Rule of 69.3 A corollary to the Rule of 72 is the Rule of 69.3. The Rule of69.3 is exactly correct except for rounding when interest rates are compoundedcontinuously. Prove the Rule of 69.3 for continuously compounded interest.
Short Answer
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Using the formula for continuously compounded interest, \(A = P e^{rt}\), we set the final amount to double the principal amount, \(2P\). This gives us the equation \(2 = e^{rt}\). Taking the natural logarithm of both sides, we get \(\ln(2) = rt\). Solving for \(t\), we obtain \(\frac{\ln(2)}{r} = t\). Since the natural logarithm of 2 is approximately equal to 0.693, the equation becomes \(\frac{0.693}{r} = t\). Multiplying by 100 to express the interest rate as a percentage, we have \(\frac{69.3}{R} = t\), where \(R\) is the annual interest rate as a percentage. This proves the Rule of 69.3 for continuously compounded interest, stating that the time it takes for an investment to double in value when interest is compounded continuously can be approximated by dividing 69.3 by the annual interest rate.
Step by step solution
01
Understand the problem
We need to prove that for continuously compounded interest, the time it takes for an investment to double in value can be approximated by dividing 69.3 by the annual interest rate. Using the continuously compounded interest formula \(A = P e^{rt}\), we will prove the Rule of 69.3.
In order to find the time it takes for an investment to double, we must set the final amount \(A\) equal to twice the principal amount (\(2P\)). Thus, the equation to solve is \(2P = P e^{rt}\).
03
Eliminate the Principal Amount and Solve for Time
Divide both sides of the equation by the principal amount \(P\). This simplifies the equation to \(2 = e^{rt}\). To solve for \(t\), we will use natural logarithms. Take the natural logarithm of both sides, which results in \(\ln(2) = \ln(e^{rt})\).Using the logarithm property that \(\ln(x^y) = y \ln(x)\), the equation becomes \(\ln(2) = rt \ln(e)\). Since the natural logarithm of \(e\) is 1 (\(\ln(e) = 1\)), this further simplifies to \(\ln(2) = rt\).
04
Divide by the Interest Rate
Now, divide both sides of the equation by the annual interest rate \(r\) to solve for the time: \(\frac{\ln(2)}{r} = t\). The natural logarithm of 2 is approximately equal to 0.693, so the equation becomes \(\frac{0.693}{r} = t\).
05
Multiply by 100 to Express Interest Rate as a Percentage
To express the interest rate as a percentage, we need to multiply the result by 100. So the equation becomes \(\frac{69.3}{R} = t\), where \(R\) is the annual interest rate as a percentage.
06
Prove the Rule of 69.3
We have derived the equation \(\frac{69.3}{R} = t\) from the formula for continuously compounded interest. This means that the time it takes for an investment to double in value when the interest is compounded continuously can be approximated by dividing 69.3 by the annual interest rate (expressed as a percentage). This proves the Rule of 69.3 for continuously compounded interest.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Continuously Compounded Interest
When it comes to understanding the growth of investments, continuously compounded interest is a key concept. Unlike simple or regular compound interest - which compounds on a fixed schedule, like annually or monthly - continuously compounded interest calculates interest on an infinitely small scale. This means that the interest is essentially being added to the principal at every conceivable moment.
Using the mathematical constant 'e', which is approximately equal to 2.71828, we express the formula for continuously compounded interest as \(A = Pe^{rt}\), where A represents the final amount, P is the principal amount, r is the annual interest rate, and t is the time in years.
To visualize, imagine that with each passing second, your investment earns a sliver of interest, which is then immediately added to the 'pot' to earn more interest. This results in an exponential growth pattern, which is what we’re diving into next.
The 'time value of money' is a fundamental financial concept that tells us money available now is worth more than the identical sum in the future due to its potential earning capacity. Essentially, a dollar today is worth more than a dollar tomorrow, because today’s dollar can be invested and earn interest.
In the context of continuously compounded interest, the time value of money is particularly significant. With each moment that passes, your money can be working harder for you, earning an incremental amount of interest on top of the interest it's already accrued. This idea is at the heart of the Rule of 69.3, illustrating how quickly an investment can double in value when the interest compounds continuously.
Exponential Growth
Exponential growth occurs when the rate of growth is proportional to the current quantity, leading to growth at an increasingly rapid rate. This concept is not just found in finance but also in populations, nuclear reactions, and more. In our financial scenario, the continuously compounded interest results in the investment balance increasing exponentially over time.
The Rule of 69.3 leverages this concept to give a quick approximation of how long it will take for an investment to double when interest is compounded continuously. Since the growth is exponential, you don't simply add a fixed amount each year; rather, the amount added grows as the investment itself grows.
Natural logarithms are a specific type of logarithms corresponding to the natural base 'e'. In finance, we use natural logarithms to reverse calculations involving exponential growth. When dealing with continuously compounded interest, natural logarithms allow us to isolate the variable representing time.
In our problem, we used the natural logarithm to transform the equation from \(2 = e^{rt}\) to \(ln(2) = rt\), making it possible to solve for the time 't'. The natural logarithm of 2 is approximately 0.693, which reveals why the Rule of 69.3 is accurate for continuous compounding. This mathematical property is essential for investors to adapt the classic Rule of 72 to fit a scenario of continuous compounding.
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Most popular questions from this chapter
Balloon Payments Audrey Sanborn has just arranged to purchase a \(\$ 450,000\)vacation home in the Bahamas with a 20 percent down payment. The mortgage hasa 7.5 percent stated annual interest rate, compounded monthly, and calls for equal monthly payments overthe next \(\mathbf{3 0}\) years. Her first payment will be due one month fromnow. However, the mortgage has an eight-year balloon payment, meaning that thebalance of the loan must be paid off at the end of year 8 . There were noother transaction costs or finance charges. How much will Audrey's balloonpayment be in eight years?Rule of 72 A useful rule of thumb for the time it takes an investment todouble with discrete compounding is the "Rule of 72." To use the Rule of 72,you simply divide 72 by the interest rate to determine the number of periodsit takes for a value today to double. For example, if the interest rate is 6percent, the Rule of 72 says it will take \(72 / 6=12\) years to double. This isapproximately equal to the actual answer of 11.90 years. The Rule of 72 canalso be applied to determine what interest rate is needed to double money in a specified period. This is auseful approximation for many interest rates and periods. At what rate is theRule of 72 exact?Balloon Payments On September 1, 2007, Susan Chao bought a motorcycle for \(\$25,000\). She paid \(\$ 1,000\) down and financed the balance with a five-yearloan at a stated annual interest rate of 8.4 percent, compounded monthly. Shestarted the monthly payments exactly one month after the purchase (i.e.,October 1, 2007). Two years later, at the end of October 2009, Susan got a newjob and decided to pay off the loan. If the bank charges her a 1 percentprepayment penalty based on the loan balance, how much must she pay the bankon November 1, 2009?Calculating Present Values You just won the TVM Lottery. You will receive \(\$1\) million today plus another 10 annual payments that increase by \(\$ 350,000\)per year. Thus, in one year you receive \(\$ 1.35\) million. In two years, youget \(\$ 1.7\) million, and so on. If the appropriate interest rate is 9percent, what is the present value of your winnings?Calculating EAR with Points You are looking at a one-year loan of \(\$ 10,000\).The interest rate is quoted as 9 percent plus three points. A point on a loanis simply 1 percent (one percentage point) of the loan amount. Quotes similarto this one are very common with home mortgages. The interest rate quotationin this example requires the borrower to pay three points to the lender upfront and repay the loan later with 9 percent interest. What rate would youactually be paying here? What is the EAR for a one-year loan with a quotedinterest rate of 12 percent plus two points? Is your answer affected by theloan amount?
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The Rule of 69 is a simple calculation to estimate the time needed for an investment to double if you know the interest rate and if the interest is compounded. For example, if a real estate investor earns twenty percent on an investment, they divide 69 by the 20 percent return and add 0.35 to the result.
The rule of 72 is only an approximation that is accurate for a range of interest rate (from 6% to 10%). Outside that range the error will vary from 2.4% to 14.0%. It turns out that for every three percentage points away from 8% the value 72 could be adjusted by 1.
The Rule of 72 states that by dividing 72 by the annual interest rate, you can estimate the number of years required for an investment to double. The Rule of 69.3 is a more accurate formula for higher interest rates and is calculated by dividing 69.3 by the interest rate.
Do you know the Rule of 72? It's an easy way to calculate just how long it's going to take for your money to double. Just take the number 72 and divide it by the interest rate you hope to earn. That number gives you the approximate number of years it will take for your investment to double.
The main difference is that Rule of 72 considers simple compounding interest, whereas Rule of 69 considers continuous compounding interest. Additionally, the accuracy of Rule of 72 decreases with higher interest rates. However, you can use Rule of 69 for any interest rate.
It's used to calculate the doubling time or growth rate of investment or business metrics. This helps accountants to predict how long it will take for a value to double. The rule of 69 is simple: divide 69 by the growth rate percentage. It will then tell you how many periods it'll take for the value to double.
For example, the Rule of 72 states that $1 invested at an annual fixed interest rate of 10% would take 7.2 years ((72 ÷ 10) = 7.2) to grow to $2. In reality, a 10% investment will take 7.3 years to double (1.107.3 = 2).
One of those tools is known as the Rule 72. For example, let's say you have saved $50,000 and your 401(k) holdings historically has a rate of return of 8%. 72 divided by 8 equals 9 years until your investment is estimated to double to $100,000.
Get a ₹50 lakh corpus with only ₹10,000 monthly investment. Learn about the 8-4-3 rule of compounding, where investments double within 8, 4, and 3 years, showcasing exponential growth. It emphasizes staying dedicated to investment plans, guarding against inflation, and adapting to market changes.
How can I double $5000 dollars? One way to potentially double $5,000 is by investing it in a 401(k) account, especially if your employer matches your contributions. For example, if you invest $5,000 and your employer offers to fully match at 100%, you could start with a total of $10,000 in your account.
At first blush, the answer is quite simple: you should start saving for retirement as soon as possible. The earlier you start, the more time your money has to grow. In fact, the amount of time you have money invested can be even more important than how much you invest.
Yes, the Rule of 72 can apply to debt, and it can be used to calculate an estimate of how long it would take a debt balance to double if it's not paid down or off.
The Rule of 69.3 leverages this concept to give a quick approximation of how long it will take for an investment to double when interest is compounded continuously. Since the growth is exponential, you don't simply add a fixed amount each year; rather, the amount added grows as the investment itself grows.
The Rule of 69 states that when a quantity grows at a constant annual rate, it will roughly double in size after approximately 69 divided by the growth rate. The Rule of 69 is derived from the mathematical constant e, which is the base of the natural logarithm.
What is the Rule of 69? The Rule of 69 is used to estimate the amount of time it will take for an investment to double, assuming continuously compounded interest. The calculation is to divide 69 by the rate of return for an investment and then add 0.35 to the result.
To calculate the percentage of a number out of the total number, just use the formula number / total number × 100. An increase or decrease in any quantity can be expressed as a percentage. This is referred to as percentage change.
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