{"id":1931,"date":"2021-06-08T00:19:49","date_gmt":"2021-06-08T04:19:49","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/?post_type=chapter&p=1931"},"modified":"2021-07-09T11:45:44","modified_gmt":"2021-07-09T15:45:44","slug":"loans","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/businessmathematics\/chapter\/loans\/","title":{"raw":"5.3 Loans and Down Payments","rendered":"5.3 Loans and Down Payments"},"content":{"raw":"
\r\n

Learning Outcomes<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nCalculate the duration, selling price and cost of financing for loans with down payments.\r\n\r\n<\/div>\r\n<\/div>\r\n\"\"The first type of debt annuity we will examine is the ([pb_glossary id=\"1978\"]loan[\/pb_glossary]) \u2014 an annuity where we borrow an initial amount of money (PV) and repay the loan with a series of equal-sized payments (PMT), at regular intervals, over the course of a fixed time period. At the end, we owe nothing (FV = 0).\r\n\r\n\r\n\r\n\r\n\r\n\r\n
PV<\/th>\r\nInterest<\/th>\r\nPMT<\/th>\r\nFV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Amount Borrowed<\/strong><\/td>\r\n+ %Charged<\/strong><\/td>\r\n=\u00a0 Regular Payments<\/strong><\/span><\/td>\r\n+ 0<\/strong><\/span><\/td>\r\n<\/tr>\r\n
+<\/span><\/strong><\/td>\r\n+<\/span><\/strong><\/td>\r\n\u2212<\/span><\/strong><\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNote: PV and PMT have opposite signs. To better understand this: PV is the initial amount we receive (loan amount) and PMT is the repayments of that loan that we must pay back after receiving the loan. The interest adds to the amount owed (we are charged interest each period on what we owe).\r\n\r\nSee the sections below for key formulas, tips and examples related to loans and down payments.\r\n

the time Required to Repay a Loan & Key Questions<\/h1>\r\nFor loans, you can be asked to calculate the time, rate, initial amount borrowed or size of the loan payments. In the example below, we will calculate the amount of time required for Zhang Min to repay her line of credit that she took out to pay for school.\r\n

EXAMPLE 5.3.1<\/h2>\r\nToday, Zhang Min borrowed $10,000 from her line of credit. Her line of credit charges 3.75%, compounded monthly. She can afford to pay $300 per month on her line of credit with the first payment one month from today. How many years will it take her to repay her line of credit?\r\n\r\nBefore we determine what to enter into your BAII Plus, let's ask a few important questions:\r\n
\r\n

Key Questions: Loans<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\n[h5p id=\"17\"]\r\n\r\n[h5p id=\"18\"]\r\n\r\n<\/div>\r\n<\/div>\r\nThis gives us the following values in the BAII Plus:\r\n\r\n\r\n\r\n\r\n\r\n
B\/E<\/th>\r\nP\/Y<\/th>\r\nC\/Y<\/th>\r\nN<\/th>\r\nI\/Y<\/th>\r\nPV<\/th>\r\nPMT<\/th>\r\nFV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
END<\/td>\r\n12<\/td>\r\n12<\/td>\r\nCPT<\/strong> 35.255<\/td>\r\n3.75<\/td>\r\n+10,000<\/td>\r\n\u2212300<\/span><\/td>\r\n0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nZhang Min will make 35 full-sized payments and a smaller final payment at the end of 36 months.\r\nTo calculate the number of years, we use the following formula:\r\n

[latex] \\textrm{Number of Years} = \\frac{\\textrm{N}}{\\textrm{P\/Y}} = \\frac{36}{12} = 3\\textrm{ years} [\/latex]<\/p>\r\nConclusion: It will take 3 years for Zhang Min to repay her line of credit.\r\n\r\n[h5p id=\"19\"]\r\n\r\n \r\n

Down Payments & Key Questions for Unknown Selling prices<\/h1>\r\nA [pb_glossary id=\"1994\"]down payment[\/pb_glossary] is a lump-sum payment made before you take out the loan. Down payments are often required when taking out a car loan or mortgage. The down payment will save money in interest charges over the duration of the loan. This is because it reduces the amount borrowed (PV):\r\n

[latex] \\textrm{Amount Borrowed (PV)} = \\textrm{Selling Price} - \\textrm{Down Payment} [\/latex]<\/p>\r\nIf we are asked to calculate the selling price when we know the value of the [pb_glossary id=\"1994\"]down payment[\/pb_glossary] and amount borrowed (PV), we can rework the above equation to solve for the selling price:\r\n

[latex] \\textrm{Selling Price} = \\textrm{Amount Borrowed (PV)} + \\textrm{Down Payment} [\/latex]<\/p>\r\n \r\n

EXAMPLE 5.3.2<\/h2>\r\nRaj wants to buy an All Wheel Drive Tesla Model S. He can take out a 5-year loan with Tesla Lending<\/a>. He must make a $10,000 down payment followed by monthly payments of $2,074\/month with the first payment one month after the car is purchased. Tesla Lending charges Raj 3.75% effective on the loan. What is the selling price of the car?\r\n\r\nBefore we determine what to enter into your BAII Plus, let\u2019s ask a few important questions:\r\n
\r\n

Key Questions: Loans with Known Down Payments and Unknown Selling Prices<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\n[h5p id=\"21\"]\r\n\r\n[h5p id=\"22\"]\r\n\r\n<\/div>\r\n<\/div>\r\nFrom the above answers and the values given in the problem, enter the following in the BAII Plus:\r\n\r\n\r\n\r\n\r\n\r\n
B\/E<\/th>\r\nP\/Y<\/th>\r\nC\/Y<\/th>\r\nN<\/th>\r\nI\/Y<\/th>\r\nPV<\/th>\r\nPMT<\/th>\r\nFV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
END<\/td>\r\n12<\/td>\r\n1<\/td>\r\n5\u00d712=60<\/td>\r\n3.75<\/td>\r\nCPT<\/strong> 113,484.44<\/td>\r\n\u22122074<\/span><\/td>\r\n0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nBecause PV equals to the amount borrowed, we know that Raj will borrow $113,484.44. We still need to calculate the selling price. In order to do this, we use the following formula:\r\n

[latex] \\begin{align*}\\textrm{Selling Price} &= \\textrm{Amount Borrowed (PV)} + \\textrm{Down Payment} \\\\ &= \\$113,484.44 + \\$10,000 \\\\&= $123,484.44 \\end{align*}[\/latex]<\/p>\r\nConclusion: The selling price is $123,484.44 for the All Wheel Drive Tesla Model S.\r\n\r\n[h5p id=\"23\"]\r\n\r\n \r\n

Cost of Financing on Loans & Practice Exercise<\/h1>\r\nWe call the interest charged on loans the [pb_glossary id=\"1979\"]cost of financing[\/pb_glossary]. The same interest formula is used as before:\r\n

[latex] \\begin{align*} \\textrm{Interest} &= \\textrm{Money Out} - \\textrm{Money In} = \\textrm{\\$ OUT} - \\textrm{\\$ IN} \\end{align*} [\/latex]<\/p>\r\nWe consider the amount borrowed, PV, to be \u201cmoney in\u201d since we are receiving this money at the start of the loan. We consider the regular payments, PMT to be \"money out.\" To calculate the total amount paid from the regular payments, calculate PMT\u00d7N since we will make N payments of size PMT. Finally, FV equals zero because nothing is owed at the end of a loan.\r\n\r\n\r\n\r\n\r\n\r\n\r\n
PV<\/th>\r\nInterest<\/th>\r\nPMT<\/th>\r\nFV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Amount Borrowed<\/strong><\/td>\r\n+ %Charged<\/strong><\/td>\r\n=\u00a0 Regular Payments<\/strong><\/span><\/td>\r\n+ 0<\/strong><\/span><\/td>\r\n<\/tr>\r\n
$ IN<\/span><\/td>\r\n$ IN<\/span><\/td>\r\n$ OUT<\/span><\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis gives us the following equation for cost of financing for loans:\r\n

[latex] \\begin{align*} \\textrm{Interest Earned} &= \\textrm{\\$ OUT} - \\textrm{\\$ IN}\\\\ &=\\textrm{Regular Payments} - \\textrm{Amount Borrowed}\\\\ &= \\textrm{PMT}\\times\\textrm{N}-\\textrm{PV} \\end{align*} [\/latex]<\/p>\r\nNotice that the down payment is not used in the above formula. Only the loan payment size and amount borrowed are used to calculate the cost of financing for a loan.\r\n\r\n \r\n

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Check Your Knowledge 5.3.2<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\n[h5p id=\"20\"]\r\n\r\n[h5p id=\"24\"]\r\n\r\n<\/div>\r\n<\/div>\r\n \r\n
\r\n

Check Your Knowledge 5.3.3<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nRaj feels like he can\u2019t afford the monthly payments for the 5-year loan (see Example 2). Instead, he takes out an 8-year loan. He will still make the $10,000 down payment followed by monthly payments with the first payment one month after the car is purchased. Tesla charges 4.25% effective on the 8-year car loan. How much extra interest will Raj pay if he takes out the 8-year loan instead of the 5-year loan (from Example 2)?\r\n\r\nFirst we calculate the size of Raj's new payments. Drag in the values in the correct calculator keys:\r\n\r\n[h5p id=\"27\"]\r\n\r\n[h5p id=\"28\"]\r\n\r\nBecause PMT = -1,392.25, we know that Raj will need to pay $1,392.25 per month if he takes out the 8-year loan. Use this payment size to calculate the amount of interest (cost of financing) on the 8-year loan:\r\n\r\n[h5p id=\"29\"]\r\n\r\n[h5p id=\"30\"]\r\n\r\nConclusion: Raj will pay an extra $9,216 in interest if he takes out the 8-year loan instead of the 5-year loan.\r\n\r\n<\/div>\r\n<\/div>\r\n

Your Own Notes<\/h1>\r\n