Current Research
Updated: August 21, 2006.
New Search Feature
I have added a new search feature at the bottom of this page. It includes an index and a site map.
Current Research Index
Here is an index with links to my previous Current Research efforts.
Current Research Index
Current Research H: The Big Project
Rob Bennett and I are joining together to put up some advanced calculators.
I have begun by putting together Calculator A. It allows you to see what happens with different allocations of stocks (i.e., the S&P500) and TIPS at a 2% (real) interest rate.
You enter a value of P/E10 and see what happens.
The optimal stock allocation at the Safe Withdrawal Rate differs from the optimal allocation at the Coin Toss Rate (i.e., with a 50%-50% chance of making it to year 30). All allocations vary with P/E10.
When P/E10 = 14, which is a typical value historically, you cannot own enough stocks. When P/E10 = 20, which defines the region of high valuations, you should cut your stock allocation to 40% to be safe. But even 60% or 70% is reasonably safe. With 100% stocks, you still have a 50%-50% chance of making it to year 30. When P/E10 = 24, as it did in the mid-1960s, the Safe allocation is 10%. The Coin Toss (50%-50%) allocation is 50% to 60%.
If you want safety at today's P/E10 of 27, you are better off out of the market. It is OK, reasonably safe, for you to allocate 10% to 20% to stocks. The Coin Toss allocation is 30% stocks.
Good things can still happen. There is a lot of volatility in the stock market. The odds of making it to year 30 are less than even.
You can download Calculator A from my Yahoo Briefcase. It is in the Big Project folder.
Yahoo Briefcase
Calculator A Details
I determined a new set of 30-year Historical Surviving Withdrawal Rates for 1923-1980 for portfolios HSWR20T2, HSWR50T2 and HSWR80T2. The portfolios consist of 20% stocks, 50% stocks and 80% stocks, respectively. The other component in each portfolio consists of TIPS at a 2.0% (real) interest rate.
I set the withdrawal amount at a fixed percentage of the portfolio’s initial balance (plus inflation). I increased withdrawal rates in increments of 0.1%. The Historical Surviving Withdrawal Rate HSWR for each start year is the highest withdrawal rate that would have maintained a zero or positive balance for 30 years.
I rebalanced allocations each year at zero cost. I set the expenses equal to 0.20% of the portfolio’s current balance. I used CPI-U for inflation adjustments.
These data extend to 2006. Previously, the data ended in 2002.
Regression Equations
These are the new regression equations.
For HSWR80T2:
The Safe Withdrawal Rate is the Calculated Rate minus 1.2%.
The Reasonably Safe Rate is the Calculated Rate minus 0.6%.
The Calculated Rate (or Coin Toss Rate) is: y=69.52/[P/E10]+1.6426.
The Likely Failure Rate is the Calculated Rate plus 0.9%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.8%.
For HSWR50T2:
The Safe Withdrawal Rate is the Calculated Rate minus 0.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.4%.
The Calculated Rate (or Coin Toss Rate) is: y=41.79/[P/E10]+2.8447.
The Likely Failure Rate is the Calculated Rate plus 0.6%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.2%.
For HSWR20T2:
The Safe Withdrawal Rate is the Calculated Rate minus 0.5%.
The Reasonably Safe Rate is the Calculated Rate minus 0.25%.
The Calculated Rate (or Coin Toss Rate) is: y=15.93/[P/E10]+3.8083.
The Likely Failure Rate is the Calculated Rate plus 0.3%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.6%.
Curve Fitting Equations
Curve fitting is especially easy when data are equally spaced. In this instance, I used allocations separated by 30%.
With equally spaced data points, it is especially easy to fit a parabola (i.e., a quadratic equation) to the data.
First, I normalized the data so that 20% stocks corresponds to x = -1, 50% stocks corresponds to x = 0 and 80% stocks corresponds to x = +1. The equation for other allocations is: x = [(stock allocation – 50%)/(30%)].
The equation for the parabola has the form: f(x) = a*x^2 + b*x + c. In this case, f(0) = c and f(+1) = a + b + c and f(-1) = a – b + c.
After manipulation:
a = ([f(+1) + f(-1)]/2) – f(0)
b = [f(+1) - f(-1)]/2
c = f(0).
I take a value from HSWR80T2 for f(+1). I take a corresponding value from HSWR50T2 for f(0). I take a corresponding value from HSWR20T2 for f(-1). I use these to determine a, b and c.
I have listed a, b and c separately. There are different values for the Safe Withdrawal Rate, for the Reasonably Safe Rate, for the Coin Toss Rate, for the Likely Failure Rate and for the Almost Certain Failure Rate. The values of a, b and c vary as P/E10 varies.
Allocations ranging from 0% stocks to 100% stocks correspond to values of x ranging from -1.667 to +1.667. I have listed the value of x that corresponds to each stock allocation.
The final step was to make sure to use $ signs and the fill handle appropriately to enter formulas easily and correctly.
The Next Step
The next step will be to calculate some more HSWRxxT2 regression equations to see how well this simple curve fit performs. If necessary, I could end up with additional, intermediate allocations for the curve fitting process.
We can get an idea of how accurate this calculator is by looking at the results with a 0% stock allocation.
We cannot get as good an indication as we might hope for. There is a hidden timing mismatch that clouds this comparison. In real life, we would expect to adjust withdrawals from an all-TIPS portfolio to match actual interest payments received. The calculator doesn’t do this. It adjusts withdrawals immediately to match inflation, before the Treasury makes its inflation adjustment. This causes portfolio lifetimes to vary slightly.
Calculator B
I have updated Calculator A with Calculator B. I have put the new calculator into my Yahoo Briefcase.
Here are the updates:
I updated the S&P500 and P/E10 references. I now display the value of the S&P500 that corresponds to whatever value of P/E10 that you enter.
The scaling formula is very simple:
S&P500 value = (P/E10 value)*[(S&P500 reference)/([P/E10 reference]).
As of December 2005, the S&P500 reference was 1262.07. The P/E10 reference was 26.4343.
Additional Regression Equations
I have calculated a complete set of regression equations with 2% TIPS.
For HSWR100T2:
R-squared = 0.6987.
The Safe Withdrawal Rate is the Calculated Rate minus 1.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.9%.
The Calculated Rate (or Coin Toss Rate) is: y=88.96/[P/E10]+0.6561.
The Likely Failure Rate is the Calculated Rate plus 1.0%.
The Almost Certain Failure Rate is the Calculated Rate plus 2.0%.
For HSWR90T2:
R-squared = 0.7103.
The Safe Withdrawal Rate is the Calculated Rate minus 1.6%.
The Reasonably Safe Rate is the Calculated Rate minus 0.8%.
The Calculated Rate (or Coin Toss Rate) is: y=79.43/[P/E10]+1.1385.
The Likely Failure Rate is the Calculated Rate plus 0.9%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.8%.
For HSWR80T2:
R-squared = 0.7172.
The Safe Withdrawal Rate is the Calculated Rate minus 1.2%.
The Reasonably Safe Rate is the Calculated Rate minus 0.6%.
The Calculated Rate (or Coin Toss Rate) is: y=69.52/[P/E10]+1.6426.
The Likely Failure Rate is the Calculated Rate plus 0.9%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.8%.
For HSWR70T2:
R-squared = 0.725
The Safe Withdrawal Rate is the Calculated Rate minus 1.0%.
The Reasonably Safe Rate is the Calculated Rate minus 0.5%.
The Calculated Rate (or Coin Toss Rate) is: y=60.14/[P/E10]+2.0664.
The Likely Failure Rate is the Calculated Rate plus 0.7%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.4%.
For HSWR60T2:
R-squared = 0.719.
The Safe Withdrawal Rate is the Calculated Rate minus 0.9%.
The Reasonably Safe Rate is the Calculated Rate minus 0.45%.
The Calculated Rate (or Coin Toss Rate) is: y=50.14/[P/E10]+2.5319.
The Likely Failure Rate is the Calculated Rate plus 0.6%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.2%.
For HSWR50T2:
R-squared = 0.7326.
The Safe Withdrawal Rate is the Calculated Rate minus 0.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.4%.
The Calculated Rate (or Coin Toss Rate) is: y=41.79/[P/E10]+2.8447.
The Likely Failure Rate is the Calculated Rate plus 0.6%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.2%.
For HSWR40T2:
R-squared = 0.722.
The Safe Withdrawal Rate is the Calculated Rate minus 0.6%.
The Reasonably Safe Rate is the Calculated Rate minus 0.3%.
The Calculated Rate (or Coin Toss Rate) is: y=32.48/[P/E10]+3.2421.
The Likely Failure Rate is the Calculated Rate plus 0.5%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.0%.
For HSWR30T2:
R-squared = 0.6949.
The Safe Withdrawal Rate is the Calculated Rate minus 0.6%.
The Reasonably Safe Rate is the Calculated Rate minus 0.3%.
The Calculated Rate (or Coin Toss Rate) is: y=23.94/[P/E10]+3.546.
The Likely Failure Rate is the Calculated Rate plus 0.4%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.8%.
For HSWR20T2:
R-squared = 0.6434.
The Safe Withdrawal Rate is the Calculated Rate minus 0.5%.
The Reasonably Safe Rate is the Calculated Rate minus 0.25%.
The Calculated Rate (or Coin Toss Rate) is: y=15.93/[P/E10]+3.8083.
The Likely Failure Rate is the Calculated Rate plus 0.3%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.6%.
For HSWR10T2:
R-squared = 0.4754.
The Safe Withdrawal Rate is the Calculated Rate minus 0.4%.
The Reasonably Safe Rate is the Calculated Rate minus 0.2%.
The Calculated Rate (or Coin Toss Rate) is: y=8.96/[P/E10]+3.9789.
The Likely Failure Rate is the Calculated Rate plus 0.25%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.5%.
For HSWR00T2:
R-squared = 0.0474.
The Safe Withdrawal Rate is the Calculated Rate minus 0.2%.
The Reasonably Safe Rate is the Calculated Rate minus 0.1%.
The Calculated Rate (or Coin Toss Rate) is: y=1.84/[P/E10]+4.1546.
The Likely Failure Rate is the Calculated Rate plus 0.15%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.3%.
Note that there is a spread of 0.5% even without stocks because of the timing mismatch.
Calculator C
I have put Calculator C into my Yahoo Briefcase.
It has a section with the exact regression equations (in allocation increments of 10%) for making comparisons. My impression: curve fitting does well.
Since it is not overwhelmingly difficult to determine a full set of equations, I am planning to collect complete sets of data at a few additional TIPS interest rates. I will use curve fitting for intermediate interest rates, but not for stock allocations.
All of this is subject to change. It is easy to become overwhelmed in a project such as this one. Should the effort become overly burdensome, I will feel free to go back to curve fitting and interpolation.
Calculator D
I have put Calculator D into my Yahoo Briefcase. It was extracted from Calculator C.
It is a compact calculator. It uses the exact equations for stocks and 2% TIPS. You simply enter a value of P/E10. It converts this into a current S&P500 index level. It displays a full set of withdrawal rates for each stock allocation in increments of 10%.
At the Safe Withdrawal Rate, you have about a 95% (one-sided) probability of reaching year 30 successfully. At the Reasonably Safe Rate, you have about an 80% chance of reaching year 30 successfully. At the Coin Toss Rate, your odds are down to 50%-50%.
All withdrawals are given as a percentage of your portfolio's initial balance (plus inflation).
Calculator E
I have put Calculator E into my Yahoo Briefcase. It is identical to Calculator D except that I have added a full set of data with 1% TIPS.
It is a compact calculator. You simply enter a value of P/E10. It converts this into a current S&P500 index level. It displays a full set of withdrawal rates for each stock allocation in increments of 10%.
It has two sets of displays. One with 2% TIPS and the other with 1% TIPS. It uses the exact equations.
Enter today's value of P/E10, which is 27.5. Notice that the best stock allocation at the highest level of safety is ZERO with 2% TIPS but 30% with 1% TIPS. Similarly, at the reasonably safe level, you should allocate 20% to stocks with 2% TIPS and 30% with 1% TIPS.
Here are the regression equations with 1% TIPS.
For HSWR100T1:
R-squared = 0.6987.
The Safe Withdrawal Rate is the Calculated Rate minus 1.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.9%.
The Calculated Rate (or Coin Toss Rate) is: y=88.96/[P/E10]+0.6561.
The Likely Failure Rate is the Calculated Rate plus 1.0%.
The Almost Certain Failure Rate is the Calculated Rate plus 2.0%.
For HSWR90T1:
R-squared = 0.7072.
The Safe Withdrawal Rate is the Calculated Rate minus 1.4%.
The Reasonably Safe Rate is the Calculated Rate minus 0.7%.
The Calculated Rate (or Coin Toss Rate) is: y=78.51/[P/E10]+1.1369.
The Likely Failure Rate is the Calculated Rate plus 1.0%.
The Almost Certain Failure Rate is the Calculated Rate plus 2.0%.
For HSWR80T1:
R-squared = 0.7207.
The Safe Withdrawal Rate is the Calculated Rate minus 1.0%.
The Reasonably Safe Rate is the Calculated Rate minus 0.5%.
The Calculated Rate (or Coin Toss Rate) is: y=68.69/[P/E10]+1.5596.
The Likely Failure Rate is the Calculated Rate plus 0.9%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.8%.
For HSWR70T1:
R-squared = 0.7278.
The Safe Withdrawal Rate is the Calculated Rate minus 1.0%.
The Reasonably Safe Rate is the Calculated Rate minus 0.5%.
The Calculated Rate (or Coin Toss Rate) is: y=58.71/[P/E10]+1.969.
The Likely Failure Rate is the Calculated Rate plus 0.7%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.4%.
For HSWR60T1:
R-squared = 0.7292.
The Safe Withdrawal Rate is the Calculated Rate minus 0.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.4%.
The Calculated Rate (or Coin Toss Rate) is: y=49.1/[P/E10]+2.3337.
The Likely Failure Rate is the Calculated Rate plus 0.6%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.2%.
For HSWR50T1:
R-squared = 0.7264.
The Safe Withdrawal Rate is the Calculated Rate minus 0.7%.
The Reasonably Safe Rate is the Calculated Rate minus 0.35%.
The Calculated Rate (or Coin Toss Rate) is: y=39.64/[P/E10]+2.6839.
The Likely Failure Rate is the Calculated Rate plus 0.45%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.9%.
For HSWR40T1:
R-squared = 0.7334.
The Safe Withdrawal Rate is the Calculated Rate minus 0.6%.
The Reasonably Safe Rate is the Calculated Rate minus 0.3%.
The Calculated Rate (or Coin Toss Rate) is: y=31.69/[P/E10]+2.9096.
The Likely Failure Rate is the Calculated Rate plus 0.5%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.0%.
For HSWR30T1:
R-squared = 0.6987.
The Safe Withdrawal Rate is the Calculated Rate minus 0.4%.
The Reasonably Safe Rate is the Calculated Rate minus 0.2%.
The Calculated Rate (or Coin Toss Rate) is: y=22.42/[P/E10]+3.212.
The Likely Failure Rate is the Calculated Rate plus 0.3%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.6%.
For HSWR20T1:
R-squared = 0.6393.
The Safe Withdrawal Rate is the Calculated Rate minus 0.4%.
The Reasonably Safe Rate is the Calculated Rate minus 0.2%.
The Calculated Rate (or Coin Toss Rate) is: y=14.78/[P/E10]+3.4011.
The Likely Failure Rate is the Calculated Rate plus 0.3%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.6%.
For HSWR10T1:
R-squared = 0.4612.
The Safe Withdrawal Rate is the Calculated Rate minus 0.3%.
The Reasonably Safe Rate is the Calculated Rate minus 0.15%.
The Calculated Rate (or Coin Toss Rate) is: y=7.77/[P/E10]+3.5409.
The Likely Failure Rate is the Calculated Rate plus 0.2%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.4%.
For HSWR00T1:
R-squared = 0.0072.
The Safe Withdrawal Rate is the Calculated Rate minus 0.04%.
The Reasonably Safe Rate is the Calculated Rate minus 0.02%.
The Calculated Rate (or Coin Toss Rate) is: y=0.32/[P/E10]+3.635.
The Likely Failure Rate is the Calculated Rate plus 0.03%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.06%.
Calculator F
I have put Calculator F into my Yahoo Briefcase. It is in the Big Project folder. It is identical to Calculator E except that I have added a full set of data with 3% TIPS.
Here are the regression equations with 3% TIPS.
For HSWR100T3:
R-squared = 0.6987.
The Safe Withdrawal Rate is the Calculated Rate minus 1.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.9%.
The Calculated Rate (or Coin Toss Rate) is: y=88.96/[P/E10]+0.6561.
The Likely Failure Rate is the Calculated Rate plus 1.0%.
The Almost Certain Failure Rate is the Calculated Rate plus 2.0%.
For HSWR90T3:
R-squared = 0.7106.
The Safe Withdrawal Rate is the Calculated Rate minus 1.4%.
The Reasonably Safe Rate is the Calculated Rate minus 0.7%.
The Calculated Rate (or Coin Toss Rate) is: y=79.95/[P/E10]+1.1797.
The Likely Failure Rate is the Calculated Rate plus 1.0%.
The Almost Certain Failure Rate is the Calculated Rate plus 2.0%.
For HSWR80T3:
R-squared = 0.7166.
The Safe Withdrawal Rate is the Calculated Rate minus 1.4%.
The Reasonably Safe Rate is the Calculated Rate minus 0.7%.
The Calculated Rate (or Coin Toss Rate) is: y=70.62/[P/E10]+1.6971.
The Likely Failure Rate is the Calculated Rate plus 0.8%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.6%.
For HSWR70T3:
R-squared = 0.7237.
The Safe Withdrawal Rate is the Calculated Rate minus 1.0%.
The Reasonably Safe Rate is the Calculated Rate minus 0.5%.
The Calculated Rate (or Coin Toss Rate) is: y=61.52/[P/E10]+2.1877.
The Likely Failure Rate is the Calculated Rate plus 0.9%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.8%.
For HSWR60T3:
R-squared = 0.7307.
The Safe Withdrawal Rate is the Calculated Rate minus 0.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.4%.
The Calculated Rate (or Coin Toss Rate) is: y=52.39/[P/E10]+2.6514.
The Likely Failure Rate is the Calculated Rate plus 0.6%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.2%.
For HSWR50T3:
R-squared = 0.7286.
The Safe Withdrawal Rate is the Calculated Rate minus 0.7%.
The Reasonably Safe Rate is the Calculated Rate minus 0.35%.
The Calculated Rate (or Coin Toss Rate) is: y=43.06/[P/E10]+3.1034.
The Likely Failure Rate is the Calculated Rate plus 0.45%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.9%.
For HSWR40T3:
R-squared = 0.7275.
The Safe Withdrawal Rate is the Calculated Rate minus 0.6%.
The Reasonably Safe Rate is the Calculated Rate minus 0.3%.
The Calculated Rate (or Coin Toss Rate) is: y=34.71/[P/E10]+3.478.
The Likely Failure Rate is the Calculated Rate plus 0.5%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.0%.
For HSWR30T3:
R-squared = 0.6988.
The Safe Withdrawal Rate is the Calculated Rate minus 0.4%.
The Reasonably Safe Rate is the Calculated Rate minus 0.2%.
The Calculated Rate (or Coin Toss Rate) is: y=25.52/[P/E10]+3.8923.
The Likely Failure Rate is the Calculated Rate plus 0.3%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.6%.
For HSWR20T3:
R-squared = 0.6431.
The Safe Withdrawal Rate is the Calculated Rate minus 0.4%.
The Reasonably Safe Rate is the Calculated Rate minus 0.2%.
The Calculated Rate (or Coin Toss Rate) is: y=17.3/[P/E10]+4.2103.
The Likely Failure Rate is the Calculated Rate plus 0.3%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.6%.
For HSWR10T3:
R-squared = 0.4733.
The Safe Withdrawal Rate is the Calculated Rate minus 0.3%.
The Reasonably Safe Rate is the Calculated Rate minus 0.15%.
The Calculated Rate (or Coin Toss Rate) is: y=9.83/[P/E10]+4.478.
The Likely Failure Rate is the Calculated Rate plus 0.2%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.4%.
For HSWR00T3:
R-squared = 0.0466.
The Safe Withdrawal Rate is the Calculated Rate minus 0.3%.
The Reasonably Safe Rate is the Calculated Rate minus 0.15%.
The Calculated Rate (or Coin Toss Rate) is: y=2.12/[P/E10]+4.7455.
The Likely Failure Rate is the Calculated Rate plus 0.25%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.5%.
Calculator G
Calculator G allows you to determine withdrawal rates for any value of P/E10 at any TIPS (real) interest rate.
I have added interpolation equations to Calculator F to create Calculator G. The interpolations are for the TIPS interest rates. The value of x is TIPS interest rate minus 2%. Otherwise, the previous discussion of curve fitting applies.
The equation is cell D21 is:
=((D36+D60)*0.5-D48)*($C$15-2)^2+((D36-D60)*0.5*($C$15-2))+D48
I used the fill handle to create all of the other equations in the top block.
I have placed Calculator G into my Yahoo Briefcase.
Caution about the Timing Mismatch
There is no general way around the timing mismatch in our calculators. The mismatch is because inflation adjustments for TIPS payments are made AFTER inflation occurs. Our calculators adjust withdrawals AS inflation occurs.
In reality, most people automatically adjust their spending habits to match their TIPS income, not inflation.
The practical effect is that the appropriate rate to use when looking at high TIPS allocations (0% or 10% stocks) is the Likely Failure rate, not the Coin Toss rate.
You can see the exact formula for 100% TIPS portfolios in Equations for Design. It has examples.
Equations for Design
Dividend Calculators A and B
These calculators allow you to estimate dividend growth. The estimates are in terms of the income stream, dividend amounts, not dividend yields, which vary with price.
All of the numbers include inflation adjustments (i.e., real dollars).
Here are the design details.
I have extracted the following formulas from the Dividend-Based Design Example:
At year 4:
y = -0.3162x+23.799 plus 30% and minus 20%.
At year 8:
y = -0.9914x+79.81 plus 60% and minus 30%.
At year 12:
y = -1.0328x+95.556 plus 100% and minus 50%.
The equations determine y = the 4-year moving average of the growth of the dividend amount and x = the payout ratio based on smoothed earnings E10 (where x is a percentage). The first year of dividends occur in Year 1 (as opposed to Year 0).
You put P/E10 into cell B13. I scale this into an S&P500 index value. The reference values are in cells C9 and C10.
The S&P500 Dividend Yield goes into cell D15. I use this value when I determine the payout ratio (in terms of E10, the average of ten years of real earnings. Cell C16 equals the dividend yield of the S&P500 index in cell D15 time P/E10 (in cell B13). That is, the payout ratio is the dividend yield in percent divided by the earnings yield as a fraction E10/P.
I used these values to fill in all of the formulas in the EXACT RESULTS for the NUMBER OF YEARS SPECIFIED section of the table. The extremes apply the confidence limits. The intermediate levels apply one-half of the confidence limits. They correspond to confidence levels close to 20% and 80%.
I scale S&P500 results by 1.725 for High Quality, High Dividend Stocks.
I applied the curve fitting equation to the number of years in both calculators. I used curve fitting equations previously discussed with x = (number of years – 8)/4. The equation in cell D26 of Dividend Calculator A is:
=(((D64+D40)*0.5-D52)*(($E$19-8)/4)^2)+(D64-D40)*0.5*(($E$19-8)/4)+D52
I used the fill handle for the other S&P500 equations.
Similarly, for the High Quality, High Dividend Stocks, I use the equations:
=(((J64+J40)*0.5-J52)*(($E$19-8)/4)^2)+(J64-J40)*0.5*(($E$19-8)/4)+J52
I interpolate among years in Dividend Calculator B. I ignore all interactions (between the number of years and the payout ratio). I used the same curve fitting equations with x = (payout ratio from cell C16 – 50%)/10%.
Cell C25 contains:
=((F37+D37)*0.5-E37)*(($C$16-50)/10)^2+(F37-D37)*0.5*(($C$16-50)/10)+E37
Cell F25 contains:
=((L37+J37)*0.5-K37)*(($C$16-50)/10)^2+(L37-J37)*0.5*(($C$16-50)/10)+K37
I used the fill handle to drag each equation down.
I have placed these calculators in my Yahoo Briefcase in the Big Project folder. I have labeled them Dividend Calc A and Dividend Calc B.
Dividend-Based Design Example
Yahoo Briefcase
I was asked about what happens to dividend growth projections based on more recent times, starting in the mid-1950s. Here is my response.
Dividend Growth Sensitivity Study
Calculator H
I have added SwAT and SwOptT portfolios to calculator G.
SwAT switches stock and TIPS allocations, subject to constraint A, which keeps both stock and bond allocations between 25% and 75%.
1) The stock allocations are 75%-40%-25%.
2) The P/E10 thresholds are 11 and 21, respectively.
SwOptT switches stock and TIPS allocations optimally.
1) The P/E10 thresholds are 9-12-21-24.
2) The allocations are 100%-50%-30%-20%-0%, respectively.
When P/E10 is below 9, the stock allocation is 100%. When the P/E10 is between 9 and 12, the stock allocation is 50%. And so on.
New Regression Equations
I have calculated regression equations for SWAT and SwOptT portfolios.
For SWAT3 with 3% TIPS:
R-squared = 0.7364.
The Safe Withdrawal Rate is the Calculated Rate minus 0.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.4%.
The Calculated Rate (or Coin Toss Rate) is: y=42.13/[P/E10]+3.6298.
The Likely Failure Rate is the Calculated Rate plus 0.6%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.2%.
For SWAT2 with 2% TIPS:
R-squared = 0.7451.
The Safe Withdrawal Rate is the Calculated Rate minus 0.7%.
The Reasonably Safe Rate is the Calculated Rate minus 0.35%.
The Calculated Rate (or Coin Toss Rate) is: y=40.91/[P/E10]+3.341.
The Likely Failure Rate is the Calculated Rate plus 0.55%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.1%.
For SWAT1 with 1% TIPS:
R-squared = 0.741.
The Safe Withdrawal Rate is the Calculated Rate minus 0.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.4%.
The Calculated Rate (or Coin Toss Rate) is: y=39.37/[P/E10]+3.0814.
The Likely Failure Rate is the Calculated Rate plus 0.6%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.2%.
For SwOptT3 with 3% TIPS:
R-squared = 0.6931.
The Safe Withdrawal Rate is the Calculated Rate minus 1.0%.
The Reasonably Safe Rate is the Calculated Rate minus 0.5%.
The Calculated Rate (or Coin Toss Rate) is: y=37.14/[P/E10]+4.1438.
The Likely Failure Rate is the Calculated Rate plus 0.5%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.0%.
For SwOptT2 with 2% TIPS:
R-squared = 0.6817.
The Safe Withdrawal Rate is the Calculated Rate minus 0.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.4%.
The Calculated Rate (or Coin Toss Rate) is: y=36.14/[P/E10]+3.7751.
The Likely Failure Rate is the Calculated Rate plus 0.5%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.0%.
For SwOptT1 with 1% TIPS:
R-squared = 0.6704.
The Safe Withdrawal Rate is the Calculated Rate minus 0.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.4%.
The Calculated Rate (or Coin Toss Rate) is: y=34.34/[P/E10]+3.4569.
The Likely Failure Rate is the Calculated Rate plus 0.5%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.0%.
Interpolation
I use the same curve fitting equations as before. The entry in cell P22 is:
=((P36+P60)*0.5-P48)*($C$15-2)^2+((P36-P60)*0.5*($C$15-2))+P48
I used the fill handle to extend this formula to fill in the block (cells P and Q, rows 22 through 26).
Downloading
You may download this calculator from the Big Project folder in my Yahoo Briefcase. It is Calculator H.
Yahoo Briefcase
CTVR Calculator A
Constant Terminal Value Rates (CTVR) are withdrawal rates that leave the balance at the end of a period identical to the initial balance plus inflation.
CTVR Calculator A is almost identical to Calculator H. It calculates 30-Year Constant Terminal Value Rates instead of rates that end with a zero balance.
I only include portfolios with 20%, 50% and 80% stocks in my initial version, CTVR Calculator A. In addition, I have renamed portfolios SwAT and SwOptT to CSwAT and CSwOptT to make it clear that the final balance calculated is no longer zero.
Date Range
I normally construct Safe Withdrawal Rate calculators based on 1923-1980 data. In this instance, I narrowed the range to 1923-1975 because of anomalies associated with incomplete sequences and switching (SwAT and SwOptT portfolios). The problem turns out to be the default level of P/E10. It is blank for the years 2006-2010. The calculator treats these blanks as zeros. It assigns a 100% stock allocation for these years. The dummy data for stocks in years 2006-2010 include heavy losses. This artifact reduced the Constant Terminal Value Rates to unreasonably low levels for 30-year sequences beginning in 1976-1980.
[Years 1921 and 1922 are consistent outliers, showing saturation, with exceedingly high earnings yields. The combination, outliers and extra weighting by Excel’s curve fitting algorithm, reduces the goodness of fit.]
New Regression Equations
Here are the new regression equations.
3% TIPS:
For CTVR20T3 with 3% TIPS:
R-squared = 0.6738.
The Safe Withdrawal Rate is the Calculated Rate minus 0.4%.
The Reasonably Safe Rate is the Calculated Rate minus 0.2%.
The Calculated Rate (or Coin Toss Rate) is: y=16.69/[P/E10]+2.435.
The Likely Failure Rate is the Calculated Rate plus 0.2%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.4%.
For CTVR50T3 with 3% TIPS:
R-squared = 0.7892.
The Safe Withdrawal Rate is the Calculated Rate minus 0.6%.
The Reasonably Safe Rate is the Calculated Rate minus 0.3%.
The Calculated Rate (or Coin Toss Rate) is: y=43.81/[P/E10]+1.6196.
The Likely Failure Rate is the Calculated Rate plus 0.5%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.0%.
For CTVR80T3 with 3% TIPS:
R-squared = 0.8017.
The Safe Withdrawal Rate is the Calculated Rate minus 0.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.4%.
The Calculated Rate (or Coin Toss Rate) is: y=73.28/[P/E10]+0.3523.
The Likely Failure Rate is the Calculated Rate plus 0.7%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.4%.
For CSWAT3 with 3% TIPS:
R-squared = 0.7371.
The Safe Withdrawal Rate is the Calculated Rate minus 0.9%.
The Reasonably Safe Rate is the Calculated Rate minus 0.45%.
The Calculated Rate (or Coin Toss Rate) is: y=45.72/[P/E10]+2.0713.
The Likely Failure Rate is the Calculated Rate plus 0.5%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.0%.
For CSwOptT3 with 3% TIPS:
R-squared = 0.6157.
The Safe Withdrawal Rate is the Calculated Rate minus 1.0%.
The Reasonably Safe Rate is the Calculated Rate minus 0.5%.
The Calculated Rate (or Coin Toss Rate) is: y=38.86/[P/E10]+2.6956.
The Likely Failure Rate is the Calculated Rate plus 0.6%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.2%.
2% TIPS:
For CTVR20T2 with 2% TIPS:
R-squared = 0.6359.
The Safe Withdrawal Rate is the Calculated Rate minus 0.4%.
The Reasonably Safe Rate is the Calculated Rate minus 0.2%.
The Calculated Rate (or Coin Toss Rate) is: y=14.73/[P/E10]+1.8117.
The Likely Failure Rate is the Calculated Rate plus 0.2%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.4%.
For CTVR50T2 with 2% TIPS:
R-squared = 0.7697.
The Safe Withdrawal Rate is the Calculated Rate minus 0.6%.
The Reasonably Safe Rate is the Calculated Rate minus 0.3%.
The Calculated Rate (or Coin Toss Rate) is: y=41.3/[P/E10]+1.3278.
The Likely Failure Rate is the Calculated Rate plus 0.4%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.8%.
For CTVR80T2 with 2% TIPS:
R-squared = 0.8011.
The Safe Withdrawal Rate is the Calculated Rate minus 0.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.4%.
The Calculated Rate (or Coin Toss Rate) is: y=72/[P/E10]+0.2628.
The Likely Failure Rate is the Calculated Rate plus 0.6%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.2%.
For CSWAT2 with 2% TIPS:
R-squared = 0.7076.
The Safe Withdrawal Rate is the Calculated Rate minus 0.9%.
The Reasonably Safe Rate is the Calculated Rate minus 0.45%.
The Calculated Rate (or Coin Toss Rate) is: y=43.59/[P/E10]+1.7051.
The Likely Failure Rate is the Calculated Rate plus 0.45%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.9%.
For CSwOptT2 with 2% TIPS:
R-squared = 0.5678.
The Safe Withdrawal Rate is the Calculated Rate minus 1.0%.
The Reasonably Safe Rate is the Calculated Rate minus 0.5%.
The Calculated Rate (or Coin Toss Rate) is: y=36.34/[P/E10]+2.2695.
The Likely Failure Rate is the Calculated Rate plus 0.5%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.0%.
1% TIPS:
For CTVR20T1 with 1% TIPS:
R-squared = 0.5943.
The Safe Withdrawal Rate is the Calculated Rate minus 0.4%.
The Reasonably Safe Rate is the Calculated Rate minus 0.2%.
The Calculated Rate (or Coin Toss Rate) is: y=13.03/[P/E10]+1.1434.
The Likely Failure Rate is the Calculated Rate plus 0.2%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.4%.
For CTVR50T1 with 1% TIPS:
R-squared = 0.7599.
The Safe Withdrawal Rate is the Calculated Rate minus 0.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.4%.
The Calculated Rate (or Coin Toss Rate) is: y=39.43/[P/E10]+0.9925.
The Likely Failure Rate is the Calculated Rate plus 0.4%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.8%.
For CTVR80T1 with 1% TIPS:
R-squared = 0.7973.
The Safe Withdrawal Rate is the Calculated Rate minus 0.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.4%.
The Calculated Rate (or Coin Toss Rate) is: y=70.68/[P/E10]+0.1631.
The Likely Failure Rate is the Calculated Rate plus 0.7%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.4%.
For CSWAT1 with 1% TIPS:
R-squared = 0.6792.
The Safe Withdrawal Rate is the Calculated Rate minus 1.0%.
The Reasonably Safe Rate is the Calculated Rate minus 0.5%.
The Calculated Rate (or Coin Toss Rate) is: y=41.09/[P/E10]+1.3467.
The Likely Failure Rate is the Calculated Rate plus 0.5%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.0%.
For CSwOptT1 with 1% TIPS:
R-squared = 0.5158.
The Safe Withdrawal Rate is the Calculated Rate minus 1.2%.
The Reasonably Safe Rate is the Calculated Rate minus 0.6%.
The Calculated Rate (or Coin Toss Rate) is: y=33.6/[P/E10]+1.8475.
The Likely Failure Rate is the Calculated Rate plus 0.7%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.4%.
Downloading
You may download this calculator from the Big Project folder in my Yahoo Briefcase. I have listed it as CTVR Calc A.
Yahoo Briefcase
HCTVR Calculator A
Constant Terminal Value Rates (CTVR) are withdrawal rates that leave the balance at the end of a period identical to the initial balance plus inflation. Half Constant Terminal Value Rates (HCTVR) leave a final balance of one-half of the initial balance (plus inflation).
HCTVR Calculator A is almost identical to CTVR Calculator A. It calculates 30-Year Half Constant Terminal Value Rates instead of 30-Year CTVRs.
Half Constant Terminal Value Rates are useful for retirees who have a reasonable chance of living longer than the 30-year planning period. They can look upon long life as a blessing even if their retirement portfolio disappoints.
For young retirees, this defines an intermediate level of risk. It is not as restrictive as a Safe Withdrawal Rate. They can relax their safety demands at the Constant Terminal Value Rate, while maintaining a high level of safety at the Half Constant Terminal Value Rate.
I have only include portfolios with 20%, 50% and 80% stocks in my initial version, HCTVR Calculator A. In addition, I have renamed portfolios SwAT and SwOptT (and CSwAT and CSwOptT) to HCSwAT and HCSwOptT.
Collecting Data
Collecting Half Constant Terminal Value Rate data on any of my calculators (including Deluxe Calculator V1.1A08a) is easy. Simply enter an initial balance of $200000 and look at columns S (date) and AB (30-Year Real Balances), rows 10-144. One-half of the initial balance is $100000. Look for the first occurrence of a balance with five digits instead of six.
Similarly, when collecting Constant Terminal Value Rates. Set the initial balance at $100000 and scroll to columns S and AB.
This technique can be used with any calculator developed originally from the Retire Early Safe Withdrawal Calculator.
I collected data for 30-year sequences beginning in 1921-1980. I only used the sequences beginning in 1923-1975 for the same reasons as before. See my description of the Big Project Constant Terminal Value Rate Calculator A.
Regression Equations
Here are the Half Constant Terminal Value Rate regression equations.
3% TIPS:
For HCTVR20T3 with 3% TIPS:
R-squared = 0.671.
The Safe Withdrawal Rate is the Calculated Rate minus 0.4%.
The Reasonably Safe Rate is the Calculated Rate minus 0.2%.
The Calculated Rate (or Coin Toss Rate) is: y=16.97/[P/E10]+3.3267.
The Likely Failure Rate is the Calculated Rate plus 0.3%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.6%.
For HCTVR50T3 with 3% TIPS:
R-squared = 0.7776.
The Safe Withdrawal Rate is the Calculated Rate minus 0.6%.
The Reasonably Safe Rate is the Calculated Rate minus 0.3%.
The Calculated Rate (or Coin Toss Rate) is: y=44.85/[P/E10]+2.2902.
The Likely Failure Rate is the Calculated Rate plus 0.5%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.0%.
For HCTVR80T3 with 3% TIPS:
R-squared = 0.7765.
The Safe Withdrawal Rate is the Calculated Rate minus 1.0%.
The Reasonably Safe Rate is the Calculated Rate minus 0.5%.
The Calculated Rate (or Coin Toss Rate) is: y=74.44/[P/E10]+0.893.
The Likely Failure Rate is the Calculated Rate plus 0.7%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.4%.
For HCSWAT3 with 3% TIPS:
R-squared = 0.7793.
The Safe Withdrawal Rate is the Calculated Rate minus 0.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.4%.
The Calculated Rate (or Coin Toss Rate) is: y=46.49/[P/E10]+2.7101.
The Likely Failure Rate is the Calculated Rate plus 0.6%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.2%.
For HCSwOptT3 with 3% TIPS:
R-squared = 0.6625.
The Safe Withdrawal Rate is the Calculated Rate minus 1.0%.
The Reasonably Safe Rate is the Calculated Rate minus 0.5%.
The Calculated Rate (or Coin Toss Rate) is: y=39.47/[P/E10]+3.3292.
The Likely Failure Rate is the Calculated Rate plus 0.6%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.2%.
2% TIPS:
For HCTVR20T2 with 2% TIPS:
R-squared = 0.6585.
The Safe Withdrawal Rate is the Calculated Rate minus 0.3%.
The Reasonably Safe Rate is the Calculated Rate minus 0.15%.
The Calculated Rate (or Coin Toss Rate) is: y=15.14/[P/E10]+2.8146.
The Likely Failure Rate is the Calculated Rate plus 0.25%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.5%.
For HCTVR50T2 with 2% TIPS:
R-squared = 0.7793.
The Safe Withdrawal Rate is the Calculated Rate minus 0.6%.
The Reasonably Safe Rate is the Calculated Rate minus 0.3%.
The Calculated Rate (or Coin Toss Rate) is: y=42.86/[P/E10]+2.0272.
The Likely Failure Rate is the Calculated Rate plus 0.5%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.0%.
For HCTVR80T2 with 2% TIPS:
R-squared = 0.7766.
The Safe Withdrawal Rate is the Calculated Rate minus 1.0%.
The Reasonably Safe Rate is the Calculated Rate minus 0.5%.
The Calculated Rate (or Coin Toss Rate) is: y=73.09/[P/E10]+0.8203.
The Likely Failure Rate is the Calculated Rate plus 0.8%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.6%.
For HCSWAT2 with 2% TIPS:
R-squared = 0.7677.
The Safe Withdrawal Rate is the Calculated Rate minus 0.7%.
The Reasonably Safe Rate is the Calculated Rate minus 0.35%.
The Calculated Rate (or Coin Toss Rate) is: y=44.45/[P/E10]+2.3984.
The Likely Failure Rate is the Calculated Rate plus 0.6%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.2%.
For HCSwOptT2 with 2% TIPS:
R-squared = 0.6349.
The Safe Withdrawal Rate is the Calculated Rate minus 1.0%.
The Reasonably Safe Rate is the Calculated Rate minus 0.5%.
The Calculated Rate (or Coin Toss Rate) is: y=37.21/[P/E10]+2.9788.
The Likely Failure Rate is the Calculated Rate plus 0.6%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.2%.
1% TIPS:
For HCTVR20T1 with 1% TIPS:
R-squared = 0.6718.
The Safe Withdrawal Rate is the Calculated Rate minus 0.3%.
The Reasonably Safe Rate is the Calculated Rate minus 0.15%.
The Calculated Rate (or Coin Toss Rate) is: y=13.6/[P/E10]+2.295.
The Likely Failure Rate is the Calculated Rate plus 0.15%.
The Almost Certain Failure Rate is the Calculated Rate plus 0.3%.
For HCTVR50T1 with 1% TIPS:
R-squared = 0.7718.
The Safe Withdrawal Rate is the Calculated Rate minus 0.7%.
The Reasonably Safe Rate is the Calculated Rate minus 0.35%.
The Calculated Rate (or Coin Toss Rate) is: y=40.64/[P/E10]+1.7758.
The Likely Failure Rate is the Calculated Rate plus 0.6%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.2%.
For HCTVR80T1 with 1% TIPS:
R-squared = 0.7772.
The Safe Withdrawal Rate is the Calculated Rate minus 0.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.4%.
The Calculated Rate (or Coin Toss Rate) is: y=72.14/[P/E10]+0.73.
The Likely Failure Rate is the Calculated Rate plus 0.7%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.4%.
For HCSWAT1 with 1% TIPS:
R-squared = 0.7458.
The Safe Withdrawal Rate is the Calculated Rate minus 0.8%.
The Reasonably Safe Rate is the Calculated Rate minus 0.4%.
The Calculated Rate (or Coin Toss Rate) is: y=42.2/[P/E10]+2.1068.
The Likely Failure Rate is the Calculated Rate plus 0.6%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.2%.
For HCSwOptT1 with 1% TIPS:
R-squared = 0.606.
The Safe Withdrawal Rate is the Calculated Rate minus 1.0%.
The Reasonably Safe Rate is the Calculated Rate minus 0.5%.
The Calculated Rate (or Coin Toss Rate) is: y=35.28/[P/E10]+2.5966.
The Likely Failure Rate is the Calculated Rate plus 0.5%.
The Almost Certain Failure Rate is the Calculated Rate plus 1.0%.
Downloading
You may download this calculator from the Big Project folder in my Yahoo Briefcase. I have listed it as HCTVR Calc A.
Yahoo Briefcase
Variable Terminal Value Rate Calculator A
I have added the Variable Terminal Value Rate Calculator A to my Yahoo Briefcase. It interpolates withdrawal rates for balances between 0% and 100%.
Constant Terminal Value Rates (CTVR) are withdrawal rates that leave the balance at the end of a period identical to the initial balance plus inflation. Half Constant Terminal Value Rates (HCTVR) leave a final balance of one-half of the initial balance (plus inflation).
I use my 30-year Safe Withdrawal Rate calculator for balances of zero at Year 30. Technically, this is not quite the same thing as a Terminal Value Rate calculator with a final balance of zero. Often, a portfolio balance falls just below its Terminal Value before recovering at Year 30.
This adds flexibility for traditional retirees who have a reasonable chance of living longer than the 30-year planning period. This allows younger retirees to look at intermediate levels of risk.
Yahoo Briefcase
Variable Terminal Value Rate Calculator B
My latest Variable Terminal Value Rate Calculator B identifies optimal stock allocations (subject to constraint B).
Constraint B is that stock and TIPS allocations are both between 20% and 80%. This is similar to Benjamin Graham’s constraint (Constraint A) that both stock and bond allocations remain between 25% and 75%.
I find it very interesting to see the strong influence of high valuations. When valuations are in the reasonable range (i.e., whenever P/E10 is less than 20), the optimal stock allocation is almost always 80%, the largest percentage that I allow. Low TIPS interest rates and high terminal value percentages also favor high stock allocations.
Today’s story is new. Today’s valuations are unusually high. Today’s P/E10 is 26.8. Requiring high levels of safety has a great influence at today’s valuations. It had minimal influence in the past.
I have just added Variable Terminal Value Rate Calculator C to my Yahoo Briefcase. It too is in the Big Project folder. I placed an extra copy of the optimal allocations and withdrawal rates in a more convenient location (columns M through Q, rows 1 through 9). Otherwise, it is identical to Variable TVR Calc B.
Curve Fitting Equations in Variable VTR Calc B
First, I normalize the allocation.
I set x = (allocation-50%)/30%.
When the allocation is 80%, x = +1. When the allocation is 50%, x = 0. When the allocation is 20%, x = -1.
Then I find the polynomial f(x) = a*x^2 + b*x + c that fits at x = +1, 0 and -1.
Setting x = +1, 0 and -1 results in the following equations:
f(1) = a + b + c
f(0) = c
f(-1) = a – b + c
Solving:
a = (f(1)+f(-1))*0.5 – f(0)
b = (f(1)-f(-1))*0.5
c = f(0)
I calculate separate polynomials f(x) for each table row: Safe Withdrawal Rate, Reasonably Safe, Coin Toss, Likely Failure and Almost Certain Failure.
I constrain both stock and TIPS allocations from 20% to 80%. I call this Constraint B.
I find the optimal value of x by setting the first derivative of f(x) equal to zero:
x(optimal) = -b/(2*a).
In terms of the data, unconstrained:
x(optimal) = -((f(1)-f(-1))/((f(1)+f(-1))*2 - f(0)*4))
This equation fails if f(x) is a straight line: a = 0. If a = 0, I set x(optimal) = 2. This equation also fails if it locates a minimum instead of a maximum: a > 0.
There are three values to check:
x = +1, x(optimal) and -1.
I calculate f(x) for all three values. I select whichever value is highest. I identify the value of x that produces the highest value of f(x).
I convert the selected value of x into an allocation:
allocation = 50% + (30%*x)
Yahoo Briefcase
Calculator I
I have added Calculator I.
It is the same as Calculator H except that I have removed withdrawal rates with 0% stocks and 10% stocks and I have added TIPS withdrawal rates at years 10, 15, 20, 25, 30, 35, 40, 45 and 50.
Previously, the timing mismatch caused confusion. It caused a spread in the TIPS-only withdrawal rates.
The TIPS-only baseline will help you interpret your results. The final balance with the TIPS-only baseline is always zero, but it comes with a Government guarantee.
Some people express concern that the Government could play games with inflation statistics, reducing the ability of TIPS to counter inflation. Keep in mind that all of the numbers are adjusted by identical amounts. The performance of a TIPS-only portfolio would still be the same relative to a portfolio with stocks.
You may download this calculator from the Big Project folder in my Yahoo Briefcase. It is Calculator I.
NOTE: I have often found it necessary to refresh my Yahoo Briefcase several times in order to get a display.
Yahoo Briefcase
Compact Variable Terminal Value Rate CVTVR Calculators
I have placed compact versions of my Variable Terminal Value Rate (Compact VTVR or CVTVR) Calculators into the Big Project folder in my Yahoo Briefcase. They are CVTVR A, B and C.
They do not contain quite as much information as the previous calculators. You are likely to find them easier to use.
AFTERNOON UPDATE:
I have now added CVTVR D, which adds a graphical display to CVTVR C. When you reflect upon these results, think about how they compare with the TIPS baselines (row 56 of CVTVR D) and think about how they would compare with a dividend-based strategy.
These calculators show the range of withdrawal rates that will leave you (at year 30) with a final balance of zero (that's right, completely depleted), 50% of your initial balance (plus inflation) and 100% of your initial balance (plus inflation). Portfolios consist of the S&P500 and TIPS. You can vary allocations between 20% and 80%.
The probabilities that separate each range of results are (roughly) 5%, 20%, 50%, 80% and 95%. Keep in mind that you still have a 10% chance of being blindsided, possibly to your benefit and possibly to your detriment.
When you look at the graph, think about what you could do with a dividend-based strategy or even a TIPS-only approach. Many people, especially younger retirees, would like to have their portfolios maintain at least one-half of the buying power at year 30 even if lightening strikes. Most dividend-based strategies do much better. They retain full value. They even grow.
Withdrawal rates are percentages of the initial balance. I have adjusted all withdrawals to match inflation.
JULY 25, 2006 UPDATE:
I have replaced CVTVR D with CVTVR Da. This removes an error when I tried to constrain the maximum display to 10%. I have added CVTVR Ea and CVTVR F. These replace the column display with bar displays. CVTVR Ea is similar to CVTVR D. It shows what happens with year 30 balances of 0%, 50% and 100% of the initial balance. CVTVR F has you specify the year 30 percentage balance. It shows you what happens to portfolios with 20%, 50% and 80% stock allocations. It also calculates the optimal stock allocations to reach the final percentage that you specify.
JULY 27, 2006 UPDATE:
I added CVTVR G, CVTVR H, CVTVR I, CVTVR J and CVTVR K this morning.
CVTVR K is my first two step calculator. In the first section, I assume that you invest entirely in TIPS for N years because of today's valuations. In the second section, you see what happens to withdrawal rates for the next 30 years. You can enter your guesses as to future P/E10 levels, such as 14 to represent typical valuations, 8 for bargain valuations and 18 for higher than normal valuations. The calculator displays percentages in terms of your initial balance.
Typical choices for N are 10 years, a conservative estimate of how long it takes to reach reasonable valuations, 15 years, a conservative extreme to reach bargain valuations, and 20 years, a pessimistic extreme.
For younger retirees, a good place to start is N = 10 years with a 50% terminal value percentage, which is 50% of the buying power of the initial balance at year 40. If you can get 2.3% interest from TIPS throughout the first decade (you can get 2.5% today), and if P/E10 falls to 14, which is historically typical, well above bargain levels: the calculator tells us that you can withdraw 4.0% (plus inflation) for the full 40 years and still end up with 50% of your initial balance (after adjusting for inflation).
July 28, 2006 UPDATE:
I have added CVTVR L. It is similar to CVTVR K, but I have added results for 100% stocks during the second period.
July 30, 2006 UPDATE:
I added CVTVR M last night. It adds switching portfolios SwAT and SwOptT to CVTVR L. These switching algorithms are optimized for starting at times of high valuations. Refer to Current Research A, B and C for details. They differ from Latch and Hold. They have no memory. They vary allocations strictly on the basis of the current level of P/E10.
August 2, 2006 UPDATE:
I have added CVTVR Ma and CVTVR N. CVTVR Ma corrects the portfolio labels in the graph. The lowest bar, which is very short, should be for TIPS. What I previously identified as TIPS is really a 100% stock portfolio.
CVTVR N adds another TIPS section. You can invest entirely in TIPS before and/or after investing for 30 years in portfolio that includes stocks. I report all withdrawal rates in terms of the initial balance. If you wish to bypass a TIPS-only section, simply enter 0 for the number of years (N or M).
I report quite a bit of information. Just remember that I include boxes for all calculator entries. For example, you need to enter P/E10 only once. [To make this easier, I translate P/E10 to the equivalent current value of the S&P500 index.]
Yahoo Briefcase
A Tip about my Yahoo Briefcase
Often, I have to press refresh several times to get my Yahoo Briefcase to display a page.
Super Variable Terminal Value Rate SVTVR Calculators
I have now developed two Super Variable Terminal Value Rate SVTVR calculators. Each consists of four Compact Variable Terminal Value Rate (Compact VTVR or CVTVR) Calculators side by side, in a single spreadsheet. Each has its own set of inputs at the top.
Making comparisons is easy.
I have placed these calculators into the Big Project folder in my Yahoo Briefcase. They are SVTVR A and B.
AUGUST 9, 2006 UPDATE:
I am continuing to make improvements. My latest is Super Variable Terminal Value Rate Calculator SVTVR I1. It is very easy to use.
AUGUST 12, 2006 UPDATE:
I have added Super Variable Terminal Value Rate Calculator SVTVR J to the Big Project folder in my Yahoo Briefcase. It is very easy to use.
It adds logic that limits the display of TIPS-only portfolios to 0% to 100%.
AUGUST 19, 2006 UPDATE:
I have added Super Variable Terminal Value Rate Calculators SVTVR K and L to the Big Project folder in my Yahoo Briefcase. They are easy to use.
Yahoo Briefcase
A Helpful Theorem
To calculate withdrawal rates for various final balances, it is sufficient to know the withdrawal rates for final balances of zero and 100% of the initial balance. Rates for intermediate balances maintain the same proportions.
A Helpful Theorem
Year 15 Calculator A
I have placed Year 15 Calculator A into my Yahoo Briefcase. It is the same as Super SVTVR Calculator L except for its 15-year timeframe.
Enter P/E10, your stock allocation, TIPS interest rate and final balance percentage for four portfolios. See how likely you will be of reaching you final balance percentage as a function of your withdrawal rate.
Year 15 Calculator Equations
Yahoo Briefcase