The state of public pension funds is at the heart of the potential default risk of states and municipalities, as the recent bankruptcy of Detroit demonstrates. There is a wide gap between common analysis at public pension funds and modern "best practice" principles of economics and finance. This note applies modern risk analysis to a hypothetical pension fund with more than 1,200 active employees, retirees, and terminated but vested employees. The fund is intentionally structured as a "young fund" with good cash flow to illustrate best practice risk analysis with accuracy and transparency. We provide a brief "risk model audit" citing common errors in public pension fund risk analysis. We address the need to use a 75-year horizon instead of a 30-year horizon. We illustrate the implications of using realistic market prices for the risk free rate in risk assessment. We show how to create a replicating portfolio of investments to minimize the volatility of the net present value of the fund. We show how to project the potential bankruptcy of the fund. We outline next steps in overlaying modern no-arbitrage asset return simulation for analysis of credit adjusted and mortality adjusted value at risk for the fund at multiple time horizons. We explain how a risky investment strategy has the potential to both reduce the probability of the bankruptcy of the fund and make the "worst case outcome" worse.

On April 30, 2013, the Sacramento Bee reported on a study that indicated that the actual debt of the State of California and local governments in California could well be $1.1 trillion instead of the official state debt total of $132.6 billion.

The study was published by the California Public Policy Center. The Sacramento Bee reports that the study is "a target of criticism by unions and other liberal groups, which accuse it of being part of a right-wing conspiracy to attack unions and public employees." The purpose of this note is not to add another estimate of the total debt of any public body. Rather it is to lay out the differences between best practice risk management and the common practices employed by public pension funds in reporting their assets and liabilities. For the most part, the reasons that "best practice" differs from common practice in public pension fund analysis are obvious and sensible. We illustrate them in this paper using a hypothetical pension plan data set of 1,000 active employees, 105 retired employees, and 109 inactive employees who are vested.

In a prior note, we posed five questions that every municipality, pension fund and pension fund board should be able to answer. We summarize those questions here:

**Question 1: Ignoring income from investments, looking only at individuals currently covered by the existing plan, what are the annual dollar amounts of cash in to the fund (from employee and employer contributions) and cash paid out of the fund (in the form of benefits to the beneficiaries) for the next 75 years?**

**Question 2: Using the data from Question 1 and the current true market value (not actuarial value) of assets, what compound annual rate of return is necessary for the current assets of the fund to meet the net cash flows projected for the next 75 years?**

**Question 3: Assume that we think the current pension fund investment strategy is too risky and that we demand that the fund's assets be shifted into U.S. Treasury securities on a matched maturity basis with the fund's projected future cash flows. This mimics the Social Security Administration strategy. What amount of money is necessary to implement this strategy today? If this amount of money is $X, what is the actual current true market value of assets (not actuarial value) as a percent of $X?**

**Question 4: Is the expected return on the pension fund's assets less than the "break-even" return in the answer to question 2? If so, assuming that the fund earns exactly the expected return with no risk, in what year will the assets of the fund be reduced to zero?**

**Question 5: Assume the fund continues to pursue its current highly risky investment strategy. Using the actual history of market returns in the U.S. and Japan, where equities remain still almost 80% below their peak in 1989, what is the probability that the fund's assets will go to zero in each of the next 75 years?**

We answer four of these five questions in this paper for our sample municipal pension fund, which we call ABC Employees Retirement System ("ABCERS"). The fifth question will be answered in another paper. Along the way, we show how deviations from the standard assumptions of modern finance and economics have created an inaccurate assessment of the current financial condition of public pension funds and an inaccurate assessment of the risks. General Treasurer Gina Raimondo of Rhode Island stressed the need for accuracy in an important report to the taxpayers and pension beneficiaries in Rhode Island.

**An Introduction to Technical Errors in Assessing**

**Pension Fund Financial Condition and Risk**

We summarize the typical analytical errors in pension public valuation and risk assessment.

1. Pension fund analysis offer blurs the distinction between obligations and income from current employees and retirees with potential obligations and income from future employees *who have not been hired*. This "open book" analysis, while common, obscures the financial position of the pension fund at the present time; the fund's present position is an inherited set of income and expenses that needs to be analyzed "as is." It is a risk management truism that one doesn't hedge the risk of assets and liabilities that it doesn't own now. A secondary analysis adds future employees but the main emphasis should be on current beneficiaries and retirees.

2. Pension fund analysis cuts off the long-term liabilities of the analysis by using a 30-year time horizon, but essentially all of the cash income to the fund will fall within the first 30 years. A large amount of the liabilities of the fund will occur after the 30-year point. Terminating the analysis at 30 years understates the liabilities of the fund. The Social Security Administration routinely reports on both 30 and 75 year horizons, and Kotlikoff and Burns (2012) explore even longer-time horizons. We show the impact of the shorter 30-year horizon in the example below.

3. Pension fund analysis assumes there is a risk-free return (typically 7.75% in recent years) that is constant and applicable to all maturities. The inconsistency of this assumption with observable market interest rates on a typical valuation date, June 30, 2012, is shown in Exhibit 1 (source: Kamakura Corporation and Board of Governors of the Federal Reserve System):

Exhibit 1

For short maturities, the assumed risk-free yield typically used by public pension funds is 7.71% higher than actual market risk-free yields. At a 30-year maturity, the 7.75% yield used by public pension funds is 4.75% higher than any risk-free yield in the marketplace. The use of true, observable market data is standard in economics and finance. For well-regarded text books in this regard, see Jarrow and Turnbull (1999), Hull (2011), and Jarrow and Chatterjea (2013).

4. Public pension funds use an unrealistic assumption of inflation, which affects employee and employer contributions, asset returns and liabilities. Public pension funds assume that inflation rate is constant and uncorrelated with interest rates and asset rates. Both of these assumptions are false. A typical inflation assumption, say a constant annual inflation rate of 3.00%, is currently well over the 10-year yield of Treasury Inflation Protected Securities (at MINUS levels at various times in 2013) and the 30-year TIPS yield (1.39% on July 25, 2013 according to the H15 statistical release of the Board of Governors of the Federal Reserve).

5. Public pension funds fail to assess their own probability of failure, which is normally the principal objective of risk management. The Trustees of the Social Security Administration directly report on the date at which the trust fund will be exhausted (estimated as 2033 in the April 2013 report). Public pension funds do no such thing. Stress testing to test survival capability is standard Federal Reserve Board practice under the 2013 "CCAR" Comprehensive Capital Adequacy and Review program. This program is being extended to all banks with more than $10 billion in assets in the years ahead.

6. Public pension funds ignore their own liabilities in determining an asset strategy, unlike an individual who is self-funding his own retirement. At an individual level, as one ages, one typically invests a greater and greater proportion of assets into fixed income investments to insure there is no short-fall due to asset losses after retirement. Public pension funds, by contrast, have invested heavily in equities, hedge funds, venture capital and other alternative assets. Such an asset strategy produces a much great risk of loss and a bigger potential failure than the fixed income strategy pursued by the Social Security Administration. Such risks are all but ignored in public pension plan and municipal financial reporting.

7. Pension funds use actuarial tables (typically 1994 or RP-2000 actuarial tables of the Society of Actuaries) that are both old and averages over long-time frames, understating the longevity of the typical pensioner as noted by King and Soneji (2012, 2013).

8. Pension funds simulate asset returns using average historical returns, even though a forward-looking simulation would allow for the very high probability that the future will not be like the past. The use of historical returns over a period of extraordinary high growth is a common source of error in risk assessment.

9. Pension funds typically fail to report any analysis that shows the impact of assumptions different than the ones they put forth themselves. This leaves the general public, taxpayers, and pension beneficiaries in the dark about the degree of "model risk" embedded in the conventional wisdom of public pension fund analysis.

We now illustrate the impact of these differences in analytical approaches on our hypothetical pension fund, ABCERS.

**An Introduction to ABC Employees Retirement System**

We introduce the characteristics of the hypothetical ABC Employees Retirement System in this section. The underlying active employee roster is a typical one of 1,000 people. There are assumed to be 105 retirees and 109 inactive but vested former employees. We use the RP-2000 mortality tables in combination with selected use of the Society of Actuaries 1994 tables and ABCERS-specific mortality tables. Retirement and termination probabilities are based in ABCERS experience. Employee and Employer contribution rates to the pension plan vary by job description consistent with the terms of the various ABCERS pension plans. We assume that the market value of the assets at ABCERS at June 30, 2012 is $11,285,930.

Because the number of active employees at ABCERS is five times larger than the combined number of retirees and vested terminated employees, ABCERS is a "young fund." This means that the upfront early cash flow is strongly positive and the fund is likely to be over-funded. The analysis of a fund in this position is extremely useful, if only because it is "de-politicized" relative to a fund in a seriously underfunded position. With that in mind, we turn to sample scenarios for ABCERS.

In aggregate, the sum of the employee and employer contributions for a single mortality/retirement/termination probability simulation is shown in Exhibit 2:

Exhibit 2

The vertical line is placed at the 30-year time horizon used by most public pension funds for "self assessment." As Exhibit 2 shows, this cuts off some of the cash flow projected to be received from the employer and employees. While the amounts are small in this case, the arbitrary termination of cash flow analysis is a potentially very serious risk management mistake. This error is typical of the way savings and loan associations viewed the risk of 30-year fixed rate mortgages in the 1980s -- using a *one-year* time horizon. The error is potentially more serious when looking at total cash flow, but first, we want to emphasize that, for actuarial risk alone, the cash flows into the pension fund are random, not constant. Exhibit 3 is one more scenario from a Monte Carlo simulation. In practice, one would analyze a large number of Monte Carlo scenarios that involve both actuarial risks and investment risks:

Exhibit 3

We now turn to net cash flows in and out of the pension fund, ignoring investment income and expenses. Exhibit 4 is just one actuarial scenario, but it illustrates very clearly the implications of terminating the time horizon of the analysis at 30 years:

Exhibit 4

By terminating the analysis at 30 years, the bulk of the negative cash flows (in nominal, not present value, terms) that stem from promises already made to current employees and retirees will be completely ignored. This could potentially result in a very serious understatement of the liabilities of the fund. For a young fund, like ABCERS, the impact is much less serious. Note that even this scenario understates liabilities given that mortality rates are probability overstated (understating longevity) and that negative cash flow may well continue beyond 75 years.

**Measuring the Net Present Value of the Fund's Assets and Liabilities**

We know that the current asset value of ABCERS is $11,285,930. When we calculate the net present value of contributions and liabilities, we use both 30 and 75 year horizons. We start with the standard 7.75% discount rate. The results for one scenario for ABCERS are given in Exhibit 5:

Exhibit 5

The results show a present value of future contributions of $77.2 million over a 30-year horizon. Combined with initial assets of the fund of $11.3 million, the present value of the assets of ACERS is $88.5 million. The present value of the first 30 years of liabilities is $44.6 million, and the net present value of the ABCERS fund is $43.9 million. In short, as predicted above, because the fund is "young," it is overfunded. More money is being contributed to the fund than it needs to provide the promised benefits to beneficiaries. Over a 75-year horizon, which is more accurate, the net present value of the fund is $41.2 million at a 7.75% discount rate.

What if the true observed market risk free rate, the U.S. Treasury curve, is used to calculate present value? The analysis is given in Exhibit 6:

Exhibit 6

The results show that the debates over the proper discount rate for public pension funds have missed an important point. For a young fund like ABCERS, the net present value of the fund at the 30-year horizon is $49.2 million, higher than the $43.9 million using the 7.75% discount rate. This comes about because the positive cash flows come in the early years and discounting at the lower real market risk free rate gives the net positive cash flows a higher present value. For older funds of course, the opposite effect will be found since negative cash flows begin much earlier, if not immediately.

Over the more accurate 75-year horizon, the true risk-free rate produces a net present value for the fund of $30.8 million in this scenario. This compares with the $41.2 million net present value found using a 7.75% discount rate over the 75-year horizon.

We now turn to the five questions that every pensioner should ask of their public pension fund boards of directors using the ABCERS example.

**Question 1: Ignoring income from investments, looking only at individuals currently covered by the existing plan, what are the annual dollar amounts of cash in to the fund (from employee and employer contributions) and cash paid out of the fund (in the form of benefits to the beneficiaries) for the next 75 years?**

The net cash flow in to the ABCERS fund was shown for two scenarios in Exhibits 2 and 3. The liabilities of the fund over 75 years are given in Exhibit 7:

Exhibit 7

Again, the red line at the year 2043 indicates the impact of terminating the analysis at the 30-year point. A large amount of nominal cash flow continues after year 30, even when analyzing a "closed book" of ABCERS employees, consistent with best practice in pension analytics.

The net cash flow for the fund over 75 years was shown in Exhibit 4.

**Question 2: Using the data from Question 1 and the current true market value (not actuarial value) of assets, what compound annual rate of return is necessary for the current assets of the fund to meet the net cash flows projected for the next 75 years?**

The answer to this question can be answered simply in common spreadsheet software using the net cash flows in Exhibit 4 and the known market value of assets on June 30, 2012, $11,285,930. ABCERS is so well funded that the internal rate of return that equates the net cash flows (excluding investments) to $11,285,930 varies between -1.00% and -2.00%, depending on the actuarial scenario being analyzed. For an underfunded pension plan, the necessary IRR will be far in excess of the 7.75% risk free rate that most public pension plans have been assuming.

**Question 3: Assume that we think the current pension fund investment strategy is too risky and that we demand that the fund's assets be shifted into U.S. Treasury securities on a matched maturity basis with the fund's projected future cash flows. This mimics the Social Security Administration strategy. What amount of money is necessary to implement this strategy today? If this amount of money is $X, what is the actual current true market value of assets (not actuarial value) as a percent of $X?**

To answer this question, we assume that the pension fund can go long and short U.S. Treasury zero coupon bonds in any amount in a perfectly efficient market. This would be the same procedure used to create a portfolio that exactly replicates the cash flows on a coupon-bearing U.S Treasury bond. Because Exhibit 6 shows that the ABCERS portfolio is overfunded, even discounting at the true market risk-free (U.S. Treasury) rate, when we create a "replicating portfolio" of U.S. Treasury zero coupon bonds, we will have extra funds left over.

Exhibit 8 shows the creation of the replicating U.S. Treasury portfolio for the first 30 years in one actuarial scenario. The full 75 year replicating portfolio is available by request from the author. The first three columns show the years to maturity, maturity date, and U.S. Treasury zero coupon bond prices prevailing on June 30, 2012. Column 4 shows the net cash flow from the non-investment activities of ABCERS. A positive number represents "cash flow out" so we need to buy a Treasury zero coupon bond with exactly that principal amount to provide the net cash flow needed to pay pensioners. If the number is negative in column 4, that means there is net cash flow coming in. Instead of investing in the Treasury rate, we borrow this amount at the Treasury rate, creating cash flow at time 0. We repay the borrowing with the cash inflow to the pension fund. At each successive maturity, we need to make sure we still have enough money to buy the Treasury bonds we need to exactly match pension fund cash inflows and cash outflows.

We start with the assets of the fund, $11,285,930. The first cash flow we see in Exhibit 8 occurs on June 30, 2013. It represents "cash in" of $8,120,564. We can borrow this amount at the U.S. Treasury rate, giving us net additional cash of $8,104,032 at June 30, 2012 with which we can buy other maturities of Treasuries. This results in net cash available of $19,389,962. We then step forward and analyze what we need to buy or borrow for June 30, 2014. We do this for each of the 75 years of cash flows. We know in advance that ABCERS has a positive net present value when discounted at the true market risk free rate, so we will end up with net "cash in" from borrowing more at the U.S. Treasury rate than "cash out" from investing at the U.S. Treasury rate. By exact matching of pension cash flows, the asset value of the fund becomes completely stable for all 75 years. We have delivered the promised pension benefits at no risk to taxpayers or pension beneficiaries.

Exhibit 8

The graph in Exhibit 9 shows that we are able to create a stable net asset position at the fund for 75 years at about $35 million. The difference versus the starting liquid assets of the fund $11,285,930 is the net present value at the risk-free Treasury discount rate that we were able to convert to cash at time zero by buying U.S. Treasury zero coupon bonds and borrowing at U.S Treasury rates. Note that the red line again indicates the end of the normal 30-year pension fund horizon.

What about a fund that is an "old fund" with many more retirees relative to active employees? What will the same analysis show? The answer in Exhibit 8 will be different. At some point, as we go to buy more U.S. Treasury zero coupon bonds that match our obligations to pensioners, we will not have enough money at time 0 to buy bonds at that maturity (we are assuming we try to postpone bankruptcy of the fund for as long as possible by buying the nearby maturities for as far out as the assets of the fund permit). Assume the maturity that we cannot afford is June 30, 2033 (the same date on which the U.S. Social Security system exhausts its assets). That is the year in which the assets of the fund will be completely exhausted and the only funds available to pay pensioners will be the current cash contributed by the employer and the current, active employees. In the case of the Social Security system, the Trustees (2013) estimated that only 77% of promised benefits could be paid in 2033. For an "older version" of ABCERS, the graph in Exhibit 9 would show the assets of the fund going to zero on a specific date. That is the bankruptcy date of the fund. Monte Carlo simulation can produce bankruptcy probabilities for each time horizon.

**Question 4: Is the expected return on the pension fund's assets less than the "break-even" return in the answer to question 2? If so, assuming that the fund earns exactly the expected return with no risk, in what year will the assets of the fund be reduced to zero?**

In the case of ABCERS, the expected return required for "break-even" net present value versus current assets of $11,285,930 was very favorable -- even if the fund loses 1.00% or 2.00% per year on invested assets, it will survive. If, by way of contrast, 12% was the breakeven internal rate of return needed, we could project the bankruptcy date in this manner. First, we assume the initial assets of $11,285,930 are invested at a (unrealistic) risk-free rate set by the pension board of 7.75%. We assume that any cash inflows to the fund are invested at the same rate. For an underfunded plan, even at this unrealistic risk-free investment return level, cash flows are so negative that the fund will run out of money. Again, the graph of Exhibit 9 would show the net assets of the fund go to zero at a specific date, which is the bankruptcy date of the fund.

**Results to be Presented in A Subsequent Paper**

Using the mortality simulations we have presented above in combination with modern no-arbitrage simulation of investment returns allows us to produce a rich set of risk analysis that helps elected officials, pension board members, pension beneficiaries, and taxpayers to set a completely transparent, accurate and realistic investment strategy. One of the questions we will answer in the next version of this paper is Question 5 above:

**Question 5: Assume the fund continues to pursue its current highly risky investment strategy. Using the actual history of market returns in the U.S. and Japan, where equities remain still almost 80% below their peak in 1989, what is the probability that the fund's assets will go to zero in each of the next 75 years?**

Standard risk analysis will show that the typical public pension fund suffers from a very high degree of potential volatility in its net present value (both positive and negative). On the one hand, a desperately underfunded "older fund" merely guarantees its bankruptcy by shifting to a low-risk investment in U.S. Treasuries like the Social Security Administration. On the other hand, an underfunded pension fund often doesn't completely understand that things can quickly go from bad to worse with a highly volatile asset strategy that has little or no correlation with the value of the fund's liabilities. Many funds have already forgotten what happened in 2008-2009, but a robust set of risk calculations will show the risk clearly.

A selected set of risk calculations would include the following:

- Historical value at risk at time zero of the net present value of the fund
- Forward looking Monte Carlo simulation of multi-period value at risk with mortality risk, default risk, and investment return risk
- Interest rate stress tests of both fund assets and liabilities
- Measures of the correlations of changes in the net present value of fund investments, contributions and liabilities

The results of such analysis would be sobering news for taxpayers, elected officials, pension board members, pension beneficiaries, and active employees. The probability of a broken promise to stake holders in the pension fund system should be a critical focus for all concerned.

**Conclusions**

This paper has outlined the application of modern valuation and risk management methods to a hypothetical pension fund that is lucky to be "young," with higher cash inflow and less cash outflow to active employees and retirees covered by the current plan. A more typical fund doing the same analysis can accurately identify its probability of survival and the time to default when following a conservative investment strategy. A more risky investment strategy, ironically, has the potential to both reduce the probability of default (from a near certainty) and increase the worst case scenario if asset returns are typical of 2008-2009 or of Japan's lost decades. Modern risk analysis provides an accurate basis for explaining risk and return in plain English to those paying for the pension plan (taxpayers and active employees) and those benefitting from it (pensioners prior to a potential failure of the plan).

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**Disclosure: **I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.