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Question 1167429: Kenneth ran into some money and decides to invest it for retirement. He has $75,000 to invest over
40 years. Find the effective rates given:
(a) 4.5% growth compounded monthly.
(b) 4.45% growth compounded continuously.
(c) Should Kenneth invest in option (a) or option (b)? Why?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the present value of the investment is 75,000.
the investment period is 40 years.
the discrete compounding growth rate is 4.5% compounded monthly.
the continuous compounding growth rate is 4.45% compounded continuously.
the formula for discrete compounding is f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period
n is the number of time periods.
the formula for continuous compounding is f = p * e ^ (r * t)
f is the future value
p is the present value
r is the interest rate per time period
t is the number of time periods.
when your initial investment is 75,000 and your time period is 40 years, your future value will be calculated as follows:
with discrete monthly compounding, you get:
f = 75000 * (1 + .045/12) ^ (40 * 12) = 452198.6287.
with continuous compounding, you get:
f = 75000 * e ^ (.0445 * 40) = 444739.2314
4.5% compounded monthly gives you more future value than 4.45% compounded continuously.
if you look at the effective annual rates, you will see why this occurs.
with discrete compounding, the effective annual growth factor becomes:
f = (1 + .045/12) ^ 12 = 1.045939825.
with continuous compounding, the effective annual growth factor becomes:
f = e ^ (.0445) = 1.045504977.
with discrete compounding, 75,000 * 1.045939825 ^ 40 = 452198.6287
with continuous compounding, 75,000 * 1.045504977 = 444739.2314
the effective annual growth rate tells you which will give you a greater future value.
the answers to your questions are:
(a) 4.5% growth compounded monthly.
effective annual growth rate is .045939825 or 4.5939825%
(b) 4.45% growth compounded continuously.
effective annual growth rate is .045504977 or 4.5504977%
(c) Should Kenneth invest in option (a) or option (b)? Why?
invest in option a because the effective annual interest rate is higher.
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