The Compound Interest and Mortgages - Report Example

One of the most important things in our society is the currency system that creates the money people use to buy things. Money allows people in a civilized society to purchase the things they need and desire. Since is so important it is wise to find a way to make money work for you in order to accumulate wealth. There is a formula that can be used by individuals and businesses to create money. This formula is called compound interest. This article discusses the compound interest formula and its applications. There are three main types of interests: simple, compound and continuous interest. Simple interest is calculated by multiplying the amount of money by the interest rate by the amount of time. Compound interest is without a doubt the most important thing in mathematical finance. The formula for compound interest is A = P(1 + r)^n where P is the initial amount of money or principal, A is after n time periods, and r will be the interest rate per time period. The formula works because, after the first time period the amount A becomes A = P + Pr = P(1 + r). At the end of the second time period the amount is A = P(1 + r) + P(1+r)r = P(1+r)^2. And so on. So A = P(1+r)n for all positive integers n is the general formula. Continuous interest is special type of compound interest in which the time variable of the formula is infinity. In order to calculate continuous interest a person must use the following formula, A=Pe^(rt). The P, r, t variables have the same meaning as in the compound interest formula. The e in the formula is not a variable, it is actually a special mathematical character called natural logarithm. In numerical terms the e equals approximately 2.7183 (Selu). A useful for method for mental computations involving compound interest is called the Rule of 72. It states that the amount of time it takes for an investment to double in value at a compound interest rate of R percent compounded annually is approximately 72/R years. For example, it will take approximately 72/5 = 14.4 years for money invested at 5 percent to double. The rule can also be used to determine approximately what interest rate is needed to have an investment double in a given number of years. For example, if one wants to know what interest rate is needed to double the principal in 10 years, divide 72 by 10 to obtain the rate of 7.2 percent. Rule: R = 72/T . The justification for the Rule of 72 is that if in the formula A = P(1 + r)^n we let A be 2 and P be 1, then n is the number of years for doubling at rate r per year. 2 = (1 + r)^n. As everyone knows, as the positive real number h gets close to 0, the value of (1+h)^(1/h) gets arbitrarily close to e from below. From 2 = ((1 + r)^(1/r)^nr) we get that 2 is approximately equal to e^(nr), with e^(nr) being an overestimate. Do ln both sides. Use 1 = ln e. Get .6931 = nr, n .6931/r, n = 69.31/R for R interest rate r in percent. But this is an underestimate for n. Replace 69.31 by 72 to get a better estimate and to use an integer with many integer divisors to facilitate mental arithmetic by the human brain working quickly on one’s feet. In order to better understand the power of compound interest and how a person can harness this power to create wealth over time there are five examples provided below with real applications. Example 1: If the compounding period is one year and the money is deposited for four years and P = $516 and r = .06, then n = 4 and the formula gives A = $651.44. If instead you go ahead and then make the compounding period to be one month in this example, then r = .005, n = 48 and A = $655.57. Example 2: $5000 is deposited in a savings account at a large bank in a small town near Chicago. The population of the town is 6500. The account pays 4% interest per year compounded monthly. The money is left undisturbed in this account for ten years. During this time the value of the U.S. dollar rises 8% against the Japanese yen. After ten years by how much, in dollars, has the account increased in value? Solution: Clearly P is 5000. It is absolutely trivial to see that r = .04/12 = .0033. So to find n you just multiply 12 by 10 and I get 120 which we know to be n and she can use this value of n in the formula. So A = 5000(1+.00333)^120 = 7451.19. So the amount is $7451.19. So the increase after 10 years is $2451.19. Example 3: If in 1790 Benjamin Franklin had invested $100 at 10% percent interest compounded annually and the amount accumulated undisturbed, then it would be worth a enormous amount of money today. Mr. Franklin would have today after 218 years earning interest on his $100 investment $105,585,763,451.61. Example 4: Estimate the amount of money that $1000 would grow to at 8 percent per year, compounded annually, for 90 years. By the rule of 72 we see that the money doubles every 9 years. So in 90 years $1000 grows to approximately $1,024,000. Example 5: The most typical case of the use of compound interest in the business world is in the banking industry. The payment people make on bank loans to purchase homes utilize compound interest. The formulas for calculating monthly payments on a mortgage are more complicated because there are more payments. The banks make their wealth by putting compound to work for them. In the case of the consumer compound interest works against them in their payments of mortgages. The actual formula might be complicated to calculate, but fortunately you can find out about this on the internet. The website mortgagecalculator.org is a good mortgage calculator. To use this tool you must put in the amount of the loan, the rate per month, and the length of the loan. To see the effect of compound interest lets compare a typical mortgage with a same size loan based on simple interest. A 30 year mortgage at 5% on a $100,000 loan has a payment of $536.82. On the other hand if mortgages did not use compound interest and instead used a simple interest formula the home owner would only have to pay $291.66. The different in the two amounts constitute interest earning for the bank derived from the use of compound interest in the mortgage formula. Compound interest is an amazing formula that can create incredible amounts of wealth for any person that has utilizes the concept to their advantage. One of the most important things to remember about compound interest is that time is critical element that in reality creates the money for a person. The more time that passes by the greater the accumulation of wealth. The Benjamin Franklin example clearly showed how a messy $100 turned into a multibillion dollar fortune after 210. Unfortunately Franklin after 218 was not around in 2009 to reap the benefits of his investment. It is also very important to remember that for compound interest to work the money invest can not be taken out of the bank or investment fund so that the compounding effect can manifest itself period after period. Even though typically the period variables is measures in terms of years, the compounding period can be shorter terms such as trimesters or months. It is important for people to know the basics of compound interest in order to utilize its power for important goals such as retirement plans. If person starts saving and investing early in life, this individual can achieve higher amounts of wealth with lower economic resources in comparison with a person that starts savings after they reach middle age. Compound interest is a great tool that can be use by anyone to create wealth and as result achieve financial independence. Works Cited Page Mortgagecalculator.org. 2009. “Mortgage Calculator.” 30 January 2009. < http://www.mortgagecalculator.org/> Selu.edu. “Continuous Compound Interest Formula.” 29 January 2009. Read More