**Journal of Risk and Insurance**

# The Cost of Pension Insurance

^{1}Defined benefit pension plans are those in which sponsors make promises to workers to pay annuities starting at some future retirement age. Firms hold the responsibility to fund these promises. These plans are to be distinguished from defined contribution plans, in which firms contribute some amount to workers' accounts each year. The latter variety (which include the 401(k) versions) is similar to tax-deferred savings accounts that belong to workers after a short vesting period. See Ippolito (1997).

^{2}We ignore the cost in a private insurance solution to protect against the possibility of insolvency, which needs to be addressed in this solution. We also ignore other factors such as taxes, administrative costs, moral hazard, and so on.

^{3}Owing to infrequent data observations, simulations of the insured event will likely yield a more accurate estimate of the loss function than past experience.

^{4}When these articles were written, the statute implied a continuous opportunity for ongoing firms to give their underfunded pensions to the PBGC in exchange for some portion of a firm's net worth. This is not how the insurance was operated; in fact, bankruptcy always has been the condition under which the underfunding could be put to the PBGC. This reality was codified in the Pension Protection Act of 1987. Most options models after this date reflect the conditional nature of the put option.

^{5}They assume that the insurance is a so-called term contract instead of a term-renewable contract. See note 54.

^{6}See note 15.

^{7}Hsieh et al. (1994) have a separate covariance for each of 194 pensions based on each plan's asset allocation at the beginning of the period. Each of these numbers is treated as a parameter.

^{8}VanDerhei (1990) calculates pension insurance risk using data for a large sample of plans, but his study is restricted to the pricing of a one-year contract on the assumption that the contract is not term renewable, and he does not consider the cost imposed by the market risk in the insurance.

^{9}One previous attempt at stochastic modeling of pension insurance is found in Estrella and Hirtle (1988). Their model does not incorporate the minimum contribution rules, nor does it address the cost imposed by the market-risk component in the insurance.

^{10}Insurance companies that cover catastrophic events, often large natural disasters, issue these bonds. They pay high yields but are subject to loss of principal if insurance losses exceed some predetermined level.

^{11}That is, by packaging many independent risks, the standard deviation of losses becomes small in relation to mean premiums, and thus, the probability of insurer insolvency is small.

^{12}This measure often differs from reported liabilities, as found in either Compustat data or Form 5500 Annual Pension Plan Reports submitted to the Internal Revenue Service (IRS), because in both of these cases, interest rates are permitted to diverge from market rates.

^{13}Upon termination, the PBGC calculates each participant's pension annuity earned to date, which is payable starting at the participant's eligible retirement age. The liability is the discounted value of this annuity. The guaranteed annuity has an overall limit equal to about $30,000 per year for an age-65 retirement date (scaled down to an actuarial equivalent at earlier retirement ages). In addition, recent increases in plan generosity over the five years before bankruptcy are scaled down.

^{14}The duration of liabilities is the percentage change of liabilities resulting from a 1 percent change in the discount rate used to convert future benefit flows to a present value. It depends on the portion of participants that are retired and on the age and service distribution of workers who have not attained retirement age. In addition to its natural sensitivity to the level of interest rates, the plan's duration depends on the growth rate of employment in the firm. If the firm loses employment over time, the plan becomes more heavily weighted by retirees and older workers, thereby reducing its duration, and vice versa. In the 417 plans used in the model below, the average duration in period zero of the simulation (calculated at the 6 percent interest rate) is 10.02; the 5th, 25th, 50th, 75th, and 95th percentile values are 6.8, 8.0, 9.5, 11.2, and 15.6, respectively.

^{15}The model incorporates most of the Internal Revenue Code to calculate required contributions for each plan. All shortfalls in assets relative to liabilities are amortized over various prescribed periods; thus when underfunding develops, it can be many years before it is fully amortized. Meanwhile, more shocks to the pension occur, creating more overlaying amortization schedules, and so on. For example, each year, the plan's actuary makes an assumption about interest rates and investment returns. When investment experience is realized, the difference between actual and assumed earnings is amortized over five years. Each year, a new amortization schedule is added as new experience is compared against assumptions, and so on.

When the actuary changes assumptions (such as the assumed interest rate, mortality table, quit rates, retirement rates, and so on), this triggers an amortization schedule over ten years in which the total amount of these amortizations is the present value of the change in liabilities implied by the altered assumptions. New benefit awards are amortized over 30 years. In addition, many kink points trigger different sets of rules, depending on funding status. For example, when underfunding becomes too high, a set of so-called deficit reduction rules is triggered, requiring faster funding.^{16}For the 417 plans used in the simulations, we calculate a correlation coefficient between assets and liabilities for each plan. The calculation is based on 100 20-year economic scenarios. The correlation coefficient has a mean of 0.66, with 5th and 95th percentile values of 0.30 and 0.92.

Similarly, the coefficient is nonstationary for the same plan across scenarios. Calculating the same correlation statistic for each plan for each 20-year scenario, then calculating the coefficient of variation on this statistic for each of 417 plans, provides a measure of the range of the pension asset–pension liability correlation for the same plans in different economic scenarios. The mean, 5th, and 95th percentile values of this statistic are 0.72, 0.32, and 1.34.^{17}Indeed, the bankruptcy court requires contributions

*less*than the minimum because it does not permit any contributions that amortize past underfunding; it permits only funding for current accruals under the plan. This policy conforms to general bankruptcy rules that prevent some creditors from receiving payment for prebankruptcy debt pending an agreement on reorganization, assuming that a reorganization is possible.^{18}For example, if the plan is small and not very underfunded, the creditors' committee might permit the plan to continue, in which case the committee might find it beneficial to the reorganization to allow funding to continue normally. We make allowance for this case below by simply assuming that any plan more than 80 percent funded will not terminate in bankruptcy.

^{19}If, however, insureds agree to a contract that requires retention of these balances and also requires funding according to the maximum allowable funding, then the insurer will issue a different policy and charge a lower price. We estimate the cost on the assumption that the current contract remains in force and so invoke the conservative assumptions that most insurers would likely follow. We revisit this issue below. In particular, we price an alternative pension insurance policy that constrains the use of credit balances. We also price a policy that constrains the use of credit balances

*and*sets a higher floor on minimum contributions. Consistent with a general theme of the model herein, these results strongly affirm the view that the cost of pension insurance is critically dependent on the contribution rules specified in the contract.^{20}Virtually all large claims are characterized by so-called due and unpaid contributions. The reason that missed contributions are limited to one year is that plans are required to report missed contributions on government reports and, in addition, firms in financial trouble attract the attention of a watch group at the PBGC. In recent years, missed contributions for more than one year are rare.

^{21}Benefit increases over the five years preceding a claim are phased in at 20 percent per year; the overall annual pension limit is $30,000 at age 65 (lower for younger retirees); and special supplemental benefits are extra payments that typically are paid during ages younger than 62 to top off pensions to Social Security amounts. These limitations are reflected in the simulation model.

In contrast, one factor works to make guaranteed benefits greater than reported liabilities. In particular, when plans are terminated in bankruptcy, even if the firm continues in existence, retirement rates at early ages typically accelerate owing to reductions in employment. Since early benefits usually are subsidized in the plan (meaning that benefits are not reduced to fully reflect the longer period of collection), the increase in early retirements increases liabilities in the plan. The statute prescribes early retirement assumptions in these cases; in effect, workers are expected to retire at the earliest attainable ages upon a claims event; this recalculation works to increase the amount of the pension liabilities.^{22}It is possible for the reverse to be true in some cases if guaranteed liabilities are unusually small compared to pension liabilities, but this situation is unusual. The simulation model evaluates each plan separately each year of every simulation, so a range of differences in these measurement concepts is reflected in the simulation results.

^{23}In particular, using data from the simulation reported below (and weighted to represent the population of all insured pension plans), insurance underfunding in 1999 is set at $93.1 billion compared to $72.6 billion in pension underfunding. These estimates are based on using a 6 percent discount rate to calculate pension liabilities.

^{24}The model herein has a reasonably full definition of guaranteed benefits that is an additional calculation not normally made by pension plans. Claims assets are easily determined from market value assets and credit balances included in annual pension reports.

^{25}The PBGC vigorously resists claims that do not fit the statutory requirements. Based on its file of insufficient terminations, except for some instances in which small plans were abandoned, it is rare that a claim of substantial dollar amount involves plans more than 60 or 70 percent funded; thus, the 80 percent filter is fairly conservative. To quantify the importance of imposing a filter, we compare claims projections based on the model described below on two assumptions, one that the PBGC takes all underfunded plans in a bankruptcy versus one that accepts claims only if underfunding is 20 percent or more of guaranteed liabilities. In the first case, average claims amount to about $1 billion per annum compared to $0.8 billion with the 80 percent filter in place. Thus, the recognition of some filter is not inconsequential and points to the importance of further empirical work in this area.

^{26}Cash flow is set equal to income before extraordinary items plus depreciation plus deferred taxes. Assets equal market value of common stock outstanding plus book value of debt outstanding. This variable is suggested by Petersen (1994).

^{27}This variable equals pension assets divided by liabilities as reported in footnotes to the sponsor's 10-k filing, as reported in Compustat. Liabilities are converted to common market discount rates. Pension funding has been included in several prior studies, including Martin and Henderson (1983), Maher (1987), and Carroll and Niehaus (1998).

^{28}With the exception of pension underfunding, the variables in the empirical work are commonly found in other studies of financial distress (for example, Altman, 1993; Gilbert et al., 1990; Iskandar and Emery, 1994; VanDerhei, 1990). We try several variations on this model. For example, we include more lags on all the variables to reflect the fact that bankruptcies typically evolve over several years and are affected by the accumulation of idiosyncratic events and aggregate movements in the economy. We also try including beta values, higher order terms, and interaction terms to mimic the nonlinearity expected in more theoretical treatments of bankruptcy. While most efforts do not disappoint (expected signs usually show up), they do not materially affect any of the results reported below. So we opt for a simple model over a more complex one.

^{29}Information about defined benefit plans is found in the pension footnote to the sponsor's annual financial report. The data include any firm that had a pension footnote in any year over the period 1980–1997.

^{30}In some instances, firms that enter bankruptcy are not recorded as bankruptcies on Compustat, but information from other sources, notably, McHugh (1997), is used to complete the Compustat information.

^{31}To determine the appropriate weights, we list all observations dropped from the analysis owing to missing data in Compustat, then look to see how many subsequently turn out to be bankrupt firms (using both Compustat listings and McHugh, 1997). This process confirms that a disproportionate number of soon-to-be-bankrupt firms are more likely to be eliminated from the analysis owing to missing data, meaning that one has a choice-based sample that, if left unadjusted, imparts a bias to all the coefficients. We correct the problem by running a weighted logit estimate, where the weights are population-to-sample bankruptcy probabilities. This adjustment yields results that reflect unbiased estimates of universe bankruptcy probabilities.

^{32}More detail on these variables is given in Appendix A. Firm assets are the total of the market value of equity plus the book value of debt.

^{33}Financial markets will discipline ratios that drift too far from other firms in either direction, suggesting mean reversion. But the data do not reject a random walk specification for firm employment.

^{34}Data on the firm financial variables are available from 1972 onward, so the covariance between these variables and interest rates and stock returns reflects that period, while the variances and covariance between interest rates and stock returns reflect the longer 1926–1997 period.

^{35}That is, at the end of each period, the firm's probability of bankruptcy is calculated; if it is, say, 3.32 percent, a draw is made from a random-number generator with numbers from zero to 10,000. If the number is 332 or less, the firm is made bankrupt; otherwise, it continues to be simulated in the next period.

^{36}Cochrane (1997) makes this point with regard to the mean of stock returns, but the same idea applies to all parameters in generating a stochastic process. Also see Samwick (1996)for an application of this principle to the model herein.

^{37}This is not an arbitrage-free interest rate model, but that is not important for this application. Arbitrage-free models require a more complex formulation but do not importantly affect interest rate volatility. Since the focus is on the impact of interest rate volatility and not bond pricing, one can do with a much more simplified version of the yield curve than normally is used in other applications. In principle, the model described herein would work similarly with a more complex yield curve. See more detail in Appendix B.

^{38}Recall that the interest rate equation herein is estimated in logs; this form already reflects larger noise terms in an absolute sense for higher interest rates. Also recall that the stock returns are based on annual data. While evidence exists of level-dependent noise in the monthly or quarterly data, the annual data do not have much support for this characterization.

^{39}The results are not dramatically altered if one ignores all the parameter uncertainty, but they add some robustness to the tails in the distribution and reflect the full uncertainty about interest rates and stock returns. We do not incorporate standard errors on other parameters in the model; none of the other parameters are as influential on the results as compared to the two major macro financial variables.

^{40}More details about the model can be found in Ippolito and Ross (1996). Also see Winklevoss (1993) and Pacelli (1996).

^{41}Most data are from Schedule B of the Form 5500 Annual Pension Plan Report submitted annually to the IRS and related attachments included in the filing. The data include the age-service matrix of covered workers, compensation levels, actuarially assumed age-specific hiring rates, age-service specific quit and retirement rates, benefit formulas, and so on. In addition, all existing amortization schedules are loaded; these schedules determine required contributions stemming from past experience in the plan.

^{42}Since the plan details are not all loaded, we set a calibration parameter to ensure a match to reported liabilities. For these purposes, we calibrate to current liabilities on Schedule B of the plan's Form 5500 Annual Report submitted to the IRS.

^{43}The pension model accommodates these changes by either increasing the number of hires or decreasing hires and increasing layoffs.

^{44}The equity return is the S&P return, which is one of the stochastic variables; the bond return in the model is determined by the stochastic interest rate and the assumption that corporate bonds have a maturity roughly equal to 20 years.

The market model is adjusted to reflect the single interest rate in the model. Estimate the following market equation for 146 plans in this model for which historical data date back to 1980; for the*i*th fund,where

*r*_{i,t}is the rate of return for the fund in period*t*,*I*_{t}is the risk-free yield in period*t*for a 30-year Treasury bond,*r*_{s,t}is the return on a stock index in period*t*, is the return on a bond return index, and α_{i}, are vectors of estimated parameters. On average, the betas sum to about 0.7; the stock beta ranges from 0.05 to 0.95; the alphas range mostly between minus and plus 0.01; and the standard error of the equation on average is 0.06. The results do not change markedly if all pensions are assumed to hold 50-50 stock and bond portfolios. In the simulation model, bonds held by pension plans are assumed to have a 20-year maturity, and they reflect a duration based on this assumption. It is easy to substitute a particular asset allocation formula. For example, substituting 0.5 for the beta values in the market equations and setting the alpha term and the error term to zero accomplish a 50-50 allocation of stocks and bonds.^{45}That is, at the start of each 20-year simulation, each plan draws a market equation at random. It retains this allocation for the 20-year run and then draws a new market equation at the start of the next 20-year simulation.

^{46}The real interest rate is equal to 2 percent in the simulations.

^{47}The simulation results reported below assume that real wages increase at the rate of 1 percent per year. In addition, the pension model incorporates some real increases in salary over tenure; these assumptions depend on assumptions reported by the plans.

^{48}First, we verify that the sample of plans in this model is representative of the distribution funding ratios in the universe of insured pensions. Second, we check the probability of bankruptcy for sponsors in the sample herein to the universe of sponsors and find that the sample has a somewhat lower probability. So we adjust the intercept in the bankruptcy model to adjust the average bankruptcy probability in the sample to reflect risks in the universe of sponsors. Third, after ensuring that the exposure characteristics of the sample reflected universe estimates, we attach weights by assigning a "partner'' to each sample plan. Each "partner'' has the same pension plans and employment size as its assigned sample plan but different financial ratios and, thus, different probabilities of bankruptcy. Partner firms are separately simulated in the model. We adjust the intercept so that the average probability of bankruptcy in the sample is consistent with the overall average probability of bankruptcy of all firms in the Compustat data that offer pensions.

^{49}The annual averages of the three firm-level stochastic variables are estimated with the stock return and interest rate equations, using seemingly unrelated least squares. The covariance matrix on the error terms is used to make draws for each of the variables from a joint distribution.

^{50}Economic scenarios are costly to run (in the sense of taking time) mostly because of the extensive number of pension calculations that must be made. Cycles run in very little time because they entail no new pension calculations. Yet cycles add information about the cost implied by any scenario. Thus, the use of cycling gives acceptable results in less time than using more scenarios and no cycling. To determine an appropriate number of cycles and scenarios, we use a criterion that the standard error on the 90th percentile loss estimate should be no more than 5 percent of the estimate. It turns out that this criterion is satisfied with ten cycles and 300 scenarios. That is, we run 20 blocks of 300-scenario simulations (ten cycles per scenario) and calculate the standard error on the 20 estimates of the 90th percentile result. This number is less than 5 percent of the grand 90th percentile estimate. As another check, we conduct a 1,000-scenario run with 100 cycles per scenario and find essentially the same distribution of claims as the one generated by the 300-scenario (ten cycles per scenario) run. Since weighting occurs by running partners for each sample sponsor, each partner is also cycled several times through each scenario. While each partner is assumed to have the same pension as the sample plan, it has separate draws of the financial ratios and separate bankruptcy draws.

^{51}In the within-scenario cycles, pension plan underfunding and sponsor size remain unchanged. However, the sponsor draws different disturbance terms for its financial ratios, thereby giving it a different bankruptcy probability in each year of the scenario and a new set of bankruptcy draws. The employment path remains the same in the resamples. By not redrawing employment levels, the pension calculations do not need to be redone, saving considerable run time in the model.

^{52}The run time for the model is pretty much proportional to the number of periods over which the scenarios are played out. Longer periods tend to deliver more extreme paths owing to cumulative negative equity shocks and interest rates changes. However, long scenarios give the minimum contribution rules time to work, which ameliorates the impact of the shocks. Very short periods have less extreme outcomes but more chance to be influenced by single-year events, as in an adverse stock return draw. These experiments suggest that one reaches sharply diminishing returns after ten or so years, in terms of learning much about the distribution of insurance losses. These effects fall off even more rapidly after 15 or so years. For example, as compared to the 20-year runs, which give annual actuarial costs of $807 million per annum, a 15-year run produces $830 million. The 95th percentile results are also close ($2.73 billion in the 20-year run and $2.58 billion in the 15-year run). Thus, by 20 years, the implications of the insurance are pretty much fully played out.

^{53}In principle, it does not matter if a third or fourth source of common risk is added to the model. One can zero it out the same way.

^{54}In this sense, pension insurance is like life insurance. Usually life insurance is purchased on a term-renewable basis, meaning that insureds can continue coverage as long as they pay premiums; the insurer cannot increase premiums just because the insured's health deteriorates. Thus, if an individual is diagnosed with cancer that might be fatal in one or more years in the future, the premium does not increase as a result, because this is the event that is being insured. These types of problems give rise to long-term contracts, as in a term-renewable contract. In contrast, term insurance covers a fixed time period. For example, a studio might purchase it to cover a key actor during the duration of the actor's work in creating a movie. In this contract, the premium changes each time a contract is written and reflects the insured's health at the start of each new contract.

In the context of pension insurance, the insurance event is bankruptcy and the insurance is term renewable, meaning that the insurer cannot raise the premium for a particular plan sponsor because it falls into poor financial condition, because this is the event for which the insurance is purchased.^{55}The parallel in life insurance is that overall rates are increased if mortality rates among insureds increase, but rates are not increased for particular insureds who become fatally ill. As in the case of life insurance (for example, smokers vs. nonsmokers), some classes of pension insureds are naturally characterized by much smaller probabilities of bankruptcy (for example, large versus small firms), and presumably, this feature could be reflected in price schedules that depend on size.

^{56}We briefly visit this issue below.

^{57}As in the case of life insurance, a term policy offers substantial insurance against factors that cause a "quick death'' but offers little protection against factors that produce a "slow death.'' See note 54.

^{58}See note 10.

^{59}An adverse equity returns draw increases overall premium assessments, and the insured's short position can offset a part or all of this exposure. A fall in interest rates causes an increase in insurance underfunding, which can be offset by a long position in bonds.

^{60}Market assets are priced daily in most pension funds; liabilities could be priced daily at least as a function of market interest rates. Since assets and liabilities are all computerized, at least for large plans, the cost of more frequent snapshots is small.

^{61}Participants include all workers and retirees (or beneficiaries) in the plan, as well as workers who have left the company but are entitled to a future vested benefit. There are about 33 million participants in the insurance program.

^{62}In the VRP calculation, pension liabilities are discounted using 85 percent of the four-year weighted average of the 30-year Treasury rate over the past four years (the weights are 4-3- 2-1, with the most recent interest rate getting the highest weight). Assets are "actuarial,'' which can differ from market assets by as much as 20 percent. Assets are also inclusive of credit balances. In addition, for some plans that are at least 90 percent funded (using 100 percent of the weighted average referenced above), the VRP is set to zero.

^{63}Extensive sensitivity testing confirmed that the general character of the distribution is robust: Average claims tend to be fairly close to premiums, but the standard deviation of claims is about the same magnitude as the median result. This means that the distribution is skewed—there is always a substantial right tail.

^{64}Since the model is evaluating ten cycles per scenario, several cycles can emerge in the same "best'' or "worst'' lists with identical combinations of equity returns and interest rates. This is why there are fewer than 300 points plotted in the figure.

^{65}Since the interest rate model is random walk, it is not surprising that the average value turns out to be approximately equal to its starting value (6.0 percent). Because inflation in the model is equal to the interest rate minus (an assumed) real interest rate of 2.0 percent, the inflation rate is fixed at 4.0 percent. Finally, since the stock return is strictly mean reverting, its value is set at its mean-reverting value of 7.4 percent (where this number is a geometric mean).

^{66}That is, there are 3,000 simulations times 20 years, for a total of 60,000 simulated years.

^{67}We calculate the present value of the insurer's total premium collections each simulated year and report key features of the resulting distribution in Table 5. The numbers in columns 2 and 3, when multiplied, do not produce the numbers in column 1, because the percentiles are calculated independently.

^{68}Each period, the model calculates the duration of pension liabilities held by the insurer. An algorithm chooses the amount of 30-year zero-coupon Treasuries (special issues from the Treasury to the PBGC) to attain a dollar-duration match of liabilities. The insurer invests the remainder in one-year T-bills. Since the yield curve in the model is flat, both fixed-income instruments have the same yield but represent the polar differences in duration.

^{69}The insurer needs to take a somewhat longer position in its bond portfolio to hedge the interest rate risk in its exposure. We ignore this hedge for purposes of this article.

^{70}The coefficient on the interest rate is still close to significance even if small, because the immunization program incorporated into the insurer's portfolio of annuities is not perfect.

^{71}This is a different rationale than those found in social contract issues surrounding issues such as Social Security (Feldstein and Samwick, 1997; Geanakoplos et al., 1998; Smetters, 1999, 2000). Unlike universal contracts across generations, the insurance issue involves subsets of the population implicitly contracting with taxpayers to provide a credible insurance function. It is very much like a private contract, except for the public guaranty. But there is no reason why the guaranty should not be fully priced.

^{72}This number is the difference between the geometric mean return on large stocks and Treasury bills (Ibbotson, 1999).

^{73}Note that the signs are opposite because claims are the dependent variable in column 1 while net worth is the dependent variable in column 2.

^{74}Divide $300 million by the average amount of insurance underfunding from Table 5 ($65 billion), which comes to $4.60 per $1,000. The quoted variable rate premium assessed by the PBGC is $9, but it is assessed against a different measure of underfunding that is only about half of insurance exposure. See the following note.

^{75}If insurance underfunding in a simulated year is

*IU*and the PBGC-measured exposure is*U*^{PBGC}, then for 60,000 simulation years, one obtains the following regression result:That is, insured exposure is higher than PBGC-measured exposure, and the r-squared is less than 50 percent. Average insured exposure is $65 billion (present-value terms) in a simulated year, while average PBGC exposure is $33. In addition, the standard deviation on insured exposure is $82.4 billion, while it is $33.8 billion for PBGC exposure.

We experiment with a variable rate charge of approximately $21 per $1,000 of PBGC-measured underfunding; this charge generates $16.1 billion in present-value premiums to match expected actuarial losses. But the market relationship shows that the coefficient on the average equity return measures $122 billion, implying substantial remaining market risk.^{76}They assume that, besides the impact of investment returns, pension assets have a net outflow (benefit payouts minus contributions) of 2.7 percent of assets, and that liabilities, besides increasing by the interest rate, have a negative drift of 3.0 percent of assets. We merely set these parameters to zero.

^{77}If one ignores contribution flows, then one must also ignore benefit outflows; otherwise, one would impart a declining funding ratio on all plans in the simulation, thereby dramatically overstating the cost of the insurance. Alternatively, if one incorporates contribution rules, one must also incorporate payouts, or else the opposite problem would develop.

^{78}In comparing these two (and any other two) policies, the simulations have the same seeds, so that the same paths of interest rates and stock returns are drawn. Also see note 77.

^{79}The ballpark hedging cost implied from the estimates in column 3 is $13 billion (see text). Thus, 3.4 times this amount is $44 billion.

^{80}Recall that the cost of the current contract is about $29 billion ($16 billion in expected claims, plus $13 billion in hedging costs).

^{81}In this alternative, we run the simulations with no contributions and benefits and set the expected equity return to the risk-free rate.

^{82}Lewis and Pennacchi (1994) report their results in a way that makes it possible to make a direct comparison with these results. Specifically, they provide an estimate of the present-value cost over the infinite horizon as a function of starting plan funding ratio and the sponsor's equity-to-debt ratio. The costs are expressed as a percentage of starting pension liabilities. Liability weights are attached to each of these cells based on the distribution of funding ratios and leverage ratios in this sample of plans (we use observed funding ratios to be comparable with Lewis and Pennacchi's (1994) data). By pursuing this exercise, one finds a present value of losses equal to $289 billion in year zero. This is an estimate of the full cost of the insurance over the infinite horizon, inclusive of implied hedging costs.

To compare to the results of the current model, we use the Martingale estimates to incorporate both actuarial losses and the cost of hedging the market risk. Since estimates are just for 20 years, assume that the average of the 20th-year claims would persist indefinitely in real terms. Because both the current model and Lewis and Pennacchi deal with a closed set of pension plans, this is a clear overstatement of future costs (because plans continue to terminate with no replacements). Expressing claims in each year of the simulation from year zero to infinity in year-zero dollars and discounting by the 2 percent real return, one arrives at a present-value estimate of roughly $100 billion, or about one-third of their estimate.

This result is not specific to Lewis and Pennacchi (1994). For example, Bodie and Merton (1993)report results on the same order of magnitude.^{83}Pension liabilities are calculated on a termination basis.

^{84}That is, any time a term-renewable insurance is canceled midstream, some insureds will absorb losses; for example, if a life insurance company cancels its term-renewable policies, those insureds with terminal insurance will lose the entire value of the insurance. In this sense, the government could make a one-time transfer to plan sponsors that are already in financial trouble at the time of privatization to compensate them for the loss of insurance value.

^{85}Note that in a Martingale approach, plans invested in all bonds have no implied hedging costs. If one constructed a proper dynamic hedging policy for this insurance, however, then one would need to short some amount of stocks because bankruptcy probabilities are correlated with overall stock returns (owing to the nonindependence of draws affecting the financial variables).

^{86}For example, as the funding ratio either falls toward zero or far exceeds 100 percent, asset allocation in the pension tends to play a smaller role in determining the price of the insurance.

^{87}Some evidence of moral hazard is provided in Niehaus (1990) and Ippolito (1989).

^{88}Currently, pensions sponsored by large firms are overcharged in favor of subsidies to smaller firms. Sponsors that carry risky portfolios in their pensions receive implicit subsidies from those that maintain less risky portfolios. The issues involved in cross-subsidies have been discussed even prior to the enactment of the insurance (Wooten, 2001).

^{89}We evaluate a sample of firms with employment data over a ten-year period; we cannot reject the random walk specification.

^{90}First estimate the equations using ordinary least squares. Then estimate the disturbance variances by regressing the logged, squared residuals from the first regressions, , against employment and a dummy variable denoting the finance industry. Let

*a*represent the predicted values from these regressions. Reestimate Equations (A1–A4) with the data inversely weighted by each observation's predicted standard deviation, .^{91}To retain symmetry in the log formulation, we eliminate percentage changes in employment above 25 percent and below 20 percent.

^{92}We also try an alternative approach for both the cash flow and employment growth equations. Two equations are specified, one with high variance and one with low variance, together with a probability of being assigned to either equation. Experimentation shows essentially the same simulation results as the simple specifications used above. We also try estimating the equations using seemingly unrelated least squares, but this procedure does not yield materially different results from those reported above.

^{93}To measure the common-year effects, we estimate the autoregressive equations using the annual average of all six stochastic variables (the four firm variables and the stock return and interest rate) over the period 1972–1997. However, we overrode the variance for equity returns and interest rates (and the covariance between these two variables) because we had data to estimate the two-by-two component of the matrix over a longer period. This override requires one to modify the other common-year covariances to maintain the necessary properties of the covariance matrix, notably that it remains invertible and positive semidefinite.

To obtain the common-industry effects for the four firm variables, we estimate the same equations using the two-digit industry averages of these variables and include year dummy variables. To obtain the idiosyncratic component of the disturbance, we estimate the same four equations using firm data differenced from the two-digit industry annual mean values. The noise terms add to the total variance in the 1972–1997 data (although the variances do not add up in Table A2, because the errors in the idiosyncratic equations are either corrected for heteroskedasticity or modeled with a mixture of normal distributions). We correct for heteroskedasticity in the idiosyncratic estimates for Equations (A1–A4).^{94}While some evidence exists of mean reversion in stock prices, it does not seem important over short periods (Fama and French, 1988; Poterba and Summers, 1988). Since the hedge herein is chosen each year (it could be rolled more often), the alternative model is not important for these purposes. Moreover, since a true random walk model in prices generates more variance in returns over time, it is suitable for a pension insurer to err on the side of assuming too much volatility than too little.

^{95}The distribution comprises 40,000 combinations of 12 monthly returns, each randomly drawn from the Ibbotson data. One such annual observation is 1+

*r*_{1}=?_{j=1}^{12}(1+ρ_{j}), where ρ_{j}is the randomly chosen monthly return from Ibbotson (1999).^{96}This specification is only one way to model the stochastic process of stock returns. Importantly, it assumes that the process that generates historical returns is stable over the observed period and is appropriate for future periods. For a good review of many issues surrounding this point, see Cochrane (1997).

^{97}That is to say, in an arbitrage-free model, one wants

*E*(*P*_{t+1})=λ*E*(*P*_{t})+ɛ , where*P*_{j}is the bond price in period*j*, λ is the present-value operator,*E*is the expectations operator, and ɛ is noise. Suppose that there is only one interest rate,*r*. Then the price of a perpetuity is . In the simulation model,*r*_{t+1}=*r*_{t}+ɛ , which implies*E*(1/(*r*+ɛ))≠*E*(1/*r*), which implies the possibility of arbitrage profits (this is an application of Jensen's inequality). Suppose that the problem is fixed by simply using a random walk model to portray the perpetuity price, so that and then derive*r*from this relation. Then for this interest rate, one could not satisfy the random walk requirement on prices for a bond with maturity less than infinity, so arbitrage opportunities again would emerge.*Ipso facto*, to eliminate arbitrage for all bond maturities, one cannot assume a flat yield curve.

Steven Boyce is a senior economist at the Pension Benefit Guaranty Corporation (PBGC). Richard Ippolito is a professor at the George Mason University School of Law. The ideas expressed in this article do not reflect the official position of the PBGC. We have benefited from many helpful comments from Ed Altman, David Babbel, Marshall Blume, Zvi Bodie, J. David Cummins, Darrel Duffie, Stephen Kane, Christopher Lewis, Olivia Mitchell, Greg Niehaus, Frank Santomero, Bill Sharpe, Irwin Tepper, Jack VanDerhei, seminar participants at Stanford Business School, the Kellogg School of Management, Dartmouth College, New York University, the Wharton School, and Innovation Second Pillar in Zug, Switzerland. Andrew Samwick and Mitchell Petersen were especially generous with their time in helping us on the project. We recognize our colleagues who helped build the stochastic simulation used herein—notably, David Freund, John Hirschmann, Martin Holmer, Hardee Mahoney, Jane Pacelli, and Bill Ross. We are also indebted to Apameh Anoushirvani and Cathleen Tracey for building the extensive database that provides the basis for the estimates herein.

## Abstract

This article estimates the cost of the federal pension insurance program. Pension insurance claims have an important market-risk component, which means that the cost of the exposure cannot be estimated by discounting future claims by the risk-free rate. Moreover, owing to the complexity of the insurance contract, its price cannot be estimated with known options formulas without introducing an error of nonquantifiable magnitude. To circumvent these problems, we model the insurance program in its full complexity and use a Monte Carlo method. By hedging the exposure with a dynamic premium policy that offloads the market risk to the insureds, one can calculate the risk-free, or actuarial, cost of that policy. One can also characterize the nature of the subsidy and its structure across insured plans. Finally, we provide an estimate of the implicit cost of the hedge function that taxpayers currently are providing for zero remuneration. The model shows that simple contingent claims models of pension insurance result in a price that is about triple the true market cost of the insurance, and that pension insurance models that ignore market risk understate the cost by half. The solution demonstrates the broad characteristics that might characterize a credible private-sector version of pension insurance.