Julie, Ivana and Janka want to invest €$1\, 000$ in the bank. Each of them will deposit the money in the bank for $5$ years.
- Julie will invest money at a monthly interest rate of $1\%$,
- Ivana will invest money at a semi-annual interest rate of $6\%$, and
- Janka will invest money at an annual interest rate of $12.5\%$.
Juraj told them that Janka will have the most money after $5$ years because she has the highest interest rate. Is Juraj right? Do not consider inflation and fees.
Yes, Juraj is right because the euros accumulated after $5$ years are:
Julie: $$1\,000\left(1+\frac{1}{100}\right)^5\doteq1\,051.01$$
Ivana: $$1\,000\left(1+\frac{6}{100}\right)^5\doteq1\,338.23$$
Janka: $$1\,000\left(1+\frac{12.5}{100}\right)^5\doteq1\,802.03$$
Janka will have the most money in $5$ years.
No, Juraj is wrong, because the euros accumulated after $5$ years are:
Julie: $$1000\left(1+\frac{1}{100}\right)^{60}\doteq1\,816.70$$
Ivana: $$1\,000\left(1+\frac{6}{100}\right)^{10}\doteq1\,790.85$$
Janka: $$1\,000\left(1+\frac{12.5}{100}\right)^5\doteq1\,802.03$$
Julie will have the most money in $5$ years.
No, Juraj is wrong, because the euros accumulated after $5$ years are:
Julie: $$1\,000\left(1+\frac{1}{100}\right)^{12}\doteq1\,126.83$$
Ivana: $$1\,000\left(1+\frac{6}{100}\right)^2\doteq1\, 123.60$$
Janka: $$1\,000\left(1+\frac{12.5}{100}\right)^1\doteq1\,125.00$$ Julie will have the most money in $5$ years.
To solve the problem, we use the following formula: $$a_n=a_0\left(1+\frac{p}{100}\right)^n$$ where $a_n$ is the future value, $a_0$ is the initial investment, $p$ is the interest rate (in percent), and $n$ is the number of compounding periods.
Julie deposits €$1\,000$ in the bank for $5$ years with a monthly interest rate of $1\%$. Since the interest is compounded every month, over $5$ years the interest will be compounded a total of $60$ times ($5$ years $\times$ $12$ months $= 60$ periods): $$1\,000\left(1+\frac{1}{100}\right)^{60}\doteq 1\,816.70$$
Ivana deposits €$1\,000$ in the bank for $5$ years with a semi-annual interest rate of $6\%$. Since the interest is compounded every half-year, over $5$ years the interest will be compounded a total of $10$ times ($5$ years $\times$ $2$ half-years $= 10$ periods): $$1\,000\left(1+\frac{6}{100}\right)^{10}\doteq1\,790.85$$
Janka deposits €$1\,000$ in the bank for $5$ years with an annual interest rate of $12.5\%$. Since the interest is compounded yearly, over $5$ years the interest will be compounded exactly $5$ times ($5$ years $\times$ $1$ compounding period per year $= 5$ periods): $$1\,000\left(1+\frac{12.5}{100}\right)^5\doteq1\,802.03$$