Question

1. how much must i deposit today if i want to be able to withdraw $620 per month for six years from an account which earns a nominal rate of three percent, compounded monthly, if withdrawals are made at the beginning of each month? round to the nearest cent. do not include any unit (if your answer is $111.11, then type 111.11 without $ sign.)

Explanation:

Step1: Identify given values

Monthly withdrawal $PMT = 620$, nominal rate $r = 0.03$, compounding periods/year $n = 12$, time $t = 6$ years, withdrawals at start (annuity due).

Step2: Calculate monthly rate

$i = \frac{r}{n} = \frac{0.03}{12} = 0.0025$

Step3: Calculate total periods

$N = n \times t = 12 \times 6 = 72$

Step4: Apply annuity due formula

Present value $PV = PMT \times \frac{1-(1+i)^{-N}}{i} \times (1+i)$
Substitute values:
$PV = 620 \times \frac{1-(1+0.0025)^{-72}}{0.0025} \times (1+0.0025)$

Step5: Compute $(1.0025)^{-72}$

$(1.0025)^{-72} \approx 0.83564491$

Step6: Compute numerator of fraction

$1 - 0.83564491 = 0.16435509$

Step7: Compute the fraction

$\frac{0.16435509}{0.0025} = 65.742036$

Step8: Multiply by $(1.0025)$

$65.742036 \times 1.0025 \approx 65.906391$

Step9: Multiply by PMT

$620 \times 65.906391 \approx 40861.96$

Answer:

40861.96