Post by Admin/YBB on Aug 30, 2021 14:09:51 GMT -6
Much of the fund literature is oriented towards accumulation. But with millions in retirement, some novel measures are needed for the SWP (Systematic Withdrawal Plan; decumulation) phase. Two factors AF and AAF can indicate how the funds would have worked for uniform withdrawals. Monthly SWP = Lump-sum/AF if the final value is 0, or Lump-sum/AAF if the final value is equal to the initial value.
Yahoo Finance data for monthly adjusted-prices is used here. For adjusted-prices, a ratio adjustment is made in the prices to account for distributions that reduces the pre-adjustment prices. Adjusted-prices for two specific times can also be used to approximate total-return between those times.
Harmonic mean h of monthly adjusted-prices p(i), i = 1, 2, …, N, can be found from Excel function HARMEAN (cell1:cellN). Calculations related to SWP (decumulation) are simplified with the use of harmonic mean. If uniform withdrawal amount is A per month for N months, then the total number of shares withdrawn S = (N/h)xA. The initial value of these shares is P = p(1)x(N/h)xA and the final value F = 0.
Let us define factor AF = p(1)x(N/h), then P = AFxA. So, the factor AF is the initial amount ($AF) needed to support $1/month SWP for N months such that the final account value F is zero.
Bond-Equivalent Adjustment. The idea is that we start with an initial amount P, withdraw a uniform amount A per month for N months, and then end up with a final amount F = P (instead of 0 previously). Let us define a Cumulative Factor CF = p(N)/p(1), or the ratio of the final (N th) and first (1 st) adjusted-prices. As an aside, the approximate annualized total return (TR) for the fund over N months is 100x12x{CF^(1/(N-1)) – 1}, where ^ is the symbol for generalized-root. We already know from the para above that AFxA is the initial amount that will be exhausted after N withdrawals of amount A. So, we must start with some initial value P > AFxA. Then the question is, when this extra initial amount (P - AFxA) becomes equal to F = P at the end? After some simple manipulations, and defining factor AAF = AFxCF/(CF - 1), we get P = AAFxA. Note that CF must be greater than 1; when CF =1, the formula will blow up; when CF < 1, then the logic fails for preserving the initial principal, and so should the math, leading to an invalid negative value (or simply, that CF becomes undefined). So, the factor AAF is the initial amount (P = $AAF) needed to support $1/month SWP for N months such that the final account value F is also P.
So, for SWP with final F = 0, P = AFxA,
and for SWP with final F = P, P = AAFxA.
Example 1 – p(i): From 1/1/1985 to 5/1/2021 (36.3 years), monthly adjusted-prices downloaded from Yahoo Finance
N = 437
VWINX SWP: AF = 124.4296, CF = 13.3475, AAF = 134.5070
VWELX SWP: AF = 121.9074, CF = 14.4477, AAF = 130.9727
VFINX SWP: AF = 104.6265, CF = 40.4904, AAF = 107.2759
Example 2 – p(i): From 6/1/2011 to 5/1/2021 (recent 10 years), monthly adjusted-prices downloaded from Yahoo Finance
N = 120
VWINX SWP: AF = 90.2243, CF = 1.7796, AAF = 205.9638
VWELX SWP: AF = 88.1583, CF = 1.9222, AAF = 183.7505
VFINX SWP: AF = 70.2101, CF = 3.8287, AAF = 95.0316
These data for factors AF and AAF are for comparative purposes only. Over 36.3 years in Example 1, the AF data show that conservative-allocation VWINX and VWELX performed quite similarly, but SP500 VFINX was significantly ahead of both of them (lower AF is better). This was due the long up-trend in equities in spite of turmoil encountered several times (1987, 2000-02, 2007-09, 2020). Over the recent 10 years in Example 2, the AF data showed that the results were similar for conservative-allocation VWINX and moderate-allocation VWELX, but SP500 VFINX was significantly ahead of both of them. Of course, it should be realized that among the three, the conservative allocation VWINX has the lowest volatility (standard deviation SD), the moderate allocation VWELX has volatility in the middle and all-equity VFINX has the highest volatility. That in these two timeframes all-equity VFINX had better results with higher volatility is an indication of the current strong market and that may not be so in other timeframes. On the other hand, the results for conservative-allocation VWINX and moderate-allocation VWELX should be more consistent in different markets. As for the factor AAF, with the requirement of also preserving the initial principal at the end, the ordering was more definitive: The all-equity VFINX was in the lead, moderate-allocation VWELX in the middle, and conservative allocation VWINX lagged.
For a historical stress test during the financial crisis 2007-09, 10 years to 3/31/2009 were used for the same three funds. The factor AF were 101.7659 (VWINX), 111.3538 (VWELX), 127.208 (VFINX), and we can see a complete reversal. In the lead was conservative-allocation VWINX, in the middle was moderate-allocation VWELX, and the worst was all-equity VFINX. As for the factor AAF, which required preservation of initial principal, the conservative-allocation VWINX had a very high value, but moderate allocation VWELX and all-equity VFINX had undefined values (failures due to CF < 1). So, when the markets are sharply down, or for specialty funds, the factor AF can always be calculated, but the factor AAF may become undefined as there may be failure to preserve the initial principal (CF < 1).
Taking an inspiration from the Sharpe Ratio, the volatility can be incorporated as follows. The factor AF can used to figure out the annual Payout Rate PR = 1,200/AF and then calculate Ratio1 = PR/SD = 1,200/(AF*SD). The factor AAF can be used to calculate annual Sustained Distribution Rate SDR = 1,200/AAF and then calculate Ratio2 = SDR/SD = 1,200/(AAF*SD). The standard deviation SD would be calculated by using the monthly returns (not monthly adjusted-prices) over the appropriate time period used for determining AF and AAF. Higher values of Ratio1 and/or Ratio2 would be better.
Factor AF to support SWP of $1/month and factor AAF to support SWP of $1/month with the preservation of initial principal are unique characteristics of funds and should be helpful to retirees in the decumulation phase. Under adverse market conditions, or for some specialty funds, the factor AAF may become undefined. It is hoped that the funds and/or fund rating organizations (M*, Lipper, AAII, etc) may provide this information in future voluntarily, or through regulation by the SEC.
Yahoo Finance data for monthly adjusted-prices is used here. For adjusted-prices, a ratio adjustment is made in the prices to account for distributions that reduces the pre-adjustment prices. Adjusted-prices for two specific times can also be used to approximate total-return between those times.
Harmonic mean h of monthly adjusted-prices p(i), i = 1, 2, …, N, can be found from Excel function HARMEAN (cell1:cellN). Calculations related to SWP (decumulation) are simplified with the use of harmonic mean. If uniform withdrawal amount is A per month for N months, then the total number of shares withdrawn S = (N/h)xA. The initial value of these shares is P = p(1)x(N/h)xA and the final value F = 0.
Let us define factor AF = p(1)x(N/h), then P = AFxA. So, the factor AF is the initial amount ($AF) needed to support $1/month SWP for N months such that the final account value F is zero.
Bond-Equivalent Adjustment. The idea is that we start with an initial amount P, withdraw a uniform amount A per month for N months, and then end up with a final amount F = P (instead of 0 previously). Let us define a Cumulative Factor CF = p(N)/p(1), or the ratio of the final (N th) and first (1 st) adjusted-prices. As an aside, the approximate annualized total return (TR) for the fund over N months is 100x12x{CF^(1/(N-1)) – 1}, where ^ is the symbol for generalized-root. We already know from the para above that AFxA is the initial amount that will be exhausted after N withdrawals of amount A. So, we must start with some initial value P > AFxA. Then the question is, when this extra initial amount (P - AFxA) becomes equal to F = P at the end? After some simple manipulations, and defining factor AAF = AFxCF/(CF - 1), we get P = AAFxA. Note that CF must be greater than 1; when CF =1, the formula will blow up; when CF < 1, then the logic fails for preserving the initial principal, and so should the math, leading to an invalid negative value (or simply, that CF becomes undefined). So, the factor AAF is the initial amount (P = $AAF) needed to support $1/month SWP for N months such that the final account value F is also P.
So, for SWP with final F = 0, P = AFxA,
and for SWP with final F = P, P = AAFxA.
Example 1 – p(i): From 1/1/1985 to 5/1/2021 (36.3 years), monthly adjusted-prices downloaded from Yahoo Finance
N = 437
VWINX SWP: AF = 124.4296, CF = 13.3475, AAF = 134.5070
VWELX SWP: AF = 121.9074, CF = 14.4477, AAF = 130.9727
VFINX SWP: AF = 104.6265, CF = 40.4904, AAF = 107.2759
Example 2 – p(i): From 6/1/2011 to 5/1/2021 (recent 10 years), monthly adjusted-prices downloaded from Yahoo Finance
N = 120
VWINX SWP: AF = 90.2243, CF = 1.7796, AAF = 205.9638
VWELX SWP: AF = 88.1583, CF = 1.9222, AAF = 183.7505
VFINX SWP: AF = 70.2101, CF = 3.8287, AAF = 95.0316
These data for factors AF and AAF are for comparative purposes only. Over 36.3 years in Example 1, the AF data show that conservative-allocation VWINX and VWELX performed quite similarly, but SP500 VFINX was significantly ahead of both of them (lower AF is better). This was due the long up-trend in equities in spite of turmoil encountered several times (1987, 2000-02, 2007-09, 2020). Over the recent 10 years in Example 2, the AF data showed that the results were similar for conservative-allocation VWINX and moderate-allocation VWELX, but SP500 VFINX was significantly ahead of both of them. Of course, it should be realized that among the three, the conservative allocation VWINX has the lowest volatility (standard deviation SD), the moderate allocation VWELX has volatility in the middle and all-equity VFINX has the highest volatility. That in these two timeframes all-equity VFINX had better results with higher volatility is an indication of the current strong market and that may not be so in other timeframes. On the other hand, the results for conservative-allocation VWINX and moderate-allocation VWELX should be more consistent in different markets. As for the factor AAF, with the requirement of also preserving the initial principal at the end, the ordering was more definitive: The all-equity VFINX was in the lead, moderate-allocation VWELX in the middle, and conservative allocation VWINX lagged.
For a historical stress test during the financial crisis 2007-09, 10 years to 3/31/2009 were used for the same three funds. The factor AF were 101.7659 (VWINX), 111.3538 (VWELX), 127.208 (VFINX), and we can see a complete reversal. In the lead was conservative-allocation VWINX, in the middle was moderate-allocation VWELX, and the worst was all-equity VFINX. As for the factor AAF, which required preservation of initial principal, the conservative-allocation VWINX had a very high value, but moderate allocation VWELX and all-equity VFINX had undefined values (failures due to CF < 1). So, when the markets are sharply down, or for specialty funds, the factor AF can always be calculated, but the factor AAF may become undefined as there may be failure to preserve the initial principal (CF < 1).
Taking an inspiration from the Sharpe Ratio, the volatility can be incorporated as follows. The factor AF can used to figure out the annual Payout Rate PR = 1,200/AF and then calculate Ratio1 = PR/SD = 1,200/(AF*SD). The factor AAF can be used to calculate annual Sustained Distribution Rate SDR = 1,200/AAF and then calculate Ratio2 = SDR/SD = 1,200/(AAF*SD). The standard deviation SD would be calculated by using the monthly returns (not monthly adjusted-prices) over the appropriate time period used for determining AF and AAF. Higher values of Ratio1 and/or Ratio2 would be better.
Factor AF to support SWP of $1/month and factor AAF to support SWP of $1/month with the preservation of initial principal are unique characteristics of funds and should be helpful to retirees in the decumulation phase. Under adverse market conditions, or for some specialty funds, the factor AAF may become undefined. It is hoped that the funds and/or fund rating organizations (M*, Lipper, AAII, etc) may provide this information in future voluntarily, or through regulation by the SEC.
