Final answer:
To maintain the purchasing power of 100,000 EUR over ten years with an annual inflation rate of 4.2% and an interest rate of 6.5% compounded monthly, approximately 79,837 EUR must be invested today, rounded to the nearest integer.
Step-by-step explanation:
To calculate how much needs to be invested today at a 6.5% annual interest rate, compounded monthly, in order for the real value of the investment to be worth 100,000 EUR in ten years, we must consider the effects of an annual inflation rate of 4.2%. First, we'll calculate the equivalent future value of 100,000 EUR accounting for inflation. This is necessary to preserve the real purchasing power of the money.
To adjust for inflation, we use the following formula, where FV is the future value, PV is the present value, r is the inflation rate, and t is the time in years: FV = PV(1 + r)^t.
To find the inflated future value (FV_i): 100,000 EUR * (1 + 0.042)^10 = 100,000 EUR * 1.042^10 ≈ 148,024 EUR.
Next, we will calculate the present value of that future amount using the formula for present value (PV) of a compound interest investment: PV = FV / (1 + r/n)^(nt), where:
- FV is the future value we just calculated,
- r is the annual interest rate (6.5% or 0.065),
- n is the number of times the interest is compounded per year (12 for monthly),
- t is the number of years (10).
Using the formula: PV = 148,024 / (1 + 0.065/12)^(12*10) ≈ 148,024 / (1 + 0.00541667)^120 ≈ 79,837 EUR.
The amount to be invested today should be to the nearest integer, which is 79,837 EUR after rounding.