By now, many of you may have heard about the country’s largest city to go bankrupt: Stockton, California. Although there are many reasons for the city’s bankruptcy (e.g., from the *New York Times*: “falling housing prices, foreclosures, … retiree pensions and hefty price tags for buildings”), skyrocketing pension costs are one of the primary ones. In this article, I would like to try and provide evidence as to why I believe the vast majority of pension funds will become underfunded (as if they aren’t already), and where we can expect the underfunding percentage to stabilize. Since my background is in engineering and not finance, I will attempt to do this with some simple mathematical models, as opposed to the usual financial models, which are somewhat difficult for the layman to understand (due to their use of specialized vocabulary, among other things).

The City of Austin Employees’ Retirement System (COAERS) will be the focus of this analysis. Please note that this analysis can be extended to other pension funds as well (with minor modifications). In my opinion, the primary difficulty encountered in designing a sustainable pension fund lies in predicting the future. If one could predict the future, one could figure out exactly how much money one would need to put into the pension fund and at what time. However, the future is inherently unpredictable. Therefore, when designing a pension fund, one should take this unpredictability into account by using conservative estimates for investment performance. Unfortunately, this has not been done in the past (for a variety of reasons, one being that overestimating investment performance enables lower payments into the pension fund). As a result, many pension funds today are underfunded.

In the case of COAERS, it is quite easy to figure out how much money the pension fund is required to pay out to individuals in retirement. The formula is straightforward and consists of the following variables:

1. The amount of money required to be paid out to individuals in retirement, M_needed

2. The wage percentage multiplier, P

3. The number of years worked, Y_worked

4. The final wage, W_final

5. The number of years retirement is drawn, Y_needed

The wage percentage multiplier, P, is an important variable in determining what percentage of the retiree’s final wage, W_final, the retiree will receive at retirement. Multiplying the wage percentage multiplier, P, by the number of years worked, Y_worked, yields the percentage of the retiree’s final wage that the retiree will receive annually during retirement. For COAERS, the wage percentage multiplier is set at 3% (historically, it was different, see pages 127-128 of this report). It is worth noting that the general trend over time has been to increase the percentage multiplier, thereby increasing the annual payout the retiree receives during retirement — in 1941, it was 1.125% — by 2002, it was 3% (many of the changes made to the percentage multiplier applied retroactively). Thus, if an employee works for 20 years, upon retirement, that retiree will receive 60% of their final wage, W_final, annually for the remainder of their life. The table below shows some example numbers:

Wage Percentage Multiplier, P, [%] | Number of Years Worked, Y_worked, [years] | Percentage of Final Wage Received Annually Upon Retirement, [%] |

3% | 20 years | 60% |

3% | 25 years | 75% |

3% | 30 years | 90% |

The way in which the final wage, W_final, is calculated varies somewhat for each pension fund. In the case of COAERS, the final wage is the “average salary of the highest 36 months of the last 10 years of service.” Therefore, it is important to account for wage raises in the analysis that follows.

The number of years retirement is drawn can be calculated by differencing the retirement age and the life expectancy, which have median values of roughly 62 and 78. Please note that the retirement age for COAERS is used here; the retirement age for Social Security is 65.

With these variables properly defined, we can now calculate the amount of money required to be paid out to the individual in retirement. It should be noted that this money is paid out over the duration of the individual’s retirement and not in a lump sum at the beginning; this is an important note to make since the remaining money in the pension fund continues to accrue gains while the individual is in retirement. The amount of money needed to be paid out to the individual in retirement is:

M_needed = P * Y_worked * W_final * Y_needed

In order to determine how much money needs to be paid out to the individual during retirement, one simply needs to know these four variables. The wage percentage multiplier, P, is set by the pension fund. The number of years worked, Y_worked, and the final wage, W_final, should be known fairly well by the pension fund (see page 121 of the COAERS report posted above). The number of years retirement is needed should also be known well due to the collection of various life expectancy statistics across the nation. Basically, what I am trying to say is that the amount of money that needs to be paid out to the individual in retirement is well-determined. The table below shows some example numbers:

Wage Percentage Multiplier, P, [%] | Number of Years Worked, Y_worked, [years] | Final Wage,W_final,[$] | Number of Years Retirement Drawn,Y_needed,[years] | Amount of Money Promised to Retiree in Retirement, M_needed, [$] |

3 | 20 | $48,807 | 16 | $468,552 |

3 | 25 | $51,297 | 16 | $615,567 |

3 | 30 | $53,913 | 16 | $776,360 |

In the above table, the final wage, W_final, is calculated using a base wage, W_base, and assuming a wage growth, W_growth, of 1% a year. Thus, W_final = W_base * W_growth ^ Y_worked. The purpose of the above exercise was to demonstrate that it is relatively straightforward to determine exactly how much money the pension fund will be required to pay out to its retirees during retirement.

The way pension funds generate the amount of money promised to its retirees is by making investments. In essence, both the employee and employer make contributions to the pension fund while the former works and the pension fund invests this money in the hope of generating a return, which will allow the pension fund to deliver on its future promises.

What I want to examine now is how much money will actually be in the pension fund, based on the performance of its investments. I will make quite a few simplifying assumptions, but the analysis will hold together nevertheless. Many pension funds require their employees to contribute a certain amount of money from their income to the pension fund yearly; these pension funds then match their contribution. In order to model this, we need to introduce several new variables.

1. The combined contribution rate, C (for both the employee and employer)

2. The investment growth rate, G_i

3. The amount of money in the retirement fund, M_actual

The combined contribution rate is defined as the percentage of one’s income that is contributed to one’s retirement. It is the sum of both the contribution rate for the employee and employer. In the case of COAERS, the contribution rate for the employee and employer is 8% and 14%, respectively. Thus, the combined contribution rate, C, is 22%. Historically, this has varied as well (see pages 127-128 of the COAERS report posted above). In 1941, it was 8% — by 2011, it was 22%. It is worth noting that as early as 2006, the combined contribution rate was only 17%. We then assume that a lump sum contribution is made to the pension fund at the beginning of each year. Please note that this is not a realistic assumption since the contributions will be spread out over the year; however, it is a conservative assumption, so it actually strengthens our case. The amount of money that is contributed to the pension fund in one year is equal to the combined contribution rate, C, multiplied by the current wage, W_current (the current wage is defined below, its functional form is similar to that of the final wage, W_final). From here, we assume that the pension fund will be making investments, so we assign an investment growth rate, G_i.

Assuming the employee is receiving pay raises every year (perhaps indexed to price inflation), we can write the current wage as a function of the years elapsed since beginning work:

W_current = W_base * W_growth ^ Years_elapsed

In the above equation, Years_elapsed is the number of years that have elapsed since the employee began working. Essentially, we are assuming that the employee will receive a pay raise of W_growth every year (we do not have to do this, but it simplifies things). In this analysis, we assume that W_growth is equal to 1% a year (as mentioned above).

We can then calculate the amount of money in the pension fund at the time of retirement as follows:

M_actual = ( M_actual + C * W_current ) * G_i

The last two equations might not make sense to those who are not programmers since the term we are solving for appears on both the left- and right-hand sides of the equation, but all we are doing is iterating to find a solution. To illustrate this, consider the first three iterations:

Iteration Number, [] | Combined Contribution Rate, C, [%] | Base Wage of Employee, W_base, [$] | Wage Growth of Employee, W_growth, [%] | Current Wage of Employee, W_current, [$] | Investment Growth Rate, G_i, [%] | Amount of Money Currently in Retirement System, M_actual, [$] |

1 | 22% | $40,000 | 1% | $40,000 | 5% | $9,204 |

2 | 22% | $40,000 | 1% | $40,400 | 5% | $19,034 |

3 | 22% | $40,000 | 1% | $40,804 | 5% | $29,411 |

Basically, on the first iteration, we set M_actual equal to 0, then solve for M_actual. On the second iteration, we use the M_actual we solved for during the first iteration on the right-hand side of the equation. If we repeat this process for the number of years that the employee works before retiring, it yields the total amount of money in the pension fund upon retirement of the employee. It should be noted that this is only the case if the employee stays with the City of Austin until they retire. If the employee were to accrue a number of years of service at the City of Austin, then start to work for another employer, the amount of money in the pension fund for them would increase while they were working for the other employer because their money would still be invested. This model does not take that into account, so one should keep that in mind; however, COAERS refunds contributions to employees that leave the City of Austin, so the net gain to the pension fund appears to be zero. Nevertheless, assuming the employee retires from the City of Austin and immediately begins to draw their retirement, the amount of money in the pension fund upon retirement is equal to the following:

M_actual = C * W_base * Poly_function

In the above equation, Poly_function is a polynomial function that is dependent on the number of years worked by the employee before retiring (in reality, it is the number of years that the money has been in the pension fund upon retirement, as mentioned above, but we will ignore this here since COAERS refunds contributions to members that leave), the investment growth rate, and the wage growth rate. In the interest of simplicity, consider a small number of years worked, like 3. The polynomial function, Poly_function, would be equal to:

Poly_function = ( G_i ^ 3 * W_growth ^ 0 ) + ( G_i ^ 2 * W_growth ^ 1 ) + ( G_i ^ 1 * W_growth ^ 2 )

If the number of years worked is equal to 5:

Poly_function = ( G_i ^ 5 * W_growth ^ 0 ) + ( G_i ^ 4 * W_growth ^ 1 ) + ( G_i ^ 3 * W_growth ^ 2 ) + ( G_i ^ 2 * W_growth ^ 3 ) + ( G_i ^ 1 * W_growth ^ 4 )

The pattern should be obvious. Thus, returning to our example values above (and assuming some constant investment growth rates), we get the following amount of money that is in the retirement system upon retirement:

Combined Contribution Rate, C, [%] | Number of Years Worked, Y_worked, [years] | Investment Growth Rate, G, [%] | Amount of Money in Retirement System Upon Retirement, M_actual, [$] | Amount of Money Promised to Retiree in Retirement, M_needed, [$] | Shortfall in Amount of Money Promised to Retiree (negative denotes shortfall, positive denotes surplus), S, [$] | Shortfall as Percentage of M_promised (negative denotes shortfall, positive denotes surplus), S_p, [%] |

22 | 20 | 5 | $331,047 | $468,552 | -$137,505 | -29.4 |

22 | 25 | 5 | $486,006 | $615,567 | -$129,561 | -21.1 |

22 | 30 | 5 | $687,015 | $776,360 | -$89,345 | -11.5 |

22 | 20 | 8 | $467,157 | $468,552 | -$1,395 | -0.3% |

22 | 25 | 8 | $755,709 | $615,567 | $140,142 | 22.8% |

22 | 30 | 8 | $1,183,221 | $776,360 | $406,861 | 52.4% |

It is worth noting that the actual shortfall is lower than this, due to continued investment growth while the individual is in retirement. However, in the current low interest rate environment, assets that are eligible for investment by pension funds for individuals in retirement (low risk) have extremely low growth rates (e.g., 10-Year US Treasury Note has a yield of around 1.5%). Since this post is already extremely long, I will not describe how this is done, but just say that the shortfall is reduced from 5-10%, depending on the case in the table above. Thus, even if this is taken into account, a large shortfall still exists.

As one can see, in all of the realistic cases (investment growth rate of 5%), a shortfall arises in the pension fund. Why is this a problem? Because somebody has to pick up the shortfall, or retirees will not receive their promised benefits. Who will likely end up paying? Taxpayers. I do not believe funding shortfalls can be addressed in the short-term by increasing contribution rates because this would simply shift the problem to a later date (i.e., kicking the can down the road). If a pension fund increases combined contribution rates, and then uses the influx of money to pay current retirees, the combined contribution rate for current employees has effectively remained the same, which means that they will face similar problems in the future if investment performance stays constant over the long-term.

In closing, I would like to focus on a key point that I believe is extremely important: underfunding of pension funds distorts the true cost of goverment, depending on how the shortfall is made up. In my opinion, if the shortfall is covered by taxpayers, this represents an increased cost of government in the past. The reason I say this is because the employees and employers (i.e., governments) underfunded their pension funds in the current, which allowed them to deploy funds elsewhere. Since these funds were actually needed in the pension fund, taxpayers will eventually have to cover them down the road. I am currently developing a more general program that allows for all of the simplifying assumptions (e.g., constant wage growth) to be variable. When that analysis is finished, I will write a follow-up post that explores my findings.

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