock.****It is like bonds because:**__Usually does not have voting rights__- Dividends usually are limited in amount and expected (like bond interest)

- Grants ownership interest
- Has no maturity date
__Does not require dividends be paid, though they are expected____Dividends are not an expense and are not tax deductible.__

**Preferred Stock Characteristics I:**Preferred Stock Characteristics:

**Preferred Stock Characteristics II:****Preferred Stock Valuation:**PV of expected cash flows.**PS Valuation:**- Preferred cash flow is preferred dividends
- Elements use to value P/S are:
- Estimated future annual dividends
- Investors' required rate of return
- An assumption that dividend stream will exist in perpetuity

Allen issues $100 par value preferred stock that is selling for $101 per share, on which the firm has to pay an underwriting fee of $5 per share sold. The stock is paying an annual dividend of $10 per share. Allen's tax rate is 40%. Which one of the following is the cost of preferred stock financing to Allen?

- Annual Dividend: $10
- Net Proceeds of Stock Issuance: $101 - $5 underwriting fee = $96
- $10 / $96
**= 10.4%**

**Preferred Value Theoretical Value (PSV) Calculation:**- PSV: Annual Dividend / Required Rate of Return
- Example:
- Annual Dividend: $4
- P/S Investors' required rate of return: 8%
- PSV: $4 / .08 =
**$50**

**Preferred Stock Expected Rate of Return (PSER) Calculation:**- PSER: Annual Dividend / Market Price
- Example:
- Annual Dividend: $4
- Market Price: $52
- PSER: $4 / $52 = 7.7%

A company recently issued 9% preferred stock. The preferred stock sold for $40 a share, with a par of $20. The cost of issuing the stock was $5 a share.

**What is the company's cost of preferred stock?**

## Chapter 10 Long Term Financing

Long-term financing is a financial plan or a debt obligation that a firm used in its operations in a time frame exceeding a year.

Three types of long-term financing:

o Bonds o Preferred stocks o Common stocks

#### 1. BONDS

o A bond is a long-term promissory note that promises to pay the bondholder a predetermined, fixed amount of interest each year until maturity. At maturity, the principal will be paid to the bondholder.

o A bondholder has a priority of claim to the firm's assets before the preferred and common stockholders in the case of a firm's insolvency.

o Bondholders must be paid interest due them before dividends can be distributed to the stockholders.

o A bond's par value is the amount that will be repaid by the firm when the bond matures, usually RM1,000.

o Coupon interest rate is the contractual agreement of the bond specifies as a percent of the par value or as a flat amount of interest which the borrowing firm promises to pay the bondholder each year. For example: A RM1,000 par value bond specifying a coupon interest rate of 9% is equivalent to an annual interest payment of RM90.

o The bond has a maturity date, at which time the borrowing firm is committed to repay the loan principal.

## Watch the video: Mathematics of Investment - Simple Interest - Promissory Notes Topic 6 (November 2021).