Caveat: The rules for rounding the results of calculations are designed to be conservative(i.e., keeping digits that might prove useful). They err on the side of spurious precision.Even if your measurements are correct and you follow the rounding rules, your last digit or two may not be accurate.Examples (see Roundoff.xls):Example: 123 + 456.Example: 123  456.Principal: You give some money to the bank.Simple interest: The bank pays you cash (“thanks for letting us hold onto your money”).Compound interest: The bank keeps the cash and adds it to your balance.APR = Annual Percentage Rate = (Interest)/(Balance), assuming annual interest.If you have $1000 in the bank at 10% simple interest, how longdoes it take before your money doubles?If you have $1000 in the bank at 10% interest compounded annually,how long does it take before your money doubles? (create Compound.xls)Rule of 70 & Rule of 72For the rest of today, we’ll assume that all interest is compound interest.Which is better: $10,000 invested at 20% interest, or $20,000 invested at 10% interest ... ten years from now? (use Compound.xls)"Annual income twenty pounds, annual expenditure nineteen six, result happiness. Annual income twenty pounds, annual expenditure twenty pound ought and six, result misery." (Mr. Micawber, in David Copperfield by Charles Dickens)This is true even if you’re lucky enough to inherit a million dollars. Suppose you have a bank account with a million dollars, paying compound interest with an APY of 10%.(See Micawber.xls.) How expensive a lifestyle can you afford?$100,000/year? $150,000/year? $50,000/year?Compound Interest Formula for Interest Paid Once a Year:(1) A = P  (1 + APR)^YwhereA = accumulated balance after Y years,P = starting principalAPR = annual percentage rate (as a decimal)Y = number of yearsWhen a bank offers “annual interest P% interest compounded monthly”, what it’s really giving you is (P/12)% interest each month.Compound Interest Formula for Interest Paid n Times per Year:(2) A = P  (1 + APR/n)^(nY)where A = accumulated balance after Y years,P = starting principalAPR = annual percentage rate (as a decimal)n = number of (equal-sized) compounding periods per yearY = number of yearsAPY = Annual Percentage Yield = (1+APR/n)^nA = P  (1 + APR/n)^(nY) = P  ((1 + APR/n)^n)^Y, so(3) A = P  (1 + APY)^YSo, if your money is compounded more often than annually, if you know the APY you canjust pretend it’s compounded annually at an interest rate of APY and use the simple formula (3), and you’ll get the same answer to the question “How fast is my money growing?” as if you used the complicated formula (2). (See APY.xls)Do Examples 8 and 9 via trial-and-error, with spreadsheet (see Examples8and9.xls)Transition to continuous compounding; define e(“What’s the APY if APR = 100% and n is really really large?”)For the Tuesday after spring break, read 4.2HW due two weeks from today (on web by early next