Chapter 4 Mathematics of Finance
Section 4.1 Simple Interest and Discount
A fee that is charged by a lender to a borrower for the right to use the borrowed funds. The funds can be used to purchase a house, a car, or goods that were charged on a credit card, for example. The interest charge typically is expressed as an annual percentage rate. Slide 1-4
Example1: Solution: Example2: To furnish her new apartment, Maggie Chan borrowed $4000 at 3% interest from her parents for 9 months. How much interest will she pay? Use the formula I Prt, with P 4000, r 0.03, and t 9 /12 3 / 4 years : I Prt 3 I 4000 0.03 90. 4 Thus, Maggie pays a total of $90 in interest. Example3: Slide 1-5
Example 1: A= 4000 + 90 = 4090 Example 2: A= 4500 +2565= 7065 Example 3: A= 500 + 90= 590 A A 2.33 Slide 1-6
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Interest (I), Future Value (FV: A), Present Value (P) Slide 1-8
Section 4.2 Compound Interest Compound interest is interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan. Compound interest can be thought of as interest on interest, and will make a deposit or loan grow at a faster rate than simple interest, which is interest calculated only on the principal amount.
Example: Solution: Suppose that $5000 is invested at an annual interest rate of 3.1% compounded continuously for 4 years. Find the compound amount. In the formula for continuous compounding let P 5000, r.031, x and t 4. Then a calculator with an e key shows that rt.031(4) A Pe 5000 e $5660.08. Slide 1-10
A=9000 e 0.04 5 1.22 A=9000*1.22= 10992.64 Slide 1-11
I= PRT= 10000*.05*3= $1500 A=P+I= 10000+1500=11500 Difference between simple and compound interest 15765.25-11500= $4265.25 Slide 1-12
quarterly 4 4 4 0.0509 5.09 Slide 1-13
Semi annually 2 2 0.0506 2 5.06 Slide 1-14
How much do you need to invest now, to get $10,000 in 10 years at 8% interest rate? PV = $10,000 / (1+0.08) 10 = $10,000 / 2.1589 = $4,631.93 So, $4,631.93 invested at 8% for 10 Years grows to $10,000 Your goal is to have $2,000 in 5 Years. You can get 10%, so how much should you start with? PV = $2,000 / (1+0.10) 5 = $2,000 / 1.61051 = $1,241.84 $1,241.84 will grow to $2,000 if you invest it at 10% for 5 years. Slide 1-15
Section 4.3 Annuities, Future Value, and Sinking Funds
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Example: Solution: A business sets up a sinking fund so that it will be able to pay off bonds it has issued when they mature. If it deposits $12,000 at the end of each quarter in an account that earns 5.2% interest, compounded quarterly, how much will be in the sinking fund after 10 years? The sinking fund is an annuity, with n 4(10) 40. The future value is n (1 i) 1 S R i R 12,000, i.052 / 4, and 40 (1.052 / 4) 1 12,000.052 / 4 $624,369.81. So there will be about $624,370 in the sinking fund. Slide 1-19
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Section 4.4 Annuities, Present Value, and Amortization
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Example: Solution: Jim Riles was in an auto accident. He sued the person at fault and was awarded a structured settlement in which an insurance company will pay him $600 at the end of each month for the next seven years. How much money should the insurance company invest now at 4.7%, compounded monthly, to guarantee that all the payments can be made? The payments form an ordinary annuity. The amount needed to fund all the payments is the present value of the annuity. Apply the present-value formula with R 600, n 7 12 84, and i.047 /12 (the interest rate per month). The insurance company should invest P n 84 1 (1 i) 1 (1.047 / 12) R 600 $42,877.44. i.047 /12 Slide 1-24
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