October 14, 2013
This time Arithmetic Growth Simple Interest Geometric Growth Compound Interest A limit to Compounding
Simple Interest Simple Interest Simple Interest is interest that is paid on the original principal only, not on any accumulated interest. Simple Interest For a principal P and an annual rate of interest r, the interest earned in t years is I = Prt and the total amount A accumulated in the account is A = P(1 + rt)
When Simple interest might be used When might simple interest be used? Private loans between individuals commercial loans for less than one year financing of corporations and the government through bonds. A bond is a loan with repayment at the fixed term and simple interest in the meantime.
Arithmetic Growth Arithmetic Growth Arithmetic growth (also called linear growth) is growth by a constant amount in each time period. Question: On November 15, 2010, you could buy a 30-year U.S. Treasury bond for $10,000 that pays 4.25% simple interest every year through November 15, 2040. How much total interest would it earn by then?
Compound Interest Compound Interest Compound interest is interest that is paid on both the original principal and accumulated interest. Compounding Period The compounding period is the fundamental interval on which compounding is based, within which no compounding is done. Nominal Rate, Effective Rate and APY A nominal rate is any stated rate of interest for a specified length of time. The effective rate is the rate of simple interest that would realize exactly the same amount of interest over the same length of time. For a year, the effective rate is called the annual percentage yield(apy)
Compound Interest Compound Interest Formula An initial principal P in an account that pays interest at a periodic interest rate i per compounding period grows after n compounding periods to A = P(1 + i) n Compound Interest Formula for an Annual Rate An initial principal P in an account that pays interest at a a nominal annual rate r compounded m times per year, grows after t years to ( A = P 1 + r ) mt m
Notation for Savings A amount accumulated, can be denoted FV for future value P initial principal, sometimes denoted PV for present value r nominal annual rate of interest t number of years m number of compounding periods per year n = mt total number of compounding periods i = r/m periodic rate, the interest rate per compounding period
Example I put $1,000 dollars in a savings account with 10% nominal interest per year. How much money will I have after 10 years? If it doesn t compounded A = 1000 +.1*1000*10 = 2000 If it compounds annually A = 1000(1 +.1) 10 = 2593.74 If it compounds quarterly A = 1000(1 +.1 4 )10 4 = 2685.06 If it compounds monthly A = 1000(1 +.1 12 )10 12 = 2707.04 If it compounds daily A = 1000(1 +.1 365 )10 365 = 2717.91
Geometric Growth Geometric Growth (Exponential Growth) Geometric Growth (also called exponential growth) is growth proportional to the amount present. Question: If a math teacher assigns one second of homework the first week of school, two seconds the second week, four seconds the third, and so on. How much homework would a student have to do in week 36 (in hours)?
Geometric Growth Geometric Growth (Exponential Growth) Geometric Growth (also called exponential growth) is growth proportional to the amount present. Question: If a math teacher assigns one second of homework the first week of school, two seconds the second week, four seconds the third, and so on. How much homework would a student have to do in week 36 (in hours)? Answer: Work = 2 36 1 (1/3600) = 34, 359, 738, 368/3600 = 9, 544, 371.77
Effective Rate Formula for Effective Rate Formula for APY effective rate =(1 + i) n 1 APY = (1 + r m )m 1 where APY = annual percentage yield (effective annual rate) r = nominal interest rate m = number of compounding periods per year
Example With nominal annual rate of 6% compounded monthly, what is the APY? Answer: APY = ( 1 + 0.06 ) 12 1 = 0.0617 = 6.17% 12
A limit to Compounding Continuous Compounding Continuous compounding is the method of calculating interest that yields what compound interest tends toward with more and more frequent compounding per period Continuous Interest Formula For a principal P deposited in an account at a nominal annual rate r, compounded continuously, the balance after t years is A = Pe rt If it compounds daily A = 1000(1 +.1 365 )10 365 = 2717.91 If it compounds continuously A = 1000e (.1)10 = 2718.28
Problems Question: I put $1,000 dollars in a savings account with 2% nominal interest per year. How much money will I have after 10 years? with Simple Interest? Compounded annually? Compounded quarterly? Compounded daily? Compounded continuously? Question: I had a CD with National City Bank through 2010 that paid 4.69% interest compounded daily. What was the APY for this rate? Question: A Paper Series EE Savings Bond is sold at half face value, to the full face value by 20 years from the issue date. What is the minimum APY for such a bond
Problem If I put a $1,000 a year into a savings account with APY 10%. How much money will I have at the end of 5 years?
Next time A Model for Saving Present Value and Inflation