A Mathematical Analysis of the Delay Revision to the U.S. Social Security System. Robert R. MacGregor Rice University MATH 211 Section 2

Transcription

1 A Mathematical Analysis of the Delay Revision to the U.S. Social Security System Robert R. MacGregor Rice University MATH 211 Section 2 November 10, 2002

2 CONTENTS 1 Contents 1 An Overview 2 2 The Development of a Mathematical Model The Differential Equations Solutions to the Differential Equations Finding r such that P 1 (t) > P 2 (t) Derivation of the Catch-up Time T if r = Conclusion 10

3 1 AN OVERVIEW 2 1 An Overview Social Security has long been a subject of debate in modern American society. In the year 2000, confusion over Social Security laws was exacerbated by a modification that allowed working people who have reached full retirement age to delay their social security benefits until a later date. By delaying, they would receive a larger monthly benefit than if they chose to begin receiving Social Security immediately. The obvious question for a working person near retirement age is when to apply for their Social Security benefits. This report aims at answering that question. The profile of an average American who would be concerned with Social Security decisions in the near future was developed. In this report, the analysis is based on a 45 year old American (hereafter referred to as John Doe; note: Jane Doe applies equally well), earning $27,871 per year. 1 Mr. Doe has the choice of applying for his Social Security benefits at the age of 66 and 6 months and receiving $12,312 per year or delaying his benefits until he is 70 years old and receiving $16, To determine which route Mr. Doe should take, a mathematical model was developed to predict the growth of two hypothetical accounts. The first account, P 1 (t), simulates the growth of money if Mr. Doe began depositing his benefits at age 66 and 6 months. The second account P 2 (t) simulates the growth of money if Mr. Doe began receiving his benefits at age 70. Both accounts are assumed to generate income based off a fixed return-oninvestment rate r. The income both accounts accrue is also taxed at some rate ρ, which is the marginal income tax rate (in Mr. Doe s case 15%). 3 Adding even more complexity, a portion denoted by q of the Social Security benefits is taxed along with the income of the account. Using this financial data, a mathematical model was constructed to predict the growth of P 1 (t) and P 2 (t) over time. It is evident that if Mr. Doe applied for his benefits at 66 and 6 months, he would possess more money for some n years than if he delayed his benefits. However, because he would 1 Per-capita income in Texas in 2000 according to the Texas Comptroller s Office. 2 These values were found at the Social Security Administration web site. 3 The marginal income tax rate is the rate at which the next dollar of one s income would be taxed. This is determined by a graduated income tax system that can be found through the Internal Revenue Service.

4 2 THE DEVELOPMENT OF A MATHEMATICAL MODEL 3 be receiving more Social Security benefits annually (A 1 and A 2 represent the annual Social Security benefits), by delaying he would eventually have more money than if applied for benefits immediately, that is, P 1 (t) > P 2 (t) after some time period T. The model was used to determine this time for varying values of return on Mr. Doe s account. When Mr. Doe or, more generally, any American concerned about working after retirement should apply for Social Security benefits is a decision that depends on numerous factors. For instance, financial debts or a terminal illness could force someone to begin receiving benefits immediately. Therefore, this data is only a guideline to be followed, and will not apply equally well in all cases. 2 The Development of a Mathematical Model Before one can go about the process of representing these financial processes mathematically, an appropriate model type must be found. A convenient way of representing compound interest (interestingly, interest is analogous to population growth) is through the use of differential equations. The principal of a differential equation is to first determine the change in something, rather than analyze the thing itself. From these equations, a model of the growth of P 1 (t) and P 2 (t) can be constructed. 2.1 The Differential Equations The general differential equation of both of the accounts, P (t), is given below: benefits {}}{ P (t) = rp (t) + A ρ(rp (t) + qa) }{{}}{{} return taxation (1) The braces show what the sections of the differential equation represent. It can be seen that rp (t) is the interest accrued on the amount of money in the account. A is simply the amount of Social Security benefits being deposited into the account annually. The total amount of taxation is represented by ρ(rp (t)+qa). The taxation depends on the amount of interest the account is accumulating plus some portion q of the Social Security benefits (remember to mind your ρ s and q s!).

5 2 THE DEVELOPMENT OF A MATHEMATICAL MODEL 4 From this differential equation, a pair of equations can be generated to represent P 1 (t) and P 2 (t) separately: P 1(t) = rp 1 t + A 1 ρ(rp 1 (t) + qa 1 ) (2) P 2(t) = rp 2 t + A 2 ρ(rp 2 (t T D ) + qa 2 ) (3) Note that in P 2(t) the time is given by t T D. This is due to the fact that there is a time delay between the two accounts, and is analagous to a phase shift of the second differential equation. These differential equations will be the basis of all further analysis. 2.2 Solutions to the Differential Equations Now that a representation of the change in the accounts has been found, it is possible to find the amount of money in each account at a specific time t. First, it is assumed that at time zero the amount in both accounts is zero, that is, P 1 (0) = 0 and P 2 (0) = 0. Now, to find the general form of Equation(1) one simply uses integration. First, P (t) and P (t) are grouped together: P (t) P (t) = A(1 ρq) (4) Next, an integration factor u defined as e to power of the integral of the coefficient of P (t) is determined: u = e r(1 ρ)dt u = e r(1 ρ)t Multiplying Equation(4) by the integration factor results in the left hand side of the equation resembling the product rule, so that the integral can be easily taken: [P (t)e rt(1 ρ) ] = A[1 ρq]e rt(1 ρ) P (t)e rt(1 ρ) = A[1 ρq] e rt(1 ρ)dt P (t) = A(1 ρq) + Ce rt(1 ρ)

6 2 THE DEVELOPMENT OF A MATHEMATICAL MODEL 5 Now the constant of integration can be found since it is known P (0) = 0: 0 = C = A[1 ρq] + C A[1 ρq] Therefore the general form P (t) and the specific solutions P 1 (t) and P 2 (t) are given by: P (t) = A[1 ρq](ert(1 ρ) 1) P 1 (t) = A 1[1 ρq](e rt(1 ρ) 1) P 2 (t) = A 2[1 ρq](e r(t T D)(1 ρ) 1) 2.3 Finding r such that P 1 (t) > P 2 (t) In determining when to begin receiving benefits it is useful to find the rate of return r max such that P 1 (t) > P 2 (t) for all t given that r > r max. This can be done by defining f(x) to be the ratio of P 2(t) P 1 and requiring f(t) < 1 as (t) t. By factoring the exponents of e we can prepare f(t) so that a limit will result in convenient cancellation: f(t) = P 2(t) P 1 (t) 1 > A 2[e r(t T D)(1 ρ) 1] A 1 [e rt(1 ρ) 1] 1 > A 2[e rt(1 ρ) e rt D(1 ρ) 1] A 1 [e rt(1 ρ) 1] Now distributing and taking the limit as t : ( ) lim 1 = lim A2 e rt(1 ρ) e rtd(1 ρ) A 2 t t A 1 e rt(1 ρ) A 1 (5) (6) (7)

7 2 THE DEVELOPMENT OF A MATHEMATICAL MODEL 6 1 > A 2e rt D(1 ρ) A 1 A 1 > e rt D(1 ρ) A 2 ln( A 1 ) > rt D (1 ρ) A 2 r > 1 T D (1 ρ) ln ( A1 We now define this quantity as r max (note the negative sign is absorbed by the natural logarithm): ( ) 1 r max = T D (1 ρ) ln A2 (8) So it is shown that if r > r max then P 1 (t) > P 2 (t). Practically, this defines some rate of return so that at r max it is certainly a better decision to begin receiving social security benefits immediately, than to postpone them. In the case of Mr. Doe, r max is found to be: r max = ( ) (1 0.15) ln r max A 2 ) A 1 So if Mr. Doe s rate of return is greater than 9.1%, he should definitely begin taking benefits immediately. However, such a high rate of return is common only in the stock market, so if Mr. Doe invests in more conservative ways, his decision is not so clear. Therefore, further analysis is needed. 2.4 Derivation of the Catch-up Time To find a general form for the time T at which P 2 (t) catches up with P 1 (t) the two equations are simply equated with one another and solved for t. P 1 (t) = P 2 (t) A 1 (e rt(1 ρ) 1) = A 2 (e r(t T D)(1 ρ) 1)

8 2 THE DEVELOPMENT OF A MATHEMATICAL MODEL 7 9 x P1 and P2 versus Time for 4% return P1 P2 7 6 Pn(t) (Dolalrs) Time (years) Figure 1: A plot of P 1 (t) and P 2 (t) for r = 0.4 A 1 e rt(1 ρ) A 1 = A 2 e r(t TD)(1 ρ) A 2 e rt(1 ρ) A 1 A 2 = A 1 A 2 e rt D(1 ρ) ( ) A 1 A 2 rt(1 ρ) = ln A 1 A 2 e rt D(1 ρ) ( ) 1 T = ln A 1 A 2 A 1 A 2 e rt D(1 ρ) This represents the elapsed time for P 2 (t) to reach P 1 (t). We can analyze what this time will be for varying values of r. A return of 2% is common for checking and money market accounts, 4% is common for bonds, and 10% is common for stocks. If rɛ{0.2, 0.4, 0.1}, then T ɛ{16.46, 18.91, }. To illustrate this more clearly, the graphs of P 1 (t) and P 2 (t) at rɛ{0.4, 0.1} are given in Figure(1) and Figure(2), respectively. The catch-up time is the value of t at the intersection of P 1 (t) and P 2 (t). We see in a qualitative sense that the catch-up time varies proportionally with r, that is, T r. Therefore, the lower the value of r, the lower the catch-up time, so that it is a greater benefit to Mr. Doe to delay Social Security the lower r is. The less risky Mr. Doe s investment (that is, the

9 2 THE DEVELOPMENT OF A MATHEMATICAL MODEL x 106 P1 and P2 versus Time for 10% return P1 P2 2 Pn(t) (Dolalrs) Time (years) Figure 2: A plot of P 1 (t) and P 2 (t) for r = 0.1 lower the value of r), the lower the catch-up time, and the greater the benefit to Mr. Doe. The life expectancy for Mr. Doe is 32.0 years (36.3 years for Ms. Doe) from age We see that if he chooses to delay his Social Security benefits, he has an estimated 7 years (11.3 years for Ms. Doe) from the time he will begin to receive benefits until his death. Therefore, it is likely that T will not be reached during his lifetime unless, perhaps, if r is very low, so almost certainly he would be best off taking Social Security benefits immediately (unless he wishes to think he will have an abnormally long life). A graph of T versus r given in Figure(3) further illustrates this. It is thereby shown that T increases exponentially with r, reaching an asymptote at r max. 4 Calculated from the mean life expectancy of American individuals. National Vital Statistics Report, 1999

10 2 THE DEVELOPMENT OF A MATHEMATICAL MODEL 9 70 T versus r T (years) return on investment Figure 3: A plot of T versus r 2.5 T if r = 0 To further show the relationship of T to r, we can solve Equation(1) assuming r = 0. This yields: P (t) = A ρqa P (t) = A(1 ρq)dt P (t) = A(1 ρq)t + C since P (0) = 0 the integration constant will also be zero, leaving: P (t) = A(1 ρq)t (9) setting P 1 (t) = P 2 (t) and solving for T algebraically: A 1 (1 ρq)t = A 2 (1 ρq)(t T D )

11 3 CONCLUSION 10 T = T DA 2 A 2 A 1 So in Mr. Doe s case if r = 0 then T Therefore, the least amount of time P 2 (t) could catch-up with P 1 (t) is approximate years. Given Mr. Doe s estimated life expectancy of 7 years (or Ms. Doe s of 11.3) it would seem prudent for Mr. Doe to begin applying for Social Security as soon as possible. 3 Conclusion A mathematical model was constructed using differential equations to predict the growth of two hypothetical accounts, thereby determining the catch-up time between the two for varying real-world return-on-investment rates. It was found that for an American earning $27,871 per year upon retirement, the catch-up time would be no less than years. Since the life expectancy of most Americans is much less than that, it can be concluded that the most logical decision for an average American would be to apply for Social Security benefits as soon as possible. However, the personal factors for such a decision are numerous, and many people will probably delay receiving their benefits. Such decisions are not always logical, but hopefully this has provided some concrete data to better inform those people affected.