12, Sam makes a deal to pay $400 a month for 3 years on a car loan at 4.2% annua
ID: 1172665 • Letter: 1
Question
12, Sam makes a deal to pay $400 a month for 3 years on a car loan at 4.2% annual interest compounded monthly to pay for a car. To calculate the present value of an ordinary annuity: 1-1+ 72 To calculate the payment needed to amortize a debt: SD PMT-PV-1 12 72 where PV-present value (S) r-annual interest rate (as a decimal) and n = number of payments per year (periods) PAMT-periodic payment (S) t-time in years To calculate the interest paid on an amortized loan 1 =mPMT-PV If part or all of the price of an item is paid off with periodic payments, then to calculate the cash value of the item: Present Value of Down Cash Value-Periodic PmntsPayment (ifary)Trade-In (ifarny)Explanation / Answer
(14) Trade in Price = T1 = $ 6000, Downpayment = D1 = $ 1000, Monthly Payments = $ 400, Interest Rate = 4.8 % per annum or 0.4 % per month, Tenure = 2 years or 24 months
(a) Present Value(PV) of monthly payments = 400 x (1/0.004) x [1-{1/(1.004)^(24)}] = $ 9136.201
Cash Value = PV + T1 + D1 = 9136.201 + 6000 +1000 = $ 16136.201
(b) Monthly Repayments = $ 400, Tenure = 24 months
Total Amount Paid = 400 x 24 = $ 9600
PV of Repayments = $ 9136.201
Interest Paid = 9600 - 9136.201 = $ 463.799
(c) Previous Loan: Interest Rate = 4.2 % per annum or 0.35 % per month, Tenure = 3 years or 36 months, Monthly Repayments = $ 400,
PV of Repayments = 400 x (1/0.0035) x [1-{1/(1.0035)^(36)}] = $ 13507.56
As the loan in part (14) has a lower PV of cash value as compared to the one in (13), the former should be chosen purely based on lower present value of payouts. Further, the loan in part(14) has a higher upfront payout which might not be convenient for someone who is short of money at present but expects his/her debt paying capacity to go up thereby making the loan in part(13) better. The latter loan makes sense only to someone who does not have immediate cash flow constraints and therefore wants to leverage the benefit of overall reduced payments.
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