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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. 2.2 Compound Interest

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3 You deposit $1,000 into a savings account. The bank pays you 5% interest, which it deposits into your account, or reinvests, at the end of each year. At the end of 5 years, how much money will you have accumulated? Let us compute the amount you have at the end of each year. At the end of the first year, the bank will pay you simple interest of 5% on your $1,000, which gives you PV(1 + r t) = 1,000( ) = $1,050.

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4 Compound Interest At the end of the second year, the bank will pay you another 5% interest, but this time computed on the total in your account, which is $1,050. Thus, you will have a total of 1,050( ) = $1, If you were being paid simple interest on your original $1,000, you would have only $1,100 at the end of the second year. The extra $2.50 is the interest earned on the $50 interest added to your account at the end of the first year.

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5 Compound Interest Having interest earn interest is called compounding the interest. We could continue like this until the end of the fifth year, but notice what we are doing: Each year we are multiplying by So, at the end of 5 years, you will have 1,000( ) 5 ≈ $1,

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6 Compound Interest It is interesting to compare this to the amount you would have if the bank paid you simple interest: 1,000( × 5) = $1, The extra $26.28 is again the effect of compounding the interest. Banks often pay interest more often than once a year. Paying interest quarterly (four times per year) or monthly is common.

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7 Compound Interest If your bank pays interest monthly, how much will your $1,000 deposit be worth after 5 years? The bank will not pay you 5% interest every month, but will give you 1/12 of that, or 5/12% interest each month. Thus, instead of multiplying by every year, we should multiply by /12 each month. Because there are 5 12 = 60 months in 5 years, the total amount you will have at the end of 5 years is

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8 Compound Interest Compare this to the $1, you would get if the bank paid the interest every year. You earn an extra $7.08 if the interest is paid monthly because interest gets into your account and starts earning interest earlier. The amount of time between interest payments is called the compounding period.

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9 Compound Interest The following table summarizes the results above. The preceding calculations generalize easily to give the general formula for future value when interest is compounded.

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10 Compound Interest Future Value for Compound Interest The future value of an investment of PV dollars earning interest at an annual rate of r compounded (reinvested) m times per year for a period of t years is or FV = PV(1 + i ) n where i = r/m is the interest paid each compounding period and n = mt is the total number of compounding periods.

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11 Compound Interest Quick Example To find the future value after 5 years of a $10,000 investment earning 6% interest, with interest reinvested every month, we set PV = 10,000, r = 0.06, m = 12, and t = 5. Thus,

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12 Example 1 – Savings Accounts In November 2011, the Bank of Montreal was paying 1.30% interest on savings accounts. If the interest is compounded quarterly, 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 with m = 4:

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13 Example 1 – Solution The total interest paid is INT = FV − PV = 2, − 2,000 = $ cont’d

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14 Compound Interest Example 1 illustrates the concept of the time value of money: A given amount of money received now will usually be worth a different amount to us than the same amount received some time in the future. In the example above, we can say that $2,000 received now is worth the same as $2, received 6 years from now, because if we receive $2,000 now, we can turn it into $2, by the end of 6 years. We often want to know, for some amount of money in the future, what is the equivalent value at present.

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15 Compound Interest As we did for simple interest, we can solve the future value formula for the present value and obtain the following formula. Present Value for Compound Interest The present value of an investment earning interest at an annual rate of r compounded m times per year for a period of t years, with future value FV, is or where i = r/m is the interest paid each compounding period and n = mt is the total number of compounding periods.

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16 Compound Interest Quick Example To find the amount we need to invest in an investment earning 12% per year, compounded annually, so that we will have $1 million in 20 years, use FV = $1,000,000, r = 0.12, m = 1, and t = 20: Put another way, $1,000, years from now is worth only $103, to us now, if we have a 12% investment available.

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17 Compound Interest We have mentioned that a bond pays interest until it reaches maturity, at which point it pays you back an amount called its maturity value or par value. The two parts, the interest and the maturity value, can be separated and sold and traded by themselves. A zero coupon bond is a form of corporate bond that pays no interest during its life but, like U.S. Treasury bills, promises to pay you the maturity value when it reaches maturity.

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18 Compound Interest Zero coupon bonds are often created by removing or stripping the interest coupons from an ordinary bond, so are also known as strips. Zero coupon bonds sell for less than their maturity value, and the return on the investment is the difference between what the investor pays and the maturity value. Although no interest is actually paid, we measure the return on investment by thinking of the interest rate that would make the selling price (the present value) grow to become the maturity value (the future value).

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19 Compound Interest Effective Interest Rate The effective interest rate r eff of an investment paying a nominal interest rate of r nom compounded m times per year is To compare rates of investments with different compounding periods, always compare the effective interest rates rather than the nominal rates.

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20 Compound Interest Quick Example To calculate the effective interest rate of an investment that pays 8% per year, with interest reinvested monthly, set r nom = 0.08 and m = 12, to obtain

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