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| − | === | + | === <math>e</math> in Compound Interest === |
| − | + | The number <math>e</math> is important in finance for calculating compound interest. Compound interest is when interest is calculated on the sum of the principal and previously accumulated interest. This is easiest demonstrated with an example. Suppose we had a principal of $100 at an interest rate of 5%. The interest we would earn the first year would be $100 * 5% = $5 and so the amount owed would become $105. The second year, the $105 would be the new principal and interest would be calculated on that amount, which would be $105 * 5 % = $5.25. This pattern would continue on for as many years as necessary. | |
| − | + | The idea of compound interest was thought about even in the earliest civilizations. For example, ancient Mesopotamians asked in 1700 B.C. how long it would take money to double at a 20 percent interest rate compounded annually (Maor, 1994, pg. 23). However, they did not know the answer to this question. As discussed in the page ''[[page_3|Defining e]]'', Jacob Bernoulli's work in studying compound interest ended up giving a formula for calculating compound interest: | |
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<math>\begin{align} | <math>\begin{align} | ||
| − | + | P' = P\left(1+\frac rn\right)^{nt} | |
\end{align}</math> | \end{align}</math> | ||
| + | where <math>P</math> is initial principle, <math>P'</math> is final principle, <math>r</math> is the interest rate, <math>t</math> is the number of periods, and <math>n</math> is the number of times the interest is compounded. We showed using <math> P = r = 1 </math> and <math> t = n </math> that this formula would converge to <math>e</math> when <math>n \to \infty</math>. | ||
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| − | + | Bernoulli's formula is the basis of financial mathematics, used to calculate loans, mortgages, annuities, etc. The value of <math>n</math> is often manipulated to account for different periods of compounding like semi-annually (2 times), quarterly (4 times), monthly (12 times), and daily (365 times). | |
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| + | When we take <math>n \to \infty</math>, it is called "continuous compounding". After <math>t</math> periods, you can calculate the final principal using this formula: | ||
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<math>\begin{align} | <math>\begin{align} | ||
| − | + | P' = Pe^{rt} | |
\end{align}</math> | \end{align}</math> | ||
| − | + | where <math>P</math> is initial principle, <math>P'</math> is final principle, <math>r</math> is the interest rate, and <math>t</math> is the number of periods. In the equation, <math>e^{rt}</math> is considered the "growth rate". | |
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| − | + | == Rule of 72 == | |
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| + | In finance, people often want to know how long it would take for their investment to double in value. For instance, how many years would it take a $100 investment to become $200 if it was compounded continuously? This question can be easily answered using the formula we stated above for calculating the final principal for continuously compounded investments. We have to solve <math>t</math> for this equation: | ||
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<math>\begin{align} | <math>\begin{align} | ||
| − | + | 2P = Pe^{rt} | |
\end{align}</math> | \end{align}</math> | ||
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| + | If we solve for <math>t</math> and <math>P = 1</math>, we get the following formula: | ||
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<math>\begin{align} | <math>\begin{align} | ||
| − | + | t = \frac {\ln 2}{r} = \frac{.693147}{r} = \frac{69.3147}{100r} ≈ \frac {72}{r} | |
\end{align}</math> | \end{align}</math> | ||
| − | + | Therefore, we can approximate the amount of time it would take our investment to double by using the above formula. So for example, if we had a $100 investment at an interest rate of 5% compounded continuously, we can say it would take about 14 years for the investment to become $200. | |
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| − | + | The "Rule of 72", as it is called, is used frequently to give a quick measure as to how long it would take for your money to double. For a more precise answer, one would use the exact formula. | |
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| − | + | == Force of Interest == | |
| − | + | When there is continuous compounding, the continuous compounding interest is called the force of interest, symbolized by <math>\delta</math>. The force of interest is often a function of time and given by this formula: | |
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<math>\begin{align} | <math>\begin{align} | ||
| − | + | \delta_t = \frac {a'(t)}{a(t)} = \frac {d}{dx} \ln a(t) | |
\end{align}</math> | \end{align}</math> | ||
| − | + | where <math>a(t)</math> is known as the "accumulation function". | |
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| − | + | The accumulation function represents the ratio between the value of the investment at time <math>t</math> and the investment at time 0. Solving for <math>a(t)</math> gives us the following formula: | |
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| − | <math> | + | <math> |
| − | + | a(t) = e^{\int_0^t ẟ\ dx} = e^{t\delta} | |
| − | \ | + | </math> |
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| + | The force of interest is used widely to calculate accumulation functions that can then be applied to other calculations within financial mathematics related to loans, mortgages, annuities, etc. | ||
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| + | <big>References</big><br> | ||
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| + | Interest Accumulation and Time Value of Money. (n.d.). In ''Financial Mathematics for Actuaries''. Retrieved from http://mysmu.edu/faculty/yktse/FMA/S_FMA_1.pdf <br /> | ||
| + | Maor, E. (1994). ''E: The Story of a Number.'' Princeton, NJ: Princeton University Press. Retrieved from http://webwork.utleon.edu.mx/Paginas/Libros/E%20-%20The%20Story%20of%20a%20Number.pdf <br /> | ||
| + | O'Connor, J. J., & Robertson, E. F. (2001, September). The number e. Retrieved from http://www-history.mcs.st-and.ac.uk/HistTopics/e.html <br /> | ||
| + | The Rule of 72. (n.d.). In ''EE204: Business Management for Electrical Engineers and Computer Scientists''. Retrieved from http://web.stanford.edu/class/ee204/TheRuleof72.html <br /> | ||
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| − | + | [[Walther_MA279_Fall2018_topic3|Back to Mysteries of the Number e]] | |
[[Category:MA279Fall2018Walther]] | [[Category:MA279Fall2018Walther]] | ||
Latest revision as of 23:54, 2 December 2018
$ e $ in Compound Interest
The number $ e $ is important in finance for calculating compound interest. Compound interest is when interest is calculated on the sum of the principal and previously accumulated interest. This is easiest demonstrated with an example. Suppose we had a principal of $100 at an interest rate of 5%. The interest we would earn the first year would be $100 * 5% = $5 and so the amount owed would become $105. The second year, the $105 would be the new principal and interest would be calculated on that amount, which would be $105 * 5 % = $5.25. This pattern would continue on for as many years as necessary.
The idea of compound interest was thought about even in the earliest civilizations. For example, ancient Mesopotamians asked in 1700 B.C. how long it would take money to double at a 20 percent interest rate compounded annually (Maor, 1994, pg. 23). However, they did not know the answer to this question. As discussed in the page Defining e, Jacob Bernoulli's work in studying compound interest ended up giving a formula for calculating compound interest:
$ \begin{align} P' = P\left(1+\frac rn\right)^{nt} \end{align} $
where $ P $ is initial principle, $ P' $ is final principle, $ r $ is the interest rate, $ t $ is the number of periods, and $ n $ is the number of times the interest is compounded. We showed using $ P = r = 1 $ and $ t = n $ that this formula would converge to $ e $ when $ n \to \infty $.
Bernoulli's formula is the basis of financial mathematics, used to calculate loans, mortgages, annuities, etc. The value of $ n $ is often manipulated to account for different periods of compounding like semi-annually (2 times), quarterly (4 times), monthly (12 times), and daily (365 times).
When we take $ n \to \infty $, it is called "continuous compounding". After $ t $ periods, you can calculate the final principal using this formula:
$ \begin{align} P' = Pe^{rt} \end{align} $
where $ P $ is initial principle, $ P' $ is final principle, $ r $ is the interest rate, and $ t $ is the number of periods. In the equation, $ e^{rt} $ is considered the "growth rate".
Rule of 72
In finance, people often want to know how long it would take for their investment to double in value. For instance, how many years would it take a $100 investment to become $200 if it was compounded continuously? This question can be easily answered using the formula we stated above for calculating the final principal for continuously compounded investments. We have to solve $ t $ for this equation:
$ \begin{align} 2P = Pe^{rt} \end{align} $
If we solve for $ t $ and $ P = 1 $, we get the following formula:
$ \begin{align} t = \frac {\ln 2}{r} = \frac{.693147}{r} = \frac{69.3147}{100r} ≈ \frac {72}{r} \end{align} $
Therefore, we can approximate the amount of time it would take our investment to double by using the above formula. So for example, if we had a $100 investment at an interest rate of 5% compounded continuously, we can say it would take about 14 years for the investment to become $200.
The "Rule of 72", as it is called, is used frequently to give a quick measure as to how long it would take for your money to double. For a more precise answer, one would use the exact formula.
Force of Interest
When there is continuous compounding, the continuous compounding interest is called the force of interest, symbolized by $ \delta $. The force of interest is often a function of time and given by this formula:
$ \begin{align} \delta_t = \frac {a'(t)}{a(t)} = \frac {d}{dx} \ln a(t) \end{align} $
where $ a(t) $ is known as the "accumulation function".
The accumulation function represents the ratio between the value of the investment at time $ t $ and the investment at time 0. Solving for $ a(t) $ gives us the following formula:
$ a(t) = e^{\int_0^t ẟ\ dx} = e^{t\delta} $
The force of interest is used widely to calculate accumulation functions that can then be applied to other calculations within financial mathematics related to loans, mortgages, annuities, etc.
References
Interest Accumulation and Time Value of Money. (n.d.). In Financial Mathematics for Actuaries. Retrieved from http://mysmu.edu/faculty/yktse/FMA/S_FMA_1.pdf
Maor, E. (1994). E: The Story of a Number. Princeton, NJ: Princeton University Press. Retrieved from http://webwork.utleon.edu.mx/Paginas/Libros/E%20-%20The%20Story%20of%20a%20Number.pdf
O'Connor, J. J., & Robertson, E. F. (2001, September). The number e. Retrieved from http://www-history.mcs.st-and.ac.uk/HistTopics/e.html
The Rule of 72. (n.d.). In EE204: Business Management for Electrical Engineers and Computer Scientists. Retrieved from http://web.stanford.edu/class/ee204/TheRuleof72.html
