Luis has $150,000 in his retirement account at his present company. Because he is assuming a position with another company, Luis is planning to "roll over" his assets to a new account. Luis also plans to put $3000/quarter into the new account until his retirement 20 years from now. If the new account earns interest at the rate of 8%/year compounded quarterly, how much will Luis have in his account at the time of his retirement?
assuming he adds 3000 at the beginning of each quarter, the account will have
1 quarter: 150000*1.02+3000*1.02
2 qtrs: 150000*1.02^2 + 3000*(1.02^2 + 1.02)
n qtrs: 150000*1.02^n + 3000(1.02 + 1.02^2 + ... + 1.02^n)
= 150000*1.02^n + 3000 (1.02^n-1)/(1.02-1)
= 150000*1.02^n + 3000(1.02^n-1)/.02
= 150000(1.02^n + 1.02^n - 1)
= 150000(2*1.02^n - 1)
so, after 20 years (80 quarters), he will have $1,312,631.75
01832436288
To find out how much Luis will have in his account at the time of his retirement, we can break the problem down into two parts:
1. Calculate the future value of his existing retirement account.
2. Calculate the future value of the additional quarterly deposits he will make until retirement.
For the first part, we can calculate the future value of Luis' existing retirement account using the formula for compound interest:
Future Value = Present Value * (1 + Interest Rate)^Number of Periods
Given:
- Present Value (PV) = $150,000
- Interest Rate (r) = 8% per year (converted to quarterly compounding, which is 2% per quarter)
- Number of Periods (n) = 20 years (converted to 80 quarters)
Using these values, we can calculate the future value of his existing retirement account:
Future Value (Part 1) = PV * (1 + r)^n
Future Value (Part 1) = $150,000 * (1 + 0.02)^80
Now, let's calculate the future value of the additional quarterly deposits he will make until retirement.
Given:
- Deposit Amount (D) = $3,000 every quarter
- Interest Rate (r) = 8% per year (converted to quarterly compounding, which is 2% per quarter)
- Number of Periods (n) = 20 years (converted to 80 quarters)
To calculate the future value of these periodic deposits, we can use the formula for future value of an ordinary annuity:
Future Value (Part 2) = D * ((1 + r)^n - 1) / r
Future Value (Part 2) = $3,000 * ((1 + 0.02)^80 - 1) / 0.02
Now, let's add the future values from both parts to find the total amount Luis will have in his account at the time of his retirement:
Total Future Value = Future Value (Part 1) + Future Value (Part 2)
You can calculate this total value by substituting the calculated values into the formula.
Oops. the formula is
150000(1.02^n + 1.02^(n+1) - 1)
so $1,327,258.06