Survey of Math: Chapter 21: Consumer Finance Savings Page 1

Transcription

1 Survey of Math: Chapter 21: Consuer Finance Savings Page 1 The atheatical concepts we use to describe finance are also used to describe how populations of organiss vary over tie, how disease spreads through a population, how ruours spread through a population, even the otion of particles suspended in a fluid, as well as any other situations. Matheatics is so beautiful because the techniques you learn to solve one type of proble typically can be used to solve other probles! Money deposited in a savings account in a bank will earn interest. The initial aount you deposit is called the principal, and the oney which is earned is called the interest. How does the oney grow? What will your balance be after one year? There is a lot that goes into answering these questions, since interest can be paid in different ways. Growth of Savings: Siple Interest Siple interest pays interest only on the principal, not on any interest which has accuulated. Siple interest is rarely used for saving accounts, but it is used for bonds. Exaple You put $98.45 in a savings account which pays siple interest of 6% a onth. How uch oney do you have in the savings account after 4 onths? Solution To answer this question, we can build fro what we know. Siple interest eans we pay interest only on the initial aount deposited principal), which was $ The interest aount will be 6%= 6/100 = 0.06 of the principal, and added to the account balance once a onth. Interest Period Date Interest Added Accuulated Aount 0 Jan 1 0 $ Feb 1 $ =$5.91 $ $5.91 = $ Mar 1 $ =$5.91 $ $5.91 = $ Apr 1 $ =$5.91 $ $5.91 = $ May 1 $ =$5.91 $ $5.91 = $ This table is the for an Excel spreadsheet would take to calculate siple interest. initialization and it is the second row that contains forulas. Notice the first row is an We see that the growth is by a constant aount $ =$5.91) every tie period onth in this case). This is the requireent for linear or arithetic growth. It gets the nae linear since the graph of the aount versus the tie is a straight line linear function).

2 Survey of Math: Chapter 21: Consuer Finance Savings Page 2 Siple Interest Forula For siple interest of r percent paid every tie period with a principal P, we get Years Accuulated Aount 0 P 1 P ) + P r 2 P + P r) + P r = P + 2P r 3 P + 2P r) + P r = P + 3P r 4 P + 3P r) + P r = P + 4P r t. P + P rt ie., for a principal of P with siple interest of r% paid every tie period, we get an accuulated aount after t years of A = P + P rt = P 1 + rt). The forula gives you another way of calculating a quantity that could be done using a spreadsheet style table. Exaple You put $98.45 in a savings account which pays siple interest of 6% a year. How uch oney do you have in the savings account after 4 years? Solution Identify P = $98.45, r = 6% and tie period of one year. So for four years, t = 4. A = P 1 + rt) = $ ) = $ Growth of Savings: Copound Interest Copound interest pays interest on the principal and the accuulated interest, not just the principal. Exaple You put $98.45 in a savings account which pays copound interest of 6% a onth. How uch oney do you have in the savings account after 4 onths? Solution To answer this question, we can build fro what we know. Copound interest eans we pay interest on the accuulated aount in the account. The interest aount will be 6%= 6/100 = 0.06 of this aount, and added to the account balance once a onth. Copounding Period Date Interest Added Accuulated Aount 0 Jan 1 0 $ Feb 1 $ =$5.91 $ $5.91 = $ Mar 1 $ =$6.26 $ $6.26 = $ Apr 1 $ =$6.64 $ $6.64 = $ May 1 $ =$7.04 $ $7.04 = $ We see that the aount of growth increases as tie increases. The aount of growth is proportional to the aount present, which is the requireent for geoetric growth.

3 Survey of Math: Chapter 21: Consuer Finance Savings Page 3 Interest Terinology Savings probles typically involve a bit ore terinology than we ve used so far. The copounding period is the tie which elapses before copound interest is paid. The tie when copounding is done effects the accuulated aount, since the current aount affects the aount of interest added, and the current aount will change if we copound ore frequently. The noinal rate is the stated rate of interest for a specified length of tie. The noinal rate does not take into account how interest is copounded! The effective rate is the actual percentage rate of increase for a length of tie which takes into account copounding. It represents the aount of siple interest that would yield exactly as uch interest over that length of tie. The effective annual rate EAR) is the effective rate given over a year. For savings accounts, the EAR is also called the annual percentage yield APY). Copound Interest Forula For a noinal annual rate r, copounded ties per year, we have i = r/ as the interest rate per copounding period. Now let s try to derive a forula for copound interest. Copounding Period Aount 0 P 1 P + P i = P 1 + i) 2 P 1 + i) + P 1 + i)i = P 1 + i) 2 3 P 1 + i) 2 + P 1 + i) 2 i = P 1 + i) 3 4 P 1 + i) 3 + P 1 + i) 3 i = P 1 + i) 4 n. P 1 + i) n ie., for a principal of P with copound interest of i = r/ paid every copounding period, we get an accuulated aount after n = t copounding periods t is nuber of years, is nuber of copounding periods per year) of A = P 1 + i) n = P 1 + r ) t.

4 Survey of Math: Chapter 21: Consuer Finance Savings Page 4 Annual Percentage Yield APY) By definition, the APY is the siple interest rate that earns the sae interest as the copound interest after one year t = 1). Copound Interest A = P 1 + r ) t = P 1 + r ) Siple Interest A = P 1 + rt) = P 1 + APY) Set these quantities equal, and solve for APY: P 1 + APY) = P 1 + r ) 1 + APY) = 1 + r ) APY = 1 + r 1 ) Exaple You put $98.45 in a savings account which pays copound interest of 6% a onth. How uch oney do you have in the savings account after 4 years? Solution Identify P = $98.45, i = 6%. So for four years, t = 4 and = 12 copounding periods per year, so n = t = 12 4 = 36. A = P 1 + i) n = $ ) 48 = $ Exaple $1000 is deposited at 6% per year. Find the balance at the end of one year, if the interest paid is a) siple interest b) copounded quarterly. Solution a) The principal is P =$1000, and the noinal rate is r =6% = After one year, t = 1. If we use siple interest, we have an accuulated balance of A = P 1 + rt) = $ ) = $ b) The noinal annual rate is r =6% = 0.06, when copounded quarterly, eans we have = 4, so i = r/ = 0.06/4 = One year corresponds to t = 1, so after one year we have n = t = 4 1 = 4, A = P 1 + i) n = $ ) 4 = $ Exaple $1000 is deposited at 7.5% per year. Find the balance at the end of one year, and two years, if the interest paid is copounded daily. What is the APY? Solution The noinal annual rate is r =7.5% = 0.075, when copounded daily, eans we have = 365, so i = r/ = 0.075/365 = One year corresponds to n = t = = 365, so after one year we have A = P 1 + i) n = $ ) 365 = $ Two years corresponds to n = t = = 730, so after two years we have A = P 1 + i) n = $ ) 730 = $ APY = 1 + r ) 1 = ) = 7.79%. 365

5 Survey of Math: Chapter 21: Consuer Finance Savings Page 5 A Liit to Copounding Sketch the graph of the accuulated aount for 10 years if the principal is P =$1000 and the annual interest rate is r = 10% for siple interest, copound interest copounded yearly, copound interest copounded quarterly, and copound interest copounded daily assue 365 days in a year). To get the values, we can use the forulas we derived. Here is the process for getting the accuulated aount after 1 year so t = 1 in all forulas); the rest are calculated in a siilar fashion using t = 2, 3, 4,.... Siple interest after 1 year: A = P 1 + rt) = $ ) = $ after 1 year. Copound interest copounded yearly = 1, i = r/ = 0.10/1 = 0.10, and n = t = 1): A = P 1 + i) n = $ ) 1 = $ after 1 year. Copound interest copounded quarterly = 4, i = r/ = 0.10/4 = 0.025, and n = t = 4): A = P 1 + i) n = $ ) 4 = $ after 1 year. Copound interest copounded daily = 365, i = r/ = 0.10/365 = , and n = t = 365): A = P 1 + i) n = $ ) 365 = $ after 1 year. black: siple interest. red: copound interest, copounded yearly. green: copound interest, copounded quarterly. blue: copound interest, copounded daily. The curves are all essentially the sae for short ties. There are ore points for copounding quarterly than yearly since interest is paid ore often during the year. There is not uch difference over 10 years to copounding quarterly and copounding daily. Copounding ore frequently leads to a larger accuulated balance, but there is a liit to this process. The liit would be if we copounded continuously.

6 Survey of Math: Chapter 21: Consuer Finance Savings Page 6 Copounding Continuously Consider a principal P = $1 and a rate of r=100% which is copounded over shorter and shorter tie periods. We are interested in how uch the accuulated aount will be after one year. Copound interest copounded n ties a year i = 1/, and n = t = to get one year, t = 1)): A = P 1 + i) n = ) after 1 year. There is a table in the text of these nubers, here is a sketch We see that the accuulated aount is approaching a nuber: ) if is very large. This nuber is siilar to π = in that it is atheatically significant and appears in any situations, and so we give it a special designation: e ) if is very large. This leads the the continuous interest forula, which is A = P e rt after t years if interest is copounded continuously at annual rate r. The function e rt is called the exponential function. The continuous interest forula is the upper liit on the accuulated aount that can accrue due to copounding interest.

7 Survey of Math: Chapter 21: Consuer Finance Savings Page 7 Review of interest forulas principal P and annual rate r) Siple interest: A = P 1 + rt) is the aount after t years. Copound interest, copounded ties over 1 year for t years: A = P 1 + i) n is the aount where i = r/. Continuously copounded interest: A = P e rt is the aount after t years. APY = 1 + ) r 1 is the annual percentage yield. The forulas allow us to answer questions which would be difficult to answer using a table, and also to answer questions quickly without a lot of calculation. However, the tables allow us to answer questions that do not atch the conditions under which the forulas were derived. Therefore, both forulas and spreadsheet tables are useful in understanding how personal finance works.