## Before investing and trading - savings and calculations (II)

Before investing and trading - savings and calculations (II)

1. Choose a bank that offers higher interest rate!

Now let's study savings in a bit more detail. If your country offers tax free savings account (TSFA), be sure to invest some every year.

Before opening a TFSA  or other types of saving account, you need to choose the major banks that offer higher interests. Normally, only online banking can give you higher interest rate. For example, ING direct.

Take Canada as an example, ING direct offers 2% interest rate for TSFA, as of the time when I write this article.

2. Calculations

Suppose your initial investment at beginning of the current year is $$p$$, annual interest is $$r$$ that compounded only once a year (actually, it normally compounded monthly or daily and you will get slightly higher rate than APR.). Now we are interested in the following question:

Suppose that we invest $p every year at the beginning of the year, how much will we have at the end of the n-th year? At the end of the first year, the amount is p(1+r). The above amount will carry interest and beomes p(1+r)(1+r) = p(1+r)2 at the end of the second year. Also, at the beginning of the second year, p$ will be deposited, so at the end of the year, it becomes p(1+r). Thus, the total at the end of the second year is

p(1+r)+p(1+r)2.

Similarly, at the end of the third year, the amount is

p(1+r)+p(1+r)2+p(1+r)3.

By induction, at the end of the  nth year, the amount is

$$p(1+r)+p(1+r)^2+\cdots +p(1+r)^n$$

use the formula for geometrical series

(1)            $$aq+aq^2+\cdots aq^n=\frac{aq(q^n-1)}{q-1}$$

we have

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

The above is the amount at the end of nth year, if deposit p dollars at the beginning of each year, with interest r compounded once a year.

Suppose you deposit $200 each year, then, assume the current 2% (0.02) rate offered by ING direct, then at the end of year 5, you will have$1061.62. This is equivalent to 6.16% increament of your total investment.

Also, the interest rate is likely to increase, since the current interest is already near the bottom. If we consider the unual monthly or daily compounding the actual rate will be slightly higher.

3. Monthly compounding and more

Still consider the same as above except that we compound monthly. Then the monthly interest will be $$\frac{r}{12}$$, the end of the n-th year will be the 12n-th month, use formula (1), take $$q=\left(1+\frac{r}{12}\right)^{12}$$  we can get the amount at the end of n-th year to be

$$\frac{p\left[(1+\frac{r}{12})^{12(n+1)}-\left(1+\frac{r}{12}\right)^{12}\right]}{\left(1+\frac{r}{12}\right)^{12}-1}.---------------- (2)$$

Use the above formula, we can calculate that (take p=1)  the rate of return at the end of the 5th year with 2% APR is 6.22%, which is slightly higer than the previous yearly compounding case, but not dramatic.

You may work a bit harder to get the following interest formula (just withour "%" sign)

$$\left(\frac{\left(1+\frac{r}{12}\right)^{12(n+1)}-\left(1+\frac{r}{12}\right)^{12}}{\left(\left(1+\frac{r}{12}\right)^{12}-1\right)* n}-1\right)*100$$

With daily compounding, the formula becomes (suppose 365 days per year)

$$\left(\frac{\left(1+\frac{r}{365}\right)^{365(n+1)}-\left(1+\frac{r}{365}\right)^{365}}{\left(\left(1+\frac{r}{365}\right)^{365}-1\right)* n}-1\right)*100$$

Still, 2% APR and 5 year investment will have return rate of 6.22596%, a tiny amount of increament from monthly compouding.

Banks are normally using the daily compounding method, but interest will be paid monthly.

As you can check, it takes 62 years for you to get double return, i.e., your total investment in the past 62 years (including the  amount you deposited at the beginning of 62nd year) doubles at the end of the year 62.

Although you have seen silver price jumped several times the value a few month ago, but if you do this kind of high risk investing, you will also have greater risk of losing all you have and get nearly 100% loss.

Put stop loss order for every trade, do not risk what you can not afford to lose, make a balance of high and low risk investments, just a few tips.

4. A bit calculus, taste of limit - continuous compounding

Let's increase compounding frequency, and keep increasing, that means, even within a fraction of a second, your interested will be calculated for infinitly many times and added on to what you already have.

Some of you may think, looks like I will have all the money in the world soon! No way!

By using the famous limit

$$\lim\limits_{x\to\infty}\left(1+\frac{1}{x}\right)^x=e,$$

where $$e$$ is an irrational number that about equal to 2.718281828459045.......

we have the continuous compounding formula that corresponding to formula (2) to be

$$\frac{p\left[e^{r(n+1)}-e^r\right]}{e^r-1}.$$

Now plug in p=1, r=0.02 and n=5 will get 5.31131. Divide by 5,the total investment and then minus 1, we get the rate of return: 0.0622613 or 6.22613%.

Unless you can compound your interest in negative time ♥, you can not get more than this rate!