5.3 Amortization and Sinking Funds

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1 5.3 Amortization and Sinking Funds Sinking Funds A sinking fund is an account that is set up for a specific purpose at some future date. Typical examples of this are retirement plans, saving money for a trip, a company accumulating capital to make a big purchase, etc. In the problems related to the sinking funds, we are usually given the amount to be accumulated by specific date in the future as the future value of an annuity (FV), and try to solve for the payment (PMT). PV will be zero. Example: A corporation creates a sinking fund in order to have $810,000 to replace machinery in 10 years. The account is at 3.4% compounded weekly. (a) How much should be placed into the account at the end of each week to meet this goal? (b) How much is the account worth after 6 years? Example: Suppose Fred wants to take a trip to Europe at the end of his college career. If Fred needs $4,500 for this trip, how much would he need to deposit into an account paying 8% year compounded quarterly if he expects to graduate 4 years from now? How much interest did he earn? Interest Earned: Interest earned on a sinking fund where N payments of P MT dollars have accumulated to F V dollars is given by, 1

2 Example: Andrea wishes to accumulate a retirement fund of $450,000. How much should she deposit each month into her retirement account, which pays interest at a rate of 3.5%/year compounded monthly, to reach her goal upon retirement 40 years from now? (Round answer to the nearest cent.) Example: Jason is currently planning to retire in 25 years and wishes to withdraw $8,000 per month from his retirement account for 25 years starting at that time. How much must he contribute each month into a retirement account earning interest at the rate of 6% per year compounded monthly to meet his retirement goal. What if the account after retirement is at 4% per year compounded quarterly? 2

3 Amortization Amortization is paying off an amount owed over time by making planned, incremental payments of principal and interest. To amortize a loan means to kill it off. The process can be shown using amortization table. This table lists the payment made each period, how much goes to interest, how much goes towards the original balance of the loan, and how much of the original balance of the loan is left. With amortization problems we will solve for the payment PMT required to amortize a loan of PV dollars. Amortization means we pay-off the loan so that FV is zero. Example: Find the annual payment needed to amortize for a $8,000 loan amortized over three years if the interest rate is 4.7% per year compounded annually. Then construct the amortization table (Round answers to the nearest cent.) Interest Paid: Interest paid on an amortization of a loan of PV dollars where N payments of PMT dollars have been made is given by, 3

4 The equity of a home (or any other piece of property) is Equity = The original value of the house What you still owe. For example, if someone owns a car worth $15,000 (original value), but owes $5,000 on a loan against that car, the car represents $10,000 of equity. Equity is a measure of how much of the property you own. When you get a loan, usually the bank wants some collateral: something that the bank can take if you dont pay your loan. Usually its a house or some other property. In order to assess how much they can take from you, they will want the equity of the property: how much that property is worth in your name. Example: A family has purchased a house for $180,000. They made an initial down payment of $20,000 and secured a mortgage with interest charged at the rate of 8%/year compounded monthly on the unpaid balance. The loan is to be amortized over 30 years. (Round answers to the nearest cent.) (a) What monthly payment will the family be required to make? (b) How much does the family still owe after 10 years (c) What will be their equity after 10 years? 4

5 Example: A group of private investors purchased a condominium complex for $4 million. They made an initial down payment for 12% of the total and obtained a loan for the rest. The loan is to be paid off over 14 years at an interest rate of 11% per year compounded quarterly. (a) What is the equity of the complex at the start of the loan? (b) What is the quarterly payment for the loan? (c) What is the equity of the complex after 10 years? Example: Johnny wants to buy a $50,000 pirate ship in the Caribbean He puts $10,000 down and gets a loan for the rest. The loan is to be amortized over the next 10 years at a rate of 12% per year compounded monthly (a) Find the monthly payment for this loan and construct the first 3 lines of the amortization table. (b) What is the equity of the ship after 6 years? 5

6 Example: George, Fred s twin brother, wants to also take this trip (need $4500). However, he does not have the same account as Fred. Instead he will put a down payment of $100, and make monthly payments into an account with 8.25% interest compounded monthly. How much should each payment be? If George decides to make $80 payment each month. How much money would he have in 4 years? 6