Sec 5.3 Compound Interest Compound Interest Formula A = P (1 + where A = Accumulated amount at the end of t years. P = Principal r = Nominal interest rate per year m = Number of conversion periods per year t = Term (number of years) Present Value Formula for Compound Interest P = A(1 + r −mt ) m r mt ) m
Continuous Compound Interest Formula A = P ert where P = Principal r = Annual interest rate compounded continuously t = Time in years A = Accumulated amount at the end of t years Present Value Formula for Continuous Compound Interest P = Ae−rt Example 1 Find the accumulated amount A if the principal P=$2500, interest rate r=7%, after t=10 years, and compounded semiannually. Solution. Compounding semiannually means that m=2, A = P (1 + r mt 0.07 (2)(7) ) = 2500(1 + ) = 2500(1.035)14 = 4046.7 m 2 1
Example 2 Find the interest rate needed for an investment of $5000 to grow to an amount of $7500 in 3 yr if interest is compounded monthly. Solution. Here P=5000, A=7500, t=3, m=12, from the formula 7500 = 5000(1 + Solve this equation, 36 ln(1 + then 1+ then r = 12(1 − e r (3)(12) ) 12
3 r ) = ln 12 2
ln 3 r 2 = e 36 12 ln 3 2 36
)=
Example 3 Find the interest rate needed for an investment of $4000 to double in 5 yr if interest is compounded continuously. Here P=4000, A=(2)(4000)=8000, t=5, from the formula 8000 = 4000e5r Then 5r = ln 2 then ln 2 5 Example 4 How long will it take an investment of $8000 to double if the investment earns interest at the rate of 8% compounded continuously? Solution. Here A=(2)(8000)=16000, and P=8000, and r=0.08, from the formula 16000 = 8000e0.08t r= then ln 2 0.08 In real application, we have options to choose the best strategy. Therefore, we have to compare the results from different plans or different formula. Either we want the shortest time or the largest accumulated compound. t=
2