The Math of Recovery in Retirement Portfolios

he primary goal of a retirement portfolio is very simple: avoid large losses. This article examines the math of gains and losses in a withdrawal portfolio-which essentially represents a retirement portfolio from which an individual is withdrawing money on a regular basis. The time frame of this analysis is the 36-year period from 1970-2005.

As shown in "Math of Recovery", a retirement portfolio-which is being drawn down by annual withdrawals-faces a much steeper climb to break-even after a loss than does a buy-and-hold portfolio. (Recall that all mutual performance data assumes a buy-and-hold situation). Where a buy-and-hold portfolio needs a 4.6% average annual return to recover from a 20% loss within 5 years, a retirement portfolio in withdrawal mode must generate a 12.5% average annual return over a 5-year period to restore the pre-loss account balance.

The withdrawal portfolio assumes a starting balance of $500,000, an initial withdrawal at the end of the first year of 5% of the starting portfolio balance (in this case, $25,000), and an annual increase in the withdrawal of 3%. Thus, the second year withdrawal in this analysis was $25,750, the third year withdrawal was $26,523, and so forth. A buy-and-hold portfolio is included in this study for comparison purposes and assumes an initial investment only (no additional investments and no withdrawals). Notice that a withdrawal portfolio is in "recovery mode" even if the return that it is recovering from is positive (i.e., 5%, 2%). This is due to the fact that the required return needs to exceed the withdrawal rate that is increasing each year, which in this case was 5% in the first year.

The assets included in the retirement portfolio include large cap US equity, small cap US equity, non-US equity, US intermediate term bonds, commodities, and US Treasury Bills. The historical performance of large cap US stocks was represented by the S&P 500 Index. Small cap US stock was represented by the Ibbotson Small Companies Index from 1970-1978, and the Russell 2000 Index from 1979-2005. The performance of non-US stock was represented by the Morgan Stanley Capital International EAFE Index (Europe, Australasia, Far East) Index. U.S. intermediate term bonds were represented by the Ibbotson Intermediate Term Bond Index from 1970-76 and the Lehman Brothers Intermediate Term Bond index from 1977-2005. The historical performance of commodities was captured by the Goldman Sachs Commodities Index (GSCI). Finally, the historical performance of cash was represented by 3-month Treasury Bills. The annual returns of each asset from 1970-2005 are shown below in "Raw Data". The Consumer Price Index was also included. The primary data source for this study was Morningstar Principia. Other data sources included "Stocks, Bonds, Bills, Inflation" by Ibbotson Associates, and Scott Berglund.

Math of Recovery

Portfolio Return from which a Recovery is Needed	Needed Average Annual % Return to Restore Original Portfolio Balance
	WITHDRAWAL RETIREMENT Portfolio   $500,000 initial balance, First Year Withdrawal of 5% of initial balance, 3% increase of annual withdrawal
	Within 1 Year	Within 2 Years	Within 3 Years	Within 4 Years	Within 5 Years
5%	5.2%	5.2%	5.3%	5.4%	5.5%
2%	8.4%	6.9%	6.4%	6.3%	6.2%
0%	10.7%	8.0%	7.2%	6.9%	6.7%
-2%	13.1%	9.2%	8.0%	7.5%	7.2%
-5%	16.8%	11.1%	9.3%	8.4%	8.0%
-10%	23.7%	14.4%	11.5%	10.1%	9.4%
-15%	31.4%	18.0%	13.9%	12.0%	10.9%
-20%	40.2%	22.0%	16.5%	14.0%	12.5%
-25%	50.2%	26.4%	19.4%	16.1%	14.3%
-30%	61.8%	31.3%	22.6%	18.5%	16.2%
-35%	75.3%	36.9%	26.1%	21.2%	18.4%
Portfolio Return from which a Recovery is Needed	BUY-and-HOLD Portfolio
	Within 1 Year	Within 2 Years	Within 3 Years	Within 4 Years	Within 5 Years
-2%	2.0%	1.0%	0.7%	0.5%	0.4%
-5%	5.3%	2.6%	1.7%	1.3%	1.0%
-10%	11.1%	5.4%	3.6%	2.7%	2.1%
-15%	17.6%	8.5%	5.6%	4.1%	3.3%
-20%	25.0%	11.8%	7.7%	5.7%	4.6%
-25%	33.3%	15.5%	10.1%	7.5%	5.9%
-30%	42.9%	19.5%	12.6%	9.3%	7.4%
-35%	53.8%	24.0%	15.4%	11.4%	9.0%

Raw Data

Year	Large US Equity	Small US Equity	Non-US Equity	Intermediate Term US Bonds	Commodities	Cash	CPI
1970	3.92	(17.40)	(11.66)	16.90	15.17	6.80	5.57
1971	14.14	16.50	29.59	8.70	20.15	4.52	3.27
1972	19.16	4.40	36.35	5.20	42.37	4.23	3.41
1973	(14.69)	(30.90)	(14.92)	4.60	74.90	7.46	8.71
1974	(26.47)	(19.90)	(23.16)	5.70	39.50	8.35	12.34
1975	37.23	52.80	35.39	7.80	(17.22)	6.08	6.94
1976	23.64	57.40	2.54	12.90	(11.94)	5.23	4.86
1977	(7.44)	25.40	18.06	3.00	10.38	5.52	6.70
1978	6.40	23.50	32.62	2.23