b1. F = Pe rt , which assumes continuous compounding, says that the Future value
ID: 3186146 • Letter: B
Question
b1. F = Pert, which assumes continuous compounding, says that the Future value (F) of an amount (P) invested today at an annual rate (r), expressed as a decimal for the time (t) in years, is given by the function. EXAMPLE: invest $100 at the annual rate of 5 1/2% for 6 years and 3 months and you should get back (at the end of the time), F = $100e(0.055)(6.25) = $100e(0.3438) = $100(1.4102) = $141.02. Some auto dealerships are advertising, “72 months, no interest, no payments” However, on the maturity date you have to pay the whole amount owed. If you are even a minute late, then you owe all the interest you avoided, too. You want a $50000 auto for which, after down payment, you qualify for a $40000 loan as advertised. Six percent per annum will be assessed if you are late paying. What total amount will you owe if you are late?
b2. Alternatively, if a borrower tells you that he needs a loan for 6 years and 3 months and will pay you an annual rate of 5 1/2% for the loan, but will only give you $141.02 back at the end of the loan term, you should only loan him $100 today. Here is a loan proposition more in line with current rates. A borrower agrees to pay you 4.5% annually for 3 years and 3 months. At the end of the term he will make a balloon payment of $9000 to repay the loan and interest. What amount (P) does the formula P = F/ert indicate you should loan this prospect?
Explanation / Answer
b1. Here P = $40000, r = 6% = 0.06, t = 6 years
Total amount, F = P.e^rt = 40000×e^0.06×6
= 40000×e^0.36 = $ 57333.18
b2) Given, F = $9000, t = 3.25 year, r = 4.5% = 0.045
Thus, P = F/e^rt = 9000/e^0.045×3 .25 = 9000/1.157
= $ 7776.05
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