Answer to Question #102104 in Financial Math for tabi

Question

1) Isaac borrowed $7000 at 15.5% compounded quarterly 4.5 years ago. One year ago he made a payment of $1800. What amount will extinguish the loan today? (Do not round intermediate calculations and round your final answer to 2 decimal places.)

2) A $4500 loan at 15.25% compounded monthly is to be repaid by three equal payments due 2, 8, and 10 months after the date of the loan. Calculate the size of each payment. (Do not round intermediate calculations and round your final answer to 2 decimal places.)


3) What single payment six months from now would be economically equivalent to payments of $650 due (but not paid) four months ago, and $950 due in 12 months? Assume money can earn 3.7% compounded monthly. (Round your intermediate calculations and final answer to the nearest cent.)

Answer 1

In modern banking practice, interest is accrued and added to fixed capital (capitalized), as a rule, not once, but several times a year - for half-year, quarter, month. Moreover, the contracts usually do not indicate the rate for the accrual period, but the annual rate j and the accrual period. The rate j in this case is called the nominal.

1) S = P (1 + j / m) N

m is the number of interest calculation periods in a year, then each time interest is accrued at the rate of N = m ⋅ n is the total number of calculation periods, n is the loan term in years.

Find the accumulated loan amount:

S = 7 000 (1 + 0.155 / 4) ^{4 * 4.5} = 7 000 (1.03875) ¹⁸ = 7 000 * 1.9824 = 13 876.8

13 876.8-1 800 = 12 076.80 left to pay

2) Find the accumulated loan amount:

S = 4 500 (1 + 0.1525 / 12) ^{12 * 1} = 4 500 (1.01271) ¹² = 4 500 * 1.1636 = 5 236.2

You need to pay in 3 equal installments: 5 236.2 / 3 = 1745.40 i.e. such every amount must be paid in 2, 8 and 10 months

3) P₁ = 650/(1 + 0.037 * 6) = 650 / 1.222 = 531.91 and P₂ = 950/(1 + 0.037 * 12) = 950 / 1.444 = 657.89

650 + 950 = 1600

Economically equivalent amount:

650 + 531.91 + 657.879 = 1839.8

We bring all the values ​​to one period: 12 months, so we exclude the unpaid amount for 4 months.