Measured by CAPE, stocks in the U.S. seem expensive now. But CAPE has an historical bias. In the early 20th century, it drifted to consistently low values; in recent decades, to persistently high ones. These trends occurred whether the stock market was booming or busting. After removing CAPE's apparent bias, stocks in 2016 seem nearly normal. Which of these opposing views is more credible?

As noted in an earlier post, CAPE is a uniquely reliable predictor of future stock prices. It tends to correctly forecast the long-term direction, up or down, although it does better for the distant future than for next month or even the next year or two. A refined version of CAPE, called LogrCAPE0.33%, is shown below for the years from 1896 through 2015.* 

Right now, the computed value of this version of CAPE is above 0.5, as indicate by the last plotted value at the right end of the blue line. That's higher than the market peaks in 1929 and the late 1960's, which were excellent times to cut back on stock-ownership. However, the long-term trend of CAPE, plotted with the red line, suggests that CAPE was depressed in value throughout the early 20th century, then rose to higher levels in recent decades. A majority of the below-zero observations occurred before 1950, yet virtually every value since the late 1980's has been above zero. An extension of the trend-line suggests that CAPE may continue to have high relative values for another decade or more.

It's reasonable to ask whether CAPE would be a better predictor if its long trend were taken into consideration. Accordingly, the next chart adjusts the LogrCAPE0.33% metric by removing the trend.** The picture is more balanced, with values above and below zero occurring in every quarter-century. And the new trend-line is flat (a statistical necessity). Looking at recent values, the selloff in 2008 and early 2009 now looks like a good buying opportunity, with the de-trended CAPE hitting a low of -0.5. The current, near-normal value raises no alarm to avoid stocks.

One reason to be skeptical of the de-trended CAPE is that many other indicators say stocks are over-valued and are thus likely to weaken in the future. But what if all the indicators, including CAPE, share the same historical bias? Perhaps something about global economies or contemporary finance has changed fundamentally since late in the last century and will persist for decades to come. Maybe investors have more confidence in long-term payoffs from owning stocks because they see value in more global trade, or in better management of national economies, or in reduced fear of world wars and pandemics. Who knows? Lending credence to this speculation is a telling statistic. In a comprehensive analysis of U.S. markets since 1896 for periods of owning stocks from 1 to 40 years, the de-trended CAPE explained more variance (26%) in real returns than did LogrCAPE0.33% (20%).***

A rational person could take either side of the argument. Our calculators strike a compromise. They report value-adjusted allocations to stocks and bonds by first averaging LogrCAPE0.33% and the de-trended CAPE, then estimating allocations and returns from that average. Additional details about the statistical machinery of the calculators are posted here.

* For improved statistical results, the refined version computes a very long-term 0.33% exponential average of companies' earnings, instead of the classical 10-year ordinary average. It then takes the natural logarithm this ratio: current, inflation-adjusted prices divided by the exponentially averaged earnings. In a slight improvement on the standard method, the current month's price adds in the inflation-adjusted value of one month of dividends.

** The trend-line is a log-linear cubic polynomial. The de-trended CAPE is simply LogrCAPE0.33% minus the value predicted by the least square fit of the cubic model.

*** The to-be-predicted variable in the analysis was the actual, inflation-adjusted, compound annual return of U.S. stocks, with dividends reinvested. The returns were computed for all holding periods from one to 20 years, plus all even-numbered holding periods from 22 to 40 years, from January 1896 through December 2015. In a least-squares regression, the observed returns were predicted from a combination of CAPE and the holding period, using a function of the form r = b * c * POWER( m, y ) + a, where r is the log-real-return; c is the de-trended CAPE or LogrCAPE0.33%; a, b, and m are fitted constants; and y is the holding period.

Shiller's CAPE is a sensible measure of whether U.S. stocks are over- or under-valued. But for statistical analysis, it has some weaknesses. Correcting them can improve the insights that CAPE may offer about future returns from stock investments.

Robert Shiller's CAPE is a well established, highly respected metric of the extent to which U.S. stocks are over- or under-valued, rooted in historical data that spans more than a century. Among its good points: CAPE is based on an idea which, on its face, makes a lot of sense. And it does a better job of predicting future stock prices than many other contenders (albeit, with some caveats). So, what's not to like?

First, consider the good points.

Essentially, CAPE compares a one-time snapshot to an historical trend. The snapshot is the price of stocks for large U.S. companies at a given time (the Standard and Poor's 500 or a reasonable surrogate). The trend is the average earnings of those companies for the preceding decade. Why use a 10-year average? Shiller's insight was that doing so smooths out the peaks and dips in earnings that companies experience during the normal boom and bust of economic cycles. Of course, over periods as long as a decade, inflation has to be considered, so CAPE adjusts all stock prices and earnings to express them in current dollars. In short, CAPE estimates whether, after making sensible adjustment for inflation, the current price of stocks is cheap or dear compared to companies' demonstrated ability to generate earnings. For a stock-buyer who wants to capture future earnings, it makes perfect sense.

​What's more, CAPE is a decent predictor, according to standard statistical tests. In Asset Management, Andrew Ang evaluated 14 possible predictors of future stock returns, over periods from 90 days to five years. CAPE was one of only two that gave statistically reliable predictions, and, of those two, only CAPE can be calculated contemporaneously. If you want to make a decent prediction, without waiting for future data, CAPE is the metric of choice.

Now, the weak points:

• Balance. When calculated as a simple ratio and used in that form to conduct statistical analysis, CAPE is imbalanced. It gives too much weight to very high values (extreme over-valuation) and too little weight to very low ones (extreme under-valuation).
• Persistence. Counting current earnings for exactly ten years, equally weighted throughout that period, is not optimal. The persistent effects of extreme conditions are better captured by averaging earnings in a different manner.
• Trend. CAPE's message may be clouded by its own long-term trends. The CAPE value that is purportedly normal today may differ from what was normal decades ago.

​For each of these problems, fortunately, there is a plausible correction.

To see how CAPE over-weights high values and under-weights low ones, consider the chart below. It shows historical values of CAPE, from January 1896 to December 2015, assigned to bins in what statisticians call a "histogram."

In this chart, the values are reasonably well balanced, although a slight elongation to the left has replaced PE10Y's extreme elongation to the right. Accordingly, the statistical skew for LogrPE10Y is -0.17, which is close to the ideal value of zero, though slightly negative. By using the median for the calculation, LogrPE10Y has exactly half its values below zero, and half above. Furthermore, a doubling of stock-prices would raise the metric from zero to about 0.7, while a halving would lower it by the same amount, to -0.7.

Notice that, in the chart for LogrPE10Y, halving of normal valuations has occurred a bit more often since 1896 than doubling of valuations. That's evident because the bars below -0.75 are somewhat higher than the ones above +0.75. From looking at the original, badly skewed chart for the ordinary PE10Y, you would never have seen that. In fact, you might have concluded just the opposite, because of PE10Y's long tail to the right. This observation is one way to appreciate the misleading nature CAPE's classical computation. 

In PE10Y, today's price is compared to ten years of annualized, inflation-adjusted earnings, with the earnings estimated each month. Every month's estimate gets the same weight. But why should ten-year-old earnings get the same weight as current earnings, while those from eleven years ago get no weight at all?

​Shiller's original idea was that a decade-long average filters out the business cycle. While it may do so ordinarily, there are exceptions. For example, a period without a recession, but slightly longer than ten years, occurred from February 1991 to April 2001. Furthermore, the impact of earnings on stock-prices may be more than economic. There may be a behavioral component, as well. Investors' memory of extreme events like the bull market from 1982 to 2000 or the crash from 1929 to 1932 may influence their perception of stock-values. Would a twenty-year CAPE, for example, be a better filter of very long bull markets? Would an extended series of weights that decline ever so gradually for a very long time resemble investors' lingering memory of fearsome crashes?

The following chart addresses these questions by depicting three different ways to compute the average earnings in CAPE.

Since 1896, CAPE has not varied randomly. Over periods of 20 years or so, it has tended to hover at negative, neutral, or positive levels, not wandering freely across the full range of potential values. The chart below demonstrates the non-random behavior, using LogrPE0.3% for CAPE. The plotted line is a fitted (cubic) polynomial. If CAPE were truly random, the line would be flat or nearly so.

The trend shown in the chart poses some ambiguity about the proper interpretation of CAPE. For example, the dot-com peak in 2000 was double the size of any previous peak, when measured on raw values of LogrPE0.3%. However, when measured by the excess above trend, the extremity of 2000 was about the same as the 1929 peak. Making the interpretation even more difficult, the trend as of 2016 could look different when the fitted line is recomputed a decade or two in the future. If the fitted line were projected into the future, it would forecast a leveling-off, with today's CAPE becoming the "new normal." But, as Mark Twain famously observed, forecasting is hazardous, especially when you are trying to predict the future. It's just not clear whether current stock-prices are almost normal (compared to the trend line) or extremely over-valued  (assuming zero to be normal).

Speculatively, one might view the long trend as a behavioral indicator. Perhaps investors get accustomed to high or low stock-valuations, causing the "normal" level of CAPE to rise during long-term bull markets and to fall during periods of prolonged economic distress.

Whatever the reason, the long trend suggests why the explained variance, reported in an earlier chart, is low. It's just 20% because the statistical model made no adjustment for the long trend. Realistically, invoking CAPE to predict future stock-returns or to design a portfolio is a rough approximation. Using it is better than ignoring it. And, although LogrPE0.3% is a "better CAPE," it offers modest improvements, not guarantees.

Follow-up posts examine how the "better CAPE" can be used judiciously to fine-tune an investment portfolio and to adjust realistic expectations of future returns. In doing so, the follow-ups give a deep-dive into the merits of de-trending CAPE and the inner workings of our calculators.

* In data of the sort used here, with overlapping time-periods, traditional statistical tests may be biased. Ang's book, in chapter 8, section 5.2, explains why. Thus, the absolute level of explained variance (R-Squared) should be taken with caution. That said, different methods of measuring CAPE were all subject to the same biases. Comparing the explained variance of the methods is probably correct directionally, if not in magnitude.
​
** Two sets of regression analyses were performed. In both sets, real (inflation adjusted) stock-returns were predicted from CAPE values, using monthly data from January 1896 to December 2015. For exponential averages, PE10Y in December 1880 was the starting value. Annualized stock-returns, with dividends reinvested, were computed for all annual holding periods from one to 20 years, and for all biennial holding periods from 22 to 40 years. In one set of analyses (reported here), the regression slopes and intercepts were forced to be constant across all holding periods. In a second set (to be covered in a future post), the regression parameters were allowed to vary for each holding period.

The Able to Pay calculators were updated in late May and early June, 2016, with an improved statistical model, built on a better method to evaluate whether stock-prices are cheap or expensive. The model currently estimates that U.S. stocks are expensive, and thus adjusts downward the percentage allocated to stocks. It does so more strongly than the previous calculators, which used good, but less robust methods. Depending on your inputs, the new calculators may adjust stock-percents by about 5% to 10% more than previously, and they may forecast lower future returns (in the current environment of relatively expensive stocks). The changes mainly effect two aspects of the calculators:

• Estimated future returns (and the future dollar values that depend on those estimates). 
• Value-adjusted percentages for portfolio composition. (Normal percentages for portfolio composition are not effected.)

Details on the new statistical model were described in other posts.

​CAPE is a paradoxical measure of over- or under-valuation of stock prices. It works best for long-term holdings, but makes the strongest predictions for short-term ones. Given the paradox, how can it best be used to craft an investment portfolio? And what is it telling us now?

About a year ago, I published a series of posts on CAPE, a metric of over- and under-valuation for U.S. stocks. It's one of the few indicators that has historically predicted future stock prices with some statistical reliability. Unfortunately, it's does so in a paradoxical manner. Over a few years, the prediction is strong but unreliable; over many years, up to 15 years or more, the prediction is weak but reliable. "Strong" means that low or high values of CAPE imply large future movements in stock prices, up or down; "reliable" means that the total movement, the cumulative rise or fall in stock prices over all the years, is very consistent. Thus, the paradox is that CAPE offers a big, high-risk bet for near term trades, and a small, low-risk bet for long-term buy-and-hold.

Since my original posts, CAPE hasn't changed much. It is still high, by historical standards, implying lower than normal returns on U.S. stocks in the future, maybe in the next year or two, or maybe not, but quite probably over the next decade or more.

In an effort to squeeze every last bit of predictive power from this paradoxical metric, I've recently completed a comprehensive re-analysis of CAPE and U.S. stock prices from 1881 through 2015. Among the questions answered by the analysis are these:

• CAPE is usually computed as a 10-year average. Would more or fewer years in the average offer better predictive power?
• Is there a holding-period for which CAPE is most useful? Does the trade-off between higher reliability and weaker effect-size mean that there's an optimal number of years to wait to capitalize on an investment decision motivated by CAPE?
• How can CAPE be used to inform the allocation to stocks in a complex portfolio? For example, in portfolios for college savings or retirement, where some investments may be spent in a few years and others perhaps a decade or more in the future, how can CAPE optimize the percentage invested in stocks?

Later posts addressed these questions, starting here. You can find an overview in Finding Value on the Portfolios tab, or by searching for "CAPE".

Does your adviser put your interests first? Are your adviser's recommendations colored by sales commissions he or she may get or by fees extracted from your account? Historically, the federal agencies responsible for overseeing the financial industry have imposed minimal requirements on investment advisers and financial professionals. Advisers were required to recommend reasonable products, not necessarily the ones that would be lowest in cost or otherwise best suited to your financial goals.

Then, early in 2015, the U.S. Department of Labor (DoL) announced a plan to tighten the requirements by imposing new regulations. The DoL drafted regulations whose objectives were to:

• Strengthen the responsibility of retirement plans and retirement advisers to act in their clients' best interest, and
• Extend fiduciary requirements to IRA providers and to rollovers between IRA and 401(k) or 403(b) plans.

The DoL solicited public comments and, after receiving nearly 400,000 of them, published the official new regulations on April 6, 2016.

Note that the DoL's jurisdiction extends only to retirement accounts. It does not cover ordinary brokerage and after-tax savings accounts. That, in itself, is an important limitation. Furthermore, the final regulations, while arguably an improvement, are weaker than the original proposal on several counts. Among the many commentaries appearing in the financial media, two articles that aptly describe the limitations of the new DoL regulations are here and here. Rather than reprise the skepticism evident in those articles, my aim in this post is to suggest how you can evaluate your own adviser, taking into consideration what the new regulations do and don't require.

Ask your adviser the following questions. Or review the website of the adviser's firm to see whether the answers are available there. A genuine fiduciary (one who puts your interests first) would answer "Yes" to all, or very nearly all, of these questions. Some of them are based on ideas in the original DoL proposal that were weakened, regrettably, after some investment firms objected.

1. Will you sign a written fiduciary oath, as recommended by National Association of Personal Financial Advisers and the Committee for the Fiduciary Standard? (Note: If the answer is, "I support the oath but my legal department won't let me sign it," that counts as a "No.")
2. Will you charge a flat fee or hourly rate for work delivered or a fixed percentage of the assets that you manage in my accounts, and not charge a sales commission of any kind? (Note: You should pay your financial adviser as you would pay your attorney or accountant.)
3. If your employer gets revenue or fees from any of the products you recommend to me, will you provide a detailed accounting of those charges, even if you personally get no commission? 
4. When you recommend products to me, will you provide a detailed list of all fees and charges I will pay for purchasing and owning those products, including total annual dollar amounts for current and future years?
5. When you recommend products to me, will you show me multiple firms that offer such products, comparing the merits of each firm?
6. When you provide financial advice about investments or savings, will you show me the potential future performance of my account, including a most-likely estimate and a worst-case estimate?
7. Will your advice show how the future performance of my investments or savings might be impacted if either inflation or deflation occurs?
8. If my financial goals or preferences change for any reason, will you be available to help me understand the impact of making the changes?

The point of asking these questions is that the federal regulations simply are not strong enough to guarantee that your adviser will put your interests first, even if he or she professes to be a fiduciary under current guidelines. To download a pdf copy of the questions, click below:

Earlier posts in this series reviewed possible explanations for what economists call "the annuity puzzle." The gist of the puzzle is that classical economic theory says retirees should buy more annuities than, in fact, they do. This series of posts can't claim to fully solve the puzzle. Instead, the more modest aim here is to use research by economists to help current and future retirees make good decisions about annuities.

A decision to buy or avoid an annuity is likely to be better made if it's properly framed. Being well framed means that the choice is stated in a way that counteracts common biases that could otherwise lead to a poor decision. That's the conclusion of a 2008 study published by Jeffrey R. Brown, J. R. Kling, S. Mullainathan, and M. V. Wrobel.

In the study, the authors surveyed 1,342 participants and asked them to evaluate decisions made by hypothetical retirees. The retirees were said to have "made permanent decisions on how to spend a portion of their money in retirement ... each person has some savings and can spend $1,000 each month from social security." Participants were told about the decisions that two such retirees had made and were asked which of the two had made a better choice.

For example, in one comparison, "Mr. Red" made a decision that allowed him to "spend $650 each month for as long as he lives in addition to social security. When he dies, there will be no more payments." He was compared to "Mr. Gray [who] invests $100,000 in an account which earns a 4% interest rate. He can withdraw some or all of the invested money at any time. When he dies, he may leave any remaining money to charity." Given this choice, most participants said Mr. Gray had made the better decision.

However, in another comparison, Mr. Red was the same, but Mr. Gray was described differently. He was able to "choose an amount to spend each month in addition to social security. How long his money lasts depends on how much he spends. If he spends only $400 per month, he has money for as long as he lives. When he dies, he may leave the remainder to charity. If he spends $650 per month, he has money only until age 85. He can spend down faster or slower than each of these options." In this case, most participants said Mr. Red had made the better decision.

Although described differently, these two cases were mathematically equivalent. In both of them, Mr. Red had a lifetime annuity, and Mr. Gray had a savings account earning 4% interest. When the savings account was characterized as in investment (the first case) it was favored over an annuity. However, when the savings account was characterized in terms of the spending pattern it would foster (the second case), it was disfavored.

The study was well designed in a manner that systematically compared a simple lifetime annuity to several other options, including interest-bearing investments and more complex annuities like the ones insurance companies may pitch to skeptical buyers. In every case, the simple lifetime annuity was logically and mathematically the better choice, but it was favored by the participants only when framed in language that clearly stated how each option would affect the retiree's budget and lifelong spending.

Thus, looking at budgets and spending led to better decisions than looking at investment factors such as interest rates and earnings. Focusing on budgets and spending counteracted people's tendencies, described in the previous post in this series, to over-simplify and to miscalculate risks.

Summing up, two points stand out from the various studies reviewed in this series of posts:

1. Consider your future health costs (for the reasons described in the first post). If you have excellent health insurance, such as a strong Medigap policy and long-term care insurance, then a simple annuity may be a reasonable way to smooth and stabilize your retirement income. If not, then you may be wise to set aside a reserve fund before purchasing an annuity.
2. Lay out your retirement income and budget without an annuity. Then consider whether purchasing an annuity might improve your lifetime spending. Do this instead of focusing on interest rates, investment returns, or payback periods.

Of course, other factors will also come into play, including your desire to leave bequests to your heirs or charity and your shared income with a spouse or partner, if you have one. For a step-by-step method that incorporates all these considerations, along with specifics regarding the best types of annuities, see our article on annuities for retirees.

Classical economic theory says that retirees should prefer annuities over investing in stocks and bonds. But they don't. Why? An earlier post reviewed one explanation, citing a study that said classical theory overlooks how health shocks reduce lifespans and impose burdensome costs. This post reviews other studies offering a second explanation. It examines how retirees think when deciding whether to buy an annuity.

In 2014, Jeffrey R. Brown, A. Kapteyn, E.  F.  P. Luttmer, and O. S. Mitchell, published a study that examined decisions to buy steady, lifelong annuity payments or to take a one-time, lump-sum payment. They noted that much of the research about such decisions "assumes a rational and fully informed individual who knows ... mortality rates, market returns, inflation, future expenditures, and income, and who can use this knowledge to optimally choose the mix of financial products to smooth" one's future budget over many years. That's a tall order, even for the brightest among us. Most of us may compensate, according to the study, by over-simplifying the decision.

To conduct the study, Brown and his colleagues generated realistic but hypothetical choices between annuities and lump-sum payments, and presented the choices to 2,112 individuals. The data indicated that most participants in the study used "simple heuristics rather than full optimization." The tendency to simplify the decision was strongest for those who had low scores on financial literacy and numerical ability, and those with less education.

A very common method of simplification, even among the more shrewd participants, was to buy an annuity or to trade one for a lump sum only when it was "an exceptionally good deal." While it might seem rational to insist on a good deal, the typical price-points in the study were extreme. On average, the participants were willing to buy an annuity only if the price was marked down by about 75% from the amount they would accept when trading an annuity for a lump sum. That's like offering to buy a new home for $60,000 but, as the owner of the same house, refusing to accept any offer below $250,000. A modest difference between the offer and list prices might make sense, but at values this extreme, a rational sale at fair value won't happen. In short, annuity buyers may have expectations that preclude reasonably priced transactions.

Another recent study offers a systematic explanation of the decision process that may guide annuity buyers. Under review for possible publication in an academic journal, the study is by Daniel Gottleib. Using a mathematical model from behavioral economics called Prospect Theory, the study analyzed consumers' decisions to buy or avoid either an annuity or life insurance. Logically, annuities and life insurance are complementary. One pays if you die; the other, if you live. As Gottleib argues, a theory that explains why retirees buy fewer or smaller annuities than is optimal (which they do) may also explain why the same retirees buy more life insurance than is reasonable (which they also do).

Setting aside the sophisticated math of Prospect Theory, one of its core ideas, as Gottleib explains, is that "individuals are risk averse over gains and risk seeking over losses." This means that decisions are based on miscalculation of the true risks. Here are some plain-language examples:

• Consumers (retirees and younger individuals alike) avoid taking losses. If, for example, they own a stock listed below what they paid for it, they avoid selling the stock at a loss and instead hold onto it, hoping that the price may recover. In doing so, they risk an even greater loss. For an annuity, the irrevocable decision to buy one may seem like a loss, if the buyer worries about dying before receiving enough payments to recoup the original cost. Just as the owner of a losing stock will hold onto it at risk, potential annuity-buyers hold onto their cash at the risk of depleting it, should they live longer than expected.
• Consumers are also over-eager to take gains. Stock-owners who have a modest gain, for example, may be inclined to sell pre-emptively and not let their profits run. They are averse to the risk that modest gains might vanish. Similarly, one who has held an annuity long enough to receive payments equal to the original purchase price may be reluctant to buy more annuities, even when doing so might be optimal. Logically, for example, it might be wise to increase an annuity if inflation has eroded the value of the original payments or if the buyer's good health has boosted the odds of a very long life. But such wisdom is atypical for retirees.

Gottleib's study offers the cold logic of Prospect Theory to explain decisions like these and similar ones regarding life insurance, including consumers' proclivity to under-purchase life insurance when young and over-purchase it when old. A point in favor of Gottleib's analysis is, thus, the breadth of decisions it covers.

The foregoing studies imply that when thinking about annuities, we are prone to over-simplify complex decisions and to miscalculate the true risks. Are there ways to counteract these misguided tendencies? Other research suggests that there are, and that better informed choices may be possible. That's the subject of Part 3 in this series.

For step-by-step methods to evaluate whether an annuity would be right for you, including methods to find a good insurance company that sells annuities, read our article on annuities for retirees.

Since a ground-breaking study by Menachem Yaari in 1965, economists have called it "the annuity puzzle." Classical economic theory says that retirees should, in a perfectly rational world, prefer annuities over investments such as stocks and bonds.* But they don't. In fact, very few retirees hold annuities. Why? Is the theory wrong? Or are retirees making bad decisions?

There's been plenty of research on the topic since 1965, with no definitive resolution. Some recent studies tackle the problem from very different perspectives. While not written as advice to retirees, the studies do, as I read them, offer useful insights that can sharpen one's thinking about annuities. In this post, I'll explain the core idea from one study. In subsequent posts, I'll discuss additional studies and review what the new research may imply about whether and when a retiree may sensibly consider an annuity.

The first paper, published in the November 2015 issue of American Economic Review, is by Felix Reichling and Kent Smetters. It aims to resolve the annuity puzzle by examining "​health shocks that simultaneously affect longevity and increase uninsured health costs." Setting aside the elegant mathematics used in the paper, the core idea is quite plausible.

To illustrate the idea, suppose you had purchased a lifetime annuity at age 65. It has paid you a steady stream of income for a few years, just as promised. When you bought the annuity, you had good information about your health. You knew that people with similar health had only a small chance of dying right away and maybe a 50% chance of dying in their 70's. But now, at age 72, you've gotten what Reichling and Smetters call a "health shock." While on vacation in a tropical paradise, you contracted a local disease that nearly did you in. You survived, after a long stay in the hospital, and your future health, alas, now looks bleak. To put a number on it, your chance of dying in your 70's is approaching 90%.

Making matters worse, you had no Medigap coverage, and you needed care from some pricey specialists. Consequently, you've withdrawn much more than planned from your retirement savings. Your personal wealth is much diminished.

The core idea articulated by Reichling and Smetters, supported by mathematics and by data, is that when both these events occur, a health shock that increases the risk of dying plus costs not easily covered by insurance or savings, then annuities become less attractive than holding bonds or other safe investments. Should you live in a world where the combination of such events is plausible, you would, according to Reichling and Smetters, be rational to avoid annuities.

Nonetheless, in the Reichling-Smetters model, annuities continue to make sense if only one of the unhappy events occurs. Thus, in certain scenarios, annuities may still be attractive. Here are two such scenarios:

• You've already had a bad health shock. For you, the next shock would be if your health improves, or a medical miracle arrives in the nick of time, and you live longer than the few years you thought you had left. Or, as a less extreme case, your health may be weaker than average because of a chronic condition. Think about it. Wouldn't an annuity be helpful, just in case you find yourself greeting the sunrise far beyond what you pessimistically expect? That's essentially what Reichling and Smetters argue.
• Your savings and insurance are ample. You can afford to buy an annuity so generous that no health shock would devastate your finances. It might wreck your health, but you would remain financially secure. Alternatively, you may be so shrewd about health insurance and long-term care that you have sufficient coverage for all conceivable costs. Then, whether or not you ever suffer a health shock that actually triggers your insurance benefits, an annuity would be a sensible way to guarantee and stabilize your future income.

Let's revisit your hypothetical self, at age 65, before you took that disastrous vacation. If you were thinking like Reichling and Smetters, you would do one of two things, maybe both.

First, you would set aside some of your retirement savings as a reserve fund for uninsured medical costs. Or you would increase your insurance coverage for medical expenses and long-term care. Doing so would make you less vulnerable to the double-whammy of a health shock that reduces your longevity combined with uncovered costs that degrade your financial viability. To get started, read our article on reserve funds for retirees, and use our Retiree Reserves calculator.

Second, you would buy annuities only to the extent that, after setting aside some reserves or increasing your insurance, you still have ample savings. You would have to review your personal circumstances, perhaps in consultation with a financial professional who receives no compensation for selling annuities or possibly with sophisticated software like ESPlanner.

Aiming to set a benchmark, Reichling and Smetters presented a chart (their Figure 7) indicating that for a healthy 65-year-old, at least $500,000 in savings would be necessary before buying any annuity. With $1,000,000 in savings, the chart says that annuitizing half or more might be defensible. Smetters offered similar advice when interviewed by the New York Times. Given the level of savings available to most retirees, this advice would rule out annuities for a substantial majority.

You should, of course, make your own decisions and consult your own adviser. As one way to get a second opinion, Part 2 in this series reviews another recent study that took a different perspective. It examined the decision process that retirees may be using, perhaps unwisely, when they avoid annuities. You can also find a step-by-step process for evaluating annuities and insurance companies in our article on annuities for retirees, which was recently updated to incorporate ideas from this series of posts.

* To make the economic theory accessible, I am necessarily glossing over some technical details. For example, the classical model assumes that annuity buyers have no "bequest motive." They do not intend to set aside some assets to be inherited by a spouse, child, or charitable organization. To dig into the technical details on your own, you might start here and then consult the home page of Prof. Smetters.

Disclaimer: Historical data cannot guarantee future results. Although a mixture of bonds, stocks, and other investments may be safer than investing exclusively in one class of assets, diversification cannot guarantee a positive return. Losses are always possible with any investment strategy. Nothing here is intended as an endorsement, offer, or solicitation for any particular investment, security, firm, or type of insurance. You are responsible for your own investment decisions. Please read our full disclosures and Fiduciary Oath.

On February 2, 2016, we released an updated and improved version of Plan to Retire, one of our calculators. Whether you are 20 or 70 or any age between, the improved calculator will enable you to test out your retirement plans. You can ...

• See your forecasted income throughout the period between ages 60 and 70, when key dates occur for Social Security, pensions, and retirement withdrawals.
• Test different scenarios, and see how they will affect your savings and potential income.
• Study options for phased retirement by independently setting several key ages: when you start taking retirement benefits from Social Security, when you stop working, when a pension might start, and when you begin withdrawing from your retirement savings. You can make these dates all the same, all different, or any combination.
• See how your retirement savings will be affected if you increase or reduce the amounts you and your employer save each year, or if you start taking withdrawals.
• Test how investing more or less in stocks will affect the growth of your savings and the amount of your retirement income.

You can find this calculator and others on the Calculators menu, above.

This post is about the fiduciary duties of investment firms and advisers to represent their clients' best interests, and about the positions taken on this matter by the investment firms recently rated in our article Best Firms 2016.

In April 2015, the U.S. Department of Labor (DoL) proposed new regulations regarding conflicts of interest in retirement plans and the definition of investment advice. Among the most important objectives of the proposed regulations are to:

• Strengthen the responsibility of retirement plans and retirement advisers to act in their clients' best interest, and
• Extend fiduciary requirements to IRA providers and to rollovers between IRA and 401(k) plans.

The DoL solicited public comments on the proposal over the summer and held hearings in September. In December, Republicans in Congress failed in an effort to modify the omnibus spending bill with a rider that would have overridden the proposed new regulations. After reviewing the public comments, the DoL is expected to issue final standards this year.

To evaluate each firm's position on the proposal, I read their official submissions to the DoL and looked for news or commentary on the firm's website. The submissions ranged from brief to profuse, making it difficult to compare, for example, one that was limited but vaguely supportive versus another that detailed its clear support in some areas and constructive opposition in others. I've opted, therefore, to judge them by comparison to comments submitted by Alicia H. Munnell and Anthony Webb of the Center for Retirement Research at Boston College. The comments of Munnell and Webb are both strongly supportive of the DoL proposal and lucidly explained. I recommend reading them.

In the list below, I've ordered the firms from best to worst according to my own judgment of the degree to which they conformed to the Munnell-Webb analysis. I also gave a firm credit for disclosing its position on its own website, in a manner retail investors would could easily find and understand.

1. Betterment submitted strongly supportive comments, covering certain aspects of the proposal applicable to individual investors and 401(k) plans. They fully disclosed the firm's position on their own website.
2. Wealthfront submitted very limited comments, mainly supporting low-fee automated services for small investors. They fully disclosed the firm's position on their own website.
3. Fidelity submitted modest comments here and here, requesting revision of some aspects of the DoL proposal, largely on grounds of compliance cost and administrative complexity, grounds that Munnell and Webb explicitly reject. Fidelity did not disclose any information about the DoL proposal or the firm's position anywhere on their retail client website.
4. Charles Schwab submitted extensive comments here and here. They supported the intent and some specifics of the DoL proposal, and constructively offered clarifications, while conforming mostly but not fully to the main points advocated by Munnell and Webb. Schwab did not disclose any information about the DoL proposal or the firm's position anywhere on their retail client website.
5. Vanguard submitted extensive comments here, here, and here. They responded constructively when in opposition but rejected some significant constraints favored by Munnell and Webb. Vanguard did not disclose any information about the DoL proposal or the firm's position anywhere on their retail client website.
6. TIAA-CREF submitted very extensive comments here and some follow-on comments here, agreeing in concept with the need for clearer fiduciary standards but opposing many significant constraints favored by Munnell and Webb. TIAA-CREF posted an article about the DoL proposal on the firm's main website but disclosed no information about the firm's position.
7. BlackRock submitted extensive comments here. They objected to the DoL proposal, including many significant constraints favored by Munnell and Webb, yet they allowed a generally favorable blog-post on the website of their subsidiary, Future Advisor, in conflict with the parent firm's strongly opposing comments to the DoL.

A full appreciation of the proposed standard requires some willingness to delve into legal and financial technicalities. That said, if you want to learn more, here are some ways to get started:

• Pensions & Investments, a publication for executives who manage pension accounts and institutional investments, reviewed the DoL proposal in August 2015, explaining who's a supporter, who's not, and why.
• The Board of Certified Financial Planners and the Financial Planning Coalition have articulated their reasons for being strong supporters. 
• A complete list of the public comments is published on the DoL website. Using control-F in your browser, you can search for the comments of a particular firm or individual.

Disclaimer: Nothing here is intended as an endorsement or solicitation of any security or product from any particular firm or financial adviser. Please read our full disclosures.

​Within categories, different firms had different strengths. The highlights are listed below. For full reviews of each individual firm, see Best Firms 2016, on the Investing menu above.

• For services, Vanguard took top honors, scoring Excellent or Good for each of the component areas. Betterment was a close second, with strong scores on all components except college savings.
• On portfolios, Betterment and Wealthfront were tied at the top, because both offer well designed, indexed solutions, with best-in-class rebalancing, and at least some element of factor investing.
• Fees were rated best at Betterment and Wealthfront, thanks to their low-cost funds, fully transparent expenses, and automated advisory tools. Vanguard lagged slightly because their recently introduced advisory fees, while low compared to traditional financial advisers, are higher than the costs for comparable advice at automated firms.
• All investors, new and experienced, whether they seek to make their own decisions or get guidance, are well served at Vanguard. Betterment was next best in this category.
• For online content, including articles, graphics, and calculators, three firms achieved perfect scores: TIAA, Vanguard, and Betterment. Their websites are excellent places to find sensible ideas on a wide variety of personal financial goals and concerns.

When reviewing this information, please be aware of our disclosures and Fiduciary Oath.

Stock markets across the globe have been plunging lately, prompting all the news media and investment firms to speculate on reasons why and what to do. One sensible journalist, for example, concluded that the reasons are, alas, unknowable. And good investment firms like Betterment and Vanguard have sprinkled their websites with reassuring advice to stay on plan and not sell in a panic.

My take on the matter centers on this line of argument:

1. Volatility is a factor intrinsic to markets and economies. They occasionally suffer steep plunges, so-called "corrections" in markets and recessions in economies, sometimes in concert, sometimes not. Between the drastic dives, they more commonly grow in a positive and orderly fashion. But the dives will happen, sooner or later.
2. Predicting when they will happen, in advance, is hard. Maybe a few prescient souls can do it accurately for a while, but logically, if many of us could do it consistently, our collective actions would neutralize the effect or at least diminish it. We would be richer than we are.
3. Although we can't predict in advance, we can respond intelligently after the fact. One rational response is to rebalance. As Andrew Ang explains in his fine textbook, Asset Management, rebalancing a portfolio is, for strictly mathematical reasons, a counterweight to volatility. One good way to implement a rebalancing strategy is to monitor your portfolio monthly or quarterly, and realign it to your target allocations if the overall percentage of stocks has drifted 3% to 5% from your objective. Another way is to invest at a low-cost firm like Betterment or Wealthfront that does so automatically. (Learn more by reading our article on rebalancing.)
4. Additionally, volatility can be countered, somewhat, by investing in funds that specialize in finding low-risk stocks. Ang's book has more on this topic, and you'll find an overview in our article on factor investing.

To illustrate these ideas, consider the chart below. It shows three portfolios, all for the period from December 31, 2013, to January 20, 2016 (essentially, the last two years, up to the day before this post).

• Custom RB3% is a custom set of five low-cost mutual funds, rebalanced at the end of the month, if the portfolio's allocations drift by 3% from their targets. It was rebalanced twice previously, at the end of January and May 2015, and meets the criterion for rebalancing now.
• Custom RB5% is the same mutual funds, rebalanced monthly if the drift is more extreme, 5% or more beyond the target allocations. It was never rebalanced over the two-year period, but just now meets its 5% criterion for the first time.
• VSMGX is an all-in-one fund similar in composition to the custom fund. It has even lower costs than the custom portfolios, and is constantly rebalanced.

One implication from the chart is that rebalancing was effective when the custom portfolio drifted more than 3% to 5%, as compared to the VSMGX baseline. Granted, this one example would scarcely be adequate evidence, by itself. It aligns, however, with extensive historical analysis as reviewed in our article on rebalancing and with mathematical analysis as explained in Ang's textbook.

A second implication from the chart is about factor investing, which was the strategy used to select funds for the custom portfolios. Employing ideas outlined in two articles, Diversify! and Factor Investing (both on the Portfolios menu above), the custom portfolios consisted of:

• Vanguard Global Minimum Volatility (15%)
• iShares All-Country World Minimum Volatility ETF (15%)
• Vanguard Strategic Equity (15%)
• Vanguard U.S. Value (15%)
• Vanguard Long-Term Government Bond Index (40%)

This combination of funds is weighted for low-risk and for value and momentum, especially in smaller companies. Overall it is invested 15% in international stocks, 45% in U.S. stocks, and 40% in long-term Treasuries. Although well diversified, it is more selective than a broadly indexed strategy, as represented by VSMGX, Vanguard's LifeStrategy Moderate Growth fund. That fund currently invests 24% in a total index of international stocks, 36% in a total index of U.S. stocks, and 40% in two bond indexes representing all U.S. bonds and international government bonds. (In 2014 and early 2015, it had somewhat smaller international allocations.)

Although both the custom and VSMGX portfolios have a 60-40 split between stocks and bonds, and both implement a rebalancing strategy, their results in the recent two-year period were quite different, as the chart illustrates. Don't jump to conclusions, however. Future years may turn out differently than the last two, and the usual cautions apply: Diversification does not guarantee a positive return; any investment strategy can lose money; and your results may differ from those depicted here. The ideas in this post are presented for information only and are not meant to endorse a particular firm or security. Please read our full disclosures.

Taxes are such a headache.

Still, despite the pain, it's dumb to be dumb about taxes on your investments.

​I've assembled a new article, Being Tax-Wise, that tries to capture the main points to consider when planning your investments, so as to make them a notch more tax-savvy.

Some of the ideas are obvious: Save for retirement in an IRA or your employer's plan. Others, not so obvious: should you have both an IRA and your employer's plan?

Some ideas may be surprising: Did you know you can be the beneficiary of a 529 college savings plan you establish for your own graduate degree, and possibly get a state-tax rebate for doing so?

And, yes, some ideas are downright convoluted: Being tax-smart about retirement spending will keep you mentally fit, if your frontal lobe is ready to be exercised.

The article closes with suggestions on where to turn for tax-advice and second-opinions. In an area like taxes, where you are the one ultimately accountable for the decisions, professional advice and second-opinions would be smart for everyone. You can find the full article on the Investing menu.

The Portfolios menu has been expanded with new and updated articles on how to build a diversified portfolio of investments. The main ideas are in Diversify!, which uses specific examples to illustrate how to diversify across stocks and bonds, domestically and globally. Factor Investing digs deeper, showing how retail investors can build simple portfolios to capitalize on factors from academic research, such as value, momentum, and low-risk. And Finding Value explains how to use the able2pay.com calculators to adjust stock and bond allocations if U.S. stocks are over-valued (as they are today) or under-valued (as might happen in the future).

The article also describes, step by step, how you can implement your savings plan at two good firms:

• Betterment gets the top rating, because of its sophisticated online apps and simple process.
• A good, low-cost strategy is also explained for Vanguard, though it requires a bit more effort.

​You can find the full article on this and other topics on the Goals menu.

The best and most popular news-posts at able2pay.com are being updated and moved. To see what's available, click on Retirement or on the Goals, Investing, or Portfolios menus at the top of this page. Included under Portfolios is an article on Basic Portfolios that amplifies the post immediately below.

Do you want  guidance on funds to buy in your 401(k), 403(b) or IRA, college-savings, or investment account? Here's an easy method from able2pay.com.

• Answer two questions to set targets for stocks and bonds, suitable for your goals and preferences.
• Allocate to three basic funds, which any firm will offer.

Step 1. Set a target percentage for stock funds.
When do you expect to spend most or all of the investments in this account?

• All in the next year or two: Set final target = 0% and stop (skip the next question).
• Most or all within five years: Set preliminary target = 30% and continue.
• Six or more years from now: Set preliminary target = 60% and continue.

What is more important to you, maximizing gains or minimizing losses?

• Maximize gains: Add 15% to your target.
• Minimize losses: Subtract 15% from your target.
• Do both: Keep your preliminary target (no change).

You now have a target of 0%, 15%, 30%, 45%, 60% or 75%. This is the amount you will allocate to stock funds. The rest will go to bond funds. (If you want more finely tuned values, you can get them with our Best-Invest calculator.)

Step 2. Allocate to suitable stock and bond funds.
Use your target to find the matching row in the table below. Then use the percentages in your row to invest in the funds shown by the column headings. Most 401(k), 403(b), IRA, college-savings, and investment accounts will offer these funds or equivalents. Below the table are descriptions that will help you find the right ones for your account.

• U.S. Stocks: Use an index fund that invests in the total U.S. stock market. If not available, use an index fund of large-cap U.S. stocks or the S&P 500. Avoid anything that's not indexed or that's more specific.
• International Stocks: Use an index fund that invests in all non-U.S. stock markets. If not available, use an index fund of EAFE stocks (developed countries outside the U.S.). Avoid anything that's not indexed or that's more specific.
• Intermediate Bonds: Use a bond fund that invests primarily in a broad mix of U.S. treasury and corporate bonds. Avoid funds with a limited focus on TIPS or on inflation-linked, international, high-yield, or short-term bonds. In a 401(k), 403(b), or IRA account, you should also avoid municipal bonds.
• Short-Term Bonds: Use a fund of U.S. treasury bonds with maturities less than five years. If not available, use instead a short-term bond index fund or a money-market fund.

Optional Step for IRA and Taxable Accounts
The following options are good improvements for an IRA or taxable account, but are unlikely to be available for your 401(k), 403(b), or college-savings plan:

• Invest at Betterment, and simply use the target from Step 1. Betterment's automated service will invest in a well-designed set of U.S. and international stocks and bonds.
• Invest  at Vanguard, using the alternate table below and these funds: Global Minimum Volatility; either Mid-Cap Value Index or Strategic Equity; Intermediate-Term Bond Index; and Short-Term Treasury.
• Invest in a low-fee brokerage account such as Fidelity or TD Ameritrade, using the alternate table below and these Exchange-Traded Funds (ETFs): iShares MSCI All-Country World Minimum Volatility (ACWV), Vanguard Mid-Cap Value (VOE) or equivalent; Vanguard Intermediate-Term Bond (BIV) or equivalent; Vanguard Short-Term Government Bond (VGSH) or equivalent.

To learn more about global low-volatility stocks, mid-to-small value-stocks, and Betterment's portfolio, see the article Diversify! on the Portfolios menu.

Part of a series on retirement, this post explains how to use our calculator, Retiree Reserves. For other articles in the series, visit the Retirement tab. 

Who should use the Retiree Reserves calculator? It is mainly intended for people who are retired or expect to retire in a year or two. However, you may also find it helpful if you are in your late 50's or early 60's and are considering whether to buy long-term care insurance.

What does the calculator do? It generates average and high-end estimates of the dollar value to hold on reserve for potential future expenses during retirement. These are potentially large expenses not included in your normal budget. In addition, the calculator recommends how to invest your reserve funds. All the calculator's estimates and recommendations take into account the life expectancy for your age and gender.

What potential expenses will the calculator estimate? Three types of expenses are covered: health care other than insurance premiums, long-term care services, and unusual housing expenses such as major home repairs.

How does the calculator estimate these unknowable costs? It uses data from various public sources on your expected longevity; on the expenditures of people over 65 years of age for medical costs, long-term care services, and household maintenance; and on home prices. If you own a home, condo, or coop, the calculator uses your home's value as input for the estimates. Optionally, the calculator allows you to personalize some inputs, such as long-term care in your area, if you wish.

What if you rent your home or apartment? In that case, the calculator will omit housing maintenance from your estimated reserve fund, since rent is part of your normal budget.

How does the calculator handle long-term care? The average estimate is based on government data regarding the amount actually paid for long-term care by people age 65 and older. The high-end estimate assumes two years of a private room in a nursing home, at rates based on a recent, nationwide survey (or a rate you specify).

This post was part of a series which is now located on the Retirement tab. The series covers:

• The finances of Rachel, a hypothetical retiree, and the decisions she had to make.
• Key goals for any retiree.
• How to estimate a retirement budget.
• Home equity as a retirement asset. 
• A reserve fund during retirement, mostly for medical expenses.
• Annuities that do (or don't) make sense for retirees.
• Managing income and payouts from your retirement savings.

Some economists argue that during retirement, you should be "dis-saving," by which they mean spending down your assets, not holding reserves or saving for the future. They might also argue that "mental accounting" is irrational. From the standpoint of objective financial analysis, they would say your decisions can go awry if you mentally set aside certain funds to be spent only for certain purposes. Money is fungible, they point out. It can be used for any purpose, so spend it where you need to, constrained only by a goal to maintain a stable standard of living.

While in principle one might agree with these notions, in practice most of us do otherwise. We feel safer if something is set aside to cover the unexpected. Whether it's called a reserve fund, an emergency fund, or a sheltered account, it's a safety cushion whose purpose is as much psychological (for peace of mind) as financial (for budget planning).

Because the fund's objective is to pay for the unexpected, it is really a form of self-insurance. During your working years, a reserve fund mainly offers protection against losing your job. Thus, it's unemployment insurance that you devise for yourself and your family. During retirement, the main risk of unplanned expenses is for health care not covered by Medicare or other insurance. To a lesser extent, there may also be risks of occasional large expenses for a home you own, if not covered by your home-owner's insurance.

To plan a reserve fund for your retirement, ask yourself three questions:

• What potential expenses should you self-insure with a reserve fund?
• What amount should you reserve to cover those expenses for your unique circumstances?
• Where should you hold and invest your reserve fund?

For worked examples that illustrate how to answer these questions, click here to go to the full article.

Question: What bias affects both the cost-of-living adjustment (COLA) from the Social Security Administration (SSA) and the oft-cited 4% Rule for retirement income?

Answer: Both use last year's inflation. They look back when they should be looking ahead.

Admittedly, inflation in recent months is somewhat correlated with inflation a year from now. Since 1914, the Consumer Price Index (CPI-U) in any given month has had a correlation of 0.54 with inflation 12 months ahead. Over the last 30 years, however, the correlation has declined to 0.21, a value so low that it's next-to-worthless for making predictions.

By looking backward to predict forward, both SSA and the 4% Rule are, in effect, adjusting in arrears.  SSA waits until inflation has happened, and then modifies payments as of January the following year. Similarly, the 4% Rule advises that late in the year, you should find the past year's inflation and increase your retirement spending by that amount in the year to come. In both cases, the inflationary train has left the station and reached the next stop on its journey. You can catch the next train, but you'll always be one station behind.

A better idea, clearly, would be to use an accurate predictor of next year's inflation, something better than simply assuming that the future will mimic the recent past. How might one do that?

One idea is to use the median (middle) estimate of experts whose livelihood depends on precisely forecasting future inflation. As it happens, this method is easy to implement and reasonably accurate. The Philadelphia branch of the Federal Reserve Bank conducts a survey of professional forecasters and publishes the results quarterly. Over the last 30 years, the correlation between this forecast and the year-ahead value of the broad CPI-U measure of inflation has been 0.52. As shown by the chart below, the survey tends to ignore brief divergences, up or down, and predict something like an average trend.

​Indeed, the survey correlates extremely well (0.92) with future core inflation, as measured by the CPI without its volatile food and energy components. In effect, the forecasters, although asked to predict the broadest measure of inflation, CPI-U, are actually estimating core inflation. That's good because, as argued previously in this series, core inflation is a good way to manage a retirement budget. Our Safe-Payout calculator therefore uses the survey forecast as the recommended COLA for those who set their retirement budget with the "collared inflation" method (a safe variant of the 4% Rule).

Is it possible to do even better than the survey forecast? After all, the Philly Fed is not Lake Wobegon, "where all the children are above average." In the survey, some forecasters must do better than others, but we aren't told who they are or whether their superior performance persists over time. Cherry-picking the best forecasters would be perilous, admittedly, because statistical theory says they will tend to regress, doing less well in the future than they have in the past.

With that precautionary observation in mind, take another look at the chart above. It offers an additional forecast based on an econometric model from academic research. Shown by the red line in the chart, the model's correlation with the broad CPI-U metric has been 0.83 since 1978. The model is a simplified version of the so-called "triangle model" of Robert J. Gordon, an economist at Northwestern University who developed the model and has applied it to data for a 51-year period from 1962 to 2013. 

In the simplified form shown here, the model says future inflation depends on:

• Inertia, or the tendency of core inflation to persist. While Gordon uses a weighted average of the previous five years of inflation, my analysis of data since 1978 found equally good results by simply using the most recent 3-month average of the trimmed mean PCE measure of inflation.
• Demand for employment, as measured by the gap between short-term unemployment and a "natural rate." If, say, only 3% of the labor force has been unemployed for six months or less, and the natural rate is higher than that, then jobs are so easily found and quickly filled that upward pressure on wages and prices will soon push inflation higher.
• Supply of critical goods, which can cause inflation to fall or rise if temporary supply-shocks boost or diminish the availability, and thus affect the price, of those goods. In the simplified model, I used just one value, which had statistical significance for the period 1978-2015: an average of the energy component of the CPI for the previous quarter.

As developed by Gordon, the triangle model was designed to predict inflation one quarter ahead (three months), not a full year. In fact, the simplified version did very well indeed, when predicting next quarter's CPI-U, both since 1978 (0.93) and for the more difficult period of the last 30 years (0.74). However, when predicting inflation a year ahead, over the past 30 years, the triangle model did no better than the middle-of-the-pack for professional forecasters.