Category Archives: Number

Happy-O-Meter

How to measure happiness? The definition of happiness is dependent on each person. For me, it is a matter of feeling. That feeling depends not only on how good I feel at the moment but also on how large the gap between my financial status and the living cost. I would like to quantify the happiness in financial term.

As long as the growth rate of number of things that I want in life is lower than the growth rate of my income, the longer I live, the more happy I am. If I can maintain a zero (or negative) growth rate for the number of things that I want while growing my income, well, it is only a matter of time to accumulate enough net worth to stop full time working and start full time living. Cool. Continue reading





How-to Retire with $ 1 Million?

Today I am going to write about an interesting topic on “How to Retire with $ 1 Million”. The topic is interesting because there are at least two interpretations that I can think of based on that title.

Firstly, it could mean how to have $ 1 million available at the time you retire. So that you could retire as a millionaire. Secondly, if you look at it at different angle, it could also mean how to retire once you have accumulated $ 1 million. As long as you are millionaire, you can retire. Same title but two ways to look at it.

I think both of the interpretations bring you to the same goal: to achieve financial independence. They are both parts of the same thing. Let’s study both cases individually.

A Little Bit of Math

Before that, I need to introduce a formula that will be used throughout the case studies.  If you are not interested, you can skip this section.

Given:

b_n=b_0(1+r)^n+12x\sum_{i=1}^{n}{(1+r)^i}

where:

  • b_n is the final balance
  • b_0 is the initial balance
  • r is the rate of return (e.g.: 0.01 = 1%) of your investment
  • n is the number of years
  • 12 is the number of months in a year
  • x is the amount of saving per month
  • \sum_{i=1}^{n}{(1+r)^i} is another way of writing (1+r)+(1+r)^2+...+(1+r)^n

The above the is famous compounding formula b_n=b_0(1+r)^n that we used to see usually but with an additional term 12x\sum_{i=1}^{n}{(1+r)^i} which is used to represent the monthly contribution of money to the final balance.

The monthly contributions are also subject to the compounding effect which is why we have to multiple them with (1+r)^i where i is the number of years they are compounded. Here, I am only looking at the balance which is compounded on a yearly basis. Therefore we only calculate the return by the end of the year with the amount 12x.

The formula above is very troublesome and impractical to apply in reality due to the fact that it has the sigma term \sum in it. With a little bit of math manipulation and rearrangement of the terms, the same formula can also be written as:

b_n=b_0(1+r)^n+12x(1+r)\frac{(1+r)^n-1}{r}

Now, you will know where are the figures in the following case studies coming from. They are all based on the above formula.

Case Studies

Case 1: How to have $ 1 million available when you retire?

In this case, it is more about how to make sure you have $ 1 million when you are reaching you age of retirement so that you could life comfortably during periods where you are no longer working.

For most people, I assume the age of retirement is around 65 years old. So I will use 65 as the age of retirement in the following chart. For example, if you are 30 years old right now, there are still 35 years ahead (to accumulate your asset) before you retire.

Scenario 1: You have no saving

How to use the chart: Assuming a return rate of 6% (which is reasonable) for your investment and assuming you are 25 years old right now without any saving, you would need to save an amount of $ 507.98 monthly till your retirement age (65) so that you could retire as a millionaire.

You can change the return rate (by selecting a different return rate from below the chart) to see the different amount that you need to save monthly according to your current age.

As you can see from the chart above, the earlier you start saving, the less you need to save monthly. You can see the power of compounding visually.

Scenario 2: You start with $ 100 k of saving

With everything remains the same (retire at age 65), what is your required monthly saving to retire as a millionaire if you now already have a saving of $ 100 k?

At age 25 with $ 100 k saving, your monthly contribution with 6% return rate is basically $ -14.51. A negative value! You don’t have to save anything each month, in fact, you can even withdraw $ 14.51 from your investment each month till you retire while still have $ 1 million available for you at age 65.

Comparison between with and without initial saving

By comparing both scenarios where 1. you start with no saving and 2. with $ 100 k saving, the difference in amount you need to save monthly is only about a few hundreds or at most thousands. The impact of having or not having saving now does not stop you from becoming millionaire. So it is really not a big deal if you still have no saving as long as you start saving regularly now and thereafter.

You need to really slow down and appreciate the wonder of compounding effect.

Before ending this case study, this is the formula used for the results:

x=\frac{r(b_n-b_0(1+r)^n)}{12(1+r)((1+r)^n-1)}

where:

  • b_n is the final balance
  • b_0 is the initial balance
  • r is the rate of return (e.g.: 0.01 = 1%) of your investment
  • n is the number of years
  • 12 is the number of months in a year
  • x is the amount of saving per month

Case 2: How to retire once you are millionaire?

Is $ 1 million enough for retirement? Let’s see how much monthly income you will have if you have $ 1 million with different return rate.

With a return rate of 6%, which is very reasonable and highly probable, you could have $ 5000 passive monthly income once you are millionaire. That is very sufficient to live a comfortable lifestyle. That amount may even be higher than the salary of the majority people.

However, there is another factor to consider: inflation. $ 5000 in today’s money is a lot more than $ 5000 in 30 or 40 years in the future.

How long it takes you to become a millionaire actually means a lot. So let’s see what you can do to speed up your journey to become a millionaire.

Scenario 1: You have no saving

With a return rate of 6%, if you contribute $ 1600 each month to your investment, you will become a millionaire in 23.57 years. Simple. If you started saving at 25, you can become a millionaire at age 48.57.

Scenario 2: You have $ 100 k in saving

With a return rate of 6%, if you contribute $ 1600 each month to your investment provided that you already have $ 100 k saving, it will take you 19.13 years to become a millionaire.

Comparison between with and without initial saving

As you can see from the chart, the difference in time between with and without initial saving is small if you contribute large enough monthly saving. The difference in time to become a millionaire becomes negligible once you contribute more than $ 3000 each month to your investment.

Before ending this case study, this is the formula used for the results:

n=\frac{log(\frac{b_nr+12x(1+r)}{b_0r+12x(1+r)})}{log(1+r)}

where:

  • b_n is the final balance
  • b_0 is the initial balance
  • r is the rate of return (e.g.: 0.01 = 1%) of your investment
  • n is the number of years
  • 12 is the number of months in a year
  • x is the amount of saving per month
  • log is logarithm

Conclusion

Hopefully this article can serve as a guide to encourage and inspire (young) people to start saving as early as possible and do so regularly as we have seen that the key to become millionaire and achieve financial independence relies on large and regular savings.

You can download the excel file to play and do the simulations yourself here.

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