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Interest: money for rent "If you believe in things you don't understand, YOU SUFFER." Stevie Wonder $ in the U.S.A. In 1690, the Massachusetts Bay Colony issued the first paper money in the colonies. In 1796, The Mint, who once considered producing doughnut-shaped coins, settled on the a solid $2.50 gold piece that was to become the standard U.S. coin until the middle 1800’s. But, by the end of the Civil War, nearly 40 % of all U.S. paper currency in circulation was counterfeit. As a result, in 1865, the Secret Service was created under the U.S. Treasury Department and in less than a decade, counterfeiting was sharply reduced. During that decade, the U.S. Treasury created the two-cent coin, the significance here is it was that coin that was the first to bear the motto "In God We Trust". Since 1877, all U.S. currency has been printed by the Bureau of Engraving and Printing, which began as a six person operation in a dark basement of the Department of Treasury. Now, 2,300 Bureau employees work in a twenty-five acre floor space spanning two buildings, where 38 million notes are printed each work day with a face value of approximately $541 million and that comes to eight billion U.S. notes each year, enough to wrap around the equator 30 times. These notes are tough, you could double-fold a U.S. currency note about 4,000 times before it would tear. Almost half of these notes printed are $1 notes. Source: US Federal Reserve and the United States Mint. Similarly for coins, over half of the coins produced in the U.S. are pennies. The composition of the one cent penny has undergone change upon change in the last 211 years. It was pure copper from 1793 to 1837. Then it was bronze, then copper and nickel, then bronze again, and today it is 97 ½ % copper, 2 ½ % zinc. The actual number of coins produced yearly in this country is roughly, by denomination, 10 billion pennies, 1.3 billion nickels, 2.3 billion dimes, nearly 2 billion quarters, and 30 million half-dollars. Who’s on this money, aside from an assortment of presidents? Four women have been portrayed on US coins, but Martha Washington is the only woman whose portrait ever appeared on U.S. paper money. While no portraits of African Americans have ever appeared on paper money, commemorative coins were issued bearing the images of George Washington Carver, Booker T. Washington and Jackie Robinson. Paper money did bear the signatures of four African American men who served as Registers of the Treasury and one African American woman who served as Treasurer of the United States. And it is Abraham Lincoln on the penny as the only U.S. president to face to the right, all the other portraits of presidents on U.S. circulating coins face to the left. The reason is less grandiose that one would think. President Lincoln is facing to the right because his likeness on the penny is simply an adaptation of a plaque bearing his resemblance. Source: http://www.factmonster.com/ipka/A0774850.html Why do we use the symbol of an “S” with a vertical line slashed through it? Many people wonder where the symbol of the dollar sign evolved from, and the answer, though mildly controversial, is the peso. Though adopted by the United States dollar in 1785, the symbol “$” was widely used before that year. From dusting off old manuscripts, one can actually see the Mexican or Spanish "P's" for pesos replaced by an “S” written over the “P”. This "S" gradually evolved as it was written over the "P" into the dollar sign symbol “$” we are all now familiar with. How important is finance in the media? In the 60 days following President Clinton's State of the Union address, the New York Times ran 27 stories regaling the President's campaign finance misfortunes and 3 stories regarding education reform, the showpiece of his acceptance speech. How important is money in this culture we live? They say the most common gift in the U.S.A. for a newborn is the savings bond. If you think this through, the choice of a savings bond is the perfect gift for the baby, one size fits all, and it grows with the child. When the child becomes school age, if the dog eats their homework, they are out of luck, but if the dog eats their savings bond, it can be replaced. (And by the way, if you had done so in July of 1976, you could have purchased as $ 25 Savings Bond for $ 17.76 instead of the usual price of $ 18.75 thanks to the bank’s celebration of the Bicentennial.) Finally, how important is money on this planet? The number of wars fought between countries that each had at least one McDonald's franchise is zero. Effective rates, Simple Interest, Compound Interest All of us see advertisements everyday where a company wants you to purchase their product with credit. They want to give you merchandise even though at this time you simply do not have all the cash. Financing is available. Interest or ’rent on money’ can be understood and calculated by incorporating some fundamental ideas of mathematics. One of the first observations we need to make is the connection between interest theory and the power of mathematics to succinctly capture numerical patterns. When studying interest we will see rigorously structured patterns and then use the familiar linear and exponential growth models to continue our journey in recognizing mathematics embedded through out our lives. Most often we think of interest in the form of a financial transaction but this is not always the case. Interest can be in the form of goods or services as well. For example, an artist or craftsman may produce a piece of work and give it to the owner of a studio or workshop in lieu of actual cash payments for shop privileges. Our presentation of the mathematics involved with the theory of interest will focus specifically on monies paid as rent for money borrowed. While turning the pages of any news paper you will see advertisements from businesses that either want to loan you money, ‘Consolidate all your debt into one low monthly payment: interest only 3.125%/3.32% APR - 30 year fixed 5.375%/5.563% APR’ or want you to invest your money with them ’11.18% Investment opportunity, 3 months 8.32% 6 months 10.51% - 12 months 11.18% minimum investment of $1000 and all rates are annual yield’ What is APR? and why the fine print? Lets start our analysis with two interest paying concepts. The first is called an effective rate of interest and the second in called a nominal rate of interest. The effective rate of interest is the actual rate at which interest is paid on principal, P (the money originally invested). We can observe the effective rate using the ratio I/P. Interest payments by themselves are calculated by taking the product of principal the interest rate and the amount of time, I Pr t . The idea of simple interest follows a linear growth pattern. That is, the original principal is enhanced with interest payments on regular intervals of time and the same amount of interest is added to the principal following each interest period. Algebraically, we see this new quantity as principal plus interest, P I . Under the simple interest scenario we add the same fixed amount of interest to an account at the end of each interest period. The value of the accounts grow according to the following pattern: P I , P 2 I , P 3I , P 4 I and so on until reaching the end of the n th interest period and the final account balance will be P nI . Lets build our first numerical example on this topic. Problem One Under the rules of simple interest how much will an initial principal of $100.00 grow over a period of 1, 2, 3, 4, 5, and 10 years at 3% simple interest paid annually? We have P 100 , r .03 and t 1, 2, 3, 4, 5 and 10 . The amount of interest paid at the end of each successive year is I Pr t 100.00 .03 1 3.00 . The account value at the end of each year increases by the same fixed amount; 100.00 1 3.00 103.00 , 100.00 2 3.00 106.00 , 100.00 3 3.00 109.00 , 100.00 4 3.00 112.00 , 100.00 5 3.00 115.00 and 100.00 10 3.00 130.00 . We need to notice two very important things here. First, the structure of each equation is that of a linear model, y mx b . Our original principal is our starting value identified with b , the amount of interest paid each year is our constant rate of change m , time in increments of years occupy the independent variable x and finally the end of the year value of the account is represented by the dependent variable y . Second, we need to examine the effective rate over time. Our technique of identifying percent changes is of new old value here. Recall, the percent change formula structure is . How much old interest with respect to the amount of money found in our account was actually paid at the end of each year in our first example? 103 100 3 106 103 3 .03 or 3% the first year, .02913 or 2.913% the 100 100 103 103 109 106 3 112 109 3 .0283 or 2.83% the third year, .02752 second year, 106 106 109 109 115 112 3 .02679 or 2.679% the fifth year and or 2.752% the fourth year, 100 112 130 127 3 .02362 or 2.362% the tenth year. The effective rate of interest being 127 127 paid is decreasing. The amount of interest being paid remained constant while the amount of money in our account grew creating the situation where we have a decreasing effective rate. Simple interest provides for a decreasing effective rate. In ordinary English, we may say since we are accumulating 3 dollars interest per year, the 3 dollars when compared to $ 100, $ 103, $ 106 and so on is a smaller percentage of each amount found in the account each year. Does this seem fair to you? The bank, which may house the money seems to be getting more money each year, but the interest, though constant, in turn is a smaller percentage of that amount as time progresses. Our effective rate ratio I/P shrinks each time interest is paid. What needs to change in this scenario so that the effective rate does not decrease but remains constant? If you are thinking that interest needs to be paid on previous interest payments into the account you are correct. Situations where interest is paid on previous interest added to the account fall in the area of compound interest. Interest paid on interest earned is the compounding of interest. In our next example we will use the same numbers for our original principal and interest but will employ the use of an exponential growth model to complete the calculations. Problem Two Again, starting with the basic concept of interest we can develop an algebraic expression to use when calculating the future value of our account. The exponential model naturally presents itself when we start to analyze this situation. At the completion of the first 1 interest period our account balance has grown from P to P Pr t P1 rt . Because the time intervals remain constant t 1 year we do not need to keep track of the variable t as we let t 1 and our expression becomes P1 r . We will need to keep count of the number of times interest is paid and we will use the variable n for the number of compounding periods. At the end of the second interest period our balance becomes 1 P1 r P1 r r P1 r 1 r P1 r . The expression used to find the account balance at the end of the third interest period is 2 2 2 1 3 P1 r P1 r r P1 r 1 r P1 r . Is the pattern obvious? The n th 1 1 1 1 2 interest payment will produce a balance of P 1 r dollars. The structure of this formula is identical to our exponential growth formula a b x . The original principal P is our starting value a and 1 r is our growth rate b . Incorporating the same numerical values as used in the previous example the year to year account values under the 1 2 compound interest scenario are 1001 .03 1 103.00 , 1001 .03 1 106.09 , n 1001 .03 1 109.27 , 1001 .03 1 112.55 , 1001 .03 1 115.92 , and 3 4 5 1001 .03 1 134.39 . Examining the effective rate we see the effective rate remains new old 103 100 3 106.09 103 3.09 .03 , second year .03 , constant. old 100 100 103 103 109.27 106.09 3.18 .02997 , fourth year third year 106.09 106.09 112.55 109.27 3.28 115.92 112.55 3.37 .03001 , fifth year .02994 and 109.27 109.27 112.55 112.55 134.39 130.48 3.91 .02996 . The effective the effective rte for the tenth year is 130.48 130.48 rate of interest paid remains constant through out the life of this account. Why, in this example does the approximate effective rate bounce around the .03 value? 10 Compound interest provides for a constant effective rate and recall, simple interest provided for a decreasing effective rate. With compound interest, not only is the money found in the bank increasing, the amount of interest paid is increasing and the interest accumulated seems to be a constant percentage of the current account value. This is strikingly different from simple interest, where the interest is constant for each successive year, but it is a smaller percentage of the amount found in the account. So, if you are investing money and the effective rate is simply a measure of the ratio of interest to the money found in the bank, which investing method would you prefer, simple or compound? In general, we have two distinct methods of accumulating interest. Either by the simple interest process or the compound interest process and there are some occurrences where both scenarios are combined to analyze the growth of interest. Simple interest incorporates a linear growth model and has the characteristic of producing a decreasing new old P nI P (n 1) I P nI P nI I I effective rate, . old P (n 1) I P (n 1) I P (n 1) I As we can see, the percent change formula when applied to the simple interest situation simplifies to a ratio where the denominator will continue to numerically grow with each passing interest payment period. The growth of the denominator and the existence of no change in the numerator of this ratio is what mathematically forces a decreasing effective rate. Now, when we look at the effective rate of interest produced by the compound interest model we see there is a constant effective rate of interest n n 1 n 1 new old P 1 r P 1 r P 1 r 1 r 1 r . Each compounding n 1 n 1 old P 1 r P 1 r period has the same interest rate applied to the value of the account. The account grows proportionally at the end of each compounding period. Example Three We need a future value of $ 5000 in three years, and can earn 6 ½ % compound interest annually, what must be our present value (or original principle). Solution We know, Future Value Principal 1 i so, Principal n Future Value 1 i n . 5000 3, 649.40 dollars. We have earned $ 5000 – $ (1 0.065)5 3649.40 = $1,350.60 as rent on our original principal. Principle or Present Value = Problem Four Suppose you invest $ 5,000 at 6 ½ % simple interest for 10 years. a) How much is found in the account after 1, 2, 3, 9 and 10 years? b) Find the effective rate of interest for year 1, 2, 3 and 10. c) How much interest is found in the account after 10 years? Solution: a) This investment is based on simple interest, and this formula has a linear structure of A P nI , where I Pr t . Let P = 5000, r = 0.065 and t = 1, then I 5000(0.065)1 325 . At the end of year 1, we have 5000 + (1)325 = 5325 At the end of year 2, we have 5000 + (2)325 = 5650 At the end of year 3, we have 5000 + (3)325 = 5975 Continuing with this process, we have At the end of year 9, we have 5000 + (9)325 = 7925 At the end of year 10, we have 5000 + (10)325 = 8250 new old . Let’s examine this ratio for years old new old 5325 5000 325/ 5000 0.065 1, 2, 3, 9, and 10. For year 1, old 5000 b) The effective rate is found by looking at new old 5650 5325 325/ 5325 0.061 . Continuing, for year 3, old 5325 new old new old 325/ 5650 0.058 . For year 9, 325/ 5975 0.054 old old And for year 10, the effective rates appear to continue this trend of plummeting fast, and new old 325/ 7925 0.041 c) The amount of interest after 10 years is 10($ we have old 325) = $ 3250 For year 2, Let’s compare this investment based on simple interest with an investment based on compound interest. We will use the same numbers as we did in Examples 3 and 4, so our comparison is easy to visualize. Problem Five Suppose you invest $5,000 at 6 ½ % compound interest for 10 years. a) How much is found in the account after 1, 2, 3, 9 and 10 years? b) Find the effective rate of interest for year 10. c) How much interest is found in the account after 10 years? Solution a) This investment is based on compound interest, so now we have A P(1 r )n , which has the exponential structure “ ab x ”. Let P = 5000, r = 0.065 and n = 1, 2, 3, 9, and 10. End of year 1, we have A 5000(1 0.065)1 5325 which matches identically with the simple interest calculation. End of year 2, we have A 5000(1 0.065)2 = $ 5,671.13 which is more than the $ 5,650 we had in simple interest. End of year 3, we have A 5000(1 0.065)3 = $ 6,039.75 which, again, is more than we had with simple interest, where we had $ 5,975. End of year 9, we have A 5000(1 0.065)9 = $ 8812.85 as compared to the $ 7925 accumulated from the simple interest investment. End of year 10, we have A 5000(1 0.065)10 = $ 9385.69 as compared to the $ 8250 accumulated from the simple interest investment. new old b) Again, the effective rate is found by looking at . Let’s examine this ratio old for year 10. 10 9 9 1 50001.065 50001.065 50001.065 1.065 1 new old 0.065 9 9 old 50001.065 50001.065 which is 6.5 % so it appears the effective rate stays constant, as we would predict. c) The interest found in the account after 10 years is $9385.69 - $5,000 = $4,385.69 which is considerably more than the interest of $3,250 that was derived from a simple interest bearing account for the same number of years, with the same amount of principle. And this was an investment that was compounded annually, once a year. Problem Six The graph below shows two different accounts each with an investment of $100 for 5 years. The horizontal axis represents time measured in years, the vertical axis represents the dollar amount an investment accumulates to after so much time has past. In one account, the $100 is compounded quarterly at 12 % annually, and in the other account, the $100 is compounded quarterly at 9 % annually. Based on this information and the graph below, which statement is the most reasonable? a) If both accounts continue for 20 years, the 9 % graph would eventually get closer to the 12 % graph. b) There is no literal difference in the amount in both accounts until after the first year. c) A small increase in the interest rate makes a substantial difference after 5 years. d) The difference between the two graphs is constant from year to year. a) b) c) d) Solution Let’s explore the reasonableness of each of the choices separately. If both accounts continue for 20 years, the 9 % graph would eventually get closer to the 12 % graph. This does not seem reasonable. Every indiaction is that the difference between the curves should increase as times passes. There is no literal difference in the amount in both accounts until after the first year. This again does not seem reasonable. Since there is a difference between the graphs in the interval 0 < t < 1, and since the money is compounded quarterly, then there should be a difference in the accounts as early as after the first 3 months (1st quarter). A small increase in the interest rate makes a substantial difference after 5 years. This appears quite reasonable. From the graph, there appears to be nearly a $25 difference in the accounts after 5 year, $180 versus $ 155. Since the investment was $100, this is 25 % percent of the original investment, which is substantial. The difference between the two graphs is constant from year to year. Let’s analyze the graph. Approximating from the graph, after 1 year there seems to a $5 difference. After two years, there seems to be a little more than $5 difference, but after 5 years, the difference is nearly $25. It is not reasonable to assume the rate of change is constant. Exercise Set: 1. Suppose you invest $ 40,000 at 5 ½% simple interest for 10 years. a) Find the effective rate of interest for year 1, 2, 3 and 10. b) How much is found in the account after 10 years? c) How much interest is found in the account after 10 years? 2. Suppose you invest $ 5,000 at 6 ½ % compound interest compounded annually for 10 years. a) Find the effective rate of interest for year 1, 2, 3 and 10. b) How much is found in the account after 10 years? c) How much interest is found in the account after 10 years? 3. Suppose you invest a quarter of a million dollars at 12 % simple interest for 10 years. a) Find the effective rate of interest for year 1, 2, 3 and 10. b) How much is found in the account after 10 years? c) After 10 years, how much interest was paid to the account? 4. Suppose you invest quarter of a million dollars at 12 % compound interest compounded annually for 10 years. a) Find the effective rate of interest for year 1, 2, 3 and 10. b) How much is found in the account after 10 years? c) After 10 years, how much interest was paid to the account? 7. $500 invested at simple interest at 10% or $500 at compound interest at 9.9% annually. 8. $500 invested at simple interest at 10% or $510 at simple interest at 9.9%. 9. $500 invested at simple interest at 10% or $490 at compound interest at 9.95% annually. 10. Compare the two investment options, $2000 at 11 ½ % compounded annually or $2250 at 11.25% compounded daily. For which investment is the effective rate larger, after a) 1 year b) 2 years c) 3 years d) 5 years e) 10 years f) 25 years 11. Over a twenty year period how much would you have to invest today to obtain a million dollars in an account a) at 10 ½ % simple interest. b) 10 ½ % annual compound interest. c) at 18 % annual compound interest. 12. The graphs below show the results of investing $100 per month into accounts For problems 5 to 9, for which investment is the effective rate is larger after the 5th year. 5. $500 invested at simple interest at 10% or $ 480 at simple interest at 10%. 6. $5,000 invested at simple interest at 10% or $480 at simple interest at 10%. One account is compounded monthly @ 6.25 % and the other account is compounded quarterly @ 9.25 %. Despite compounding more often, the monthly account lies beneath the account that compounds money quarterly. Based on this information, which statement below is true. a) There is no difference in the return of the two accounts until after the 2nd year. b) The higher the interest rate will always yield a greater return regardless of the compounding periods. c) The longer the money sits in the two accounts, the greater the difference in return for the account that pays 9.25 %. d) There is no significant difference in return in the two accounts after 25 years. Compound Interest A question is knocking loudly on our door. For the investment from Problem Five, would the interest have been greater for this investment if the principle was compounded more frequently than once a year? How about if the interest rate was raised and the compounding periods stayed the same? How about if the compounding periods were more often, but the rate was lowered? Or if the compounding periods were less frequent, but the rate was raised? Let’s begin answering these questions by observing an investment based on compound interest, where the compounding is not done once a year (annually). We will use $100 at 3 % annual compounded daily for a year. The first natural question is do we do we compound the money 3 % each and every day? Well, if this were true, after one year, we would have 100(1 0.03)365 $ 4,848272.45 . Cool huh. Got a $ 100? Run to the bank – go do this… . Oh, wait a minute, let’s think this through a little more carefully, if we compound 3 % every day for 365 days, then that is 365 times you compound the money in a year at 3 %. Loosely, this means the 3 percent is not an annual percentage anymore. Let’s not run to the bank quite yet. We need to increment the 3 percent over the 365 days. Simply put, we apply 1/365th of the 3 percent each day. Our formula will need to account for this incrementing process. This process of dividing the Annual Percentage Rate (APR) by the number of times you compound per year is referred to as the nominal rate. The nominal rate for this problem is 0.03/365 or 0.000082 each day. After one year, we would have 365 0.03 100 1 $103.05 . This makes more sense. Now, go run to the bank. Or walk. 365 The sense of urgency seems to have dissipated greatly. For compound interest, we use the formula, A P 1 i , where P is the principal, ‘i’ is the nominal rate or the interest rate per compounding period, and N is the number of interest conversion periods over the life of the investment. Notice in our last investment, we were not given ‘i’ and N out right, we had to calculate these values. N nt N r So, we augment our compound interest formula to look like A P 1 i P 1 , n where P is the principal, ‘i’ is replaced with ‘r/n’, where r is the annual interest rate and n the number of times compounding occurs each year. Also, observe that N is replaced with the product n t where t is the time, in years and n is the number of interest payment periods. Now, when the interest accruing portion of the formula is raised to the n t power, it is raised to the number of times the account is compounded or paid interest over the life of the investment. Compounding Periods If we compound monthly, n = 12, meaning we will have 12 interest compounding periods per year and if the interest rate is 6 % annual, we find the nominal rate by taking 1/12 th of 6 % or 0.5 % and apply this rate to each compounding period. Continuing this thought process to correctly construct the nominal rates, if we are compounding quarterly, n = 4, we would divide the APR by 4, if we compounded daily, n = 365, we divide the APR by 365. There are some businesses which operate on the terms of a Banker’s year and this requires one to follow the rule of 360 days in each year with each of the 12 months having 30 days. So, under the banker’s rule nominal rates involving daily interest conversion periods requires using a value of n = 360. A Partial Period Not all financial situations begin and end on pristine timeframes suitable for our N compound interest formula P 1 i . What happens when an account is quietly growing at a comfortable rate and then needs to be interrupted at some point in the middle of an interest conversion period? For example, say our account is compounded quarterly and all the money needs to be withdrawn on a day in the middle of the quarter. What is best for the investor? When we mathematically model the growth of monies in a compound N interest accruing account we do so with an exponential model, a b x or P 1 i . If we were to look at a graph representing the growth of money in our account over time we describe the rate of change as increasing at an increasing rate. The fact our account is increasing at an increasing rate tells us the curve representing this growth is concave up. Now if we were to pick any two points along this curve and then connect those points with a line, the line would be above the curve over the entire interval between the points selected. Thus, if we use our compound interest formula to calculate the growth over the first k whole interest conversion periods and then use our linear growth rate model for the last partial period we will have calculated the investment growth in favor of the investor. This can be seen in the graph above. Recall the vertical axis represents money accumulated in the account and simply put the line is above the graph of the exponential model thus, representing more interest growth over this partial period time frame. Our k Future Value of the account is now arrived at using the formula FV P1 i 1 it where t represents the partial increment of time where the last interest conversion period was interrupted. Consider the following scenario. We invest $1,000,000.00 into an account where interest is convertible monthly. Lets assume the APR for the account is 6%. This provides for a nominal rate of .06 / 12 .005 and for each complete interest conversion period the account balance would grow by a factor of 1.005 . Say, the money is left to accrue interest for one year. The account balance at the end of the year would be 12 $1,000,0001.005 $1,061,677.81 . Now suppose the money was left in the account for an additional 21 days. Because the account is set up to pay interest on monthly intervals we will use a combination of both the linear and exponential formulas to determine the account balance. Notice we will use the same nominal rate and we will assume there are 30 days in this month. We have the following 21 12 $1,000,0001.005 1 .005 $1,065,393.68 . Compared to simply adjusting the 30 30 $1,065,390.90 the technique exponent in our P 1 i model; $1,000,0001.005 of including a single linear growth factor pays the investor more money, $2.78 more. Similarly, if an account received interest payments on a convertible quarterly basis and N 12 21 the investment time falls in the middle of a quarter the increment of time used for the linear growth factor may be approximated using the banker’s rule. Your ratio of time under the quarterly scenario should incorporate a denominator of 90 days and the numerator will be the actual number of days. So, 21 days into the quarter our time variable would look like 21 90 . Exercise Set 1. How much would you have to invest today to obtain 25,000 dollars in the bank in 5 ½ years at 6 7/8 % annual compound interest. 2. How much would you have to invest today to get a 1000 dollars in the bank in 8 months at 12% annual compound interest. 3. It is March 3rd , 2004. You want $6,000 on April 4th, 2006. How much would you have to invest on 3/3/04 in an account yielding a) 8 % simple interest. b) 7 ¾ % annual compound interest. 4. It is March 3rd , 2004. You deposit $16,000. How much would you have on April 4th, 2006 if you invested in an account yielding a) 8 % simple interest. b) 7 ¾ % annual compound interest. 5. It is January 1st, 2004. You deposit $1,000. How much would you have on April 4th, 2006 if you invested in an account yielding a) 8 ½ % annual compound interest. b) 5 % annual compound interest. 6. What is the difference between an effective rate for simple interest investments compared to compound interest investments? 7. If you have a choice of investing $10,000 into two accounts. How long would you have to have the money in the bank for it to be worth it if you choose an account that yields 11.10% compounded annually over one that yields 11 ¼ % simple interest. 8. You invest $100 into an account yielding 6 ¾ % simple interest for 2 years 5 months. You then invest the accumulated total into a new account yielding 6 % compounded annually for 1 year 7 months. How much money is in the new account after those 4 years? 9. You invest $100 into an account yielding 5 ½ % simple interest for 5 years 2 months. You then invest only the accumulated interest into a new account yielding 6% compounded annually for 1 year 10 months, you leave the original principal in the other account. How much money do you have total in both accounts after those 7 years? 10. You invest $100 into an account yielding 11 % simple interest for 10 years. You withdraw the money and reinvest it all into an account that yields 12 % simple interest for another 10 years. How much interest did you earn on that original investment of $100? 11. You borrow $7,500 for 2 years 9 months simple interest. You pay back $9,000. Find the interest rate. Annual Yield Let’s compare two investments, one with an annual percentage rate of 7 ¼ % compounded quarterly, the other with an annual percentage rate of 7 1/8 % compounded daily; which would be the better choice if we were to invest? One has a higher rate, the other compounds more frequently. The Annual Yield is the simple interest equivalent to the compounded interest rate. It is sometimes referred to as the Annual Percentage Yield, or APY. The APY allows us to compare different rates with differing compounding periods. It allows the consumer or investor to fairly differentiate between different investments strategies. How does a person decide which credit card to choose or which loan to secure? And if one must compare exactly such as the question we posed initially, where the choice is between two investments, where the investment with the higher rate is also the investment that compounds less frequently, we search for the APY. To find the annual yield, we revert the compound interest to its equivalent simple interest form. This means we simply set a simple interest investment equal to a compound n interest investment, or more precisely, P(1+rt) = P 1 i . Dividing through by P, and renaming rt as Y to stand for annual yield, we have: n 1 + Y = 1 i 4 0.0725 For 7 ¼ % quarterly: 1 + Y = 1 , therefore we have 4 4 0.0725 Yield 1 1 0.0744 . 4 0.07125 For 7 1/8 % daily: 1 + Y = 1 365 365 , therefore we have 365 0.0725 Yield 1 1 0.0738 . 365 So, the 7 ¼ % annually compounded quarterly will grant a higher annual yield, 7.44 % APY as compared to the 7.38 % APY, despite compounding less frequently. For example, a CD (Certificate of Deposit) may provide an annual percentage rate of 3.87 % compounded monthly. You will see the words 3.87 % APR in the verbiage. Next to these words, you will see the words 3.93 % APY, which means 3.93 % is the equivalent interest rate when reverted back to simple interest over 1 year. This information is provided for you because it allows you to compare this CD to another CD with differing terms. Problem One We invest $1000 into an account convertible monthly at 6 percent annual interest. How much money would we have accumulated after 2 years 5 months? What is the nominal interest rate? What is the annual yield? Solution The principle is $1000, the time is 29 months, which is the number of compounding periods, dividing the interest rate by 12 months the nominal rate is 0.06/12 = 0.005 or 0.5 percent per month. So, we have A P1 i nt N r .06 P1 10001 n 12 29 1155.62 dollars. n 12 r .06 The annual yield is found by Y 1 1 1 1 0.062 6.2% . n 12 Problem Two We need $10,000 in 3 years and 3 months. We deposit a sum now into an account that will compound monthly at 6 percent annual. How much should we deposit today to obtain a future value of $ 10,000? What is the annual yield? Solution The Future Value is $10,000 and again, the nominal rate is 0.06/12 = 0.005 or 0.5 percent. We rewrite our formula inputting these values and have N A P1 i nt 39 r .06 P1 $10,000 P1 and now solving for the variable P n 12 39 .06 we have $10,0001 P $8,232.35 . That is, we will need to invest $8,232.35 12 today. The annual yield is exactly the same as calculated in the previous example, why? Y 6.2% . Problem Three Invest $ 3000 at 5 ¼ % annual compounded daily. a) Find the future value after 13 years and then after 14 years. b) Divide 5.25 into 72 and compare this number with the results from part a). c) Determine the Annual Yield. d) Explicitly state what the Nominal Rate means. Solution a) To find the future value after 13 years, we have nt N r A P 1 i P 1 n 0.0525 FV 3000 1 365 (365)(13) $5,936.16 To find the future value after 14 years, we have nt N r A P 1 i P 1 n 0.0525 FV 3000 1 365 (365)(14) $6, 256.11 Notice that somewhere between the 13th year of our investment and the 14th year of our investment, our investment doubled in value. b) 72/5.25 = 13.7. The Rule of 72 tells us if we divide 72 by our annual interest rate, we will know how long it will take to double our investment at 8 % compounded annually. An account paid 9 8% interest compounded annually will double in value at 9 years, 1 .08 2 . If we invest in an account with an interest rate below 8 % annual, it will require more than 9 years to double our investment. If we invest into an account with an interest rate above 8 % annual, we will require less than 9 years to double our investment. In our example we see the account balance doubled sometime between the 13th and 14th year as the rule of 72 predicted. c) The annual yield may be calculated as follows: 0.0525 1 + Y = 1 365 5.4 %. 365 4 0.0525 1 0.05389 or about , therefore we have Yield 1 4 d) The nominal rate is a portion of the annual percentage rate applied when calculating interest to be paid at the end of each interest conversion period. Exercise Set 1. How much money is required to be invested today to obtain $2000 in 9 months if your account pays 5 ½ % compounded quarterly? What is the nominal interest rate? 2. How much money is required to be invested today to obtain $20,000 in 4 years 5 months if your account pays 6 3/8 % compounded monthly? What is the nominal interest rate? 3. How much money will you have in 5 years 2 months if you invest $2000 into an account that pays 12 ½ % compounded quarterly? What is the nominal interest rate? 4. How much money will you have in 20 years if you invest a quarter of a million dollars into an account that pays 11 ½ % compounded monthly? What is the nominal interest rate? 5. How much money will you have in 7 years if you invest $ 100 into an account that pays 4 5/8 % compounded daily? What is the nominal interest rate? 6. Would you have more money if you invested $ 1000 into an account that paid 5 1/3 % compounded monthly for 6 years or if you deposited $ 1000 into an account that paid 5 % compounded daily for 6 years? Use the Annual Yield for each account to determine your answer. 7. Would you have more money if you invested $ 1000 into an account that paid 5 1/3 % compounded monthly for 6 years or if you deposited $ 1000 into an account that paid 5 % compounded daily for 6 years? 8. Would you have more money if you invested $ 1000 into an account that paid 6 % compounded quarterly for 6 years or if you deposited $ 990 into an account that paid 5.9 % compounded monthly for 6 years? 9. Would you have more money if you invested $ 1000 into an account that paid 6 % compounded quarterly for 6 years or if you deposited $ 1100 into an account that paid 5.9 % compounded quarterly for 6 years? 10. Compare the annual yield of two investments, one at 1.49 % compounded daily to 1.54 compounded monthly. Hint: You may want to take the calculation to the thousandths place to accurately make the comparison. 11. Compare the annual yield of two investments, one at 2 % compounded quarterly to 1.99 % compounded monthly. 12. Compare the annual yield of two investments, one at 1.64 % compounded daily to 1.7 compounded monthly. Annuity An annuity is created when you make equal periodic payments into an account that will compound the money deposited. Any type of investment like this is called an annuity. You are probably thinking this seems like an IRA or how a retirement plan works. You are correct. These are types of annuities. In reality, this type of plan is an investment strategy most of us should employ. Most of us don’t have a large lump sum of money to invest in a savings account and then have the comfort level to leave it there for years. But, most of us do have a small amount of money, say $100, that we may invest periodically, say monthly, into an account that compounds the money at some annual rate, say 3 % , for a period of time, say 10 years. This is an annuity. To visualize what actually happens in terms of growth to the money, imagine the first month where you place $100 into an account. For the second month, one-twelfth of 3 % interest is added into your account, which reflects the nominal rate applied to the first $100 and then you have added another $100 into the account. For the third month, we apply one- twelfth of 3 % interest to the balance of what was in the account already and then we deposit (or add) another $100 into the account. Let’s stop right here. If we do this for 10 years, we have a long sum, with one-twelfth of 3 % applied to our account balance over and over again. We will come back to this elongated sum in a moment. We need to briefly clarify the nature of sums, first. Why? Because if we clarify the nature of this type of sum we can take advantage of this repetitive pattern and arrive at a formula for the sum that will not have all of the middle terms, in this case, the 120 terms representing the amount of money for each month of this investment. Sums There are two fundamental types of sums we will look at, Arithmetic and Geometric. Arithmetic sums are found where you add many numbers together that differ by the same amount, like 1+2+3+ … + 98+99+100. This is an arithmetic sum or more formally known as an ‘arithmetic series’ because each term in this sum, 1, 2, 3 and so on differs by 1. Now, we could shorten this addition by realizing we could add the first term to the last term, the second term to the second to last term, 1 + 100 and 2 + 99 are both 101, and so on. In fact, there are 50 pairs of 101 that comprise this sum, and 50(101) = 5,050. The sum (series) adds up to 5050. We can find the sum of the first n counting numbers using the following formula we will derive here. Let S be the value of the sum for the first n integers, S 1 2 3 n 1 n . And if we mimic the pattern described above by adding S to itself we will have reduced the solution to a small formula. We have: S 1 2 3 n 1 n + S n n 1 3 2 1 2S n 1 n 1 n 1 n 1 n 1 2S nn 1 S nn 1 . 2 Checking our formula for the sum of the first 100 integers we have, 100100 1 100 101 S 50 101 5050. 2 2 The sum found in our annuity has terms that specifically do not differ by the same amount. However, each term does differ by the same proportion (the application of interest defined by the nominal rate which produces a constant effective rate). Mathematical sums of this nature are referred to as a Geometric sum or series. Appealing to the structure of each term in our annuity we N see the exponential model a b x P1 i repeated over and over. we have a series of terms following a very rigid pattern. Consider the following expanded sum, a b n a b n 1 a b 2 a b 1 a b 0 . And using the distributive property we can rewrite this sum as a b n b n 1 b 2 b1 b 0 . Now, if we focus just on the sum of the b’s to various powers we will be able to rewirte this large sum as a small formula similar to our arithmetic sum formula. Let S be equal to the value of the sum S b n b n 1 b 2 b 1 b and rather then adding S to itself we will subtract a multiple of S , bS b n 1 b n b 2 b 1 then simplify by collecting like terms and solving for S . S b n b 2 b1 b 0 bS b n 1 b n b 2 b 1 S 1 b b n 1 b 0 b n 1 b 0 1 b n 1 . and S 1 b 1 b And the task of summing all the terms in our geometric series has been reduced to applying the formula we just derived, a b n a b n 1 a b 2 a b 1 a b 0 = a b n b n 1 b 2 b1 b 0 1 b n 1 = a 1 b Let’s return to our scenario of an annuity. We are making equal periodic payments of $100 each month into an account that converts interest at an APR of 3 % compounded monthly for 10 years. If we observe structure of the monthly payments and their growth over the ten year period we see, the first $100 is compounded 10x12 or 120 times. The next month, the next $100 is compounded one less number of times or 119 times. The third month’s $100 payment is compounded 118 times. We continue this process of compounding each $100 until 10 years of $100 deposits and interest payments have past. All of the money in the account is ours, how much is there? We need to add up all of these $100 payments and their associated interest accumulations. We have a long sum with 121 terms, too long to write in a short text like this, so we abbreviate our sum as follows: 120 119 118 1 0 0.03 0.03 0.03 0.03 0.03 100 1 100 1 100 1 ... 100 1 100 1 12 12 12 12 12 Even if we write each interest growth factor as 1+0.03/12 = 1.0025, we still have a lengthy sum of 121 terms that looks like the following: 120 119 118 1 0 1001.0025 1001.0025 1001.0025 1001.0025 1001.0025 This is a lot of symbols to write, so we will sum all of these terms by using our geometric series formula. Over the ten year life span of our annuity our savings strategy has netted us 1 b n 1 1 1.0025121 14109.08 dollars. This is 14109.08-12000 = 2109.08 dollars 100 a 1 1.0025 1 b paid to us as rent on our monthly deposits of $100 for ten years. If we made equal periodic payments of $ 100 per month for 20 years instead of 10 years, we would 1 1.0025 241 $33,012.28 in the account after those 20 years. We deposited into the have $100 1 1.0025 account 12(100)(20) = 24,000 dollars and earned $33,012.28 – $24,000 = $9012.28 in interest Lets relate the future value of an annuity formula to a generic timeline. Time line Pymt Pymt Pymt 0 1 2 Pymt N-1 time at zero is present day Pymt N there are N interest conversion periods One observation that needs to be made first is the number of payments on our time line and the total number of interest conversion periods. These two numbers are not the same. The total number of payments is N+1 and the total number of interest conversion periods is N. Now any annuity with these properties can be calculated fairly quickly and without much difficulty. The sum of all the terms in our annuity will look like N r r Pymt1 Pymt1 n n Pymt1 i Pymt1 i N Pymt 1 i 1 i N N 1 N 1 N 1 1 r r Pymt1 Pymt1 n n Pymt1 i Pymt1 i 1 1 i 1 i 1 0 0 0 and finally, this geometric sum can be rewritten as 1 1 i N 1 1 1 i N 1 1 1 i N 1 1 i N 1 1 Pymt Pymt Pymt 1 1 i Pymt i i 1 1 i So, the future value of an annuity where the payment periods coincide with the interest conversion 1 i N 1 1 periods is Pymt . i Present Day Annuity F. Scott Fitzgerald once said the very rich are different from you and me. In 2000, Alex Rodriquez signed with the Texas Rangers for a quarter of a billion dollars in a deal that doubled the previous richest contract in sports history. Specifically, they lured the shortstop with a 10-year, $252 million contract, to include a $ 10 million dollar signing bonus. Rangers owner Tom Hicks said "Alex is the player we believe will allow this franchise to fulfill its dream of continuing on its path to becoming a World Series champion". Hicks had paid $250 million to buy the entire franchise three years prior from the group headed by George W. Four years later, Rodriguez was playing for the NY Yankees as he watched the Boston Red Sox foil the curse and win the World Series. But, signing bonuses are not just for the rich, the athletes, the authors, the musicians, the broadcasters. Many trades and professions now use signing bonuses as an attraction to lure better employees. Nurses are needed and are being enticed with an unprecedented assortment of signing bonuses, relocation reimbursements and incentive packages. From undersized hospitals in Maine to understaffed large ones in Houston, recruiters are involved in bidding wars for nurses. In Massachusetts as far back as 1998, to combat the teacher shortage, teachers were offered a $20,000 signing bonus as an incentive. For the past five years, recent pharmacy graduates had their choice of jobs when they graduated, starting salaries were $75,000 each year, with the added plus that many employers offered signing bonuses. According to Pharmacy Week, a national pharmacists' employment publication, signing bonuses of more than $10,000 are reported by 15 percent of pharmacists; the average bonus is over $5,000. What do people do with their new signing bonus? Do they supplement their monthly income with it? They could. Suppose we reverse our ‘annuity’ investment strategy. What would this mean? Instead of making equal periodic payments into an account for a period of time and watch it accumulate to a lump sum, let’s take a lump sum of money and withdraw from it equal payments until the original deposit is zero dollars. A present day annuity looks like an investment strategy where we invest, say one million dollars, into an account that pays an annual interest rate, say 3 %, compounded every so often, say monthly, and we withdraw the same amount, equal periodic withdrawals, of say so much every month for a period of time, say 10 years. How much money could we withdraw each month? Where do you see investment strategies like this? An athlete signs a million dollar signing bonus and then the fairy tale ends as they endure a career ending injury. Naturally, they want to live off of that signing bonus for the next 30 years. An author signs a 5 million dollar signing bonus and their book does not sell, so they live off that bonus for 10 years. Also, a couple has saved one million dollars from age 23 to 59 in annuities and savings accounts. They reinvest the lump sum and want to live off it for 10 years, while they are still relatively young, until their retirement and Medicare kicks in. Here, the notion of withdrawals is replaced with monthly payments, but the mathematics is the same. These are all present day annuities. The present day annuity equation can be derived from the following discussion and example. Suppose you have a Lump Sum of money you wish to withdrawal from periodically until all the money is gone. While you are making withdrawals the remaining portion of your lump sum stays in the bank and earns interest at the nominal rate, i . Here we let LS represent our Lump Sum, W represents the amount of our withdrawals and time is measured in units of interest conversion periods. Time 0 1 2 Account Balance LS 1 LS 1 i W LS 1 i W 1 i W 1 LS 1 i W 1 i W 2 3 LS 1 i 2 1 W 1 i W 1 i W 1 LS 1 i W 1 i W 1 i W 3 2 1 4 LS 1 i 3 W 1 i W 1 i W 1 i W 2 LS 1 i W 1 i W 1 i W 1 i W 4 N 3 LS 1 i W 1 i N N 1 2 1 W 1 i W 1 i 0 1 0 After the N th withdrawal the account balance is zero. It is from this last line where we fully develop formulas to use when considering situations that represent a Present Day Annuity. The N N 1 1 0 equation LS 1 i W 1 i W 1 i W 1 i 0 and a little algebra is all we need. By pushing all of the negative withdrawals and the interest factors over to the other side of the equation we now have some recognizable expressions. The left hand side of the equation represents the future value of a lump sum and the right hand side of the equation is in the familiar form of an annuity N N 1 1 0 LS 1 i W 1 i W 1 i W 1 i . We can rewrite the equation and then solve for the withdrawal variable, W . This will give us exactly what we were looking for, a formula that tells us how much money can we periodically withdraw from a lump sum earning interest over a specified period of time. 1 i N 1 N N 1 1 0 LS 1 i W 1 i W 1 i W 1 i W i i N W LS 1 i and with a little algebra massage this can be further simplified to N 1 i 1 i . W LS N 1 1 i Problem One You land a new job and accept a bonus of $ 10,000 on the first day. What do you do with the signing bonus? You have choices. Let’s think this through carefully. a) You could supplement the income with the bonus because some starting salaries are just not enough to live on. You could use the $10,000 signing bonus to supplement our income over the next three years. How much could you withdraw each month to help supplement your income if the bonus was invested in a present day annuity that pays 6% annual compounded monthly. b) You could invest the bonus to save for a purchase three years in the future because the starting salary may not provide a means to save for that home or vehicle you want. You can invest the $10,000 signing bonus today so that the accumulated amount may be used for a one time purchase three years into the future for say, a down payment on a home or a vehicle. How much money will you have in three years if your bonus was invested into an account that pays paid 6 % annual convertible monthly. c) You could use the bonus to supplement our retirement 30 years in the future because one’s retirement plan may be below one’s desire for more money in those retirement years. You could use the $10,000 signing bonus to supplement your retirement plan. Invest the bonus into an account that pays 6 % annual compounded monthly and leave it there for 30 years. Then in addition, open another account which is an annuity and pay an affordable and manageable $60 per month for 30 years at 6 % annual, compounded monthly. How much would we have accumulated in 30 years to add to our retirement? Solution For each of the three choices our nominal rate i .06 12 .005 because the annual interest rate is 6 % and it is compounded monthly. a) This is an example of a present day annuity. We will have 12(3) = 36 compounding periods, so N = 36. We will deposit a lump sum and withdraw from it equal periodic payments for three years. i .005 W LS 10,000 304.22 dollars. N 36 1 1 i 1 1.005 Thus, you would be able to supplement your monthly income by $304.22. Actually, you would have supplemented your income with 12(3)($304.22) = $10,951.92 over the three years, as opposed to just the original $10,000 signing bonus. b) This is a familiar example of compounding a lump sum of money. You will have 12(3) = 36 compounding periods, so N = 36 and the future value of this account is FV 10000(1 .005) 36 11,966.81 dollars. You would have $11,966.81 available for a down payment on a car or home in three years. c) This is an example of compounding a lump sum and then combining this future value with that of an annuity that has also matured over 30 years. We will have 12(30) = 360 compounding periods, so N = 360. From the $10,000 bonus, a future value of FV 10000(1 .005) 360 60,225.75 dollars is to be added to the $60 a month annuity. The future value of this annuity is 1 i N 1 1.005360 1 FV Pymt 60 60,270.90 dollars. Thus, in thirty years, you would i .005 have $ 60,270.90 + $ 60,225.72 = $120,496.62 in addition to any other retirement plan. Problem Two Now that you have remained steady and faithful to your investments of the $10,000 signing bonus and the $60 a month annuity, how many dollars can you withdraw monthly from this lump sum over the next fifteen years into retirement? How many dollars will you have invested in total? How much money did your personal retirement plan pay you in the form of rent on your money? Solution Keeping the same nominal rate, your nest egg of $120,496.62 spread out in monthly withdrawals .005 1016.82 dollars over fifteen years (Present Day Annuity) will pay you 120,496.62 180 1 1.005 each month. In the end a total of 180 (1016.82) = 183,027.60 dollars would have been paid out and the account balance would be zero. Your total investment in this plan would have been the original 10,000 dollar signing bonus and 60 (360) = 21,600 dollars paid into the annuity. Summing these two dollar amounts gives us a total investment of $31,600. In return you would have earned $183,027.60 - $31,600.00 = $151,427.60 as rent paid to you on your money. Problem Three Let’s examine the age-old question, if you were handed $1 million dollars (you won the Survivor game that is shown on TV, you won the lottery, you received a signing bonus for the book you just wrote, you pick the scenario …) could you live off the money for the rest of your life? That is, are you set for life if you invest this lump sum in a present day annuity at 5 7/8 % annual compounded monthly for the next 30 years until social security kicks in? Solution Since you are compounding monthly at 5 7/8 %, the nominal interest rate, “i” = 0.05875/12 and there are 12 months in a year and we are compounding for 30 years, that is, 12(30) = 360 compounding periods. You would have .05875 i 12 W LS 1,000,000 5915.38 dollars per month to live off of if N 360 1 1 i 1 1 .05875 12 you were handed $1,000,000 and needed to make it last for 30 years. We should note, your $5915.38 monthly allowance when summed over the 30 years is more than two million dollars; $2,129,536.80. Could you do this? Exercise Set Problems 1-10. Signing Bonus; what to do with the money? Many choices face the recent college graduate who has been offered a job with a salary and a signing bonus. Specifically, what should be done with the signing bonus? The choices for what to do with the signing bonus is the core issue addressed in problems 1-10. Problems 1-3. Supplement the income with the bonus. “Starting salaries are never enough to live on.” For each profession below, use the indicated signing bonus for that profession to supplement one’s income over the next three years. How much could one withdraw each month to help supplement their income if they invested the bonus in a present day annuity that pays paid 5 1/8 % annual compounded monthly. 1. In 1998, Massachusetts instituted a $20,000 signing bonus to teachers to address the concern about attracting quality teachers. 2. In 2002, the San Jose Mercury News reported the recent graduates from nursing school were offered starting salaries up to $65,000, plus generous signing bonuses of as much as $5,000. 3. 15 percent of recent pharmacy graduates have signing bonuses offered to them of at least $10,000. Problems 4-6. Use the bonus to save for a purchase three years in the future. “Starting salaries rarely provide a means to save for that home or vehicle you want.” For each profession below, use the indicated signing bonus for that profession to invest today so that the accumulated amount may be used for a one time purchase three years away, say a down payment on a home or a vehicle. How much money would one have in three years if they invested their bonus into an account that pays paid 5 1/8 % annual compounded monthly. 4. Teachers: Bonus = $20,000. 5. Nurses: Bonus = $5,000 6. Pharmacists: Bonus = $10,000 Problem 7-9. Use the bonus to supplement one’s retirement 30 years in the future. “Retirement plans wrapped up in salaries rarely provide enough money for one’s retirement.” For each profession below, use the indicated signing bonus for that profession to supplement your retirement plan. Invest each bonus into an account that pays 5 1/8 % annual compounded monthly and leave it there for 30 years. Then in addition open another account which is an annuity and pay an affordable and manageable $50 per month for 30 years at 5 1/8 % annual, compounded monthly. How much would one have accumulated in 30 years to add to their retirement? 7. Teachers: Bonus = $20,000. 8. Nurses: Bonus = $5,000 9. Pharmacists: Bonus = $10,000 10. If you were granted a modest bonus, say $ 10,000 upon accepting a job, which investment strategy would you favor and why? The choices are as follows: use the bonus to invest into a present day annuity to help supplement your income short term for three years, invest the bonus into a savings account that compounds the money so that in three years you have a lump sum to put down on a home or car, or use the money combined with small payments to help you when you retire. Write a paragraph telling the story of your choice and include the mathematics you used to arrive at your decision. 11. In 2004, Jevon Kearse became one of the richest athletes. Kearse landed in Philadelphia, signing an eight-year, $66 million contract that makes him the highest paid defensive lineman in NFL history. The Eagles gave Kearse a $16 million signing bonus, this for a lineman who had missed 14 games the past two seasons and whose sack totals had decreased in each of his first full seasons in the league. If Kearse endured a career ending injury his first game of the 2004 season, how much could he live off each month if he invested the full $16 million into a present day annuity that paid 5 1/2% annual compounded monthly for thirty years? 12. Find the amount of an IRA annuity after 6 years if you paid into it $600 per year at 12 % compounded annually. 13. Find the amount of an IRA annuity after 6 years if you paid into it $50 per month at 12 % compounded annually. 14. Find the amount of an IRA annuity after 6 years if you paid into it $50 per month at 12 % annual compounded quarterly. 15. Find the amount of an IRA annuity after 6 years if you paid into it $50 per month at 12 % compounded monthly. 16. Find the amount of an IRA annuity after 6 years if you paid into it $50 per month at 12 % compounded daily. 17. Find the amount of an IRA annuity after 6 years if you paid into it $ 12.50 per week at 12 % compounded monthly. Hint: Convert the interest rate to a nominal rate that coincides with the 12 .12 52 payment schedule, 1 j 1 . 12 18. Find the amount of an IRA annuity after 6 years if you paid into it $50 per month at 14 ½ % compounded monthly. 19. Mr. and Mrs. Allen have purchased a $250,000 home. They have made a 20% down payment. The balance was amortized at 6.5 % compounded monthly for 30 years. a) What is their monthly payment? b) What is amount the Allen’s would pay in interest at the end of the term of their loan? c) How much did the home actually cost the Allen’s, including interest, if they paid off the home in the required 30 year period, assuming no late payments? d) After 8 years, how much equity have the Allen’s built up? e) Half way into the term of their loan, after 15 years, how much equity have the Allen’s built up in their $250,000 home? f) After 24 years, after 80 percent on their loan’s term has come and past, how much equity have the Allen’s built up? 20. If you pay $50 each month into an extended Christmas Club account, paying 8 ½ % interest compounded monthly, what amount do you have after 8 months? How much did you put into the account? 21. You know you are not making ends meet, but help is on the way. Your tax refund came in last week, so you have some extra money. You know an extra $200 per month would balance your monthly budget. In order to withdraw $200 per month to help make ends meet for the next 2 ½ years how much should you invest today into an account earning 12 % compounded monthly. 22. You have earned a million dollar signing bonus for taking your new job. You invest it in a present day annuity at 8 ¾ % annual compounded monthly. How much money would you be able to live off if this turned out to be your sole income for 10 years. Assume you withdrew equal periodic payments each month for 10 years? 23. You make $100 payments each month into an annuity for 20 years at 7 ¾ % annual compounded monthly. After 20 years, you deposit the entire amount in the account in a present day annuity that earns 8 ½ percent compounded monthly. How much could you withdraw each month if you needed the money to live off of for another 20 years? 24. Make up numbers tell a story of how you saved $1,000,000. 25. Preparing for when Murphy’s Law hits - Career ending injuries. Here is a list of the top 10 fortunate athletes. Source: Sports Illustrated and cnn.com, http://sportsillustrated.cnn.com/2004/writers/pete_mcent egart/05/14/money.list/ From the “Sport’s Illustrated List of the 50 Fortunate -- the best-compensated athletes ranked by current annual income – May 2004” For each athlete, take only their endorsements and appearance fees and invest those fees into the same present day annuity, one that invests 8 ½ % annual compounded monthly for 30 years. How much could each athlete live off monthly for 30 years using only their ‘glamour money’, the endorsement and appearance fees, if their career was to end tomorrow? The Fortunate 50 1 Tiger Woods Salary or winnings (including bonuses) Endorsements and appearance fees Total income 2 Shaquille O'Neal Golf $6,673,413 $70,000,000 $76,673,413 Los Angeles Lakers Salary or winnings (including $26,517,858 bonuses) Endorsements and appearance fees $14,000,000 Total income $40,517,858 Cleveland 3 LeBron James Cavaliers Salary or winnings (including $4,018,920 bonuses) Endorsements and appearance fees $35,000,000 Total income $39,018,920 4 Peyton Manning Indianapolis Colts Salary or winnings (including $26,900,000 bonuses) Endorsements and appearance fees $9,500,000 Total income $36,400,000 Minnesota 5 Kevin Garnett T'wolves Salary or winnings (including $29,000,000 bonuses) Endorsements and appearance fees $7,000,000 Total income $36,000,000 6 Oscar De La Hoya Boxing Salary or winnings (including $30,000,000 bonuses) Endorsements and appearance fees Total income 7 Andre Agassi Salary or winnings (including bonuses) Endorsements and appearance fees Total income 8 Kobe Bryant $2,000,000 $32,000,000 Tennis $2,530,929 $24,500,000 $27,030,929 Los Angeles Lakers Salary or winnings (including $13,498,000 bonuses) Endorsements and appearance fees $12,000,000 Total income $25,498,000 New York 9 Derek Jeter Yankees Salary or winnings (including $19,000,000 bonuses) Endorsements and appearance fees $6,000,000 Total income $25,000,000 10 Grant Hill Orlando Magic Salary or winnings (including $13,279,250 bonuses) Endorsements and appearance fees $11,000,000 Total income $24,279,250 Different Payment and Interest Conversion Periods One situation requiring a little adjustment in the nominal rate used to calculate the future value of an annuity type account occurs when the payment time intervals and the interest conversion periods do not coincide. The development of all of our financial formulas has been dependent on the notion that both the payment and interest enter the account on the same day. Suppose you have constructed for yourself a monthly payment structure and you discover a fabulous deal on an interest bearing account where the interest rate is convertible quarterly? What interest rate should you use for your calculations? Lets call it j. To derive this pseudo nominal rate you will need to set both the known or offered growth rate expression and your unknown growth rate expression equal to each other and then solve for the unknown rate. The equation you will have developed looks like: n 1 j 1 r . n Once you solve or isolate the j variable you can use this to accurately calculate earning due to interest contributions for accounts such as annuities or payout annuities. Working with the scenario discussed a 6% compounded quarterly account receiving monthly payments would require finding the appropriate nominal rate, j to employ in your calculations. The conversion formula is K n 4 rk .06 12 j 1 1 and the nominal rate we are looking for is j 1 1 .004975 . 4 n An annuity receiving monthly payments of $100 where interest is converted quarterly over a ten 1.0049751012 1 year period of time will produce an account value of 100 16,361.28 dollars. If .004975 we were to have run the calculations without first converting to incorporate the appropriate nominal rate we would have produced a future value of $33,128.82. Although this would seem to be a very popular proposition from the investors point of view the near tripling of funds over the ten year time period should present to us a red flag that something in our calculation was not right. Exercise Set For problems 1-5 convert the stated interest rate to the appropriate nominal rate required to accurately calculate the future value of an account. Then determine the future value of each account. 1. $100 is deposited weekly for two years with an APR of 4 1 % 8 convertible monthly. 2. $50 is deposited weekly for two years with an APR of 4 1 % convertible 8 daily. 3. $250 is deposited monthly for 18 years with an APR of 6% convertible daily. 4. $1000 is deposited annually for 18 years with an APR of 5 7 % 8 convertible quarterly. 5. $1000 is deposited monthly for five years with an APR of 3 1 % 3 convertible daily. The Mortgage Payment Equation Because you are the one seeking to rent money for say, a Home Mortgage the ‘Bank’ plays the role of investor. The bank views you as an investment. Hence, two words, ‘Credit Check’. Your basic home mortgage works just like a payout annuity with a twist as to the perspective with which we consider the equation. We need to think how the equation thinks. Since the bank views the barrower as an investment the right hand side of this equation will represent the bank’s interest in this transaction. The total quantity of money involved from the bank’s perspective is described nt r using the formula Loan Amount 1 which we write more succinctly as n N Loan Amount1 i . Now, the left hand side of the equation will represent your contributions or payment structure for the mortgage. This is simply an annuity formula and our mortgage payment 1 i N 1 N equation looks like, Pymt Loan Amount 1 i . Applying a little algebra we see the i periodic payment for this loan will be i i N Pymt Loan Amount 1 i Loan Amount . N N 1 i 1 1 1 i Example A $200,000 mortgage that has been financed over 30 years at 6% APR and is compounded monthly will have a monthly payment i .005 Pymt Loan Amount Pymt 200,000 1199.10 . N 360 1 1 i 1 1.005 The monthly payment for a 200,000 dollar mortgage loan at 6% APR convertible monthly is 1,199.10 dollars. Each mortgage payment can be divided into two distinct pieces the interest portion and the principal portion, Payment = Interest Due + Principal. Recall, the amount of interest paid following one interest conversion period under an compound interest scenario is I P1 i P Pi . Throughout the life of a mortgage the interest portion of the payment will decrease as the remaining principal portion increases. This is the result of paying down the outstanding balance. Interest due at the end of each interest conversion period is found by multiplying the outstanding balance by the nominal rate, (Outstanding Balance) i . The interest portion owed to the bank from the first payment of a loan is (Loan Amount) i . For our $200,000 loan the interest due after one month is ($200,000)(. 005) $1000.00 . Given our monthly payment for this loan was $1199.10 $1000.00 $199.10 this leaves only $199.10 for the principal portion of the payment which reduces the outstanding balance. Outstanding Balance, Interest and Principal portions of a Mortgage Payment The outstanding balance on a mortgage after making any regularly scheduled payment can be derived from the mortgage payment equation. After some algebraic manipulation we can arrive at 1 1 i n N the formula Outstanding Balance Pymt . Notice in the exponent position there is i the term n N . The lower case n represents the payment number and the capital N represents the total number of payments to be made throughout the life of the loan. The interest portion of the n th payment is found by multiplying the outstanding balance after the previous payment n 1st by the nominal rate. Using our outstanding balance formula we see the portion of our n th payment going to interest due is, 1 1 i n 1 N n 1 N (Outstanding Balance after n - 1 payments) i Pymt . The i Pymt 1 1 i i n 1 N amount of interest owed at the time of the n th payment is I n Pymt 1 1 i . Expanding this formula we have I n Pymt 1 1 i which we read as: Interest Due Pymt Pymt1 i = Payment - Principal portion of the Payment. And for free we have derived the principal portion n 1 N of the n th monthly payment, Pymt1 i . Putting all of the pieces together we can now construct what is called an Amortization schedule. Time 0 1 n 360 n 1 N n 1 N Payment Interest Due Principal $1199.10 $1000.00 $199.10 N 1 1 i Loan Amount $1199.10 i Pymt 1 1 i n 1 N $5.97 Balance $200,000.00 $199,800.90 Pymt 1 i n 1 N 1 1 i n N Pymt i $1193.13 $0.00 Problem One Good Credit verses Poor Credit Now that you have landed your dream job and are planning for your future you create a plan to purchase your first home. First, you need to save for a down payment on your purchase. Today you invest your $3000.00 nest egg into an account that pays 3% compounded monthly and start a $50.00 a month annuity into an account that pays 6% both accounts remain intact over the next six years while you work on your credit. In today’s market an individual with good credit can obtain a 15 7 year mortgage loan for about 4 % APR compounded monthly and someone with poor credit may 8 obtain a mortgage with an interest rate of 9%. Is it worth maintaining a good credit rating? Solution After six years your two savings accounts have produced a total of $8132.67; .06 72 1 1 72 12 .04 $4320.44 . You have also found the perfect $30001 $3812.23 and $50 .06 12 12 little condo for sale; $120,000.00. Using your savings for the down payment you look for a bank willing to invest in you. You need another $120,000 - $8132.67 = $111,867.33 to make the purchase. Under the good credit scenario your 15 year mortgage payment will be .04875 12 $877.37 . Over the entire life of this loan you will have Pymt $111,867.33 180 1 1 .04875 12 paid the bank $877.37 180 $157,926.60 . The bank would have collected $157,926.60 $111,867.33 $46,059.27 as rent on the money you borrowed. Now, suppose you neglected to take care of your credit rating and were considered to be a bit more of a risk in the eyes of the bank. The best interest rate you can obtain for the purchase of your first home is 9%. Your .09 12 $1134.63 . Over the entire life of monthly payment will be Pymt $111,867.33 180 .09 1 1 12 the loan you will have paid the bank $1134.63 180 $204,233.40 . And the bank would have collected $204,233.40 $111,867.33 $92,366.07 as rent on your loan. By maintaining a good credit rating you will have saved $92,366.07 $46,059.27 $46,306.80 . Problem Two In Problem One we see maintaining good credit really pays. Based on the two different cases how much difference will there be in Equity after 5 years of payments? Solution In this problem we will consider Equity as the portion of the home you actually have paid off at a specific time. In both cases we will determine the outstanding balance after five years of payments and subtract this from the sale price of the home. After 60 payments the good credit scenario produces an outstanding balance of 1 1 .04875 60180 12 $83,198.07 $877.37 .04875 12 and the equity is $120,000 $83,198.07 $36,801.93 . Five years of payments under the poor credit scenario provides for an outstanding balance of 1 1 .09 60180 12 $89,569.61 $1134.63 .09 12 and the equity is $120,000 $89,569.61 $30,430.39 . There is more than $6000.00 difference in equity after five years between the good and poor credit scenarios. The good credit loan pays less each month and produces more equity! Because of your diligence in maintaining a good credit rating in these past five years you have acquired 5% more in equity. Exercise Set For problems 1-4 use the following information. In November of 2003 the maximum amount of money one could borrow under the terms of a conventional loan was $322,700.00. 1. Find the monthly payment for the maximum conventional loan if you received the loan and agreed to repay the monies over the next 30 years at an APR of 5 3 % convertible monthly. 8 2. What dollar amount of the first payment goes to interest? 3. What dollar amount of the first payment goes toward paying down the outstanding balance? 4. After the first payment, find the new outstanding balance. For problems 5-8 continue to use the maximum conventional loan amount of $322,700.00. 5. Find the monthly payment for the maximum conventional loan if you received the loan and agreed to repay the monies over the next 15 years at an APR of 4 7 % convertible monthly. 8 6. What dollar amount of the first payment goes to interest? 7. What dollar amount of the first payment goes toward paying down the outstanding balance? 8. After the first payment, find the new outstanding balance. For problems 9-17 use the following information. A 30 year home mortgage has been acquired for $200,000.00 under the terms monthly payments will be made to satisfy an annual percentage rate of 6%, compounded monthly. 9. After five years of payments, how much of the 61st payment will go directly to paying interest due? 10. After five years of payments, how much of the 61st payment will go to paying down the outstanding balance? 11. What is the outstanding balance on the mortgage immediately following the 60th payment? 12. After 15 years of payments, how much of the 181st payment will go directly to paying interest due? 13. After 15years of payments, how much of the 181st payment will go to paying down the outstanding balance? 14. What is the outstanding balance on the mortgage immediately following the 180th payment? 15. After 25 years of payments, how much of the 301st payment will go directly to paying interest due? 16. After 25years of payments, how much of the 301st payment will go to paying down the outstanding balance? 17. What is the outstanding balance on the mortgage immediately following the 300th payment? For problems 18-22 use the following information. You have enough money saved to make a 10% cash down payment on a $200,000 home or an 8% cash down payment on a $250,000 home. In both cases the APR is 6% compounded monthly. For the $200,000 home you can afford to amortize the loan over 20 years and for the $250,000 home you need to extend the life of the loan to 30 years. 18. Calculate the monthly payment for each loan. 19. Calculate and compare the total amount of interest paid over the life of each loan. 20. Determine the amount of equity accrued in each home after 5 years of payments have been made. 21. Determine the amount of equity accrued in each home after 10 years of payments have been made. 22. Determine the amount of equity accrued in each home after 15 years of payments have been made. After graduating from college you decide its okay to marry your soul mate. The two of you receive $1000 in cash from friends and family for the wedding and you also have received $500 in cash as graduation gifts. Being diligent in college has paid off as well. The two of you as you have also landed that first job and are able to save $75 each month to use as a down payment on your first home 4 years after graduation. The first home you purchase is a $117,500 condo. You have been saving for the down payment with both the lump sum deposit of $1500 into a CD paying 5 7 % 8 compounded daily and an annuity with $75 monthly payments into a regular savings account paying 2 7 % 8 compounded daily. Four years have passed and you are ready for your first 30 year mortgage. 23. Considering both accounts, how much money has been saved for a down payment? 24. Given that the sale price of the condo is $117,500 what are the monthly payments after the down payment has been applied? 25. Over the 30 year life of the mortgage how much interest would have been paid to the bank? 26. How much equity will you have accrued in the home after the first year? 27. How much equity will you have accrued in the home after the fifth year? 28. How much equity will you have accrued in the home after the 15th year? 29. How much equity will you have accrued in the home after the 25th year? 30. How much equity will you have accrued in the home after the 29th year?
