Compound interest introduction | Interest and debt | Finance & Capital Markets | Khan Academy
How to Learn About Compound Interest and Investing Using the One‐Penny Trick
Amazing Your Friends
Present your friends with a seemingly simple question."Would you rather have right now ten million dollars or one penny?" Stipulate that if they take the penny, they would then get to double the amount of money they have every day for two months. In fact, tell them they can have two months with 31 days in them (like July and August), a total of 62 days.
- Most people will take the ten million dollars. It seems like an obvious choice!
Warn them, however, that it would be a mistake to take the ten million.Tell them to take their time and think their decision through before giving you their final answer. Act like you think it's a tough choice to make.
- Your friends will probably stick with the million rather than the penny that doubles in value every day for two months.
Explain why they made the wrong choice.The math is quite simple, but your friends will be amazed at how much even a small sum will grow by doubling every day. On a sheet of paper, begin with a single penny and multiply by two each day for 62 days:
- Day 1, one penny.
- Day 2, multiply the amount of day 1 by 2, making 2 pennies.
- Day 3, multiply the amount of day 2 by 2, making 4 pennies.
- Day 4, multiply the amount of day 3 by 2, making 8 pennies.
- Day 5, multiply the amount of day 4 by 2, making 16 pennies.
- Day 6, multiply the amount of day 5 by 2, making 32 pennies.
Keep going, even if your friends think you’re insane.They'll change their mind as you keep multiplying by 2. You're going to need a calculator at some point, because by the time you get to day 31, your penny will have grown to 2,147,483,648! That's right — over2000 million pennies.
Keep going into the second month.By this point you've already convinced your friends that they made a big mistake choosing the ten million dollars. You don't need to go through the math for the second month to drive the point home. Instead, jump ahead to the final amount they would have earned if they'd taken the doubling penny: a staggering 2,305,843,009,213,693,952!
- That number’s so large your friends might not even know how to say it!
- In the U.S. the number is called 4 quintillion.
Convert pennies into dollars.The amounts you get from the calculations above tell you how many pennies you have, not how many dollars. Divide those amounts by 100 to see how many dollars you will have earned at the one- and two-month marks. It's still an amazing amount of money.
Try powers of 2. After doubling 1, double 2, 4, 8, 16, 32, etc.
Explain that this problem shows how an exponent works on 2 cents after one penny is doubled, and you keep doubling from there on.Exponents of a number (the base) means multiply it by itself the number of times stated in the exponent (the power).
Look at day two so the amount that starts doubling is 2 pennies.Because 1 multiplied by itself to any power is still 1, you have to start with a base of 2. Okay now double base of 2 instead of 1. So using base of 2 you can see that 2^3 = 2 x 2 x 2 = 4 x 2 = 8 . So then 2^4 = 2 x 2 x 2 x 2 = 8 x 2 = 16, etc. So, raising 2 to an exponent of 30 for the first month (of course the 31st power is on day one of the 2nd month...), and to 61 for the second month solution will give you your number of pennies after two 31 day months.
Use the exponent function on a scientific calculator to see the doubling calculation after day one instantly.For example, open the Windows calculator accessory, then you can clickViewand selectScientificand type 2 then press the [xy] key, then type 30 and press the [=] key to see the amount of pennies for the 1st month. To get the feel of using the calculator for that [xy] key, try the calculator for 2 to the 3rd power which is 8, and the 4th power which is 16, the 5th power of 2 is 32, etc. So, 32 pennies is exactly what you would have on day 6 of doubling, using the 5th power, since we had to start using the powers of 2 on day two.
Real Life Application
Admit to your friends that this scenario could never actually happen.The investment that doubles your money every single day doesn't exist. The example you gave them was purely hypothetical. However, we can all learn something important from it.
Explain the power of compounding and the use of incremental investing.Compounding refers to earning additional interest paid on earlier-earned interest (much like doubling pennies day after day). Compounding takes your first investment (the principal), adds all the earned interest (plus any newly added principal) before figuring the next interest payment. This means you get "interest on all the earlier interest," not just on the principal. This is true for each compounding period (such as a month). The shorter the compounding period, the more often your investment compounds. This means "the more often your investment compounds, the faster it grows."
- The most common compounding periods are daily, monthly, quarterly and yearly.
- The concept of compounding is important when considering investments like savings accounts and mutual funds. Look for accounts in which interest compounds daily.
Stress to your friends the importance of additional deposits to their account(s).After the frequency of your compounding period, this is the most important factor in safe and fast growth. Add as much money as possible to your investment as often as you can. This might be a month, 0 per paycheck — whatever you can afford.
- The key is to save first and spend later.
Explore different investment scenarios online.Bankrate.com has an interest calculator that shows how different choices can affect an investment.With the calculator you can figure different compounding periods, interest rates, and contribution amounts and frequencies.
- Enter an initial investment, but leave the "monthly deposit" option blank, like you're not going to add any more money. Leave the interest rate the same as well.
- Change the compounding periods, and check the results over time — say, ten years. You can see how important frequent compounding periods are for long-term investments.
- Now add regular monthly deposits — even small ones — and see how much the investment grows over time.
Encourage everyone to save money for retirement.You can do this through special accounts with tax benefits such as 401(k), IRA, and Roth IRA accounts. These three types of retirement accounts all benefit greatly from compounding and regular contributions. They are similar in that they are free to open, and they allow small initial deposits.
- Open a 401(k) plan directly through your employer.
- You can open IRA accounts through banks and brokers.
Consider when your retirement contributions will be taxed.Will your contributions be pre-tax or post-tax? Pre-tax contributions receive a tax deduction during the year in which you make them. For example, if you earn k and contribute ,000 toward your IRA, you get a ,000 tax deduction for the year. You will pay taxes on that money when you withdraw it in the future. You don't receive a deduction for post-tax contributions, but you also don't pay taxes on future withdrawals.
- Most 401(k) programs have both options available. Regular IRAs are for pre-tax contributions and Roth IRAs for post-tax contributions.
- It may be convenient to have both pre-tax and post-tax accounts, because your financial situation can change periodically. Contribute each year to the account most appropriate for you at the time.
Withdraw money from retirement funds early if needed.In certain cases (self-loans, first mortgages, medical expenses), you can withdraw money from an IRA or 401(k) with no penalty. Usually, though, there's a 10% penalty for withdrawal before retirement age.
Take advantage of payroll deductions.Automatic payroll deductions make saving easier by putting money into savingsbeforeyou get paid. This stops you from having to rely on what's left at the end of the month for your monthly deposits. It ensures regular deposits of the same amount.
Find out if your employer offers a contribution match.While they probably won't match your investment completely, they might match it up to a certain point. For example, a company might match 50% of what an employee contributes annually to their IRA or other investment up to ,000.
- If you add 0 per month, by year's end you'll have added 00 to your retirement savings. Your employer will have added another 0 as well. That results in 00 in just one year, not counting compound interest.
- If you could afford 00 per month, you would not have ,000 at the end of the year. You'd only have ,000 because the employer's match was limited to 00 per year in this example.
QuestionI don't know how to save.wikiHow ContributorCommunity AnswerKeep a mason jar with a slot in the top. When you want to input money to save, put it in the jar. When it is full, bring it to your bank and deposit it.Thanks!
QuestionIf this could happen in reality, how much would I have in a year?Top AnswererYou can use a scientific calculator to find your answer. On a practical level, however (because the answer would be 2 raised to the power of 365), it would just be an incomprehensible amount of money.Thanks!
||Might help you understand the idea.|
- You can use the exponent function on a scientific calculator to make the job even faster.
- The idea discussed above of doubling your money every day is fun but not realistic. The real point is to add money to your compounding investments as often as possible for a large payoff.
- If you understand exponents, use them as a faster method of doing the above calculations.
- Read the original legend that suggested the above example.It's a story about the reward offered by a wealthy ruler to the man who invented chess. While the legend takes several forms, using either grains of wheat or rice, the story is basically the same.
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