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1 Chapter 6 Finance 6.1 Simple Interest and Sequences Review: I = Prt (Simple Interest) What does Simple mean? Not Simple = Compound I part Interest is calculated once, at the end. Ex: (#10) If you borrow $1600 for 2 years at 14% annual simple interest, how much must you repay at the end of the 2 years? I -_ Prt or 5- Pt Prt I= = $448 =P( Hrt ) = = ) 5= / Ex: (#14) Jenny Reed bought SSX stock for $16 per share. The annual dividend was $1.50 per share, and after 1 year, SSX was selling for $35 per share. Find the simple interest rate of growth of her money.

2 6.2 Compound Interest and Geometric Sequences Ex: If $3000 is invested for 4 years at 10% compounded annually, how much interest is earned? a) Use a table to solve: Year Beginning 10% Interest Ending Principal Principal = I = P + I = P = = = = b) How much more interest did you earn, compared to $3000 invested for 4 years at 10% simple interest?

3 Compound Interest Formula: S = P(1+r) n S = Future Value P = Principal or Present Value r = Interest rate per period n = Total number of periods Ex: Use the formula to answer the previous problem. ($3000 is invested at 10%, compounded annually, for 4 years.) Future Value (Periodic Compounding) If P is invested for t years at a nominal interest rate r, compounded m times per year, then the number of compounding periods is given by n = mt, the interest rate per compounding period is i = r/m, and the future value is! = #(% + ') ) = # % + +,,- Ex: (Ex 3) Jim and Eden want to have $200,000 in Maura s college fund on her 18 th birthday, they want to know the impact on this goal of having $10,000 invested at 9.8%, compounded quarterly, on her first birthday. To advise Jim and Eden on this, find a) the future value of the $10,000 investment b) the amount of compound interest that the investment earns

4 Continuous Compounding Frequency # of Periods per Year (m) Future Value of $1 Annual / / / = 2 Semi-Annual / 1 1 = 2.25 Quarterly / 4 4 = Monthly / Weekly / /1 :; /1 = :; = Daily Hourly / <=; / >=4; <=; = >=4; = Each Minute 518, / :/>4;; :/>4;; = What if you keep going? As m gets bigger, S gets bigger up to a point. In fact, no matter how big m is, that dollar will never be worth more than = e. lim D F G D = H Future Value with Continuous Compounding: In general, if P is invested for t years at a nominal rate r (expressed as a decimal,) compounded continuously, then the future value is given by: S = Pe rt

5 Ex: Find the future value if $1000 is invested for 20 years at 8%, compounded continuously. Ex: What amount must be invested at 6.5 %, compounded continuously, so that it will be worth $25,000 after 8 years? Doubling Time Ex: How long does it take an investment of $10,000 to double if it is invested at a) 8 %, compounded annually b) 8%, compounded monthly c) 8%, compounded continuously c) Does the initial investment amount matter in your calculations?

6 6.3 Future Values of Annuities Annuity: Future Value: Ex: I want to invest $100 at the end of every year at 10%. How much will I have at the end of 5 years? To calculate this, I keep track of each deposit separately: End of 1 st Year End of 2 nd Year End of 3 rd Year End of 4 th Year End of 5 th Year (1.10) 100 (1.10) (1.10) (1.10) (1.10) 100 (1.10) (1.10) (1.10) 100 (1.10) (1.10) 100 The Value at the End of the 5 th Year = (1.10) (1.10) (1.10) (1.10) 4 (This is called a Geometric Series) S = 100 [ /./; J K/] ;./;

7 The Future Value of an Annuity In general, if payments are made at the end of each period, S = M [ /NO P K/] O, where R = The investment amount each period n = The number of periods I = The interest rate per period Future Value of an Annuity Due: Annuity Due refers to the fact that these annuities are paid at the beginning of the period, rather than at the end. S = M [ /NO P K/] O (1 + Q) Ex: Twins Problem (Ex #1 & 2) 1) Twin #2 is 29 years old. He wants to deposit $2000 per year in an account paying 10%, compounded annually, for 36 years (until he s 65.) How much will he have in 36 years?

8 2) Twin #1 starts early and deposits $2000 in an account paying 10% every year, starting from the moment she s 21. She follows this plan for 8 years, and then stops making deposits. The money continues to earn interest for an additional 36 years. How much will she have at the end of the 36 years? Sinking Fund: Periodic deposits that will produce a sum on a specified date, usually to save up for a purchase or to pay off a debt.

9 6.4 Present Value of Annuities Ordinary Annuity: An annuity where the payments are made at the end of each period. Present Value of an Ordinary Annuity: The sum of money required to purchase an ordinary annuity. Present Value of an Ordinary Annuity: A = M /K(/NO)RP O Ex: Find the lump sum that one must invest in an annuity in order to receive $1000 at the end of each month for the next 16 years, if the annuity pays 9%, compounded monthly.

10 Bonds A financial instrument with fixed payments, called coupons, and a fixed value at maturity. Par Value: The original price of a new bond. Coupon: Interest payment to the person who holds the bond. Maturity Value: The value of a bond at the end of its term, also known as the par value. Yield Rate: The rate determined by the price paid for the bond. Selling at a Discount: When the Coupon Rate is less than the yield rate (ie the market price is less than the maturity value.) Selling at a Premium: When the Coupon Rate is more than the yield rate (ie the market price is more than the maturity value.) Ex: Suppose a 15-year corporate bond has a maturity value of $10,000 and coupons at 5% paid semiannually. If an investor wants to earn a yield of 7.2% compounded semiannually, what should he or she pay for this bond? Step 1: Find the coupon payment i = 0.05/2 = ($10,000) (0.025) = $250 Step 2: Find the present value of the $10,000 maturity value Here, we look at the desired yield: 7.2% = annually, which means that I = 0.072/2 = and n = 15 * 2 = 30, since it s compounded semiannually over 15 years. S = P(1 + i) n => 10,000 = P ( ) 30 so P =

11 Step 3: Find the present value of the coupon (interest) payments From Step 1, the coupon payments are $250. There are 30 of them, so A = M /K(/NO)RP O A = 250 /K(/N;.;<=)RST ;.;<= = Step 4: Add the results of Steps 2 and 3 Price = Present value of the maturity value + Present value of the coupon payments = 6.5 Loans and Amortization Amortize: Repaying a loan with equal payments (Literally killing a loan.) Amortization Schedule: See pages 406 & 407 If the present value of an ordinary annuity is given by A = M /K(/NO)RP O then what s the formula for the payment amount for a given present value?

12 Amortization Formula For a debt of A with interest at a rate of I per period, amortized by n equal periodic payments (made at the end of each period), the size of each payment is given by: R = A O /K(/NO) RP Ex: You buy a house for $800,000. You make a $50,000 down payment and agree to amortize the rest of the debt with monthly payments over the next 30 years. If the interest on the debt is 5%, compounded monthly, find the following: a) The size of the payments: b) The total amount of the payments c) The total amount of interest paid.

13 Ex: You make payments for 10 years, and would like to know how much you still owe on the loan. How do you calculate it? The unpaid balance, or payoff amount, after k payments have been made is the present value of an ordinary annuity with n k payments: Payoff Amount Unpaid balance = A = R /K /NO R PRU O

14 (If time) Ex 4 in the book: Four years ago Benencorp decided to expand its production capacity and borrowed $1.3 million for 20 years at 5.2% compounded quarterly. After making 16 quarterly payments of $26,235.37, Benencorp is considering refinancing this loan for 15 years at 4.8% compounded quarterly, with refinancing charges of $5000 added to the amount of the new (refinanced) loan. a) Find the payoff amount of Benencorp s original loan b) Find the amount of the new loan and the new quarterly payment. c) Should Benencorp refinance?