Simon invests $\$6000$ and it's compounded semi-annually for ten years

Question

Simon invests $\$6000$ and it's compounded semi-annually for ten years, at $8\%$ per annum. What is the amount of the investment at maturity?

I did $(6000)(1.08)^{20}$, and got a completely different answer from the book - $\$13,146$. How did they get this? I don't understand.

Thank you.

Answer 1

$=p\left(1+\frac{r}{n}\right)^{nt}$

You need to divide the rate by the number of times it's compounded per year. And raise to the power of n*t or the total number of times it's compounded over the full time.

$=6000\left(1+\frac{.08}{2}\right)^{(2*10)}$

Answer 2

I am answering what you actually asked, instead of what you wanted to ask.

The investment you are describing is an annuity. Simon makes payments/investments of $6000$ each year, and earns interest compounded semiannually. We'll start with what that means.

Semiannual compounding means that the investment account earns interest twice per year, at the "nominal rate" $r\backslash 2$.

The simplest way to calculate the accumulated value is to find how much a single dollar of principle accumulates in one year, and find the effective yearly interest based on that. Then, we can use the standard annuity formula $s_n$ to find the total accumulated value.

Since a dollar deposited at time $0$ earns $4\%$ in the first six months, the account earns $1.04\times 1.04 = 1.0816$ over the course of the year (notice we applied compound interest). This means the annual effective interest rate is $r = 8.16\%$.

Now, we can use the formula

\begin{equation*} s_n = \frac{(1 + r)^n}{r}, \end{equation*}

where $n$ is the number of years that investments are made. In this case, it is 10. You can derive this formula using the geometric sum. If you make yearly investments of $1$ earning $8.16\%$ per year, the total value in the account at the end of $10$ years is approximately $13.45$. The total amount accumulated if you invest $6000$ per year under the same conditions is about $80699.81$.