This problem deals with comparing different investment options based on their interest rates and compounding frequencies. The question provides three choices:

Option A: 12% compounded annually. Option B: 11.7% compounded semiannually (twice a year). Option C: 11.5% compounded continuously. The answer uses formulas and calculates the future value (C) of each investment after 1, 5, and 20 years for comparison. Let's break down the explanation:

A) Investment with 12% annual compounding:

Formula: Ct = PV (1 + r)t Ct: Future value after t years. PV: Present value (initial investment, assumed $1 here). r: Annual interest rate (12% expressed as a decimal = 0.12). t: Time in years. Calculations: C1 = $1 (1 + 0.12)1 = $1.1200 (year 1). C5 = $1 (1 + 0.12)5 = $1.7623 (year 5). C10 = $1 (1 + 0.12)10 = $9.6463 (year 20). B) Investment with 11.7% semiannual compounding:

Formula: Ct = PV (1 + r / m) mt m: Number of compounding periods per year (2 for semiannual). Calculations: C1 = $1 [1 + (0.117 / 2)2 1 = $1.1204 (year 1). C5 = $1 [1 + (0.117 / 2)2 5 = $1.7657 (year 5). C10 = $1 [1 + (0.117 / 2)2 20 = $9.7193 (year 20). C) Investment with 11.5% continuous compounding:

Formula: Ct = PV e^mt e: Base of the natural logarithm (approximately 2.71828). Calculations: C1 = $1 e^(0.115 1) = $1.1219 (year 1). C5 = $1 e^(0.115 5) = $1.7771 (year 5). C10 = $1 e^(0.115 20) = $9.9742 (year 20). Comparison and Conclusion:

The calculations show that although option A has the highest annual interest rate (12%), it earns slightly less than option B (11.7% compounded semiannually) after 1 and 5 years due to the more frequent compounding in B.

However, for longer periods like 20 years, compounding plays a more significant role, and option C (11.5% continuously compounded) emerges as the winner with the highest future value due to its continuous reinvestment of earned interest.

Therefore, the choice between these options depends on your investment timeframe and risk tolerance. If you prioritize maximizing returns over a long period, even a slightly lower interest rate with continuous compounding might be preferable. For shorter terms, higher nominal interest rates with more frequent compounding might be better choices.

Why ?

PV =1 and in A) C10 = $1 (1 + 0.12)10 = $9.6463 (year 20). Why I should use 10 and not 20 ((1 + 0.12)^10)