The 4% Rule for Retirement Withdrawals. Preliminary Version Floyd Vest, Sept. 2012

Transcription

1 The 4% Rule for Retirement Withdrawals Preliminary Version Floyd Vest, Sept It is reported that for the 4% rule of estimation for withdrawals from a broadly diversified and balanced portfolio of stocks and bonds, there is a probability as low as four percent of running out of money before 30 years. The first withdrawal is 4% of the of the value of the portfolio and withdrawals increase each year at the annual rate of inflation. These conclusions come from back testing from 1926 to 2009 using 55 successive periods of 30 consecutive years of stock and bond returns and inflation (Pfau, p. 1). They studied the interaction of yearly inflation and returns from five different proportions of stocks and bonds. (To read about this, go to Wikipedia and Search: Trinity Study and follow the references including Pfau s update.) The 4% rule is the most frequently cited retirement income strategy (Money magazine, Oct. 2011). Interest in examining this rule of thumb has been renewed with the poor performance and volatility of the stock market beginning in 2000 and extremely low current returns on bonds. (See the Exercises, Side Bar Notes, and References.) This concern leads to examination of examples and periods with threatening interaction between inflation and investment performance. Example 1. The minimum sustainable rate. What is the minimum average real rate of return R after inflation I that could sustain the withdrawals for 30 years? We will use Formula 1. 1 (1 R) x (1) P W where we will let P = $1000 as the initial balance in the R portfolio, W = 4% of P which is $40, R is the average annual real rate of return on the investment after inflation I, and x is the number of years at which the portfolio is exhausted. r I R = where r is the average rate of return on the portfolio. 1 I Withdrawals increase each year at the rate I. P is the amount of money that will provide for the withdrawals for x years. In the above discussion, the target x is 30 years. (For the derivation of this formula, see the References in this course.) This discussion gives Formula (1 R) (2) Using the TVM Solver on the TI83/84, solving R Formula 2 gives the minimum R =.0122 = 1.22% yielding x = years. You Try It #1 (a) If inflation averages I = 3.23%, what rate of return r, for the portfolio, is required for minimum survival. (b) By the 4% rule, for a retiree in 2012 that needs $100,000 per year income from a retirement fund, how much should be in the fund? (b) By the 4% rule, for a retiree in 2012 that needs $100,000 per year income from a retirement fund, how much should be in the fund? 1

2 Example 2. A historical example. For a twenty year period from 1901 to 1921, the average real rate of return R for stocks was.2% (Henry, David). Under these conditions, how long would the 4% schedule be sustained? This question gives Formula 3. x 1 (1.002) (3) Solving for x gives years. Using averages.002 only provide approximations to the effects of annual data. You Try It #2 For twenty years from the beginning of 1929, R =.4% for stocks. Average inflation I = 1.67%. (a) How long is the 4% program sustained under these conditions? (b) What is the average rate of return for stocks? (c) Why is this period of stock performance of interest? (d) In this period, was there ever a short sequence of consecutive annual returns that reduced the 4% schedule to zero? (e) Solve Formula 3 by algebra and a scientific calculator pad. Show your work. Example 3. Graphing the balance of the remaining retirement funds. Consider the following historical example with R =.0284, I =.032, and 5% withdrawals using Formula 4. (30 x) x 1 (1 R) (4) Y1 W (1 I) for x = 0, 5, 10,.,30 which gives the R remaining funds for each year up to 30 years. (See the References in this course for the derivation of this formula.) This gives us Formula 5. (30 x) x 1 (1.0284) (5) Y1 50(1.032). Using Window x: 0, 30, 5; y: 0, 2000, , Trace reveals that for x = 0, Y1 = The curve reaches maximum at approximately x = 7.34 years at $ in the fund, and the fund is exhausted at exactly 30 years. You Try It #3. What was the average rate of return for the above investment in Example 3? Example 4. How much money was spent in Example 3, where W(1+I) was the first withdrawal occurring at the end of year one? We will use Formula 6. x (1 I) 1 (6) S W (1 I). Using the numbers from Example 3 gives I 30 (1.032) 1 (7) S=50(1+.032) = $2536. With R =.0284 and I =.032, we have.032 r = so r =.0613 = 6.13% which sustained the payout

3 You Try It #4. (a) Derive a formula for S where the first withdrawal is W which occurs at the beginning of the first year. (b) Show how to use this formula and Formula 7 on the TI83/84 TVM Solver. Example 5. Applying the 4% rule to planning investing for retirement. From 1913 to 2012, inflation averaged 3.23%. Consider the following example for a twenty year old in 2012, retiring at age 70 with a target of age 100. With 50 years until retirement and estimated living expenses at $50,000 per year in today s dollars, retirement income at age 70 should be 50,000( ) 50 = $245,057 for the first year. This amount is 4% of $6,126,415 for a target in retirement resources. How much is needed to supplement $15,000 of social security in today s dollars, assuming Social Security is inflated at 3.23%? Example 6. A convenient savings program. Examining market history suggests a conservative estimate of 5% earnings rate. A formula to use is n n 1 (1 y) 1 r I (8) Z (1 I) J where y = and J is the first savings installment y 1 I with installments to be increased each year at the rate of inflation and Z is to total amount after n years. (See the References in this course for the derivation.) Substituting we have (1 y) (9) 6,126, 415 (1.0323) J with y = y Calculating gives the first year s savings of J = $16,514. What is J considering social security? If you can t accomplish this savings, you can make a serious effort to do what you can, and you can research alternatives which provide financial security for retirement. You Try It #5 Solve Formula 9 by algebra and a scientific calculator pad. Show your work. Side Bar Notes: Vanguard.com did a study of the 4% rule. To construct a worst case scenario, they used market returns from rolling periods from 1926 through With all the money in the stock market, the earliest short fall year (zero balance) was year 28. If average inflation is 3.246% per year, what was the rate of return on the stocks? See this article at Vangaurd.com, Search: Vanguard s Investment Philosophy, We Believe #7. Studies published elsewhere include Monte Carlo Simulation, and Smoothing. How long will you live? In Money magazine, Jan./Feb it was stated that for a couple age 60, chances are 40% that one or both will live to age 94. In the last fifteen years, longevity of 65 year olds has increased by five years. For a couple age 65, there is a 25% chance that one or both will live to age 97 (Kiplinger s, July 2007, p. 24). 3

4 The RMD withdrawl plan. Anthony Webb and Wei Sun reported in their article that a RMD strategy (IRS requires mandatory withdrawals. See IRS Publication 590.) outperformed the 4% rule. (Kiplinger s Personal Finance, 8/2012, p. 54). Recent decline in interest rates. As recently as the year 2000, five-year Treasuries yielded 6.15%. As of July 2012, they yield 0.7% (Scott Burns, Denton Record Chronicle, July 8, 2012). The interest rates on bonds and CDs have also declined to very low rates. For CDs, see bankrate.com. For bonds, see Vanguard brokerage. You may not be able to work longer. One solution to lack of retirement resources is to work longer. Half of the retirees surveyed by EBRI said they left the workforce earlier than planned. (Kiplinger s Personal Finance, 9/2012, p.57.) Health care costs during retirement. A 65 year old couple who live until age 95 can expect to incur $400,000 in unreimbursed medical costs, assuming annual health care inflation of 6% per year. (Kiplinger s Personal Finance, 7/2012, p. 53) Do the math to calculate the beginning annual expenses which increase at 6% a year. A 4% rule annuity. For a 65 year old, $100,000 buys a lifetime inflation adjusted annuity which pays more than $4000 the first year. Does such an annuity exist? Smart Money, May, 2010, p. 57 reports for a 65 year old, a $300,000, lifetime annuity with a 3% annual increase pays $1434 per month the first year. Does this beat 4%? If you lived to age 95, what was the return on the investment? What about someone dying at age 80? The chasm between starting salaries for college graduates. Petroleum engineers $120,000; Pharmaceutical sciences $105,000; Math/Computer sciences $98,000; Early childhood education $36,000; Counseling/psychology $29,000. Money magazine, March 2012, p. 38. How much would a teacher make in a life time if they got a 3% step raise for the first 20 years and an inflation adjustment of 3% for 40 years from age 25 to age 65? What is a master s degree worth? If they invested 10% a year at 7%, how much would they have? Many teachers who get ahead find a second income. It is called the lost decade. Go to Wikipedia and Search S&P500 and get the total annual returns from the beginning of 2000 to the end of 2011 or later. Notice three large successive loses in 2000, 2001, 2002, and then a big loss in Calculate the average return for this period. Research inflation at usinflationcalculator.com, and assess the survivability of the 4% rule under these conditions. Free or near free college courses. At times of their choosing students use internet videos of lectures by respected professors, and complete interactive quizzes and regular homework. Student can conduct discussions online. Of the 160,000 people who enrolled in a Stanford artificial intelligence course, only 23,000 finished. Stanford, Duke, Princeton, and Johns Hopkins have partnered with Coursera to offer more than 100 online courses. MIT, Harvard, and University of California, Berkley offer courses through edx (THE WEEK, Sept. 7, 2012). 4

5 The TIAA 4 1 % study. A Monte Carlo analysis was used to examine the 2 1 survivability of a 4 % rule from historical data for different portfolios of 2 stocks, bonds, and cash for 20 to 40 years. For 30 years, conservative (20%,50%,30%) has the probability of 32.6% of survival and aggressive (100%) 91.6%. They examined the effect of a fixed lifetime annuity paying 7% for a 65 year old. For 50% invested in the annuity, and conservative investment of the remaining money, the probability was 81.3%, and for aggressive was 97.5%. Exercise: Do an example to mathematically demonstrate TIAA s claim that for only withdrawals from a portfolio, In fact [for aggressive portfolios, if] the rate of growth [for the portfolio] exceeded the withdrawal rate,, [it could happen that] the longer the life span, the more money was left in the portfolio at the end of the investor s life Exercise: Assume half of $500,000 was used to purchase a 7% lifetime annuity. The article points out that annuity payments in this study are fixed and an investor must withdraw a greater amount from the non-annuity nest each year to maintain an inflation adjusted income of.045(500,000)(1+.03 ) n for each year beginning with n = 0, 1, 2,, 29, with inflation at 3%. Assume annuity payments and withdrawals occur at the beginning of each year. (a) What is the yearly annuity income? What is the remaining nest egg? (b) For the first three years, and the 1 last year, calculate the withdrawal from the remaining nest egg to maintain a 4 % 2 income rule. (c) Discuss how to calculate the average rate of return required to provide the withdrawals. Making Retirement Income Last a Lifetime, 2012 (tiaa-cref.org/institute/events/web_conferences/wc_valueadded2.html) (Research by John Ameriks, Ph. D.). The optimistic years. For the twenty years, from 12/1979 to 12/1999, $1000 invested in stocks grew to $26,817. What was the rate of return? Nearly free investing. Invest in a broad selection of commission free Electronically Traded Funds with expense ratios as low as.04%. Exercise: Do an example to illustrate that Over a 35 year contribution, low cost investing [could] accumulate about 45% more than investments with expense ratios of 2% (Nearly free at last, A new opportunity at Schwab, Scott Burns, Denton Record Chronicle, Oct. 28, 2012). Do a graph or proof to demonstrate that as the rate of return x increases, the percent difference increases. For lifetime annuities, for the same $1000, some insurance companies guarantee as much as 65% more per month than others. For variable annuities, the expense ratio can range from more than to 2% to.2%. Surrender charges can range from 12% to 0%, usually descending annually. (The TACT Bulletin, Aug. 1985, p. 22). The strengths of insurance companies are rated by A. M. Best, Standard and Poors, Duff and Phelps, and Weiss. In 2008 for an age 65 male, from the AnnuityShopper website, average payout on $100,000 for a lifetime annuity was $8,460 per year. For a 3% yearly increase, $6,470 per year. 5

6 Diversification beyond stocks and bonds. Look at EverBank Wealth Management, Enhanced Yield Portfolio to see fourteen sectors, national and international. You can diversify by your own selection of low cost mutual funds and ETFs. Real estate returns have shown little correlation with stock and bond returns. Shopping for a life-time annuity, called an immediate annuity. See imediateannuities.com, which includes 16 of the best insurance companies. Retirement planning advice from the experts. Assuming retirement is at age 67 and they live 25 years: For funding retirement, one needs 8 to 12 times the final salary. Consider yearly retirement withdrawals of 85% of final after tax income less social security. Safe withdrawal rates are 4% to 6% of beginning balance increasing each year at the rate of inflation. The chance that 6% will last 25 years is 50% to 70%. See Fidelity Retirement Income Planner or T. Rowe Price Retirement Income Calculator. (Money magazine, Dec. 2012, page 32.) 6

7 Exercises: #1. For stocks, for the twenty year period from the beginning of 1966, average R = 1.9% (Henry, David) and I = 6.28%. (a) How long will the 4% program last? (b) What is the average rate of return on stocks? (c) Why is this period of interest? #2. (a) If a retiree has a portfolio of $1,000,000 and followed the 4% rule for 30 years, and the average rate of return on his investment was 9% (a long term average for stocks) and inflation averaged 3.23% (a long term average for inflation), how much was left on the table at the end of 30 years? How much is the total withdrawal from the fund? (b) If they spend $500,000 on a 7% life time annuity and for thirty years have withdrawals from the remaining nest egg, increasing adequately, to maintain the 4% rule, will they have less left on the table? (See the tiaa article referenced in the TIAA 4 1 % 2 study in the Side Bar Notes.) Wrestle with showing mathematically that this is true. One approach is to calculate that left on the table. Another is a much simpler argument. They bought this annuity for security not knowing that they would earn 9%. #3. In Money magazine, Sept. 2012, p. 97, an expert is quoted as saying that for funding retirement, a person needs 25 times the residual annual costs. Residual costs equals cost of living expenses minus consistent income, for example social security. Consider a cost of living of $60,000 a year and social security of $22,800 per year. (a) What is 25 times the residual? (b) With withdrawals increasing each year at an average inflation rate of 3.2%, what rate of return on the investment is required for 30 years? #4. Assume that a retiree at age 65 has a portfolio of $1,500,000 and follows the 4% rule. For the first 15 years his investment earns 0% per year and inflation is 3.2%. (a) How much money is left after 15 years at age 80? (b) If costs increase at the rate of inflation, what is the cost of living at age 80? If he earns 6% for the next several years, when is he broke? #5. There are many uncertainties reflecting on the 4% rule including major unplanned expenses or other developments. List 15 or more major unplanned developments. #6. From Vanguard s Investment Philosophy, We Believe #7 : Strong returns in the early years of spending enhance the portfolio s final value far more than the same level of returns in later years. Do an example to verify this. #7. Do the math to check this: Consider a 55-year-old pre-retiree with no retirement savings. If the person earns $80,000 a year, makes the maximum annual $22,500 in 401K contribution, gets a 3% employer match and a 3% per year raise, and earns a 6% return, his balance could top $400,000 by age 65. (Kiplinger s Personal Finance, 9/2012, p. 57) #8. With $500,000 in a retirement fund, and withdrawing 4% the first year increasing each year at 3% inflation, by Monte Carlo software, there is an 88% change that the 7

8 money will last for 30 years. For the 88% chance, what is the rate of return on the portfolio (50% in a stock mutual fund and 50% in a bond fund)? (Money magazine, Oct. 2011) Assume that the money lasts exactly 30 years and 20,000 is collected at the beginning of the first year. #9. According to THIS WEEK, Sept. 7, 2012, since 1985, U. S. college tuition and fees have grown 559%. (a) At what annual compounded rate has it increased? (b) At what annual simple interest rate has it increased? #10. Use usinflationcalculator.com to calculate average inflation from 1913 to #11. Ameriks (Vanguard.com, Insights) did a 1926 to 1979 study of portfolios lasting 30 years or longer, and found that for most portfolios that started increasing withdrawals around 1967, they lasted less than 30 years. (a) Us usinflationcalculator.com to calculate the average inflation rate for each of the three successive ten year periods starting in (b) From a Vanguard bar graph, starting in 1966, the first ten year approximate average return on 70% stocks, 24% bonds, and 5% cash was about -2.3%. (See The Truth About Risk, and Does the 4% Rule Hold Up? ) What was the effect of high inflation during early parts of the 30 years? What was the effect of low returns during the early part, and the interaction between the two? Make up and test examples. See References. #12. A guaranteed 5% for life. A reverse mortgage was advertised as paying 5% per year fixed, tax free. (a) For a $100,000 house, in twenty years you get your money back. But the payments are spread over 20 years in the future. If money is worth 6%, what is the present value of the payout compared to the $100,000 value. If money is worth 5%, how long does the $100,000 last? (b) At 4%, how long does it take to get the money back? (c) If the house appreciates at 3% a year, what is its value in twenty years? If money is worth 6%, what is the present value of the future appreciated value at the time of contract? What rate of return does the mortgage company make on the investment not counting expenses? (d) For a person in the 15% income tax bracket, who lives 30 years, what is the equivalent taxable rate of return? #13. The Department of Labor standards for full disclosure of fees charged for 401(k) retirement investment plans went into effect in Total fees ranged from over 2% to 1/10 of one percent. Do the math to check the following: Scott Burns, Denton Record Chronicle, Oct. 14, 2012: Suppose twins Joe and John choose identical careers with salaries of $50,000 at age 30. Suppose both save 10 percent of their income and both receive a raise of 3 percent a year. Suppose both also target retiring at age 65 and have plans that have identical returns of 7 percent before fees. The only difference is that one plan has expenses of 1.5 percent while the other has expenses of 0.5 percent. How big a difference will costs make? The twin in the high-cost plan will accumulate 6.59 years of final income, while the twin with the lower cost plan will accumulate 8.85 years of income, a difference of 34 percent. Reduce the cost to 0.1 percent, and the amount accumulated will rise to 9.6 years of final income. 8

9 Suggestion: Calculate the total saved by age 65 and compare the percentage difference. Assume that retirement income starts at $50,000 inflated at 3% a year and withdrawals are from the same 401(k)s and increase at 3% a year. Calculate how long the money lasts and compare the percentage difference. You may not get Scott s numbers but yours may be better. #14. Denton Record Chronicle, Oct. 17, 2012: Social Security payments average $13,572 a year. Since 1975, the annual COLA has average 4.2%. This October, a 1.7% increase was announced. (a) If a retiree at age 66 starts at $14,572 a year which increases as 4.2% a year to age 95, how much do they receive at age 95? (b) What is the total they collect through the years? (c) If money earns 7%, how much is required to fund it? (What is Social Security worth?) (d) Graph the history of the fund. When is there a maximum balance and how much is it? #15. (a) According to Scott Burns (Denton Record Chronicle, No. 11, 2012), for a 60/40 equities/fixed income portfolio, 69.83% of 6% withdrawal programs will survive 30 years. Sixteen percent of 65-year-old couples will have one person survive 30 years. Estimate on average how many of these 100 couples will face running out of money? Explain why this is an estimate. (b) He says that for a 60/40 equities/fixed portfolio, the 4% program the failure rate is 2.55%. Estimate what percent of couples will run out of money. #16. Do a case study for a friend s clients: Retired married school teachers at age 79 in 2012 have a retirement income of $21,144 per year social security, and $71,616 in teacher retirement. They have $1,076,655 in investments other than a paid for home and personal possessions. Their 2012 living expenses are $110,000 a year (including income taxes). They hope to live to age 100 and don t want to run out of assets. The $1,076,655 investments include $170,136 in I-bonds and EE-bonds which have averaged paying 4% per year. There is $301,726 in an insurance company fixed IRA paying a minimum of 3%. There are $321,798 in CDs paying 1.52% and $91,493 in money market accounts paying.80%. Investments in tax shelter annuities and mutual fund IRAs are $123,974 in US and international stocks, and $67,528 in real-estate. Their teacher retirement is net of premiums for health insurance. They do mandatory withdrawals which was $19,045 for Their home is on the tax rolls at $152,000. They have long term care insurance, and no debts. They could adjust their cost of living downward by $20,000 a year. What is the possibility that the retired teachers have enough assets to meet their goals? Make and justify assumptions about inflation, social security COLA, a teacher retirement COLA of no more than 1%, returns on individual investments, cost of living, and the value of their home. What is the probability that one or both would live to age 100? What investments would you advise? What would be the consequences of running out of investment money? 9

10 Answers to Exercises: 2. The formulas you use and the numerical answers depend on the time line you set up for your derivation. One formula for the amount left on the table is 30 (1 y) S = 1,000,000(1+.09 ) 30-40,000( ) 30 where y =. y S = $5,630,135. One figure for the total withdrawal is $1,975, Using the TI84 TVM Solver gives R = Solving = r =.0435 = 4.35%. r gives 9. (a) Inflation rate = 7.2% compounded per year. (b) A simple interest rate was 21% per year. 11. Using the average inflation rate of 5.5% and average returns of -2.3%, for inflating withdrawals starting at $40 from a beginning $1000 in1966, $ would fund first ten years of withdrawals from the fund. The amount withdrawn is $515.01, and the amount remaining is approximately $277.39, depending on time lines. 14. (c) The Social Security is worth $270, (a) Estimate on average: five couples or less, 5%. This is an estimate because some of those couples who died before end of 30 years might have run out of money before they died. (b) Estimate: 2.55% of 16 =.4 or less,.4%. 10

11 References: Vest, Floyd, The Mathematics of Financial and Social Responsibility, Mathematics and Computer Education, Vol. 28, No.2, Spring Has derivations of the formulas used in this article. In this course. Vest, Floyd, Taking the Long View of Life, Consortium, HiMap Pull-Out, No. 83, Fall/Winter Has derivations of some of the formulas used in this article. In this course. Vest, Floyd, Funding Retirement During a Worst Market Scenario, to appear in this course. Based on The truth about risk, at Vanguard.com. Vest, Floyd, Calculating the Value of Social Security Benefits at Different Ages, 2011, in this course. Vest, Floyd, Living and Investing with Inflation, Fisher s Effect, 2011, in this corse. Cooley, Phillip L., Carl M. Hubbard, Daniel T. Walz, Retirement Savings: Choosing a Withdrawal Rate That is Sustainable, AAII Journal, Feb. 1998, pp (This is called the Trinity Study, and presents the test of the 4% rule.) Henry, David, Yale professor predicts doom, gloom for stocks, USA Today, March 16, At Vanguard.com, do a search for Does the 4% rule hold up? Do a search for The truth about risk, and other articles. See Americks at Vanguard. For information on the stock market earnings, see Wikipedia, Search: S&P 500. Pfau, Wade, Trinity Study, Retirement Withdrawal Rates and the Chance of Success, Updated Through 2009, found in Retirement Researcher Blog. Money magazine, Nov. 2012, p. 98, I Hear I Can Take Only 4% From My Portfolio. Why? also Money magazine, Sept. 2012, p. 97. COMAP.com offers a free download course in financial mathematics with over 40 articles, for upper high school and undergraduate college, with emphasis on personal finance. See if your school will give you tests and credit. See Simply register, click on an article in the annotated bibliography, download it, and study it or teach it. Unit 1: The Basics of Mathematics of Finance Unit 2: Managing Your Money Unit 3: Long-Term Financial Planning Unit 4: Investing in Bonds and Stocks Unit 5: Investing in Real Estate Teachers Notes: A class project. Have the class collect the characteristics of the average pre-retiree and retiree. Ask the question, Who wants to be average? Characteristics could include such things and financial condition, longevity, health, education, earning power, knowledge and attitudes toward finance, and others. Scott Burns, Denton Record Chronicle, April 17, 2011: The average social security benefit is $14,088 per year. For the average, their home is their largest investment. In the last five years of their lives, 43% of Medicare recipients spend out of pocket more than their total assets minus the value of their residence (released from Mount Sinai School of Medicine, Sept. 2012). 11

12 Forty percent of American families have $500 or less in savings (Fox News, ). For families age 65 to 69, other than home equity, only 45% had more than $20,000 in non-retirement account assets. Including retirement accounts, the median total holdings was $52,000. For ages 51 to 61 in 1992, the median home equity was $150,000. ( For more troubles of pre-retirees, see Kiplinger s Personal Finance, p. 57, Jan. 2013, and Money magazine Dec. 2012, page