Chapter 21. Savings Model
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1 Chapter 21. Savings Model
2 Arithmetic Growth and Simple Interest
3 What s important Bring your calculator. Practice the correct way to use the calculator. Remember the formula and terminology. Write your reasoning( formula you use) in details when you take any test. Answer without any reasoning will get zero point. I won t allow you to share calculator with others. You cannot use your cell phone as a calculator.
4 Terminology Principal The initial balance of the savings account. Example: You are opening a savings account today. You deposited $50. Then, $50 is principal. Interest Money earned on a savings account or a loan. Example: If at the end of the year, your savings account has $51, then the interest is $1.
5 Simple Interest The method of paying interest only on the initial balance in an account, not on any accrued interest.
6 Simple Interest Example You deposited $1000. An annual simple interest rate is 1%. Then, Your Account Balance: After a year: 0.01 X = After two years: 2X(0.01X1000)+1000=1020 After n years: n X (0.01X1000)+1000
7 Simple Interest Example You deposited $1000. An annual simple interest rate is 1%. That is, at the end of each year, you get $10 as an interest no matter how many years you deposit.
8 Simple Interest For a principal P and an annual rate of interest r, after t years, Interest I = P r t Total amount A = P + I = P + P r t = P 1 + r t = P(1 + rt)
9 Use of Simple Interest Private loans between individuals (easy to calculate) Commercial loans for less than a year (Doesn t make much differences from the compound interest)
10 Example Let s suppose that you have exhausted the amount that you can borrow under federal loan programs and need a private direct student loan for $10,000. National City Corporation quoted a rate in May 2008 of 5.7% for the school year. It offers an interest-only repayment option, under which you make monthly interest payments while you are in school and pay on the principal only after graduation. Under this plan, National City earns simple interest from you while you are in school. How much monthly interest would you pay for such a $10,000 loan?
11 Answer Student loan= $10,000. Annual rate= Simple interest 5.7% Monthly interest= Prt = $10, = $
12 Arithmetic Growth(Linear Growth) Growth by a constant amount in each time period Example: simple interest
13 Geometric Growth and Compound Interest
14 Compound Interest Interest that is paid on both the original principal and accumulated interest
15 Compound Interest Suppose that you deposited $10,000 into the savings account with annual compound interest rate of 10%. After.year later Interest 1 10% 10,000 = ,000 = % 11,000 = ,000 = 1,100 Savings total 10, ,000 = 11,000 11, ,100 = 12,100
16 After.year later Interest Compound Interest Savings total 1 10% 10,000 = ,000 = , ,000 = 11, % 11,000 = ,000 = 1,100 11, ,100 = 12, % 12,100 = ,100 = 1,210 12, ,210 = 13,310
17 Nominal Rate Any state rate of interest for a specified length of time. Nominal rate have broad( vague ) meaning. It doesn t indicate or take into account whether or how often interest is compounded.
18 Annual Interest Rate of 10% (Nominal Rate) That might mean Annual interest rate of 10% compounded quarterly. ( every three months, 4 times a year ) Annual interest rate of 10% with simple interest
19 Annual interest rate of 10% compounded quarterly with $1,000 Principal It is compounded every three months, that is, 4 times a year. Annual interest rate of 10% in this case means 2.5% quarterly.
20 Annual interest rate of 10% compounded quarterly with $1,000 Principal After 0 month 3 months 6 months 9 months 12 months Balance 1, , ,000 = 1, , ,025 = 1, , , = 1, , , = 1,103.82
21 Annual interest rate of 10% After a year later, with simple interest 0.1 1, ,000 = 1,100
22 Annual interest rate of 10% with simple interest vs. compound interest with $1,000 a year later A year later Interest amount Effective interest compared to the principal Simple interest $1, % Quarterly Compound interest $1, %
23 Effective Rate and Annual Percentage Rate If we know the effective rate and annual percentage rate for the compound interest, then it is easier to know how much you earn interest. Effective rate: the rate of simple interest that would realize exactly as much interest over the same length of time. Annual percentage yield(apy): Effective rate for a year the money after a year the principal 100 % Principal Annual interest amount = 100(%) Principal
24 Rate per Compounding Period Terminology Meaning Number of compounding per year Rate when nominal annual rate is r. Annually Every year 1 r Bi-annually Twice a year 2 r 2 Quarterly Four times a year 4 r 4 Monthly Every month 12 r 12 Daily Every day 365 r 365
25 Rate per Compounding Period Terminology Periodic rate when nominal annual rate is 12% Annually 12% Bi-annually 12 2 = 6% Quarterly 12 4 = 3% Monthly = 1% Daily = 0.03%
26 Nominal annual interest rate of 10% After years with Principal $1000 Balance in dollars = 1000( ) = = 1000( ) ( ) ( ) = 1000( ) 2 (1+0.1)=1000( ) 3 n 1000( ) n
27 Nominal annual interest rate of 10% with Principal $ ( ) n Number of years Principal Interest rate per year Generalization Interest rate per compounding period Number of compounding
28 Compound Interest Formula An initial principal π in an account that pays interest at a periodic interest rate π per compounding period grows after π compounding periods to π π¨ = π·(π + π) = π·(π + π π ) π Remark: π, π Periodic interest rate π = where r is an annual rate of interest and m is the number of compounding per year.
29 Compound Interest Formula for an annual nominal rate r, t years later π π¨ = π·(π + π) = π·(π + π π π ππ ) = π·(π + ) π π π‘: Number of years m: Number of compounding per year r: nominal annual interest rate n: Number of compounding
30 Geometric Growth ( Exponential Growth ) Growth proportional to the amount present( Not to the principal )
31 Example Suppose that you have a principal of P=$1000 invested at 10% nominal interest per year with annual compounding. How much balance will you have 10 years later?
32 Answer Suppose that you have a principal of P=$1000 invested at 10% nominal interest per year with annual compounding. How much balance will you have 10 years later? $1000( )1 10 = $ = $
33 Use of Calculator( Scientific ) (1) Calculate what are in the parenthesis. (2) Calculate the exponent and then raise to the power. (3) Multiply by the principal. However, there are many other correct ways to use the calculator.
34 Example Suppose that you have a principal of P=$1000 invested at 10% nominal interest per year with quarterly compounding. How much balance will you have 10 years later?
35 Answer Suppose that you have a principal of P=$1000 invested at 10% nominal interest per year with quarterly compounding. How much balance will you have 10 years later? $1000( ) =$1000(1.025)40 = 4 $
36 Example Suppose that you have a principal of P=$1000 invested at 10% nominal interest per year with monthly compounding. How much balance will you have 10 years later?
37 Answer Suppose that you have a principal of P=$1000 invested at 10% nominal interest per year with monthly compounding. How much balance will you have 10 years later? $1000(1 + ) = $1000(1.0083)120 = $
38 Basic Useful Formula If a = b, then a 1 n = b 1 n. (a m ) n = a mn (a + b) n a n + b n
39 Example A man borrowed $29,000 for two years under simple interest. At the end of the two years his balance due was $31,900. What annual simple interest rate did he pay?
40 Answer A man borrowed $29,000 for two years under simple interest. At the end of the two years his balance due was $31,900. What annual simple interest rate did he pay? 5%
41 Example Suppose you invest $6000 and would like your investment to grow to $8000 in five years. What interest rate, compounded monthly, would you have to earn in order for this to happen?
42 Answer Suppose you invest $6000 and would like your investment to grow to $8000 in five years. What interest rate, compounded monthly, would you have to earn in order for this to happen? 5.77%
43 Group Activity You have $3500 that you invest at 7% simple interest. How long will it take for your balance to reach $4235?
44 Answer You have $3500 that you invest at 7% simple interest. How long will it take for your balance to reach $4235? Three years
45 Group Activity Merrie borrowed $1000 from her parents, agreeing to pay them back when she graduated from college in five years. If she paid interest compounded quarterly at 5%, how much would she owe at the end of the five years?
46 Answer Merrie borrowed $1000 from her parents, agreeing to pay them back when she graduated from college in five years. If she paid interest compounded quarterly at 5%, how much would she owe at the end of the five years? $1282
47 Group Activity Merrie borrowed $500 from her parents, agreeing to pay them back when she graduated from college in four years. If she paid interest compounded daily at 16%, how much would she owe at the end of the four years?
48 Answer Merrie borrowed $500 from her parents, agreeing to pay them back when she graduated from college in four years. If she paid interest compounded daily at 16%, how much would she owe at the end of the four years? $948
49 Effective Rate What percent of interest do you earn after n compounding period compared to the principal? Effective rate: (π + π)π π the money after the principal = π·ππππππππ π: number of compounding π i: interest rate per compounding period ( ) π r: nominal interest rate
50 Annual Percentage Yield (APY) What percent of interest do you get after a year compared to the principal?(i.e. as an annual simple interest rate) Effective rate = (1 + r m )m 1 m: number of compounding per year
51 APY for 4% of nominal interest with $10,000 deposit( Principal) Account type Number of compounding Balance after a year Interest amount % of increase after a year (APY) Simple interest 1 $ % Compounding bi-annually Compounding quarterly 2 $10404= 10000( ) 2 4 $ from %(= ) % Compounding monthly 12 $ from %
52 Example With a nominal annual rate of 6% compounded monthly, what is the APY?
53 Answer With a nominal annual rate of 6% compounded monthly, what is the APY? ( ) 1 = = 6.17%
54 Example Suppose that the monthly statement from the fund reports a beginning balance(p) of $ and a closing balance(a) of $ for 28 days(n). What is the effective daily rate?
55 Example Suppose that the monthly statement from the fund reports a beginning balance(p) of $ and a closing balance(a) of $ for 28 days(n). What is the effective daily rate? Effective rate= (1 + i) n 1 n: number of compounding, i: interest rate per compounding period ( r m ) Difference of money after 28 days = Principal 1 + i n 1 What is i?
56 Example Beginning balance(p): $ , Closing balance(a): $ for 28 days(n). Difference of money after 28 days = Principal 1 + i n = [( 1 + i 28 1] = (1 + i) = (1 + i) (1 + i) 28 = i = ( ) 1 28 = i = So, the daily effective rate is %.
57 Example Angela invests in a savings account that pays 4% interest compounded monthly. What is the APY for this account?
58 Answer Angela invests in a savings account that pays 4% interest compounded monthly. What is the APY for this account? 4.07%
59 Example You have $4300 that you invest at 5% simple interest. How long will it take for your balance to reach $7525?
60 Example You have $4300 that you invest at 5% simple interest. How long will it take for your balance to reach $7525? 15 years
61 Example Suppose you invest in an account that pays 5% interest, compounded quarterly. You would like your investment to grow to $5000 in 16 years. How much would you have to invest in order for this to happen?
62 Answer Suppose you invest in an account that pays 5% interest, compounded quarterly. You would like your investment to grow to $5000 in 16 years. How much would you have to invest in order for this to happen? $2258
63 Simple Interest Versus Compound Interest
64 Continuous Compounding A = Pe rt
65 About Number e e The number (1 + 1 m )m approaches when m gets larger and larger.
66 Continuous Compounding Continuous compounding is the method of calculating interest in which the amount of interest is what compound interest tends toward with more and more frequent compounding.
67 Continuous Interest Formula A = Pe rt A: Balance after t years when a principal P is continuously compounded r: nominal annual rate
68 Example What is the balance after 1 year if the principal is $1000 and 10% continuous compounding?
69 Example What is the balance after 1 year if the principal is $1000 and 10% continuous compounding? 1000 e 0.10 = $
70 Example What is the balance after 5 years if the principal is $1000 and 10% continuous compounding?
71 Example What is the balance after 5 years if the principal is $1000 and 10% continuous compounding? 1000 e (5) (0.10) =
$496. 80. Example If you can earn 6% interest, what lump sum must be deposited now so that its value will be $3500 after 9 months?
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