Notes & discounting · Reviewer

Notes & discounting

A concise reviewer for Slater Chapter 17.

Use actual days/360 for notes and weeks/52 for the labeled Treasury-bill questions. Keep intermediate maturity values and discounts unrounded. Round final money to cents and annual rates to 0.01 percentage point. Effective rates on proceeds use simple annualization. No fees are included.

How a promissory note works

A note is a promise to pay a specified amount at a specified future time.

The maker is the borrower who signs the promise. The payee is the party entitled to payment. The term is the duration; the maturity date is when payment comes due.

For an interest-bearing note, the maker repays principal plus interest. A non-interest-bearing note has maturity value equal to face value. A simple discount note can still have a borrowing cost because interest is withheld up front.

• Interest-bearing: I = PRT; M = P(1 + RT)
• Non-interest-bearing: M = face value

Worked example: One payment at maturity

A $5,000 note earns 6% simple interest for 120 days. Find its maturity value.

1. I = 5,000 × 0.06 × 120/360 = $100.00.
2. M = 5,000 + 100 = $5,100.00.
3. The initial cash received is $5,000; the one-payment settlement is $5,100.

$5,100.00 at maturity

Watch out: Face value and maturity value are different for a positive-interest note. For a simple discount note, face value already represents maturity value.

Slater Ch. 17, pp. 450–454

Bank discount & proceeds

The bank takes its discount first. Proceeds are the usable cash left for the borrower.

Bank discount B is based on maturity value M, bank discount rate D, and time T. Proceeds Pr are maturity value minus the bank discount.

The annualized effective cost uses proceeds as its money base. With positive discount and positive proceeds, this rate exceeds the stated discount rate. It is a simple annualization, not compound APY.

• B = M × D × T
• Pr = M − B = M(1 − DT)
• Effective rate = B / (Pr × T) = D / (1 − DT)

Worked example: What does the borrower really receive?

A $12,000 simple discount note at 8% runs 90 days. Find bank discount, proceeds, and effective rate.

1. B = 12,000 × 0.08 × 90/360 = $240.00.
2. Pr = 12,000 − 240 = $11,760.00.
3. Effective rate = 240 / (11,760 × 90/360) = 8.16%.

$240.00 discount · $11,760.00 proceeds · 8.16% effective

Watch out: Do not add bank discount to face value at maturity. The borrower already pays that charge by receiving less cash at issue.

Slater Ch. 17, pp. 451–453

Find the amount you need

Desired proceeds are not the same as the note’s face value.

To obtain a specific amount of usable cash, solve the proceeds equation for maturity value. Positive discount means the required face value exceeds desired proceeds.

You can also isolate D or T from B = MDT. Keep the denominator unrounded. Day-based notes use T = days/360; convert a computed T back to days when needed.

• M = Pr / (1 − DT)
• D = B / (MT)
• T = B / (MD)
• Simple-interest note: P = M / (1 + RT)

Worked example: Receive $9,000 after discount

You need $9,000 net proceeds at a 6% bank discount for 120 days. What face value is needed?

1. DT = 0.06 × 120/360 = 0.02.
2. M = 9,000 / (1 − 0.02) = 9,000 / 0.98.
3. M = $9,183.67, rounded to the nearest cent.

Computed face value: $9,183.67

Watch out: Adding 2% of the desired proceeds does not solve the equation exactly. The discount is applied to the larger maturity value.

Slater Ch. 17, pp. 454–457

Treasury-bill yield

In the textbook model, a bill is bought below face value and redeemed at face value.

The discount is the difference between what the investor pays and what is received at maturity. Effective yield measures this gain against the purchase price.

The chapter’s 13-week example uses weeks/52. The mock-exam questions state this convention explicitly. Keep it separate from the days/360 convention used for the other notes.

• Purchase price = face value − discount
• Annualized yield = discount / (price × time)

Worked example: A hypothetical 13-week bill

A $20,000 Treasury bill has a 6% annual discount rate for 13 weeks. Use weeks/52.

1. T = 13/52 = 0.25 year.
2. Discount = 20,000 × 0.06 × 0.25 = $300.00.
3. Purchase price = 20,000 − 300 = $19,700.00.
4. Effective yield = 300 / (19,700 × 0.25) = 6.09%.

Price: $19,700.00 · Effective yield: 6.09%

Watch out: Yield uses the amount invested, not face value, in its denominator. These are hypothetical chapter calculations, not current quoted market yields.

Slater Ch. 17, pp. 452–453

Sell a note before maturity

Calculate the original maturity value first, then apply the bank’s discount to the remaining term.

The original rate and full note term determine maturity value. The bank uses its own discount rate for the remaining days between sale and maturity. Proceeds equal maturity value minus bank discount.

In the chapter’s arrangement, the seller has a contingent liability: the seller may have to pay if the maker defaults. For a non-interest-bearing note, skip original interest and use face value as maturity value.

• M = P[1 + R(original days / 360)]
• Remaining days = original term − elapsed days
• B = M × D × remaining days / 360
• Pr = M − B; seller gain = Pr − P

Worked example: An early sale

A $15,000, 180-day note at 6% is sold after 120 days to a bank charging 8% discount.

1. Original I = 15,000 × 0.06 × 180/360 = $450.00. M = $15,450.00.
2. Discount period = 180 − 120 = 60 days.
3. Bank discount = 15,450 × 0.08 × 60/360 = $206.00.
4. Proceeds = 15,450 − 206 = $15,244.00. Seller gain above principal = $244.00.

$15,244.00 proceeds · $244.00 above original principal

Watch out: Discount maturity value, not original principal. Use the bank’s remaining holding period, not the seller’s elapsed holding period.

Slater Ch. 17, pp. 458–460