How many years will it take an investment to triple in value if the interest rate is 3% compounded continuously?

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Sharon S.

Algebra

6 months, 1 week ago

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Video Transcript

this is a continuously compounded. This is compounding every second of every hour of every day. What kind of formula is this? Well this is my my output is going to equal my um my original starting value. E. To the art of the tea. Okay so I want an investment to triple so my output will be three times my original My interest rate is going to be 3%. Remember whenever I have interest rate I always convert to a decimal 0.03 and I'm looking for the time. First thing I'm gonna solve this equation I could divide by P to both sides Which will eliminate piece. I just have three equals to 0.03 times t. Okay how do I get rid of the exponential function? But what's the inverse of the exponential function? The logarithmic function Specifically log base E. Which is the same thing as the natural log. So I could take the natural log of both sides. Melanie will cancel And then I'm gonna divide by 0.03 to get T by itself. Since it's multiplying Ln of three All divided by .03. And my t. value will be about 36.62 years. Okay assuming it's in years Round to nearest 10th of the year. Is that to round up at 6? It does not. So it will take 36.6 years investment investment to triple

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In this section we cover compound interest and continuously compounded interest.

Use the compound interest formula to solve the following.

Example: If a $500 certificate of deposit earns 4 1/4% compounded monthly then how much will be accumulated at the end of a 3 year period?

Answer: At the end of 3 years the amount is $576.86.

Example: A certain investment earns 8 3/4% compounded quarterly.  If $10,000 is invested for 5 years, how much will be in the account at the end of that time period?

Answer: At the end of 5 years the account have $15,415.42 in it.

The basic idea is to first determine the given information then substitute the appropriate values into the formula and evaluate.  To avoid round-off error, use the calculator and round-off only once as the last step.

• Annual  n = 1
• Semiannual n = 2
• Quarterly n = 4
• Monthly n = 12
• Daily n = 365

One important application is to determine the doubling time.  How long does it take for the principal in an account earning compound interest to double?

Example: How long does it take to double $1000 at an annual interest rate of 6.35% compounded monthly?

Answer:  The account will double in approximately 10.9 years.

The key step in this process is to apply the common logarithm to both sides so that we can apply the power rule and solve for time t.  Use the calculator in the last step and round-off only once.

Example: How long will it take $30,000 to accumulate to $110,000 in a trust that earns a 10% return compounded semiannually?

Answer: Approximately 13.3 years.

Example: How long will it take our money to triple in a bank account with an annual interest rate of 8.45% compounded annually?

Answer: Approximately 13.5 years to triple.

Make a note that doubling or tripling time is independent of the principal. In the previous problem, notice that the principal was not given and that the variable P cancelled.

Use the continuously compounding interest formula to solve the following.

Example: If a $500 certificate of deposit earns 4 1/4% annual interest compounded continuously then how much will be accumulated at the end of a 3 year period?

Answer: the amount at the end of 3 years will be $576.99.

Example: A certain investment earns 8 3/4% compounded continuously.  If $10,000 dollars is invested, how much will be in the account after 5 years?

Answer: The amount at the end of five years will be $15,488.30.

The previous two examples are the same examples that we started this chapter with.  This allows us to compare the accumulated amounts to that of regular compound interest.

As we can see, continuous compounding is better, but not by much.  Instead of buying a new car for say $20,000, let us invest in the future of our family.  If we invest the $20,000 at 6% annual interest compounded continuously for say, two generations or 100 years, then how much will our family have accumulated in that time?

The answer is over 8 million dollars. One can only wonder actually how much that would be worth in a century.

Given continuously compounding interest, we are often asked to find the doubling time.  Instead of taking the common log of both sides it will be easier take the natural log of both sides, otherwise the steps are the same.

Example: How long does it take to double $1000 at an annual interest rate of 6.35% compounded continuously?

Answer: The account will double in about 10.9 years.

The key step in this process is to apply the natural logarithm to both sides so that we can apply the power rule and solve for t.  Use the calculator in the last step and round-off only once.

Example: How long will it take $30,000 to accumulate to $110,000 in a trust that earns a 10% annual return compounded continuously?

Answer: Approximately 13 years.

Example: How long will it take our money to triple in a bank account with an annual interest rate of 8.45% compounded continuously?

Answer: Approximately 13 years.

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