Question

II. Profit Reinvestment inflation is compounded monthly. Then the required down payment will be 20% of T.C.Hardware Store wants to construct a new building at a second location. If construction could begin immedi- ately, the cost would be $500,000. At this time, however the owners do not have the required 20% down payment, so they plan to invest $2000 per month of their profits until they have the necessary amount. They can invest their money in an annuity account that pays 6% compounded monthly, but they are concerned about the 3% average inflation rate in the construction industry. They would like you to give them some projections about how 3% inflation will affect the time required to accrue the down payment and the building's eventual cost. They would also like to know how their projected profits, after the new building is complete, will affect their schedule for paying off the mortgage loan. Specifically, the owners would like you to prepare a report that answers the following questions 0.03 500,000 1 12 where n is in months. Thus you must find n such that this amount equals the future value of the owners' annuity. The solution to the resulting equation is difficult to obtain by algebraic methods but can be found with a graphing utility 2. If the 3% annual inflation rate is accurate (compound- ed monthly), what will T. C. Hardware's projected construction costs be (to the nearest hundred dollars) when it has its down payment? 3. If T. C. Hardware borrows 80% of its construction costs and amortizes that amount at 7.8% compounded monthly for 15 years, what will its monthly payment be? 4. Finally, the owners believe that 2 years after this new building is begun (that is, after 2 years of payments on the loan), their profits will increase enough so that they'll be able to make double mortgage payments each month until the loan is paid. Under these assumptions, how long will it take T. C. Hardware to pay off the loan? 1. If the 3% annual inflation rate is accurate, how long will it take to get the down payment? Hint: Assume that

Answer 1

1). Taking inflation into consideration, the amount required for the down-payment (D) will be:

D = 20%*total cost*(1+ annual inflation/12)^n = 20%*500,000*(1+3%/12)^n

The unknown here is number of months n. If the owners invest $2,000 per month for n months then it should equal the down-payment amount.

Present Value of monthly investment of $2,000 at 6% p.a.(r) compounded monthly for n months is:

PV = 2,000*(1 - (1+r)^-n)/r = 2,000*(1 - (1+6%/12)^-n)/6%/12

To find n, we need to solve D = PV

20%*500,000*(1+3%/12)^n = 2,000*(1 - (1+6%/12)^-n)/6%/12

Using excel solver for the above equation, n = 71.1 months or 71 months (rounding off)

2). The inflation adjusted projected construction cost of $500,000 after 71 months will be:

500,000*(1+3%/12)^71.1 = 597,135.97 or 597,100 (rounding off to the nearest hundred dollar)

3). Amount to be borrowed A = 80%*597,100 = 477,680

Monthly payment: PV = 477,680; r = 7.8%/12 = 0.65%; N = 15*12 = 180, solve for PMT

PMT = PV*r/(1 - (1+r)^-n) = 477,680*0.65%/(1 - (1+0.65%)^-180) = 4,509.98 (monthly payment)

Note: This can also be calculated easily using the PMT function.

4). Total loan to be paid off T = PMT*N = 4,509.98*180 = 811,795.96

Amount paid off in two years A1 = PMT*2*12 = 4,509.98*24 = 108,239.46

Amount remaining AR = T - A1 = 811,795.96 - 108,239.46 = 703,556.50

Monthly payment M now becomes 2*PMT = 2*4,509.98 = 9,019.96

Number of months needed to pay off AR will be AR/M = 703,556.50/9,019.96 = 78 months

Thus, the company will take 2 years + (78/12) years = 2 + 6.5 = 8.5 years to pay off the total loan.