You want to have an income of $50,000 per year in retirement, and you think you will be alive for 30 years in retirement. How much do you need to have invested the day you retire, in real dollars, assuming a 3% real rate of return?
I just need to make sure I'm doing this right.. I'm worried I'm not handling the r right. Does "real rate of return" = r?
PVA = ?
FV = 50,000 a year
r = .03
n= 30
PVA = 980,022.07
Yes, your answer is correct, as follows.
P=A[P/A,0.03,30]
=50000[P/A,0.03,30]
=50000((1+0.03)^30-1)/(.03*(1+0.03)^30)
=980022.067
As for the interest, there is no mention of the compounding rate, and 3% is qualified as "real" rate or return, or EAR (effective annual rate), whatever the APR might be. So using 3% as EAR is correct.
To calculate the amount you need to have invested the day you retire, you can use the Present Value Annuity (PVA) formula. The PVA formula helps calculate the amount you need to invest upfront to receive a future series of cash flows. In this case, the cash flow is $50,000 per year, and you plan to receive it for 30 years in retirement.
The formula for PVA is:
PVA = FV * (1 - (1 + r)^(-n)) / r
Where:
PVA is the present value annuity
FV is the future value (annual income you want) = $50,000
r is the real rate of return, expressed as a decimal = 0.03
n is the number of years in retirement = 30
By plugging in these values into the formula, you can calculate the required investment:
PVA = 50,000 * (1 - (1 + 0.03)^(-30)) / 0.03
PVA = 980,022.07
So, based on a 3% real rate of return, in order to have an income of $50,000 per year for 30 years in retirement, you will need to have approximately $980,022.07 invested on the day you retire.