SYDE 331
MANAGERIAL AND ENGINEERING ECONOMICS
Spring 2002
Assignment #2
Maximum marks: 30
1. You have decided to buy a new stereo system and have agreed to make 36 equal monthly payments at 6% interest compounded monthly. Each payment will be for $45.63. What was the approximate original cost of the stereo system? (3 marks)
2. A couple decides on the following savings plan for their child's university education. When the child is six months old, and every six months thereafter, they will deposit $350 into a savings account paying 8% interest compounded semiannually. Immediately after the child's tenth birthday, they will stop making deposits and let the accumulated money earn interest, at 6% interest compounded quarterly, for the next eight years (until the child=s 18th birthday), at which time the child will withdraw the total amount for his/her university education.
a. How much money will be in the account when the child withdraws the accumulated amount on his/her 18th birthday? (5 marks)
b. Suppose that, on his/her 18th birthday, the child withdraws the accumulated amount from the savings account, then immediately purchases an annuity that pays him/her a certain amount of money at the beginning of each of his/her five years at university. Assuming that the child begins university on his/her 18th birthday, how much will the annual payment be if this annuity is paying 6% interest compounded annually (3 marks)
3. A home was purchased twelve years ago for $75,000. The home was financed by paying a 20% down payment and signing a 25year mortgage at 8% interest compounded monthly on the balance owing. Mortgage payments are made at the end of each month.
The market value of the house is now $90,000. How much equity does the owner have in the house after twelve years? (5 marks)
4. Mr. A borrows $8000 at 18% interest from the Hi-Interest Finance Company, and agrees to repay the loan with monthly payments over four years. The finance company allows him to repay the entire loan early, but if he does so, he must pay a penalty equal to three months= payments.
Immediately after the 20th payment, Mr. A realizes that he could re-finance the loan by borrowing enough money from the Lo-Interest Finance Company to repay the loan from the Hi-Interest Finance Company. He would then repay the Lo-Interest Finance Company by making 28 monthly payments at 13.5% interest. Should he refinance the first loan? Support your answer with calculations. (5 marks)
5. Do Problem 3.31 from the textbook. (3 marks)
6. Do Problem 3.32 from the textbook. (3 marks)
7. Do Problem 3.38 from the textbook. (3 marks)
SYDE 331
MANAGERIAL AND ENGINEERING ECONOMICS
Spring 2002
Solutions to Assignment #2
Maximum marks: 30
1. 3 marks; deduct 1 mark if interest rate is incorrect, deduct 1 mark if 36 months was not used
general approach: find PV of a 36-month ordinary annuity, given the PMT
answer: $1499.90
PV = PMT [PV of 36-month ordinary annuity at 0.5% interest per period]
= $45.63 [ 1 - (1+0.005) -36) / 0.005 ] = $1499.90
2. Part a. 5 marks: 3 marks for future value of annuity after twenty payments and two marks for future value of the single amount
general approach:
Step 1: calculate FV of an ordinary annuity given payment
Step 2: calculate FV of a one-time deposit
answer: $16,783.24
solution:
(future) value of annuity after 20 payments of $350
= $350 [(1.0420 -1) / 0.04] = $10,422.30
(future) value of $10,422.30 at child=s eighteenth birthday
= $10,422.30 (1.015)32 = $16,783.24
Part b. 3 marks; deduct two marks if treated as an ordinary annuity rather than as an annuity due
general approach: find payment of an annuity due given PV
answer: $3758.76
PV of an annuity due = $16783.24 = present value of 4 annual payments of an ordinary annuity plus present value of one payment immediately (ie., at time zero)
$16783.24 = PMT + PMT [ 1 - (1+0.06) -4) / 0.06 ]
PMT = $16783.24 / (1 + 3.4651) = $3758.76
OR:
PV = PMT [PV of an ordinary annuity factor] [1 + i]
$16783.24 = PMT [ 1 - (1+0.06) -5) / 0.06 ] [1+0.06]
PMT = $3758.76
3. 5 marks
general approach:
Step 1: calculate payment given PV of an ordinary annuity
Step 2: calculate amount owing after 12 years using payment from Step 1
Step 3: calculate equity
answer: $45, 173.61
solution:
monthly payment = [75000 x 0.80] / [ 1 - (1+0.0067) -300) / 0.0067 ] = $463.09 (1 mark)
amount owing after 12 years
= $463.09 [ 1 - (1+0.0067) -156) / 0.0067 ] = $44826.39 (3 marks)
Notice that although the term of the mortgage is now almost 50% finished, the home owner still owes about 74% of the amount borrowed!!
Equity = current net market value minus any unpaid loan balance
Net market value = amount received after subtracting all costs associated with selling the house; we will assume the costs of selling the house are zero (in reality, they probably would be about $5500 on this house).
equity = $90000 - 44826.39 = $45173.61 (1 mark)
4. 5 marks
Hi-Interest Finance Company:
! monthly payment is $235 (1 mark)
! amount owing immediately after the 20th payment is $5340.78 (calculated as the present value of the remaining 28 monthly payments at 18% interest (1 mark)
! penalty is $705
Lo-Interest Finance Company:
! Mr. A would borrow $5340.78 + 705 penalty = $6045.78 (1 mark)
! Mr. A would make 28 monthy payments at 13.5% interest of $252.91 (1 mark)
Conclusion: Mr. A should not refinance the loan because doing so would increase his monthly payments (1 mark)
5. 3 marks: 2 marks for present value of geometric gradient annuity, 1 mark for for finding future value of PV of geometric gradient annuity
general approach: find PV of a geometric gradient annuity given payment, then find FV of the PV after ten years
answer: about $3.6 million
solution:
! base amount = $100,000
! interest rate = 20%
! growth rate = 10%
! growth adjusted interest rate = (1.20) / (1.10) - 1 = 9.09%
! geometric gradient factor = 6.3923 / 1.1 = 5.81118
! present value of geometric gradient annuity = 100,000 (5.81118) = 581,118
! FV of the PV after ten years = 581,118 (1 + 0.2)10 = 3,598,131.60
6. 3 marks: 2 marks for present value of geometric gradient annuity, 1 mark for for finding future value of PV of geometric gradient annuity
general approach: find PV of a geometric gradient annuity given payment, then find FV of the PV after ten years
answer: about $3.6 million
solution:
! base amount = $100,000
! interest rate = 10%
! growth rate = 20%
! growth adjusted interest rate = (1.10) / (1.20) - 1 = -8.333%
! geometric gradient factor
= [ ((1 - 0.08333)10 -1) / (-0.08333 ( 1 - 0.08333)10) ] [0.83333]
= [ -0.581096 / -0.034909 ] [0.83333]
= 13.871685
! present value of geometric gradient annuity = 100,000 (13.8716) = 1,387,168
! FV of the PV after ten years = 1,387,168 (1 + 0.1)10 = 3,597,953.10
7. 3 marks: 1 mark for calculating monthly payment from the annuity, 1 mark for estimating daily amount, 1 mark for conclusion
general approach: find payment of an ordinary annuity due given PV
answer: Tina would have about $3.68 to spend each day. She should not retire.
amount available for annuity = $18,800
the monthly payment from the annuity would be:
PMT = $18,800 / [1 - (1.005833)-660 / (0.005833)] = 18,800 / 167.73958 = $112.08
the average daily amount can be estimated as $112.08 x 12 / 365 = $3.68
Some students did this analysis using an annuity due, rather than an ordinary annuity (assuming that Tina needs money at the beginning of each month). This assumption is acceptable.