SECTION 5.7 Financial Models 347 Exploration To observe the effects of compounding interest monthly on an initial deposit of $1, graph Y r 1 12 x 1 12 ( ) = + with r 0.06 = and r 0.12 = for x 0 30. ≤ ≤ What is the future value of $1 in 30 years when the interest rate per annum is r 0.06 = (6%)? What is the future value of $1 in 30 years when the interest rate per annum is r 0.12 = (12%)? Does doubling the interest rate double the future value? Comparing Investments Using Different Compounding Periods Investing $1000 at an annual rate of 10% compounded annually, semiannually, quarterly, monthly, and daily will yield the following amounts after 1 year: EXAMPLE 2 Annual compounding ( ) = n 1 : ( ) ( ) = ⋅ + = + = A P r 1 $1000 1 0.10 $1100.00 Semiannual compounding ( ) = n 2 : ( ) ( ) = ⋅ + = + = A P r 1 2 $1000 1 0.05 $1102.50 2 2 Quarterly compounding ( ) = n 4 : ( ) ( ) = ⋅ + = + = A P r 1 4 $1000 1 0.025 $1103.81 4 4 Monthly compounding ( ) = n 12 : ( ) ( ) = ⋅ + = + = A P r 1 12 $1000 1 0.10 12 $1104.71 12 12 Daily compounding ( ) = n 365 : ( ) ( ) = ⋅ + = + = A P r 1 365 $1000 1 0.10 365 $1105.16 365 365 For example, to rework Example 1, use = = = P r n $1000, 0.02, 4 (quarterly compounding), and = t 1 year to obtain ( ) ( ) = ⋅ + = + = ⋅ A P r n 1 1000 1 0.02 4 $1020.15 nt 4 1 The result obtained here differs slightly from that obtained in Example 1 because of rounding. Now Work PROBLEM 7 From Example 2, note that the effect of compounding more frequently is that the amount after 1 year is higher.This leads to the following question:What would happen to the amount after 1 year if the number of times that the interest is compounded were increased without bound? Let’s find the answer. Suppose that P is the principal, r is the per annum interest rate, and n is the number of times that the interest is compounded each year. The amount A after 1 year is ( ) = ⋅ + A P r n 1 n Rewrite this expression as follows: ( ) ( ) = ⋅ + = ⋅ + ⎛ ⎝ ⎜⎜ ⎜⎜ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ = ⋅ + ⎛ ⎝ ⎜⎜ ⎜⎜ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ = ⋅ + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ A P r n P n r P n r P h 1 1 1 1 1 1 1 n n n r r h r / (3) ↑ =h n r
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