Economics

Redistributional consequences

Because the defined benefit does not depend on contributions, but on the highest salary, it is possible that there are two members who made the same contributions, but one of them receives a higher pension than the other. The redistribution goes from members who (relatively) receive most of their rises when they are young, towards members who (relatively) receive more raises when they are old. To understand why it can happen, we need to remember that the contributions to the plan are proportional to the salary earned in any given year . The pension is proportional to the highest (average) salary (see pension formula). If your salary profile is pretty flat, then your pension is proportional to your salary (which did not change because your salary profile is flat), which in turn is proportional to your contribution. On the other hand, if your salary was very low for most of the time of your employment, but jumped up during the last 3 years, you will pay relatively little in contributions, but receive a large pension.

Example: Mary and Joe are hired by the University at the same time, with the same starting salary of $100,000. Both of them will retire after 25 years. Both of them pay pension contribution of 10% of their salary each year and both of them will receive pension equal to

25 years × (2% × Highest Salary).

(For the purpose of presentation, we simplify things relative to the true pension formula)

• • Mary works for a unit where merit is valued. She publishes lots of papers during her first 5 years. At the end of the 5 years, she receives a raise of $25,000. She never gets any more raise. Mary’s overall contribution to pension is equal to

    • 5 years × 10% × $100,000 + 20 years × 10% × $125,000 = $300,000.

  • Her pension will be equal

    • 25 years × 2% × $125,000 = $62,500.

  • Joe works for the unit, where experience, measuered by the seniority valued. After 15 years, Joe gets a seniority related raise of $50,000. Joe’s overall contribution to pension is equal to

    • 15 years × 10% × $100,000 + 10 years × 10% × $150,000 = $300,000,

  • and it is exactly equal to Mary’s contribution. His pension will be equal

    • 25 years × 2% × $150,000 = $77,500,

  • which is higher than Mary’s.

In this example, Mary’s contributions end up subsidizing Joe’s pension.

There is also another type of “redistribution”. We are going to receive the same pension for each year that we are alive. In particular, members who die early receive much less money from the plan than members who live for long. Such “redistribution” is inherent to all social insurance schemes, and, perhaps after some thought, most people find such a “redistribution” acceptable. (Survivor or death benefit are some ways of dealing with it.)

Inter-generational transfer of risk

A defined benefit plan provides security of pensions to current employee. Because the pensions are financed from investments in risky assets (like stock market), there is a risk that the amount of contribution in the plan is not enough to pay the benefits. (The risk goes both ways, so there is also an upside risk, where there is too much money in the plan.) In a defined benefit plan, this risk is absorbed by future generations. We will illustrate it on an example.

Consider Joe, who started employment with the university at 1990 and plans to retire after 30 years at 2020. Joe’s pension contribution was deducted every month, and together with the employer’s contribution, added to the plan funds. The total contribution made by Joe and the employer is equal to $0.5 mln. This contribution was invested (partially on the stock market) and it is expected to double to $1 mln, which would give Joe a comfortable pension of $70K a year.

Because Joe’s savings were invested on the market, they can turn out to be different from what is expected, i.e, $1 mln. They may go up, and turn out to be $1.2 mln, or go down and end up being only $0.8 mln. On average (or in expectation), the money that Joe saved are exactly enough to cover his pension. However, there is a risk that they will be too high or too low.

Because Joe is in a defined benefit plan, he does not bear this risk. He is guaranteed pension of $70K no matter what happens on the stock market. The risk is absorbed by somebody else. In this case, the somebody else is Mary who started to work in 2020 and expects to retire after 30 years at 2050. If Mary is lucky, the stock market just went up, and her pension fund is flush with extra $0.2 mln. That means that her contribution to her own pension can be reduced by $1.2-$1=0.2 mln. (Who exactly benefits, depends on another important feature of the plan, namely sponsors. Under single-sponsor plan, the employer takes the surplus. Under a jointly sponsored plan jointly sponsored plan, the surplus is shared 50-50 between Mary and the employer.)

However, if the market goes down, Mary starts her employment with a need to pay $1-$0.8=$0.2 mln for Joe’s pension. There is nothing that she can do about it (expect, of course, not accepting the employment.).

Joe’s pension security is paid by the risk imposed on Mary.

Is there anything in store for Mary? Well, she knows that when she retires in 2050, she is going to receive a defined benefit, that is risk-free and that is guaranteed by contribution of Lua, a new hire of 2050 recruiting. Her own pension risk will be transferred to the next generation.

Is there a different way of organizing Joe’s pension, in which he does not burden Mary (and in which Mary does not burden Lua)? Yes, at least two:

• • There is a risk-free way of guaranteeing Joe’s pension without burdening Mary. That would be if Joe (or his pension fund) invested all his money into risk–free assets, like long-term government bonds. The trouble is that such assets bring much lower return. In order to be guaranteed the same pension, Joe would have to save much more. In the case of the current plan, such a risk–free plan could cost up to 40% of the salary (instead of 20% that the current plan costs now).

  • Another alternative is a defined contribution, in which all the risk of Joe’s pension is born by him.

Actuarial calculations

Below, we explain in a bit more detail how going concern and solvency valuations are done. The explanation emphasizes the role of two key economic variables: the performance of the stock market, and the rates of return. Moreover, we explain how the difference in the assumptions about the rate of return leads to different values of the deficits.

In both cases, the actuary wants to compare assets (i.e., how much money the plan has) with liabilities (how much it owes to beneficiaries).

The assets part is easy. The plan owns some bonds, some stocks, some cash. All of it can be sold on the market, hence it has a value. Currently, the value of assets is equal to roughly $4 bln and it does not really depend on the type of valuation.

The liabilities are tricky. First, because they are uncertain. Second, because they are in future. Which leads to more uncertainty. Here is roughly what the actuary does to compute their value.

Future liabilities. First, the actuary needs to figure out what is going to be the pension. Because the pension will depend on the highest salary (see Pension formula↑), it is not known at this moment. The actuary will make some reasonable assumptions about the rate at which salaries grow to forecast future salaries (and the size of the pension) from today’s salaries. For example, suppose that Katie’s salary this year is equal to $80,000. She has been recently years ago, and she has 30 years before retirement. Using the past trends as a basis, the actuary assumes that the salaries will grow at the rate 4% each year (see Actuarial evaluation of the University of Toronto pension plan). That means that Katie’s salary next year will be equal to

$80, 000 × 1.04 = $83, 200.

In order to calculate Katie’s salary at the retirement (which will determine her pension), we need to multiply the above by 1.04 29 more times, which makes it

$80, 000 × (1.04)30 = $259, 471.

Given this number, and Katie’s years of service the actuary can figure out what will be her pension. The actuary can use mortality tables to determine how long she is going to live and collect the pensions. There will be bunch of other related assumptions. These assumptions essentially try to predict what is going to be the value of liabilities in future. These assumptions are quite stable and they do not vary too much from one year to another. They are not what is important for our problem. The next issue is more important.

Present value of future liabilities. The actuary needs to compare the value of the assets with liabilities. The problem is that the assets are valued today, and, at least till now, we only figured out what is the value of the liabilities in future. Comparing assets today and future liabilities would be comparing apples to oranges, which is not a good idea and not helpful.

For this reason, the actuary wants to compute the present value of future liabilities. One way to think about it is how much money we need to have today so that if we invest them, the investment will give us exactly how much we need and exactly at the right moment. So suppose that we figured out that Katie’s pension 30 years from now is going to be $180,000 in year 2047. How much money do we need today to make sure that we have $180,000 in year 2047? Well, let’s call this number Katie’s Next Egg. If we invest Katie’s Nest Egg partially on the stock market and partially with bonds, with the rate of return equal to 5.75%, then after 30 years we will get

Katie's Nest Egg × (1.0575)30.

We want to figure out how big Katie’s Nest Egg should be to make sure that the above pot of money is equal to exactly

$180, 000.

Here are calculations:

= $33, 640.

She needs only $33,640 dollars today to pay for $180,000 pension 30 years from now. Not bad. The $33,640 is the Going Concern Present Value of the Katie's pension for year 2047. (Of course, she will need more money for her pension in 2048 and for each subsequent year until she dies. We can calculate the next eggs for other years in similar way. We need to remember that the length of Katie's live is uncertain.)

In the above calculations, we assume that Katie’s Nest Egg will be invested with return rate 5.75%. This is an assumption that Actuarial evaluation of the University of Toronto pension plan makes for the Going Concern valuation. If the plan exists and continues operating, the manager of the plan will invest funds in a mixture of stocks and bonds, as all managers of all pension funds do. The stock market returns are uncertain, they can go up and down. If they go up, the manager will reap benefits. If they go down, the manager will have to cover the loses. Because with going concern, we assume that the manager exist and cooperates, there will be somebody to cover the losses (or reap benefits).

When the actuary does solvency valuation, she assumes that there is no more manager to invest the assets, and there is nobody to cover future losses. For this reason, the law requires that the valuation is done as if all the assets were invested in maximum security instruments, so that to minimize the risk of not being able to cover future benefits. Bonds are considered safe. That’s why bond rate is used.

Because the bond rate is smaller (essentially the investment is safe, but not as good), the amount of assets necessary to cover the future liability is larger. Suppose, as in Actuarial evaluation of the University of Toronto pension plan, that the bond rate is equal to 2.85%, which is much smaller than the return rate 5.75% used in the going concern valuation. Using the smaller rate, Katie’s Nest Egg should be equal to

Katie's Nest Egg =

= $77, 471.

Then, $77,471 is the solvency Present Value of the Katie’s Liabilities. Note that it is twice as much as her going concern value. (The twice as much comes from the fact that Katie looks 30 years ahead. If she were looking only 15 years ahead, the difference would be much smaller. If she were looking 0 years ahead, there would be no difference.)

Riskiness of a pension plan

For the purpose of this section, we are going to think about a pension plan in terms of how it affects my income, where income is defined as a salary minus contribution to the pension plan before the retirement, and a pension after the retirement. The contributions and the pensions are determined by the plan. Some plans (like our current plan) specify a precise amount of contributions and pensions. Such plans do not involve any risk. Other plans may specify rules how the contributions and/or benefits are determined. For example,

• • a jointly sponsored defined-benefit plan fixes the amount of pensions in roughly the same way like our current plan. Thus, pensions are not risky. However, the contributions are risky as they depend on the past performance of the plan. More precisely, the plan has to maintain funds to pay off future liabilities. If the performance of investments is below expectation, we get a deficit, which needs to be filled with increased contributions. Similarly, if the performance of investments is above expectations, there is a surplus, that leads to lower contributions in the future.

  • under defined contribution, the contributions are not risky (because they are defined). However, the pensions are risky. This is becayse the contributions are invested (partially) on the stock market and, depending on the performance of the market, may lead to lower or higher pensions.

We are going to compare plans that deliver the same expected income in each period. Thus, on average, each plan gives me the same amount of money. But they differ with respect to how risky the income is.

A standard statistical measure of a risk is variance. Variance measures how scattered our income is going to be around its average value. The more scattering, the larger value. Let It be the t-period random income induced by the pension plan. Then, Var(It) measures how much the true income (i.e., the one that we are going to see) can differ from its expected value (i.e., from what we think the income is going to be equal, on average).

We want to know the riskiness over the entire life. The idea is to add the variances from different periods. (Almost. The problem is that when we are really old, we might have a large variance, but who cares if we are unlikely to live so long. We will correct for mortality.) We define a riskiness of a plan as

ℛ(pension plan) =

),

where mt is the probability that an individual survives till age x.

The idea of using the sum of variances as a measure of the riskiness has an economic interpretation. In each period, if the amount of risk is not too much, then the income variance is proportional to the utility loss from a risky, rather than deterministic income. The proportionality constant is equal to the second derivative of the utility function. If the consumption levels are chosen optimally, and the per-period utility functions have constant and equal across periods risk aversion, then the second derivatives are equal (at the expected values of income). Thus, the above measure is proportional the utility loss caused by the risk of the plan.

More technical details about how the riskiness of different plans is calculated can be found here.