2-You start to save for a major purchase. You can invest $320 every three months
ID: 3145467 • Letter: 2
Question
2-You start to save for a major purchase. You can invest $320 every three months for 3 years and 9 months. You make the first payment in three months fro today. You are able to earn 5.34% compounded semi-annually. What is the amount of interest that you earn during the entire term? 3. What is the future value eight years from now of each of the following cash-flow streams if money can earn 9% compounded semiannually? a)A single payment of $5000 today. b)An ordinary annuity starting today with eight annual payments of $900. c) An ordinary annuity starting in three years with 20 quarterly payments of $400 4. Ken sold his car to Carolyn for $2000 down and monthly payments of 70 for 3 ½ years, including interest at 12% compounded monthly. $160. What was the selling price of the car?Explanation / Answer
Formula for investment with interest compounded semi-anually is:
A=P(1+r/2)2n where A=Final Amount, P=Principal, r=rate of interest, n=number of years.
For the 1st installment of $320, n=3.5 since total period is 3 years 9 months and we are paying after 3 months from today, hence 3 years 6 months.
r=5.34%, Putting the values in formula A=320(1+5.34/200)7, =320(1+2.67/100)7
Similarly next installment of $320 will have n=3.25
Again, putting values for Installment no.2, A=320(1+5.34/200)6.5, =320(1+2.67/100)6.5
Similarly if we put all the quarterly installments and add them we the final value of our investment at the end of the period of 3 years 9 months.
The n for all 15 installments goes as below:
n=3.5,3.25,3,2.75,........,0.25,0
Hence, we get a series as shown below:
A=320[1+(1+2.67/100)0.5 + (1+2.67/100)1 + ..........+ (1+2.67/100)7]
This is a GP series with 1st term a=1, n=15
common ratio r= (1+2.67/100)0.5
Its Sum Sn is given by Sn= [a(1-rn)]/(1-r)
Putting the sequence in Sn formula,
Sn = [1- (1+2.67/100)15/2]/ 1-(1+2.67/100)0.5 solving......
=(1-1.22)/(1-1.013)
= 0.22/0.013
= 16.92
Therefore,
A=320*16.92
A=5414.4
Final amount after the period ends = $5414.4
Total investment = 15*320=$4800
Interest received for the period = 5414.4-4800 = 614.4
Final Ans= 614.4
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