Time_is_Money

Photo I took at the Grand Bazaar in Istanbul, Turkey

Since today is International Pi Day, I thought a bit of mathematics would be in order to honor the occasion. Nothing to do with Pi the constant really, but perhaps some knowledge to help you get a ‘piece of the pie’ so to speak! A good question in life is how to build wealth. For most of us this amounts to saving and good investment, but how does it work out mathematically and what are the important variables involved? Let’s find out… Don’t worry if you are not mathematically inclined and I will try to keep it as simple as possible, but the important point is to understand the variables, and the concepts they represent for you the investor. If you wish to skip over the mathematics, you can just refer to the boxed equations, which represent the core results.

Starting with a simple application will help to define some variables and introduce the concepts. We begin by asking a simple question: If you invest a given amount of money (P) called the principal, what is the total return amount (T1), that you will have at the end of 1 year, given a yearly rate of return (i) called the interest rate. The answer is simple:

T_{1} = P(1+i)

Now, we ask the question, what about the total return amount, T2, after 2 years? Well, after the first year, T1 becomes the principal for the 2nd year, that is:

T_{2}=T_{1}(1+i)\\ {\quad}=P(1+i)(1+i)\\ {\quad}=P(1+i)^{2}

And for the 3rd year:

T_{3}=T_{2}(1+i)\\ {\quad}=P(1+i)^{2}(1+i)\\ {\quad}=P(1+i)^{3}

and so on… The general answer for N years is:

\boxed{T=P(1+i)^{N}}  Eq. I

This equation embodies the concept of compound interest and shows the total growth (T) with an initial investment (P), assuming an average interest rate (i) over a number of years (N).

Let’s try something a little more advanced now and ask the more complicated question regarding the growth of your money if, instead of just investing one lump sum, you invest a certain amount per year. So, suppose you invest (P) in the first year, then your total after that year is simply T1=P(1+i), as was initially stated at the beginning. In the second year you again invest (P) on top of T1, so what is the total, T2 after the second year? It is:

T_{2}=(T_{1}+P)(1+i)\\ {\quad}=[P(1+i)+P](1+i)\\ {\quad}=P(1+i)[1+(1+i)]

And for the 3rd year:

T_{3}=(T_{2}+P)(1+i)\\ {\quad}=[P(1+i)[1+(1+i)]+P](1+i)\\ {\quad}=P(1+i)[1+(1+i)+(1+i)^{2}]

The trend is clear and we have what is called a geometric progression. Writing the terms in brackets as a geometric series:

1+(1+i)+(1+i)^{2}+...=\displaystyle\sum_{x=0}^{N-1}\left(1+i\right)^{x}

where N is the number of years. It gets a little complicated here, so bear with me. We can rewrite the summation as:

\displaystyle\sum_{x=0}^{N-1}\left(1+i\right)^{x}=\displaystyle\sum_{x=0}^{N}\left(1+i\right)^{x}-\left(1+i\right)^{N}

At this point I can use a trick with summations of the form S_{n}=\displaystyle\sum_{n=0}^{N}a^{n}

aS_{n}=\displaystyle\sum_{n=0}^{N}a^{n+1}

aS_{n}-S_{n}=\displaystyle\sum_{n=0}^{N}a^{n+1}-\displaystyle\sum_{n=0}^{N}a^{n}=a^{N+1}-1

We can now write:

\displaystyle\sum_{x=0}^{N-1}\left(1+i\right)^{x}={{(1+i)^{N+1}-1}\over{(1+i)-1}}-(1+i)^{N}={{(1+i)^{N}-1}\over{i}}

So, in general, we have:

\boxed{T=P(1+i){{(1+i)^{N}-1}\over{i}}}  Eq. II

Illustrating again the power of compound interest and shows the total growth (T) with a yearly investment (P), assuming an average interest rate (i) over a number of years (N).

As a check, Eq. II should reduce to Eq. I when N = 1. In this case [(1+i)-1]/i = 1 and indeed it does! So, what does all this mean anyway?

Eq. I – Suppose you invest $10,000, what can you expect after N years assuming an average interest rate of 10 percent?

P-onetimeP-onetime-pie.

For the last 25 years the stock market return has been around 10% per year (or i = 0.1). Even historically since 1929 (after the Crash) the return is about the same during that time. All this despite lower returns the last 5 to 10 years. For the last 10 years the returns have been around 4.5% and for the last 5 years about 2.3%. This data is as of 2012.

Eq. II – Suppose you invest $1000 per year, what can you expect after N years assuming an average interest rate of 10 percent?

P-yearlyP-yearly-pie

I chose these examples as they are essentially equivalent methods of investing money to make a million. You can start with 10K and let it ride from the start or you can invest 1K per year over the same time span. There are other combinations of investment strategy and you must choose the one that suits you best. I have provided the framework in calculation ways, but you must decide what works for you. If you are young then time is on your side, but if you are older, you will have to play catch up, so to speak. Play around with the numbers in the equations I provided. There are calculators online that do such things too. Here is one from Dave Ramsey you might find useful. His calculator does a monthly compounding for an average yearly interest rate, which comes out a bit more than the equations I derived since I only consider the yearly average compounding. Have fun and play around with it now that you may understand the principles a little better, which was the primary objective of this blog in quantitative and qualitative ways. In that spirit,  may your investing ways serve you well in latter days…

good news-rich

None of these calculations account for inflation, which is a factor in considering the total amount calculated in today’s dollars vs. what it might be worth in the future. Inflation (like most interest rates) varies year to year, but has an historical average of about 2-3% per year. Conceptually, this calculation should be analogous to interest gain over N years. The illustration here was to show the power of numbers in geometric progression that is reflected in compound interest gains over time. This is a great advantage as shown here and it can be said that indeed, as the old adage goes (and mathematically verified here), Time is Money!

Happy \pi  Day!PIDAY

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