Compound Interest - Revision history

Khanh at 22:03, 24 October 2021

2021-10-24T22:03:06Z

Khanh at 16:24, 11 October 2021

2021-10-11T16:24:31Z

Johnraymond.yanez: Added Links to materials

2020-08-17T14:29:11Z

Added Links to materials

New page

* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Simple_and_Compound_Interest/MAT1053_M7.1Simple_and_Compound_Interest.pdf Simple and Compound Interest], Book Chapter
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Simple_and_Compound_Interest/MAT1053_M7.1Simple_and_Compound_InterestGN.pdf Guided Notes]

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-Compound interest includes interest earned on the interest that was previously accumulated.
+'''Compound interest''' includes interest earned on the interest that was previously accumulated.
 Compare, for example, a bond paying 6 percent semiannually (that is, coupons of 3 percent twice a year) with a certificate of deposit (GIC) that pays 6 percent interest once a year. The total interest payment is $6 per $100 par value in both cases, but the holder of the semiannual bond receives half the $6 per year after only 6 months (time preference), and so has the opportunity to reinvest the first $3 coupon payment after the first 6 months, and earn additional interest.
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 * [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Simple_and_Compound_Interest/MAT1053_M7.1Simple_and_Compound_Interest.pdf Simple and Compound Interest], Book Chapter
 * [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Simple_and_Compound_Interest/MAT1053_M7.1Simple_and_Compound_InterestGN.pdf Guided Notes]

← Older revision		Revision as of 16:24, 11 October 2021
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		+	Compound interest includes interest earned on the interest that was previously accumulated.
		+
		+	Compare, for example, a bond paying 6 percent semiannually (that is, coupons of 3 percent twice a year) with a certificate of deposit (GIC) that pays 6 percent interest once a year. The total interest payment is $6 per $100 par value in both cases, but the holder of the semiannual bond receives half the $6 per year after only 6 months (time preference), and so has the opportunity to reinvest the first $3 coupon payment after the first 6 months, and earn additional interest.
		+
		+	For example, suppose an investor buys $10,000 par value of a US dollar bond, which pays coupons twice a year, and that the bond's simple annual coupon rate is 6 percent per year. This means that every 6 months, the issuer pays the holder of the bond a coupon of 3 dollars per 100 dollars par value. At the end of 6 months, the issuer pays the holder:
		+
		+	:<math>\frac {r \cdot B \cdot m}{n} = \frac {6\% \times \$10\,000 \times 1}{2} = \$300</math>
		+
		+	Assuming the market price of the bond is 100, so it is trading at par value, suppose further that the holder immediately reinvests the coupon by spending it on another $300 par value of the bond. In total, the investor therefore now holds:
		+
		+	:<math>\$10\,000 + \$300 = \left(1 + \frac{r}{n}\right) \cdot B = \left(1 + \frac{6\%}{2}\right) \times \$10\,000</math>
		+
		+	and so earns a coupon at the end of the next 6 months of:
		+
		+	:<math>\begin{align}\frac {r \cdot B \cdot m}{n}
		+	&= \frac {6\% \times \left(\$10\,000 + \$300\right)}{2}\\
		+	&= \frac {6\% \times \left(1 + \frac{6\%}{2}\right) \times \$10\,000}{2}\\
		+	&=\$309\end{align}</math>
		+
		+	Assuming the bond remains priced at par, the investor accumulates at the end of a full 12 months a total value of:
		+
		+	:<math>\begin{align}\$10,000 + \$300 + \$309
		+	&= \$10\,000 + \frac {6\% \times \$10,000}{2} + \frac {6\% \times \left( 1 + \frac {6\%}{2}\right) \times \$10\,000}{2}\\
		+	&= \$10\,000 \times \left(1 + \frac{6\%}{2}\right)^2\end{align}</math>
		+
		+	and the investor earned in total:
		+
		+	:<math>\begin{align}\$10\,000 \times \left(1 + \frac {6\%}{2}\right)^2 - \$10\,000\\
		+	= \$10\,000 \times \left( \left( 1 + \frac {6\%}{2}\right)^2 - 1\right)\end{align}</math>
		+
		+	The formula for the '''annual equivalent compound interest rate''' is:
		+
		+	:<math>\left(1 + \frac{r}{n}\right)^n - 1</math>
		+
		+	where
		+
		+	:r is the simple annual rate of interest
		+	:n is the frequency of applying interest
		+
		+	For example, in the case of a 6% simple annual rate, the annual equivalent compound rate is:
		+
		+	:<math>\left(1 + \frac{6\%}{2}\right)^2 - 1 = 1.03^2 - 1 = 6.09\%</math>
		+
		+	==Resources==
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Simple_and_Compound_Interest/MAT1053_M7.1Simple_and_Compound_Interest.pdf Simple and Compound Interest], Book Chapter		* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Simple_and_Compound_Interest/MAT1053_M7.1Simple_and_Compound_Interest.pdf Simple and Compound Interest], Book Chapter
	* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Simple_and_Compound_Interest/MAT1053_M7.1Simple_and_Compound_InterestGN.pdf Guided Notes]		* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Simple_and_Compound_Interest/MAT1053_M7.1Simple_and_Compound_InterestGN.pdf Guided Notes]