Lesson Problems on paying debts in Finance
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<H2>Problems on paying debts in Finance</H2> <H3>Problem 1</H3>Payments of $5,000 due in 3 months and $6,000 due in 9 months are to be paid off with simple interest allowed at 13%. How much would be required to pay off the loan today? ( Use today as the focal date). <B>Solution</B> <pre> Since the problem tells nothing about compounding, I will assume that 13% is simple annual interest rate. To answer the question, we should add the present values of the two accounts. Present value of the first account is {{{5000/(1 + (3/12)*0.13)}}} = 4842.62 dollars (rounded). Present value of the second account is {{{6000/(1 + (9/12)*0.13)}}} = 5466.97 dollars (rounded). The total to pay today is the sum 4842.62 + 5466.97 = 10309.59 dollars. <U>ANSWER</U> </pre> <H3>Problem 2</H3>A farmer needs to make the following payments against a loan on his truck: R10000 six months from now, R20000 one year from now, R40000 two years from now. Due to drought, he could not make the first two payments. After 18 months, the farmer immediately made a payment of R50000 against the loan. What single payment should he make two years from now to settle his debt if a simple interest rate of 17% is applied on all the amounts. <B>Solution</B> <pre> Timeline Here's a simple representation of the timeline: 0 months: Loan Disbursement 6 months: Payment of R10,000 due (missed) 12 months: Payment of R20,000 due (missed) 18 months: Down payment of R50,000 made 24 months: Payment of R40,000 due, and the farmer makes his last payment to settle his debt. Single Payment Calculation (a) His accumulated debt 18 months from now is base the interest base the interest scheduled accrued scheduled accrued payment for not paid amount payment for not paid amount not paid "after 18 months from now" not paid "after 18 months from now" at the time 18-6 = 12 months overdue at the time 18-12 = 6 months overdue against the schedule against the schedule 10000 + 10000*(18-6)*(0.17/12) + 20000 + 20000*(18-12)*(0.17/12) = 33400. (b) The balance "at 18 months from now" after paying 50000 against the loan is 33400 - 50000 = -16600, so the farmer is now ahead of his schedule. (!) Now THE BANK owes 16600 to the farmer. The problem does not describe what the bank does in this situation. Formally, at this point I should stop in bewilderment. OK, let's assume that the bank makes nothing, so the farmer's money (16600) lie and wait "to 24 months from now" with no change. (c) Then the single payment "24 months from now" is 40000 - 16600 = 23400. <U>ANSWER</U> </pre> <H3>Problem 3</H3>Julia owes Jose the following amounts: (a) $55,000 at the end of 4 years, (b) $48,000 at the end of 5 years, and (c) $75,000 at the end of 3 years. Jose has an account in the bank, which pays 9% annually compounded quarterly. Therefore, Jose agrees to accept a single payment from Julia at the end of 2 years. to deposit it to this account and to settle Julia's obligations this way. What amount should pay Julia to Jose in 2 years from now ? <B>Solution</B> <pre> In this problem, we should calculate the value of the debt at the end of the two years from today. It is all what should be calculated, since this value represents the Julie's obligation at the end of 2 years. 1. Calculate the value of each obligation at the end of 2 years from now: Obligation 1 at the end of 2 years from now is (rewind two years back from 4 years) Value = {{{55000 / (1 + 0.09/4)^(2*4)}}} Value = 46031.61 Obligation 2 at the end of 2 years from now is (rewind three years back from 5 years) Value = {{{48000 / (1 + 0.09/4)^(3*4)}}} Value = 36752.04 Obligation 3 at the end of 2 years from now is (rewind one years back from 3 years) Value = {{{75000 / (1 + 0.09/4)^(1*4)}}} Value = 68613.25 2. Calculate the Total of All Obligations two years from now Total of the three obligations: 46031.61 + 36752.04 + 68613.25 = 151396.90 <U>ANSWER</U>. The single payment of 151396.90 will settle Julia's obligations at the end of 2 years from now. </pre> <H3>Problem 4</H3>Julia owes Jose the following amounts: (a) $55,000 at the end of 4 years, (b) $48,000 at the end of 5 years, and (c) $75,000 at the end of 3 years. Jose has an account in the bank, which pays 9% annually compounded quarterly. Therefore, Jose agrees to accept a single payment from Julia at the end of 2 years to deposit it to this account and to settle Julia's obligations this way. Julia has an account in other bank, which pays 12% annually composed quarterly. What deposit should make Julia to her account to settle her obligations paying Jose in 2 years from now ? <B>Solution</B> <pre> In this problem, we should calculate the value of the debt at the end of the two years from today. It is all what should be calculated, since this value represents the Julie's obligation at the end of 2 years. 1. Calculate the value of each obligation at the end of 2 years from now: Obligation 1 at the end of 2 years from now is (rewind two years back from 4 years) Value = {{{55000 / (1 + 0.09/4)^(2*4)}}} Value = 46031.61 Obligation 2 at the end of 2 years from now is (rewind three years back from 5 years) Value = {{{48000 / (1 + 0.09/4)^(3*4)}}} Value = 36752.04 Obligation 3 at the end of 2 years from now is (rewind one years back from 3 years) Value = {{{75000 / (1 + 0.09/4)^(1*4)}}} Value = 68613.25 2. Calculate the Total of All Obligations two years from now Total of the three obligations: 46031.61 + 36752.04 + 68613.25 = 151396.90 3. Now the last step is to find the Present value of this sum, 151396.90, in Julia's bank two years back at 12% compounded quarterly. Present Value = {{{151396.90/(1+0.12/4)^(2*4)}}} = 119514.11 (rounded). <U>ANSWER</U>. The single payment of $119,514.11 to Julia's bank now will settle Julia's obligations at the end of 2 years from now. </pre> My other lessons on Finance problems in this site are - <A HREF=https://www.algebra.com/algebra/homework/Finance/Problems-on-simple-interest-accounts.lesson>Problems on simple interest accounts</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Problems-on-discretely-compounded-accounts.lesson>Problems on discretely compounded accounts</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Problems-on-continuously-compounded-accounts-.lesson>Problems on continuously compounded accounts</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Find-future-value-of-an-Ordinary-Annuity.lesson>Find future value of an Ordinary Annuity</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Find-a-value-of-a-regular-deposits-for-an-Ordinary-Annuity.lesson>Find regular deposits for an Ordinary Annuity</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/How-long-will-it-takes-for-an-ordinary-annuity-to-get-an-assigned-value.lesson>How long will it take for an ordinary annuity to get an assigned value?</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Find-future-value-of-an-Annuity-Due-saving-plan.lesson>Find future value for an Annuity Due saving plan</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Regular-withdrawals-from-an-annuity-account.lesson>Regular withdrawals from an annuity account</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Ordinary-annuity-account-with-non-zero-initial-deposit-as-a-combined-total-of-two-accounts.lesson>Ordinary annuity account with non-zero initial deposit as a combined total of two accounts</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Annual-depositing-and-semi-annual-compounding-in-ordinary-annuity-saving-plan.lesson>Annual depositing and semi-annual compounding in ordinary annuity saving plan</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/21-add-Regular-withdrawals-from-a-compounded-account.lesson>Variable withdrawals from a compounded account (sinking fund)</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Present-value-of-an-ordinary-annuity-saving-plan.lesson>Present value of an ordinary annuity cumulative saving plan</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Starting-value-of-a-sinking-fund.lesson>Problems on sinking funds</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Find-the-compounding-rate-of-an-ordinary-annuity.lesson>Find the compounding rate of an ordinary annuity</A> - <A HREF=https://www.algebra.com/algebra/homework/Finance/Accumulate-money-using-ordinary-annuity-then-spend-money-via-sinking-fund.lesson>Accumulate money using ordinary annuity; 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