To calculate the monthly withdrawals you can make from your retirement savings for 15 years, you can use the formula for the monthly payment of an annuity:
�
=
�
⋅
�
⋅
(
1
+
�
)
�
(
1
+
�
)
�
−
1
M=
(1+r)
n
−1
P⋅r⋅(1+r)
n
Where:
M is the monthly withdrawal amount.
P is the principal amount (your savings), which is $500,000.
r is the monthly interest rate, which is the annual interest rate divided by 12 (9% / 12).
n is the total number of payments, which is 15 years * 12 months per year (180).
Now, plug these values into the formula:
�
=
500
,
000
⋅
(
0.09
/
12
)
⋅
(
1
+
0.09
/
12
)
180
(
1
+
0.09
/
12
)
180
−
1
M=
(1+0.09/12)
180
−1
500,000⋅(0.09/12)⋅(1+0.09/12)
180
Calculating this will give you the monthly withdrawal amount:
M ≈ \frac{500,000 \cdot 0.0075 \cdot (1.0075)^{180}}{(1.0075)^{180} - 1} ≈ \frac{3,750 \cdot 7.186}{6.186} ≈ \frac{26,922.75}{6.186} ≈ $4,354.07
So, you can withdraw approximately $4,354.07 per month to make your savings last for 15 years, assuming a 9% annual interest rate.