## The Unfunded Liabilities of the Social Security System

The Trustees of the Social Security System issue a report each year on the overall state of the system. The 2015 report is available publicly (https://www.ssa.gov/oact/tr/2015/trTOC.html). Consider the following statement in the Overview section of this recently-released report.

The open group unfunded obligation for OASDI over the 75-year period is $10.7 trillion in present value….

In the above statement, OASDI stands for Old Age and Survivors Disability Insurance, the formal name for the Social Security program. The statement seems shocking because it suggests an enormous payment of benefits owed to individuals for which there is no funding source. This staggeringly large number is what is often reported in the media. But my guess is that most readers have no way of knowing how this number arises. With a bit of arithmetic, it is possible to show how it is obtained.

**Present Values**

Undergraduates are taught how to compute the present values of future payment streams. This concept is needed to understand the dollar amount above. Suppose that an amount today, A(0), is placed in an interest bearing account earning an interest rate, i. Then we can find what that initial amount is worth in one year from the formula

A(0)(1 + i) = A(1)

where A(1) is what is in the account after one year. Rewrite this equation as

A(0) = A(1)/(1 + i).

In this form, the formula tells us that we can determine what a future amount, one year from now, is worth today by dividing the future amount by one plus the interest rate. This is an example of a present value formula. The present value today, A(0), of an amount a year from now, A(1), is A(1)/(1 + i).

If we want to know what we would have in the account two years from now, recall that after one year we have A(0)(1 + i) and so, leaving that amount in the account gives us

A(0)(1 + i)(1 + i) = A(0)(1 + i)² = A(2)

where A(2) is the amount we have in the account after two years. Rewrite this as

A(0) = A(2)/(1 + i)²

which tells us how to determine what an amount, two years from now, is worth today.

Now you can see a pattern emerging. Computing a present value for many years requires a formula such as

A(0) = A(1)/(1 + i) + A(2)/(1 + i)² + A(3)/(1 + i)³ +……

and the $10.7 Trillion dollar number above is A(0) in a present value computed for payments stretching out 75 years. Employees of the Social Security system estimate the benefits that must be paid over 75 years, choose an interest rate, and compute a present value.

**An Alternative Viewpoint**

There is another way to think about this large number in the trillions of dollars which is also correct and has the advantage of providing some perspective on how large these unfunded liabilities really are. It is easy to show that if the government placed $10.7 Trillion in an interest-bearing account paying the rate of interest, i, used in our formulas, the government could make all the required payments from this initial principle and the interest that it would earn in an interest-bearing account.

No matter how you slice it, there is an enormous unfunded liability in the Social Security system (and there are similar problems in the Medicare System). Without any changes in the system, there will be large benefit cuts in the future to Social Security recipients. I will soon write more about this system to illustrate how payments will change without any legislation to change the design of Social Security.