Lesson 9: Frequency of CompoundingThis lesson discusses the frequency of compounding and its affect on the present and future values using the compound interest functions presented in Assessors’ Handbook Section 505 (AH 505), Capitalization Formulas and Tables. The lesson: Show
Intra-Year Compounding
Compounding interest more than once a year is called "intra-year compounding". Interest may be compounded on a semi-annual, quarterly, monthly, daily, or even continuous basis. When interest is compounded more than once a year, this affects both future and present-value calculations. With intra-year compounding, the periodic interest rate, instead of being the stated annual rate, becomes the stated annual rate divided by the number of compounding periods per year. The number of periods, instead of being the number of years, becomes the number of compounding periods per year multiplied by the number of years.
Calculating a FW$1 Factor Given Monthly CompoundingIn lesson 2, we calculated the annual FW$1 factor at a stated annual rate of 6% for 4 years with annual compounding. The resulting factor was 1.262477. Now let’s calculate the FW$1 for an annual rate of 6% for 4 years, but with monthly compounding. In this case, the periodic monthly rate is 0.5% (one-half of one percent per month, 6% ÷ 12), and the number of monthly compounding periods is 48 (12 periods/year × 4 years). In order to calculate the FW$1 factor for 4 years at an annual interest rate of 6%, with monthly compounding, use the formula below:
The FW$1 factor with monthly compounding, 1.270489, is slightly greater than the factor with annual compounding, 1.262477. If we had invested $100 at an annual rate of 6% with monthly compounding we would have ended up with $127.05 four years later; with annual compounding we would have ended up with $126.25. AH 505 contains separate sets of compound interest factors for annual and monthly compounding. Factors for annual compounding are on the odd-numbered pages; factors for monthly compounding are on the even-numbered pages.The FW$1 factor for 4 years at an annual interest rate of 6%, with monthly compounding, is in AH 505, page 32 (monthly page). Link to AH 505, page 32 Calculating a PW$1 Factor Given Monthly CompoundingIn lesson 3, we calculated the PW$1 factor at an annual rate of 6% for 4 years with annual compounding. The resulting factor was 0.792094. Let’s calculate the PW$1 factor for 4 years at an annual interest rate of 6%, with monthly compounding. In this case, the periodic monthly rate is 0.5% (one-half of one percent per month, 6% ÷ 12), and the number of monthly compounding periods is 48 (12 periods/year × 4 years). In order to calculate the PW$1 factor for 4 years at an annual interest rate of 6%, with monthly compounding, use the formula below: The PW$1 factor for 4 years at an annual interest rate of 6%, with monthly compounding, can be found in AH 505, page 32. The amount of the factor is 0.787098. Link to AH 505, page 32 GeneralizationsThe following two generalizations can be made with respect to frequency of compounding and future and present values:
Most appraisal problems involve annual payments and require the use of annual factors. Monthly factors are also useful because most mortgage loans are based on monthly payments, and it is often necessary to make mortgage calculations as part of an appraisal problem. For other compounding periods, the factors for which are not included in AH 505, the appraiser can calculate the desired factor from the appropriate compound interest formula. As noted, AH 505 contains factors for annual and monthly compounding only. What is the effective annual rate of 12% compounded semi annually?The effective annual interest rate is 12.36%.
What nominal rate converted quarterly could be used instead of 12% compounded semi annually?Find the nominal rate, which if converted quarterly could be used instead of 12% compounded semi-annually. Explanation: 512.
What is the rate per interest period of 12% compounded quarterly?The correct answer is c) 12.55%.
What is the effective rate equivalent to 12% compounded monthly?Now, let's solve for the effective annual rate for 12% compounded monthly. To do this we simply plug in (1+. 01)12 – 1, which equals 12.68%. Notice how this rate is higher when we have more frequent compounding.
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