Equations for Design

These are equations that we use repeatedly when we design our retirement portfolios.

TIPS equations

The TIPS Equivalent Safe Withdrawal Rate TESWR is the amount that you would withdraw each year to end up with exactly zero dollars. The number of years is N. The interest rate is r.

TESWR = r / [1 – (1/[1+r]^N) ]

Here are some examples when r is 2%.

TESWR = 11.133% when N = 10 years.
TESWR = 6.116% when N = 20 years.
TESWR = 4.465% when N = 30 years.
TESWR = 4.000% when N = 35 years.
TESWR = 3.656% when N = 40 years.
TESWR = 3.391% when N = 45 years.
TESWR = 3.182% when N = 50 years.

When you withdraw at a rate WR with an interest rate of r and a TIPS Equivalent Safe Withdrawal Rate of TESWR at year N, the remaining fraction rf, which is the balance at year N and the initial balance, is given by this equation:

rf = [TESWR – WR] / [TESWR – r]

Here are some examples when r is 2% and N = 10 or 20 years. When N = 10 years, TESWR = 11.133%, as listed above. When N = 20 years, TESWR = 6.116%, as listed above.

At year 10, if the withdrawal rate is 4.0%, the remaining fraction is rf = 78.10%.
At year 10, if the withdrawal rate is 3.8%, the remaining fraction is rf = 80.29%.
At year 10, if the withdrawal rate is 3.6%, the remaining fraction is rf = 82.48%.

At year 20, if the withdrawal rate is 4.0%, the remaining fraction is rf = 51.41%.
At year 20, if the withdrawal rate is 3.8%, the remaining fraction is rf = 56.27%.
At year 20, if the withdrawal rate is 3.6%, the remaining fraction is rf = 61.13%.

Continuing the Withdrawal Level

To maintain a withdrawal rate equal to 4.0% of the initial balance (plus inflation), the withdrawal rate in the second period has to be 4.0% / (the remaining fraction) = 5.12% after 10 years and 7.78% after 20 years.

To maintain a withdrawal rate equal to 3.8% of the initial balance (plus inflation), the withdrawal rate in the second period has to be 3.8% / (the remaining fraction) = 4.733% after 10 years and 6.753% after 20 years.

To maintain a withdrawal rate equal to 3.6% of the initial balance (plus inflation), the withdrawal rate in the second period has to be 3.6% / (the remaining fraction) = 4.365% after 10 years and 5.889% after 20 years.

Portfolio Equations

These equations are for portfolio lifetimes of 30 years. In every case, I set the annual expenses equal to 0.20% of the portfolio’s current balance. This is the approximate cost of an S&P500 index fund.

NOTE: I do not charge expenses to portfolios consisting entirely of TIPS.

This is the equation for the Calculated Rate of portfolio SwOptT2. The lower confidence limit is the Safe Withdrawal Rate. The upper confidence limit is the High Risk Rate. The term x is the percentage earnings yield 100/[P/E10].

y = 0.3276*x + 3.9729 plus 1.3% and minus 0.7%.

SwOptT2 consists of stocks (S&P500) and TIPS at a 2% interest rate. Allocations vary in accordance with P/E10. The P/E10 thresholds are 9-12-21-24. The stock allocations are 100%-50%-30%-20%-0%. This is the Optimized Switching portfolio with 2% TIPS.

This is the equation for the Calculated Rate of portfolio SwAT2. The lower confidence limit is the Safe Withdrawal Rate. The upper confidence limit is the High Risk Rate. The term x is the percentage earnings yield 100/[P/E10].

y = 0.3776*x + 3.5222 plus 1.3% and minus 0.7%.

SwAT2 consists of stocks (S&P500) and TIPS at a 2% interest rate. Allocations vary in accordance with P/E10. The P/E10 thresholds are 11-21. The stock allocations are 75%-40-25%. This is the optimized Switching portfolio with 2% TIPS subject to constraint A. Constraint A limits stock and bond allocations to 25% to 75%.

This is the equation for the Calculated Rate of portfolio HSWR50T2. The lower confidence limit is the Safe Withdrawal Rate. The upper confidence limit is the High Risk Rate. The term x is the percentage earnings yield 100/[P/E10].

y = 0.4031*x + 2.9478 plus 1.5% and minus 0.8%.

HSWR50T2 consists of stocks (S&P500) and TIPS at a 2% interest rate. I rebalance allocations annually. The stock allocation is 50%. The TIPS allocation is 50%.

This is the equation for the Calculated Rate of portfolio HSWR80T2. The lower confidence limit is the Safe Withdrawal Rate. The upper confidence limit is the High Risk Rate. The term x is the percentage earnings yield 100/[P/E10].

y = 0.6758*x + 1.7538 plus 2.0% and minus 1.1%.

HSWR80T2 consists of stocks (S&P500) and TIPS at a 2% interest rate. I rebalance allocations annually. The stock allocation is 80%. The TIPS allocation is 20%.

This is the equation for the Calculated Rate of portfolio CTVR50 The lower confidence limit is the Safe Withdrawal Rate. The upper confidence limit is the High Risk Rate. The term x is the percentage earnings yield 100/[P/E10].

y = 0.3279*x + 1.4254 plus 0.72% and minus 0.72%.

CTVR50 (Constant Terminal Value Rate) consists of stocks (S&P500) and commercial paper. I rebalance allocations annually. The stock allocation is 50%. The commercial paper allocation is 50%. At the Calculated Rate, the final portfolio balance equals the initial balance.

This is the equation for the Calculated Rate of portfolio CTVR80. The lower confidence limit is the Safe Withdrawal Rate. The upper confidence limit is the High Risk Rate. The term x is the percentage earnings yield 100/[P/E10].

y = 0.7033*x + 0.2133 plus 1.10% and minus 1.10%.

CTVR80 (Constant Terminal Value Rate) consists of stocks (S&P500) and commercial paper. I rebalance allocations annually. The stock allocation is 80%. The commercial paper allocation is 20%. At the Calculated Rate, the final portfolio balance equals the initial balance.

This is the equation for the Calculated Rate of portfolio HFWR50 The lower confidence limit is the Safe Withdrawal Rate. The upper confidence limit is the High Risk Rate. The term x is the percentage earnings yield 100/[P/E10].

y = 0.3408*x + 2.0804 plus 0.88% and minus 0.88%.

HFWR50 (Half Failure Withdrawal Rate) consists of stocks (S&P500) and commercial paper. I rebalance allocations annually. The stock allocation is 50%. The commercial paper allocation is 50%. At the Calculated Rate, the lowest portfolio balance equals (or exceeds minimally) one-half of the initial balance in throughout the entire 30 years.

This is the equation for the Calculated Rate of portfolio HFWR80. The lower confidence limit is the Safe Withdrawal Rate. The upper confidence limit is the High Risk Rate. The term x is the percentage earnings yield 100/[P/E10].

y = 0.6822*x + 0.5298 plus 1.864% and minus 1.864%.

HFWR80 (Half Failure Withdrawal Rate) consists of stocks (S&P500) and commercial paper. I rebalance allocations annually. The stock allocation is 80%. The commercial paper allocation is 20%. At the Calculated Rate, the lowest portfolio balance equals (or exceeds minimally) one-half of the initial balance in throughout the entire 30 years.

Have fun.

John Walter Russell
July 28, 2005