Simple discount: moratorium, bills and their net value

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Simple commercial (bank) and rational (true) discount, face and present value, moratorium and the link between discount and interest rates.

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Rivisto daLeon Diniz

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Face value and present value

The value of a debt on its due date is usually called its face value (or nominal value). The value that, invested on a date before the due date, produces an amount equal to the face value is called the present value, current value or real value.

Compared with the formulas we have already learned, the face value plays the role of the amount, and the present value plays the role of the capital (the initial value).

The distinction exists to avoid confusion about which value is being presented. It is often more useful to work with the present value, because it can be compared directly with today's values, which we already know.

The same reasoning works in the opposite direction. When the debtor asks to postpone payment, a moratorium (an extension of the term) is granted, and the creditor charges interest for the extra time: the face value of the debt is carried forward to the new date. For example, a debt of R$ 10,000 (Brazilian reais) due today, extended by 3 months at simple interest of 2% p.m., becomes 10,000 · (1 + 0.02 · 3) = R$ 10,600 on the new date.

Simple discount

The idea of discount is associated with a reduction granted on a sum of money under certain conditions. For example, when a purchase is made in large quantities, the seller commonly grants a discount on the unit price.

Another situation occurs when a company sells a product on credit. In this case, the seller usually issues a duplicata (a Brazilian trade bill, a receivable backed by a sale), which gives it the right to receive the agreed amount from the buyer on a future date. If the seller needs the money earlier, it can go to a bank and discount the bill: the company assigns to the bank the right to collect the bill and, in exchange, receives the money in advance, minus a discount.

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Likewise, a company can discount promissory notes or post-dated cheques at a bank. Because it is so common, this type of discount has its own calculation method, called commercial discount or bank discount, which we will now study.

Commercial or bank discount (bank discount on face value)

Commercial discount uses a rate similar to the interest rate, the discount rate, which is applied to the face value of the bill. That is why in Brazil it is also called discount "por fora" (from the outside). The notation is:

  • D: amount of the discount
  • N: face value of the bill (the value at maturity)
  • Vd: net value of the bill (what is received when discounting it)
  • d: discount rate
  • t: number of periods until maturity

\[ D = N \cdot d \cdot t \]

\[ V_d = N - D \]

Hence:

\[ V_d = N - N \cdot d \cdot t \]

\[ V_d = N \cdot (1 - d \cdot t) \]

\[ \frac{V_d}{N} = 1 - d \cdot t \]

\[ d \cdot t = 1 - \frac{V_d}{N} \]

Examples

A bill of R$ 18,000.00 was discounted at a bank two months before maturity, at a commercial discount rate of 2.5% p.m. Find the discount and the net value paid to the company.

\[ D = N \cdot d \cdot t = 18{,}000 \cdot 0.025 \cdot 2 = 900 \]

\[ V_d = 18{,}000 - 900 = 17{,}100 \]

In this case, the bank pays R$ 17,100 today and receives R$ 18,000 two months later.

A promissory note of R$ 12,000.00 was discounted at a bank 42 days before maturity, at a commercial discount rate of 2% p.m. What is the discount? What net value did the company receive, given that the bank charged a service fee of 0.5% of the note's value, paid on the day of the discount? What is the effective interest rate of the transaction for the period?

\[ D = 12{,}000 \cdot \frac{0.02}{30} \cdot 42 = 336 \]

\[ \text{Service fee: } 0.005 \cdot 12{,}000 = 60 \]

\[ \text{Net value received by the company: } 12{,}000 - 336 - 60 = 11{,}604 \]

\[ \text{Effective interest rate: } i = \frac{12{,}000}{11{,}604} - 1 = 0.0341 = 3.41\% \text{ for the period (42 days)} \]

Note that the effective rate (3.41% in 42 days) is higher than the proportional discount rate (2.8% in 42 days): the discount rate is applied to the face value, but the company receives less than that, and the service fee makes the transaction even more expensive.

When discounting a bill with 72 days to maturity, a bank wants to earn an interest rate of 6% for the period. What monthly discount rate should it charge?

Earning 6% for the period means the face value is 6% higher than the net value paid by the bank. Taking N = 100:

\[ \frac{100}{V_d} - 1 = 0.06 \]

\[ V_d = \frac{100}{1.06} = 94.34 \]

So the discount for the period is 100 - 94.34 = 5.66, that is, d · t = 5.66%. Since 72 days correspond to 72/30 = 2.4 months:

\[ d \cdot 2.4 = 0.0566 \]

\[ d = \frac{0.0566}{2.4} = 0.0236 = 2.36\% \text{ p.m.} \]

A cheque of R$ 5,000 is discounted 3 months before maturity, with a commercial discount of 3% p.m. How much does the company receive?

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Answer: R$ 4,550 — D = 5,000 · 0.03 · 3 = 450 and V = 5,000 - 450 = 4,550. R$ 4,850 discounts only one month; R$ 4,587.16 would be the rational discount (5,000 / 1.09), which is calculated on the present value, not on the face value.

Rational discount (true discount)

In rational discount, also called true discount or discount "por dentro" (from the inside), the rate is applied to the present value, not to the face value. This is exactly the reasoning of simple interest: the present value Vr, invested at rate i for t periods, produces the face value N.

\[ N = V_r \cdot (1 + i \cdot t) \]

\[ V_r = \frac{N}{1 + i \cdot t} \]

\[ D_r = N - V_r = \frac{N \cdot i \cdot t}{1 + i \cdot t} \]

Going back to the R$ 18,000.00 bill discounted two months early, now with a rational discount of 2.5% p.m.:

\[ V_r = \frac{18{,}000}{1 + 0.025 \cdot 2} = \frac{18{,}000}{1.05} = 17{,}142.86 \]

\[ D_r = 18{,}000 - 17{,}142.86 = 857.14 \]

At the same rate, the rational discount (R$ 857.14) is smaller than the commercial one (R$ 900.00), because it is applied to a smaller base. That is why banking practice uses the commercial discount, which is more advantageous for the bank.

A bill of R$ 22,000 is discounted 5 months before maturity at 2% p.m. What is the difference between the commercial discount and the rational discount?

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Answer: R$ 200 — commercial: 22,000 · 0.02 · 5 = 2,200; rational: present value = 22,000 / 1.10 = 20,000, so the discount = 2,000. The difference is 2,200 - 2,000 = 200: the commercial discount is applied to the face value (22,000), the rational discount to the present value (20,000).

Relation between the discount rate and the simple interest rate

Since rational discount is simply simple interest, some books use the word interest where others say discount "por dentro". Comparing the two notations:

  • D (discount) corresponds to J (interest)
  • N (face value) corresponds to M (amount)
  • Vd (net value) corresponds to C (capital)
  • d: commercial discount rate (por fora)
  • i: interest rate (por dentro)
  • t: number of periods

For the two rates to be equivalent, the amount the bank pays and the amount it receives must be the same in both calculations, that is, the interest on the net value must equal the commercial discount:

\[ J = D \]

\[ V_d \cdot i \cdot t = N \cdot d \cdot t \]

\[ i = \frac{N}{V_d} \cdot d \]

\[ i = \frac{N}{N - N \cdot d \cdot t} \cdot d \]

\[ i = \frac{d}{1 - d \cdot t} \]

The effective interest rate is always higher than the commercial discount rate, and the gap grows with the term.

Examples

If the commercial discount rate is 4% p.m. and a bill matures in three months, what is the monthly simple interest rate of the transaction?

d = 4% and t = 3:

\[ i = \frac{0.04}{1 - 0.04 \cdot 3} = \frac{0.04}{0.88} = 0.0455 = 4.55\% \text{ p.m.} \]

A bill maturing in two months was discounted at a bank, giving the bank an effective simple interest rate of 3% p.m. What discount rate was used?

i = 3% and t = 2:

\[ 0.03 = \frac{d}{1 - 2d} \]

\[ 0.03 \cdot (1 - 2d) = d \]

\[ 0.03 - 0.06d = d \]

\[ 0.03 = 1.06d \]

\[ d = \frac{0.03}{1.06} = 0.0283 = 2.83\% \text{ p.m.} \]

A bank discounts 5-month bills at a commercial discount rate of 2% p.m. What monthly simple interest rate does it actually earn?

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Answer: 2.22% p.m. — i = d / (1 - d · t) = 0.02 / (1 - 0.10) = 0.02 / 0.90 = 0.0222. The effective rate is never equal to the discount rate, because the bank charges 2% on the face value but only lends the net value, which is smaller. 1.82% would be the reversed calculation, d / (1 + d · t).

Rounding errors in real life

Be careful when rounding intermediate values! Keep all decimal places until the end of the calculation and round only the answer: in high-value transactions, or ones repeated many times, small roundings become large differences.

Here are some rounding and approximation errors that caused historic disasters (text in Portuguese):

Exercises

  1. Calculate the monthly interest rate (por dentro) used in a 60-day discount transaction on a bill whose redemption value is R$ 10,000.00 and whose principal is R$ 9,750.00.
  2. Calculate the simple discount on a bill of R$ 1,000.00 maturing in 60 days, given that the discount rate “por dentro” (rational) is 1.2% per month.
  3. A bill with a redemption value of R$ 1,000.00 and 80 days to maturity is traded at simple interest, with a discount rate “por fora” (commercial) of 15% per year. Calculate:
    1. the principal of this bill;
    2. the simple discount; and
    3. the monthly return of this bill until maturity.
  4. Imagine the bill from Problem 3 is sold with a repurchase guarantee in three days, and that this three-day transaction guarantees a return of 1.2% per month. Calculate:
    1. the value of the bill at the time of repurchase; and
    2. the monthly return and the annual discount rate (“por fora”) of this bill for its remaining 77 days to maturity.
  5. An investor invested a principal of R$ 1,000.00 to receive an amount of R$ 1,300.00 in 36 months. Calculate, under simple interest:
    1. the investor's quarterly return; and
    2. the monthly discount rate (“por fora”) corresponding to the return in item a.
  6. A commercial bank lends R$ 15,000.00 to a client for three months at a rate of 1% per month, simple interest, charged in advance. Thus the net amount released by the bank is R$ 14,550.00, and the client must pay R$ 15,000.00 at the end of the 3rd month. In addition, the bank requires an average balance of R$ 1,500.00 throughout the term of the loan. Calculate the bank's monthly rate of return on this transaction, at simple interest.
  7. A company wants to discount bills at a commercial bank that operates with a commercial discount rate of 1% per month, simple interest. The first bill is worth R$ 10,000.00 and matures in 90 days. The second is worth R$ 10,000.00 and matures in 180 days. Calculate the amount the bank will credit to the company's account for discounting these bills.
  8. A company obtains a loan of R$ 10,000.00 from a commercial bank at a rate of 1.2% per month (discount “por dentro”), simple interest, which can be settled at the end of any month. After three months, the company decides to settle this loan with funds from a new loan at the same bank, at a rate of 1% per month, also simple interest. A few months later, the company decides to settle the second loan and finds that the total interest accumulated on the two loans is R$ 981.60. Calculate:
    1. the value of the second loan, enough to settle the first;
    2. the final payment to settle the second loan;
    3. the term of the second loan; and
    4. the average monthly simple interest rate paid by the company, considering both loans together.
  9. An investor deposits a certain sum at a financial institution. At the end of four months, when closing the account, the investor finds that the accumulated amount totals R$ 10,480.00. This amount is then deposited at another financial institution for five months. At the end of that period, the accumulated amount at the second institution is R$ 11,108.80. Given that both institutions use simple interest at the same rate, calculate:
    1. the monthly simple interest rate of the two institutions; and
    2. the initial deposit at the first institution.
  10. A commercial bank discounts bills at a discount rate of 1.2% per month (“por fora”), but requires an average balance of 20% of the transaction value as a form of banking reciprocity. A company asks this bank to discount R$ 100,000.00 in bills, all maturing in 90 days. Considering a 30-day month, calculate the amount to be credited to the company's account and the bank's monthly return, at simple interest, without and with the average balance.

Solutions

  1. i = 1.282% per month
  2. D = R$ 23.44
  3. a) PV = R$ 966.67; b) Discount = R$ 33.33; c) i = 1.293% per month
  4. a) FV = R$ 967.83; b) i = 1.295% per month; d = 15.04% per year
  5. a) i = 2.5% per quarter; b) d = 0.641% per month (equivalent to 1.92% per quarter)
  6. i = 1.1494% per month
  7. PV = R$ 19,100.00
  8. a) PV2 = R$ 10,360.00; b) FV2 = R$ 10,981.60; c) n2 = 6 months; d) average i = 1.0907% per month
  9. a) i = 1.2% per month; b) PV1 = R$ 10,000.00
  10. Amount credited: R$ 96,400.00. Return without the average balance: 1.2448% per month; with the average balance: 1.5707% per month

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