= $22, = $143,211. These considerations yield the following time line.

Size: px

Start display at page:

Download "= $22, = $143,211. These considerations yield the following time line."

Dwayne Lucas
5 years ago
Views:

1 1 America s Retirement Challenge, (A Mathematical Model) (Preliminary Version) Floyd Vest, August 2013 This article is based on a speech entitled Meeting America s Retirement Challenge, given by Ronald P. O Hanley, President of Asset Management and Corporate Services of Fidelity Investments, which was presented to the U.S. Chamber of Commerce Capital Markets Summit, Washington, D.C., April 10, The speech was reported in an article Meeting America s Retirement Challenge at fidelity.com/viewpoints/personal-finance/retirementinsecurity. Mr. O Hanley recommended significant reforms to the U.S. retirement savings system so that Americans can maintain their standard of living for 30 or more years of retirement. He said that nearly four in ten retired households do not have sufficient income to cover their expenses. Well over half of Americans have less than $25,000 in total savings, not counting their home and pension plans. (This term may refer to Social Security, and to defined benefit pension plans which are employer sponsored and benefits are based on salary and tenure with the company.) Also, 28% have put aside less than $1000. Mr. O Hanley said that a general rule of thumb is that workers need to save at least eight times their annual salary to provide for retirement. The term annual salary may mean the annual salary at retirement. These savings probably provides retirement funds to pay necessary income to supplement Social Security benefits. To include Social Security in the calculations, we went to ssa.gov/oact/quickcalc/index.html and entered the date of birth of 6/15/1949 and retirement at 9/2011. We selected estimates in today s dollars, and entered $40,000 as 2011 income, then clicked Calculate. The calculator responded that you are retiring at age 62 and 3 months at $874 per month. For retirement at age 66, we entered retirement at 9/2015 and got $1235 per month. The calculator assumed that the person worked until 9/2015. Social Security offers an extra benefit for delayed retirement to age 70 and uses a benefit formula of 1.32 times the normal age 66 retirement benefit (ssa.gov/oact/quickcalc/early_late.html). We estimated the age 70 benefit to be 1.32(14,820)( ) 4 = $22, Example 1, The Rule of Eight Times. In building an example, we will choose the retirement at age 66 and use age 30 income of 40,000( ) 4 = $45, and Social Security benefit at = $14,820 per year. Savings should supply 45, ,820 = $30,621 per year living expenses to supplement Social Security. We will assume that the at retirement age 66 salary is 45,441( ) 36 = $143,211. These considerations yield the following time line.

2 2 Age Year Salary $40,000 $45,441 $143,211 SS $10,488 $14,820 $46,706 Figure 1. Retirement planning time line. The article recommend a rule of thumb for retirement assets of eight times salary for which we use 8 143,211 = $1,145,688. These funds should supply the living expenses necessary to supplement Social Security benefits at age 66 which would be (45,441 14,820)( ) 36 = $96,504 for the first year of retirement. We will assume withdrawals from the savings increase each year at the rate of inflation for 30 years, and that Social Security benefits increase at this rate. We will apply Formula 1 to the above considerations: 1 (1 y) n r I (1) P R y where y. For our example P = $1,145,688, R = $96,504, 1 I n = 30, and I = 3.24% = This formula provides for the withdrawals to increase each year at the rate I and the first withdrawal is R at the end of the first year. P gives total retirement assets, and I is the inflation rate, and r is the rate of return on the investment. (For derivations of this and other formulas, see the article in this course, The Mathematics of Financial and Social Responsibility. ) Substitutions give Formula 2: 30 1 (1 y) r.0324 (2) 1,145,688 = 96,504 y with y. We will solve for y and then for r which is the rate of return required for the retirement fund to provide the increasing r.0324 withdrawals for 30 years. Calculating yields y = 7.45% =.0745 and.0745 = giving r = 10.9%. (See the Side Bar Notes for the calculations on the TI 83/84.) If they could retire on 70%, we calculate (x ) =.70(45,411) with income to supplement Social Security = x = $16,989 and 16,989( ) 36 = $53,541 at age 66. Calculating with this value for R yields y = 2.335% = and r giving r = 5.65% which looks more attainable than 10.9% above Example 2, Savings Programs. The article recommends that employees should save at least 6% of their salary and savings should increase over time. Currently the default rate in the PPA program is 3%. Roughly 88% of employees accept the current 3% default rate. Other

3 3 beginning savings rates were mentioned including 10%, 15%, 18%, and 20%. To investigate, we will use Formula 3: x (1 y) 1 P r I (3) D x1 y with y. This formula for a savings program allows (1 I) 1 I for the first saving D to occur at the end of the first year and savings increase at the rate I for x years accumulating to the amount P. The rate of return earned by the savings is r. For an example at 6% savings, D =.06(45,441) = $ , x = 36 years from age 30 to age 66, P = $1,145,688, I = Substitution yields x (1 y) 1 1,145,688 r.0324 (4) y = 375,307. We solve for y = = (1.0324) giving r = 10.1%. This is the rate of return required for increasing savings to accumulate to P, beginning at 6% of $45,441. We will build: Table 1, Required rate of return r to meet the goal of P. Percent 3% 6% 10% 15% 18% 20% D $1363 $2727 $4544 $6816 $8179 $9088 y r 13.2% 10.01% 7.68% 5.61% 4.6% 4.03% These calculations suggest that the person in our example should start savings with at least 10% and increase each year at I = 3.24% assuming salaries increase at this average long term rate of inflation. The 4% Rule says that the first year s withdrawal is 4% of the retirement assets and withdrawals increase at the rate of inflation. This says that the amount of retirement assets equals 25 times the first withdrawal. In the case of our example we have 25(96,504) = $2,412,600. Historical studies have shown that these funds have better than a 5% chance of falling short of providing withdrawals for 30 years. Comparing these results, the Eight times rule provides less security. (See the article in this course, The 4% Rule for Retirement Withdrawals. ) Example 3. We will experiment with our own numbers and use the 4% Rule. Let P 1 = 25(96,503) = 2,412,600, and r = 7% and using Formula 3 we have 36 (1 y) 1 2, 412, (5) D 35 y = 790,325 with y = =.0364 = 3.64% and (1.0324) solving for D gives D = $10,970 which is 24% of age 30 salary. To get ahead in life, many people have a second or third income. Example 4. President Obama s 2014 budget proposal includes a $3.4 million cap on accumulated tax preferred retirement savings. You couldn t contribute more after you have

4 4 reached total accumulations of $3.4 million (Kiplinger s Personal Finance, 9/2013, p. 9). Consider a person age 25 who wants to retire at age 70 on $100,000 in age 25 dollars. This is a very reasonable target. Many well paid recent college graduates earn over $150,000 a year. Millions of current retirees have this income and wouldn t want less. According to the 4% Rule which is commonly recommended, retirement assets should be 25 times the beginning income and income increases each year at the rate of inflation. Research has shown that funds calculated by this formula have more than a 5% chance of not lasting 30 years. The long term average inflation rate is 3.24% per year. At age 70, our 25 year old would need 100,000( ) 45 = $419,913 for first year retirement income, and retirement assets of ,913 = $10,497,825. This number makes Obama s recommendation questionable. We will calculate further with: P = $10,497,825, R = $419,913, N = 30 years, I = 3.24% r.0324 =.0324, in Formula 1. Calculating gives y = 1.219% = and r = 4.5% as the minimum rate of return for survival of funds for 30 years. We will calculate needed savings with Formula 3 and use r = 7%, N = 45 years, and the above numbers, and get the first years savings of D = $23,503. For well-paid recent college graduates earning $150,000, this savings is 15.7% of income and could be managed comfortably. Hopefully, their salary will increase at the rate of inflation or better. In 2013, For a Roth 401(k), there are no income limits on who can contribute. The total amount you may contribute to your 401(k) accounts traditional, Roth, or a combination is $17,500. For IRAs, a single taxpayer with MAGI over $112,000 and up to $127,000 can contribute a reduced amount. The customary limit for IRA contributions in 2013 is $5500 (Kiplinger s Personal Finance, 9/2013, p. 52). They could contribute to a tax deferred annuity, but these don t work the same as a Roth, and a person needs to be careful for expense ratios. See the article in this course, Investing in Tax Deferred Variable Annuities, Fall Investigates mathematically the effect of high expense ratios of variable annuities and the possibility of their out weighing the tax deferral advantage. Based on a Vanguard study. Government EE bonds and I-bonds are tax deferred. Also look at tax advantaged programs for saving for educational expenses. See the article in this course Some 529 Plans and Other Options for Saving for College, Dec Conclusion: Meeting America s retirement challenge is a vital national security issue (from the article). Side Bar Notes: TI 83/84 code for solving Formula 2 for y: N=30, PV = , PMT = , FV=0, P/Y=1, PMT:END and solve for I% to get 7.45% = y giving r = 10.9%. TI 83/84 code for solving Formula 4 for y: N=36, PMT = , PV=0, FV= , and solve for I% to get y = 6.651%, giving r = 10.1%.

5 5 Derivation of Formula 3. Formula 3 is slightly different from the savings formula in The Mathematics of Financial and Social Responsibility, in this course. Consider D(1 r) D(1 I )(1 r)... D(1 I) (1 r) D(1 I) gives n1 n2 n2 n1 = P. Dividing by (1+I ) n1 n1 n2 n1 n1 1 r 1 r 1 r P 1r r I D D... D D = D n1 D1 1 I 1 I 1 I (1 I ) x0 1 I x0 1 I From the formula for the sum of an ordinary annuity, this gives Formula 3, n (1 y) 1 P (3) D n1 y (1 I) where r I y. 1 I x. Employees are not typically equipped to make investment decisions including savings adequacy, longevity, asset allocation, and investment management (from the article). The six legged stool for the retirement system: Social Security, defined benefit pension plans, defined contribution plans, personal savings, purchasing life time annuities, and other investments. The article had a four legged stool. National security. If millions of Americans reach retirement with insufficient resources, the impact on our national security could be catastrophic (the article). Retirement savings have propelled capital formation for the private and public sectors of business on a scale never seen before (from the article). No access. Thirty-five percent of workers have no access to an employer sponsored retirement plan or do not enroll (from the article). Poor financial literacy is a threat on the horizon. Obstacles to financial education should be removed (the article). Serious advanced financial education should be required. One obstacle is the lack of training of high school mathematics teachers in the area of financial mathematics, and the reticence for financial mathematics in the mathematics curriculum on the part of managers of curriculum. Student enrollment should be required. The Pension Protection Act (PPA), 2006 has enabled automatic enrollment, and an automatic default savings rate of 3%. These features are automatic unless the employee opts out (from the article). Some facts about the young. Student debt now exceeds consumer debt. How could student debt be avoided? Only ten percent of the elderly live in poverty while twenty one per cent of children of young parents live in poverty. There are many more broken homes and single parents than in past generations. In certain groups, 70% of children are born out of wedlock. The young pay twice as much for housing as a multiple of their income as that paid by older people. A growing number of states spend more on prisons than they spend on education. The low-income-young pay no income tax (Scott Burns, The real Doomsday, Denton Record Chronicle, April 22, 2012). Discuss where you place yourself on these issues. x

6 6 Standard and Poors downgrades US and other debt. The downgrades include France, Austria, Italy, Spain, and Portugal (Denton Record Chronicle, Jan. 15, 2012). What do you think happened to Greece? Discuss what a downgrade means? What is the cause of the downgrade? What is the cause of the excess debt? Retirement stats. When doing long term financial planning keep these stats in mind: In 2011, 85% of older workers think they will work past 62, vs. 71% in % left work earlier than they d planned. Age 64 is average retirement age for men (62 for women). 74% of Americans expect to do some work post-retirement. 39% say they ll work to make ends meet, 35% because they want to ( Money magazine, March 2012, p. 108, Employee Benefit Research Institute, Boston College Center for Retirement Research, Wells Fargo 2011 Retirement Survey). Unfunded pension liabilities. An Associated Press survey found that 50 states have a combined $690 billion in unfunded pension liabilities and $418 billion in retiree health care obligations (Denton Record Chronicle, Dec. 11, 2011). Describe what unfunded liabilities means. How much to save for retirement. According to research, to supplement Social Security, people need to save enough to replace 40% of their earning power at retirement (Scott Burns, The upside of uncertainty, Denton Record Chronicle, Aug. 4, 2010). This says we should plug in R = $57,084 in Formula 1. Check this figure. Try it out in Example 1. This figure depends on certain averages, expectations, and probabilities which you may not accept. Name at least six of them.

7 7 Exercises: Show your work. Label answers, variable, and numbers. Show formulas with numbers when appropriate. #1. Derive Formula 1. #2. Do a long term financial planning calculation for someone your age including Social Security. #3. Derive the formula for the future value of an ordinary annuity. #4. Describe ten different ways people commonly acquire a second and third income. #5. Describe and study the articles in this course which you think would be helpful for the topic in this article. Some of them supply answers to the exercises. #6. Look up and study the article which is the basis for this study at Fidelity.com, and other articles in viewpoints/personal-finance and study them. Write an article for your class and send it to COMAP.com. #7. For an example, graph the amount remaining in retirement fund after x years with withdrawals increasing each year, where the funds last n years and x n. You should get a nice recognizable retirement curve. #8. Find a formula which provides funds which last forever under the terms of Formula 1. #9. The article referred to target mutual funds. Look up some target funds and describe how they work and how they survived the bear market beginning in How do they work for people beginning their investment at different ages? What are some uses for target funds other than retirement savings? What is the theory behind the way target funds work? What problem do they propose to solve? What percent of employee enrollees chose target funds? Kiplinger s Personal Finance, Aim for the Best Target Funds, 9/2013 reported on Fidelity Freedom, Vanguard Target Retirement, and five others. They reported returns for 2015 and 2050 targets. It is interesting to see how the 2015 funds survived the down years of 2001, 2002, 2008, and #10. Discuss how Traditional IRAs and 401ks work. Discuss how Roth IRAs and Roth 401ks work. Which would you prefer? Can you trust the government not to put new taxes on them just as they did on Social Security benefits. #11. Describe how Social Security benefits are taxed for retirees. See the IRS 1040 booklet or a computerized income tax calculation program. #12. Discuss what you plan to do about retirement calculations and savings when you are employed. Where would you invest savings? What is the history of stock and bond investments? What other types of investments are available? What is the history of inflation? Have there been long periods of high inflation? #13. How often would high inflation double the cost of living for a retiree? Some may be on a fixed income.

8 8 #14. The article said that if millions of Americans reach retirement with insufficient resources, the impact on national security could be catastrophic. Discuss, from your point of view, how this could happen. Discuss some viewpoints other than yours. #15. Make up a problem which follows Example 1 and calculate y then r. Name your calculating device and give some code. #16. Make up a problem which follows Example 2 and calculate y then r. Name your calculating device and give some code. #17. Make up a problem which follows Example 3 and calculate y then r. Name your calculating device and give some code. Choose your own retirement income. Many people currently easily retire on $100,000 per year. They wouldn t want to retire on less. How much would this be when you reach age 70? #18. Make and solve a problem which includes a defined benefit retirement asset. Solve for the needed savings. Look up figures on longevity, and you may plan for more than thirty years of retirement. Give an argument for retiring at age 70. For couple age 65, there is better than a 20% chance that one or both will live to age 95. On average which high school and college students come out ahead financially? #19. Investigate the financial education in your high school or college and write a report. What level is taught and by whom? How many students participate? What do your mathematics teachers think? What do you think about it? What teachers are qualified to teach this financial mathematics course? What are some of the obstacles to such a course? Why did Mr. O Hanley not suggest a financial mathematics course in high school? #20. Why would people want to be 95% confident that they could live comfortably during a 30-year retirement when there is only an 80% chance that they would live that long? Write your opinion. Is the expense and inconvenience worth it? #21. Spenders can usually be recognized by their vacations, their expensive cars, the expensive house, the high-priced California wines, and their debts. Write a description of savers. Write your opinions. #22. The ten or so years beginning in 2000, referred to as the lost decade, were difficult years for the stock market. If you had $10,000 in the SPDR S&P 500 (SPY) at the start of 2001, by the end of 2010 you would have $11,519. Calculate the average rate of return. But if you had pulled out at the start of 2001, 2002, and 2008, by the end of 2010 you would have $26,316. Calculate the average rate of return. This is called market timing. Discuss what you think of market timing. #23. T. Rowe Price compared an all stock portfolio from 1985 through 2012 and found that a portfolio of 60% stocks, 30% bonds, and 10% cash returned 9.8% annualized about 93% of an all stock portfolio with 62% of the risk. For $10,000 at the beginning of 1985 through 2012, what was the final total return accumulation for the mixed portfolio and the all stock portfolio? (Ki linger s Personal Finance, 4/2013). What does the term risk mean? See the articles in this course about risk.

9 9 #24. S&P 500 Index funds and similar index funds are popular among investors. They outperform from 70% to 80% of managed funds. The S&P 500 index includes the largest 500 companies according to market capitalization of their stocks. Value investing indexes chose companies by their concrete measures such as profits, sales, and book value. Mr. Arnott back tested his ETF, FTSE RAFI US 1000 (PRF) for the ten years from the end of 1999 through His fund earned 4.75% annualized and the S&P 500 lost.95% per year on average (Scott Burns, Fundamental investing passes test. Denton Record Chronicle, May 2, 2010). Calculate the value of $10,000 invested in each. The advantage of the RAFI index was what percent of the $10,000? There are different kinds of RAFI indexes. #25. Check the calculations in Example 4. What do you think of our 25 year old? What do you think of Obama s budget recommendation. How much is left on the table if returns average 9%? At what age would their savings reach $3.4 million? Has there ever been 40 consecutive years averaging 4.57% inflation or greater? #26. The advantage of money in your pocket: If a person increases the deductible on their home owners insurance policy from $1000 to $10,000, they can save about $1500 per year in premiums (Kiplinger s, 9/2013). At r = 7%, I = 3.24%, and for 30 years, what is the future value of savings starting at $1500 and increasing each year at the rate I? People with skills can manage repairs at a low cost. Buy class four shingles which are hail and wind resistant and you will get a discount on your insurance.

10 10 References in this course: Employer sponsored programs usually involve tax sheltered retirement savings such as Traditional IRAs, and 401Ks. There are around sixty articles in this course related to this article. Comparing Taxed, Tax Deferred, and Tax Sheltered Investing, HiMAP Pull-Out, Consortium 50. Does T M Need a Roth? HiMAP Pull-Out, Consortium 72. The Mathematics of Conversion of a Traditional IRA to a Roth IRA, For a free course in financial mathematics, with emphasis on personal finance, for upper high school and college, see COMAP.com. Register and they will you a password. Simply click on an article in the annotated bibliography, download it, and 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 Unit 6: Solving Financial Formulas for i.

= Solving gives r = For purchasing power, the monthly rate. (1 + r ) 12

= Solving gives r = For purchasing power, the monthly rate. (1 + r ) 12 1 Penny Pinching (Preliminary Version) Floyd Vest, July 2013 Scott s Example: Honey I hocked the car but its worth it. Denton Record Chronicle, Aug. 5, 2013. Scott says that the credit union offered loans