1. First of all, today is my 25th birthday and I begin thinking about retirement. My goal is to retire on my 67th birthday with enough money saved to meet my needs. The first payment is $350,000 and it should be increased by 2% annually during the thirty years of retirement. Thus, assuming the first payment is S, the second payment will be:
S2 = S * (1+0,02) = 350,000 * 1,02 = 357,000;
while the third payment is:
S3 = S2 * (1+0,02) = S*(1+0,02)*(1+0,02) = 350,000*(1+0,02)*(1+0,02)=364,140;
Similarly, the fourth payment will be:
S4 = S3*(1+0,02) = S*(1+0,02)*(1+0,02)*(1+0,02) = 371,422.8
Thus, the key formula is:
It can be considered as a geometric progression. Thus, its sum is calculated by using the following formula:
Thus, the total sum of payments during the thirty years is
S30 = 350,000*((-0,02)^30)/-0,02 = 14,198,827.72
Taking into account that I want to purchase a new car for $75,000 on my 75th birthday, the minimum amount of money I must have in my account on the 67th birthday is 14,198,827.72 + 75,000 = 14,273,827.72.
2. Secondly, I have to calculate the minimum annual deposit that should be made into my account, assuming that, the annual interest rate is 10%. The first sum should be made on my 26th birthday, while the last one will be made on 67th birthday. Thus, the total period of deposit is 42 years. Thus, the minimum amount of deposit can be calculated from the following formula:
14,273,827.72 = b*42*(1+0,1);
where b is a minimum amount of deposit. Thus, b = 14,273,827.72 / (1,01*42) = 308,957.31
That is why the minimum annual deposit that should be made is $308,957.31.