Amortization based withdrawal formulas

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These are the formulas used for Amortization Based Withdrawal (ABW).

Amortization Formula

If is the lump sum value, is the amount paid over periods, and is the interest rate, then

Note that this formula assumes that the periodic payment begins next period, not immediately. The lump sum is the value today (period 0), and the periodic payments are being made in periods 1 through . This is natural in the context of a loan: the loan is taken out today and the repayments only start next period.

The formula does not work for . You can input close to zero in the above formula to get an approximate answer. Or you can use simple division to get the exact answer.

In Excel, the PMT function can be used for this calculation:

Amortization Based Withdrawal Formula

To calculate portfolio withdrawals, set current portfolio value, number of years over which withdrawals are to be spread out, and expected return of the portfolio.

The amortization formula above assumes that payments begin next year, not immediately. If withdrawals are to start this year, the formula needs to be divided by :

In Excel, the PMT function can be used for this calculation:

Adding a Terminal Balance

Instead of fully depleting the portfolio, the amortization can be modified to leave behind a terminal balance. If the target terminal balance is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} dollars, withdrawal is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \frac{(P-\frac{B}{(1+r)^n})*r}{1-\frac{1}{(1+r)^n}}*\frac{1}{1+r} \; }

In Excel, the PMT function can be used for this calculation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=\operatorname{PMT}(r,n,-P,B,1) }

Allowing for Rising or Falling Withdrawal Schedules

Instead of a constant withdrawal schedule, the amortization can be modified to generate a withdrawal schedule that grows at a rate of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} per year. First year's withdrawal is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g>0} generates a rising withdrawal schedule, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g<0} generates a declining withdrawal schedule, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g=0} generates a constant withdrawal schedule.

The formula does not work for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=g} . You can input Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} close to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} in the above formula to get an approximate answer. Or you can use simple division to get the exact answer.

In Excel, the PMT function can be used for this calculation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=\operatorname{PMT}(\frac{1+r}{1+g}-1,n,-P,\frac{B}{(1+g)^n},1) }

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