Let your money make you more money.

Intuitively we understand the concept of compounding your savings or investments. In practice, impatience and lack of discipline interrupts any compounding we have achieved in our wealth before it becomes a force of nature.

We speak of compounding in terms of money, and perhaps it is best illustrated as such. However, as you will see, the concept of compound interest transcends money. It is one of the basic concepts that underpins almost everything we do that grows in scale.

## What is Compound Interest?

Once your investment generates interest, you have a choice. You can take the earned interest out and spend it elsewhere, or you can choose to reinvest the interest and increase your capital. Interest reinvested to generate more interest is called compound interest.

## Compound Interest Formula

If P is the principal invested, r is the rate of interest per term period, then the interest received after 1 period = P *x* r

Thus, at the end of 1 period, you will have the original Principal plus the 1 period interest, or P + P *x* r.

This can be expressed as P *x* (1 + r)

If you reinvest the entire value for another period at the same rate of interest, at the end of the period 2, you will end up with P *x* (1 + r) *x* (1 + r), or P *x* (1 + r)^{2}

Extending this further, the total value at the end of term t equals P *x* (1 + r)^{t}.

## Compound Interest Examples

When using the compound interest formula, it is important to match the rate of interest to the term of investment. Let me illustrate this with examples.

### Example 1:

Suppose we invest $1000 for 1 year paying 12% annual interest, compounded monthly. We wish to find out the future value of the investment at the end of 1 year.

Since this is monthly compounding, we have 12 compounding terms in the year. Therefore, t = 12.

The rate of interest needs to be expressed as the monthly rate of interest. Here we can use 12%/12 = 1% per month.

Therefore, the Future Value at the end of the year = $1000 *x* (1 +1%)^{12} = $1126.83

### Example 2:

Now suppose the same $1000 was instead invested for 1 year at 12% interest, with quarterly compounding.

In this case, t = 4 and r = 12%/4 = 3%.

Future Value = $1000 *x* (1 + 3%)^{4} = $1125.51

### Example 3:

Consider the same principal that is invested at the same rate of interest for 1 year, but it is compounded only once at the end of the year.

In this case, t = 1 and r = 12%.

Therefore, Future Value = $1000 *x* (1 + 12%)^{1} = $1120.00

As you read this example and see how compound interest works, you will notice a key principle at work here.

*PRINCIPLE 1: All things being equal, more frequent compounding grows wealth faster*

## Continuous Compound Interest Formula

As we increase the frequency of compounding, the number of terms become larger and the size of the terms grow smaller. If more frequent compounding results in faster and greater growth in capital, it is no surprise to know that continuous compounding can maximize the capital growth.

When money is compounded continuously, it is always earning interest and the interest immediately gets reinvested for futher growth.

The formula for continuous compounding is

FV = P *x* e^{rt}, where e is the mathematical constant with approximate value of 2.7183.

Derivation of the formula is out of scope for this article.

I want to reiterate that the matching of the term is important. If t is expressed in years, than the r, or the rate of interest should be expressed as an annual rate.

## Importance of Compound Interest

Compound interest is often called the 8th wonder of the world as it has the power to generate an unstoppable momentum when applied consistently for a long period of time. The astute reader has no doubt noticed that the compound interest formula is exponential in nature. Therefore, each additional growth comes in bigger and faster spurts.

A close inspection of the compound interest formula shows a few more principles at work.

*PRINCIPLE 2: All things being equal, wealth grows exponentially if the compunding takes place over longer period of time*

and,

*PRINCIPLE 3: All things being equal, higher interest rate grows wealth faster*

## Illustrating the Power of Compounding

The following chart shows the power of compounding over a long period of time. The rate of interest is 1%/month, or 12% annually. $1000 is compounded for 30 years, which approximates a typical investment horizon. At the end of 30 years, the $1000 investment grows to $35,949 or about 36 times the originial investment.

Worth noting here is that at the end of 20 years, or 2/3rd into the 30 year investment period, the value of the investment was merely $10,893. In the next 10 years, the investment added about $25K to its value. This illustrates how powerful it can be to let your investments compound for long periods of time.

Most of the gains come at the back end of the investment period. The gains accelerate as the time goes on. An investor who gets frustrated in the early years when not much seems to be happening never gets to benefit. Long term orientation and the discipline to stay with the investment makes tremendous difference in the wealth growth.

What if the investment earned a slightly higher return? Let’s say, 1.25% instead of 1%/month. This is a 15% return per year instead of 12% return per year. You wouldn’t think this makes a huge difference, but you would be wrong.

After 30 years, an investment earning 15% year compounding monthly will be worth $86,460. This is 86 times the original investment instead of 36 times. An improvement of 3% in returns per year results in a massive difference in the final wealth after 30 years of investing.

## How to Compound Money? – Simple Acts That Destroy Your Wealth (and Acts that Increase Your Wealth)

We know now that a small change in annual returns can have disproportionate effect on the future value of our investment. So any thing that decreases the annual return should be avoided, if the goal is to maximize future wealth. Similarly, every action that either increases the annual return, or eliminates drag on the annual returns, should be undertaken.

Here are a few actions that destroy wealth:

- Investing in low return assets and not investing in higher return assets over a long period of time – for example, you buy CDs and bonds and avoid investing in stocks
- Investing for a smaller period – this happens if you start investing very late in your career
- Chasing more speculative investments with uncertain returns – focusing on consistent dividend payers may be boring but it can seriously compoound your wealth over time
- Paying taxes when they can be avoided or deferred – for example, not utilizing tax advantaged accounts

Here are some of the actions that can help increase your wealth by compounding:

- Move your investments that generate highly taxed income (interest/dividends) to tax free or tax deferred accounts
- Focus on capital gains in your taxable accounts
- Municipal bonds and other government bonds that are free from federal or other taxes should be purchased in the taxable accounts
- Proper tax planning can help eliminate taxes or defer taxes for later
- Longer holding periods can reduce capital gain taxes – so plan to hold investments for longer periods
- Capital gains can be completely avoided if the investments are not sold. This is hard to do in practice, but a long term orientation can certainly help in deferring capital gains taxes further out in the future
- Focus on risk management so losses can be avoided. Prioritizing loss avoidance (over chasing returns) can significantly improve your wealth. Perhaps counterintuitive to some investors. It literall pays to keep in mind that you can compound losses over time, just as well as you can compound winnings.

Ultimately profitable investing is a series of choices you make. Each choice, while it may seem insignificant at the time, has a very profound impact on the wealth you end up with over time. Choose well, and you will do very well. Choose unwisely, and you will lose.

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