The compound discount formula (bank discount)In compound commercial discount, the discount of each period is obtained by applying the discount rate d to the value at the beginning of that period, and not always to the face value. We will use the same notation as for simple discount:D: amount of the discountN: face value of the billVd: net value (present value) of the billd: discount ratet: number of discount periodsLet us discount the bill period by period, starting from maturity:a) In the 1st discount period (t = 1):value at the beginning of the period = Ndiscount for the period = N · dvalue at the end of the period = N - N · d = N · (1 - d)b) In the 2nd discount period (t = 2):value at the beginning of the period = N · (1 - d)discount for the period = N · (1 - d) · dvalue at the end of the period = N · (1 - d) - N · (1 - d) · d = N · (1 - d) · (1 - d)and therefore:\[ V_d = N \cdot (1 - d)^{2} \]c) Similarly, the net value after the 3rd discount period is:\[ V_d = N \cdot (1 - d)^{3} \]d) And, after t discount periods:\[ V_d = N \cdot (1 - d)^{t} \]\[ D = N - V_d \]ExamplesA bill of $ 10,000.00 with 60 days to maturity is discounted under compound interest, with a discount rate “por fora” (bank discount) of 1.2% per month. Calculate the present value of the bill and the compound discount, in $.Since the rate is monthly, 60 days correspond to t = 2 months:\[ t = 2;\ d = 1.2\% \text{ p.m.};\ N = 10{,}000;\ V_d = ?;\ D = ? \]\[ V_d = N \cdot (1 - d)^{t} = 10{,}000 \cdot (1 - 0.012)^{2} = 10{,}000 \cdot 0.976144 = 9{,}761.44 \]\[ D = N - V_d = 10{,}000 - 9{,}761.44 = 238.56 \]With simple commercial discount at the same rate, the discount would be 10,000 · 0.012 · 2 = 240.00: with compound discount it is slightly smaller, because the second month's rate is applied to a value that has already been discounted.A bill of R$ 20,000 (Brazilian reais) is discounted 3 months before maturity with a compound bank discount of 2% p.m. What is the net value?Answer: R$ 18,823.84 — V = 20,000 · (1 - 0.02)³ = 20,000 · 0.941192 = 18,823.84. R$ 18,800 is the simple commercial discount (20,000 · (1 - 0.06)); R$ 18,846.45 is the compound rational discount, 20,000 / 1.02³.Present value and face value at compound interest (rational discount)The face value (N) is the value of the obligation on its due date. The present value is the value that, invested at compound interest from today until maturity, produces an amount equal to the face value. This calculation is the compound rational discount, or discount "por dentro" (true discount). Using the same letter Vd for the present value:\[ V_d \cdot (1 + i)^{t} = N \]\[ V_d = \frac{N}{(1 + i)^{t}} \]For the discount with discount rate d and the discount with interest rate i to give the same present value:\[ V_d = \frac{N}{(1 + i)^{t}} = N \cdot (1 - d)^{t} \]\[ \frac{1}{(1 + i)^{t}} = (1 - d)^{t} \]\[ \frac{1}{1 + i} = 1 - d \]\[ d = 1 - \frac{1}{1 + i} = \frac{i}{1 + i} \]or, solving for the interest rate:\[ i = \frac{d}{1 - d} \]Unlike simple discount, this relation does not depend on the term t.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.?\[ V_d = \frac{N}{(1 + i)^{t}} = \frac{10{,}000}{(1.015)^{3}} = 9{,}563.17 \]A commercial bank makes loans at an interest rate of 6.00% per semester, but the interest is paid in advance, when the funds are released. Thus, for each $ 1,000.00 loan, to be repaid in six months, the bank releases $ 940.00. Calculate the effective monthly rate charged on these loans, under compound interest.Interest charged in advance works like a discount rate of 6% per semester:\[ d = 6\% \text{ per semester} \]\[ 0.06 = 1 - \frac{1}{1 + i} \]\[ \frac{1}{1 + i} = 0.94 \]\[ 0.94 + 0.94 \cdot i = 1 \]\[ 0.94 \cdot i = 0.06 \]\[ i_{\text{semester}} = \frac{0.06}{0.94} = 0.06383 = 6.383\% \text{ per semester} \](Check: 940 · 1.06383 = 1,000.) Converting to the equivalent monthly rate:\[ i_{\text{monthly}} = (1 + 0.06383)^{\frac{1}{6}} - 1 = 0.01037 = 1.04\% \text{ p.m.} \]A bank charges interest in advance: on a one-month loan of R$ 1,000, it releases R$ 980. What is the effective monthly interest rate?Answer: 2.04% p.m. — the discount rate is d = 20 / 1,000 = 2%, and the interest rate is i = d / (1 - d) = 0.02 / 0.98 = 0.0204. Check: 980 · 1.0204 = 1,000. The 2% is the discount rate, calculated on the R$ 1,000, but the client only received R$ 980.Under compound discount, what bank discount rate per period is equivalent to an interest rate of 25% per period?Answer: 20% — d = i / (1 + i) = 0.25 / 1.25 = 0.20. Check: discounting 20% from 100 gives 80, and 80 · 1.25 = 100. The 33.33% would be the reverse calculation, d / (1 - d) with d = 25%.Exercises1) The ratio between the future value and the present value of a financial transaction at compound interest is 2. If the term of this transaction was 730 days, calculate the interest rate per 30 days.2) $ 120,000 was invested in shares. After 276 days, the shares were sold for $ 140,981.02. Calculate the interest rate per 365 days.3) A debt of R$ 80,000.00 is due in 5 months. Considering an interest rate of 1.3% p.m., find its present value on the following dates:Today.Two months from now.Two months before maturity.4) How much should I invest today, at compound interest of 1.5% p.m., to meet a commitment of R$ 27,000.00 two months from now?5) How much should I invest today, at compound interest of 3% p.m., to meet a commitment of R$ 27,000.00 two months from now?6) A debt of R$ 50,000.00 is due in two months, and another, of R$ 60,000.00, in 4 months. How much should I invest today, at compound interest 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 in 3 months and another of R$ 80,000.00 in 4 months. How much should I invest today, at compound interest of 0.8% p.m., to meet these commitments?8) A piece of equipment is sold for R$ 50,000.00 to be paid two months from now. For cash payment, there is a 3.5% discount. Which is the best payment option for a buyer who can invest money at 1.8% p.m.?9) A piece of equipment is sold for R$ 50,000.00 to be paid two months from now. For cash payment, there is a 3.5% discount. Which is the best payment option for a buyer who can invest money at 1.4% p.m.?10) What is better for a buyer: paying an amount 45 days from now or paying in cash with a 3% discount? Assume the buyer can invest money at 1.3% p.m.11) What investment rate makes a buyer indifferent between the alternatives: paying an amount 45 days from now or paying in cash with a 3% discount?12) Rogério holds a debt security maturing in 6 months with a face value of R$ 85,000.00. Ernesto offers to swap it for a security maturing in three months, worth R$ 82,000.00. With a market compound interest rate of 1% p.m., check whether the swap is advantageous for Rogério.Answer key1) 2.8895% per 30 days2) 23.75% per 365 days3) a) R$ 74,996.80; b) R$ 76,959.40; c) R$ 77,959.874) R$ 26,207.875) R$ 25,450.096) R$ 107,327.297) R$ 204,888.328) Pay later (investing R$ 48,247.46 today at 1.8% p.m., the buyer will have the R$ 50,000 in two months, slightly less than the R$ 48,250.00 cash price)9) Pay in cash (to have R$ 50,000 in two months at 1.4% p.m., the buyer would need to invest R$ 48,628.86, more than the R$ 48,250.00 cash price)10) Pay in cash11) 2.05% p.m.12) It is not advantageous. Today, Rogério's security is worth 85,000 / 1.01⁶ = R$ 80,073.84, and Ernesto's is worth 82,000 / 1.01³ = R$ 79,588.39; Rogério would be swapping a security for one of lower value.
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