Rule of 72 vs Rule of 69

How fast will your money double? That’s what both the Rule of 72 and the Rule of 69 try to do in a hurry with a back of the envelope calculation that we’ll share below.

These shortcut formulas are used to estimate how long it will take for an investment to double in value assuming a fixed annual rate of interest. So, if you’re looking to make mental math easy when dealing with compound interest, these rules can come to the rescue.

The Rule of 72:

• The Rule of 72 is an easy way to estimate the number of years needed to double an investment when you know the interest rate.
• The simple formula involves dividing 72 by the annual interest rate.
• For example, if your investment grows at an annual rate of 6%, then divide 72 by 6 and you’ll discover that in 12 years your Investment will double. 
• This rule works best for interest rates between 6% and 10%, where the approximation is quite accurate.

Real-World Investment Examples:

1. Stock Market: Historically, the average annual return of the U.S. stock market is around 10%. Using the Rule of 72, you can forecast that it would take approximately 7years for an investment in a diversified stock portfolio to double.
2. Bonds: If a bond yields 4% per year, it would take approximately 18 years to double because 72 divided by 4 is 18. So if you’re considering this investment avenue, you can now set proper expectations for when you’ll generate a 2x return.
3. High-Yield Savings Accounts: Suppose you have a high-yield savings account offering 2% interest then it would take 36 years for your savings to double at that rate.

The Rule of 69 (or Rule of 69.3):

• The Rule of 69 is slightly more precise and is generally used for continuously compounded interest rather than simple annual compounding. In certain financial situations, especially in professional and academic contexts continuous compounding is preferred.
• If you have a calculator divide 69.3 by the annual interest rate for a more exact estimate with continuous compounding.
• Example: If you have a continuously compounded annual interest rate of 6%, then it would take 11.55 years to double.

Real-World Investment Examples:

1. Some more advanced types of financial products, like certain money market funds or even derivatives, calculate returns using continuous compounding. The Rule of 69 would give a more accurate result in these types of cases.
2. Among University professors this rule is popular, not least because of its application in pricing options via the Black-Scholes model, where continuous compounding plays a significant role.

So, When Should You Choose The Rule of 72 vs Rule of 69?

1. When an investment compounds annually, such as many bonds and fixed-rate accounts, run with the Rule of 72 as a rule of thumb.
2. The Rule of 69 is better suited to continuously compounded interest that is evident more in theoretical finance and some specific investment instruments.

What Is The Difference Between The Rule of 72 and Rule of 69?

1. Type of Compounding:
   • The Rule of 72 is used for annual compounding.
   • The Rule of 69 is used for continuous compounding.
2. The Rule of 72 can be done simply using mental arithmetic while the Rule of 69 is mathematically more precise but less frequently used in everyday investing and usually needs a calculator.

Practical Strategies and Applications

1. When to Use the Rule of 72:
   • You can use the Rule of 72 for a typical savings accounts as well as stock market returns where interest is compounded yearly.
   • It’s also good for quick, on-the-go financial calculations in order to set realistic investment goals.
2. When to Use the Rule of 69:
   • If you’re using financial derivatives like options or working on an academic finance problem set, you’ll probably want to use the Rule of 69. 
   • Because it is more specialized, it is often used by finance professionals and academic researchers.

Advantages & Benefits

• Rule of 72: One obvious benefit of the Rule of 72 is how quick and easy it is to use when planning or estimating returns for typical investments.
• Rule of 69: For a deeper understanding of continuous compounding effects in more complicated financial scenarios, the Rule of 69 is better.

Disadvantages & Risks

• A challenge of the Rule of 72 is it’s less accurate when the interest rate veers away from the 6%-10% range.
• The Rule of 69 has a drawback too in that it demands a deeper understanding and is less intuitive for most everyday investors.

When Is It Best To Use Each?

• Use the Rule of 72 for most basic investment calculations.
• Apply the Rule of 69 for scenarios where you really need to be precise and when continuous compounding matters.

Common Mistakes to Avoid

• Don’t use the Rule of 72 for continuously compounded investments or the Rule of 69 for simple annual compounding.
• Don’t forget that these rules are approximate and so they shouldn’t take the place of more precise financial analysis, especially when making large investment decisions.

When Will You Double Your Money At Different Interest Rates?

1. Low Interest Rates (1%-3%):
   • For an investment growing at 1% annually, the doubling time using the Rule of 72 is a full $$72\text{years (72 divided by 1).}$$
   • At 3%, the doubling time is cut dramatically to just 24 years (72 divided by 3).
   • If you ever wondered why it takes so long for an investment to show progress when the interest rate is low, you can see now why. A lot of time is needed for the low interest rate to have a meaningful impact.
2. Moderate Interest Rates (5%-10%):
   • With a 5% interest rate, your investment would double in 14.4 while at an 8% rate the time slides to just 9 years.
   • In this 5-10% range, the Rule of 72 is generally considered most reliable and provides clarity into when you can expect a 2x return.
3. High Interest Rates (15%-20%):
   • At a 15% rate, the doubling time is very short at just 4.8 years while for a 20% return just 3.6 years is needed.
   • Of course, when you are looking at investments with 15-20% interest rates, the risks start to rise commensurately, so buyer beware is sage advice.

How Continuous Compounding Works in Advanced Finance:

Continuous compounding is an extreme form of compound interest where the frequency of compounding is infinite because instead of interest being calculated yearly, monthly, or even daily, it’s calculated at every possible moment.

1. Example 1: Investment in a Savings Account
   • Imagine that you invest $10,000 at an annual interest rate of 5%, compounded continuously. After a decade of holding the investment, your investment would grow to approximately $16,487.20.
2. Example 2: High-Yield Investment Scenario
   • Now consider instead that you invest $5,000 at an annual interest rate of 10%, compounded continuously for 7 years. In that case, the investment would double and grow to about $10,068.75 in 7 years.
3. Example 3: Derivative Pricing in Finance
   • In advanced finance, continuous compounding is essential when pricing financial derivatives. Take as an example the Black-Scholes model for option pricing where the risk-free interest rate is compounded continuously to account for fluctuations over infinitesimally small time periods.
   • Now further imagine that a stock option is priced using a risk-free rate of 3% with a term of 2 years. Continuous compounding ensures the time value of money is accurately represented in the model.

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