Compound Discount FormulaUnder compound interest, the discounts for each period are obtained by applying the discount rate d per period to the capital existing at the beginning of the discount period. Following the same notation:- Total discount amount: D- Total face value of the note: N- Net value of the note: Vd- discount rate: d- Number of periods: tThus, we have:a) in the 1st discount period (n = 1):capital at the beginning of the period = Ndiscount for the period = N x dcapital at the end of the period = Vd = N – N x d = N (1 – d)b) in the 2nd discount period (n = 2):capital at the beginning of the period = N (1 – d)interest for the period = N (1 – d) x dcapital at the end of the period = Vd = N (1 – d) – N (1 – d) x d = N (1 – d) x (1 – d)and therefore,\[ Vd = N (1 – d)^{2} \]c) The expression for the present value N, at the end of the 3rd discount period, can be deduced in an analogous way:\[ Vd = N (1 – d)^{3} \]d) In the n-th discount period:\[ Vd = N (1 – d)^{t} \]ExamplesA note with a value of $ 10,000.00, with 60 days to maturity, is discounted under compound interest, with an “outside” discount rate equal to 1.2% per month. Calculate the present value of the note and the value of the compound discount, expressed in $.\[ t = 60\text{ days}; d = 1.2\% p.m. ; N = 10.000; Vd = ?; D = ?; \]\[ Vd = N (1 – d)^{t} = 10.000 · (1-0,012)^{2} = 10.000 · 0,97614 = 9.761,44 \]\[ D = N - Vd = 10.000 - 9.761,44 = 238,56 \]Present and nominal value in compound interestNominal value (N) is the value of the obligation on its maturity date, while present value (Vd) would be its present value, where, applied at compound interest from this date to its maturity, it produces an amount equal to the Nominal value.Thus:\[ Vd (1 + i)^{t} = N \]\[ Vd = \frac{ N}{(1 + i)^{t}}\]If we make a comparison with the discount rate:\[ Vd = \frac{ N}{(1 + i)^{t}} = N (1 – d)^{t} \]\[ \frac{1}{(1 + i)^{t}} = (1 – d)^{t} \]\[ \frac{1}{(1 + i)} = 1 – d \]\[ d = 1 - \frac{1}{(1 + i)} \]ExamplesA person has a debt of R$ 10,000.00 due in three months. What is its present value today, considering an interest rate of 1.5% p.m.?\[ Vd = \frac{ N}{(1 + i)^{t}} = \frac{10.000}{(1,015)^{3}} = 9.563,17 \]A commercial bank carries out its credit operations with an interest rate of 6.00% per semester. However, the interest is paid in advance, when the funds are released. Thus, for each $ 1,000.00 loan, to be settled within six months, this bank releases a principal of $ 940.00. Calculate the effective monthly rate charged in these operations, under compound interest.\[ d = 6\% p.s. \]\[ 0,06 = 1 - \frac{1}{(1 + i)} \]\[ - 0,94 = - \frac{1}{(1 + i)} \]\[ 0,94 + 0,94 · i = 1 \]\[ 0,94 · i = 0,06 \]\[ i_{\text{semestral}} = \frac{0,06}{0,94} = 6,38\% \]\[ i_{\text{mensal}} = (1 + 0,0638)^{\frac{1}{6}} = 1,036\% \]Exercises1) The relationship between the present and future value of the financial operation with two capitals under compound interest is equal to 2. If the term of this operation was 730 days, calculate the interest rate with a period of 30 days.2) $120,000 were invested to buy shares. After 276 days the shares were sold for the amount of $140,981.02. Calculate the interest rate with a period of 365 days.3) A debt of R$ 80,000.00 is due in 5 months. Considering an interest rate of 1.3% p.m., obtain its present value on the following dates:a) Today.b) In 2 months.c) Two months before maturity.4) How much should I invest today, at compound interest at a rate of 1.5% p.m., to meet a commitment of R$ 27,000.00 in two months?5) How much should I invest today, at compound interest at a rate of 3% p.m., to meet a commitment of R$ 27,000.00 in two months?6) A debt of R$ 50,000.00 is due in two months and another of R$ 60,000.00 is due in 4 months. How much should I invest today, at compound interest at a rate of 0.8% p.m. to meet these commitments?7) A debt of R$ 60,000.00 is due in two months, another of R$ 70,000.00 is due in 3 months and R$ 80,000 due in 4 months. How much should I invest today, at compound interest at a rate of 0.8% p.m. to meet these commitments?8) An equipment is sold for R$ 50,000.00 for payment in two months. Cash payment has a 3.5% discount. Which is the better payment option for a buyer who can invest their money at a rate of 1.8% p.m.?9) An equipment is sold for R$ 50,000.00 for payment in two months. Cash payment has a 3.5% discount. Which is the better payment option for a buyer who can invest their money at a rate of 1.4% p.m.?10) Which is better for a buyer: paying an amount in 45 days or paying cash with a 3% discount? Suppose the buyer can invest their money at a rate of 1.3% p.m.11) What investment rate makes a buyer indifferent between the alternatives if they are: paying an amount in 45 days or paying cash with a 3% discount.12) Rogério has a debt note due in 6 months with a face value of R$ 85,000.00. Ernesto offers to exchange it for a note due in three months, with a value of R$ 82,000.00. Given a 1% per month compound interest market rate, verify whether the exchange is advantageous for Rogério.Answer Key1) 2.8895% at 30 days2) 23.75%3) a) R$ 74,996.80 ; b) R$ 76,959.40; c) R$ 77,959.87;4) R$ 26,207.875) R$ 25,450.096) R$ 107,327.297) R$ 204,888.328) On credit9) Cash10) Cash11) 2.05% p.m.12) It is advantageous.
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