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	31.56	7.67	9.02
1979	18.30	43.07	4.75	6.59	33.78	10.86	13.29
1980	32.22	38.60	22.58	6.65	11.06	12.71	12.52
1981	(5.08)	2.03	(2.28)	10.79	(22.98)	15.58	8.92
1982	21.46	24.95	(1.86)	25.42	11.57	11.66	3.83
1983	22.46	29.13	23.69	8.22	16.23	9.24	3.79
1984	6.26	(7.30)	7.38	14.29	1.03	10.33	3.95
1985	31.74	31.05	56.16	18.00	10.02	7.97	3.80
1986	18.68	5.68	69.44	13.06	2.05	6.29	1.10
1987	5.26	(8.80)	24.63	3.61	23.76	6.13	4.43
1988	16.61	25.02	28.27	6.40	27.92	7.06	4.42
1989	31.68	16.26	10.54	12.68	38.25	8.67	4.65
1990	(3.12)	(19.48)	(23.45)	9.56	29.13	7.99	6.11
1991	30.48	46.04	12.13	14.11	(6.13)	5.68	3.06
1992	7.62	18.41	(12.17)	6.93	4.41	3.59	2.90
1993	10.06	18.88	32.56	8.17	(12.32)	3.12	2.75
1994	1.31	(1.82)	7.78	(1.75)	5.28	4.45	2.67
1995	37.53	28.45	11.21	14.41	20.32	5.79	2.54
1996	22.94	16.49	6.05	4.06	33.90	5.26	3.32
1997	33.35	22.36	1.78	7.72	(14.09)	5.31	1.70
1998	28.57	(2.55)	19.93	8.49	(35.71)	5.01	1.61
1999	21.04	21.26	27.03	0.49	40.89	4.87	2.68
2000	(9.10)	(3.02)	(14.19)	10.47	49.71	6.32	3.39
2001	(11.88)	2.49	(21.42)	8.42	(31.91)	3.67	1.55
2002	(22.09)	(20.48)	(15.94)	9.64	32.03	1.68	2.38
2003	28.67	47.25	38.59	2.29	20.68	1.05	1.88
2004	10.71	18.33	20.25	2.33	17.28	1.43	3.64
2005	4.91	4.55	13.54	1.68	25.69	3.34	3.84
36-Year Average Annualized Return	11.06	11.95	10.45	8.19	12.33	6.37	4.72
36-Year Standard Deviation of Return	17.03	22.13	22.02	5.53	23.87	3.14	3.15
Growth of $10,000 in Buy-and-Hold Portfolio	$436,088	$582,504	$358,362	$170,256	$657,349	$92,360	$52,613

Having stated the obvious-that retirement portfolios should avoid large losses-let's take a look at the historical record of the six assets in this analysis and how well they accomplish that goal in a withdrawal portfolio and in a buy-and-hold portfolio.

Four different portfolios were included in this study: (1) 100% large US equity, (2) 100% intermediate US bond, (3) traditional 60% equity/40% bond, (4) a global multi-asset portfolio.

We first consider a 100% large US equity portfolio. Understandably, this portfolio is not representative of a retirement portfolio. Rather, it is presented as a comparison against more traditional retirement portfolios. Its 36-year average annualized return was 11.06% with a standard deviation of just over 17%. In a 100% large-cap US equity buy-and-hold portfolio there was an 81% chance of recovering from a 10% portfolio loss within five years (using performance data from 1970-2005). Conversely, in a withdrawal portfolio the chance of recovery from a 10% loss within five years was only 66% (as shown in "Four Portfolios").

For a portfolio consisting entirely of intermediate term US bonds (also not a likely retirement portfolio asset mix) the chance of recovery from a 10% loss was 100% within 5 years in a buy-and-hold portfolio but only 31% in a withdrawal portfolio. An offsetting virtue of an all-bond portfolio is that the one-year returns are almost always positive. Thus, an all-bond portfolio doesn't recover well from losses, but it very rarely had losses during the 36-year period of this study. The only negative return over this 36-year period was -1.8% in 1994.

Four Portfolios

Retirement Portfolio	36-Year Annualized Return (%)	36-Year Standard Deviation of Return (%)	Probability of Recovery from a 10% loss within 5 Years in a Buy-and-Hold Portfolio	Probability of Recovery from a 10% loss within 5 years in a Withdrawal Portfolio
100% Large Stock	11.06	17.03	81%	66%
100% Bond	8.19	5.53	100%	31%
Traditional 60% Equity/40% Bond   (50% Large US Stock, 5% Small US Stock, 5% Non US Stock, 35% Bond, 5% Cash)	10.26	10.61	94%	66%
Global Multi-Asset   (15% Large US Stock, 15% Small US Stock, 15% Non US Stock, 15% Commodities, 35% Bond, 5% Cash)	11.06	7.82	100%	75%

The next portfolio represents a traditional balanced portfolio with an asset mix of 60% equity and 40% bond/cash (patterned after the Vanguard Balanced Index or the Dodge and Cox Balanced fund). The specific allocations were 50% large US equity, 5% small US equity, 5% non-US equity, 35% intermediate term bond, and 5% cash. It had a 10.26% annualized return over the 36-year period and a 10.61% standard deviation of return (this assumes annual rebalancing to keep the portfolio proportions constant over time). In a buy-and-hold situation, the traditional balanced portfolio had a 94% chance of recovery from a 10% loss within 5 years and a 66% chance of recovery in a withdrawal portfolio. This portfolio had slightly lower return than the 100% large stock portfolio but dramatically reduced volatility of return. Unlike the 100% large stock portfolio, the traditional balanced portfolio entirely avoided large losses during the time frame of this study. There were six years in which the portfolio had negative returns compared to eight years in the 100% large-cap US equity stock portfolio. However, the largest single year loss was -26.5% in the 100% large cap US equity portfolio, but only -13% in the traditional 60/40 balanced portfolio.

Finally, a global multi-asset portfolio was analyzed. As in the traditional 60/40 portfolio, this portfolio also allocated 60% to equities and 40% to bonds/cash. However, unlike a traditional 60/40 portfolio, the equity allocation in the global multi-asset was equally divided among large-cap US equity, small-cap US equity, non-US equity, and commodities-with each equity asset representing 15% of the portfolio. The remaining portion of the portfolio was 35% intermediate term bonds and 5% cash. The portfolio was assumed to be annually rebalanced so as to keep the portfolio allocations constant over the 36-year period.

The 36-year average annualized return of the global multi-asset portfolio was 11.06%, exactly the same as the 100% large US stock portfolio and 80 bps higher than the traditional 60/40 balanced portfolio. Its standard deviation of return over the 36-year period was only 7.82%, or 921 bps lower than the 100% large stock portfolio, 279 bps lower than the traditional 60/40 balanced portfolio, and only 229 bps higher than the all-bond portfolio. Several of the assets in this portfolio have high individual return volatility, but when assets with low cross correlation are blended together in a portfolio, the overall portfolio volatility was markedly reduced. In this case, the volatility of the global multi-asset portfolio was only slightly higher than a 100% bond portfolio. This serves as a reminder that the risk/return characteristics of a portfolio are more important than the risk/return behaviors of the individual components.

The probability of recovery from a 10% loss within five years in a global multi-asset buy-and-hold portfolio was 100%--equal to the 100% bond portfolio. In a withdrawal portfolio, the global multi-asset portfolio had a 75% chance of recovery from a 10% loss within 5 years-dramatically higher than the all-bond portfolio and an appreciable improvement from the all-stock portfolio and the traditional retirement portfolio. In addition to excellent recovery probabilities, the multi-asset withdrawal portfolio had impressive one-year return frequencies. Slightly over 40% of the returns over this 36 year period were in the range of +15% to +20%. While it did have three negative one-year returns over the 36-year period, the average loss in those three years was only 2.96%.

A summary of the recovery probabilities from various portfolio losses is presented in "The Recovery Race". The probabilities of recovery within five years in a global multi-asset withdrawal portfolio are higher than those in an all-bond portfolio, but slightly lower than in a 100% large stock portfolio or traditional 60/40 balanced portfolio in the loss range of -35% to -15%. The global multi-asset portfolio has superior recovery probabilities (compared to the other three portfolios) when the portfolio loss is less than -10%.

The Recovery Race

Several cautions are in order. First, as with any analysis of portfolio survival probabilities, the results for a withdrawal portfolio are affected dramatically by the portfolio performance during the first several years of the period being studied. As shown in "Raw Data", the performance of US large equities from 1970-1974 was hampered by significant losses in 1973 and 1974. Indeed, the 5-year annualized return of the US large equities from 1970-1974 was -2.38%. Conversely, the annual returns from 1970-1974 in the GSCI Index were stunning. It is because of this very issue-timing of performance sensitivity-that the analysis in this paper examined the probabilities of recovery in buy-and-hold portfolios and withdrawal portfolios over 32 rolling five-year periods. There were 32 unique starting years and 32 unique ending years. Unlike Monte Carlo approaches, this approach does not simulate performance hundreds of thousands of times. This analysis used historical data to create a set of actual historical recovery probabilities, while a Monte Carlo approach uses historical data to estimate performance and recovery probabilities going forward.

Second caution: This analysis used 5% as the withdraw rate with a 3% rate of inflation in the subsequent annual withdrawals. Higher or lower withdrawal rates change the results dramatically. The probabilities of recovery within five years in a global multi-asset withdrawal portfolio using six different withdrawal rates are shown in "Withdrawal Variation". If, for example, the withdrawal rate (WR) is 3% (blue line) there is a 75% chance of recovery from a 20% loss. Alternatively, if the withdrawal rate is 8% (brown line) there is a 4% chance of recovery with 5 years from a 20% loss. Thus, this type of analysis is very sensitive to withdrawal rates and the inflation of the withdrawal rate.

Withdrawal Variation

Third caution: The time frame of this analysis (1970-2005) was a period of robust returns across the board. Equities averaged in excess of 10%, intermediate bonds averaged over 8%, and commodities generated a 36-year average annual return of over 12%. Going forward we cannot know if these returns will be replicable over the next 36 years. It may be prudent to have more modest performance expectations over the next 7-10 years.

In summary, there are several "bottom line" conclusions:

(1) All-bond or all-stock withdrawal portfolios during retirement won't get the job done. Bonds don't have sufficient performance potential as evidenced by low recovery probabilities in a withdrawal portfolio (refer back to "The Recovery Race"), while an all stock portfolio has too much return volatility (i.e., standard deviation) which manifested itself in a relatively low recovery probability in a buy-and-hold portfolio (81% within 5 years).

(2) The traditional 60/40 retirement portfolio of 60% equities and 40% bonds/cash represents a dramatic improvement from a 100% stock or 100% bond portfolio. It delivers nearly the same return as the 100% stock portfolio, but with dramatically lower volatility of return. It has good recovery probabilities in both a buy-and-hold portfolio and a withdrawal portfolio. More importantly, it avoids large losses.

(3) A global multi-asset retirement portfolio that had larger allocations in small cap US equities, non-US equities, and commodities (while maintaining a traditional 60% equity/40% bond & cash mix) demonstrated the best performance characteristics. Its historical performance equaled the 100% stock portfolio, while its volatility of return was nearly that of a 100% bond portfolio. In general, it had the best recovery probabilities among all the portfolios examined (particularly within historically relevant loss ranges of -2% to -10%). Even better, the high recovery probabilities were not needed very often as the global multi-asset portfolio was largely resistant to losses over the 36-year period. It had only three years with negative returns compared to six in the traditional 60/40 portfolio and eight negative years in the 100% large stock portfolio. Loss avoidance turns out to be the real benefit of combining assets with low correlation to each other. Modest exposure to assets that have low correlation to equities and bonds (such as commodities or REITs) turn out to be valuable components in retirement portfolios. At least, they create better math.

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Craig L. Israelsen, Ph.D. is an associate professor at Brigham Young University. He teaches family finance in the Department of Home and Family Living. His research interests include mutual fund analysis. He writes monthly for Financial Planning magazine. Learn more about Craig Israelsen at http://familyliving.familylife.byu.edu/faculty/israelsen.htm

Special thanks for Scott Berglund for editorial and analytical assistance.