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Message: <blockquote data-quote="Airtel" data-source="post: 3119004" data-attributes="member: 124625"><a href="http://www.elakiri.com/forum/newreply.php" target="_blank">Mathematics of interest rates</a>  [<a href="http://en.wikipedia.org/w/index.php?title=Compound_interest&action=edit&section=4" target="_blank">edit</a>] Simplified Calculation Formulae are presented in greater detail at <a href="http://en.wikipedia.org/wiki/Time_value_of_money" target="_blank">time value of money</a>. In the formulae below, i or r are the interest rate, expressed as a true percentage (i.e. 10% = 10/100 = 0.10). FV and PV represent the future and present value of a sum. n represents the number of periods. These are the most basic formulae: <img src="http://upload.wikimedia.org/math/e/0/c/e0ca87a82c591a0e0610792963751fd5.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> The above calculates the future value of FV of an investment's present value of PV accruing at a fixed interest rate of i for n periods. <img src="http://upload.wikimedia.org/math/4/b/5/4b5b30b41d1b085e20fc5a5a3825e936.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> The above calculates what present value of PV would be needed to produce a certain future value of FV if interest of i accrues for n periods. <img src="http://upload.wikimedia.org/math/3/6/b/36b79c738b8ba926854c017a0544c239.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />or<img src="http://upload.wikimedia.org/math/0/1/2/012f25957f86a752650507e4b04fdda1.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> The above two formulae are the same and calculate the compound interest rate achieved if an initial investment of PV returns a value of FV after n accrual periods. <img src="http://upload.wikimedia.org/math/8/f/d/8fd9a90e0a838d6af25128576643ba62.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> The above formula calculates the number of periods required to get FV given the PV and the interest rate i. The log function can be in any base, e.g. natural log (ln)  [<a href="http://en.wikipedia.org/w/index.php?title=Compound_interest&action=edit&section=5" target="_blank">edit</a>] Compound Formula for calculating compound interest: <img src="http://upload.wikimedia.org/math/3/c/6/3c61f664e4b9ae0ea85f89dff6b52548.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> Where, <ul> <li data-xf-list-type="ul">P = principal amount (initial investment)</li> <li data-xf-list-type="ul">r = annual nominal interest rate (as a decimal)</li> <li data-xf-list-type="ul">n = number of times the interest is compounded per year</li> <li data-xf-list-type="ul">t = number of years</li> <li data-xf-list-type="ul">A = amount after time t</li> </ul>Example usage: An amount of $1,500.00 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Find the balance after 6 years. A. Using the formula above, with P = 1500, r = 4.3/100 = 0.043, n = 4, and t = 6: <img src="http://upload.wikimedia.org/math/a/3/d/a3d51ac5f6c04328671922571dc543d2.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> So, the balance after 6 years is approximately $1,938.84.  [<a href="http://en.wikipedia.org/w/index.php?title=Compound_interest&action=edit&section=6" target="_blank">edit</a>] Translating different compounding periods Each time unpaid interest is compounded and added to the principal, the resulting principal is grossed up to equal P(1+i%). A) You are told the interest rate is 8% per year, compounded quarterly. What is the equivalent effective annual rate? The 8% is a nominal rate. It implies an effective quarterly interest rate of 8%/4 = 2%. Start with $100. At the end of one year it will have accumulated to:$100 (1+ .02) (1+ .02) (1+ .02) (1+ .02) = $108.24We know that $100 invested at 8.24% will give you $108.24 at year end. So the equivalent rate is 8.24%. Using a financial calculator or a <a href="http://members.shaw.ca/RetailInvestor/futurevaluetables.pdf" target="_blank">table</a>is simpler still. Using the Future Value of a currency function, input <ul> <li data-xf-list-type="ul">PV = 100</li> <li data-xf-list-type="ul">n = 4</li> <li data-xf-list-type="ul">i = .02</li> <li data-xf-list-type="ul">solve for FV = 108.24</li> </ul>B) You know the equivalent annual interest rate is 4%, but it will be compounded quarterly. You need to find the interest rate that will be applied each quarter.  <img src="http://upload.wikimedia.org/math/0/b/2/0b2d31c8885f746410c62c5106554faf.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />  $100 (1+ .009853) (1+ .009853) (1+ .009853) (1+ .009853) = $104The mathematics to find the 0.9853% is discussed at <a href="http://en.wikipedia.org/wiki/Time_value_of_money" target="_blank">Time value of money</a>, but using a financial calculator or <a href="http://members.shaw.ca/RetailInvestor/futurevaluetables.pdf" target="_blank">table</a> is easier. Input <ul> <li data-xf-list-type="ul">PV = 100</li> <li data-xf-list-type="ul">n = 4</li> <li data-xf-list-type="ul">FV = 104</li> <li data-xf-list-type="ul">solve for interest = 0.9853%</li> </ul>C) You sold your house for a 60% profit. What was the annual return? You owned the house for 4 years, paid $100,000 originally, and sold it for $160,000.$100,000 (1+ .1247) (1+ .1247) (1+ .1247) (1+ .1247) = $160,000Find the 12.47% annual rate the same way as B.) above, using a financial calculator or <a href="http://members.shaw.ca/RetailInvestor/futurevaluetables.pdf" target="_blank">table</a>. Input <ul> <li data-xf-list-type="ul">PV = 100,000</li> <li data-xf-list-type="ul">n = 4</li> <li data-xf-list-type="ul">FV = 160,000</li> <li data-xf-list-type="ul">solve for interest = 12.47%</li> </ul>[<a href="http://en.wikipedia.org/w/index.php?title=Compound_interest&action=edit&section=7" target="_blank">edit</a>] Example question: In January 1970 the <a href="http://en.wikipedia.org/wiki/S%26P_500" target="_blank">S&P 500</a> index stood at 92.06 and in January 2006 the index stood at 1248.29. What has been the annual rate of return achieved? (ignoring dividends).  <img src="http://upload.wikimedia.org/math/9/d/0/9d0f85bedc9191014f8b03f59998332b.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> <img src="http://upload.wikimedia.org/math/4/5/9/45961ffa8fb9c3c07c35166975e220f3.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> <img src="http://upload.wikimedia.org/math/4/2/e/42e7504ea01258341a9271abfdb6d3c7.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />   [<a href="http://en.wikipedia.org/w/index.php?title=Compound_interest&action=edit&section=8" target="_blank">edit</a>] Answer: <img src="http://upload.wikimedia.org/math/f/9/3/f93a1686770c91a12c07afce83839094.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />   [<a href="http://en.wikipedia.org/w/index.php?title=Compound_interest&action=edit&section=9" target="_blank">edit</a>] Doubling The number of time periods it takes for an investment to double in value is <img src="http://upload.wikimedia.org/math/9/a/6/9a67f278e3a45d9793475c572c071784.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> where <img src="http://upload.wikimedia.org/math/3/7/1/3714878e9e07938379ca367c604d2b04.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> is the interest rate as a fraction. Let p be the interest rate as a percentage ( i.e., 100 i ). Then the product of p and the doubling time t is fairly constant:   interest doubling time product   [<a href="http://en.wikipedia.org/w/index.php?title=Compound_interest&action=edit&section=10" target="_blank">edit</a>] Periodic compounding The amount function for compound interest is an exponential function in terms of time. <img src="http://upload.wikimedia.org/math/0/9/8/098237749fc74c8599ba82d9fb3d4b50.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> <ul> <li data-xf-list-type="ul">t = Total time in years</li> </ul> <ul> <li data-xf-list-type="ul">n = Number of compounding periods per year (note that the total number of compounding periods is <img src="http://upload.wikimedia.org/math/2/d/8/2d81da8f4c1baf35e1ffce8a5d2a9970.png" alt="" class="fr-fic fr-dii fr-draggable " style="" />)</li> </ul> <ul> <li data-xf-list-type="ul">r = <a href="http://en.wikipedia.org/wiki/Nominal_interest_rate" target="_blank">Nominal annual interest rate</a> expressed as a decimal. e.g.: 6% = 0.06</li> </ul>As n increases, the rate approaches an upper limit of er. This rate is called continuous compounding, see below. Since the principal A(0) is simply a coefficient, it is often dropped for simplicity, and the resulting <a href="http://en.wikipedia.org/wiki/Accumulation_function" target="_blank">accumulation function</a> is used in <a href="http://en.wikipedia.org/w/index.php?title=Interest_theory&action=edit&redlink=1" target="_blank">interest theory</a> instead. Accumulation functions for <a href="http://en.wikipedia.org/wiki/Simple_interest" target="_blank">simple</a> and compound interest are listed below: <img src="http://upload.wikimedia.org/math/9/8/8/988022ad48425e5532a6ca5ed8350536.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /><img src="http://upload.wikimedia.org/math/7/9/6/79692aac6f7d2bb15b2dee9d9fde32a5.png" alt="" class="fr-fic fr-dii fr-draggable " style="" /> Note: A(t) is the amount function and a(t) is the accumulation function.</blockquote>

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