Term Life Insurance and How to Calculate Savings

Transcription

1 The Mathematics of Life Insurance Decisions Floyd Vest, November 20 (Preliminary Version) There are different types of life insurance such as term life insurance, whole life insurance, and universal life insurance. Young people often buy term life insurance for ten or twenty years until they have acquired adequate investments to provide for family needs. Different types of term life insurance are straight term for a fixed number of years, renewable term can be renewed, and convertible term can be converted to whole life. Whole life insurance provides insurance for the entire life. Cash value increases with age. According to Gitman, the actual cash value earnings are far below what could be earned on alternative investments (Gitman, p. 95). Some financial advisors recommend purchasing term life insurance, and investing in other areas the additional premiums for the whole life policy. Consider the following example of a $50,000, ten year term policy and a whole life policy. Example. The following table gives the premiums for the whole life policy and the term policy. It gives the accumulating cash value for the whole life policy and the annual savings with term insurance. () (2) (3) (4) (5) Annual Premiums Accumulated Cash Value Savings Year Whole Life Term of Whole Life with Term , , , , , , Table. Premiums, Cash Value, and Savings with Term. (This table was adapted from a textbook by changing the numbers and supplying additional columns.) In Table, premium values are calculated from the beginning of each year. At the end of the tenth year, the Accumulated Cash Value of the Whole Life policy is $32,592. If the Savings are invested at 8%, the compounded amount is $40, at the end of the tenth year. The advantage of the term insurance and investments is $ The whole life cash value accumulated 4.2% on the annual Savings with Term. The above interest rate calculations were done in about 988 and don t reflect interest rates at

2 the time of writing this article. See the Side Bar Notes and Exercises for calculation methods, TI84 code, and more recent interest rates. Investing the Savings with Term at 8%. For the TI84, write () Y = 2893 x x x. To evaluate Y at 8%, enter.08 for x. Code and commentary: From the home screen: Math Vars > Enter You see Y on the home screen. Write (.08 ) Enter You see for a compounded amount at 8% of $40, To calculate the rate of return on Savings with Term to the cash value accumulation of $32,592, let (2) Y = 2893 x x x Use the TI84 Solver to solve for x with Y ( ) 0 x. Then x = + i and i = 4.2%: Code and commentary: Math Select 0 for Solver You see on the home screen: Equation Solver eqn:0= Press Vars Y-Vars Enter Select Y and press Enter. The expression is pasted to the equation solver. You see eqn:0= Y Press and you see x= Enter an estimate for x such as.04. Then Alpha Solve You see x =.042 so i = 4.2%. The whole life cash value accumulation paid 4.2% on the Savings with Term. Investing the Savings with Term in an IRA. The savings could be invested in a Roth IRA in which the family invests after tax dollars and pays no income taxes at withdrawal, or in a Traditional IRA in which they invest pretax dollars, and the money is taxed at withdrawal. Consider investing a pretax equivalent in a Traditional IRA. Let A = after tax dollars and B = the equivalent before tax dollars. For the 25% marginal A income tax rate, B -.25B = A. B =. (See the 200 IRS Tax Rate Schedule in the.75 Side Bar Notes.) Using Formula above gives Y = x x... x = = $54,8.2 at 8%, before taxes. (Life Insurance Cash Value is ordinarily not taxed. Tobias, Andrew, Managing Your Money, Brady, 989, p.5-9) (See the articles in this course on investing in IRAs and tax deferred investments.) How much to save each year to replace the $50,000. Consider investing at 8% at the end of each year for ten years. The formula is 0 (.08) 50,000 = R and R = $0, per year..08 If inflation averages 3% per year and income keeps up with inflation, a person could start out saving less and increasing each year for ten years. The ASAVE formula is N ( y) N r I (3) D Z ( I) where y. Let r =.08, I =.03, y I 2

3 Z = 50,000, and N = 0. Solving for D gives $ or about $ per month increasing each year at 3%. If a family can t save this much, they could buy 5 or 20 years of term life insurance. If you have large enough policy you can sell it. (See the Exercises. See the articles in this course on long term financial planning for derivations of formulas.) Side Bar Notes: Historical inflation. See usinflationcalculator.com. There have been periods of 40 consecutive years of inflation at 4.5% or higher. From Aetna Insurance. The term life insurance portion of your group benefits may not be an economical choice. Alternative term life insurance can save you over 60%, and coverage will not terminate if you retire or separate from service. Your current group rates are based on age only. If you are healthy and non-smoking you can save thousands of dollars on an individual policy because you get a discount. Your rates depend on health, non-smoking, and sex. A very healthy non-smoking female pays the same rate as a very healthy smoking male. Social Security Survivor Benefits, when considering life insurance: Millions of families are receiving Social Security survivor benefits. Within a family, surviving children and spouse may receive survivor benefits. A child may receive up to 75 percent of the deceased parent s basic Social Security Benefit. The family maximum could be from 50 to 80 percent of the parent s full benefit amount. If the family benefit exceeds this amount, each person s benefit is reduced proportionately (except the parent s). (See ssa.gov.) Insurance premiums vary widely with the companies and marketing. Shop rates on the internet using a quote service. Write a report on the extent of differences in insurance rates and share it with your class. Start with the internet and with door-to-door salesmen. You will find different fee schedules for the same coverage. The way to compare them is to calculate the present value. See the TI84 manual pages4-8 and -3. Example: Consider the premium stream at the beginning of each year -2000, -500, -400, Use npv(interest rate, CF0, CFList). Put -500, -400, -500 in a List. Code on the home screen: 2 nd { (-)500, (-)400, (-)500 2 nd } Sto 2 nd List 2 Enter for L 0 Use any List name. You see {-500, -400, -500} L0 To get npv(, 2 nd Finance 7 You see npv( Write 4, for 4%. (-)2000, for CF0. 2 nd List 2 for L 0 Enter. You see as the present value at 4%. Write the formula and do the calculations to check if the present value at 4% is $ Interest-Adjusted Cost Indexes for permanent life insurance, $00,000 issue, a male, age 35, based on premiums, cash value, and dividends. The lower the index the better. Index values ranged from 8.0 to The arithmetic mean was 3.56 for twenty largest insurance companies. (Martin, John D., Personal Financial Management, McGraw Hill, 980) 3

4 Are you underestimating your life span? For a 65 year old, they have a 58% chance of living to age 85, a 38% chance of living to age 90, and a 9% chance of living to age 95. (TIAA Participant, Nov. 990) It was recently reported that in the last fifteen years, the longevity of 65 year olds has increased by five years (Nov. 20). In 20, there are triple the number of 90 year olds than three decades ago. The National Institute of Aging said personal savings can be a problem if people don t plan for the longer life. For the 90 year old, living expenses can be greater. Among 95 year olds, 9 per cent are disabled. Of 65 year olds, half expect to live to age 90. Ninety years olds often live busy lives driving, shopping, visiting, and volunteering. (Denton Record Chronicle, Nov. 8, 20) If for a 65 year old, they have a 9% chance of living to age 95, for a 65 year old couple, what is the probability that one or both will live to age 95, to age 00? Disability Insurance verses Life Insurance. You are probably more likely to become disabled than to die. Disability insurance is sold to provide five years or less in earnings replacement, assuming that after a few years of disability, a person would qualify for Social Security disability benefits. (See ssa.gov.) A few years of living expenses on hand would replace disability insurance. Ratings of Life Insurance Companies. The insurance companies are rated by A. M. Best, Standard & Poor s, Moody s Investors Service, Duff & Phelps, and Weiss Research. You can find the latest Standard & Poor s ratings at Use Insurance Company Ratings Lookup. AAA is very strong, then AA, A, BBB is good, and so on. You can find ratings and reviews of products in Best s Reviews in your local public or college library. Consider Term Insurance Comparison, in the December 99 issue and perhaps more recent. You will be informed of the wide range of prices. If you find a good website, let us know. Floyd Vest, at info@comap.com. The English teacher who became a millionaire by age 40. His method: invest as much as you can in low cost index Bond and Stock funds with an international flavor. Rule No. : Spend Like You Want to Grow Rich, not like you are one of the poor phony rich who spend like they are rich. (Denton Record Chronicle, Nov.3, 20. See Scott Burns.com for the Couch Potato portfolio.) (See Millionaire Teacher. You Should Have Learned in School, 80 pages, multiple good research citations, $.0 used at amazon.com.) 200 Income Tax Rate Schedule, Married filing jointly If your taxable income is But not over The tax is of the amount over $0 6,750 0% 0 6,750 68,000, % 6,750 68,000 37, % 68,000 37, ,250 26, % 37,300 The effect of government economic policy. How are economic growth, inflation, and interests rates affected by government policy such as government spending, tax rate changes, or changes in the fed funds interest rates? Two American professors, Thomas 4

5 Sargent and Christopher Sims won the Nobel economics prize for research that sheds light on the cause-and-effect between economic policy and growth, inflation, and interest rates, ie: the interest rates insurance companies pay, mortgage and loan interest rates, interests rates on savings such as CDs and bonds, bond prices, stock prices, and how households and companies adjust their expectations. It is not without exaggeration that Professors Sargent s and Sim s mathematical methods are used daily in all central banks in the developed world. (Denton Record Chronicle, Oct., 20) Exercises:. (a) If a family needed to save $00,000 in ten years at 4% how much a year would they need to save each year. (b) With savings increasing each year at the rate of inflation of 3%, how much should they save the first, second, and last year? (c) How much at 6%? (d) Where could they invest their money for ten years to average 6%? 2. In the discussion of Table, 8% paid $40, and 4.2% paid $32,592 in ten years. This may not be as much difference in accumulations as expected from the difference between 8% and 4.2%. (a) The 4.2% amount is what percent less than the 8% amount? (b) Calculate the value in ten years of a single payment of $00,000 at 8% and at 4.2%. The 4.2% amount is what percent less than the 8% amount? (c) Try $2500 per year savings at 8% and 4.2% and compare. (d) Calculate the value in ten years of a single deposit of $2500 at 8% and 4.2% and compare. (e) What generalizations can you reach? 3. For families, the reason for accumulating financial assets and life insurance is for financial security and long term financial planning for retirement. Consider the following problem. Fill in the blanks on other paper. Show the meaning of the variables and the numbers substituted in the formulas. Let us assume that a 25 year old married couple needs $40,000 a year to live in 996 dollars and they both expect to retire at age 70. They estimate that inflation will average 4.5% a year. (See the Side Bar Notes for history of inflation.) The husband has a retirement program (EMPH) which pays $30,000 a year in 996 dollars with a COLA (Annual Cost of Living Adjustment) of %. The wife s retirement (EMPW) will be $20,000 a year in 996 dollars with a % COLA. The following calculations are based on the assumption that benefits and salaries increase at the rate of inflation during working years. They are not planning to have Social Security. (See the Social Security Offset Rule for certain government employees, at ssa.gov.) Both retirement programs last as long as either lives. With inflation at 4.5%, retirement living expenses would rise at 4.5% a year. Their supplemental retirement savings are expected to earn 7% a year. They assume the husband will live to age 85 and the wife will live to age 95. 5

6 The cost of living at age 70 would be. The cost of living at age 95 would be. We can calculate the total retirement assets P based on $40,000 per year at age 25, needed to provide increasing withdrawals at the rate of inflation from retirement age, to wife s age 95, to be P =. We can calculate, at retirement age, the present values EMPH and EMPW of the retirement programs for as long as either shall live to be EMPH = and EMPW =. Then we ask the amount Z in inflated dollars they will need at retirement to supplement the above benefits and the $2000 savings currently on hand. This calculation gives Z =. To calculate the needed savings from age 25 to retirement to accomplish these goals, we use the ASAVE formula for an increasing annual savings program, to get D = which is to be. For a level savings program, we get J = which is to be. (See the following articles in this course for the formulas and their derivations: Taking the Long View of Life, and The Mathematics of Financial and Social Responsibility. ) (This question was on a test given in about 990 in a financial mathematics course taught by the author in the Mathematics Department, University of North Texas.) 4. By investing regularly, the family is participating in Dollar Cost Averaging. Let s assume that they invest $00 in a mutual fund every six months. In the first six months, they bought shares at $0, and bought 00 shares. In the second six months, they bought shares at $20, and bought 50 shares. Then later share prices increased to $5. (a) How much have they invested? (b) How many shares do they hold? (c) What is their account value at $5 per share? (d) What is their profit? (e) What is the average share price? (f) What is the average share cost? 5. (a) A problem for the TI84: For security, families make investments. If they had a $00 investment that paid.085 monthly (8.5% per year paid to them monthly) and 2 electronically invested the monthly interest in a savings account paying 7.75% compounded monthly, what would be the end-of-year gain over $00, and thus giving the APY? (b) What is the APY for 8.5% compounded monthly? 6. Prove that in equivalent after tax terms, the Roth and Traditional IRAs give the same yield but the Roth IRA allows a larger investment. Each has its advantages. See IRS Publication

7 Answers to Exercises:. (a) At 4%, for end of year savings, the required savings is $8, per year. (b) The end of first year saving should be $ The second The last should be (+.03 ) 9. (d) After building an emergency fund of six months living expenses, For long term investing they could see mutual fund families on the internet. Consider a balanced fund of stocks and bonds. Research loads, expense ratios, standard deviation, and earnings history. 2. (a) $40, ,592 = is 9.7% of 40, The 4.2% amount is 9.7% less than the 8% amount. (b) In ten years at 8%, the future value of $00,000 is $25, At 4.2% it is $50, The 4.2% amount is 30.% less than the 8% amount. (c) Saving $2500 per year, the 4.2% amount is 6.3% less than the 8% amount. 3. The cost of living at age 70 would be 40,000(+.045 ) 45 = $289, The cost of living at age 95 would be $87, per year. We can calculate the total retirement assets P (based on the $ per year ) to provide increasing withdrawals at the rate of inflation from retirement age to wife s age 95, to be P = $5,408, We can calculate, at retirement age, the present values EMPH and EMPW of the retirement programs for as long as either shall live to be EMPH = $2,795, and EMPW = $,863, Then we ask the amount Z in inflated dollars they will need at retirement to supplement the above benefits and the $2000 savings currently on hand. This calculation gives Z = P EMPH EMPW 2000( +.07 ) 45 = $707,030. To calculate the needed savings from age 25 to retirement to accomplish these goals, we use the ASAVE formula for an increasing annual savings program, to get D = $285 which is to be increased each year at 4.5%. For a level savings program, we get J = $ which is to be saved each year. 4. (a) $2000 (b) 50 shares (c) 50 5 = $2250 (d) Profit = $250. (e) Average share price = = $5 (f) Average share cost = = $3.33 Notice that the harmonic mean $3.33 is less than the arithmetic mean $5. By investing the same amount regularly, the family is participating in dollar cost averaging. They buy more shares when prices are down and less when prices are up. There is statistical evidence that time diversification (dollar cost averaging) reduces risk and improves total return. Other kinds of diversification are helpful. (See the article in this course entitled Evaluating Investments: irr, mirr, npv, AM, TR, and HM for mathematical proofs and more examples.) 5. (a) On the TI84: = PMT, = I%, 2 = N, solving, FV = The gain = $ APY = 8.446% (b) The APY for 8.5% compounded monthly is 8.46%. Write the mathematical derivations and formulas to explain these calculations. 7

8 References Gitman, Lawrence J. and Michael D. Joehnk, Fundamentals of Investing, Second Edition, Harper & Row Pub., 984. Vest, Floyd, Mortality and Money, Mathematics and Life Insurance, HiMAP Pull-Out, Consortium, Summer 993 (In this course, contains informative Side Bar Notes on buying life insurance.). Winger, Bernard J. and Ralph R. Frasca, Investment Introduction to Analysis and Planning, Merrill, 988. For more on insurance companies, see Investing in Tax Deferred Variable Annuities and other articles in this course. Teachers Notes For a free download course in financial mathematics, including the basic of mathematics of finance and for upper high school and undergraduate college, see COMAP.com. Register and they will you a password. Click on an article, download it, and teach it. Unit : 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. See several articles in this course related to life insurance. For a copy of the TI 84 manual, see 8