Nominal value and real (or present) valueThe value of a debt on its maturity date is usually called the nominal value; the value applied on a date prior to the maturity date, which results in an amount equal to the nominal value, is called the current value or real value or present value.It is noted that, by applying the formulas already learned, the real value would be the initial value or principal.The distinction between nominal and real value exists to avoid confusion about the value that will be presented. Often it is more interesting to use the present value because it can serve as a comparison with current values that are already known.Simple discountThe idea of discount is associated with a reduction given to a monetary amount under certain conditions. For example, when a purchase is made in large quantity, it is common for the seller to grant some discount on the unit price.Another situation involving the concept of discount occurs when a company sells a product on credit; in this case, the seller usually issues a bill of exchange that gives them the right to receive the agreed amount from the buyer on a future date. If the seller needs money, they can go to a bank and make the so-called discount of the bill of exchange. In short, the following happens: the company transfers to the bank the right to receive the bill of exchange in exchange for money received in advance.Analogous to the discounting of bills of exchange, a company can discount promissory notes or post-dated checks at a bank. Since this type of discount is quite common, it has a well-defined calculation system, called commercial or bank discount, which we will study.Commercial or bank discountFor discount calculation, a new rate similar to the interest rate is used, called the discount rate or discount "outward".- Total discount value: D- Total value of the title: N- Net value of the title: Vd- discount rate: d- Number of periods: t\[ D = N · d · t \]\[ V_d = N - D \]\[ V_d = N · (1 - d · t) \]Therefore:\[ V_d = N - N · d · t \]\[ V_d = N · (1 - d · t) \]\[ \frac{V_d}{N} = 1 - d · t \]\[ d · t = 1 - \frac{V_d}{N} \]ExamplesA bill of exchange for R$ 18,000.00 was discounted at a bank two months before maturity, at a commercial discount rate of 2.5% per month. Find the discount and the net amount paid to the company.\[ D = N · d · t = 18.000 · 0,025 · 2 = 900 \]\[ V_d = 18.000 - 900 = 17.100 \]In this case, the bank will pay 17,100 reais and will receive 18,000 two months laterA promissory note for R$ 12,000.00 was discounted at a bank 42 days before maturity, at a commercial discount rate of 2% per month. What is the discount? What is the net amount received by the company, knowing that the bank charged a service fee of 0.5% of the value of the promissory note, paid on the day the company discounted it? What is the effective interest rate of the operation over the period?\[ D = 12.000 · (\frac{0,02}{30}) · 42 = 336 \]\[ \text{Service fee:} 0,005 · (12.000) = 60 \]\[ \text{Net amount received by the company: } 12.000 - 336 - 60 = 11.604 \]\[ \text{Effective interest rate: } i = \frac{12.000}{11.604} - 1 = 3,41\% a.p. \]When discounting a bill of exchange with 72 days until maturity, a bank intends to earn an interest rate of 6% over the period. What monthly discount rate should it charge?\[ \frac{100}{V_d} - 1 = 0,06 \]\[ V_d = \frac{100}{1,06} = 94,34 \]Thus:\[ d = 100 - 94,34 = 5,66 \% a.p. \]\[ 5,66\% = 100 · d · \frac{72}{30} \]\[ d = 5,66\% · \frac{30}{72 · 100} = 0,0236 = 2,36\% a.m. \]Relationship between discount rate and simple interest rateIn this case, there are books that make a distinction of interest for discount. Instead of interest, they use the term discount "inward".- Total discount value: D or J (Interest)- Total value of the title: N or M (amount)- Net value of the title: Vd or C (capital)- discount rate: d- interest rate: i (discount "inward")- Number of periods: t\[ D = N · d · t \]\[ J = N · d · t \]\[ i · t · V_d = N · d · t \]\[ i = \frac{N}{V_d} · d \]\[ i = \frac{N}{N-D} · d \]\[ i = \frac{N}{N-N · d · t} · d \]\[ i = \frac{1}{1-d · t} · d \]\[ i = \frac{d}{1-d · t} \]ExamplesIf the commercial discount rate is 4% per month and the maturity term of a bill of exchange is three months, what is the monthly simple interest rate of the operation?- d = 4%- t = 3\[ i = \frac{0,04}{1-0,04 · 3} = 0,0455 = 4,55\% a.m. \]A bill of exchange with a maturity term of two months is discounted at a bank, providing it with an effective simple interest rate equal to 3% per month. What discount rate was used?- i = 3%- t = 2\[ 0,03 = \frac{d}{1-d · 2} \]\[ 0,03 · (1 - 2 · d) = d \]\[ 0,03 - 0,06 · d = d \]\[ 0,03 = 1,06 · d \]\[ d = \frac{0,03}{1,06} = 0,0283 = 2,83\% \]Approximation errors in real lifeBe careful when approximating!Links to errors that caused historical disaster:- http://livrozilla.com/doc/358992/pequenos-erros-podem-ter-grandes-consequ%C3%AAnciasExercisesCalculate the monthly interest rate “inside” used in a 60-day discount operation on a bill whose redemption value is R$ 10,000.00 and whose principal value is R$ 9,750.00.Calculate the value of the simple discount on a bill for R$ 1,000.00, due in 60 days, knowing that the “inside” discount rate is 1.2% per month. A bill with a redemption value of R$ 1,000.00, with 80 days remaining until maturity, is being negotiated at simple interest, with an “outside” discount rate of 15% per year. Calculate: the principal value of this bill; the value of the simple discount; and the monthly return on this bill, until maturity. Imagine that the bill in Problem 3 is sold with a repurchase guarantee within three days, and that in this three-day operation a return of 1.2% per month is guaranteed. Calculate: the value of the bill at the time of repurchase; and the monthly return and the annual discount rate (“outside”) of this bill for its remaining term of 77 days until maturity. An investor applied a principal of R$ 1,000.00 to receive a total of R$ 1,300.00 after 36 months. Calculate, under simple interest: the investor’s quarterly return; and the monthly discount rate (“outside”) that corresponds to the return in item a. A commercial bank lends R$ 15,000.00 to a client for a period of 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 the 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 loan term. Calculate the bank’s monthly return rate on this operation, using simple interest. 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 has a value of R$ 10,000.00 and matures in 90 days. The second bill has a value of R$ 10,000.00 and matures in 180 days. Calculate the amount to be credited by the bank to the company’s account for discounting these bills. A company obtains a loan of R$ 10,000.00 from a commercial bank, at a rate of 1.2% per month (discount “inside”), simple interest, which can be settled at the end of each month. After three months, the company decides to settle this loan with funds obtained, at the same bank, through a new loan, at a rate of 1% per month, also at simple interest. After some months, the company decides to settle the second loan and finds that the total interest accumulated in the two loans is R$ 981.60. Calculate: the value of the second loan sufficient to settle the first; the amount of the final payment to settle the second loan; the term of the second loan; and the average monthly rate, at simple interest, paid by the company, considering the two loans together. An investor deposits a certain amount in a financial institution. At the end of four months, when closing the account, he finds that the accumulated amount at that date totals R$ 10,480.00. This same amount is then deposited in another financial institution for a period of five months. At the end of that period, the accumulated amount in the second institution is equal to R$ 11,108.80. Knowing that both institutions operate with simple interest and remunerate their deposits at the same rate, calculate: the monthly simple interest rate of both institutions; and the value of the initial deposit in the first institution.A commercial bank carries out its bill discount operations with a discount rate of 1.2% per month (“outside”), but requires an average balance of 20% of the transaction value, as a form of banking reciprocity. A company approached 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, using simple interest, without the average balance and with the average balance.Solution 1.282% R$ 23.44 PV = R$ 966.67; b) Discount = R$ 33.33; c) i = 1.293% per month a) FV = R$ 967.83; b) i = 1.295% per month; d = 15.04% per year a) i = 2.5% per quarter; b) d = 1.92% per quarter i = 1.1494% per month PV = R$ 19,100.00 a) PV 2 = R$ 10,360.00; b) FV 2 = R$ 10,981.60; c) n 2 = 6 months; d) average i = 1.0907% per month a) i = 1.2% per month; b) PV 1 = R$ 10,000.00 1.5707%
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