Question 1167429
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.