Excess long real rate volatility

Journal of Multinational Financial Management 9 (1999) 155 – 176

Excess long real rate volatility H.J. Smoluk * Uni6ersity of Southern Maine, School of Business, Box 9300, 96 Falmouth Street, Portland, ME 04104 -9300, USA Received 31 May 1998; accepted 30 September 1998

Abstract Variance and comovement bounds tests are performed on riskless real interest rates for the USA, Canada, UK, Germany, and Japan. Each country’s long real rate exhibits excess volatility relative to its fundamental long real rate derived under the rational expectations theory of real term structure. Internationally, each country’s long real rate relative to the USA exhibits excess comovement relative to their corresponding fundamental long real rates. The excess volatility clouds the arbitrage-induced link between long and short real rates. This noise hinders the monetary transmission mechanism and the ability of central bankers to influence long real rates by managing short real rates. © 1999 Elsevier Science B.V. All rights reserved. JEL classification: E43; G15 Keywords: Real term structure; Rational expectations; Variance bounds

1. Introduction Since the level of economic activity in any country is highly dependent on real rates, monetary officials frequently attempt to manage their business cycles by influencing the real term structure. In theory, changes in short real interest rates, brought about by monetary policy, affect long real rates, and eventually real economic activity. An analysis of this link between short and long real rates, in terms of their second moments, can provide valuable insight into the effectiveness of monetary policy. The purpose of this paper is to examine this link using variance and comovement bounds tests for real interest rates under the rational expectations * Tel.: +1-207-780 4407; fax: + 1-207-780 4662; e-mail: [email protected]. 1042-444X/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S 1 0 4 2 - 4 4 4 X ( 9 8 ) 0 0 0 5 3 - X

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theory of real term structure (RETRTS). The RETRTS represents a synthesis of the rational expectations theory of nominal term structure (RETNTS) and the Fisher (1930) theory of interest rates. Tests of nominal term structure generally reject market efficiency due to excess volatility in long rates relative to short rates. For instance, Shiller (1979), Mankiw and Summers (1984), Mankiw (1986) suggest that nominal long rates are excessively volatile since attempts to explain their behavior with expected future nominal short rates generally fail. Shiller (1979) rejects the rational expectations theory of nominal term structure and states that time-varying term premiums may be one source of the rejection. Mankiw (1986) examines several theories on time-varying term premiums, but rejects each of them as an explanation for all the excess nominal rate volatility1. This paper makes two contributions to literature by employing variance and comovement bounds tests on riskless real interest rates2. First, it shows that actual long real interest rates exhibit excess volatility relative to long real rates derived under the RETRTS3. The results are robust across countries in that excess volatility is found in each of the five countries sampled: the US, UK, Canada, Germany, and Japan for the period of January 1964 to May 1996. These results suggest that the source of the excess volatility found by Shiller (1979), Mankiw and Summers (1984), Mankiw (1986) in long nominal interest rates is within the real rate component, and not in the time-varying term premium4. The second contribution this paper makes is that it demonstrates excess comovement in actual long real rates between the US and each of the other four countries sampled relative to fundamental real rates derived from the RETRTS model. These results have significant domestic and international policy implications for term structure. On the domestic side, one area that most economists agree on is that long real rates play a significant role in the demand for housing, consumer durables, and capital expenditures. Exploiting the RETRTS, monetary officials attempt to manage short real rates in order to influence long real rates, and hence, the direction of their economies. The noisier this transmission mechanism, the less

1

Under the context of the pure RETNTS, investors are risk-neutral so that long and short nominal rates are perfect substitutes. Begg (1982) rejects the idea that investors are risk-neutral and claims that the time-varyiag term premium discussed by Shiller (1979) actually represents investor uncertainty about future infliction so that markets are essentially efficient. 2 The riskless real rate in this paper refers to a real interest rate that is derived by subtracting not only expected inflation from nominal bond yields, but also an estimate for inflation risk. The resulting rate is ‘riskless’ in the sense that it is free of an estimated inflation uncertainty risk premium. 3 In this paper, the actual long real interest rate is equal to the long Treasury bond yield less both an expected inflation premium and an unexpected inflation (term) premium. 4 Mankiw (1986) states that the literature lacks an operationally explicit theory for the term premium, so that it is difficult to conclude that it is the source of nominal excess volatility. Accordingly, I suggest that without an operationally explicit model that can disentangle long real rates from time-varying term premiums, it is difficult to make conclusions about the source of the excess volatility.

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effective is monetary policy. One conclusion that can be drawn from this paper is that excess volatility in long real rates undermines the monetary transmission mechanism in that long real rates may be strongly influenced by variables other than short real rates. Internationally, excess volatility and comovement in real rates suggests the possibility of large foreign asset risk premiums. The relative variability and comovement in real returns to foreign assets plays a significant role in the determination of foreign asset risk premiums under mean-variance portfolio theory. This paper shows that the magnitude of excess volatility in real rates is different across the countries sampled. Furthermore, the magnitude of excessive comovement between the US and each foreign country sampled also varies. These results have international policy implications that focus squarely on countries with not just volatile markets, but with countries that exhibit significant excess real rate volatility. Countries with significant excess real rate volatility are likely to exhibit excess covariability with other nations, and therefore are forced to receive discounts on their investments from foreigners5,6. According to Fisher (1930), nominal interest rates are composed of a real rate plus an expected inflation premium. This decomposition facilitates an examination of the RETRTS. The RETRTS is analogous to the RETNTS except that real rates are used in place of nominal rates. In other words, the theory suggests that long real rates are composed of a weighted average of expected short real rates. An unbiased representation of expected future short real rates is obtained by substituting ex-post short real rates into the model to derive a fundamental (perfect-foresight) long real rate series. The pure (i.e. no risk premiums) RETNTS implies that long nominal rates are perfect substitutes for the series of expected future short rates via arbitrage conditions. The RETRTS, therefore, implies that long real rates are perfect substitutes for the series of expected future short real rates. The expectations theory of real term structure is a hypotheses about real rates. Any rejection of market efficiency based on such an asset pricing model is inconclusive due to the joint hypothesis problem. The joint hypothesis problem states that any rejection of market efficiency may be due to either model misspecification or market inefficiency and, therefore, the true reason for the rejection indeterminable. The model may be misspecified for the following reasons: (1) the ‘wrong’ form of present-value relation is used; (2) ex-post short real rates or inflation rates are in some way biased estimates of expectations; or (3) the time-varying inflation risk premium adjustment to actual nominal interest rates is misspecified.

5

Excess volatility within a country generally translates into excess comovement between countries provided that the return correlations are sufficiently positive (or the excess volatility is significant) since cov(x,y) =corr(x,y)[var(x)var(y)]1/2. 6 Another international policy concern is real exchanges. Real interest rate parity conditions suggest that the volatility in real interest rates translates into volatile real exchange rates.

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The real rate volatility tests derived in this paper offer a significant advantage over nominal interest rate volatility tests in terms of the joint hypothesis problem. By examining real rate volatility, rather than nominal, the time-varying term premium issue noted in Shiller (1979), Mankiw and Summers (1984), Mankiw (1986), is addressed. Interestingly, Shiller does not attempt to model time-varying term premiums and Mankiw’s work on term premiums is done at the short end of the maturity spectrum to avoid problems with modeling long-term inflationary expectations. This paper attempts to disentangle long-term expected inflation and unexpected inflation (risk) premiums from the long real rate using a constant discount model so that any rejection of market efficiency due to the term premium is minimized7. Section 2 of this paper reviews the RETNTS and Fisher theory of interest rates. A synthesis of these two theories (RETRTS) is developed where fundamental long real rates are composed of a weighted average of expected future short real rates. Both variance and comovements bounds using actual and fundamental long real rates are derived. The variance bounds method examines domestic market efficiency. It states that the variance of the fundamental real rate should place an upper bound on the variance of the actual long real rate. Similarly, the international efficient markets hypothesis suggests that the comovements in actual real long rates between countries i and j should be accounted for by the comovements in fundamental long real rates between these same countries. In other words, it tests whether the comovements in fundamental long real rates are strong enough to account for the comovements in actual long real rates. Section 3 examines variance and comovement bounds tests on actual and fundamental real rates. The results show that the excess volatility in nominal long rates cannot be solely attributed to time-varying inflation risk premiums, so that excess real rate volatility is a highly probable source for the excess nominal volatility. Section 4 concludes the paper.

2. Real term structure

2.1. The Fisher/expectations theory of nominal interest rates Fisher (1930) defined that any one-period nominal interest rate is composed of an expected real rate of return, rt, and a premium for expected inflation, pt, that guarantees the real rate of return to the investor8. Rt =Et [rt +pt +rtpt ]

7

(1)

As Begg (1982) points out, the term premium that Shiller and Mankiw are discussing may actually represent a risk premium for inflation uncertainty. In any case, as discussed below in footnote 14, the premium modeled in expression 25 can take into account the term (maturity) of the bond. 8 The guarantee exists only if the weighted average of future inflation equals pt, the expected inflation rate.

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or equivalently, Rt =Et [(1 + rt )(1 +pt )] −1.

(2)

Et represents expectations conditional on information at time t. Under the expectations theory of term nominal structure an n-period long rate represents an average of n expected future nominal short rates each of which, in turn, can be decomposed into a short real rate and an inflation premium. A geometric average representation is9, LR t(n) =Et (1 + lr t(n) )(1 +p t(n) )] −1

(3)

lr t(n) =Et [(1 + srt )(1 +srt + 1) … (1+ srn )]1/n − 1

(4)

p t(n) =Et [(1 + pt )(1 +pt + 1) … (1 + pn )]1/n − 1.

(5)

Substituting expressions 4 and 5 into expression 3 yields, LR t(n) =Et [(1 + srt )(1 +pt )(1 +srt + 1)(1+ pt + 1) … (1 + srn )(1+ pn )]1/n − 1. (6) Expression 6 states that long nominal rates are composed of an average of expected future short real rates, srt, and an average of expected future inflation rates, pt. An unbiased representation of expected future one-period short rea1 rates is computed by subtracting the ex-post inflation rate from the ex-post nominal short rate, srt =SRt −pt.

(7) 10

The result is an ex-post short real rate . Mathematically, the problem with expression 6 is that geometric averages impose a nonlinear relation between nominal long rates and both real rates and inflation rates. A linear approximation of the n-period nominal long rate is as follows, (n) LR t(n) =lr (n) t +p t

(8)

From expression 8 a fundamental long real rate, under the rational expectations framework, can be calculated where ex-post short real rates are substituted for expected short real rates. By assuming a perpetuity, n= , the fundamental long real rate, lr*t and fundamental inflation component, p*t are,

 



n

lr*t = (1 − gSR) % g kSRsrt + k . k=0

n

(1 − gSR) % g kSRpt + k . p*= t k=0

9

(9) (10)

The RETNTS assumes long and short rates are perfect substitutes thereby implying investors are risk neutral so that inflation risk premiums are not a component of long bond yields. 10 The variance and comovement bound methods require ex-ante real interest rates. Since observable ex-ante real rates are unavailable for the sample period used and the countries involved, ex-post real rates are employed as a proxy to ex-ante real rates.

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where, gSR =

1 1 +SR

.

Here the discount rate is the mean of the short rate series SR. A similar linear model is examined by Shiller (1979), Mankiw and Summers (1984), McFadyen et al. (1991).

2.2. Variance and como6ement bounds 11 In the following analysis, investors are only concerned with real rates. Under the assumption of market efficiency, the real rate earned from investing in a long bond, lrt, equals the expected long real rate derived from the series of expected short real rates, lr*t . Therefore, under rational expectations, an optimal predictor of the fundamental long real rate is the actual long real rate, lrt =Etlr*t.

(11)

expression 11 can be rewritten so that the fundamental long real rate equals the actual long real rate plus an error term, lr*t , =lrt +et

(12)

Variance decomposition results in, var(lr*t ) = var(lrt ) +var(et )

(13)

where the forecast error, under rational expectations, is uncorrelated with both actual short and long real rates implying, var(lr*t ) ] var(lrt )

(14)

Therefore, market efficiency states that the variance of the fundamental long real rate is greater than or equal to the variance of the actual long real rate given that the variance of the error term must be positive. Domestic market efficiency tests compare the variance of the fundamental long real rate to the variance of the actual long real rate for a single country. International market efficiency tests, on the other hand, compare the covariance of fundamental long real rates between two (or more) countries to the covariance of actual long real rates between these same countries. The international tests here are motivated by considering the bond markets of two countries. Long real rate column vectors (in bold) are expressed as lr%t =(lrit, lrjt ) and lr%* t = (lr* it , lr* jt ) so that, lrt =Etlr*t .

(15)

Given that lrt is an optimal predictor of lr*t , lr*t =lrt +et 11

For the development of this method of market efficiency testing see Shiller (1989), ch. 10.

(16)

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where et is a 2 × 1 vector under a two country case. The variance–covariance matrices of expression 16 imply, var(lr*t ) = var(lrt ) +var(et ).

(17)

The ith main-diagonal element reveals the variance terms for the ith country, var(lr*it ) =var(lrit ) + var(eit ).

(18)

The cov(lr*it , eit ) term equals zero since the optimal predictor, lrit, includes both current long and short real rates in the information set at time t. Positive-semidefiniteness of the main-diagonal of the error term variance–covariance matrix implies, var(lr*it ) ]var(lrit ).

(19)

The off-diagonal element of expression 17 reveals the covariance terms between countries i and j, cov(lr*it, lr*jt ) =cov(lrit, lrjt ) +cov(eit, ejt ).

(20)

While the efficient markets hypothesis says nothing about the sign of cov(eit, ejt ), if empirical evidence shows it is positive then12, cov(lr*it, lr*jt ) ]cov(lrit, lrjt )

(21)

Therefore, a testable null hypothesis of market efficiency in this paper is, H0: var(lr*it ) ]var(lrit )

(22a)

cov(lr*it, lr*jt ) ]cov(lrit, lrjt )

(22b)

The first part of the null hypothesis implies domestic market efficiency. It states that the variance of the fundamental long real rate is greater than or equal to the variance of the actual long real rate. The second part of the null hypothesis implies international market efficiency and asserts that the covariance of the fundamental long real rates should be greater than or equal to the covariance of the actual long real rates (provided that cov(eit, ejt ) \0 in expression 20). The theoretical variance bounds require that short real rates are stationary. There is theoretical and empirical support for such an assumption. From a theoretical perspective, real interest rates are considered to be a function of the business cycle and its mean reverting tendencies. Empirically, however, the support is more controversial. Using a variety of tests, such as the Dickey and Fuller (1979, 1981), Phillips and Perron (1988), Perron (1989), many researchers have found that real rates are nonstationary. Recently, though, several researchers have cast some doubt on these conclusions, see for example, DeJong et al. (1992), Campbell and Perron (1991), Cochrane 12 If cov(eit, ejt ) is negative, market efficiency may still hold under the concept of positive information pooling. Positive information pooling implies that it is more efficient to predict the combined value of the long real rates rather than the individual components. This issue is not a concern in this paper since each covariance term is positive. See Table 3 and Eq. (5).

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(1991). The problem with these tests is that have very low power in finite samples and they often fail to reject the null hypothesis of a unit root. Zu and Zhang (1997) question the nonstationarity conclusions of many researchers by employing more powerful unit root tests based on cross-maturity Treasury bill yields. They conclude nominal interest rates are stationary and that the primary reason for many researchers failing to reject the unit root hypothesis may be the low power of the tests used. Therefore, given the low power of unit root tests, the recent evidence that nominal interest rates appear stationary, and theoretical considerations, short real interest rates are assumed stationary. A general linear model for the actual long real rate is as follows, lrt =LRt −p*t −IRPt

(23)

where IRPt represents an inflation uncertainty risk premium and p*t . represents a fundamental inflation premium (expected) based on ex-post inflation rates. A linear model for the fundamental long real rate is also required. Transforming expression 9 into an operable moving average, starting backwards from a terminal value of sr*T equal to the mean of the actual short real rate series results in13, lr*t =[gSRlr*t + 1 +(1 −gSR)srt ],

(24)

where lr*t + 1 represents the fundamental long real rate.

2.3. A Time-6arying inflation risk premium Under the pure expectations hypothesis investors are risk-neutral so that risk premiums do not exist. Empirically, however, a time-varying risk premium may be the cause of the rejection of variance bounds tests employing the RETNTS. Assuming that a risk premium exists, the source of this premium remains controversial in the literature. Begg (1982) claims that a time-varying term premium, which reflects inflation uncertainty, is embedded in nominal bond yields, also see Kandel et al. (1996). Smoluk (1997) examines the issue and suggests that investors appear to be overreacting to current inflation so that market efficiency is rejected under the RETNTS. Graphically, and using regression analysis, Smoluk shows a statistically significant relationship between the forecast error in nominal yields (similar to error term in expression 12) and current inflation rates for each country. In this paper it is assumed that the source of the risk premium reflects investors’ concern about uncertain future levels of inflation. Consequently, when buying

13

The transformation, without the SR subscript notation for convenience, is as follows: lr*t =(1− g)[srt + g 1srt + 1 + g 2srt + 2 + … +g ksrt + k ] = (1 − g)srt + (1−g)g 1srt + 1 + (1−g)g 2srt + 2 + … +(1 − g)g ksrt + k =(1 − g)srt + g(1− g)[srt + 1 + g 2srt + 2 + … +g ksrt + k ] = (1 − g)srt + glr*t + 1.

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bonds, investors incorporate not only an expected inflation premium, but also an inflation uncertainty risk premium into nominal yields. The data in this paper reveal that by not accounting for this inflation uncertainty premium, the actual long real rate for each country is higher than its corresponding fundamental real rate. One possible explanation for this result is that by merely subtracting ex-post inflation rates from nominal long rates the total inflation premium built into long nominal bond yields is understated. Omitting the inflation risk premium, therefore, results in an overstating of the actual long real rate relative to the fundamental long real rate. Accordingly, expression 23 should have an IRPt that captures both the inflation ‘overreaction’ premium and/or the inflation uncertainty premium that may be present in the actual long real rate for country i, but is not in its fundamental long real rate. Such an inflation risk term premium is estimated as follows, IRPit =ai

)

)

Pit −Pit − 1 . Pit

(25)

expression 25 captures investors’ ‘overreaction’ to inflation by adding an inflation uncertainty risk premium to the expected inflation term. This inflation risk premium equals a percentage (ai ) of the absolute value of the inflation rate for the last month, where Pi, represents the consumer price index for country i. This premium also captures inflation volatility, which arguably is the primary source of inflation uncertainty. Notice that this premium becomes more volatile as the price index becomes more volatile. Furthermore, the risk premium is zero if the inflation rate is zero, and is constant if the recent inflation rate is constant. Alpha is a constant that adjusts IRPi,t so that the mean of actual long real rate equals the mean of fundamental real rate14. A lag of only 1 month for the price index is used for several reasons. First, the inflation risk premium based on the last month’s price index captures the inflation volatility that concerns of investors. Second, using time periods that are more than 1 month reduces the premium volatility and, hence, increases the volatility of the actual long real rate. An inflation premium based on recent inflation produces less volatile actual long real rate and, therefore, more conservative variance and comovement bounds tests. Finally, it results in a loss of only one observation in the data set. Kandel et al. (1996) use a similar, but more narrowly defined, proxy for the inflation uncertainty premium. The inflation risk premium in expression 25 is only one of many possible functional forms. Mankiw (1986) uses a similar form, except the Pt is replaced with long nominal rates. The hypothesis that he tests is whether the spread between long and short rates is positively related to the volatility in long nominal yields. Using regression analysis, Mankiw finds a negative relation between long nominal yield vol and spread, thereby rejecting the hypothesis that spread is a function of long yield volatility. 14 In addition, a may be considered to incorporate any maturity related effects that a term premium may, in theory, reflect.

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3. Bond market variance and comovement bounds tests

3.1. The data Details, such as descriptions and sources, for all data series are found in the Appendix A, data sources. All data are monthly. The long yields for the US, Canada, Germany, and Japan are composites of long-term government bonds with varying maturities and, therefore, represent averages. These averages tend to smooth the actual long series relative to a particular maturity yield series and, therefore, provides a more conservative (less volatile) series when compared to the fundamental yield derived from a single maturity short rate. For the UK, composite long bonds yields are used except for the period January 1992–May 1996 where 10-year constant maturity bonds are used15. It is important to point out that while the variance bounds derived here are for infinite-term bonds, the use of finite term bonds is still valid. As mentioned earlier, the variance bounds methodology applies to finite term bonds so long as the maturity of the long bond is substantially greater than the maturity of the short instrument. This paper employs short nominal rates that are of 3 months or less compared to long nominal rates that are at least 7 years in maturity. The short nominal rate for the US, UK, and Canada is the 3 month Treasury bill rate. German Treasury bill rates for much the of sample period were controlled by the German government when the Bundesbank fixed both the selling and repurchase rates. Consequently, the call money rate, which represents the price of money lent or borrowed without security among German banks to meet temporary shortages or excess supplies of liquidity, is used. Japanese Treasury bill rates, if not strictly controlled by the government, are not available for parts of the sample period. In addition, the Treasury bill rates were, for the most part, fixed by the Ministry of Finance and the market was frequently thin. Therefore, the call money rate, that represents the price of money lent or borrowed without security among Japanese banks to meet temporary shortages or excess supplies of liquidity, is used. Theoretically, Treasury bond yields are not based on a weighted average of expected future call money rates. However, for variance and comovement bounds tests, the use of call money rates represents a reasonable substitute for Treasury bill rates because of their short-term nature and low default risks. Inflation rates are calculated from consumer price indices found in various OECD publications. These indices attempt to include prices for all consumer goods and are not seasonally adjusted. Sample statistics for each variable appears in Table 1 for the sample period January 1964 – May 1996. It is worth noting that there are substantial differences in the variation of inflation rates across countries. Inflation variances play a key role 15 The OECD stopped publishing the 2 1/2% consol in 1991. Splicing together different default-free long rates does not present a problem here. The variance and comovements methods are robust to actual long rates of any maturity, provided its maturity is greater than the maturity of the short rate used to compute the fundamental rate.

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in the discussion that follows. Also note that the sample means for the fundamental real and actual rates are set equal based on ai (expression 25).

3.2. The results with the actual long real rate adjusted for IRPt A summary of the variances and covariances for the bond yields and their components by country is shown in Table 2. The results show that the variance of the actual long real rate is more volatile than the nominal rate for each country. Graphically, the fundamental long real rate, for the USA, UK, Canada, Germany and Japan along with the actual long real rate are depicted in Fig. 1a–e for the sample period January 1964–May 1996. All countries show that actual long real rates are significantly more volatile than their corresponding fundamental real rates. The corresponding sample variance–covariance matrices for riskless real rates are depicted in Table 3. The results in Table 3 show a violation in both expressions 22a and 22b and, therefore, the efficient markets hypothesis is rejected. The variance –covariance matrices show excess volatility within each country as well as excess comovement in actual long real rates among countries. Table 1 Summary sample statisticsa Variable

USA

UK

Canada

Germany

Japan

SR lr¯ , lr¯ * p¯ * IRP LR LR* p¯ gSR a i(n) Vaˆr(SRt ) Vaˆr(pt )

6.603 1.497 5.168 1.381 8.046 6.665 5.227 0.993 3.229 7.241 18.321

9.242 1.803 7.696 .426 9.925 9.499 8.073 0.994 0.648 10.429 92.620

8.059 2.903 5.403 .821 9.127 8.306 5.517 0.995 1.750 11.786 30.479

5.927 2.571 3.400 1.577 7.548 5.971 3.540 0.995 4.939 6.442 13.766

6.351 1.794 4.208 0.563 6.565 6.002 5.199 0.995 0.921 6.499 110.583

a The sample period covers January 1964–May 1996. The term SR denotes the mean of the nominal short rate. The short rate for the USA, UK, and Canada is the 3-month Treasury bill rate, for Germany and Japan it is the call money rate; lr¯ denotes the mean of the long actual real rate from expression 23 and lr¯ * denotes the mean of the fundamental long real rate from expression 24; the mean of lr¯ reflects an inflation uncertainty risk premium adjustment of IRPt so that lr¯ = lr¯ *. The statistic p¯ * denotes the fundamental inflation premium based on a weighted average of ex-post inflation rates with a terminal value equal to the mean of the inflation series; IRPt denotes the inflation uncertainty risk premium based on expression 25; LR denotes the mean of the actual long rate, which for all countries, is the long government bond yield, with the exception of Japan for the period January 1964–September 1966 where the official discount rate is used; LR* denotes the mean of the nominal fundamental yield based on a weighted average of ex-post short rates with a terminal value equal to the mean of the short rate, LR*= lr*t +p*t ; gSR denotes a constant discount factor and is equal to 1/(1+SR)) where SR is the mean of the monthly short rate; a is a constant needed to adjust the actual long real rate so that lr¯ =lr¯ *. Annualized returns.

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Table 2 Summary of sample variances and covariancesa

Vaˆr(LRt ) Vaˆr(lrt ) Vaˆr(p*t ) Vaˆr(IRPt ) Coˆv(lrt, p*t ) Coˆv(lrt, IRPt ) Coˆv(p*t , IRPt )

USA

UK

Canada

Germany

Japan

5.769 8.104 0.459 1.048 −1.620 −0.405 0.104

5.584 7.943 2.966 0.169 −3.035 0.143 0.145

5.948 9.281 0.987 0.468 −2.360 −0.157 0.123

1.634 2.428 0.125 1.593 −0.189 −1.122 0.055

2.811 4.292 1.283 0.317 −1.505 −0.160 0.124

a Long bond yields are modeled: LRt = lrt+p*t +IRPt. Variance decomposition results in var(LRt ) = var(lrt )+var(p*t )+var(IRPt )+2cov(lrt, p*t )+2cov(lrt, IRPt )+2cov(p*t , IRPt ). LRt denotes the actual long bond yield, lrt the actual long real rate, p*t . the fundamental inflation premium, and IRPt the inflation uncertainty premium. Actual long real rates are modeled: lrt =LRt−p*t −IRPt.

3.3. Accounting for biases According to Flavin (1983), Kleidon (1986), Marsh and Merton (1986), the variance and covariance analysis above contains several biases that severely limit any conclusions concerning market efficiency. Flavin (1983) demonstrates that small sample bias is significant and can push the results toward rejecting market efficiency. Kleidon (1986), Marsh and Merton (1986) address out-of-sample bias. Out-of-sample bias in this paper arises from the use of a terminal rate for lr*t equal to the sample the short real rate series. In other words, this ‘arbitrary’ terminal rate does not capture expected short real rates beyond the sample period and may be significantly different from the market’s expectations of future events. The small sample bias, as discussed by Flavin (1983), is likely to cause a rejection of the efficient markets hypothesis (expressions 22a and 22b). The source of the bias comes from calculating the variance of the highly autocorrelated fundamental real rate around its sample mean. Since the fundamental real rate series is highly autocorrelated relative to the actual long real rate series, its sample mean is likely to be more volatile for small samples. The relatively higher volatility in the fundamental sample mean induces a downward bias in its estimated variance compared to the actual rate. Flavin states that once small sample bias is corrected for, much of excess volatility disappears. Consequently, following her estimating procedure and, consistent with Mankiw et al. (1985), the variances and covariances of both the fundamental long real rate and the actual long real rate are calculated around zero instead of their sample means. The results are the sample noncentral second moments and the noncentral covariances of the variables lr*t and lrt as follows, T

T

t=1

t=1

VaˆrNC(lr*it ) = (1/T) % (lr*it )2 ](1/T) % (lrit )2 = vaˆrNC(lrit )

(26)

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Fig. 1. The actual long real rate (volatile series) and the fundamental long real rate (smooth series) for sampled countries. The fundamental long real rate is derived under the rational expectations theory of real term structure where long rates are composed of a weighted average of ex-post short real rates and the terminal rate is equal to the mean of the short real rate. The efficient markets hypothesis suggests that the variance of the fundamental long real rate should place an upper bound on the variance of the actual long real rate. The actual long real rate is lrt =LRt −p*t −IRPt. Annualized returns.

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T

t=1

t=1

CoˆvNC =(lr*,lr *jt ) =(1/T) % (lr*lr ˆ vNC(lrit,lrjt ) it it * jt )] (1/T) % (lritlrjt )= Co (27) The use of these moments are important in side-stepping any questions of small

Fig. 1. (Continued)

H.J. Smoluk / J. of Multi. Fin. Manag. 9 (1999) 155–176

169

Fig. 1. (Continued)

sample bias that may exist in the long real rates developed in this paper since the sample means from each series are removed from the variance computations. The sample noncentral second moments (Panel A) and covariances (Panel B) for the entire sample period, January 1964 to May 1996, are shown in Table 4. The results indicate excess volatility in actual long real rates relative to fundamental real rates for each country. Furthermore, the results show excess comovements in actual long real rates relative to fundamental real rates for each country relative to the USA. To avoid out-of-sample bias, the noncentral second moments and noncentral covariances are calculated for the sample period January 1964–May 198616. The results in Table 4 still show excess volatility and covariance in actual long real rates and that out-of-sample events are not strong enough to over-turn the initial rejection of market efficiency.

3.4. The results without an adjustment for IRPt The inflation risk premium adjustment can be removed from the actual long real rate in expression 23 to provide an interesting alternative model for the

16

The 10 years dropped off the sample roughly approximates the half-life of the terminal rate for an autoregressive process of order 0.993–0.995 (see gSR in Table 1). Mathematically, 0.993x =0.50 and 0.995x =0.50 implies x= 98.7 or x=138.3 months.

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actual long real rate relative to the fundamental long real rate. The results for both the traditional variance and comovements bounds tests as well as the noncentral second moments tests are presented in Tables 5 and 6. The results in Table 5 show that the variance and covariance terms for the actual long real rate typically increase across the countries as a result of adding back the inflation risk premium. Germany and Japan, however, show slight decreases in variance. The results are more noticeable in Table 6 where the noncentral statistics generally increase substantially depending on the magnitude of the inflation risk premium.

4. Conclusion This paper investigates real interest rate volatility using variance and comovement bounds tests. The variance bounds tests show that the volatility of actual long real rates is too large compared to the volatility of short real rates for each Table 3 Variance–covariance matrices, rational expectations theory of real term structure with lrt =LRt−p*t − IRPt a (S*USj )= (SUSj )+(S eUSj )



n

5.932 6.221 + 7.943 1.736

n

1.736 3.120

n



n

8.103 6.221 + 9.281 5.083

n

5.083 5.127

n



n

1.851 6.221 + 2.428 0.962

n

0.962 2.229

n



n

USA 6s. UK (January 1964–May 1996) 0.250 0.674 8.104 B 0.674 2.349 5.932 USA 6s. Canada (January 1964–May 1996) 0.250 0.441 8.104 B 0.441 0.970 8.103 USA 6s. Germany (January 1964–May 1996) 0.250 0.164 8.104 B 0.164 0.136 1.851 USA 6s. Japan (January 1964–May 1996) 0.250 0.314 8.104 B 0.314 0.450 3.102

n

3.102 6.221 + 4.292 1.302

1.302 2.622

n

a The equation S*USj = SUSj+S eUSj represents both domestic and international market efficiency. S*USj denotes the estimated variance–covariance matrix beween the USA and country j’s fundamental long real rates, SUSj, denotes the estimated variance–covariance matrix between the USA and country j’ s actual long red rates with an adjustment for inflation risk, S eUSj denotes the estimated variance–covariance matrix between the USA and country j’s error terms. See expressionss 18 and 20. The results shown below, in the variance–covariance matrices, indicate both domestic and international market inefficiency for all individual and pairs of countries sampled. Annualized returns.

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171

Table 4 Estimated noncentral second moments and noncentral covariances with lrt =LRt−p*t −IRPt a January 1964–May 1996 Country

lrt

USA UK Canada Germany Japan USA/UK USA/Canada USA/Germany USA/Japan

lr*t

January 1964–May 1986 lrt

lr*t

Panel A: noncentral second moments 10.325 2.490 11.720 5.592 17.687 9.398 9.033 6.750 7.499 3.668

12.105 12.880 16.188 9.380 9.646

2.681 5.432 8.883 6.675 3.860

Panel B: noncentral co6ariances 8.615 3.371 12.430 4.786 5.696 4.013 5.780 3.000

9.672 13.199 5.489 7.262

3.352 4.853 4.123 3.190

a In Panel A, the column lr*t denotes the sample noncentral second moments, vaˆrNC(lr*t ), of the monthly fundamental long real rate, and column lrt, the denotes the sample noncentral second moments, vaˆrNC(lrt ), of the monthly actual long real rate with an adjustment for an inflation risk premium. See expression 26. In Panel B lr*t denotes the sample noncentral covariances, for countries i and j, of the monthly fundamental long real rate, and lrt the denotes the sample noncentral covariances, coˆvNC(lrit, lrjt ), of the monthly actual long real rate with an adjustment for an inflation risk premium. See expression 27. Within each panel, a total of 389 observations for each country covers the entire sample period, from January 1964 to May 1996. A total of 269 observations, from January 1964 to May 1986, controls for out-of-sample bias by reducing the impact of the terminal fundamental real rate which is set equal to the sample mean of the actual short real rate for the period January 1964–May 1996. Annualized returns.

country sampled, the USA, UK, Canada, Germany, and Japan. The comovement bounds test shows that the covariability of long real rates between the USA and each of these countries is too large compared to the covariability of short real rates. Hence, both the domestic and international market efficiency hypotheses are rejected. Unlike many previous studies (Shiller, 1979; Mankiw and Summers, 1984; Mankiw, 1986), this paper shows that the source of the excess volatility in nominal bond yields is within the real rate component rather than a time-varying term premium that accounts for inflation uncertainty. However, since actual long real rates, actual short real rates, fundamental real rates, and inflation risk premiums are all unobservable, and subject to misspecification, the conclusion of excess long real rate volatility and comovement is easily open to challenge. Although some of the excess volatility can be attributed to changing expectations that are not captured in the fundamental real rate, the magnitude of the excess volatility detected here makes it difficult to attribute all of it to changes in expectations. This level of excess volatility poses a concern to central bankers who attempt to influence long real rates by managing short real rates. A substantial portion of the volatility in long real rates cannot be attributed to fundamentals.

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Acknowledgements I thank Geraldo M. Vasconcellos, David L. Muething, Richard J. Kish, Larry W. Taylor, and the participants of the 1996 Economics and Finance Seminar at Lehigh University for their valuable contributions. The comments of an anonymous referee and assistance from the Editor, Ike Mathur, greatly improved this paper.

Appendix A. Data Sources A.1. USA Inflation rates, monthly, for January 1964–December 1991 are calculated from OECD (1992b), ‘Consumer prices: all items’. For January 1992–June 1996 from OECD (various monthly issues), ‘consumer prices’, Table R.2/07.

Table 5 Variance–covariance matrices, rational expectations theory of real term structure with lrt =LRt−p*t a (S*USj )= (SUSj )+(S eUSj )



n

6.864 6.110 + 8.398 2.428

n

2.428 3.593

n

0.441 8.342 B 0.970 8.662

n

8.662 6.110 + 9.434 5.294

n

5.294 4.988

n



n

1.930 6.110 + 1.777 0.901

n

0.901 1.604

n



n

3.788 6.110 + 4.290 1.775

n

1.775 2.691

n

USA 6s. UK (January 1964–May 1996) 0.250 0.674 8.342 B 0.674 2.349 6.864



USA 6s. Canada (Janaury 1964–May 1996) 0.250 0.441

USA 6s. Germany (January 1964–May 1996) 0.250 0.164 8.432 B 0.164 0.136 1.930 USA 6s. Japan (January 1964–May 1996) 0.250 0.314 8.342 B 0.314 0.450 3.788

a The equation S*USj = SUSj+S eUSj represents both domestic and international market efficiency. S*USj denotes the estimated variance–covariance matrix between the USA and country j’s fundamental long real rates, SUSj denotes the estimated variance–covariance matrix between the USA and country j’s actual long real rates without an adjustment for an inflation risk premium, S eUSj denotes the estimated variance–covariance matrix between the USA and country j’s error terms. See expressionss 18 and 20. The results show both domestic and international market inefficiency for all individual and pairs of countries sampled. Annualized returns.

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173

Table 6 Estimated noncentral second moments and noncentral covariances with lrt =LRt−p*t a January 1964–May 1996

January 1964–May 1986

Country

lrt

lrt

lr*t

USA UK Canada Germany Japan

Panel A: noncentral second moments 16.602 2.490 13.343 5.592 23.282 9.398 18.980 6.750 9.836 3.668

18.732 15.337 22.575 20.668 12.710

2.681 5.432 8.883 6.675 3.860

USA/UK USA/Canada USA/Germany USA/Japan

Panel B: noncentral co6ariances 13.260 3.371 19.358 4.786 13.863 4.013 10.562 3.000

14.829 20.387 14.468 13.252

3.352 4.853 4.123 3.190

lr*t

a In Panel A, the column lr*t denotes the sample noncentral second moments, vaˆrNC(lr*t ), of the monthly fundamental long real rate, and column lr, the denotes the sample noncentral second moments, vaˆrNC(lrt ), of the monthly actual long real rate without an adjustment for an inflation risk premium. See expression 26. In Panel B lr*t denotes the sample noncentral covariances, for countries i and j, of the monthly fundamental long real rate, and lrt the denotes the sample noncentral covariances, coˆvNC(lrit, lrjt ), of the monthly actual long real rate without an adjustment for an inflation risk premium. See expression 27. Within each panel, a total of 389 observations for each country covers the entire sample period, from January 1964 to May 1996. A total of 269 observations, from January 1964 to May 1986, controls for out-of-sample bias by reducing the impact of the terminal fundamental real rate which is set equal to the sample mean of the actual short real rate for the period January 1964–May 1996. Annualized returns.

Long bond rates, monthly, for January 1964–December 1991 from Ibbotson Associates (1995), ‘long-term government bonds: yields’, from 1 month-end to the next month-end. For January 1992 – May 1996 OECD (various monthly issues), ‘composite — over 10 years’, Table R.2/07. Short rates, monthly, for January 1960–December 1987 from McCulloch and Shiller (1987) ‘Table A-3, McCulloch par bond yield curve series, continuous compounding, end of month data’. For January 1988–December 1991 from US Department of Commerce (1992) ‘yield on US government securities (taxable) 3-month bills (rate on new issues.) For January 1992–May 1996 from OECD (various monthly issues), ‘treasury bills, 3 months’, Table R.2/07. A.2. UK Inflation rates, monthly, for January 1964–December 1991 are calculated from OECD (1992b), ‘consumer prices: all items’, (including seasonal items.) For January 1992– June 1996 from OECD (various monthly issues), ‘consumer prices’, Table R.2/17.

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Long bond rates, monthly, for January 1964–December 1978 from OECD (1984), ‘yield of government bonds: 2 1/2% consols’, last Friday of the month. Monthly, for January 1979 – December 1991 from Central statistical office (1992) ‘2 1/2 Consols, gross flat yield’, based on the mean of the middle open and middle closing prices each day, excluding accrued interest. For January 1992–May 1996 from OECD (various monthly issues):, ‘yield on government bonds, 10 years’, Table R.2/17. Short rates, monthly, for January 1964–December 1979 from Capie and Webber (1985), Table 3 (10), ‘treasury bills yields’, column IV. For January 1980–December 1991 from Central statistical office, ‘treasury bills yields, 91 day bills’, last Friday of the month. For January 1992 – May 1996 from OECD (various monthly issues), ‘yield on 3-month treasury bills’, Table R.2/17. A.3. Canada Inflation rates, monthly, for January 1964–December 1968 are calculated from OECD (1984), for January 1969 – December 1988 from OECD (1989), ‘Consumer prices: all items’. For January 1989 –June 1996, from OECD (various monthly issues), ‘Consumer prices’, Table R.2/04. Long bond rates, monthly, for January 1964–December 1988 from OECD (1984, 1989), ‘yield of long term government bonds, per cent per annum’, based on average of buying and selling closing prices on last Wednesday of month. For January 1989 – December 1991 from OECD (1990–1992a), ‘yield on long term government bonds’, last Wednesday of the month. For January 1992–May 1996 from OECD various monthly issues), ‘on the secondary market, federal government bonds, over 10 years’, Table R.2/04. Short rate, monthly, for January 1964–December 1988 from OECD (1984, 1989), ‘treasury bill rates (3 months) per cent per annum’, average yield of last weekly issue in month. For January 1989 – May 1996, ‘treasury bills (91 days or 3 months)’, average allotment rate of the last issue of the month, OECD (various monthly issues). A.4. Germany Inflation rates, monthly, for January 1964–December 1991 are calculated from OECD (1992b), ‘consumer prices: all items’, (including seasonal items.) For January–March 1992 from OECD (1992 – 1996), ‘consumer prices: all items’, (including seasonal items.) For January 1992 – June 1996 from OECD (various monthly issues) ‘consumer prices’, Table R.2/01. Long bond rates, monthly, for January 1964–December 1988 from OECD (1984, 1989), ‘yield of long term government bonds, per cent per annum’, monthly data are daily averages. For January 1989–December 1991 from IMF (1989–1992), ‘government bond yields’. For January 1992–May 1996 from OECD (various monthly issues,) ‘on the secondary market, public sector bonds, 7–15 years’, Table R.2/01.

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175

Short rates, monthly, for January 1964–December 1988 from OECD (1984, 1989), ‘call money rate (Frankfurt), per cent per annum’. For January 1989–May 1996 from OECD (various monthly issues), ‘day-to-day loans’, Table R.2/01. Monthly data are daily averages. Note: call money and day-to-day loans represent the same interest rate; it represents money lent or borrowed without security among Germany banks to meet temporary excess or shortages of liquidity. A.5. Japan Inflation rates, monthly, for January 1964–December 1969 are calculated from OECD (1984), ‘consumer prices: all items (Tokyo)’. For January 1970–December 1991 from OECD (1984), ‘consumer prices: all items’. For January 1992–June 1996 from OECD (various monthly issues), ‘consumer prices’, Table R.2/21. Long bond rates, monthly, for January 1964–September 1966 from OECD (1984) ‘official discount rate’. For October 1966–December 1983 from OECD (1984), and for January 1984 – December 1988 from OECD (1989), ‘yields of central government bonds’. For January 1989–May 1996 from OECD (various monthly issues), ‘on the secondary market, central government bonds’, Table R.2/21. Note: Long term government yields were unavailable for the period January 1964–September 1966 so the official discount rate was used as a substitute. Although it is controlled by the Bank of Japan, the rate does exhibit long-term volatility. In any case, the rate provides a conservative (less volatile) estimate for the Japanese long bond yield. Short rates, monthly, for January 1964–December 1991 from OECD (1984, 1989) ‘call money rate’. For January 1992–May 1996 from OECD (various monthly issues) ‘call money rate’, Table R.2/21.

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