- Email: [email protected]
On forecasting interest rates
Joumd of Monetary JZ.c~~ics
8 (1981 b 305-3 18. North-Holland Publishing Company
ve
movements in
of the martingale approxiwarion for two sets of Canadian interest rates data, emphasizing the iInportanW of the interval. The paper then examines three !Bet!? of reccxdcd foreca!Jtsof finds results consistent with the theoretical discussion. Canadian interest
In view of the substantial empirical evidence which supports the proposition that capital markets are efficient in their use of publicly available information, surprisingly little attention has been directed to the question of whether economic agents can successfully forecast interest rate movements. Impressed perhaps by the early evidence in favour of :he random walk hypothesis,’ most sophisticated agents tend to regard published forecasts of the future level of egate stock prices with a healthy degree of skepticism. As yet, however, such skepticism does not appear to extend to published forecasts of movements in either short- or long-term rates of interest, nor is it clear to many whether or not such skepticism is warranted. research program in Financial Markets those of the author and not those of the is indebted to Michael Huberman for
am
than for common stocks,
, where c1 is an in en equilibrium retu of drifi (i.e., P, -P,_ I =G, c kP, _ 1), c statement in the text is meant to of common stocks which translate sified portfolio are appropriately to apply to forecasts of changes easily translated into statements arket effiiency is viewed as less persuasive for bonds
with mean
J.E. Pesando, On farecasting
306
interest rates
At the theo&cal level, the question of *:Nhethereconomic agents can successfully anticipate interest rate movements in an efficient market has ishkin (1978) and esando (1979) caution that received increased attention. there is nothing in efficient markets theory which requires that short-term interest rates follow :a random walk. Thus agents without access to inside information may well succeed in forecasting movements in short-term rates. As Sargent (1976) and Pesando (1978) note, however, anticipated changes in long-term interest rates over brieJ:firrecast intervals must -- in the absence of time-varying term premiums -- be very close to zero. Long-term interest rates when observed over Fhort intervals will capproximately follow a martingale sequence and will thus exhibit random walk characteristics. Agents without access to inside information are not likely to succeed in forecasting short-run movements in long-term rates if the term premium accor&d long-term bonds is indeed time-invariant. This paper briefly reviews the theoretical results, paying particular attention to the importance of the forecast interval. Using two sets of Canadian data, the paper first illustrates the closeness of the martingale approximation. The paper then examines three sets of recorded forecasts of interest rates in Canada and finds results consistent with the theoretical discussion. The analysis highlights the fact that many of the forecast changes in long-term interest rates are so large (in absolute value) that they immediately lack ci-edibility when translated into predictions of hGld;ngperiod returns on long-term bondc.
Theoretical overview
Let R,,t denote the in crest rate on an n-period, non-coupon bond in pc ;iod & the information available to the marker in period t, and ,+Ji t the forward rate at time t for the one-period bond rate in period t + i. A&ume
that the expectations model of the term structure is valid, and that the long term rate is satisfactorily represented as an arithmetic (rather than a geometric) average of current and future short-term rates. Then, Sargent (1976) and Pesando (1978), the ex ante change in the IO can be written as follows:
E(R*tl#t-l)--R~**~,=~-*rE(*+,-~~* .tI4t-r)-~l,t-Il* n The term on the right-hand side of eq. (I), which represents the nonod rates, clearly approaches zero as n gets Iar overlapping onerepresents the ter
premium built into the interest rate on the
J.E. Pesado,
On forecasting interest rates
307
bond, then (I) becomes
+E(~(cplt-*P--ll/r-** tf this term premium is constant, martingale approximation em
t
(2)
reduces
to
(1)
(2) and
hence
the
(3) at the ex ante change in the n-period rate is Eq. (1) roximately zero if n is 1 so long as the one-period rate is not ‘tcjo -stationary’. The unit of rvation or forecast interval, which eff?&eIy identifies the one-period rate, is crucial in determining whether n is lartge and hence whether (3) is likely to be a ciose approximation. If one is forecasting the (annual) yield on a ten-year bond one quarter ahead, n equals 40. If one is forecasting the same yield one month ahead, n equals 120. In general, the shorter is the forecast interval, the smaller is the ex ante change in the yield on a gil,.en bond and hence the closer is the martingale approximation. Equivalently, it is short-pun movements in long-term rates which will not be predictable under the joint hypot esis of market efficiency and a timeinvariant term premium. For coupon bonds, as noted by Pesando (1979), the expression analogous to (1) i,
308
J.E. Pedo,
On forecasting interest
totes
end-of-the-quarter data on 90-day Treasury bills and 9Oday finance ion company paper, so as to eliminate possible biases due to time aggr [Working (1940)], alternative time series models can be fitted the Canadian data. For the sample period 1957: l-1979: 1, an ARMA(1,l) or ~tl AR(2) representation is superior to the martingale model for both short-ter~~ rates.’ The fact that the short-term rate need not follow a martin sequence is ‘rep:onciled’with the martingale property of the long-term rate when it is remembered that the latter is only approximate. I‘hs: long-term rate would exactly follow a martingale if and only if the short-term rate followed a martingale. In this case, the e:c ante capital gain or loss on longterm bonds would always equal zero, and the long-term rate would simply equal the short-term rate plus the time-invariant term premium. Finally, the martingale approximation for the longterm rate is derived on the assumption that the term premium sceorded long-term bond invariant. Many - if not most - of those who have conducted studies of the determinants of term premiums have concluded that they may well be time-invariant. McCulloch (1975) and Pesando (1975a) have presented evidence that the term premium accorded long-term government bonds in the United States and Canada, respectively, may be time-invariant. Attempts to enrich the traditional term structure equation of h4odigh&= Stitch ( 19B’f) or Modigliani-Shiller (1973) with additional variable capture the spirit of the ‘preferred habitat’ model have, on the whole, unsuccessful.Indeed, empirical support for the existence of timclvaryin premiums is largely c:bnfined to structural models of the long-t market, such as those estimated for the United States by Friedman (1977, 1979) and for Canada by Ma,sson (1978). In these models, demand funct5ns for long-term bonds are estimated for each major class of investor, such banks, life insurance companies and so forth, When the demands is set equal to supply, the models yield a (restri equation for the long-term rate. Although these redu contain asset stocks, Buancial flows and other variables varying term premiums, the repeated failure of comparable effwts in the traditional term structure equations is I” disconcerting. Clearly, the careful and successful inclusion of stock and variables into demand and supply functions is no&suf&iant to guu the implicit equation for the long-term interest rate will better charw ‘AR(l), AR(2) and ARMA(1,l) mQdels fox the Trwu -0.327B-0.905B3]r,~u,, (i-a930R)r,~(l+A367B)y, standard errors, in basis @nts, are 78.32, 75.98 and company paper rate, the models are: (l -0.934JI)r,~w,, -0.9LUBIr, = (1 +O.l54B)~. The standard errors are 94.08 93M and 92 AR(l) model, the null hypothesis that the coefkient of the be rejected. The standard errors for the martingale or AR this we&ient to equal unity, must be larger than those na
J
F t
of t
ml. Then
tian, this valuation
310
I.E. Pesado, On forecmting interest rata
r&e bond, or AR,” t, is the anticipated or ex aatte changej in the qxriod under conditi&s of market efficiency and the assumptions noted above. Let 6-I represent the capital gain or loss on the n-period or (L- 1 +&I --Kti ). One can show, in a straightfo
Again, the unit of observation or forecast interval is crucial in (6). If the forecast interval is one quarter, then n equals 40 if the maturity of the Ian term bond is 10 years. In addition, both (5) and (6) would refer to quart interest rates. If the forecast interval is monthly, then n = 120 for the bond, while (5) and (6) would refer to monthly rata. In indicates that ceteris pnribus AR& -- when expressed at an will be approximately one-third as large (in absolute value) if the intervzil is monthly rather than quarterly, and 90 forth? The reEISOnis quite straightforward: the briefer the holding-period, the smaller (in absolute value) is the capital gain or loss - and hence the smaller the change in the longterm rate - required to equilibrate the ex ante returns on short- and Ion term bonds in (5). ‘Because of Jensen’s inequality, expectation>; of bond prices and interest rates - whkh are inverseiy related - cannot both be unbiased. As formukwd in (S), the imptie unbi of future prices implies that t\e forward rate on the (n - t )=perlod bond to commenceat tim I is not an unbiased estimate _,f the corresponding spot rate, Lintner (1969) prr~idm cvicS?nce that this bias is small, and the issue is not pursued further in this paper. %ifferentiating (4) and dividing the resulting expression by pke yields
If the bond is selling at par, so ththt c/F equals R,, this expression rakxza to dP/P = -dR,/R,
* (I- t/(1 + R,)“).
Solving (F.2) for dR,, noting that dP/P=& as dcfin in the text, an6t?then in appropriate time subscripts yields (6). ‘Consider, for example, the case of a consal (II= a) when and RUM==0.12. For simplicity. assume #ul ==0. The esrrcspon R are, respectively, 0.015 and 0.03. Thus (6) implies =%‘kIO45. At an annual rate, this equals [email protected]) corresponding monthly rates for kl,, and R,., m Q.OWi monthly forecast interval, dR$,, s=- (0.01) * (-Qo05) 12 * (O.OOOOS) =0.0006 or 6 basis points. Because the bond is a on the right-hand side of (6) is simply equal to one, the PX uw interval is exactly one-third of that fcr the yuarterlv formnm! annual rates.
omit variabks
on ,t t rate foreaasi~
premiumsbonds, and 138 basis
: 1. These
able 1, indicate that the quite small. The mean
Table 1A Actual and anticipated quarterly Chan;a
Mean absolute change Maximum absolute change Mean change Standard deviation
in long-tetnr int 1999: 1.
23,w
2.09 7.21 - 0.03 2.71
26.38 127.00 6.93 3450
I l9,og
s*J)o 30.23
“All numbers ar\“: in basis points at annual rates, calculation of the anticipated changes.
Table lb Cumulative frequcwy dl’tribution of absolute values Ot anticipated &I interest t&es?
inll
Government bonds Absolute change No. I.__D__y__mm _..I 1__1_1 - _..“_.._ Less than Less than Less than Less than Less than
2 4 6 8 10
P-S
51 79 84 89 89
?eFsent 59 88 v4 100 100
“AI1 numbers are in basis points at annual calculation of the anticipated changes.
By contrast, the actual changes in quite large. Of partieuhv interest is the variances of the ex antarthan and th ratio of the former 80(x.
long-term rates must be u imply that 99.38’J$and 98, Canada and corporate bands,
regressions of the current &an known at the besinni percent ebr so of th explained variance would s suggest that the observed rel
I -I Px_usBm-e-
&
4 9 82 -
313
and (2) several of the series (such as the prime lending rate of the banks) are not determined in auction markets, only three appropriate for this study: 9Qday finance company pa corporate bonds (MYW 10 industrizris) and Ion (MYW 10 provincials), In comparing the recorded forseasts to the predkz model, care must be made to dtlsure that the current value of the relevant interest rate) do not not available at the time the reorded forecmte published data on the several interest rates series are availa& only c)~1a monthly basis, the martingale forecasts were set txlual to the interest rate at the end of the latest month which unambiguously rtxx2.4 tk month in which the recorded forecasts were made. In so doin an informational advantage to the recorded fo typically consisted of about two weeks, but in so month. The results of the forecast comparisons,’ summarized in table 2, indicate that the recorded forecasts of long-term rates in general failed to outpxform the no-change prediction of the martingale model, while this wus not the case for the recorded forwasts of short-term rates. :n view of the theoreticA discussion, this result is not surprising. Only the two-quarter ahead forecast by DRI of the long-term Canada rate proved superior to the martingak forecasts. Even this modest success merits qualification, however, in view of (1) the informational advantage accorded the recorded’forecasts, and (2) the inferior performance of the one-quarter ahead forecasts relative to those of the martingale model. rhis latter result sugce;sts that the improvement sf the two-quarter ahead forecasts may be somewnat of a statistical artifact and not likely to obtain over a longer sample period. Both the DRI and MYW survey i’orecasts of the short-term prove significantly superior to the martingale modal. Further, the fe, the short-term rate by the Conference Board were about as acxura martingale model, while their forecasts of the long-term rate were distinctly ingerior. These results, in general, enhance th with respect to the l’oreeastsof’the Ion term rates. In reinains - es;xcially with su:vey data figures do not accurately reflect the exp behavior. In this regard, it is interesting ta note that- Priedmrrn (n.&) provides evidence that the Goldsmith-Nagan int pass a now standard battery l>f tests designed to As notbxl by Pesando (1975b), such fa#~res rrri:sethe *The forecasts differ in their relecsadates, and pertain to both quarterly I& Resourcesand the Conference Board) rendend-&ho-quarter [MYWsurro result,theaccuracy of the forecastscannot be compared across forecasting
f
c
Q
316
J.E. Pesando, On forec&ing
interest
rates
whethe,r the recorded forecasts do indeed reflect those of active economic agents, in which case the latter fail to form ‘rational’ expectations, or whether such recorded forecasts must not be those of the market on the main hypothesis that the market efficiently processes performs this standard battery of tests -- th.t: un the orthogonality of forecast errors to costlessly efficiency and consistency in the use of autoregressive information - on the MYW survey data and, on the whole, the results are favourable to the hypothesis that the expectations are rationally formed. Finally, the magnitude of the changes in the long-term interest rates implicit in many of the recorded forecasts merits comment. The maximum (absolute) change forecast for the corporate bond rate, based on the twos period compared to the one-period prediction (so as to eliminate the uncertainty regarding the date - and hence the value of this rate QLI_ when the forecasts were made), equalled 22 basis points in the MYW survey data, 30 basis points for the DR.1 forecasts and 70 basis points (!) for the Conference Board forecasts. The mean absolute changes were 11.2, 12.7 and 13.3 basis points, respectively. In view of the magnitylde of the ex ante changes summarixed in table 1, these recorded forecasts clearly imply substantial variation in the ex ante return on long- relative to short-term securities. For the extreme forecast of the Conference Board, for example, the implied ex ante return on long-term bonds is 32.60x? Du this same quarter, the Cotierence Board was predicting a 6.86% retu on !&day finance company paper. On a prima facie basis, this result appears to be unreasonable and ierves to highlight the conceptual importance of translating predict& changes in long-term rates into comparable statements about their ex mte holding-period return. Although correspond& calculations for tJ:c DRI and MY W survey forecasts are less dramatic, the also imply a very sharp divergence between the ex ante returns on shortlong-term securities. I0 Both forecasters and those who monitor for would be well advised to t’ocus*on ex ante returns in assessing the lik of long-term interest rates during their forecast horizons, 5. Summary and conclusions The failure of the recorded forecasts of lo outperform the no-change prediction of the mart this result for short-term rates, is not surprising on ‘These returns ale cahxlatd by usiag the recorded foraclasts of thtt than in ion&tern interest rates to determine the capital gain or loss on long-term bond (@, aftd then adding the interest component to determine the total return to holdins lot “The maximum change predicted by DRI implied an GXmta return tin ‘lomter_&rrn bandsof 17.71% at a time when DRI forecast a 9.49 % return on finaxe companq paper. For the MYW survey, the corresponding figures are 17.23 % and9.29 %.
not appear to be well understood by many pants. These results, for example, mirror those found ts in the United States. The led to fully understand the Goldsmith-Nagan survey he recorded forecasts of Eurodollars and Way Treasury ion, while those the long-term rids) did not. A similar, although less p ise, analysis by of the National Association of Business ric Forecasting Associates and Chase sts comparable results. In rise to the inability of rapolation n”orlong-term ults are not surprising in To concludt, economic agents ought to regard published for terns movements in long-term interest rates with a heal skepticism, although the same does not necessarily apply to future movements in short-term rates. Those seeking to forecast movements in longtezm rates over longer horizons ought to explore further the use of the ex implicit in tr’ieterm structure itself as a iction of the martingale model. In this long-term rates in the recorded data which imply substantial ex ante variation in the t-term bonds - merit note. These results evidence regarding the existence of at economic agents ought tc observe term rates must be ‘small’, at least to be credible; and (2) predicted t to be recast into statements regarding ex in part to facilitate comparison with the ex ante retturns (i.e.. the int st rate) on short-term securities.
y and u~fulR~s
of interest rate forecasts, Business Ekonomics
and the short-run determination of long85, no. 4, Aug., 661-689. tations effects 011 long-term borrowing ., 1979, Substitution an of Money, Credit b.?d Ranking I 1, no. 2, May, 131-d., survey evidence on the rationality of interest rate expectations, ~~o~~i~ forthcoming.
318
J.E. Pesando, On forecasting interest rates
Lintner, Job?. 1969,The aggregation of investors’ diverse judgements and preferences iii purely competitive security markets, Journal of Financial and Quantitative Analysis IV, Dee., 347400. Lynch, Kevin, 1979, A test of the rationality of interest rate expect&w using survey datk April, Mimeo. Masson, Paul R., 1978, Structural models of the demand for bonds a $5 the term strUCrureof interest rates, Economica 45, Nova5363-377, McCulloch, J, Huston, 1975,An estimate of the liquidity premium, Journal of Politicai &onomy 83, no. 1, Feb., 95-119. Mishkin, Frederic S., 1978, Efficient-markets theory: Implications for monetary policy, Brookings Papers on Economic Activity 3, 707-752. Modigliani, Franc0 and Robert Shiiler, 1973, Inflation, rational expectations and the term structure of interest rates, Economica 40, no. 157,Feb., 12-43. Modigliani, France and Richard Sutch, 1967, Debt management and the term structure of interest rates: An empirical analysis of recent experience, Journal of Political Economy 75, no. 4, pt. 2, Aug., 569-589. Pesando, James E., 1975a, The impact of the conversion loan on the term structure of interest rates in Canada: Some additional evidence, Canadian Journal of Economics 8, no. 2, May, 281-288. Pesando, James E., 1975b, A note on the rationality of the Livingstone price expectations, ,loumal of Political Economy 83, no. 4, Aug., 849-W. Pesando, James E,, 1978, On the efficiency of the bond market: Some Canadian evidence, Journal of Political Economy 86, no. 6, Dec., 1057-1076. Pesando, James E., 1979, 0. the random walk characteristics of short- and long-term interest rates in an ezfrcient market, Journal of Money, Credit and Banking XI, no. 4, Nov., 457+466, Prell, MJ., 1973, How well do the experts forecast interest rates?, Federal Reserve Bank of Kansas City, Monthly Review, Sept.-&t., 3-13. Sargent, Thomas J., 1976, A classical macroeconometric model for the United States, Journal of P&itical Economy 84, no. 2, April, 207-237. Shiller, Robert J., 1979, The volatility of long-term interest rates and exptations models of the term structure, Journal of Political Economy 87, no. 6, Dec., 1190-l 219. Working, Holbrook, 196@,Note on the correlation of first differences of averages in a random &ain, Econometrica 28, no. 6, Oct., 916918.
8 (1981 b 305-3 18. North-Holland Publishing Company
ve
movements in
of the martingale approxiwarion for two sets of Canadian interest rates data, emphasizing the iInportanW of the interval. The paper then examines three !Bet!? of reccxdcd foreca!Jtsof finds results consistent with the theoretical discussion. Canadian interest
In view of the substantial empirical evidence which supports the proposition that capital markets are efficient in their use of publicly available information, surprisingly little attention has been directed to the question of whether economic agents can successfully forecast interest rate movements. Impressed perhaps by the early evidence in favour of :he random walk hypothesis,’ most sophisticated agents tend to regard published forecasts of the future level of egate stock prices with a healthy degree of skepticism. As yet, however, such skepticism does not appear to extend to published forecasts of movements in either short- or long-term rates of interest, nor is it clear to many whether or not such skepticism is warranted. research program in Financial Markets those of the author and not those of the is indebted to Michael Huberman for
am
than for common stocks,
, where c1 is an in en equilibrium retu of drifi (i.e., P, -P,_ I =G, c kP, _ 1), c statement in the text is meant to of common stocks which translate sified portfolio are appropriately to apply to forecasts of changes easily translated into statements arket effiiency is viewed as less persuasive for bonds
with mean
J.E. Pesando, On farecasting
306
interest rates
At the theo&cal level, the question of *:Nhethereconomic agents can successfully anticipate interest rate movements in an efficient market has ishkin (1978) and esando (1979) caution that received increased attention. there is nothing in efficient markets theory which requires that short-term interest rates follow :a random walk. Thus agents without access to inside information may well succeed in forecasting movements in short-term rates. As Sargent (1976) and Pesando (1978) note, however, anticipated changes in long-term interest rates over brieJ:firrecast intervals must -- in the absence of time-varying term premiums -- be very close to zero. Long-term interest rates when observed over Fhort intervals will capproximately follow a martingale sequence and will thus exhibit random walk characteristics. Agents without access to inside information are not likely to succeed in forecasting short-run movements in long-term rates if the term premium accor&d long-term bonds is indeed time-invariant. This paper briefly reviews the theoretical results, paying particular attention to the importance of the forecast interval. Using two sets of Canadian data, the paper first illustrates the closeness of the martingale approximation. The paper then examines three sets of recorded forecasts of interest rates in Canada and finds results consistent with the theoretical discussion. The analysis highlights the fact that many of the forecast changes in long-term interest rates are so large (in absolute value) that they immediately lack ci-edibility when translated into predictions of hGld;ngperiod returns on long-term bondc.
Theoretical overview
Let R,,t denote the in crest rate on an n-period, non-coupon bond in pc ;iod & the information available to the marker in period t, and ,+Ji t the forward rate at time t for the one-period bond rate in period t + i. A&ume
that the expectations model of the term structure is valid, and that the long term rate is satisfactorily represented as an arithmetic (rather than a geometric) average of current and future short-term rates. Then, Sargent (1976) and Pesando (1978), the ex ante change in the IO can be written as follows:
E(R*tl#t-l)--R~**~,=~-*rE(*+,-~~* .tI4t-r)-~l,t-Il* n The term on the right-hand side of eq. (I), which represents the nonod rates, clearly approaches zero as n gets Iar overlapping onerepresents the ter
premium built into the interest rate on the
J.E. Pesado,
On forecasting interest rates
307
bond, then (I) becomes
+E(~(cplt-*P--ll/r-** tf this term premium is constant, martingale approximation em
t
(2)
reduces
to
(1)
(2) and
hence
the
(3) at the ex ante change in the n-period rate is Eq. (1) roximately zero if n is 1 so long as the one-period rate is not ‘tcjo -stationary’. The unit of rvation or forecast interval, which eff?&eIy identifies the one-period rate, is crucial in determining whether n is lartge and hence whether (3) is likely to be a ciose approximation. If one is forecasting the (annual) yield on a ten-year bond one quarter ahead, n equals 40. If one is forecasting the same yield one month ahead, n equals 120. In general, the shorter is the forecast interval, the smaller is the ex ante change in the yield on a gil,.en bond and hence the closer is the martingale approximation. Equivalently, it is short-pun movements in long-term rates which will not be predictable under the joint hypot esis of market efficiency and a timeinvariant term premium. For coupon bonds, as noted by Pesando (1979), the expression analogous to (1) i,
308
J.E. Pedo,
On forecasting interest
totes
end-of-the-quarter data on 90-day Treasury bills and 9Oday finance ion company paper, so as to eliminate possible biases due to time aggr [Working (1940)], alternative time series models can be fitted the Canadian data. For the sample period 1957: l-1979: 1, an ARMA(1,l) or ~tl AR(2) representation is superior to the martingale model for both short-ter~~ rates.’ The fact that the short-term rate need not follow a martin sequence is ‘rep:onciled’with the martingale property of the long-term rate when it is remembered that the latter is only approximate. I‘hs: long-term rate would exactly follow a martingale if and only if the short-term rate followed a martingale. In this case, the e:c ante capital gain or loss on longterm bonds would always equal zero, and the long-term rate would simply equal the short-term rate plus the time-invariant term premium. Finally, the martingale approximation for the longterm rate is derived on the assumption that the term premium sceorded long-term bond invariant. Many - if not most - of those who have conducted studies of the determinants of term premiums have concluded that they may well be time-invariant. McCulloch (1975) and Pesando (1975a) have presented evidence that the term premium accorded long-term government bonds in the United States and Canada, respectively, may be time-invariant. Attempts to enrich the traditional term structure equation of h4odigh&= Stitch ( 19B’f) or Modigliani-Shiller (1973) with additional variable capture the spirit of the ‘preferred habitat’ model have, on the whole, unsuccessful.Indeed, empirical support for the existence of timclvaryin premiums is largely c:bnfined to structural models of the long-t market, such as those estimated for the United States by Friedman (1977, 1979) and for Canada by Ma,sson (1978). In these models, demand funct5ns for long-term bonds are estimated for each major class of investor, such banks, life insurance companies and so forth, When the demands is set equal to supply, the models yield a (restri equation for the long-term rate. Although these redu contain asset stocks, Buancial flows and other variables varying term premiums, the repeated failure of comparable effwts in the traditional term structure equations is I” disconcerting. Clearly, the careful and successful inclusion of stock and variables into demand and supply functions is no&suf&iant to guu the implicit equation for the long-term interest rate will better charw ‘AR(l), AR(2) and ARMA(1,l) mQdels fox the Trwu -0.327B-0.905B3]r,~u,, (i-a930R)r,~(l+A367B)y, standard errors, in basis @nts, are 78.32, 75.98 and company paper rate, the models are: (l -0.934JI)r,~w,, -0.9LUBIr, = (1 +O.l54B)~. The standard errors are 94.08 93M and 92 AR(l) model, the null hypothesis that the coefkient of the be rejected. The standard errors for the martingale or AR this we&ient to equal unity, must be larger than those na
J
F t
of t
ml. Then
tian, this valuation
310
I.E. Pesado, On forecmting interest rata
r&e bond, or AR,” t, is the anticipated or ex aatte changej in the qxriod under conditi&s of market efficiency and the assumptions noted above. Let 6-I represent the capital gain or loss on the n-period or (L- 1 +&I --Kti ). One can show, in a straightfo
Again, the unit of observation or forecast interval is crucial in (6). If the forecast interval is one quarter, then n equals 40 if the maturity of the Ian term bond is 10 years. In addition, both (5) and (6) would refer to quart interest rates. If the forecast interval is monthly, then n = 120 for the bond, while (5) and (6) would refer to monthly rata. In indicates that ceteris pnribus AR& -- when expressed at an will be approximately one-third as large (in absolute value) if the intervzil is monthly rather than quarterly, and 90 forth? The reEISOnis quite straightforward: the briefer the holding-period, the smaller (in absolute value) is the capital gain or loss - and hence the smaller the change in the longterm rate - required to equilibrate the ex ante returns on short- and Ion term bonds in (5). ‘Because of Jensen’s inequality, expectation>; of bond prices and interest rates - whkh are inverseiy related - cannot both be unbiased. As formukwd in (S), the imptie unbi of future prices implies that t\e forward rate on the (n - t )=perlod bond to commenceat tim I is not an unbiased estimate _,f the corresponding spot rate, Lintner (1969) prr~idm cvicS?nce that this bias is small, and the issue is not pursued further in this paper. %ifferentiating (4) and dividing the resulting expression by pke yields
If the bond is selling at par, so ththt c/F equals R,, this expression rakxza to dP/P = -dR,/R,
* (I- t/(1 + R,)“).
Solving (F.2) for dR,, noting that dP/P=& as dcfin in the text, an6t?then in appropriate time subscripts yields (6). ‘Consider, for example, the case of a consal (II= a) when and RUM==0.12. For simplicity. assume #ul ==0. The esrrcspon R are, respectively, 0.015 and 0.03. Thus (6) implies =%‘kIO45. At an annual rate, this equals [email protected]) corresponding monthly rates for kl,, and R,., m Q.OWi monthly forecast interval, dR$,, s=- (0.01) * (-Qo05) 12 * (O.OOOOS) =0.0006 or 6 basis points. Because the bond is a on the right-hand side of (6) is simply equal to one, the PX uw interval is exactly one-third of that fcr the yuarterlv formnm! annual rates.
omit variabks
on ,t t rate foreaasi~
premiumsbonds, and 138 basis
: 1. These
able 1, indicate that the quite small. The mean
Table 1A Actual and anticipated quarterly Chan;a
Mean absolute change Maximum absolute change Mean change Standard deviation
in long-tetnr int 1999: 1.
23,w
2.09 7.21 - 0.03 2.71
26.38 127.00 6.93 3450
I l9,og
s*J)o 30.23
“All numbers ar\“: in basis points at annual rates, calculation of the anticipated changes.
Table lb Cumulative frequcwy dl’tribution of absolute values Ot anticipated &I interest t&es?
inll
Government bonds Absolute change No. I.__D__y__mm _..I 1__1_1 - _..“_.._ Less than Less than Less than Less than Less than
2 4 6 8 10
P-S
51 79 84 89 89
?eFsent 59 88 v4 100 100
“AI1 numbers are in basis points at annual calculation of the anticipated changes.
By contrast, the actual changes in quite large. Of partieuhv interest is the variances of the ex antarthan and th ratio of the former 80(x.
long-term rates must be u imply that 99.38’J$and 98, Canada and corporate bands,
regressions of the current &an known at the besinni percent ebr so of th explained variance would s suggest that the observed rel
I -I Px_usBm-e-
&
4 9 82 -
313
and (2) several of the series (such as the prime lending rate of the banks) are not determined in auction markets, only three appropriate for this study: 9Qday finance company pa corporate bonds (MYW 10 industrizris) and Ion (MYW 10 provincials), In comparing the recorded forseasts to the predkz model, care must be made to dtlsure that the current value of the relevant interest rate) do not not available at the time the reorded forecmte published data on the several interest rates series are availa& only c)~1a monthly basis, the martingale forecasts were set txlual to the interest rate at the end of the latest month which unambiguously rtxx2.4 tk month in which the recorded forecasts were made. In so doin an informational advantage to the recorded fo typically consisted of about two weeks, but in so month. The results of the forecast comparisons,’ summarized in table 2, indicate that the recorded forecasts of long-term rates in general failed to outpxform the no-change prediction of the martingale model, while this wus not the case for the recorded forwasts of short-term rates. :n view of the theoreticA discussion, this result is not surprising. Only the two-quarter ahead forecast by DRI of the long-term Canada rate proved superior to the martingak forecasts. Even this modest success merits qualification, however, in view of (1) the informational advantage accorded the recorded’forecasts, and (2) the inferior performance of the one-quarter ahead forecasts relative to those of the martingale model. rhis latter result sugce;sts that the improvement sf the two-quarter ahead forecasts may be somewnat of a statistical artifact and not likely to obtain over a longer sample period. Both the DRI and MYW survey i’orecasts of the short-term prove significantly superior to the martingale modal. Further, the fe, the short-term rate by the Conference Board were about as acxura martingale model, while their forecasts of the long-term rate were distinctly ingerior. These results, in general, enhance th with respect to the l’oreeastsof’the Ion term rates. In reinains - es;xcially with su:vey data figures do not accurately reflect the exp behavior. In this regard, it is interesting ta note that- Priedmrrn (n.&) provides evidence that the Goldsmith-Nagan int pass a now standard battery l>f tests designed to As notbxl by Pesando (1975b), such fa#~res rrri:sethe *The forecasts differ in their relecsadates, and pertain to both quarterly I& Resourcesand the Conference Board) rendend-&ho-quarter [MYWsurro result,theaccuracy of the forecastscannot be compared across forecasting
f
c
Q
316
J.E. Pesando, On forec&ing
interest
rates
whethe,r the recorded forecasts do indeed reflect those of active economic agents, in which case the latter fail to form ‘rational’ expectations, or whether such recorded forecasts must not be those of the market on the main hypothesis that the market efficiently processes performs this standard battery of tests -- th.t: un the orthogonality of forecast errors to costlessly efficiency and consistency in the use of autoregressive information - on the MYW survey data and, on the whole, the results are favourable to the hypothesis that the expectations are rationally formed. Finally, the magnitude of the changes in the long-term interest rates implicit in many of the recorded forecasts merits comment. The maximum (absolute) change forecast for the corporate bond rate, based on the twos period compared to the one-period prediction (so as to eliminate the uncertainty regarding the date - and hence the value of this rate QLI_ when the forecasts were made), equalled 22 basis points in the MYW survey data, 30 basis points for the DR.1 forecasts and 70 basis points (!) for the Conference Board forecasts. The mean absolute changes were 11.2, 12.7 and 13.3 basis points, respectively. In view of the magnitylde of the ex ante changes summarixed in table 1, these recorded forecasts clearly imply substantial variation in the ex ante return on long- relative to short-term securities. For the extreme forecast of the Conference Board, for example, the implied ex ante return on long-term bonds is 32.60x? Du this same quarter, the Cotierence Board was predicting a 6.86% retu on !&day finance company paper. On a prima facie basis, this result appears to be unreasonable and ierves to highlight the conceptual importance of translating predict& changes in long-term rates into comparable statements about their ex mte holding-period return. Although correspond& calculations for tJ:c DRI and MY W survey forecasts are less dramatic, the also imply a very sharp divergence between the ex ante returns on shortlong-term securities. I0 Both forecasters and those who monitor for would be well advised to t’ocus*on ex ante returns in assessing the lik of long-term interest rates during their forecast horizons, 5. Summary and conclusions The failure of the recorded forecasts of lo outperform the no-change prediction of the mart this result for short-term rates, is not surprising on ‘These returns ale cahxlatd by usiag the recorded foraclasts of thtt than in ion&tern interest rates to determine the capital gain or loss on long-term bond (@, aftd then adding the interest component to determine the total return to holdins lot “The maximum change predicted by DRI implied an GXmta return tin ‘lomter_&rrn bandsof 17.71% at a time when DRI forecast a 9.49 % return on finaxe companq paper. For the MYW survey, the corresponding figures are 17.23 % and9.29 %.
not appear to be well understood by many pants. These results, for example, mirror those found ts in the United States. The led to fully understand the Goldsmith-Nagan survey he recorded forecasts of Eurodollars and Way Treasury ion, while those the long-term rids) did not. A similar, although less p ise, analysis by of the National Association of Business ric Forecasting Associates and Chase sts comparable results. In rise to the inability of rapolation n”orlong-term ults are not surprising in To concludt, economic agents ought to regard published for terns movements in long-term interest rates with a heal skepticism, although the same does not necessarily apply to future movements in short-term rates. Those seeking to forecast movements in longtezm rates over longer horizons ought to explore further the use of the ex implicit in tr’ieterm structure itself as a iction of the martingale model. In this long-term rates in the recorded data which imply substantial ex ante variation in the t-term bonds - merit note. These results evidence regarding the existence of at economic agents ought tc observe term rates must be ‘small’, at least to be credible; and (2) predicted t to be recast into statements regarding ex in part to facilitate comparison with the ex ante retturns (i.e.. the int st rate) on short-term securities.
y and u~fulR~s
of interest rate forecasts, Business Ekonomics
and the short-run determination of long85, no. 4, Aug., 661-689. tations effects 011 long-term borrowing ., 1979, Substitution an of Money, Credit b.?d Ranking I 1, no. 2, May, 131-d., survey evidence on the rationality of interest rate expectations, ~~o~~i~ forthcoming.
318
J.E. Pesando, On forecasting interest rates
Lintner, Job?. 1969,The aggregation of investors’ diverse judgements and preferences iii purely competitive security markets, Journal of Financial and Quantitative Analysis IV, Dee., 347400. Lynch, Kevin, 1979, A test of the rationality of interest rate expect&w using survey datk April, Mimeo. Masson, Paul R., 1978, Structural models of the demand for bonds a $5 the term strUCrureof interest rates, Economica 45, Nova5363-377, McCulloch, J, Huston, 1975,An estimate of the liquidity premium, Journal of Politicai &onomy 83, no. 1, Feb., 95-119. Mishkin, Frederic S., 1978, Efficient-markets theory: Implications for monetary policy, Brookings Papers on Economic Activity 3, 707-752. Modigliani, Franc0 and Robert Shiiler, 1973, Inflation, rational expectations and the term structure of interest rates, Economica 40, no. 157,Feb., 12-43. Modigliani, France and Richard Sutch, 1967, Debt management and the term structure of interest rates: An empirical analysis of recent experience, Journal of Political Economy 75, no. 4, pt. 2, Aug., 569-589. Pesando, James E., 1975a, The impact of the conversion loan on the term structure of interest rates in Canada: Some additional evidence, Canadian Journal of Economics 8, no. 2, May, 281-288. Pesando, James E., 1975b, A note on the rationality of the Livingstone price expectations, ,loumal of Political Economy 83, no. 4, Aug., 849-W. Pesando, James E,, 1978, On the efficiency of the bond market: Some Canadian evidence, Journal of Political Economy 86, no. 6, Dec., 1057-1076. Pesando, James E., 1979, 0. the random walk characteristics of short- and long-term interest rates in an ezfrcient market, Journal of Money, Credit and Banking XI, no. 4, Nov., 457+466, Prell, MJ., 1973, How well do the experts forecast interest rates?, Federal Reserve Bank of Kansas City, Monthly Review, Sept.-&t., 3-13. Sargent, Thomas J., 1976, A classical macroeconometric model for the United States, Journal of P&itical Economy 84, no. 2, April, 207-237. Shiller, Robert J., 1979, The volatility of long-term interest rates and exptations models of the term structure, Journal of Political Economy 87, no. 6, Dec., 1190-l 219. Working, Holbrook, 196@,Note on the correlation of first differences of averages in a random &ain, Econometrica 28, no. 6, Oct., 916918.