## Before investing: Interest and saving (I)

Before investing and trading - savings and calculations

Foreign currency and interest rate!

Australian Dollar (AUD) has the highest interest rate among commonly traded currencies. For example, April 2011, the central bank rate was 4.75%, while US dollar (USD) almost has no interest rate compared with AUD.

So which currency do you like to hold? This is not an easy question. Foreign currency exchange rates can fluctuate wildly, although AUD has high interest rate, but if you  live in a country other than Australia, say, USA, then there is always a risk to hold AUD, while you use USD for your daily life. Currently, 1 AUD is still worth more than 1 USD after AUD set numerous record highs against USD since the end of 2010. Thus, there is always a risk for AUD stop making record highs against USD and may even suffer a big dive. If this happens and you need USD to make purchases urgently, you will be forced to sell AUD and buy USD and thus suffer a huge loss.

Foreign currency market is always full of risks and unexpected events so that 80% or more traders actualluy lose money or making almost no profits.

APR and APY - calculate your interest

Now suppose that you live in Australia, so go ahead, put money in your bank! Assuming the rate is 4.75%. This rate is actually APR - Annual Percentage Rate (simple-interest rate before compounding). This rate will normally be compounded every month (we can assume that each month have the same number of days). To do calculations, we need to converte the percentage to 0.0475.

Compounded monthly, that is, each month, your interest rate will be $$\frac{0.0475}{12}$$, which is roughly 0.00395833. You do not need a calculator, you can just open Google.com and type in 0.0475/12, the Google will give you the answer.

Suppose initially you have 1000 AUD at the beginning of 2011, then after the first month you will have interest $$1000\times\frac{0.0475}{12} = 3.958$$. To calculate by using Google, you just type: 1000*(0.0475/12)

So you earned 3.96AUD. At the end of the first month, you will have $$1000+1000\times\frac{0.0475}{12}$$. After you factor the common factor 1000, it is $$1000(1+\frac{0.0475}{12})$$, which is roughly 1003.96. Of course, you already know this is just 1000 +3.96, but we are trying to derive a formula and want to see some pattern to show up.

The second month, the interested earned will also earn interest, so your total will be:

$$1000(1+\frac{0.0475}{12}) +1000(1+\frac{0.0475}{12})\times \frac{0.0475}{12}=1000(1+\frac{0.0475}{12})(1+\frac{0.0475}{12})=1000\left(1+\frac{0.0475}{12}\right)^2.$$

Where in the above, we factored out the factor $$1000(1+\frac{0.0475}{12})$$ in the first step.

Similarly, by induction, we can get by the end of the year, the total amount will be

$$1000\left(1+\frac{0.0475}{12}\right)^{12}.$$

This is about 1048.55 AUD. To use Google, you can simply type: 1000*(1+0.0475/12)^(12).

But suppose 1 AUD = 0.95 USD by the end of the year, this will be only 996.12 USD, so by holding AUD you would actually suffer a loss. We will generalize this result after the following AD.

In general, the total amount after n months will be $$1000\left(1+\frac{0.0475}{12}\right)^{n}.$$

If $$n$$ is the number of years, then the formula should be $$1000\left(1+\frac{0.0475}{12}\right)^{12n}.$$

More generally, if the initial amount is $$p$$, APR is $$r$$ compounded monthly, then after $$n$$ years the total will be

$$p\left(1+\frac{r}{12}\right)^{12n}.$$

Question: How many years are needed to initial investment to double?

Obviously we need to solve the equation $$p\left(1+\frac{r}{12}\right)^{12n}=2p.$$

$$p$$ cancels out, so$$\left(1+\frac{r}{12}\right)^{12n}=2.$$

Now take natural log on both sides to get

$$12n\ln \left(1+\frac{r}{12}\right)=\ln 2.$$

Solve for $$n$$, we get

$$n=\frac{\ln 2}{12\ln\left(1+\frac{r}{12}\right)}$$

In the case of our AUD example, plug in $$r$$=0.0475, we can get the result(use calculator) to be about 14.62 years. So even with this kind of high interest, it takes you about 15 years to double your principal.

Now let's look at the price of silver. The price as of April 21, 2011 is about 46.6 USD. Back a year before, it is about 18 USD even, on April 21, 2010. $$46.6\div 18 = 2.5888\cdots$$. That is, the price of silver is about 2.6 times that of a year before, much more than doubled. See the following picture taken from FXCM trading station.

So you can see why doing trading and investing may get higher return. But the risk of losing all your principal is extremely easy, especially if you use high leverage and do not use good money management skills. In worst cases, you may lose everything you got in a matter of a fraction of one second!

Note: shortly after this article was modified, siver price went up to 47.66 and then 49.78 USD, before it starts to drop to below \$28 (as of July, 2012).

The next topic will extend on this one and focus on low risk investments.