Tuesday, July 23, 2013

The Value of Public Sector Pensions

The unfunded promises of public sector pensions are in the news, with the Detroit bankruptcy. Josh Rauh at Stanford and Hoover has a nice blog post on the subject titled "Public Sector Pensions are a National Issue''. (Josh and Robert Novy-Marx wrote a very influential paper (ssrn manuscript) alerting us to the size of the state and local pension bomb.)

Josh's baseline number for the value of underfunded pensions: $4 trillion. Why so big, and why is this a surprise? Because many governments calculate their funding by assuming they will earn 8% per year. Discounting a riskless liability (pensions) at a risky rate is a basic error in finance. It's made all the time. University presidents are notorious for demanding their endowments "reach for yield" in order to "make our rate of return targets."

Reading this piece sparks a few thoughts about the risks posed by pensions and other unfunded liabilities.

Let's report risks

How to make the error clearer? Perhaps focusing on present values and arguing about discount rates obfuscates the issue. Let's talk about risk. Maybe it would clear things up if pensions had to report a "shortfall probability" or "value at risk" calculation like banks do. OK, you are assuming an 8% discount rate because you're investing in stocks. What's the chance that your investments will not be enough?   Coincidentally, when I saw Josh's piece I was putting together a problem set for my fall class that illustrates the issue well.

Here is the distribution of how much money you will have in 1, 5, 10, and 50 years if you invest in stocks at 6% mean return, 20% standard deviation of return. I added the mean in black, the median (50% of the time you earn more, 50% less) and the results of a 2% risk free investment in green. (The geometric mean return is 4% in this example.)
(Note: there is a picture here. I've noticed this blog is getting reposted here and there in text-only form. Go to the original if you want the pictures)

The mean return looks pretty good. After 50 years, you get $20 for every dollar invested, or contrariwise an accountant discounting a promise to pay $20 of pensions in 50 years reports that the present value of the debt is only $1. But you can see that stock returns (these are just plots of lognormal distributions) are very skewed. The mean return reflects a small chance of a very large payoff.

In these graphs the chance of a shortfall is 54, 59, 62, and 76% respectively. As horizon increases, you are almost guaranteed not to make the projected (mean) return! The median returns -- with 50% probability of shortfall, in red -- are a good deal lower. And the modal "most likely" return is below the riskfree rate in each case.

How is it that people get this so wrong? Let's look at the distribution of annualized returns in each case. Remember, these are exactly the same situations, we're just reporting a different number.

In these pictures, the distribution of annualized returns is symmetric, the mean and median are the same, and the distributions get narrower and narrower for longer horizons.

Comparing the two graphs, you see that annualized returns are profoundly misleading about the risks you're taking. Annualized returns have a standard deviation that goes down at the square root of horizon. But the actual return has a standard deviation that goes up at the square root of horizon, and exponentiating makes it skewed with the larger and larger chance of underperformance. Money matters, not annualized returns.

So as usual, when arguments are getting nowhwere, perhaps we need to shift the question: please report your shortfall probabilities. And your plans for what you do with shortfalls.

In  many of those cases, the plan for shortfall  comes down to "the Federal Government bails us out" (or ERISA bails out private plans.) Well, if that's true, then we have a different and interesting discounting question. Maybe 8% is the right number if someone else pays the losses!

Finance also teaches us to think about "state contingent payoffs." What does the whole world look like in the bad events? If cities and states can't pay their pensions, this very likely because stocks have performed badly, and because we've had 20 years of sclerotic growth, no growth in tax revenues, to fund the pensions. Stock returns are not uncorrelated with other aspects of state, municipal, and corporate finance. Investing in stocks to fund pensions is like selling fire insurance on your house, rather than buying it. If the house burns down, then you pay the insurance company.

What debt really matters?

Even $4 trillion is not all that huge in the grander scheme of things.  The official Federal debt is $18 trillion. But if you add the present value of unfunded pensions, social security, medicare, Obamacare, and so on you can get numbers like $50 trillion or more. Which, it should be perfectly obvious, are not going to get paid, especially if we stay on the current slow growth trajectory.  But how important is this present-value observation?  Should we routinely add up all the unfunded promises, discount them properly using the Treasury yield curve, and report the grand total?

I worry most about runnable debt. Promises to pay people trillions in the far off future are a different thing than rolling over marketable debt every year. If it looks likely we won't be able to pay pensions in 20 years, there's not all that much pensioners can do about it. If it looks like we won't pay off formal short-term debt, markets can fail to roll over, leading to an immediate financial crisis.

So, much as I value Josh's calculation, and zinging those who want to minimize the necessity of ever paying off debt, it does seem there is a difference between marketable debt that needs to be rolled over every year and promises to pensioners and social security that may eventually be defaulted on, but can't cause an immediate crisis.

The cash flows do matter. If the government has promised to make pension and other payments that on a flow basis drain all its revenues, something has to give. As it has in Detroit.

A too-clever thought

A good response occurred to me, to those cited by Josh who want to argue that underfunding is a mere $1 trillion. OK, let's issue the extra $1 trillion of Federal debt. Put it in with the pension assets. Now, convert the pensions entirely to defined-contribution. Give the employees and pensioners their money now, in IRA or 401(k) form. If indeed the pensions are "funded," then the pensioners are just as well off as if they had the existing pensions. (This might even be a tricky way for states to legally cut the value of their pension promises)

I suspect the other side would not take this deal. Well, tell us how much money you think the pension promises really are worth -- how much money we have to give pensioners today, to invest just as the pension plans would, to make them whole. Hmm, I think we'll end up a lot closer to Josh's numbers.


I used a geometric Brownian process, dp/p = mu dt + sigma dt with mu = 0.06 (6%) and sigma = 0.20 (20%). The T year arithmetic return is then lognormally distributed R_T = exp( mu - 1/2sigma^2)T + sigma root T e) with e~N(0,1). It has mean E(R_T) = exp(mu*T)=exp(0.06*T), median exp[mu-1/2sigma^2)T] = exp(0.04*T) and mode exp[(mu-3/2*sigma^2)T] = exp(0)=1.


  1. Professor - how does the need for pensions to drawdown at a relatively constant rate affect these numbers? A small cash draw could become quite burdensome after a couple years of depressed asset prices...

    1. A lot. I kept it simple and focused on the rate of return to a target date, but of course the problem of a pension is to finance a stream of payments, not a big bullet. Discount them at the treasury yield curve, but otherwise you have to think about dynamic simulations and when the cashflow crunch gets severe.

    2. I ran simulations like that (a long time ago at a company far gone) and found that even if a pension enjoyed a good average return over its life, there was still a significant chance of failure. Severe early losses, combined with steady drawdowns, can make it difficult for the pension to recover. This was in the context of a hypothetical pension bond, where all of the proceeds are invested at the beginning, and no further contributions are expected.

  2. I believe that Paul Krugman has opined that pension liabilities are not an issue in the Detroit bankruptcy, according to the NY Times. I'm glad to hear that from such an unbiased oracle.

  3. Someone is being sneaky, and it's not Krugman.

    A good response occurred to me, to those cited by Josh who want to argue that underfunding is a mere $1 trillion. OK, let's issue the extra $1 trillion of Federal debt.
    "ou see, the Boston College study doesn’t just estimate assets and liabilities; it also estimates the Annual Required Contribution, defined as

    normal cost – the present value of the benefits accrued in a given year – plus a payment to amortize the unfunded liability
    And it compares the ARC with actual contributions.

    According to the survey, the ARC is currently about 15 percent of payroll; in reality, state and local governments are making only about 80 percent of the required contributions, so there’s a shortfall of 3 percent of payroll. Total state and local payroll, in turn, is about $70 billion per month, or $850 billion per year. So, nationwide, governments are underfunding their pensions by around 3 percent of $850 billion, or around $25 billion a year."


    Ok, let's just triple that and say state and local governments are running short 75 billion. The Federal government could chip in and cough up 75 billion dollars, which is almost a rounding error in the federal budget.

    Farm subsidies are about 25 billion a year, the hedge fund loophole on carried interest is worth another couple billion, and you can dig out the rest from tax loopholes.

    1. Crickets... People hate bailouts but if we enter a period of sustained low returns to pension funds then a bailout would be the preferred solution of most progressives. And as you note here the size of the pension bailout when put in yearly terms is modest.

    2. If you went back in time that might be a workable solution. The problem is they have been under funding for decades so the accumulated amount is much greater. Plus many of the plans have used bond issues to meet funding requirements, so now you have the double whammy of meeting current pension funding while simultaneously paying on the bonds.

  4. The "too-clever thought" is clever but not original: it was phrased already by Milton Friedman in his book "Capitalism and Freedom", as a way to immediately convert Social Security to a private pension system, without needing a lengthy phase-out period.

  5. For a "conservative" investor(saver), why invest in stocks if the modal return is less than the risk-free return?

    1. Because the modal return is very unlikely

    2. Because the *median* return is higher. Guesstimating from the T=50 chart, the area under the curve between zero and 2% is smaller than the area under the curve between 2% and "median return", which means your odds of underperforming the median, but still beating the 2% return are better than your odds of underperforming 2%; and there is still a 50% probablity that you will outperform the median.

  6. Your example is very strange Sir. When you show us an example of a rate of return of 6%, you decide to use a "geometric" rate of return of 4%. Then you show us, correctly, that the median return of a lognormal distribution with an interest of 4% compounded for 50 years is 7.3 and you compute the mean of that distribution to be 20 - when in reality 20 is the median of a lognormal distribution with interest 20 and the mean of the lognormal distribution with mu=0.04 and sigma=0.2 compounded 50 years is completely of the chart.

    So for the sake of the transparency of the example, can you give us a few more details about it?

    1. Updated accordingly, and I hope typo-free

  7. I think this is one of your better posts. The subtleties of normal distributions and log normal distributions is often lost on people.

  8. Professor Cochrane,

    Actual stock market real returns from 1871 to 2013 were relatively high. Does your model suggest that this was a low probability event? Perhaps equal to mean expected returns in 1871 but well above median expected returns?

    Thank you

  9. Your implicit assumption of a random-walk for returns does not, I think, fit with the historical data. Andrew Smithers, Robert Shiller, Jeremy Grantham and others have shown that returns have mean reverting tendencies. Please check with your friend Cliff Asness on this, I think he would agree. What I believe Smithers has found is that the longer the period of analysis, the lower the variance of geometrically-compounded returns.

    1. One of the prettiest facts about stocks is they are predictable from variables such as P/D and P/E -- and thus not random walks -- but nonetheless the long-term variance is almost exactly the same as the short term variance so they are not, in fact, "safer in the long run." As a analogy, suppose weather were iid but the weather forecast knew the weather one day ahead. You could forecast the weather, but the weather is still iid. Stock returns are uncorrelated in their univariate representation though forecastable in the multivariate representation.

    2. I think there is a genuine disagreement as to the data. I will try and link to a citation of Smithers or Siegel.

      Separately, I agree with the larger point, that "reaching for yield" often behaviorally leads to greater risk taking. Also, note that Fed QE policies make these assumed 8% returns much harder to achieve. Both directly (because many pension funds are 60/40 bonds/stocks, so the bond portion of the return is decreased artificially buy Fed buying) and through the "portfolio channel" where the forward return on all risky investments (equities, real estate, etc.) is bid down in response to the QE policy. The Fed makes it harder for pension funds to achieve their stated goals; in real economic terms, it increases the PV of pension obligations/liabilities by lowering the discount rate.

    3. "Smithers has found is that the longer the period of analysis, the lower the variance of geometrically-compounded returns."

      The reason for this result is an elementary error in statistical analysis. The statistics in this case is applied to the "annualized" returns, not to the returns over a full period (e.g. 20 years). It is an elementary property of random walk that the volatility of the "annualized" returns decreases as the square root of time, while the volatility of the full period returns actually increase as the square root of time.

      For an investment portfolio, this means that the dollar risk of a shortfall actually increases with time, contrary to common wisdom.

      This is the old debate about "Time Diversification" originally discussed by Samuelson.

    4. Again, I have discussed this topic extensively in this paper:

      Personally, I think there has been a critical misunderstanding about the properties of strategic asset allocation, which has lead to overconfidence in the properties of diversification as a risk management tool, at the expense of paying attention to secular changes in asset returns. There is some mean reversion in stocks, but it is not sufficiently strong to prevent large variations of final outcomes even over 20+ year time horizons.

      The only way to take advantage of the cycles is through a more tactical portfolio construction approach.

      I'd be interested in opinions

  10. Suppose that instead of a 50 year return on a fixed investment, we're looking at a 50 year return on a series of 50 annual "contributions." More importantly, suppose each annual contribution is adjusted based on how actual returns have turned out relative to the expected rate of return. This is more similar to actual pensions, where an 'actuarial required contribution' is calculated annually.

    Would your analysis change?

    1. Yes. I was trying to keep it simple for a blog post. What would not change is, you do not earn 8% returns without taking on risk, a lot of risk. Quantify that risk as you will, but there is risk.

    2. True. Clearly an important issue is "who bears the risk." Let's say the whole contribution comes from the participants. Now, is the risk they bear different in important ways than what they would take on as individual equity investors?

      In other words, is you basic point that equities are simply too risky for any retirement savings? Or is it that the structure of defined benefit pensions somehow multiplies the inherent risk?

    3. Adjusting contributions in practice (e.g. in a 401k plan) is going to be very difficult for most salaried employees who are already stretched to make the basic contribution and may not be prepared to handle fluctuations in their cash flow.

      As a general note, a continuous stream of contributions over 50 years has a relatively short "duration", less than 20 years, which means that the average investment horizon of monies invested is much shorter than the ever elusive "long term".

    4. You (AM) are right about the practical problems of adjustable contributions. I use this as a simplifying assumption to try to uncover the core complaint about pensions.

      It seems to me that the underlying complaint is simply that equities are too risky. Shouldn't use them, except maybe for gambling. This is a reasonable argument, but clearly applies to all retirement savings , not just pensions.

      Or do pensions somehow multiply a risk that is OK for individual retirement plans?

  11. Over what 50 year period have we seen a 6% return on stocks? Just curious... And in most cases, plans will change the discount rates based on "experience," much like they do for inflation, mortality, etc. Plans have been scaling down their assumptions, to where now 8% is on the high side as of 2013. I understand both sides of the argument, but financial economists are presenting a rather tired argument, especially given that RFR's are at historical lows. Much, much, more interesting and ground-breaking would be the implications if public plans liabilities were discounted along the treasury curve. How then would they invest? What would rising rates mean to funded status? How would funded status volatility (presumably caused now by interest rate volatility) affect local municipalities--now funded status will drop in times of crisis.

  12. The real scary thing is that your hypothetical is actually understating the risk for tractability. Normal distributions are fully characterized by their mean and variance while we know that there is in fact substantial kurtosis and skew in returns. Add time varying factors (a la stochastic volatility) and you get results that have convinced me of the futility of equal weighting the discount factor for return scenarios.

    There are other more fundamental macro-economic issues such as if we are able to tax (in the economic sense) the marginal returns to work enough to fund these schemes. There are limits to the systems ability to transfer surplus and still be viable....

  13. The government will sooner bring back slavery so the bureaucrats can be paid in full. I expect the prison population to surge to provide a massive pool of unpaid labor.

  14. The comments about the nature of pensions having to make many periodic payments are definitely interesting, but I wonder how to factor in another periodic payment: dividends.

    A boring US stock index fund like an S&P tracker has historically paid about a 2% dividend on top of any capital gains. This element of the return, while variable, is much less variable than the cap gains element. If we break out the return into 4% cap gains (subject to normal standard deviation) and 2% dividends (which are a function derived from cap gains) annually, how does that impact the analysis?

  15. wouldn't it be better not to pre-fund public pensions; put them at risk; your city fails, you don't get paid

  16. A note on your math... not the math itself but the way the numbers are presented.

    While the modal probability is indeed less than the risk free rate, the probability that you earn less than the risk free rate is still in fact rather small. The chance that you will earn less than the risk free rate after 50 years is 0.078% Which is not insignificant, and sup-par returns to this degree should still be planned for by every treasurer, city manager, and pension manager.

    But, the way you present the numbers makes it look like investing is a suckers game, when, in fact, that is not the case.

  17. Professor Cochrane,

    this topic is very dear to my heart since I have been talking about similar issues for quite some time (see e.g. here: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1904992)

    The one perspective I can offer is that at this stage of the game the problem is largely behavioral, not financial. Acknowledging the problems you highlight can open a can of worms in terms of defaults, lawsuits and more. Furthermore, there are strong commercial interests in NOT exposing this problem which kind of dents the whole Strategic Asset Allocation paradigm. While there may be operators who do not fully appreciate the problems, for many others ignoring the issue is largely a matter of career survival. Sometimes denial is the only remaining strategy, while hoping for some miracle ...

  18. Prof. Cochrane:

    What we're all dying to know is, who's your pick? Summers or Yellen?

  19. There is a federal program for retirees that has more than doubled in the last 10 years to $152.7 billion in outlays (FY2014). This program has benefits that would make French railway workers blush.
    One needs to be a federal employee only for 20 years to receive full lifetime pension and medical care under this program. Due to the aging population of ex-employees already receiving benefits, outlays will inevitably soar even more in the future.
    I am speaking of the Department of Veterans Affairs.
    Unlike Social Security or Medicare, this program is not funded by payroll taxes on the people who will also become beneficiaries one day.
    The VA is purely redistributive. It takes money from corporate and personal incomes taxes and gives the money to VA beneficiaries. Call socialism GOP-style.
    VA outlays will easily surge past $200 billion in just a few more years. The value of the just the pension portion of the VA program is placed at $2-3 million per beneficiary. It would be cheaper to give each ex-employee $500k upon retirement.
    But the VA is a sacred cow. Forget about cutting this monstrosity. Just pay up.

  20. Prof Cochrane, applying this insight to private individual pension portfolios, would you say that the notion of investing in stocks over the long run because "time diversification reduces risk" (measured as annualized return) is nonsense because risk measured as probability of shortfall rises exponentially? Does this make sense to you: http://www.norstad.org/finance/risk-and-time.html ?

  21. Reporting shortfall probabilities to asses funding status is a teaching idea that may ultimately backfire. Shortfall probabilities are neither necessary nor sufficient. It's not necessary, since once you know the present value shortfall you already know everything you need to know. It's not sufficient because the probability of the loss independent of the magnitude of the loss and the state of the world in which it occurs is meaningless.

    Reporting shortfall probabilities and magnitudes is a good idea in
    assessing the investment policy of the fund. If you have real liabilities growing with inflation plus some fraction of real GDP growth, why would you be invested in nominal bonds instead of say a portfolio of inflation indexed bonds and stocks? But that's a question
    about asset-liability matching, not about the funding status.

  22. When you "divide by zero" you shouldn't be surprised when the results get weird.

  23. Prof Cochrane,
    understand the median shift reducing the actual outcome by 1/2 of variance. Can you please throw some more light on why the mode has to be lower than the median? also, how this curve would move given the practical non-iid and mean reverting tendencies (specially after herd/forced actions during euphoria and despair)? Thank you Ramadoss AXP2 Alumni

  24. Prof Cochrane,
    understand the median shift reducing the average expected outcome by 1/2 of variance. Can you please throw some more light on why the mode has to be lower than the median? also, how this curve would move given the practical non-iid and mean reverting tendencies (specially after herd/forced actions during euphoria and despair)? Thank you Ramadoss AXP2 Alumni


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