# Simon invests $\$6000$and it's compounded semi-annually for ten years 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. • I'm assuming the interest p.a. is 8%? – Malcolm Jun 17 '15 at 21:24 • Oops, yes. I forgot to include that in. – user164403 Jun 17 '15 at 21:25 • If the rate of interest is 8% then you have calculated 20 years at that rate. If you want semiannual compounding at 8% p.a., calculate $$6000\left(1 + \frac{0.08}{2}\right)^{20}$$ – Simon S Jun 17 '15 at 21:26 • Well firstly, you need to realize that the interest rate of 8% is per annum Remember that the time units here are half years, so all you need to do is do the exact same calculation, but with 1.04 instead of 1.08. – Malcolm Jun 17 '15 at 21:29 • @user164403 There has been a drastic edit to your question to remove "Per Annum" from the investment schedule. (Which might be correct). So are you investing 6000 once and letting it compound, or investing 6000 per year, all the while everything compounds? – muaddib Jun 17 '15 at 22:47 ## 2 Answers$=p\left(1+\frac{r}{n}\right)^{nt}$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.$=6000\left(1+\frac{.08}{2}\right)^{(2*10)}$• That would be the result of the first investment. What about the other$54,000 that Simon invests? There are 9 other investments he makes as I understand the question. – JB King Jun 17 '15 at 21:44
• No. He has a typo in his question. Since the answer is \$13,146, there are no additional deposits. Back of envelope math, rule of 72, money doubles after 9 years at 8%, plus a bit more. \$13K is it. – JTP - Apologise to Monica Jun 17 '15 at 22:04

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 $80699.81$.

• Why the down vote? OP's question is for an investment of $6000$ per annum, earning interest semi-annually. – nomen Jun 17 '15 at 22:11
• I haven´t downvote. But the invvestment is made only once. You should make an edit. – callculus Jun 18 '15 at 4:30