In a

previous post, I revealed how doomsday portfolio returns are being used to attack VPW.

Here's a formal proof that the

*cumulative* time-weighted total portfolio return over 34 years, for a 50/50 stocks/bonds portfolio, must have been

-21% (or very close to it) for VPW to end up delivering a ridiculously tiny $11,952 last withdrawal at age 99 from an initial $1,000,000 at age 65.

The proof is very easy to understand (and verify). It only involves arithmetic operations and some very basic algebra.

**PROOF**
All amounts and returns are expressed in inflation-adjusted terms.

The

VPW Table percentage (in our wiki) for a 50/50 stocks/bonds allocation at age

**65** is

**4.8%**. (Note that the

VPW Worksheet, in its calculations, uses a more precise 4.7991255% percentage, but to keep this post readable, I'll use the rounded percentages found in our wiki.)

So, at age 65, the retiree withdraws 4.8% of $1,000,000. After withdrawal,

**($1,000,000 X (1 - 4.8%))** remains in the portfolio. During the retiree's 65th year, the portfolio experiences an

**R1** total return (with dividends and coupons reinvested).

When reaching age 66, just before withdrawal, the portfolio balance is equal to

**($1,000,000 X (1 - 4.8%) X (1 + R1))**. At age

**66** the

VPW Table percentage is

**4.9%**. After withdrawal,

**($1,000,000 X (1 - 4.8%) X (1 + R1) X (1 - 4.9%))** remains in the portfolio. During the retiree's 66th year, the portfolio experiences an

**R2** total return (with dividends and coupons reinvested).

When reaching age 67, just before withdrawal, the portfolio balance is equal to

**($1,000,000 X (1 - 4.8%) X (1 + R1) X (1 - 4.9%) X (1 + R2))**. At age

**67** the

VPW Table percentage is

**4.9%**. After withdrawal,

**($1,000,000 X (1 - 4.8%) X (1 + R1) X (1 - 4.9%) X (1 + R2) X (1 - 4.9%))** remains in the portfolio. During the retiree's 67th year, the portfolio experiences an

**R3** total return (with dividends and coupons reinvested).

When reaching age 68, just before withdrawal, the portfolio balance is equal to

**($1,000,000 X (1 - 4.8%) X (1 + R1) X (1 - 4.9%) X (1 + R2) X (1 - 4.9%) X (1 + R3))**. At age

**68** the

VPW Table percentage is

**5.0%**. After withdrawal,

**($1,000,000 X (1 - 4.8%) X (1 + R1) X (1 - 4.9%) X (1 + R2) X (1 - 4.9%) X (1 + R3) X (1 - 5.0%))** remains in the portfolio. During the retiree's 68th year, the portfolio experiences an

**R4** total return (with dividends and coupons reinvested).

You see the pattern. This continues until age 99.

Assuming a pure VPW Table depletion model, without the 10% cap on withdrawal percentage, when reaching age 99, just before withdrawal, the portfolio balance is equal to:

**($1,000,000 X**

(1 - 4.8%) X (1 + R1) X (1 - 4.9%) X (1 + R2) X (1 - 4.9%) X (1 + R3) X (1 - 5.0%) X (1 + R4) X (1 - 5.1%) X (1 + R5) X

(1 - 5.2%) X (1 + R6) X (1 - 5.3%) X (1 + R7) X (1 - 5.4%) X (1 + R8) X (1 - 5.5%) X (1 + R9) X (1 - 5.7%) X (1 + R10) X

(1 - 5.8%) X (1 + R11) X (1 - 6.0%) X (1 + R12) X (1 - 6.1%) X (1 + R13) X (1 - 6.3%) X (1 + R14) X (1 - 6.5%) X (1 + R15) X

(1 - 6.8%) X (1 + R16) X (1 - 7.0%) X (1 + R17) X (1 - 7.3%) X (1 + R18) X (1 - 7.6%) X (1 + R19) X (1 - 7.9%) X (1 + R20) X

(1 - 8.3%) X (1 + R21) X (1 - 8.8%) X (1 + R22) X (1 - 9.3%) X (1 + R23) X (1 - 10.0%) X (1 + R24) X (1 - 10.7%) X (1 + R25) X

(1 - 11.6%) X (1 + R26) X (1 - 12.7%) X (1 + R27) X (1 - 14.0%) X (1 + R28) X (1 - 15.8%) X (1 + R29) X (1 - 18.1%) X (1 + R30) X

(1 - 21.4%) X (1 + R31) X (1 - 26.3%) X (1 + R32) X (1 - 34.5%) X (1 + R33) X (1 - 50.8%) X (1 + R34))
In the above formula, we replace (1 - 4.8%) with 95.2%, (1 - 4.9%) with 95.1%, and so on :

**($1,000,000 X**

95.2% X (1 + R1) X 95.1% X (1 + R2) X 95.1% X (1 + R3) X 95.0% X (1 + R4) X 94.9% X (1 + R5) X

94.8% X (1 + R6) X 94.7% X (1 + R7) X 94.6% X (1 + R8) X 94.5% X (1 + R9) X 94.3% X (1 + R10) X

94.2% X (1 + R11) X 94.0% X (1 + R12) X 93.9% X (1 + R13) X 93.7% X (1 + R14) X 93.5% X (1 + R15) X

93.2% X (1 + R16) X 93.0% X (1 + R17) X 92.7% X (1 + R18) X 92.4% X (1 + R19) X 92.1% X (1 + R20) X

91.7% X (1 + R21) X 91.2% X (1 + R22) X 90.7% X (1 + R23) X 90.0% X (1 + R24) X 89.3% X (1 + R25) X

88.4% X (1 + R26) X 87.3% X (1 + R27) X 86.0% X (1 + R28) X 84.2% X (1 + R29) X 81.9% X (1 + R30) X

78.6% X (1 + R31) X 73.7% X (1 + R32) X 65.5% X (1 + R33) X 49.2% X (1 + R34))
As multiplication is commutative (A X B = B X A), we can reorganise the terms as follows:

**(($1,000,000 X**

95.2% X 95.1% X 95.1% X 95.0% X 94.9% X 94.8% X 94.7% X 94.6% X 94.5% X 94.3% X

94.2% X 94.0% X 93.9% X 93.7% X 93.5% X 93.2% X 93.0% X 92.7% X 92.4% X 92.1% X

91.7% X 91.2% X 90.7% X 90.0% X 89.3% X 88.4% X 87.3% X 86.0% X 84.2% X 81.9% X

78.6% X 73.7% X 65.5% X 49.2%) X

((1 + R1) X (1 + R2) X (1 + R3) X (1 + R4) X (1 + R5) X (1 + R6) X (1 + R7) X (1 + R8) X (1 + R9) X (1 + R10) X

(1 + R11) X (1 + R12) X (1 + R13) X (1 + R14) X (1 + R15) X (1 + R16) X (1 + R17) X (1 + R18) X (1 + R19) X (1 + R20) X

(1 + R21) X (1 + R22) X (1 + R23) X (1 + R24) X (1 + R25) X (1 + R26) X (1 + R27) X (1 + R28) X (1 + R29) X (1 + R30) X

(1 + R31) X (1 + R32) X (1 + R33) X (1 + R34)))
Using a calculator or a spreadsheet, it's easy to calculate that ($1,000,000 X 95.2% X ... X 49.2%) is equal to

**$15,241**. Note that the small difference with the $15,147 calculated in my

previous post is due to using rounded percentages in this post for calculations (to allow readers to easily replicate my calculations).

As a consequence, the portfolio balance, at age 99, before withdrawal is equal to:

**($15,241 X**

((1 + R1) X (1 + R2) X (1 + R3) X (1 + R4) X (1 + R5) X (1 + R6) X (1 + R7) X (1 + R8) X (1 + R9) X (1 + R10) X

(1 + R11) X (1 + R12) X (1 + R13) X (1 + R14) X (1 + R15) X (1 + R16) X (1 + R17) X (1 + R18) X (1 + R19) X (1 + R20) X

(1 + R21) X (1 + R22) X (1 + R23) X (1 + R24) X (1 + R25) X (1 + R26) X (1 + R27) X (1 + R28) X (1 + R29) X (1 + R30) X

(1 + R31) X (1 + R32) X (1 + R33) X (1 + R34)))
The last withdrawal, at age

**99**, depletes the portfolio and, obviously, the

VPW Table percentage is

**100%**. As a consequence, the withdrawal amount, at age 99 is necessarily equal to the above formula.

The attack against VPW claimed that, at age 99, the withdrawal amount could be

**$11,952** (or less) for every 1 out of every 20 projections of the attackers's prediction-driven withdrawal tool. This gives use the following equation:

**($15,241 X**

((1 + R1) X (1 + R2) X (1 + R3) X (1 + R4) X (1 + R5) X (1 + R6) X (1 + R7) X (1 + R8) X (1 + R9) X (1 + R10) X

(1 + R11) X (1 + R12) X (1 + R13) X (1 + R14) X (1 + R15) X (1 + R16) X (1 + R17) X (1 + R18) X (1 + R19) X (1 + R20) X

(1 + R21) X (1 + R22) X (1 + R23) X (1 + R24) X (1 + R25) X (1 + R26) X (1 + R27) X (1 + R28) X (1 + R29) X (1 + R30) X

(1 + R31) X (1 + R32) X (1 + R33) X (1 + R34)))

= $11,952
Dividing both sides of the equation by $15,241 give us:

**((1 + R1) X (1 + R2) X (1 + R3) X (1 + R4) X (1 + R5) X (1 + R6) X (1 + R7) X (1 + R8) X (1 + R9) X (1 + R10) X**

(1 + R11) X (1 + R12) X (1 + R13) X (1 + R14) X (1 + R15) X (1 + R16) X (1 + R17) X (1 + R18) X (1 + R19) X (1 + R20) X

(1 + R21) X (1 + R22) X (1 + R23) X (1 + R24) X (1 + R25) X (1 + R26) X (1 + R27) X (1 + R28) X (1 + R29) X (1 + R30) X

(1 + R31) X (1 + R32) X (1 + R33) X (1 + R34))

= ($11,952 / $15,241)
My calculator tells me that ($11,952 / $15,241) is equal to 0.7842. Replacing this and subtracting 1 from both sides of the equation gives us:

**((1 + R1) X (1 + R2) X (1 + R3) X (1 + R4) X (1 + R5) X (1 + R6) X (1 + R7) X (1 + R8) X (1 + R9) X (1 + R10) X**

(1 + R11) X (1 + R12) X (1 + R13) X (1 + R14) X (1 + R15) X (1 + R16) X (1 + R17) X (1 + R18) X (1 + R19) X (1 + R20) X

(1 + R21) X (1 + R22) X (1 + R23) X (1 + R24) X (1 + R25) X (1 + R26) X (1 + R27) X (1 + R28) X (1 + R29) X (1 + R30) X

(1 + R31) X (1 + R32) X (1 + R33) X (1 + R34)) - 1

= 0.7842 - 1
The left part is equal to the cumulative portfolio growth, over 34 years. As for (.7842 - 1), it's equal to

-21.58%.

Q.E.D.

Variable Percentage Withdrawal (bogleheads.org/wiki/VPW) | One-Fund Portfolio (bogleheads.org/forum/viewtopic.php?t=287967)