Skip to 0 minutes and 12 secondsHello, and welcome back to "A step in practice". Today, we are dealing with money. Well, after all, the study of interest rates is one of the basic motivations to exponentials. We have 10,000 euros, so our initial capital, C of 10 000 euros, and we invested with an interest rate of 3% compounded every 6 months. So, after 6 months, we get the initial capital multiplied by 1 plus 1.5 divided by 100. So this is after 1 semester, that is 6 months. So, after 1 year, we've got 2 semesters, so our initial capital is multiplied by 1 plus 1.5 divided by 100 to the square. This is after 1 year.
Skip to 1 minute and 7 secondsAnd, after 15 years, there are exactly 30 semesters, so our capital is multiplied by 1 plus 1.5 divided by 100 to the 30. And this gives us, with C equal to 10 000 euros, we get, more or less, 15 000 euros, 630.8 euros. So this is exercise 1.
Skip to 1 minute and 44 secondsHere, we are asked to compare two different interest rates.
Skip to 1 minute and 53 secondsThe first interest rate is 5.05% per year, compounded every 6 months. So, after one year, we get that initial capital multiplied by 1 plus the 1/2 of 5.05 is 5.05 divided by 2 divided by 100 to the square, because there are 2 semesters in 1 year. And this gives, more or less, C multiplied by 1.05114. In the second case, the interest is compounded continuously. The interest of 5% per year is compounded continuously. This means, as we saw in Francis' talk, that the capital is multiplied by e to the 5%.
Skip to 3 minutes and 0 secondsAnd this is, more or less, C multiplied by 1.05127. So, if we compare the two numbers, while the first three digits, four digits are the same, and then we get something a little bit something greater in the second case. So this wins. It is better to have an interest of 5% compounded continuously. And this ends exercise 2.
It's your turn on the Euler number
Do your best in trying to solve the following problems. In any case some of them are solved in the video and all of them are solved in the pdf file below.
Exercise 1.
If 10 000 euros is invested at an interest rate of 3% per year, compounded semiannually (1.5% per semester). Find the value of the investment after 15 years.
Exercise 2.
Which of the given interests rates and compounding periods would provide the better investment after 1 year?

\(5.05\%\) per year, compounded semiannually.

\(5\%\) per year, compounded continuously (so that the capital is multiplied by \(e^{5/100}\) at the end of each year).
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