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Class 4 Jan 16th Completed

C bank c offers an effective quarterly rate of r q 3

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C) Bank C offers an effective quarterly rate of r Q = 3% D) Bank D offers an effective monthly rate of r mo = 1% I What do all these rates have in common Each rate, multiplied by the number of times it is compounded per year is equal to 12%. I We always call effective per period rates ‘r k ’, where k indicates the period length.
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14 Compounding Frequeny: Comparing Returns (1) I How do you compare these interest rates? We can we calculate what $1 is worth after one year if invested at the different banks. Then we can solve for the Effective Annual Rate (EAR) that would give the investor the same return. I Bank A: r Yr = 12% After one year: you have – $1*(1+r Yr )=$1(1+0.12) = $1.12 = $1(1+r Yr ) So we found that r Yr = r Yr = EAR (to state the obvious) I If interest is compounded once per year, the stated rate is the EAR.
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15 Compounding Frequeny: Comparing Returns (2) I Bank B: r semi = 6% This means that you earn 6% every six months. – Timeline: 1 Years 0 1 2 Periods – --------|------------------|-------------------|------------------ $1 $1(1+r semi ) $1*(1+r semi ) 2 After six months, you have: After one year, you have: 06 . 1 $ ) 06 . 0 1 ( * 1 $ ) 1 ( * 1 $ = + = + semi r 1236 . 1 $ ) 06 . 1 ( * 1 $ ) 1 ( * 1 $ ) 1 )( 1 ( * 1 $ 2 2 = = + = + + semi semi semi r r r
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16 Compounding Frequeny: Comparing Returns (3) I Bank B: r semi = 6% continued What effective annual rate of return (EAR) would give you the same return as Bank B s offer of 6% every 6 month? I So putting things together, we get the following general rule: This shows us how to compute the EAR r Yr that corresponds to a semi-annual effective rate of return of r semi . $1*(1 + r Yr ) = $1.1236 ! r Yr = 1.1236 " 1 ! r Yr = 0.1236 = 12.36% $1*(1 + r semi ) 2 = $1*(1 + 0.06) 2 = $1.1236 = $1*(1 + 0.1236) = $1*(1 + r Yr ) ! (1 + r semi ) 2 = (1 + r Yr ) ! EAR = r Yr = (1 + r semi ) 2 " 1
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