# Amortization based withdrawal

Amortization based withdrawal (ABW) uses amortization techniques to calculate portfolio withdrawals during retirement. Examples of withdrawals that use amortization include Variable Percentage Withdrawal (VPW) and the Required Minimum Distribution (RMD) rules for traditional retirement accounts.

## Amortization

### Understanding amortization

Amortization is the process of converting the value of a lump sum today into a series of payments over time. Most people encounter it when they take out a mortgage or car loan. Amortization is used to calculate the monthly payment required to pay back the loan. The value of the monthly payments has to be made equal to the value of the initial loan.

If the interest rate is zero, amortization is just simple division: If \$100,000 has to be paid back over 10 years, then the payment is \$100,000 ÷ 10 = \$10,000 per year.

But if the interest rate is positive, as is usually the case, then the payment in the above example would have to be more than \$10,000 per year. That's because \$10,000 in the future is worth less than \$10,000 today. This is referred to as the time value of money. If the interest is 5% per year, then \$10,000 today is worth:

• ${\displaystyle \10,000*1.05^{1}=\10,500}$ one year from now
• ${\displaystyle \10,000*1.05^{2}=\11,025}$ two years from now

and so on. We will have to adjust for the differences in value. The amortization formula makes the necessary adjustments and tells us that if the interest rate is 5% per year, then the payments will have to be \$12,950 per year for the next ten years for the total value of the payments to be equal in value to the \$100,000 that was borrowed today.

## Amortization based withdrawal

To calculate portfolio withdrawals, enter the following inputs in the ABW calculator:

• ${\displaystyle P=}$ current portfolio value
• ${\displaystyle n=}$ number of years over which withdrawals are to be spread out
• ${\displaystyle r=}$ expected return of the portfolio.

### Example

A 65 year old retiree with a \$1 million portfolio earning 3% per year and seeking to spread out withdrawals over 36 years (ages 65 to 100), would enter ${\displaystyle P=\1,000,000,n=36,r=3\%}$. The amortization calculator says that this retiree can withdraw \$44,470 per year. This means that if the the portfolio rate of return remains constant at 3% per year and the retiree withdraws \$44,470 every year, then the portfolio will be fully depleted after 36 years.

### Real vs nominal rate of return

If the real (inflation adjusted) rate of return is used, the withdrawal strategy is automatically adjusted for inflation. In the above example, if the real rate of return is 3%, the retiree will be able to take out \$44,470 (inflation adjusted) each year. If instead, the retiree enters a nominal interest rate of 5%, the calculator says that they can take out \$60,434 per year. But that's in nominal dollars, so real consumption will decline each year by the inflation rate.

Withdrawals are recalculated each year with updated information about portfolio value, expected rate of return, and number of years left. This allows withdrawals to continually adapt to new information and mitigates the impact of past incorrect assumptions. Amortization spreads the course correction required across all remaining years, thereby minimizing the impact on any one year.

#### Example

Consider what happens in our earlier example if, instead of gaining 3% as expected, the portfolio loses 10%. The shortfall relative to expectations is ${\displaystyle (1-.10)/(1+.03)-1=-12.6\%}$. Our retiree had expected to start age 66 with \$984,196 in the portfolio. Instead, the portfolio contains only \$859,977, which is 12.6% less. Consider what happens in the following two scenarios:

##### Scenario 1: Expected return stays the same

Supoose that the retiree believes that this market drop has not impacted future expected return, which in the retiree's opinion, remains at 3%. The retiree enters the new reduced portfolio value, ${\displaystyle P=\859,977}$, number of years remaining ${\displaystyle n=35}$, and expected real return ${\displaystyle r=3\%}$. The withdrawal is now reduced to \$38,857 per year. That is a 12.6% reduction relative to the previously planned withdrawal of \$44,470. So the 12.6% portfolio loss is being matched by a 12.6% spending cut across all remaining years.

##### Scenario 2: Expected return increases

If the market drop was due to a pure valuation change and the underlying earnings of the portfolio are not impacted, then the retiree might expect a higher rate of return going forward. If expected return is based on an earnings yield model (e.g. 1/CAPE) for stocks and yield to maturity for bonds, a 12.6% reduction in valuation will result in a higher expected real return of ${\displaystyle .03/(1-.126)=3.43\%}$. With the higher expected return, the withdrawal becomes \$41,181. That is a reduction of only 7.4% relative to the original plan of \$44,470. A 12.6% reduction in the portfolio led to only a 7.4% reduction in income, because the higher expected return partially offset the loss in portfolio value. So to the extent that market volatility is caused by valuation changes and gets incorporated into expected returns, withdrawals from the portfolio will be less volatile than the portfolio itself.

Instead of fully depleting the portfolio, the amortization can be modified to leave behind a terminal balance. This could be used to fund end of life care or a bequest, or it could simply serve as a general financial cushion.

In the ABW calculator, enter:

• ${\displaystyle B=}$ terminal balance

### Example

The 65 year old retiree in the earlier example wants to have \$200,000 left at the end for a bequest. Enter ${\displaystyle B=\200,000}$ in the calculator to get a withdrawal of \$41,401 per year. If the actual rate of return matches expectations, then after withdrawing \$41,401 for 36 years (ages 65 to 100), the retiree will have \$200,000 remaining in the portfolio at the end of age 100.

## Rising or falling withdrawal schedules

Instead of using a constant withdrawal schedule, some retirees may prefer a rising or falling withdrawal schedule. Retirees worried about incurring unexpected expenses or running out of money in later years may want to use a rising withdrawal schedule--take less early so they can have more later. Those wanting to fund an active early retirement may want to use a declining withdrawal schedule--take more early but have less later. (Note the similarity with savings. This can be viewed as a decision about how much to save in the early retirement years in order to fund late retirement consumption.)

In the ABW calculator, enter:

• ${\displaystyle g=}$ rate of growth of the withdrawal schedule

${\displaystyle g>0}$ generates a rising withdrawal schedule, ${\displaystyle g<0}$ generates a declining withdrawal schedule, and ${\displaystyle g=0}$ generates a constant withdrawal schedule.

### Example

If the 65 year old retiree in the earlier example wants withdrawals to grow at 1% per year, set ${\displaystyle g=1\%}$ to get a withdrawal of \$38,349. Withdrawals are scheduled to grow 1% per year, rising to \$54,325 at age 100.

If instead the retiree wishes to spend more in early retirement and wants withdrawals to fall by 1% per year, set ${\displaystyle g=-1\%}$ to get a withdrawal of \$51,118. Withdrawals are scheduled to fall 1% per year, declining to \$35,959 at age 100.

## Asset allocation

The amortization formula assumes a constant rate of return on the portfolio. This is easier to justify if the retiree plans to maintain a fixed AA. If instead the retiree is planning to change AA over time (e.g. glide down from 60/40 to 30/70 over the course of the retirement), then the expected return is not constant. Amortization can still be done, but it will require manual modeling. The formulas given here cannot be used. Entering the average expected return in the formula may provide a rough estimate, but due to the potential for error, more precise modeling is recommended.

### Fixed asset allocation

Fixed asset allocations are optimal under the following conditions:

• The retiree has a constant relative risk aversion (CRRA) utility function, at least for the consumption funded by portfolio withdrawals.
• Returns are independent and identically distributed. That implies no mean reversion, for example.

If these conditions hold, the retiree will choose to place the same fraction of the portfolio at risk every year. So the allocation is fixed.

### Fixed asset allocation with sub-portfolios

Fixed asset allocations can be adapted to more general (non CRRA) utility functions by dividing the portfolio into multiple sub-portfolios and maintaining a different fixed allocation on each sub-portfolio.

Suppose the retiree is more risk averse over certain categories of consumption (e.g. food and housing), and less risk averse over other categories (e.g. vacations). The portfolio can be subdivided into a safer portfolio to fund food and housing and a riskier portfolio to fund vacations. The safe portfolio would get a safe fixed AA like 30/70, and the riskier portfolio would get a riskier fixed AA like 80/20. Multiple sub-portfolios could be created to fund different consumption goals, each with a different AA that matches the risk-reward considerations for that goal. Withdrawals would be calculated separately for each sub-portfolio.

An example of a strategy that uses sub-portfolios is the Liability Matched Portfolio (LMP) strategy. The portfolio is divided into a "liability matched portfolio" and a "risk portfolio". The liability matched portfolio has an AA of 0/100 and is used to fund non-negotiable consumption. The risk portfolio is given a riskier AA and is used to fund discretionary "nice to have" consumption. This strategy is consistent with a utility function that exhibits infinite risk aversion for consumption below some threshold (funded by the liability matched portfolio) and finite risk aversion for consumption above the threshold (funded by the risk portfolio).

## Including other sources of income

The retiree can choose to include income from social security, pensions, annuities, and other sources into the portfolio and incorporate it into the withdrawal strategy. The payouts promised by these income sources can be viewed as bond payouts. The present value of the income is calculated by discounting the promised payouts using bond interest rates, and is included in the portfolio as a bond. Asset allocation and withdrawal calculations are done on this total portfolio.

One way that these income sources differ from actual bonds is that they cannot be sold. If stocks perform badly, the retiree would need to sell bonds to maintain the target AA. If stock returns are poor enough, the retiree may reach a point where all of the actual bonds have already been sold and rebalancing to the target AA requires selling the income stream that is being counted as a bond. Since that is not possible, rebalancing cannot proceed and the AA will remain too safe relative to the target AA.

A retiree pursuing the sub-portfolio approach described earlier may find it convenient to use these income streams to fund the safe sub-portfolios. The income streams are already duration matched. Gaps remaining can be filled with duration matched bonds. Depending on the AA, there may be little to no rebalancing needed in the safe sub-portfolios.

## ABW calculator

The Excel spreadsheet to calculate withdrawals in on Google Drive: ABW Calculator

User:Ben Mathew/Formulas for Amortization Based Withdrawal (ABW) describes the formulas used in the calculator.