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Compound interest and Euler's number

The interest rate is one of the main variables used to calculate the price of a futures contract. The futures price tends to be higher than the current price (known as "spot”) because of the interest rate applied to the period until maturity, reflecting the opportunity cost of keeping the capital invested. When buying a futures contract, the investor does not need to pay out the capital immediately, as would happen when buying a stock index or the shares that make up the index today. In this way, he can invest this capital at a given interest rate. The longer the term until maturity of a futures contract for a given financial asset, the greater the effect of the interest rate, since it is applied over a longer period. When buying futures contracts for stock indexes or individual shares, the dividends to be paid are also a variable to consider.
According to the current value of the S&P 500 of 6,000 points (November 11, 2024), considering an interest rate of 4.60% (SOFR, Secured Overnight Financing Rate, the overnight rate guaranteed by US government bonds), a discount of 6 points in dividends and the 40 days remaining until maturity, the theoretical price of the closest underlying futures contract, for December 20, 2024, is 6,024.25 (more precisely 6,024.29, but the tick size, that is, the minimum variation of the contract is 0.25 points). The price of the futures depends on the current value weighted by the interest rate and the dividends to be paid in the period in question, using the exponential function in the calculation, as it captures the continuous variations that occur in highly liquid markets and quoted continuously. A discrete variable is not advisable because it does not capture continuous variations.
The calculation formula is , where "F” is the futures contract price, "S” the spot price, "e” the exponential function, "r” the interest rate, "E” the dividends and "T” the time period. Thus, , resulting in 6024.29. However, in the case of a discrete variable, the interest would be paid only once, at the maturity of the contract, that is, on December 20, with the formula being . Replacing the variables with the values, , the futures contract price would be 6024.216. The difference is minimal, but it would become substantial for interest rates above two digits and over much longer terms.
One of the Bernoulli brothers, Jacob, when trying to understand how money grows when interest is compounded continuously (that is, it is capitalized indefinitely), discovered the Euler number. Jacob Bernoulli realized that by applying interest infinitely in one year and at an interest rate of 100%, the interest grew at a decreasing rate, but the total value did not grow indefinitely, having stabilized at a constant — Euler's number, 2.7182818284…. This number was later formalized by Leonhard Euler, becoming one of the fundamental constants of mathematics, like p (pi), 3.1416…. Johann Bernoulli, the youngest and longest-lived Bernoulli brother, would later have Leonhard Euler as his disciple, who formalized Euler's identity, , famous for being made up of five fundamental constants of mathematics: e , i, p , 1 and 0, that is, Euler's number (2.718…), the imaginary number i (, p (3.14…, the ratio between the perimeter of the circumference and its diameter), 1 and 0.
Therefore, on a one-euro loan for one year with an interest rate of 100%, if the interest is paid only once at maturity, we will receive 2 euros at the end (1 euro of our capital, plus 1 euro of interest). If the rate is compounded twice (semi-annual payment), then we would repay 50 cents after six months, compounding one and a half euros for another half year, generating an interest of 75 cents, pocketing in the end a total of 2.25 euros. The initial capital is always one euro, but now the interest was 1.25 euros, because the 50 cents received at the end of the semester were capitalized, yielding 25 cents. And so on…
PAULO MONTEIRO ROSA, Senior Economist at Banco Carregosa