Executive Summary

"The only two sure things in life are death and taxes." -Will Rogers

Add inflation and interest to this quote and it describes the forces that affect a retiree's nest egg. A nest egg is a reserve fund that a retiree has accumulated over a lifetime of saving. This fund must last from the retirement date through the rest of the retiree's life. In calculating how to spend this nest egg, the worst mistake a retiree can make is prematurely exhausting this fund. Planning for a financially secure retirement is difficult. A short-sighted calculation could leave a retiree without money to live, causing a burden on family members or the government. A conservative approach will help to prevent this mistake.

This paper will show the effect that interest rates, time, withdrawal needs, and inflation rates have on a retiree's nest egg. Section I explains how the calculation inputs affect each other and the output. Section II examines different investment horizons given specific interest rates and withdrawal levels. Section III shows the different interest rates needed to satisfy different investment horizon and withdrawal needs. Section IV discusses the effects on the annual withdrawal due to changes in the investment horizon and interest rate. Section V examines the effects of inflation rates on the nest egg.

I. The Calculation

The calculation used to make the estimates in this paper is a combination of a few basic concepts. Central to the calculation is the future value formula:

FV=PV(1+i)t where FV is the future value; PV is the present value (current amount); i is either the after tax return or the inflation rate; andt is the number of periods. This model assumes a single annual compounding period and annual withdrawal.

Inputs

The inputs to the calculation are the value of the nest egg, after tax return rate, inflation rate, age of the retiree, and withdrawal amount. Because there are so many inputs in this calculation, many will be fixed so that the calculation can be manipulated without confounding the effects.

The initial value of the nest egg has a great effect on the amount that can be withdrawn each year and the length of time that it will last. The value of the nest egg will be fixed at $200,000 for this model.

After tax return rate is the interest rate that investments earn less the tax rate. For example, a 9% interest rate with a 12% tax rate will net a 7.9% after tax return rate (.09 * (1-.12)=.079). Tax rates vary from state to state. For simplicity, the tax rate in this calculation will be fixed at 12%. It is important to understand that an increase in the tax rate would cause a decrease in the after tax return rate and that a decrease in the tax rate would cause an increase in the after tax return rate. In this calculation, however, adjustments to the after tax return rate are made by adjusting the interest rate. The annual yield of a bond is an example of an interest rate used in this model.

The nest egg grows at a rate of FV=PV*(1+i)t where PV is the value of the nest egg and i is the after tax return rate. A $1000 nest egg will grow to $1100 in one year and to $1210 in two years. Exponential growth will continue due to compounding after tax return rates (assuming 10% after tax return rate and no withdrawal).

The inflation rate is the rate at which the purchasing power of the annual withdrawal diminishes. Each dollar will purchase fewer goods or services next year than it does this year. In order to maintain standard of living, the annual withdrawal must increase at the rate of FV=PV*(1+i)t where PV is the withdrawal amount and i is the inflation rate. A $500 annual withdrawal becomes a $550 withdrawal after one year and increases to a $605 withdrawal in two years to maintain the same purchasing power. Purchasing power is diminished exponentially (assuming 10% inflation). The inflation rate will be fixed at 5.4% except for INFLATION EFFECTS section.

The age input for this calculation is the retiree's age. Age is an essential input because it establishes the time frame that the nest egg must last. A nest egg must last longer for a younger retiree. For this model, the age will be fixed at 65 years of age.

The withdrawal is simply the amount of money that a retiree needs to withdraw from the nest egg to meet expenditures. In this model, the annual withdrawal will occur on January 1 and be held constant with adjustments made for inflation.

The inputs result in the following calculation for a person aged 65 (Figure 1). The $200,000 nest egg is depleted by a $20,000 withdrawal the first year. The residual $180,000 grows by $14,256 in after tax return to a value of $194,256 at the end of the year. The process repeats the next year, with the remaining $194,256 of original nest egg being depleted by an increased withdrawal of $20,840. Thus the nest egg struggles to grow while being depleted by an ever increasing annual withdrawal. Eventually, the entire value of the nest egg will be depleted.

Figure 1:  (assuming 9% interest rate)

Nest egg value at the start of the year	Principal withdrawn	Residual nest egg value	Nest egg value at the end of the year
$200000	$20000	$180000	$194256
$194256	$20840	$173416	$187151

Output

As the calculation repeats, eventually the nest egg is depleted. The year that the nest egg is depleted is the output of the calculation. For the above calculation, the output or investment horizon is age 79. This means that the nest egg is depleted in the year of the retiree's 78th birthday. Higher interest rates, lower inflation rates, and a smaller withdrawal will all contribute to greater investment horizon.

Many of the tables in this paper were generated by manipulating one variable to achieve a specific investment horizon outcome. The maximum values for the different variables were determined by setting the investment horizon to a specific number while holding the other variables at some constant.

II. How long will my money last?

Perhaps the most important concern for a retiree is how long the nest egg will last. Figure 2 examines the effect that the annual withdrawal and interest rate have on investment horizon. A $5,000 annual withdrawal leads to a long lifespan regardless of the interest rate. Notice that when the interest rate is above 10% that there are no values. There are also no values for a $10,000 withdrawal at 13% interest rate. At these points, the nest egg accrues interest faster than inflation, taxes, and withdrawal strip it away. The investment horizon is at least three generations away, if not infinite. Unless a retiree is creating an endowment, these values are too extreme. A $25,000 or greater annual withdrawal does not accrue interest fast enough to replenish the principle at a rate that greatly affects the investment horizon.

Figure 2:

How does time effect the calculation? Recall that the nest egg grows at FV=PV*(1+i)t. Interest is raised to the t power. Since smaller withdrawals deplete the principle at a slower rate, the power of t is increasingly influential. Figure 3 shows a plot of investment horizon and withdrawal. The scatter plot has a noticeable curve. The linear trendline has an R2 of .44 and clearly is missing a key point. The non-linear power form is a much better estimator with an R2 of .77. This confirms a relationship that is explicit in the formula for the calculation. The longer an investment compounds, the greater the effect on the principle.

Figure 3:

III. How much interest do I need to earn?

If a retiree has specific withdrawal needs and a specific investment horizon, he or she will need to know what interest rate will satisfy the equation. Figure 4 provides a table of interest rates needed to reach different investment horizons given different withdrawal levels. An annual withdrawal of $5,000 does not need to earn interest until an investment horizon of 90. $10,000 and $15,000 withdrawal levels do not need to earn interest to achieve an investment horizon of 75.

As investment horizon and annual withdrawal increase, interest rates needed to reach those goals become increasingly higher. This creates a paradox for retirees. Typically, higher risk investments yield higher interest rates. Risk implies a greater potential for loss. A retiree has very little tolerance for risk (spelled loss). There should be a shift to a more conservative portfolio.

A limit of 25% interest was set to reduce unrealistic values. Withdrawal levels of $30,000 above an investment horizon of 90, $35,000 above 85, and $40,000 above 80 all returned values well above the 25% limit. The Long-term Dow Jones Yield Chart in Appendix 1 shows the long-term moving averages for the Dow as an indicator of interest rates. Figure 3 should be analyzed in the context of reasonable interest rates. The best ten year average was 14.3%. The worst average was -2.2%, supporting the notion of a balanced portfolio to limit risk. It would be foolish for a retiree to count on an historic high return to reach investment horizon.

Figure 4:

IV. How much can I draw?

Retirees are concerned with how much they can draw annually. Figure 5 provides a clear table of annual withdrawal levels given different investment horizons and interest rates. At any given investment horizon, increasing the interest rate one percent causes a noticeable increase in the withdrawal level. The 75 year investment horizon averages a 3.1% increase in value for each 1% increase in interest rates. At the 100 year investment horizon, however, the average increase is 12.1%. Once again, the time factor greatly increases the compounding effect.

Figure 5:

How do interest rates and investment horizon affect withdrawal levels? Certainly, the calculation itself shows that increasing the interest rate at a given investment horizon allows for a greater withdrawal level. Increasing the investment horizon at a given interest rate reduces the possible withdrawal level. Figure 6 shows a regression analysis for y = annual withdrawal, x1 = investment horizon, and x2 = interest rate. Although a regression equation is artificially forced on the given nest egg equation, it does return a high R2 of .87. The regression confirms what is known intuitively from the calculation. Investment horizon and interest rate are strong explanatory variables for maximum annual withdrawal levels.

Figure 6:

V. Inflation effect

The inflation rate variable diminishes the purchasing power of the withdrawal. For the pervious models, the inflation rate was fixed at 5.4%. This is the worst inflation rate for a thirty year moving average. It is a conservative estimate of inflation. Underestimating inflation could be disastrous for a retiree since the nest egg could be striped away faster than estimated. The nest egg would be depleted sooner than the desired investment horizon. 5.4% proved to be a conservative estimate for inflation. Figure 7 shows the effect that the worst, assumed, median, and best inflation rates have on an annual withdrawal of $20,000 to reach a desired investment horizon. Figure 8 shows that the moving average range narrows over time. Appendix 2 contains a table of long-term moving averages since 1947.

Figure 7:

Figure 8:

Conclusion:

Understanding the nest egg calculation is difficult. There are many variables that can affect the outcome of the calculation. Investment horizon, interest rates, withdrawal needs, and inflation all affect the value of the nest egg. Retirees face many complicated choices in deciding how to spend their nest egg. Many resources are available to retirees or those planning for their retirement. Financial planners and other investment professionals can help explain the process and establish a strategic plan. The internet, books, and magazines can help any retiree brave enough to try planning on their own. Trends in interest and inflation rates and long-term withdrawal needs greatly affect the value of the nest egg and warrant a conservative planning approach to ensure success.