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Present Value of an Annuity; Amortization Section 3-4 Prof. Nathan Wodarz Math 109 - Fall 2008 Contents 1 Present Value of an Annuity 2 1.1 Present Value of an Ordinary Annuity . . . . . . . . . . . . . . . 2 1.2 Problem Solving Strategy . . . . . . . . . . . . . . . . . . . . . . 3 2 Amortization 4 2.1 Amortization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Equity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1 1 Present Value of an Annuity 1.1 Present Value of an Ordinary Annuity Present Value of an Ordinary Annuity • Last section: Paid into an account gradually, accumulated savings • This section: One lump sum deposited at beginning, slowly paid back – Loans – Annuities (as an insurance product) Present Value of an Ordinary Annuity 1 − (1 + i)−n PV = PMT i • PV = present value (amount) (often denoted S ) • PMT = periodic payment (at end of each period) (often denoted R) • i = rate per period • n = number of payments (periods) • To solve for the payment: i PMT = PV 1 − (1 + i)−n • We will not solve for i 2 Present Value of an Ordinary Annuity Problem 1. Find the present value of the ordinary annuity, with payments of $50 made quarterly for 10 years at 8% interest compounded quarterly. A. $490.90 B. $1345.13 C. $1367.77 D. $1376.77 E. None of the above. Present Value of an Ordinary Annuity Problem 2. Tammy borrowed $10,000 to purchase a new car at an annual interest rate of 11%. She is to pay it back in equal monthly payments over a 5-year period. How much total interest will be paid over the period of the loan? Round to the nearest dollar. A. $92 B. $1435 C. $3045 D. $3630 E. None of the above. 1.2 Problem Solving Strategy Problem Solving Strategy • In general, single payments will be simple or compound interest – Look for hints as to whether simple or compound interest is used – Shorter time periods are often (but not always) simple interest • Continuing payments involve annuities 3 – If account is increasing in value - future value problem – If account is decreasing in value - present value problem – Amortization problems (below) are always present value problems 2 Amortization 2.1 Amortization Amortization • Borrow money from bank • Repay it in equal installments • View as bank buying annuity from you • After last payment back to bank, loan is amortized (literally “killed oﬀ”) • Payments determined by earlier formula i PMT = PV 1 − (1 + i)−n Amortization Problem 3. Find the payment necessary to amortize a loan of $10,100 at 12% compounded monthly, if there are to be 48 monthly payments. A. $261.74 B. $265.97 C. $266.16 D. $1217.28 E. None of the above. 4 Amortization Problem 4. The monthly payments on a $73,000 loan at 13% annual interest are $807.38. How much of the ﬁrst monthly payment will go toward the principal? A. $16.55 B. $104.96 C. $702.42 D. $790.83 E. None of the above. Amortization Schedules • How can we compute outstanding loan balances? • Not as simple as just subtracting payments – This ignores interest • Suppose there are n payments left. Outstanding balance is present value of an annuity with same payments as before, but with the fewer number of payments. Amortization Schedules Problem 5. A $7,000 debt is to be amortized in 15 equal monthly payments of $504.87 at 12% annual interest on the unpaid balance. What is the unpaid bal- ance after the second payment? A. $5,990.26 B. $6,860.00 C. $6,971.87 D. $8,126.78 E. None of the above. 5 2.2 Equity Equity • For an asset that you take out a loan on, equity measures how much of that asset you actually “own” • Equity = (current net market value) - (unpaid loan balance) • A home equity loan is a loan taken out using the equity in your house as collateral – Essentially, taking out a second mortgage • Equity can be negative - you owe more than the asset is worth Equity Problem 6. A home was purchased 14 years ago for $70,000. The home was ﬁnanced by paying a 20% down payment and signing a 25 year mortgage at 8.5% compounded monthly on the unpaid balance. The market value is now $100,000. The owner wishes to sell the house. How much equity (to the nearest dollar) does the owner have in the house after making 168 monthly payments? A. $22,913 B. $51,127 C. $51,768 D. $61,414 E. None of the above. Summary Summary You should be able to: • Calculate present value of an annuity • Understand amortization • Be able to compute unpaid balances and equities 6

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posted: | 3/18/2010 |

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