US Government Thrift Savings Plan (TSP) Floyd Vest, Jan. 2013 Preliminary Version Millions of civilian government employees and US military participate in the US Government Thrift Savings Plan (TSP). (See www.tsp.gov.) The TSP is managed by the US Federal Government (FRTIB) Board. To understand the program, a person would carefully read the details of the program, and verify that they understand the numbers. This article is an example of using financial mathematics to verify understanding of the numbers. In this article, we will assume knowledge of the formulas used to examine information from the tsp website. For a review of the mathematics with derivations of formulas, see Unit 1, The Basics of Mathematics of Finance in the course. Earnings Potential for Your TSP Account. In the TSP website, click on Plan Participation, then About TSP. Click Benefits of Participation and under it, click Earnings Potential of Your TSP Account. You will see the table of earnings growth including $54,699 after ten years. The table assumes an annual salary of $40,000, employee and agency contributions of 5% of salary each, and a 6% average annual rate of return. We should do the calculations to see if we understand the $54,699. Assume the annual contribution is 2(.05)(40,000) = $4000. Use the formula 10 (1.06) 1 (1) S1 4000 to get $52,723.18 which is too small. We could assume.06 beginning of year savings and calculate (1+.06)(52,723.18) = $55,886.57 which is too large. To continue, we could average the above numbers, we could consider monthly contributions at 6%, or monthly contributions based on 6% effective. We could try contributions of 5% plus 5% of 95%. (See the Exercises.) But, even if we got the target value of $54,699, it wouldn t prove that we have the correct interpretation. By these trials, the closest we came was $54,625.90. For those readers who are using a financial calculator, see some sample code in the Side Bar Notes. In any case, the TPS number seems reasonable and we have an understanding of it (providing we understands the mathematics). Under Benefits of Participation, try How Much Will My Savings Grow. In the form, we clicked FERS, entered $4000 = 10% of $40,000 yearly income, at 6% compounded annually for ten years. We got the contributions grew to $82,049. The above discussed Formula 1, with table, was giving $54,699 for ten years. There must be here a government match. We went back and checked CSRS, and got $54,699 which is the same as in the above discussion. We will try to estimate the government match included in the $82,049 given above. We calculate 10 (1.06) 1 (2) 82,049 = R and solve with R = $6224.89. This gives.06 1
6224.89 4000 = $2224.89 match which is 5.56% of the $40,000 salary. This 5.56% may approximate a 5% match. We could make several calculations attempting to get the 5%. The calculation table says The FERS calculation...includes matching contributions. Annuity Calculator. Under Planning & Tools, click Calculators, then Annuity Calculator. We select Single life, Begin age 65, $100,000 purchase, Level, No additional features, then Calculate. The calculator gives $513 per month for life, or about (12)513 = $6156 per year. We ask the question, if someone lives to age 95, what return do they make on their money? Then solve 1 (1 i) (3) 100,000 = 6156 which requires a financial calculator. The return i on the investment i = 4.52%. We wonder if this could be improved upon. (See the Side Bar Notes, References, and Exercises.) We select Increasing payments in the Annuity Calculator, and are informed that payments increase at 3% per year starting at $351 per month, and after ten years the payments are $458 per month. To check this we calculate (3.1) 351(1+.03 ) 10 = $472. So we try 351(1+.03 ) 9 = $458 as reported by the calculator. Again if someone lives from age 65 to age 95, what is the rate of return r on the investment? We solve 1 (1 y) r.03 (3.2) 100000 12(351) where y. Using a financial calculator, y 1.03 r.03 we get y = 1.58% =.0158, and.0158 = with r = 4.63%. Again we wonder if we 1.03 could do better. (See the Side Bar Notes and References.) How Much Should I Save? Ballpark Estimate. Click Interactive Ballpark. We put in the form: Age 64, $40,000, 65, 100%, Age of death 95, Inflation = 3.2%, 6% rate of return before and after retirement. On #10 select No does not include Social Security, and on #15, Savings $0, and 0 on #7, #13, #14. Under Social Security we put in $1000 per month. Then press Calculate. The Calculator informs that Have enough to replace 31% of final wages. Need to save 1444% of salary. Need to save $577,480. We check to see if this figure is reasonable: 40,000 12,000 = $28,000 per year from funds. 1 (1 y).06.032 (4) P = 28,000 where y =. Solving gives P = $569,731. If y 1.032 payments are at the beginning of the year we use 569,731(1+y) = $585,189. Averaging 569,731 585,189 the two gives = $577,460 which is close the calculator s $577,480. 2 So the Ballpark Estimate is reasonable, providing we understand the mathematics. Contributions Comparison Calculator, for Roth TSP and Traditional TSP. We put in: How many year to start of retirement, 1 year. Current salary $40,000. Contribution 2
10%. Tax rate 15%. 30 years of retirement. 6% return. Retirement income tax rate 15%. Pay frequency monthly. Then Calculate. The Calculator responds: Traditional Roth Pay check 283.33 per check 283.33 per check impact 3,400 per year 3,400 per year Contribution percentage 10% 8.5% Contribution 333.33 per check 283.33 per check amount 4,000 per year 3,400 per year This table has to do with Traditional TSP contributions being before income tax and Roth contributions being after tax. The $4,000 per year before tax is equivalent to $3,400 after tax since the taxes on the $4,000 are 15% of $4,000 = $600, and $4,000 - $600 = $3,400. The same is true in per check contributions, and the Contribution percentage. In the Calculator under Account Balance: Traditional TSP $4,111.85 Roth TSP $3,495.08 The tax is to be paid on withdrawals from a Traditional TSP but no tax on withdrawals from a Roth TSP. So the two account balances are tax equivalent since 4,111.85(1-.15) = $3,495.08. Other things being equal, in terms of tax equivalence, the Roth and Traditional pay the same. The after tax annual income for both the Traditional and Roth is reported as $251.46 per year. To check this figure, we calculate for the Roth 1 (1.06) (5) P 251.46 to get P = $3461.30 which differs by $33.78 from.06 the $3,495.08 balance for the Roth. So the Calculator s figures are understood and reasonable (providing we understand the mathematics). The L-2010 Lifecycle Fund. Under Investment Funds, click on Retired Funds and you the get the L 2010 Lifecycle Fund which as it approached maturity, change its allocation of G, F, C, S, and I Funds to predominately more conservative G Fund and I Fund. You notice that it had a loss of 10.53% in 2008. We wonder how well it recovered. (6) Calculate (1-.1053)(1+.1003)(1+.0565) = (1 +R ) 3 and solve for the total return R for the last three years to get R = 1.32% which indicates a recovery and 1.32% average gain (Total Return). Also in 2008, the C Fund had a loss of 37%. What percentage gain is required to recover the loss? (7) Calculate (1 -.37)(1 + g) = 1 and get the required gain g = 58.7%. 3
The L 2020 Lifecycle Fund. Under Investment Funds, click on Fund Options and then the Lifecycle Fund L 2020. You can read the annual performance chart and study the interactive chart to see the changes in allocation. You will notice that for the last five years, the average return was low. A large portion of the fund was in the C Fund (simulates the S&P 500 Index of Stocks). See the C Fund for the difficult years beginning in 2000 and notice a big loss in 2008. You see a graph of Growth of $100 since inception and the growth of inflation from 8/05 to 12/11 (at the time of this writing). You can check the average inflation rate during this approximately 7.5 years by solving (8) 100(1+I ) 7.5 = 115 to get average inflation I = 1.9% from 8/05 to 12/11. This is low compared the long term average of 3.2%. (See www.usinflationcalculator.com.) You can calculate the average total return for the fund by solving (9) 100(1 + R ) 7.5 = 128 to get R = 3.35%. Compare this to the total return of the C Fund during these years. Expense Ratios. The expense ratios on the TSP funds range from.025% to.024%. The average expense ratios for mutual funds is about 1.25% and for some insurance products 3%. We ask the question, What is the effect on earnings of expense ratios? Consider investing $50,000 from age 30 to age 70 at 6% with different expense ratios. At the expense ratio of.025% =.00025 we have (10) 50,000[(1-.00025)(1+.06) ] 40 = $509,168. At 1.25%, we have 50,000[(1-.0125)(1+.06) ] 40 = $310,949. At 3%, we have 50,000[(1-.03)(1+.06) ] 40 = $152,081. The.025% expense ratio earns 235% more than the 3% expense ratio. Try the approximation formula 50,000(1+.06-e) 40 where e is the expense ratio. Side Bar Notes: Lifetime annuities called an immediate annuity. To shop for annuities, see immediateannuities.com which includes 16 of the best insurance companies. For the same $1000, some insurance companies guarantee 65% more per month than others. The strength of insurance companies is rated by A. M. Best, Standard and Poors, Duff and Phelps, and Weis. 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 for the first year. Smart Money magazine, May, 2010 reports for a 65 year old, a $300,000 lifetime annuity with a 3% annual increase, pays $1434 per month the first year. How do these figures compare with those for TSP annuities? History of S&P 500 portfolio of stocks. See Wikipedia.org, Search S&P 500. For the lost decade from 2000 to 2008, calculate the total return R for the S&P 500. 4
The Federal Thrift Savings Plan (TSP) was authorized by congress in 1986 and military personnel were admitted in 2001. How long will you live? In Money magazine, Jan./Feb. 2011 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). 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.) The L Income Fund. If interest rates rise, the L Income Fund can lose money. (See the References.) The G Fund s assets are managed by the Federal Reserve Retirement Thrift Investment Board. The G Fund buys nonmarketable U. S. Treasury securities that are guaranteed by the U. S. Government. This means that the G Fund will not lose money. 5
Exercises: Show you work. Label numbers, variables, and answers. Give formulas. #1. For a history of a S&P 500 portfolio of stocks, (See the TSP C fund.) go to http://www.measuringworth.com (which gives a long term Total Return table which simulates the S&P 500 Index of stocks), click on Savings Growth-US $, and do the following: (a) In Amount of Initial Savings, put in 1 for $1. In the other two blanks, put in 1900 and 2000. Select Appreciation of stock market investment. Deselect the others. Click Calculate. The site reports $19,600.27. Calculate the average total return R. Were dividends reinvested? (b) Look below for the The stock assets used..., click on The Annual Standard and Poors.... Select all three options and put in 1900 to 2000, Retrieve. Print out the given four lists. Report the patterns over time in all three columns. (c) From a 1929 high, when did the portfolio recover its former high, how long? When did the total return recover its former high? Explain the difference. (d) When did the S&P 500 Index start? How did the site find earlier data? (e) Calculate the average appreciation of stocks in the portfolio beginning in 1958. Calculate the average total return R. (f) Examine the available data and report results that are of interest to you and why. #2. 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). Forty percent of American families have $500 or less in savings (Fox News, 10-20-12). 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 (www.tiaa-cref.org/institute). For more troubles of pre-retirees, see Kiplinger s Personal Finance, p. 57, Jan. 2013, and Money magazine Dec. 2012, page 126. Project: Combine the probability of living 30 years (or n years) with the probability of the money lasting 30 years (or x years) to calculate the probability of running out of money before death. What assumptions are required? People in the top half of wage earners live 4.5 years longer than people in the bottom half (Scott Burnes, Denton Record Chronicle, Dec. 16, 2012). #3. On tsp.gov, under Investment Funds, click Retired Funds. (a) Calculate the average total return R for the six year given for the L 2010 Fund. (b) Notice a 10.53% loss. What percentage gain would be required to recover the loss? 6
#4. (a) As in Expense Ratios in the above discussion, calculate the effect of expense ratios at an 8% return on investments. Compare with the effect at 6%. (b) Derive the formula being used. #5. Use Formulas 4 and 8 to calculate the funds needed for yearly increasing retirement withdrawals at 3.2% inflation from age 65 to age 95 with an interest rate of 6%. Consider a cost of living at age 30 of $50,000 per year. What is the cost of living at age 65? #6. (a) If social security is $12,000 per year at the time of age 30, use 3.2% inflation to estimate its benefit at age 65. (b) Calculate the present value of social security benefits from age 65 to age 95 at 6%. (c) Subtract this number from the total funds needed calculated in #5. #7. Use Formula 2 to calculate how much you need to save each year at 8% from age 30 to age 65 to obtain $2,328,532. Do any further calculations you wish such as using retirement saving funds on hand at age 30, and other retirement benefits such as a government pension. (See the References.) #8. (a) Go to usinflationcalculator.com and calculate the average inflation rate from 1950 to 1990. Calculate the recent average inflation rate from 2000. (b) How long does it take costs to double at 5% inflation? #9. Compare the annual lifetime pay for a 65 year old from the TSP Annuity Calculator and the examples given in Lifetime Annuities, page 7. #10. From age 65 to 95, what is the rate of return on the investment of $100,000 in the 3% increasing annuity starting at $6,470. 7
Answers: #3. (a) For the L 2010 Fund, for the six years, for the total return R, solve (1+.0299)(1+.1109)(1+.0640)(1-.1053)(1+.103)(1+.0565) = (1+R ) 6 to get R = 4.05%. (b) Let g be the percent gain required to recover the 10.53% loss. Solve (1-.1053)(1+g) = 1 to get g = 11.77%. #4. (a) 50,000[(1-.00025)(1+.08) ] 40 = $1,075,417. 50,000[(1-.0124)(1+.08) ] 40 = $656,757. 50,000[(1-.03)(1+.08) ] 40 = $321,210. (b) Derivation: Let e = expense ratio. r = rate of return before expenses. P = the amount invested for n years, with S = the total accumulation. For Year 1, after earning we have P(1+r) from which expenses at the rate e is subtracted to give P(1+r)(1-e) at the end of year one. For Year 2, we invest P(1+r)(1-e) at the rate r so at end of Year 2 we have [P(1+r)(1-e)](1+r) and subtracting expenses gives [P(1+r)(1-e)](1+r)(1-e) = P[(1+r)(1-e) ] 2 at the end of Year 2. At the end of Year n we have P[(1+r)(1-e) ] n. #5. The cost of living at age 65 = 50,000(1 +.032 ) 35 = $150,577 per year. 1 (1 y) To calculate the needed retirement funds, P = 150,577 where y.06.032 y which gives P = $3,063,872. Notice that the first withdrawal is about 5% 1.032 of the retirement funds. According to several long term historical studies, even with withdrawals starting at 4%, there is a 5% or greater chance of running out of money before 30 years. Of course, having social security, retirement programs, and lifetime annuities helps preserve funds. (See the article The 4% Rule for Retirement Withdrawals to appear in the course.) According to the 4% rule, how much is required to fund withdrawals increasing at 3.2% per year starting at $150,577? (A popular radio and TV financial planner in his book, 2007, suggests 12% return on investments and retirement withdrawals beginning at 8%. What do you think of that?) #6. Using 3.2% inflation, the social security benefit at age 65 = 12,000(1+.032 ) 35 = $36,139 per year. The present value of social security benefits from age 65 to age 95, at 1 (1 y).06.032 age 65 = 36,139 where y. This gives $735,340. Subtracting y 1.032 from the total retirement funds needed we have $3,063,872 735,340 = $2,328,532. 8
35 (1.08) 1 #7. Solving 2,328,532 = R we get R = $13,513 per year for 35 years..08 #8. For 1950 to 1990, put in 1950 and 1990, click Calculate. You get For the $20 item, the same item would cost $108.46. Solve 20(1+I ) 40 = 108.46 to get I = 4.3%. Notice that the 1990 price is five times larger than the 1950 price. Apply this to an example for the next 40 years from your time. (b) We solve 20(1 +.05) n = 40. Using log 2 logs we have (1+.05 ) n = 2. n log 1.05 = log 2. n = = 14.2 years to double. log1.05 1 (1 y) #10. To determine the rate of return r, we solve 100,000 = 6,470 where y r.03 y =, and get I% = 4.95%. TI83/84 code: To access the TVM Solver: 2 nd 1.03 Finance Enter. For PV enter (-) 100000, for PMT enter 6470. For N, enter 30. For I% enter 0. Then put the cursor on I% and press Alpha Solve and read I% = 4.95%. This gives r = 7.73%. 9
References in this course, articles which include derivation of formulas: For long term financial planning, see Taking the Long View of Life and The Mathematics of Financial and Social Responsibility. For Traditional and Roth IRAs see Comparing Taxed, Tax Deferred, and Tax Sheltered Investing, Does T M Need a Roth? and The Mathematics of Conversion of a Traditional IRA to a Roth IRA. For the effects of inflation, see Living and Investing with Inflation, Fisher s Effect, and other articles. For interest rate risk on a bond funds, see The Mathematics of Bond Pricing and Interest Rate Risk. 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 http://www.comap.com/floydvest/course/index.html. 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 Unit 6: Solving Annuity Formulas for Interest Rate, Iteration Programs Unit 7: Related Topics Unit 8: More Advanced or Technical Articles There are, at the time of this writing, another 30 articles ready to mount. 10