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Lecture #1 5.1, 5.2: Investments, Compound Interest, and Value

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Steven Rayan

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MATA32 September 6: 5.1, 5.2: Investments, Compound Interest, and Value • When money is invested, one expects to earn interest on the investment. “making a return” • The amount to be invested is called the the principal, or the “present value.” • Usually, an interest rate is agreed at the time of the investment. • There are 2 critical pieces of information: ◦ a length of time ◦ and a fraction (or percent) The length of time is the interest period. (eg. 3 months -quarterly, semiannually, annually) The fraction is the periodic interest rate. It is the fraction of the principal that is added back to the principal at the end of a period. Eg. The value of an investment today is $600.00. The negotiated interest period is 2 months and the periodic rate is 1%. What is the value of the investment in 4 months. 4 monthsst 2 interest periods will pass. After 1 2 month period: new value = old value + (old value x 0.01) = 600 + (600x0.01) = 600+6 =$606.00 After 2 two month period: new value = old value + (old value x 0.01) = 606 + (606x0.01) = 600+6.06 =$612.06 Let P = principal Let r = periodic rate Let n = # of periods Let S = value of the investment after n periods (compound amount or “accumulated amount” or “future value”) After 1 period value is P + Pr = P(1+r) After 2 periods value is P(1+r) + P(1+r)r = P(1+r)(1+r) 2 =P(2+r) 2 After 3 periods: = P(1+r) + (1+r) =P(1+r) 3 After n periods, the compounded amount (i.e.Accumulated amount) of the investment is S= P(1+r) n The interest earned after n periods (“compound interest”) is S – P Almost always, the interest rate is quoted as an annual rate (aka nominal rate or the APR:Annual percentage rate) If R is the annual rate, and there are m interest periods in a year, then the periodic rate is r = R/m. E.g. a) Suppose a student takes out a credit card and charges $3000.00 to it, in the first st month of class. Interest begins to accumulate on October 1 , 2012 at an annual rate of 24% compounded monthly. What is the balance owing on the card on October 1, 2013, assuming no payments have been made? n S= P(1+r) R=24% = 0.24 m=12 (12 periods per year) r= 0.24/12 = 0.02 st st n = 12 (because 12 periods between October 1 , 2013 and October 1 , 2014). S = 3000(1+0.02) 12 = 3000(1.02) 12 =$3804.73 The effective rate: 804.73/3000 = 0.268 (the effective rate) or use formula: r = [e+(0.24/12)] = 0.268 Therefore the “actual annual rate was 26.8% Something to keep in mind: The bank is investing in you when you deposit money. b) Suppose you want to put the $3000.00 on C.C.And want to accumulate no more than $650.00 interest in 12 months assuming no payments and monthly compounding. What is the maximum nominal rate you should agree to? Solving for little r in S = P(1+r) n S ≤ 3000 + 650 n r ≤ [12(3650/3000)] – 1 but S = P(1+r)
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