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Journal Of Investment Management, Vol. 17, No. 2, (2019), pp. 1–23

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A MODEL OF BOND VALUE*EXPLAINING YIELDS WITH GROWTH AND INFLATION

Thomas Shevlina

This paper looks to establish a new heuristic for investors, giving them a simple, intuitiveway to relate bond yields to prevailing trends in growth and inflation. The model offersan alternative to forecasting surveys, which have been over-estimating 10-year Treasuryyields for decades and continue to project yields above 4% in the long run. The model doeswell in in-sample and out-of-sample tests used in the literature to evaluate other measuresof value. The model can be used on its own or in conjunction with other models to forecastyields and also as a benchmark to evaluate yield forecasts. The model is consistent withsome of the more advanced economic models of interest rates that suggest that the lowbond yields of recent years are in line with broader economic trends, rather than due totemporary factors that are likely to reverse quickly.

1 Forecasting bond yields

The prediction of future bond yields is a perennialproblem in active management. The difficulty itpresents is well described by Fong (1983), whostates: “it is at best, extremely difficult to fore-cast the future direction of rates, much less theirmagnitude. . . (But,) regardless of one’s convic-tion, rate anticipation must be dealt with if for noother reason than that it is usually the dominant

∗Disclaimer: The views expressed herein are those of theauthor and do not necessarily represent the views of theauthor’s employer.aLiberty Mutual Investments, 175 Berkeley St., MS T21A,Boston, MA 02116, USA. Phone: (857) 224-8470. E-mail:[email protected].

source of incremental return. . .” for bond port-folios. In looking to anticipate yields, investorscould turn to published surveys of bond yieldforecasts. However, over the past two decades,forecasters have generally over-estimated bondyields. For example, every December the Liv-ingston survey provides a 2-year forecast of the10-year bond yield.

Livingston is a survey of professional forecast-ers published by the Federal Reserve Bank ofPhiladelphia and is considered to be of highquality. The survey’s median forecast has over-estimated the 10-year yield for the past 18 con-secutive years, and has only under-estimated bondyields once during the entire 24 years the survey

Second Quarter 2019 1

2 Thomas Shevlin

Figure 1 The figure shows Livingston 2-year forecasts of 10-year bond yields vs actual bond yields 2 yearslater. The figure shows a clear upward bias in forecasts.

has been conducted. On average, each fore-cast has over-estimated bond yields by 119 basispoints, and if the forecasts were the output of aregression, its R-squared would only be 0.22.1

As Figure 1 demonstrates, the forecasts exhibita clear upward bias, because the purple dots (theex-ante forecasts) are well above the blue line (theex-post bond yields).

Over-estimating bond yields is a common themeamong different forecasting surveys. For exam-ple, the President’s Council of Economic Advis-ers (CEA) published a paper on the topic in 2015.It includes a chart to demonstrate that economists’forecasts have been substantially over-estimatingyields for several decades. The chart showsthat forecasters still generally expected yields toreturn to the 4–5% range in the long run.2 This isconsistent with many long-term projections usedin the industry. For example, BNY Mellon, inits 10-Year Capital Market Return Assumptionsfor Calendar Year 2016 states, “In the US, wesee Treasury yields rising until they reach a nor-malized level in six years.”3 This statement isfollowed by a chart showing 10-year yields risingabove 4% in that time frame.

Although it does not give annual forecasts out thatfar, the Survey of Professional Forecasters (SPF)does give a 10-year average projection as well asannual forecasts out several years. In Figure 2,by combining long- and short-horizon forecastsfrom the SPF, we can see implied expected path-ways in yields that are very similar to those in theCEA’s chart and consistent with the BNY Mel-lon’s 10-year projection for 2016. The SPF 2017forecasts also imply yields rising above 4% in theout years. Like the CEA chart, we can see that theforecasts appear upwardly biased. We can also seea continuing belief that yields will eventually riseabove 4%.

Given the poor performance of forecasts, invest-ment decision-makers have good reason to beskeptical. Investment decision-makers may beable to benefit from a heuristic or rule of thumbthat gives a simple expression of bond yield“value” in terms of broad, fundamental economictrends. Although interest rate models vary in theirinputs, the Council of Economic Advisors (2015)lays out both inflation and economic growth (orcomponents of economic growth, including pro-ductivity, population, and technology growth) as

Journal Of Investment Management Second Quarter 2019

A Model of Bond Value 3

Figure 2 The figure shows Survey of Professional Forecasters’ projected pathways of 10-year bond yieldsover 10-year horizons vs actual bond yields that occurred. It shows a clear upward bias and also shows thatforecasters still expect yields to rise above 4%.

the principal drivers of interest rates supportedby economic theory.4 Moreover, in their sur-vey of the drivers of interest rates, Rachel andSmith (2015) note that “. . .changes in global trendgrowth are probably the most commonly-citeddriver of changes in real interest rates.”5 If wewant to provide a simple metric, establishing anempirical link to inflation and growth would seemlike a good place to start.

2 Literature review

Inflation and growth both have deep founda-tions in economic theory as drivers of yields.Fisher (1930) is commonly cited as linking nom-inal interest rates to inflation with the followingapproximation:

i ≈ r + π (1)

where: i ≡ nominal interest rate

r ≡ real interest rate

π ≡ inflation.

Solow (1956) presents the Solow Growth Modelwhich relates growth to real interest rates.

Ramsey (1928) relates per capita GDP growth toreal interest rates rather than total GDP growth,but Baker et al. (2005) point out that the reasonRamsey uses per capita GDP is due to unrealisticsimplifying assumptions. Although there is somedisagreement in the literature as to whether therelationship should be with growth of total GDPor per capita GDP, a link between interest ratesand growth appears well grounded in economictheory.

In the case of equity prices, there are broadmeasures of long-run value available. For exam-ple, Wright (2004) proposes several measuresof Q based on the original concept of Tobin’sQ, and also analyzes dividends and cash-flowyield. Likewise, Campbell and Shiller (1998)examine dividend–price ratios as well as whathas become known as the “Cyclically AdjustedPrice to Earnings Ratio” or “CAPE.” The CAPErelates equity prices to trailing 10-year averagereal earnings. By using a 10-year average in thedenominator, the method reduces the volatilityin earnings coming from the business cycle. Intheir study, Campbell and Shiller find that both the

Second Quarter 2019 Journal Of Investment Management

4 Thomas Shevlin

price–dividend ratio and CAPE were predictive offuture equity market prices. They used the CAPEto argue that equity prices at the time were veryhigh relative to historical norms which they sawas an omen of low returns.

Both Q and CAPE are an attempt to link prevail-ing equity prices, which are in principal forwardlooking, to concrete, backward looking, funda-mental measures of intrinsic value in order toallow one to judge whether prices are “too high”or “too low.” In both cases, these measures maydeviate significantly from their average or “fairvalue,” for extended periods of time. Neverthe-less, those investors focused on long-run returns,or long-run measures of value, find these mea-sures useful, even if their ability to forecast equityreturns over short investment horizons is limited.

Asness, Ilmanen, and Maloney (2017) examinethe effectiveness of the CAPE for forecastingreturns on an “out-of-sample” basis. They find“a puzzling gap between encouraging in-sampleevidence and disappointing out-of-sample per-formance of value timing strategies” using theCAPE and other value indicators. In the case ofCAPE, they find that a long-term trend towardhigher valuation causes a value-weighted strategyto be under-invested, on average, over a 60-yearperiod when equity valuations were generally ris-ing. They show that it is possible ex post to removethis effect and demonstrate the benefit of the valuesignal if it does not lead to a bias to be underex-posed in markets exhibiting a trend toward highervaluation, but note that this adjustment would beimpossible to do ex ante. Nevertheless, they findthat incorporating a momentum component intothe trading strategy does allow one to achievesuperior returns, so the value measure is still ofbenefit, even if by itself it does not ensure supe-rior returns. In the Appendix, the trio also lay outa bond value measure akin to the measures forequities. Using expected inflation, they calculate

the real interest rate of nominal 10-Year Trea-suries. They use inflation forecast surveys fortheir expected inflation variable as far back asthe series go and then use a statistical techniqueto approximate expectations in prior decades. Inthis case, they find a value timing strategy toexhibit superior returns to a buy-and-hold strat-egy but also that adding a momentum componentimproves returns still further.

Although imperfectly analogous, Taylor (1993)presents the “Taylor Rule,” which attempts todescribe the levels of central bank target rates.Central bankers and investors can use this frame-work as a relatively straightforward way of judg-ing whether short-term rates are “too high” or “toolow.” This judgment is based on how these rateshave typically related to prevailing economic con-ditions in the past. Although Taylor acknowledgesa lack of consensus about the size of the coeffi-cients for policy rules, he nevertheless sets fortha representative framework for such rules usinground numbers as representative parameters. Tay-lor relates the Fed Funds Rate positively to therate of inflation over the previous four quarters,the deviation of GDP from potential GDP, thedeviation of inflation from a target of 2%, anda 2% constant term which is in effect a neu-tral short-term real interest rate. Taylor presentsa chart to demonstrate how the parameters hechose for the rule do a good job of describing thelevel of central bank rates over the 6-year periodof 1987–1992. Although deliberately simple innature, the framework has had a profound impact.Unlike the long-term equity value measures wehave discussed, however, the Taylor Rule doesseek to adjust for some cyclical factors by includ-ing in its formulation both deviation from theinflation target and economic growth relative topotential.

Laubach and Williams (2003) present a modelframework for determining the appropriate



“natural rate of interest” or “r*”, which representsthat real Fed Funds rate that is consistent with sta-ble inflation and economic output being equal toits potential. This rate corresponds to the interceptterm in feedback rules such as the Taylor Rule andallows economists to study how this parametermay vary through time. The Laubach–Williamsmodel figures prominently in the economic liter-ature on interest rates. The natural rate of interest(or r*) is a cycle-neutral concept, and in this sensewould be more akin to CAPE than theTaylor Rule.However, the model is extremely sophisticatedand its recreation would be beyond the technicalexpertise of many investors.

There are different models for calculating theinflation-neutral short-term real interest rate. Inan economic letter from the Federal ReserveBank of San Francisco, Bauer and Rudebusch(2016) show three such models (including theLaubach–Williams model), charting them overthe 2000 to 2016 time period. The chart showsthat the different models of the neutral short-term real interest rates can be as much as 2percentage points apart at a given point in time.Directionally, however, the chart shows broadconsistency across the different measures, withall three falling considerably from 2000 to 2016,with the Laubach–Williams model falling fromabout 3% to about zero.

In looking for a simple rule or heuristic for defin-ing “normal” or “fair value” at the long end, onemight consider the Golden Rule, which uses theSolow Growth Model framework. Having takenthe care to lay out the Golden Rule in a paperwritten as a fairy tale, Phelps (1961) gives apleasant read. The paper shows that under certainassumptions, utility is maximized across genera-tions when the real interest rate related to capitalinvestment is equal to the potential growth rate. Itwas in the sense of “doing unto others as to one-self” because each generation following the rule

is maximizing consumption across all generationsrather than just its own generation.

The reasoning behind the Golden Rule is essen-tially that in equilibrium, the marginal product ofcapital is equal to the natural growth rate. Firmswould in all likelihood expect a return at leastequivalent to the rate of GDP growth when choos-ing to undertake an investment. Otherwise theywould seek investment opportunities elsewhere.Since the marginal product of capital will also beequal to the marginal cost of capital in equilib-rium, this implies that the marginal cost of capitalfor the economy as a whole will equal the naturalrate of growth. So as long as the marginal investorcan borrow at or near the risk-free rate, then it isnot unreasonable to expect real long-term ratesand real GDP growth to gravitate toward eachother over time. This leads us to the followingapproximation:

i10Y ≈ g + π (2)

where: i10Y ≡ nominal 10-year government

bond yield

g ≡ real growth rate

π ≡ inflation.

Bordo and Dewald (2001) examine 13 countriesand use the Golden Rule framework to arguethat inflation expectations were higher during theperiod of 1962–1995 than during the period of1881–1913. Looking at bond yields, they useaverage GDP growth rates across the two peri-ods to isolate inflation expectations by assumingthat “in longer-term equilibrium the real rates ofinterest would equal the growth rate of real out-put.” They go on to note that “this is a commonassumption in both theoretical and empirical stud-ies.” Examining differing periods of rising andfalling inflation, they find that bond yields tend tofollow inflation trends up and down, but do a poorjob of predicting rising or falling inflation trends.


6 Thomas Shevlin

Another study which looks at the relationshipbetween yields and growth is Lilico and Ficco(2012). They analyze the ability of inflation-linked bond yields to predict economic growth.They note that the Golden Rule is a “key equi-librium condition” both in economic theories ofsustainable growth and corporate finance theory.However, they go on to note that the rela-tionship between growth and real rates shouldnot necessarily be one-to-one but that “. . .oneshould expect both that there is some relation-ship between the levels of the risk-free rateand the growth rate. . . and that, over a suffi-ciently long timescale, changes in the risk-freerate to be correlated with changes in the sustain-able growth-rate.” Indeed, Laubach and Williams(2003) estimate the relationship between potentialgrowth and the neutral real short-term interest rateto be near, but not precisely, one.

Even in the case of inflation, there is researchwhich would support the view that the rela-tionship between yields and inflation is notnecessarily one-to-one. For example, Feldsteinand Summers (1978) argue that the relation-ship between bond yields and expected inflationshould be greater than one due to tax distortions.A slope greater than one is also consistent withthe work of Campbell et al. (2013). They breaknew ground by modeling the covariance of equi-ties and bonds and relating it to the evolvingrelationship between inflation and growth. Theirmodel finds a positive nominal bond risk premiumduring the early 1980s and a negative nominalbond risk premium in 2000 and especially duringthe 2007–2009 financial crisis. This is becausebond prices tended to be positively correlated withequities during the periods of higher inflation butwere negatively correlated during periods of lowinflation or especially when there was a threatof deflation. During these periods the bonds pro-vided a hedge against equity volatility. Althoughan over-simplification of their work, one could

characterize it as finding a positive risk premiumin bond yields when inflation is high and a nega-tive risk premium when inflation is low or thereis a risk of deflation, which would be consistentwith a slope coefficient greater than one in thiscontext.

Although the Golden Rule may be a commonsimplifying assumption, the evidence clearlysuggests that it may be too simple for our pur-poses. Given the widespread presumption thatyields should have a one-to-one relationship withgrowth and inflation, however, it may be rationalto expect the relationship to be near one-to-one,if not exactly one-to-one.

3 Bond value model

For the purposes of creating a heuristic to explainbond yields broadly in economic terms, giving usa measure of “normal” or “fair value,” let us beginwith the well-established drivers of GDP growthand inflation in the following equation:

i10Y = β0 + β1 ∗ π + β2 ∗ g (3)

where: i10Y ≡ nominal 10 − year government

bond yield

g ≡ real growth rate

π ≡ inflation.

Just as one can look at price-to-earnings ratiosusing both forward and backward earnings, onecould try to construct a measure of fair value foryields using forward or backward measures ofgrowth and inflation. Although forecasts of infla-tion and growth may appear to be an obviousinput to a forward looking model, these surveyshave a number of drawbacks. Ideally, one woulduse 10-year projections of growth and inflationto project 10-year yields, but these are infrequentand have a limited history. This would reduce thesample size, eliminate the 1980s and earlier fromthe field of study, and only allow for infrequent



model updates. No doubt the limited availabilityof expectations data is a principal reason manyeconomic studies, including Bordo and Dewald(2001) and Rachel and Smith (2015), use realizeddata instead. Using realized inflation and growthdata would also be consistent with Bordo andDewald’s (2001) findings that bond yields tendedto follow inflation trends up and down but did apoor job of anticipating “impending changes ininflation trends.” Another, perhaps minor, pointis that whereas past GDP and CPI are objec-tive measures, expectations may vary, and onecannot be certain that the expectations that aredriving the bond market are identical to those ofpublic forecasters. Nevertheless, there are twoforecast-based models discussed in theAppendix.

The original Taylor Rule and Laubach–Williamsuse 1-year trailing inflation as a proxy forexpected inflation. Although an approach suchas this would have the benefit of being simple,timely, and frequent, it would have the drawbackof being noisy. The Taylor Rule and Laubach–Williams also use measures of potential GDP intheir formulation of equilibrium interest rates.However, creating estimates of potential GDP isdifficult, especially in real time, and the resultsone gets will depend upon the construction ofthe model. In considering backward lookingapproaches, therefore, the Campbell and Shiller’s(1998) use of a 10-year moving average hasappeal. It is also timely and easy to calculate,and recent trends in growth and inflation wouldbe important components in the formulation ofexpectations. Plus, the length of time shouldlargely eliminate the impact of the economiccycle. It is also not uncommon in economics touse such averages when studying the relationshipbetween broad economic conditions and inter-est rates. For example, Rachel and Smith (2015)note, “One reason market measures of the globalreal rate may deviate from the long-run equilib-rium (neutral) rate is due to cyclical factors. To

sidestep this issue, we focus on very low fre-quency movements in the data.” They use both 5-and 10-year averages in their study to avoid theseproblems.

As we would like a heuristic which gives a broadsense for whether current bond yields are indeedconsistent with longer-term economic trends, letus use the trailing 10-year compound annualgrowth rate (CAGR) for both real GDP and CPI.Even though economists sometimes use “trendgrowth” synonymously with “potential growth,”for the purposes of simplicity please allow us torefer to the 10-year trailing CAGRs of real GDPand CPI as “trend growth” and “trend inflation”in this paper. Results of using these two variablesin a regression to model bond yields are displayedin Figure 3 and Table 1.

Using nothing but the compound annual growthrates of GDP and CPI, this basic econometricmodel explains 89%6 of the level of 10-yearbond yields since 1960 (red line in Figure 3).Because of serial correlation, it is necessary touse Newey-West standard errors to ensure weare not over-estimating our t-statistics, but evenusing these, we can see that all three coefficientsare significant at the 99.9% level of confidence(see Table 1). Also using Newey–West standarderrors, we can create a 95% confidence intervalfor the 10-year bond yield’s “intrinsic value” (reddotted lines in Figure 3). Although there is sub-stantial volatility within this confidence band, wecan see that yields were only briefly outside thisband during the sample period.

Although the red line may do a good job explain-ing bond yields from a long-term perspective, itmay strike readers as less useful for explainingbond yields in the short run. This is naturally trueof many measures of value. For those interestedin forecasting yields over short time horizons(e.g., 1 year or less), one could naturally con-struct a forecasting model with a shorter sample


8 Thomas Shevlin

Figure 3 The figure shows the Bond Value Model’s estimates of equilibrium bond yield as well as error bands,vs the actual 10-year bond yield.

Table 1 Results of bond value model.

(Sample Period: 1960–2017)

Latest Value Coefficient T-Stat (Newey-West) Significance Level

Trend Growth 1.43 1.1 14.7 99.9%

Trend Inflation 1.61 1.2 21.3 99.9%

Intercept N/A −1.8 −6.3 99.9%R-Squared 0.89

period, which would broaden the number of timeseries available for the model. Moreover, giventhe autocorrelation in bond yields, one might alsoprefer a model with an autoregressive componentor to forecast changes in yields rather than levelsof yields. However, for those interested in under-standing broad interest rate regimes and having anintuitive sense for how bond yields today reflectbroader economic trends, such models may notbe completely satisfying. After all, as Achen(2001) points out “lagged dependent variables(i.e., yields of the prior month or year in this case)have no obvious causal interpretation”7 leadingresearchers to omit them. He also points out that

although including autoregressive components inthe model can lead to a very high R-squared, itcan also suppress the explanatory power of theindependent variables (in this case growth andinflation) and can cause these independent vari-ables to have “implausible” coefficients. Finally,if one’s objective is a heuristic that tells onewhether current bond yields are too high orlow to be consistent with economic trends, thenincluding an autoregressive component wouldundermine that objective.

Although deliberately simplistic, this model doesdo surprisingly well forecasting out-of-sample.



For example, using the Livingston Survey, onecould combine 2-year forecasts of growth andinflation with 8-year trailing CAGRs in order toproject where the CAGRs would be in 2 years’time. If one used only data available at the timeto estimate the coefficients, one could use thisas a forecast for bond yields. The results are theazure dots in Figure 4. Over the 24-year sample,the azure dots only over-estimate bond yields by41 basis points on average. In contrast, the pur-ple dots, which represent the 2-year forecasts foryields taken directly from the Livingston Survey,overestimate bond yields by 119 basis points onaverage. Using the methodology of Campbell andThompson (2005) to calculate “out-of-sample”R-squareds, we get a result of 0.63 by combiningthe 2-year CPI and GDP forecasts with the 8-yearCAGRs and using the bond value model. Thiscompares favorably with the 0.22 we get whenusing the 2-year bond yield projections takendirectly from the survey.

Looking at results “out-of-sample,” or projectedforward using information only available at thetime of projection, is extremely important. This is

because many models appear to be a good fit whenlooking at data in the sample but prove themselvesutterly useless at predicting the future. Comparingthe out-of-sample R-squareds, it seems clear thatinvestors would have been better off using thisbond value model approach, combined with infla-tion and growth forecasts available at the time,rather than using the median bond yield forecastsavailable. This result is surprising, as the bondvalue model only uses trend growth and inflationas inputs, whereas forecasters would be privy notonly to this information, but also a great deal moreinformation for use in formulating their forecasts.For investors, of course, the accuracy of projec-tions has implications for investment results, andso using the best possible projections for makinginvestment decisions is a matter of importance.Of course, in evaluating forecasters or forecastingmodels, it would be helpful to have a benchmark.

Investment decision-makers may find the bondvalue model to be a useful benchmark for evalu-ating forecasters or forecasting models. If a givenforecast or model cannot do at least as well as thisheuristic in out-of-sample tests, then the investor

Figure 4 The figure shows that using the Bond Value Model with Livingston 2-year projections for growthand inflation (azure dots) would have given a much better forecast of 10-year yields than Livingston’s actual2-year forecasts of 10-year yield (purple dots).


10 Thomas Shevlin

may prefer not to use that forecast or model.Moreover, the bond value model can also be a use-ful input when formulating a forecast or as a sanitycheck on a forecast. By starting with a broad senseof what bond yields would be consistent with pre-vailing trends in growth and inflation, one canthen add cyclical factors, supply and demand fac-tors, and/or prevailing yields in order to arriveat a more comprehensive forecast. Furthermore,investors can use Newey–West consistent confi-dence intervals to give a sense for how far yieldsare likely to deviate from the model.

4 Comparisons with other measures of value

A key motivation of the bond value model out-lined in this paper is to create a simple metricthat gives a broad sense of value, similar to thoseused to assess equities. In many cases, measuresof equity value are ratios of the equity price tosome variables such as earnings, dividends, orcash flows. The sense of “cheap” or “rich” or “fairvalue” often comes from how far this ratio hasdeviated from its average. The bond value modeloutlined here generates a measure of the equilib-rium yield. The sense of over- or under-valued inthis case would come from how far the currentbond yield is from the equilibrium yield.

As noted earlier, Campbell and Shiller (1998)study the ability of the price-to-dividend ratio andthe CAPE to explain future changes in real equityprices. To demonstrate the value of these mea-sures, they showed scatter plots of both CAPEand dividend ratios against future changes in thereal price of the S&P 500 over 1- and 10-yearhorizons. They showed trend lines in these scat-ter plots and used the R-squared of these trends toshow the percentage of variation of future equityprices explained by the ratio. They found thatboth measures did better at predicting the changein real equity prices over 10-year horizons thanover 1-year horizons, but that CAPE’s predictivepower was superior to that of the dividend ratio.

In order to determine the value of this bondmodel as a long-term value measure analogousto those available for equities, let us use thesame approach as Campbell and Shiller (1998).Fortunately, updated versions of these measuresare available on Shiller’s website so we can usedata that match their methods exactly over thesame sample period. In comparing the bond valuemodel, we can estimate the model on a monthlybasis beginning in 1980 in order to have at least 20years of data to estimate the coefficients, and thenexpand the sample with each new period. Sinceeconomic data are released with a lag, let us alsocalculate the model with a 3-month lag. We canthen compare the prevailing yield at the end ofeach month with bond value model’s estimatedyield with a 3-month lag. We can then see if thedifference is negatively related to future changesin bond yields, the same way we can see a negativerelationship between CAPE and future changes inreal equity prices. The results over 1-, 2-, 5-, and10-year horizons are shown in Figure 5.

Table 2 shows the results for the dividend–priceratios and CAPE as well as the bond value modelover all four horizons. As we can see, the bondvalue model appears to do a better job of explain-ing future moves in bond yields over the 1-, 2-,and 5-year horizons than our two measures ofequity value do explaining future changes in realequity prices. However, the dividend–price ratiodoes as well or better than the bond value modelover the 10-year horizon and the CAPE does con-siderably better over the 10-year horizon. Thisgives us a sense that these equity value measuresare somewhat longer term in nature than the bondvalue model, but that all have the potential ofproviding a useful measure of value.

As previously discussed, Asness et al. (2017)examine the out-of-sample performance of theCAPE. They find that using it for timing purposesfails to do as well as a simple buy-and-hold



Figure 5 The figure replicates the methodology of Campbell and Shiller (1998) used to demonstrate CAPE’srelationship with future changes in equity prices and then applies it also to the Bond Value Model’s relationshipwith future changes in yields. A high R-squared is used to demonstrate predictive power. The figure shows thatR-squareds over 1-, 2-, and 5-year horizons are higher for the Bond Value Model, but higher for CAPE over the10-year horizon.


12 Thomas Shevlin

Figure 5 (Continued)

strategy. However, when combined with amomentum indicator, they can create a tradingstrategy that improves upon a buy-and-hold strat-egy on an out-of-sample basis. In the Appendix,they also suggest a bond value model based on realyields, calculated by subtracting expected 10-yearinflation from the nominal yield. Let us thereforeexamine their bond value model in comparisonwith our own by approximating their methodsusing publicly available data. They compare theirtrading strategy with a buy-and-hold approach byvarying Treasury exposure from 50 to 150% ofNAV with either investing or borrowing at thecash rate to bring total NAV to 100%. Using arange from the 5th and 95th percentiles of realyields in an expanding window sample, they com-pare the current real yield with that range of prioryields to calculate current percent exposure on the50 to 150% range.8 Exposure to Treasuries capsat 150% when at the 95th percentile of real yieldsand is 50% when at the 5th percentile or lower.

For a series of inflation expectations, the Sur-vey of Professional Forecasters provides 10-yearinflation forecasts twice a year beginning in 1990and also provides forecasts from BlueChip twice ayear for 1979–1989, giving us an extended datasetfor expected inflation. Given that we want tocalculate weights based on percentiles, a higherfrequency of data is necessary. We could make

the semi-annual data forecast monthly either byinterpolating the data or by simply using theprior forecast until the next one is published. Theresults for the trading strategy turn out to be thesame, so let us use the method of changing ourinflation expectation only when a new survey ispublished, since it would not be possible to inter-polate between the previous and next forecast ona real-time basis. We will use an expanding win-dow sample. Table 3 shows that the Sharpe Ratioof this trading strategy is superior to the buy-and-hold strategy over the sample period, which isconsistent with Asness, Ilmanen, and Maloney’sresults.

To calculate the bond value model in the sameway, we simply take the difference from the modeland the prevailing yield in each month, as we didin the comparison for the CAPE. As we can seein Table 3, the Sharpe Ratio compares favorablywith both the buy-and-hold strategy and the realyield approach suggested by Asness, Ilmanen,and Maloney. Unlike the trading strategy usingCAPE, both these trading strategies outperformthe buy and hold. It is interesting that both thesetrading strategies find more signals to under-weight than over-weight 10-year Treasuries vs.Tbills, with the real yield approach having anaverage of 66% exposure to Treasuries (and there-fore 34% average exposure to Tbills) vs. 86%



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14 Thomas Shevlin

Table 3 Out-of-sample trading strategies.

Buy and hold Real yield Bond value model

Compound annual growth rate 8.26 7.66 8.17Volatility 7.94 6.18 6.63Sharpe ratio 0.54 0.60 0.64Average allocation to 10-year treasuries 100% 66% 86%∗Sample Period, Jan 1982–Dec 2017.

average exposure for the bond value model. Giventhe trend toward lower yields over the sampleperiod, a higher exposure may have been desir-able. However, the trading strategy based on thebond value model manages to have nearly iden-tical returns despite this handicap and has lowervolatility, so it is superior from the perspective ofthe Sharpe Ratio over the 36-year sample period.

5 Medium to long-run implicationsfor yields

Whether or not the low yields in recent years aretemporary or more permanent has become a topicof significant debate. Clearly the view that yieldswere likely to rise again in the near future has beencommon among forecasters. The findings of thebond value model support the alternative viewthat yields are actually consistent with broadereconomic trends and are not likely to rise to thelevels seen in the past, unless of course trend infla-tion and growth rise to the levels we have seen inthe past as well. Although some investors may beskeptical that this simple model has uncovered thetrue nature of interest rates, it is consistent with agrowing body of thought on interest rates in theeconomic literature.

Holston et al. (2017) use the Laubach–Williamsmethodology and find that the natural rate of inter-est in the United States fell to zero during theGlobal Financial Crisis and remained there into2016. Although the rate they are modeling is thereal Fed Funds rate consistent with a constant

rate of inflation, the decline of this rate by 2percentage points has clear implications for the10-year bond yield. They attribute the interestrate decline to “shifts in demographics, a slow-down in trend productivity growth, and globalfactors affecting real interest rates.” Likewise,Rachel and Smith (2015) find a secular declinein long-term real interest rates which they thentry to attribute to different factors. Given theirviews of the sources of the change, they predictthat real yields are likely to only recover a smallportion of this decline and that they will settleat or slightly below 1% in the medium to longrun. This coupled with a credible inflation tar-get of 2% would lead to equilibrium bond yieldsaround 3%.

As mentioned earlier, Bauer and Rudebusch(2016) review models for measuring the neutralshort-term real rate. They also review three mod-els for term premium. Combining the models forshort rates and term premia, they find the evidenceto suggest a “low new normal” for interest ratesand note, “while the ongoing economic recov-ery and normalization of monetary policy in theUnited States should lead to some increases inlong-term interest rates, these seem likely to befairly moderate for the foreseeable future.”

For investors seeking a way to incorporate theview that rates are likely to remain lower thanin previous decades, the bond value model mayprove a useful tool. In order to try to comparethese other models directly with it, let us start



by taking the three models of short-term interestrates that Bauer and Rudebusch (2016) show intheir chart and combine them with the measuresof term premium. The short-term real rate esti-mates are from ∼2 ½ to 3% in 2000 and 0–1%in 2016. The estimates of term premium are allabout 1¼% in 2000 and −½% in 2016. Couplingthese with inflation expectations of about 2%, wehave equilibrium nominal yields of about 5¾ to6¼% at the beginning of 2000 vs. 1½ to 2% atthe beginning of 2016. The actual yield at thebeginning of 2000 was 6.7%, so a bit higher thanthe range of these estimates, and 1.9% in 2016,which was in the middle of the range of theseestimates. The bond value model estimate of equi-librium yield is 5.3% at the beginning of 2000 vs.1.9% at the beginning of 2016. It is therefore a bitlower than the other combined models’ estimatesin 2000, and is about the same in 2016. So thebond value model appears consistent with theseother more sophisticated models as well as theyields themselves.

For many investors, there may be significantappeal of having a simple model such as thisbond value model for approximating equilibriumnominal yields in this way. This could be doneeither in lieu of, or in addition to, deriving themore complex models of real short-term rates andterm premia, and then combining those with anexpected inflation rate, in order to estimate anequilibrium yield or broad measure of value.

6 Intuition behind coefficients

That the slope coefficients for both growth andinflation are close to one is encouraging, giventhe strong theoretical support for the slopes beingequal or close to unity. The finding that the infla-tion coefficient is greater than one is consistentwith the Feldstein and Summers (1978) argumentthat tax distortions should lead the coefficient tobe higher than one. It is also consistent with thefindings of Campbell et al. (2013), whose model

would be consistent with a positive nominal bondyield premium when inflation is high and a nega-tive premium when inflation is low or when thereis a chance of deflation. The estimated slope coef-ficient for trend CPI is 1.23, which is statisticallygreater than 1 at the 99.9% level of confidenceusing Newey–West standard errors. This givesus strong empirical evidence that the inflationcoefficient is indeed greater than 1.

The intuition for the growth coefficient beinggreater than 1 would be similar to that of theCPI coefficient (i.e., a “growth premium”). Whengrowth expectations are high, investors maydemand more to substitute away from otherinvestments, whereas when growth expectationsare low they may prefer the default protectionof Treasuries and will flee to the safety of theseassets. However, in the case of growth, the coeffi-cient estimate of 1.06 is not statistically differentfrom 1 if we use Newey–West standard errors tocalculate our t-statistics. This suggests that thetrue coefficient for growth may indeed be 1, whichwould be consistent with the Golden Rule.

The estimate for our constant coefficient, or inter-cept, is −1.8. It is statistically lower than zerowith 99.9% confidence. This implies that if wewent through a 10-year period of zero growth andzero inflation, yields would be negative. If the USeconomy were to go through such a depressedstate over such a long period of time, then it islikely that there would be abundant capital withlittle demand for investment. Investors would inall likelihood be seeking ways to protect theirprincipal from outright loss, rather than seeking apositive return. Although the US has never experi-enced such low growth and inflation trends undera fiat currency system, the experience of negativeyields in Europe even with positive growth andinflation trends suggests that the intercept coef-ficient is not unreasonable.9 It is worth noting,however, that unlike the other two coefficients,


16 Thomas Shevlin

the 95% confidence interval for the intercept coef-ficient is rather wide, ranging from −1.2 to −2.4.Though the estimate is less precise than for theother two coefficients, the negative incept appearsintuitive and is statistically significant at a veryhigh level of confidence.

7 Statistical issues

The bond value model regresses 10-year gov-ernment bond yields on the 10-year compoundannualized growth rates of GDPin chain weighted2009 US dollars and the Consumer Price Index forAll Urban Consumers (CPI-U). Data are monthly,with monthly GDP interpolated from quarterlydata. The use of overlapping time periods of suchlong length is appropriate for trying to quantify arelationship with long and variable lags. We useOrdinary Least Squares (OLS) regression. Themodel gives rise to two statistical issues: serialcorrelation in the residuals and stationarity. Ofthe two statistical issues, stationarity is the greaterconcern.

The presence of serial correlation in the errorterms means that the estimates of the coefficientsmay not be efficient (i.e., it may be possible to esti-mate a coefficient with a lower standard error), butthe coefficients are still unbiased and many econo-metricians view lack of efficiency as a minorconcern. Serial correlation also means that theestimated standard error of the coefficients maybe too low, which means that the t-statistics maygive a false positive. However, this is easy to cor-rect by using Newey–West standard errors, as wehave done in this paper. This is a common practiceamong researchers.

If time series are not stationary, models thatappear to be powerful and fit well can in fact beentirely spurious. It is therefore incumbent uponthe econometrician to prove that the time seriesinvolved in the regression are either stationaryor cointegrated. For purposes of completeness,

this paper includes five different tests for cointe-gration, with two ways to optimize the test lags(both Akaike and Schwarz information criteria),yielding a total of 16 test statistics for cointe-gration. All 16 test statistics find in favor ofcointegration at the 97.5% level of confidenceor greater and 8 of the 16 at the 99.9% levelof confidence or greater (see Table 4). The teststherefore give very strong evidence of cointegra-tion, affirming that the OLS model should not bespurious for lack of stationarity.

To consider the robustness of using OLS relativeto other regression models, this paper includesseven more advanced forms of regression mod-els on the same data set (see Table 5). Theseother models correct for serial correlation, coin-tegration, or both. All of these models havesimilar coefficient estimates, especially for trendinflation, and all models find all coefficients sta-tistically significant at well above the 95% levelof confidence.

Given the diversity of tests used for cointegrationand the diversity of more sophisticated modelsthat have confirmed the results, we have verystrong evidence that our model is statisticallyrobust. Moreover, given the homogeneity of themodel outputs, and even the coefficients, thereappears to be no need to explore the benefits ofmodels more sophisticated than OLS for quantify-ing our relationship. Given the desire to provide aheuristic accessible to a broad range of investors,the ability to use OLS is welcome.

8 Stability of the model

As we have mentioned, in many cases modelswith coefficients that seemed to fit the histori-cal data well are useless for predicting the futurebecause the coefficients change. Of course, evenif coefficients do change somewhat, the frame-work of the bond value model may still be ofuse. After all, there is healthy debate about the



Table 4 Various tests for cointegration.

Test Test statistic p-value

Augmented Dickey Fuller, AIC OptimizedAugmented Dicky Fuller −5.59 0.0000

Augmented Dickey Fuller, SIC OptimizedAugmented Dicky Fuller −5.00 0.0000

Engler-Granger AIC Optimizedtau-statistic −5.59 0.0001z-statistic −120.73 0.0000

Engler-Granger SIC Optimizedtau-statistic −5.00 0.0009z-statistic −49.57 0.0004

Phillips-Ouliaris AIC Optimizedtau-statistic −4.76 0.0021z-statistic −43.87 0.0015

Phillips-Ouliaris SIC Optimizedtau-statistic −4.77 0.0021z-statistic −44.00 0.0014

Johansen, 12 Lags, AIC OptimizedTrace 41.72 0.0014Maximum Eigenvalue 33.66 0.0005

Johansen, 2 Lags, SIC OptimizedTrace 32.83 0.0217Maximum Eigenvalue 25.87 0.0100

ARDL Bounds Test, AIC OptimizedF-Statistic 7.89 0.0010

ARDL Bounds Test, SIC OptimizedF-Statistic 5.20 0.0085

Table 5 Bond value model outputs using other forms of regression.

Ordinary General Fully Canonical Dynamic Robust ARDL ARDL Averageleast least modified cointegr ordinary least AIC SIC other

Coefficients squares squares LS regress LS squares optimized optimized Models

Trend inflation 1.23 1.21 1.24 1.24 1.24 1.20 1.22 1.23 1.23Trend growth 1.06 0.87 1.09 1.09 1.08 1.06 1.03 0.99 1.03Constant −1.80 −1.15 −1.89 −1.90 −1.86 −1.71 −1.66 −1.55 −1.69


18 Thomas Shevlin

parameters of many models. Nevertheless, asnoted earlier, one could have used the modelto forecast 2 years forward using Livingstongrowth and inflation forecasts and had consider-ably better results than using the actual forecastsof bond yields in the same survey. This is becausethe coefficients are very stable over 2-yearhorizons.

The green line in Figure 6 shows what projectionswould look like if a forecaster had perfect fore-sight over growth and inflation over the next 2years. The coefficients are estimated only usingdata available at the time in order to project yieldsforward 2 years. What it shows is that at any pointin time, the out-of-sample coefficients producenearly identical results as coefficients calculatedfrom the entire sample. This demonstrates that formaking assessments about interest rates on a 2-year horizon or less, instability in the coefficientsis de minimis.

We can also use 10-year lagged coefficients tosee how useful the model would be for making10-year forecasts. In Figure 7, we can see that for

nearly two decades the model does surprisinglywell. Even starting with only 20 years of data(1960–1979), we can see that coefficients calcu-lated in January 1980 would have given almostthe same estimate of yields in January 1990 ascoefficients calculated using the entire sample(1960–2017). However, it does not do as well forprojecting over horizons that include the GlobalFinancial Crisis. Nevertheless, the model does not“break down” due to the financial crisis. Indeed,the model appears to actually fit the data betterwith the inclusion of the crisis and recovery inthe sample.

As readers can see in Figure 7, beginning in2009, the 10-year old coefficients lead to yieldprojections that are too high. However, the goldline, which uses 5-year lagged coefficients andalso shows a divergence at the time of the cri-sis, fully converges on the red line by 2017. Thissuggests that as the data over the crisis becomeincorporated into the model, it may again forecastwell out-of-sample for 10-year horizons. Furtherreason to expect a potential improvement in the

Figure 6 The figure shows full sample Bond Value Model estimates vs the same estimates using coefficientsestimated with only prior data and a 2-year lag. The nearly identical results show coefficients are stable enoughfor 2-year forecasts.



Figure 7 The figure shows full sample Bond Value Model estimates vs the same estimates using coefficientsestimated with only prior data and with 5- and 10-year lags.

out-of-sample robustness of the model is that themodel’s statistics of fit improve with inclusion ofthe crisis and post-crisis years in the sample.

Table 6 compares the model using only 1960–2007 as the sample period with the full sampleperiod of 1960–2017. One can see that the full

sample has a better R-squared (0.89 vs. 0.86)and that its coefficients have tighter confidenceintervals. Although the confidence interval andcoefficient for trend inflation are roughly thesame, the confidence interval for the growth coef-ficient is less than half as wide and the t-statisticrises from 4.9 to 14.7. The improvement in the

Table 6 Comparing model outputs across sample periods.

95% Confidence interval*

Coefficient St Error* t-Statistic Confidence Low High Range

Dataset 1960–2017Trend inflation 1.23 0.06 21.3 99.9% 1.12 1.35 0.23Trend growth 1.06 0.07 14.7 99.9% 0.92 1.20 0.28Constant −1.80 0.29 −6.3 99.9% −2.36 −1.24 1.12R-Squared 0.89

Dataset 1960–2007Trend inflation 1.18 0.06 19.0 99.9% 1.06 1.30 0.24Trend growth 0.74 0.15 4.9 99.9% 0.45 1.04 0.59Constant −0.46 0.64 −0.7 52.5% −1.71 0.80 2.50R-squared 0.86∗Using Newey-West Heteroskedasticity Autocorrelation Consistent Standard Errors


20 Thomas Shevlin

Figure 8 The figure shows the Bond Value Model calculated over 1960–2007 and 1960–2017. The Newey-West error bands are much smaller over the 1960–2017 period, indicating that including the 2008–2017 periodsubstantially improves the precision of the model.

stability of the coefficient estimate may comefrom including a greater dispersion in growth out-comes in the sample. After all, prior to 2008 trendgrowth was always above 2% and was above 3%for 87% of the time. From 2008 onward, how-ever, trend growth has been below 2% for 88% ofthe time. In Figure 8 we can see 95% confidenceintervals of the equilibrium yield for both sampleperiods. These error bands are only half as widefor the full sample period as for the sample end-ing in 2007, due to the greater precision in thecoefficient estimates.

Given that the 10-year out-of-sample forecaststhat were too high were calculated when theNewey–West error band was twice as wide, itseems likely that future errors from using 10-year old coefficients will be smaller. With a fewmore years of data, it should become appar-ent whether the 10-year out-of-sample forecastsare again identical to the full sample forecasts.This should give more clarity about the long-termstability of the model.

9 10-Year Forecast for 10-Year Yields

We can generate a 10-year forecast for yieldsby combining 10-year forecasts of inflation andgrowth with our model. Using the Livingston Sur-vey from December 2017 gives us a yield forecastof 3.36%, as shown in Table 7. However, somemight think these forecasts a bit high. After all, forthe 10 years ending in December 2017, the CAGRfor GDP was just 1.43% and for CPI just 1.61%.Also, the projections for growth and inflation haveexhibited an upward bias in the past, with the pro-jections overestimating GDPby 46 basis points onaverage and overestimating CPI by 56 basis pointson average. If we assume just half this bias existsin today’s estimates, then this would give us a pro-jected yield of 2.80%. This is considerably lowerthan general consensus among forecasters, whichappears to be that yields will move above 4% inthe next 10 years. Readers may recall, however,that the bond value model’s projections wouldbe consistent with Rachel and Smith’s views onreal yields rising to about 1%, as well as with the



Table 7 Using Intrinsic value model for 10-yearforecast of 10-year yield.

Dec-2017 10-YearLivingston forecasts

Coefficient Actual Adjusted

GDP 1.06 2.18 1.95CPI 1.23 2.34 2.06Constant −1.80

Yield projections 3.39 2.80

95% Confidence Interval Projections:Lower bound 2.26 1.73Upper bound 4.53 3.87

more advanced models of the natural rate, whichsuggest that today’s low yields are mostly due tostructural rather than temporary factors that arelikely to reverse quickly.

10 Conclusions

The bond value model laid out in this paper givesinvestors a simple construct that may allow themto understand why yields are so low today, givethem an intuitive way to relate yields today toother regimes of higher growth and inflation, andalso help them make reasonable judgments aboutfuture yields as well as yield forecasts. The modelis statistically robust and its findings of equilib-rium bond yields are consistent with some of thecutting edge work which has shown a decline inthe natural rate of interest and term premium inrecent years. It is also consistent with longstand-ing economic theory on the formulation of interestrates, in that the coefficients for trend growthand inflation are close to, though not precisely,one. The model compares well with the CAPEas a long-term value measure, using the sametest presented by the authors of that model at itsintroduction. Plus, using the same out-of-sampletest Asness, Ilmanen, and Maloney apply in their

paper, the bond value model compares favorablywith both a buy-and-hold strategy and the alter-native bond value approach they suggest. Finally,given the simple methodology, the approach canbe accessible to many practitioners and requiresonly occasional updates to estimate the coeffi-cients. Although no model or approach can be allthings to all people, for many investors focused onlong-term value, this model may be a very usefultool.

Appendix: Forecast-based models

As mentioned in the main body of this paper, onecould also try to model bond yields more directlyon expected future growth and inflation by usingforecasting surveys. Such an approach is perhapsmore consistent with the efficient markets hypoth-esis. Regrettably, the two models attempted forthis paper have results that appear problematic.To begin, let us consider a model created using10-year forecasts for growth and inflation. Theseshould be the most theoretically consistent fore-casts since they are for the same time horizon asthe bond.

Livingston gives 10-year forecasts of GDP andCPI twice a year beginning in 1990, for a sam-ple size of just 55. Although we have a ratherhigh R-squared of 0.73, the coefficient for CPIis 3.00, which would appear inconsistent witheconomic theory (see Table A1). Moreover, theconfidence intervals for the coefficient estimatesare extremely wide and the intercept coefficientlooks implausibly low, at −7.81.

It is also possible to gather Livingston projec-tions of growth and inflation for 1 year forward(displayed in Table A1 as 1 × 2). At 1.40, thecoefficient for CPI appears more reasonable thanthe 10-year forecast-based model. However, thecoefficient for GDP is not significant and has thewrong sign. Plus, the intercept suggests that ifexpectations for growth and inflation were both


22 Thomas Shevlin

Table A1 Results of forecast-based models.

95% Confidence interval

Variable Coefficient N-W t-stat Low High Range (%) Model Stats

10-Year Trailing CAGRsGDP 1.06 14.73 0.92 1.20 27% R-Squared 0.89CPI 1.23 21.34 1.12 1.35 18% Sample 696Constant −1.80 −6.31 −2.36 −1.24 62% Begins 1960

10-Year ForecastsGDP 1.52 3.08 0.53 2.50 130% R-Squared 0.73CPI 3.00 9.06 2.34 3.67 44% Sample 55Constant −7.81 −4.46 −11.32 −4.30 90% Begins 1990

1 × 2-Year ForecastsGDP −0.30 −0.60 −1.31 0.71 668% R-Squared 0.74CPI 1.40 6.23 0.94 1.85 65% Sample 44Constant 1.87 1.21 −1.26 4.99 335% Begins 1974

zero, then the equilibrium bond yield would be1.87, which looks implausibly high. As with theother forecast-based model, the confidence inter-vals around the coefficient estimates are verywide, denoting a very low level of precision.

The issues with these two models could stem frominadequate samples, from the surveys not beingreflective of the expectations that drive the bondmarkets, or from some other issue or combina-tion of issues. Whatever the cause, the modeloutputs do not inspire confidence and appearas an inadequate basis for making investmentdecisions.

Notes

1 Using the methodology outlined by Campbell andThompson (2005), we could have an out of sample R-squared of 0.22 if the forecasts were the output of aregression model that attempt to forecast bond yields overa 2-year horizon.

2 Council of Economic Advisors (2015), p. 11.3 Rausch (2016), p. 4.4 Council of Economic Advisors (2015), p. 13.5 Rachel and Smith (2015), Executive Summary, p. 1.

6 The interpretation of the R-squared, or coefficient ofdetermination, is that the model explains 89% of the levelof bond yields over the sample period.

7 Parenthetical statement added.8 They use the expanding window sample until they have

60 years of data and use a 60-year rolling windowthereafter.

9 For example, in June of 2016, Switzerland had trendgrowth of 1.8% and inflation of 0.2%. The coefficientsfrom the Bond Value Model would have implied a bondyield of 0.2% had we had those same outcomes in theUnited States. Nevertheless, Swiss yields fell as low as−0.7% that month, 90 basis points lower than these coef-ficients would have predicted. This would be consistentwith an intercept coefficient at or below −1.8.

Acknowledgments

Joseph Gagnon, Ric Thomas, Ingo Walter,Nicholas Fok, Andreas Dische, and RobertBlauvelt.

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Keywords: Treasury yield; yield forecast; bondvalue; equilibrium yield; natural rate of interest;golden rule


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