The Expanding Lily Pad: A Retirement Riddle

There is an old riddle about a lily pad in a pond. The lily pad doubles in size every day and after 30 days it completely covers the pond.  On what day does the lily pad cover half the pond? 

If you answered without really thinking about it you probably answered “day 15”. Our mind automatically goes to this answer because we are comfortable with linear thinking.  If the lily pad covers the entire pond in 30 days than it must cover half the pond in 15 days. 

Of course, if you stopped to think about it you would realize that the lily pad only covers half the pond on day 29. It then doubles one final time and covers the entire pond on day 30. 

A more interesting question is, how much of the pond would the lily pad have covered on day 15? Take a second and write down an estimate.  

The correct answer is that the lily pad will only have covered .0031% (3 thousandths of one percent) of the pond on day 15. In fact, the lily pad will only cover more than one percent of the pond on day 24.

What makes this riddle difficult is that the lily pad’s growth is exponential, not linear. Our minds are comfortable with linear growth.  Linear growth is easy to envision and we are fairly good at estimating things that grow linearly.  However, we struggle to envision exponential growth and our estimating skills often fail us. 

Here is a graph showing the growth of the lily pad:

Expanding Lily Pad Graph

The Lily Pad and Retirement Planning

What does any of this have to do with retirement planning? Unfortunately money doesn’t double every year.  If it did retirement planning would be much easier.  However, due to the power of compounding money does grows exponentially.

Benjamin Franklin described compounding as follows: “The money that money makes, makes more money.”

In other words, as with the lily pad, the growth in later years doesn’t just operate on your initial investment, but also on all of the earlier growth. This makes the growth of our money, like the size of the lily pad, difficult for us to envision.  We overestimate how fast we think it should grow during the early years and underestimate its growth in later years.    

Let’s say you have a retirement savings goal of $1 million dollars. You have been told you can reach this goal by saving $700 a month for 30 years if you can earn an 8% annual return on your investments.  You set up a diversified portfolio capable of earning this rate of return and start saving.

After 15 years of diligent saving you check on your progress. You are alarmed to find that the value of your account is less than $250,000.  You are half way through the 30 years but are less than a quarter of the way to your goal.  You become discouraged, cash in your retirement account and go to Vegas where you lose all your money gambling.   

This would be a huge mistake, because in reality you are right on target to meet your goal. Due to compounding most of the growth is still ahead of you.  In fact, you will not reach $500,000 until after year 22, but by the end of 30 years will have saved $1,043,252. 

Here is a graph of what your savings would look like given the scenario above:

Lily Pad_Retirement Growth Graph

When Can I Retire?

One of the most common questions asked of financial planners is, “When can I retire?” Money manager and author Ric Edelman, in his book, The Truth About Retirement Plans and IRAs, has an interesting answer.  He tells people, “You will be able to retire 30 years after you begin saving.” 

The reason by now should be clear. Just as it took the lily pad 30 days to cover the pond it will take you 30 years of saving 10-15 percent of your income to accumulate a nest egg big enough to cover your current standard of living during retirement.  It simply takes about that long to get to the years of fast growth necessary to reach your goal.    

You can speed up this process by inheriting a lot of money, winning the lottery, or increasing your savings rate to a percentage most of us can’t maintain. For the rest of us properly preparing for retirement will take about 30 years of disciplined, patient, saving and investing.  The keys are persistence, patience, and time.  The sooner you start the sooner you will reach those exciting years of powerful growth, and ultimately your goal.  There is no better time to start than now.


  10 comments for “The Expanding Lily Pad: A Retirement Riddle

  1. PhysicianOnFIRE
    February 21, 2016 at 10:36 am

    The lily parable reminds me of the penny that doubles every day for a month. If I could offer you $100,000 each and every day for 31 days versus a penny on day 1 and give you double the previous day’s each day, which would you choose?

    The first option gives you $3.3 million. If you go with Door #2 and start with the penny, you end up with over $21 million at the end of the month. Same concept with different round object, same powerful message.

    • Brent Esplin
      February 21, 2016 at 11:14 am

      That is another great analogy. Thanks for sharing.

  2. Go Finance Yourself
    January 13, 2017 at 7:45 am

    Yes, the miracle of compounding. Something we should be teaching kids in school at a young age.

    I agree with your premise here. It takes time and persistence to grow something big. However, don’t sell yourself short. If you don’t want to work for 30 years, then set a goal to retire after 20 years. Set your goals high, figure out a plan to meet them, and go get it! You might surprise yourself at what you can attain.

    Thanks for sharing!

    • Brent Esplin
      January 13, 2017 at 4:48 pm

      I agree. It is certainly possible to cut time off the 30-years by increasing your savings rate, but most people aren’t willing to sacrifice that much. For most people 30 years is a realistic goal. Thanks for your thoughts.

      • January 14, 2017 at 9:05 am

        I like this analogy. I really wish they would teach compounding interest in schools.

        • January 19, 2017 at 1:49 am

          They do (my son’s in 5th grade and they are teaching simple interest and compound interest in Math). What happens is they also teach lots of other stuff that this important concept gets blurred amidst all the bookish teaching.

          • Brent Esplin
            January 19, 2017 at 6:42 am

            Glad to hear that it is being taught. I think we need to reinforce this teaching at home. If someone internalizes the concept of the time value of money early it can transform their life. Thanks for sharing.

  3. ninjapiggy1
    January 13, 2017 at 8:13 am

    Thanks for sharing this! I’ve been thinking about this concept a lot lately as it relates to early retirement. As your numbers show, early retirement is very difficult to achieve because we need to give our investments time to grow.

    A more likely and reasonable scenario for most people is to get to a point where they can switch into a more meaningful work, even if it means a cut in pay (provided they saved more aggressively up front to be able to sit back more and let compounding do most of the work).

    Kudos to all of you who have been able to retire early! It’s a very impressive feat and something to be proud of, especially given the numbers above.

    • Brent Esplin
      January 13, 2017 at 4:51 pm

      Yes, being able to cut back on hours or move into more meaningful or enjoyable work is a great alternative to retiring completely, and often brings people more happiness than actually retiring. Thanks for contributing.

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