2.6 2.4-

1.2-

1 J-

[ - DEM -NLG NLGPIus

Figure 12.1 Daily German mark-US dollar and Dutch guilder-US dollar exchange rates, from January 1986 to December 1992.

Correlation is Correlation reflects co-movements in returns, which are liable to great intrinsically a short-run instabilities over time. It is intrinsically a short-run measure, so correlation-measure based hedging strategies commonly require frequent rebalancing. Investment management strategies that are based only correlations cannot guarantee long-term performance because there is no mechanism to ensure the reversion of the hedge to the underlying. And there is nothing to prevent the tracking error from behaving in the unpredictable manner of a random walk.

Since high correlation is not sufficient to ensure the long-term performance of hedges, there is a need to augment standard risk-return modelling methodologies to take account of common long-term trends in prices. This is exactly what cointegration provides. Cointegration measures long-run co-movements in prices, which may occur even through periods when static correlations appear low. Therefore hedging methodologies based on cointegrated financial assets should be more effective in the long term. Moreover, the cointegration methodology loses none of the traditional analysis. It merely augments the basic correlation model to include a preliminary stage in which the multivariate price data are analysed, and then extends the correlation model to include a dynamic analysis of the lead-lag behaviour between returns.

12.1.2 Common Trends and Long-Run Equilibria

When asset price time series are random walks, over a period of time they may have wandered virtually anywhere, because a random walk has infinite unconditional variance. There is little point in modelling them individually, since the best forecast of any future value is the just value today plus the drift, but when two or more asset prices are cointegrated a multivariate model will be worthwhile because it reveals information about the long-run equilibrium in

Hedging methodologies based on cointegrated financial assets should be more effective in the long term. Moreover, the cointegration methodology loses none of the traditional analysis

the system. For example, if a spread is found to be mean-reverting we know that, wherever one series is in several years time, the other series will be right there along with it.

Cointegrated log asset prices have a common stochastic trend (Stock and Watson, 1988). They are tied together in the long run even though they might drift apart in the short run because the spread or some other linear combination is mean-reverting. A simple example that illustrates why cointegrated series have a common stochastic trend is

x, = w, + ext,

y, = wt + syt, (12.1)

w, = w, i + e„

where all the errors are i.i.d. and independent of each other. In (12.1) the 1(1) variables x and are cointegrated because x - ~ 1(0). They also have a common stochastic trend given by the random walk component w. Note that the correlation between Ax and Ay is going to be less than 1, and when the variances of exl and/or eyt are much larger than the variance of e, the correlation will be low.2 So, as already mentioned above, cointegration does not imply high correlation. Of course, this example is very theoretical. It is unlikely that cointegrated series will conform to this model in practice, but it useful for illustration.

When asset prices are random walks there is little point in modelling them individually, but when two or more asset prices are cointegrated a multivariate model will provide insight

The linear combination of 1(1) variables that is stationary is denoted z. It is called the disequilibrium term because it captures deviations from the long-run equilibrium in the error correction model (ECM) (§12.3). The expectation of z gives the long-run equilibrium relationship between x and y, and short-term periods of disequilibrium occur as the observed value of z varies around its expected value. The cointegration vector is the vector of weights in z. So in the case of two 1(1) variables x and y, where x - ay ~ 1(0), the cointegration vector is (1, -a). When only two integrated series are considered for cointegration, there can be at most one cointegration vector, because if there were two cointegration vectors the original series would have to be stationary.

More generally, cointegration exists between n integrated series if there exists at least one cointegration vector, that is, at least one linear combination of the 1(1) series that is stationary. Each stationary linear combination acts like glue in the system, and so the more cointegration vectors found the greater the co-dependency between the processes. Yield curves have very high cointegration.

2This follows since V(Ax) = rx2 + 2rx2, V(Ax) = rx2 + 2a2, and cov(Ax, Ay) = a2, where ex2, ex2, and rx2 denote the variances of e, gx and gv respectively. So the correlation is rx2/y/[(rx2 + 2rx2)(rx2 + 2rx2)] which is small when rx2 is much smaller than rx2, and/or cr2-.

Yield curves have very high cointegration. Often each of the n - 1 independent spreads is mean-reverting, so there are n - 1 cointegration vectors, the maximum possible number

Table 12.1: Augmented Dickey-Fuller tests on the spreads of the US yield curve

Monthly data, 1944-1992

When there are n maturities in a yield curve, often each of the n - 1 independent spreads is mean-reverting, so there are n - 1 cointegration vectors, the maximum possible number.

Consider the US yield curve data shown in Figure 6.1 that was used to illustrate some of the stylized facts about principal components in §6.2.1. Looking at the same data from a cointegration perspective now, note that each individual yield series is integrated and so it has infinite unconditional variance (§11.1.3). This means that, given enough time, a yield could be almost anywhere (except, of course, negative). In 1944, the beginning of the period shown in Figure 6.1, who would have guessed what the 1-month yield would be at the end of the period, almost 50 years later? However, if these yields are cointegrated it is known that whatever the 1-month yield is in 1992, the 3-month yield will be very close to it.

Applying a unit root test (§11.1.5) to the spread will tell us if it is stationary, that is, whether the 1-month yield and the 3-month yield are cointegrated. Table 12.1 reports ADF(l) statistics for each of the spreads with the 1-month rate. Since the 1% critical value for this test is -3.46, each spread is stationary at the 1% level.

It is important that a sufficiently long period of data is used, in order that the common long-run trends can be detected

The findings of stationarity in spreads, and consequently cointegrated yields, are quite robust to changes in data period, and to different frequency of data. But it is important that a sufficiently long period of data is used, in order that the common long-run trends can be detected. Of course, it is not necessary to use almost 50 years of data as in this example, but even when the ADF tests are performed on subsets of these data they invariably give the same type of result, provided several years of data are used.