Certainly! To determine how much money the person needs to invest at the beginning of each year to reach the goal of Rs. 1,00,000 in 18 years, we will use the concept of the Future Value of an Annuity Due.
The formula for the Future Value of an Annuity Due (FV) is:
[tex]\[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \times (1 + r) \][/tex]
where:
- [tex]\( FV \)[/tex] is the future value, which is Rs. 1,00,000.
- [tex]\( P \)[/tex] is the annual payment (the amount to be invested at the beginning of each year).
- [tex]\( r \)[/tex] is the annual rate of return, which is 8% or 0.08.
- [tex]\( n \)[/tex] is the number of years, which is 18.
We need to solve for [tex]\( P \)[/tex]. Rearranging the formula to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{FV}{\left( \frac{(1 + r)^n - 1}{r} \right) \times (1 + r)} \][/tex]
Plugging in the given values:
[tex]\[ FV = 100000 \][/tex]
[tex]\[ r = 0.08 \][/tex]
[tex]\[ n = 18 \][/tex]
Now, calculate the components step-by-step:
1. Calculate [tex]\( (1 + r)^n \)[/tex]:
[tex]\[ (1 + 0.08)^{18} = 1.08^{18} \][/tex]
Using a calculator:
[tex]\[ 1.08^{18} \approx 3.996 \][/tex]
2. Calculate [tex]\( (1 + r)^n - 1 \)[/tex]:
[tex]\[ 3.996 - 1 = 2.996 \][/tex]
3. Divide by the rate [tex]\( r \)[/tex]:
[tex]\[ \frac{2.996}{0.08} \approx 37.45 \][/tex]
4. Multiply by [tex]\( (1 + r) \)[/tex]:
[tex]\[ 37.45 \times 1.08 \approx 40.456 \][/tex]
5. Finally, divide the future value [tex]\( FV \)[/tex] by the result of the above calculation to find [tex]\( P \)[/tex]:
[tex]\[ P = \frac{100000}{40.456} \approx 2471.45 \][/tex]
Therefore, the person would need to invest approximately Rs. 2,471.45 at the beginning of each year, starting today, to achieve the goal of having Rs. 1,00,000 in 18 years with an average annual rate of return of 8%.
