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?

I tried to go off of the example provided but I don't know where the 1.402 comes from in the first example. Please give detailed feedback. I am trying to understand, just having a difficult time.

Answer 1

When it comes to payment of interests, there can be two broad classifications of interest calculation types Simple and Compound.

When interest is compounded continuously, then the final amount (A) or the future value (F) is expressed as A (or F) = Pe^rt where P is the principal amount, r is the per annum rate of interest and t is the time expressed in years.

When we say compound interest, it actually means that not only the principal sum, but also the accumulated interests are utilized to calculate the total interest payable or due. Continuous compounding refers to the most frequently compounded returns. It is the mathematical limit for a compound interest, where varies from situation to situation as the variables, P, r and t take different values. To spread the interest accrued over the years evenly, concepts of ‘natural log’ and ‘exponential function’ are used because some of their characteristics are apt for achieving the desired purpose. Sometimes when rates are different for different periods, use of natural log and exponential function helps in proper scaling over multiple periods and additionally, it is time consistent. Hence the formula, A= Pe^rt .

In our given example, our principal amount (P) = $100.

Time (t) is 6 years 3 months which is 6.25 years (because 3 months is 3/12 years = ¼ years = 0.25.)

Rate of interest = 5.5% which is 5.5/ 100 = 0.055.

Putting this into the formula, A (or F) = Pe^rt , we get A (or F)= $100 e^(0.055*6.25)

0.055*6.25= 0.34375 and e^0.34375 = 1.4102. This is where the 1.4102 comes from and when multiplied by $100, gives us $141.032 as the future value.

b1) When money is borrowed for auto dealerships, the time is given as 72 months = 72/12 = 6 years. The initial amount paid as down payment is $10000 and the remaining $40000 (because auto dealership costs $50000) is taken as a loan for 6 years at 6 % per annum. The condition says that if the due amount is paid on the maturity date, without being late then only $40000 should ideally be returned. However on being even a minute late, future value compounded continuously at the given rate and time should be returned.

Putting the numbers into the formula A (or F) = Pe^rt we get, A (or F) = $40000e^0.06*6

Here e^0.06*6 = 1.4333 and the amount is $40000*1.4333 = $57332.

Therefore on being even a minute late, the amount payable becomes $57332 instead of $40000.

b2) The above example was been from the point of view of a borrower. The same formula applies when seen from the point of view of the lender. The only difference is that, instead of calculating how much he will pay, he needs to judge whether the amount that he receives on maturity is profitable enough with the interests due, such that it returns not only the principal amount, but also covers the cost of risks undertaken by the lender by lending money. If the borrower specifies the amount that he can pay after a certain number of years, at a certain rate of interest per annum, the lender should calculate how much he should lend so as to make the deal profitable and cover all his costs of lending. This essentially means that from the agreed upon future value receivable (A or F), the lender should calculate the present loan able principal amount (P).

Since the borrower returns $9000 (A or F) at 4.5% per annum = 0.045 (r) after 3 years and 3 months which is 3.25 years (t), the formula is: $9000= P e^(0.045*3.25)

Since we know the value of F ($9000) but don’t know P, by making P the subject of the formula

F= Pe^rt we get P= F/e^rt.

Therefore from our example, P= 9000/ e^(0.045*3.25) = 9000/1.1575= $7775.38 (rounded off to 2 decimal places). We get 1.1575 because e^0.14625 = 1.1575.

Thus, ideally, the lender should lend $7,775.37 or $7,775 when the amount payable is $9000.