In a
previous post, I explained that VPW is, by design, immune to
sequence of returns risk (SORR), but VPW withdrawals are exposed to market risk (MR).
Here's a detailed illustration.
Chapter 3 of the
Pensionize your nest egg book, by Moshe Milevsky and Alexandra Macqueen, uses a triangle with three returns, 7%, -13%, and 27% which have a 7% arithmetic average. The book shows how starting from one or the other of these three returns changes the outcome from retirement success to the ruin of the retiree when making constant (inflation-adjusted) withdrawals.
I'll use this idea to illustrate how the
7th withdrawal remains unaffected by the sequence of annual returns leading to it, when using VPW, despite the intervening annual withdrawals, as long as the cumulative time-weighted return from retirement until this 7th withdrawal remains unchanged.
Let's pick a 65 years old retiree with a $1,000,000 balanced (60/40 stocks/bonds) portfolio, and $1,403/month Social Security. Stock returns are (7%, -13%, 27%) repeated twice, but staring at a different point in each of the three cases. Bond returns are 0.8% (without any fluctuation, for simplicity). All calculations are done in constant 2023 dollars (so that there's no need to change Social Security amounts).
The
VPW table has the following withdrawal percentages for a 60/40 stocks/bonds allocation from ages 65 to 71: 5.0%, 5.1%, 5.1%, 5.2%, 5.3%, 5.4%, and 5.5%.
Note how the
7th withdrawal, at age 71, is
$50,739 in all three cases, resulting into a total retirement income of $65,575. This is because VPW withdrawals are unaffected by the
sequence of returns leading to them.
Actually, if we look carefully, we see that at ages 65, 68 and 71, the withdrawals are identical for the three cases. This is because over each 3-years period, the cumulative time-weighted portfolio return is 12.677% in all three cases. This is the immunity against
sequence of returns risk at work.
But, when we look at other years, withdrawals differ. In the unlucky 3rd case, the retiree withdraws less ($44,826) in the second year after stocks lose -13%, giving a chance for the portfolio to recover. This is
market risk at work. Lower returns lead to lower withdrawals. Higher returns lead to higher withdrawals. That's the "variable" in
variable percentage withdrawal.
Case 1: Starting with 7%
Code: Select all
Age Portfolio VPW Withdrawal Remaining Stocks Bonds Social Security Total income
65 $1,000,000 5.0% $50,000 $950,000 7.0% 0.8% $16,836 $66,836
66 $992,940 5.1% $50,640 $942,300 27.0% 0.8% $16,836 $67,476
67 $1,097,968 5.1% $55,996 $1,041,972 -13.0% 0.8% $16,836 $72,832
68 $964,032 5.2% $50,130 $913,903 7.0% 0.8% $16,836 $66,966
69 $955,211 5.3% $50,626 $904,585 27.0% 0.8% $16,836 $67,462
70 $1,054,022 5.4% $56,917 $997,105 -13.0% 0.8% $16,836 $73,753
71 $922,521 5.5% $50,739 $871,783 $16,836 $67,575
Case 2: Starting with 27%
Code: Select all
Age Portfolio VPW Withdrawal Remaining Stocks Bonds Social Security Total income
65 $1,000,000 5.0% $50,000 $950,000 27.0% 0.8% $16,836 $66,836
66 $1,106,940 5.1% $56,454 $1,050,486 -13.0% 0.8% $16,836 $73,290
67 $971,910 5.1% $49,567 $922,342 7.0% 0.8% $16,836 $66,403
68 $964,032 5.2% $50,130 $913,903 27.0% 0.8% $16,836 $66,966
69 $1,064,879 5.3% $56,439 $1,008,441 -13.0% 0.8% $16,836 $73,275
70 $933,009 5.4% $50,382 $882,627 7.0% 0.8% $16,836 $67,218
71 $922,521 5.5% $50,739 $871,783 $16,836 $67,575
Case 3: Starting with -13%
Code: Select all
Age Portfolio VPW Withdrawal Remaining Stocks Bonds Social Security Total income
65 $1,000,000 5.0% $50,000 $950,000 -13.0% 0.8% $16,836 $66,836
66 $878,940 5.1% $44,826 $834,114 7.0% 0.8% $16,836 $61,662
67 $871,816 5.1% $44,463 $827,353 27.0% 0.8% $16,836 $61,299
68 $964,032 5.2% $50,130 $913,903 -13.0% 0.8% $16,836 $66,966
69 $845,543 5.3% $44,814 $800,729 7.0% 0.8% $16,836 $61,650
70 $836,922 5.4% $45,194 $791,728 27.0% 0.8% $16,836 $62,030
71 $922,521 5.5% $50,739 $871,783 $16,836 $67,575
EXPLANATION
Mutiplication is commutative (e.g. 5 X 3 = 3 X 5). Taking a 5.0% withdrawal is equal to multiplying the portfolio balance by
(1 - 5%) = 0.95. Similarly, taking a 5.1% withdrawal is equal to multiplying the portfolio balance by
0.949. And so on. After the withdrawal is taken, the portfolio grows (or shrink) by the annual return. For a 7% stock return and a 0.8% bond return, this is equal to multiplying by
(1 + ((60 % X 7%) + (40% X 0.8%))) = 1.0452. The other two factors are
1.1652 and
0.9252.
So, in case 1, the portfolio balance after 6 years, just before taking the 7th withdrawal, is equal to:
- $1,000,000 X (.95 X 1.0452) X (.949 X 1.1652) X (.949 X 0.9252) X (.948 X 1.0452) X (.947 X 1.1652) X (.946 X 0.9252)
In case 2, the portfolio balance after 6 years, just before taking the 7th withdrawal, is equal to:
- $1,000,000 X (.95 X 1.1652) X (.949 X 0.9252) X (.949 X 1.0452) X (.948 X 1.1652) X (.947 X 0.9252) X (.946 X 1.0452)
Case 3 can be expressed similarly.
As multiplication is commutative, the terms can be reordered without changing the result. In all three cases they can be arranged as follows, resulting into an identical portfolio balance after 6 years, just before taking the 7th withdrawal:
- $1,000,000 X (.95 X .949 X .949 X .948 X .947 X .946) X (1.0452 X 1.0452 X 1.1652 X 1.1652 X 0.9252 X 0.9252)
Variable Percentage Withdrawal (bogleheads.org/wiki/VPW) | One-Fund Portfolio (bogleheads.org/forum/viewtopic.php?t=287967)