Jenny wants a monthly retirement income of $12,000. she will retire on her birthday at age 70 with a $3,000 per month social security monthly benefit and a $4000 per month defined benefit pension.
she expects to die on her birthday at age 95 and would like to leave $75,000 to each of her 6 children.
she expects a 7.9% annual return on her roth 401k. she expects a 2.6% annual rate of inflation.

how much will she need to have saved to invest when she retires?

Explanation:

Since she wants to receive the income per month, change the interest rate and duration on investment variables to monthly basis;

Out of the $12,000, find Jenny's own savings after deducting social security income & Pension benefit;

= 12,000 - 3,000 - 4,000 = $5,000

Since the 5,000 is recurring, it will be the PMT in annuity calculation.

If marginal tax rate = 28%, find the aftertax nominal rate;

Pretax nominal rate = 7.9% or 0.079

After tax nominal rate = (1-0.28) *0.079

After tax nominal rate = 0.05688 or 5.688%

Next, find the real interest rate using Fisher equation that applies the nominal rate and inflation rate

Real rate = [(1+Nominal) / (1+inflation) ] -1

=[(1+0.05688) / (1+0.026)] -1

= 1.0301 -1

= 0.0301

Real rate = 3.01%

Next, using financial calculator, enter the following inputs;

N = 95 - 70 = 25 years, but convert to months = 25*12 = 300

I/Y = 3.01% /12 = 0.2508%

PMT = 5,000

FV = 75,000*6 = 450,000

then CPT PV = $1,265,460.78

Therefore, she need to have saved $1,265,460.78

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