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Chapter 2

Chapter 2. Simple Discount. START. EXIT. Chapter Outline. 2.1 Simple Discount 2.2 Simple Discount vs. Simple Interest 2.3 Secondary Sales of Promissory Notes Chapter Summary Chapter Exercises. 2.1 Simple Discount. Definition 2.1.1

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Chapter 2

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  1. Chapter 2 Simple Discount START EXIT

  2. Chapter Outline 2.1 Simple Discount 2.2 Simple Discount vs. Simple Interest 2.3 Secondary Sales of Promissory Notes Chapter Summary Chapter Exercises

  3. 2.1 Simple Discount Definition 2.1.1 A loan that is made on the basis of a fixed maturity value is called a discount loan. The tender subtracts an amount, called the discount, from the maturity value, and pays the result, called the proceeds, to the borrower.

  4. 2.1 Simple Discount • Example 1 Jennifer knows she will be getting a paycheck for $500 at the end of the week, but she needs money now. She takes out a loan against this paycheck, borrowing as much as she can pay off with her check when it arrives.

  5. 2.1 Simple Discount • Example 2 When you file your federal income taxes, you find that you are owed a refund of $737.15. Your tax preparer offers the option of getting your money right away, instead of waiting for the IRS to process your return and send your refund check. In exchange for agreeing to sign over your refund check when it arrives, the preparer agrees to lend you $707.15 today.

  6. 2.1 Simple Discount • Example 3 The Cedar Junction Central School District is scheduled to receive a state aid check for $72,500 on April 1. The district needs the funds by mid-February to meet expenses, but unfortunately the state is unwilling to make the payment early. In order to avoid a cash crunch, the district borrows $71,342 from First Terrapin National Bank, to be repaid on April 1 when the state aid arrives. The amount borrowed is based on the $72,500 that the district will have available for repayment when the aid check arrives.

  7. 2.1 Simple Discount A quick review… • If we look at the last example, we would call • $71,342 the principal • $72,500 -- $71,342 = $1,158 the interest • $72,500 (principal + interest) the maturity value • April 1 the maturity date • However, when a promissory note is based on discount, the face value is not the same as its principal (as seen before). Definition 2.1.2 The face value of a discount note is its maturity value.

  8. 2.1 Simple Discount • One common example of a discount note is U.S. Savings Bonds. A savings bond is actually a promissory note issued by the U.S. government so, when you buy a savings bond, you are actually loaning money to Uncle Sam. • Although there are several types of savings bonds, the most familiar type is sold for half of its face value. • Therefore, a “$50 U.S. Savings Bond” actually costs only $25. The $50 face value reflects the amount the government guarantees that it will pay for the bond on maturity, which is usually many years away (usually 20 years).

  9. 2.1 Simple Discount • Treasury bills (known as T bills) are another common example of a discount note. They are short-term loans to the U.S. federal government, carrying terms ranging from a few days to 6 months, though the most common terms are 4, 13, or 26 weeks. • You might buy a $1,000 face value T bill for $985, a $15 discount from the maturity value.

  10. 2.1 Simple Discount • Formula 2.1 The Simple Discount Formula D = MdT where • D represents the amount of SIMPLEDISCOUNT for a loan (equivalent to interest) • M represents the MATURITY VALUE • d represents the INTEREST DISCOUNT RATE • T represents the TERM FOR THE LOAN

  11. 2.1 Simple Discount Example 2.1.1 • Problem • A $10,000 face value discount note has a term of 4 months. The simple discount rate is 6%. Find the amount of the discount. • Solution D = MdT D = $10,000 x 6% x 4/12 D = $10,000 x 0.06 x 4/12 D = $200

  12. 2.1 Simple Discount Example 2.1.2 • Problem • A $5,000 face value note has a term of 219 days. The simple discount rate is 9 3/8%. Find the proceeds of the note. • Solution D = MdT D = $5,000 x 9 3/8% x 219/365 D = $281.25 Proceeds = Maturity Value – Discount Proceeds = $5,000 -- $281.25 = $4,718.75

  13. 2.1 Simple Discount Example 2.1.3 • Problem • A 3-month note is discounted by $28.75. The simple discount rate is 5 ¾%. Determine the maturity value and the proceeds of the note. • Solution D = MdT $28.75 = M(5 ¾%)(3/12) $28.75 = M(5.75%)(3/12) $28.75 = M(0.0575)(3/12) $28.75 = M(0.014375) M = $2,000 Proceeds = Maturity Value – Discount Proceeds = $2,000 -- $28.75 = $1,971.25

  14. 2.1 Simple Discount Example 2.1.4 • Problem • When Nestor filed his federal income taxes, he was happy to learn that he was due a refund of $799.45. He was less happy to learn that it would take 45 days for his refund check to arrive. His tax preparer offered to give him $775.00 on the spot, in exchange for Nestor’s tax refund check when it arrives. What simple discount rate does this offer equate to? • Solution Discount = $799.45 -- $775.00 = $24.45 D = MdT $24.45 = $799.45 x d x 45/365 $24.45 = $98.56232877 x d d = 0.248066379 d = 24.81%

  15. 2.1 Simple Discount Example 2.1.5 • Problem • A $10,000 T bill with 182 days to maturity sold at auction for $9,753.16. What is the simple discount rate? • Solution Discount = $10,000 -- $9,753.16 = $246.84 D = MdT $246.84 = $10,000 x d x 182/365 $246.84 = $4,986.3013699 x d d = 0.04950363 d = 4.95%

  16. 2.1 Simple Discount Example 2.1.6 • Problem • Killawog Financial Corp invested $49,200 in discount bonds with face values totaling $50,000. The discount rate was 4%. How long will it be until the notes mature? • Solution Discount = $50,000 -- $49,200 = $800 D = MdT $800 = $50,000 x 4% x T $800 = $50,000 x 0.04 x T $800 = $2,000 x T T = 0.4 years T = 0.4 x 365 = 146 days

  17. 2.1 Simple Discount Example 2.1.7 • Problem • The Reeds Corners Central School District borrowed $4,959,247 on March 7, 2005, in anticipation of receiving a state aid payment of $5,000,000. The loan was based on a simple discount rate of 4 ¼%. On what date will the district receive its state aid? • Solution Discount = $5,000, 000 -- $4,959,247 = $40,753 D = MdT $40,753 = $5,000,000 x 4 ¼% x T $40,753 = $5,000,000 x 0.0425 x T T = 0.1917788235 years T = 0.1917788235 x 365 = 70 days March 7 is the 65th day of the year so the note matures on the 66 + 70 = 136th day of the year. The state aid will be received on May 16, 2005.

  18. Section 2.1 Exercises

  19. Problem 1 • Decatur County Chamber of Commerce issued discount notes to finance a downtown renovation and beautification project. Each discount note has a face value of $2,000 and a term of 10 years. The notes were sold at a simple discount rate of 4 ½%. • Find the proceeds of each note sold. CHECK YOUR ANSWER

  20. Solution 1 • Decatur County Chamber of Commerce issued discount notes to finance a downtown renovation and beautification project. Each discount note has a face value of $2,000 and a term of 10 years. The notes were sold at a simple discount rate of 4 ½%. Find the proceeds of each note sold. • D = MdT • D = $2,000 x 4 ½% x 10 • D = $2,000 x 4.5% x 10 • D = $2,000 x 0.045 x 10 • D = $900 • Proceeds = Maturity Value – Face Value • Proceeds = $2,000 -- $900 = $1,100 BACK TO GAME BOARD

  21. Problem 2 • Find the maturity value of a 9-month discount note if the discount is $70 and the discount rate is 11.9%. CHECK YOUR ANSWER

  22. Solution 2 • Find the maturity value of a 9-month discount note if the discount is $70 and the discount rate is 11.9%. • D = MdT • $70 = M x 11.9% x 9/12 • $70 = M x 0.119 x 9/12 • $70 = M x 0.08925 • M = $784.31 BACK TO GAME BOARD

  23. Problem 3 • Lynn’s Hair Cuttery signed a $10,000 discount note with a term of 120 days and received proceeds of $9,500. What is a simple discount rate for this note? CHECK YOUR ANSWER

  24. Solution 3 • Lynn’s Hair Cuttery signed a $10,000 discount note with a term of 120 days and received proceeds of $9,500. What is a simple discount rate for this note? • D = MdT • $500 = $10,000 x d x 120/365 • $500 = $3,287.67 x d • d = 0.15208 • d = 15.21% BACK TO GAME BOARD

  25. Problem 4 • On May 11, 2007, you bought a $2,000 T bill for $1,850. The simple discount rate was 5%. What is a term? CHECK YOUR ANSWER

  26. Solution 4 • On May 11, 2007, you bought a $2,000 T bill for $1,950. The simple discount rate was 5%. What is a term? • D = MdT • $50 = $2,000 x 5% x T • $50 = $2,000 x 0.05 x T • $50 = $100 x T • T = 0.5 years • T = 0.5 x 365 = 183 days BACK TO GAME BOARD

  27. 2.2 Simple Discount vs. Simple Interest • Suppose that Lysander Office Supply borrowed $38,000 for 1 year from Van Buren Capital Funding Corp. The maturity value of the note was $40,000. Was this loan based on simple discount or simple interest? • Simple interest? Principal = $38,000 Interest = $2,000 Maturity Value = $38,000 + $2,000 = $40,000 • Simple discount? Maturity Value = $40,000 Discount = $2,000 Proceeds = $38,000 • Actually, interest and discount are just two different ways of looking at the same thing. • However, we can take a look at the deal both ways and compare simple interest and simple discount rates.

  28. 2.2 Simple Discount vs. Simple Interest Example 2.2.1 • Problem • For the transaction described on a previous slide, find (a) the simple interest rate and (b) the simple discount rate. • Solution (a) I = PRT $2,000 = $38,000 x R x 1 R = 0.0526 = 5.26% Therefore, the simple interest rate is 5.26%. (b) D = MdT $2,000 = $40,000 x d x 1 d = 0.05 = 5% Therefore, the simple discount rate is 5%.

  29. 2.2 Simple Discount vs. Simple Interest • The results on a previous slide may be surprising. However, the simple interest and simple discount rates are not the same thing. • A rate is a percent, and a percent must be of something. • For simple interest, that something is the principal ($2,000 as a percent of $38,000). • For simple discount, that something is the maturity value ($2,000 as a percent of $40,000). • Since the principal and maturity value are always different, for a given loan the simple interest rate and simple discount rate will never be same. • Because the principal will always be less than the maturity value for any loan, the simple interest rate will always be larger than the simple discount rate.

  30. 2.2 Simple Discount vs. Simple Interest Example 2.2.2 • Problem • Killawog Financial invested $49,200 in bonds whose maturity values totaled $50,000. The remaining term of the bonds was 146 days. The simple discount rate was 4%, but what would the equivalent simple interest rate be? • Solution I = PRT $800 = $49,200 x R x 146/365 R = 4.07%

  31. 2.2 Simple Discount vs. Simple Interest Example 2.2.3 • Problem • An investment manager is weighing a choice between two possible investments for a fund that she manages. She originally had planned to invest in a $10,000 face value, 9-month simple discount note issued by the Levy Pants Company, which she was offered at a simple discount rate of 8%. On the other hand, the company has offered to borrow the same amount of money from her fund by signing a note carrying a simple interest rate of 8 ¼%. Which is the better deal for the investment fund? • Solution • On the face of it, this looks like a pretty simple question; 8 ¼% is higher than 8%, and so obviously a lender would prefer the higher rate. • Discount D = MdT, D = $10,000 x 8% x 9/12 = $600 Therefore, the fund would pay $10,000 - $600 = $9,400 for the note • Simple Interest I = PRT, $600 = $9,400 x R x 9/12, R = 8.51% • The 8% simple discount rate is actually equivalent to earning an 8.51% simple interest rate.

  32. 2.2 Simple Discount vs. Simple Interest • When rates quoted for investments are based on discount, often both the interest and discount rate will be given. • Note that the rates differ, depending on the maturity date for the note.

  33. 2.2 Simple Discount vs. Simple Interest • A payday lender is an individual or business that will offer to give you immediate cash in exchange for your agreement to sign over your check to the lender when you receive it. • Suppose that such a lender offers to make this deal with you for a fee of 2% of the paycheck. • It would be quite easy to think of that 2% as a rate per year. • However, the fee is “2% of the paycheck.” So the lender will take a discount of 2% x $750 = $15.00, leaving you with $735. • What is the equivalent simple discount rate if the term of the loan is 4 days? D = MdT, $15 = $750 x d x 4/365, d = 1.825 = 182.50% • A 2% fee doesn’t sound like all that much to pay, but putting it in terms of an annual simple discount rate puts a new perspective on it. • What is the equivalent simple interest rate? I = PRT, $15 = $735 x R x 4/365, R = 1.8622 = 186.22%

  34. 2.2 Simple Discount vs. Simple Interest Example 2.2.4 • Problem • Ginny is expecting a $795 paycheck in 8 days. A payday lender offers to give her cash now for this check. The lender’s fee is 1.5% of the amount, plus a $10 service fee. Find the equivalent simple interest and simple discount rates. • Solution • First, we need to determine how much Ginny will be giving up: 1.5% of her paycheck is $11.93. Adding in the $10 service fee, the total is $21.93. She will receive the difference, $773.07. • Simple Discount D = MdT $21.93 = $795 x d x 8/365 d = 1.2586 = 125.86% • Simple Interest I = PRT $21.93 = $773.07 x R x 8/365 R = 1.2943 = 129.43%

  35. Section 2.2 Exercises

  36. Problem 1 • Elaine loans her brother $150 and he pays her back $160 thirty days later. • Find (a) the simple interest rate and (b) the simple discount rate. CHECK YOUR ANSWER

  37. Solution 1 • Elaine loans her brother $150 and he pays her back $160 ninety days later. Find (a) the simple interest rate and (b) the simple discount rate. • Simple Interest Rate I = PRT $10 = $150 x R x 90/365 $10 = 36.98630 x R R = 0.27037 = 27% • Simple Discount Rate D = MdT $10 = $160 x d x 90/365 $10 = 39.45205 x d d = 0.25347 = 25% BACK TO GAME BOARD

  38. Problem 2 • A $2,000 maturity value with a 180-day term is sold at a simple discount rate of 7.99%. Find the simple interest rate that would be equivalent to the stated simple discount rate. CHECK YOUR ANSWER

  39. Solution 2 • A $2,000 maturity value with a 180-day term is sold at a simple discount rate of 7.99%. Find the simple interest rate that would be equivalent to the stated simple discount rate. • D = MdT D = $2,000 x 7.99% x 180/365 D = $2,000 x 0.0799 x 180/365 D = $78.81 • I = PRT $78.81 = $1,921.19 x R x 180/365 $78.81 = $947.44 x R R = 0.08318 = 8.32% BACK TO GAME BOARD

  40. Problem 3 • Christy will receive a $1,500 paycheck in 14 days. However, she needs money now. A payday lender offers to give her cash today for a fee of 2% of the check amount. • Find (a) the equivalent simple discount rate and (b) the equivalent simple interest rate. CHECK YOUR ANSWER

  41. Solution 3 • Christy will receive a $1,500 paycheck in 14 days. However, she needs money now. A payday lender offers to give her cash today for a fee of 2% of the check amount. Find (a) the equivalent simple discount rate and (b) the equivalent simple interest rate. • Christy will be giving up $1,500 x 2% = $30, so she will receive $1,470. • D = MdT $30 = $1,500 x d x 14/365 $30 = $57.53 x d d = 0.52146 = 52.15% • I = PRT $30 = $1,470 x R x 14/365 $30 = $56.38 x R R = 0.53210 = 53.21% BACK TO GAME BOARD

  42. Problem 4 • An annual subscription to your favorite magazine costs $45.99. Although subscription renewal isn’t due for another month, the publisher is offering you a 5% off the regular price if you renew now. What would be the simple interest rate? CHECK YOUR ANSWER

  43. Solution 4 • An annual subscription to your favorite magazine costs $45.99. Although subscription renewal isn’t due for another three months, the publisher is offering you a 5% discount off the regular price if you renew now. What would be the simple interest rate? • Since you will save $45.99 x 5% = $2.30 and actually pay $45.99 -- $2.30 = $43.69 • I = PRT $2.30 = $43.69 x R x 3/12 $2.30 = 10.9225 x R R = 0.21057 = 21.06% BACK TO GAME BOARD

  44. 2.3 Secondary Sales of Promissory Notes • Suppose that you borrow $1,000 from Friendly Neighborhood National Bank (FNNB). You sign a 1-year note with a simple interest rate of 8%. By now, it is a simple matter for us to calculate that this means that you will be required to pay back the original $1,000 plus $80 simple interest for a total of $1,080 one year from the loan date.

  45. 2.3 Secondary Sales of Promissory Notes • Now suppose that one month after you sign the note, the bank decides that it wants to be repaid early. Does it have the right to demand that you repay the loan before the maturity date? • In most cases, the answer would be no. The lender does have other options, though, because notes can be bought and sold. • Although FNNB may not be able to collect from you before the maturity date, it probably can sell your note to someone else. Such a transaction is often referred to as a secondary sale of the note.

  46. 2.3 Secondary Sales of Promissory Notes • Suppose FNNB sells your note to Cheery Community Savings and Trust (CCST). On the maturity date your $1,080 will go to CCST instead of FNNB. This really doesn’t make any difference to you. • If FNNB does sell the note to CCST, how much should CCST pay? Obviously, CCST needs to make a profit and so won’t pay $1,080, full maturity value. • For example, the two banks might agree to take $60 off the maturity value. The note would then change hands for $1,080 -- $60 = $1,020 • When you first borrowed the money, the situation looked like simple interest, but matters look different when a note that already exists is being sold; now it’s a discount note.

  47. 2.3 Secondary Sales of Promissory Notes Example 2.3.1 • Problem • Suppose that John loans Paul $20,000 for 1 year at 8% simple interest. Three months later, John sells the note to Ringo at a simple discount rate of 7 ¾%. How much does Ringo pay for the note? • Solution • Original Loan I = PRT I = $20,000 x 8% x 1 = $1,600 Maturity Value = $20,000 + $1,600 = $21,600 • Secondary Sale D = MdT D = $21,600 x 7 ¾% x 9/12 = $1,255.50 $21,600 -- $1,255.50 = $20,344.50

  48. 2.3 Secondary Sales of Promissory Notes Example 2.3.2 • Problem • Tinker loaned Evers $997.52 for 235 days at a simple interest rate of 6.54%. But 74 days later, he sold the note to Chance using a simple discount rate of 9.95%. How much did Chance pay for the note? • Solution • Original Loan I = PRT I = $997.52 x 6.54% x 235/365 = $42.00 Maturity Value = $997.52 + $42.00 = $1,039.52 • Secondary Sale (161 days – the remaining term) D = MdT D = $1,039.52 x 9.95% x 161/365 = $45.62 $1,039.52 -- $45.62 = $993.90 • In this example, Tinker sold the note for less than the original principal, $997.52, so he actually lost money on the deal.

  49. 2.3 Secondary Sales of Promissory Notes Example 2.3.3 • Problem • Suppose that John loans Paul $20,000 for 1 year at 8% simple interest. Three months later, John sells the note to Ringo at a simple discount rate of 7 ¾%. • Calculate the simple interest rate that: • John actually earns • Ringo actually earns (assuming he doesn’t sell the note) • Paul actually pays

  50. 2.3 Secondary Sales of Promissory Notes Example 2.3.3 Cont. • Solution • John Principal = $20,000 Interest = $344.50 (see Example 2.3.1) Term = 3 months I = PRT $344.50 = $20,000 x R x 3/12 R = 6.89%

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