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Copyright © Cengage Learning. All rights reserved. 2 The Mathematics of Finance

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Copyright © Cengage Learning. All rights reserved. 2.1 Simple Interest

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3 You deposit $1,000, called the principal or present value, into a savings account. The bank pays you 5% interest, in the form of a check, each year. How much interest will you earn each year? Because the bank pays you 5% interest each year, your annual (or yearly) interest will be 5% of $1,000, or 0.05 1,000 = $50. Generalizing this calculation, call the present value PV and the interest rate (expressed as a decimal) r. Then INT, the annual interest paid to you, is given by INT = PVr.

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4 Simple Interest If the investment is made for a period of t years, then the total interest accumulated is t times this amount, which gives us the following: Simple Interest The simple interest on an investment (or loan) of PV dollars at an annual interest rate of r for a period of t years is INT = PVr t.

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5 Simple Interest Quick Example The simple interest over a period of 4 years on a $5,000 investment earning 8% per year is INT = PVr t = (5,000)(0.08)(4) = $1,600.

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6 Simple Interest Given your $1,000 investment at 5% simple interest, how much money will you have after 2 years? To find the answer, we need to add the accumulated interest to the principal to get the future value (FV) of your deposit. FV = PV + INT = $1,000 + (1,000)(0.05)(2) = $1,100 In general, we can compute the future value as follows: FV = PV + INT = PV + PVr t = PV(1 + r t ).

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7 Simple Interest Future Value for Simple Interest The future value of an investment of PV dollars at an annual simple interest rate of r for a period of t years is FV = PV(1 + r t ). Quick Example The value, at the end of 4 years, of a $5,000 investment earning 8% simple interest per year is FV = PV(1 + r t ) = 5,000[1 + (0.08)(4)] = $6,600.

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8 Example 1 – Savings Accounts In November 2011, the Bank of Montreal was paying 1.30% interest on savings accounts. If the interest is paid as simple interest, find the future value of a $2,000 deposit in 6 years. What is the total interest paid over the period? Solution: We use the future value formula: FV = PV(1 + r t ) = 2,000[1 + (0.013)(6)] = 2,000[1.078] = $2,156.

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9 Example 1 – Solution The total interest paid is given by the simple interest formula. INT = PVr t = (2,000)(0.013)(6) = $156. Note To find the interest paid, we could also have computed INT = FV − PV = 2,156 − 2,000 = $156. cont’d

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10 Simple Interest We often want to turn an interest calculation around: Rather than starting with the present value and finding the future value, there are times when we know the future value and need to determine the present value. Solving the future value formula for PV gives us the following. Present Value for Simple Interest The present value of an investment at an annual simple interest rate of r for a period of t years, with future value FV, is

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11 Simple Interest Quick Example If an investment earns 5% simple interest and will be worth $1,000 in 4 years, then its present value (its initial value) is

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12 Simple Interest Here is some additional terminology on treasury bills: Treasury Bills (T-Bills): Maturity Value, Discount Rate, and Yield The maturity value of a T-bill is the amount of money it will pay at the end of its life, that is, upon maturity. Quick Example A 1-year $10,000 T-bill has a maturity value of $10,000, and so will pay you $10,000 after one year.

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13 Simple Interest The cost of a T-bill is generally less than its maturity value. In other words, a T-bill will generally sell at a discount, and the discount rate is the annualized percentage of this discount; that is, the percentage is adjusted to give an annual percentage. Quick Example A 1-year $10,000 T-bill with a discount rate of 5% will sell for 5% less than its maturity value of $10,000, that is, for $9,500.

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