What could be the value of A?

Question

The village of Kieler completes an exploratory study and finds the current village water tower will need a replacement in 10 years at a cost of $290,000. To finance this amount, the village board will at this time asses it's 958 homeowners with a one time surcharge and then invest this amount in a ten year CD(cash deposit) paying 8 1/4% interest compunded ssemiannually.

a. How much will the village of Kieler need to invest at this time in this CD in order to raised the $290,000 in ten yeras?

b. What amount should each homeowner pay as a surcharge.

After seeing the village of Kieler raised the money to invest in a new water tower. The city board of East Dobuque decide to adopt the similar plan. However since East Dobuque is a much larger community they will need to raised $783,000 to build  three new water towers in 15 years. At this time the city board plans to asses its 2682 homeowners with a one time surcharge , and then invest the money received in a money market account paying 9% interest compunded monthly.

a. how much money will the city board need to raise at this time to meet the city's water tower need at the end of 15 years.

b. before applying the surcharge, the city board decids to use a $50 000 benefactor gift toward the water tower investment. Taking this gift into account, how much should the surcharge be on each homeowner?

Answer 1

PART 1

a. To raise $290000 in 20 periods of 6 months at 4.125% semiannually would need a principal investment of P:

290000=P(1.04125)^20=2.2444P approx. P=290000/2.2444=$129210.47 approx.

b. The surcharge would be 129210.47/958=$134.88 per homeowner.

PART 2

a. Assuming $783000 is needed in total for the water towers (rather than this amount each), 15 years is 180 months at 9/12=0.75% per month. The principal amount, P, is given by 783000=P1.0075^180=3.838P. The number 3.838 is the growth rate of the investment over 15 years. So P=$204010.21 approx.

b. We deduct $50000 from 783000 to give $733000 as the effective cost of the towers to the homeowners. So P=733000/3.838=$190982.73 to be divided between 2682 homeowners=$71.21 surcharge.