2.1. The dividend yield model

2.1.1. We start the investment variables with the dividend yield, Y(t), for which the formulae are as follows:

with parameters suggested in Part 1 (Wilkie et al., Reference Wilkie and Şahin2011):

2.1.2. This is basically an AR(1) model for YN(t), with the addition of a relatively small influence from I(t). As we did with WL(t) in Part 3B, we can split YL(t) into two parts:

We can investigate, and we can model, these parts separately, but since we assume that I(t) and hence YL1(t) are already given, this means investigating both YL and YL2. The natural bridging model for either of them is an OU one, to correspond with the annual AR(1) model for YL2, but we wish to look at them also as BBs, since, as we showed in section 2, there may not be much difference between these.

2.2. The dividend yield data

2.2.1. We have data available at apparent monthly intervals from 1923 to 2014, but in fact the data for 1923 is interpolated between December 1922 and December 1923, so we shall exclude this year and start in June 1924. There is also a small lacuna in 1929, where we have only quarterly figures, and have filled in the gaps with interpolated values. Since this was a rather eventful year, this is a pity, but overall it makes no noticeable difference. We show these monthly values in Figure 1. The graph is very well-behaved compared with those for some of the data series used in Part 3B.

Figure 1

Dividend yield %, Y(t), monthly from December 1922 to December 2014.

2.2.2. We analyse YL and YL2, both as BBs and as OU bridges. The calculation of YL2 does not have the same difficulties as that of WL2, because it is not cumulative. The initial estimates of σ _m for these four combinations are shown in Table 1. They are not very different. Those for YL2 are a little larger than those for YL, showing that, although the rate of inflation is important in the annual model, dividend yields do not respond monthly to the values; indeed they could hardly do so in detail, since the rate of inflation for the year ending in a particular month is not known till the middle of the following month, and our dividend yields are those observed on the last day of the month stated; nevertheless, the market might pay attention to the forecast forthcoming rate, and for 11 out of the 12 months the change in retail prices is already known. However, there seems to be little advantage in considering YL2 further.

Table 1

Estimates of σ _m for YL(t) and YL2(t), 1923–2014.

2.2.3. There is also rather little difference between the values of σ _m for the BB and OU models for YL. But there is a noticeable difference between the annual standard deviations, since for the BB, σ _y =√12.σ _m =0.17253 and for the OU σ _y =σ _m √{(1−α _y ²)/(1−α _m ²)}=0.01397, with α _y =0.63, and α _m =α _y ^1/12=0.962229. This, however, is quite consistent; a certain value for σ _m has different effects on the annual values σ _y with the two models. We note also that the annual standard deviation from the annual model, calculated as √(YW ².QSD ²+YSD ²)=0.16694, is not very different from the value estimated from the BB.

2.2.4. We continue analysing the YL series as if it were a BB. In Table 2 we show statistics for the forwards deviations, first unadjusted, and then standardised by dividing each value of Df _j by the estimated value of σ _m for the same year calculated from the values of Df _j in that year. We consider first the unadjusted values. We see that the T-ratios for four months are significantly large, three negative, December, January and April, and one positive, May. But we see also very high kurtosis for almost every month (a normal distribution has kurtosis of 3.0) and rather high skewness for several months. We can identify specific months with large values of Df _j : January 1975, −0.41478; February 1975, −0.20019; October 1988, +0.29222; and June 1940, +0.20896 (note that a negative sign implies that share prices rose, a positive sign that they fell). Each of these values is a large multiple of the overall standard deviation.

Table 2

Statistics for forwards deviations, Df _j , of YL, from June 1924 to June 2014; unadjusted and standardised.

2.2.5. When we standardise the values, the kurtosis is now almost always small, so we have possibly compressed the values too much, but more of the T-ratios appear significant. Standardising by the expected values of σ _m according to whatever regression formula we might choose might be better, but we have not got there yet.

2.2.6. We calculate the same table of unadjusted statistics assuming that the model is in fact an OU bridge, with α _y and µ _y as in the annual model. The values are almost the same as those shown in Table 2, so we do not show them.

2.2.7. We calculate the same statistics for the sideways deviations and show them in Table 3. For eight months out of eleven, the means are negative, and for four of these are significantly so. But the kurtosis for almost all months is very large, and the skewness for several also is; we do not show them, but they are similar to those of the forwards ratios. The M-ratios are reasonably close to unity, so although some individual values are very large, the overall tendency is not too different from a BB.

Table 3

Statistics for sideways deviations, Df _j , of YL, from June 1924 to June 2014.

2.2.8. We calculate the same statistics for the 33 years when there are upwards deviations, the 29 years when there are downwards ones, and the 28 years when there are medium ones. We have 62 years contributing to reversed deviations. We show the means in Table 4. If the data followed OU bridges more closely than BBs, the means of the upwards deviations would tend to be positive; in fact there are three positive and eight negative means; the means of the downwards deviations would tend to be negative; in fact there are seven positive and four negative means; the means of the reversed deviations would all tend to be positive; in fact all 11 are negative.

Table 4

Means for upwards, downwards, medium and reversed deviations of YL, from June 1924 to June 2014.

2.2.9. As further evidence, we show in Figure 2 the monthly values of YL from June 1924 to June 2014, marked with small dots, and with the June values marked with large dots, along with the expected values assuming BBs, with straight solid lines, and the expected values assuming OU bridges, with dotted lines. It is almost impossible to distinguish the dotted lines, even on a larger scale than shown in print, except in one or two places, where the values are well away from the middle, such as 1933–1934 and 1934–1935, or 1974–1975 and 1975–1976. In each case, the scatter of the actual values is so wide that any separation of a BB model and an OU model is impossible.

Figure 2

Monthly values of YL, from June 1924 to June 2014, with expected values assuming Brownian bridges (BBs) and Ornstein–Uhlenbeck (OU) bridges.

2.2.10. We conclude that it would, with this evidence, be perverse to suggest either that OU bridges fitted the yield data better than BBs, or vice versa, although the evidence of the annual data is very strongly in favour of an AR(1) model, as one can see by how often the actual data crosses the mean, which is taken as ln(YMU)+YW.QMU=−3.21676.

2.2.11. Just as for QL and WL there is some evidence that the value of σ _m in each year is connected with the change in YL over the year. We put YLD(t)=YL(t+1)−YL(t) and we investigate the same eight regression possibilities as for QL, regressing σ _m and σ _m ² against Abs(YLD(t)), YLD(t)², Abs(YLD(t)−YSC) and (YLD(t)−YSC)², where YSC is the calculated mean value of YLD(t)=−0.00092, which is very close to 0. The correlation coefficients are shown in Table 5.

Table 5

Correlation coefficients from regressions.

2.2.12. The correlation coefficients are smaller than those for QL, and not very different from those for WL, and are rather similar for σ _m against any of the four options. The value of YSC is so small that there is little difference between the results for the first four regressions (with YSC=0) and the last four. The correlation coefficients are very similar, as also are the values of A and B for corresponding regressions; the T-ratios for B for the σ _m regressions are all about 2.6 so adequately significantly different from zero. Since the theoretical long-term mean of YLD (as for all the autoregressive series) is 0, we prefer to use YSC=0. We show the same chart in Figure 3 as for the previous series, with the actual values of σ _m (t) plotted against those for YLD(t) as dots, and two lines for the expected values when regressed against Abs(YLD(t)) (formula 1) and against YLD(t)² (formula 3). These two are quite similar in their effect as well as in the correlation coefficients.

Figure 3

Standard deviations, and estimates of σ _m in each year, related to YLD(t) in each year, from 1924 to 2014.

2.2.13. The quadratic formula gives a slightly higher correlation coefficient

where YSA=0.040788 and YSB=0.097117. We include this variation of σ _m .

2.2.14. We next consider autoregression within the series of Df _j values or of the standardised values Df_jZ. There is almost no correlation from one year to the next in Df _j Z for individual months, unlike the situation for QL. There is a small autocorrelation within the two series for lags of 2, 5, 6 and 7 months, the (absolutely) largest correlation coefficient of−0.16 being for DF _j for a two months lag, but we consider that this can be ignored.

2.2.15. The remaining possible connection is a cross-correlation between the values of Df _j Z for YL, and the corresponding values for QL or WL. Most of the coefficients, simultaneous or for small lags, are very small. The biggest are a simultaneous correlation between the changes in the Df _j values for YL and for WL of 0.1556 and for the corresponding Df _j Z values of 0.1193. As we have noted in section 2.1.4, it is not likely that there would be direct simultaneous correlation between changes in YL over the month, and changes in QL over the month, and direct correlation with changes in WL seems less likely. We note the correlations, but they are still quite small and we ignore them.

2.3. Simulations

2.3.1. All this analysis leaves us with a slightly simpler model for YL than for QL or WL. YL is modelled over the year with a BB, with σ _m (t) modelled with the quadratic formula in section 2.2.13 and no other complications.

2.3.2. We show simulated values of Y(t)% monthly, for 50 years, in Figure 4, simulated as described above, and put onto the same vertical scale as Figure 2. When we compare these with the graph of actual values of Y(t)% in Figure 2, they look reasonably similar, though the extreme peak in 1974–1975 is not repeated.

Figure 4

Stochastic yearly and monthly simulation for 50 years of Y(t)%.

3.1. The dividend index model

3.1.1. The dividend index model resembles the Wages model, in that it involves the dividend index, D(t), its logarithm DL(t) and the annual rate of dividend increase, K(t)=lnD(t)−lnD(t−1). K(t) is dependent on the series for the annual rate of inflation, I(t), with a long moving average structure. There are small influences from the dividend yield series and the residual for the dividend index in the previous year DE(t−1). The basic formulae for K(t) and D(t) are as follows:

The parameters suggested in Part 1 (Wilkie et al., Reference Wilkie and Şahin2011) were as follows:

3.1.2. The relationship between, K(t), DL(t) and D(t) exactly parallels that between I(t), QL(t) and Q(t), as did J(t), WL(t) and W(t), so we can again conclude that we should apply bridging to DL(t) and not to K(t), deriving the simulated monthly values of K(t) from the simulated values of DL(t).

3.1.3. As for J and WL, we can partition K and DL into two parts, one depending on I(t) and the other not

We can apply the partitioning to DL(t) as

We can then investigate both DL and DL2. Although DL2(t) includes an influence from YE(t−1) and from DE(t−1) we can take these into account by investigating autocorrelations and cross-correlations of the resulting Df _j or Df _j Z series. However, in practice the calculation of DL2, being based on an accumulation of values of K2, has the same problems as that of WL2, and the results are not satisfactory.

3.2. The dividend index data

3.2.1. The dividend index data comes from the same sources as the dividend yield data, and its values have been calculated by multiplying the given share price index by the given dividend yield. Those who construct indices must calculate a dividend index in order to calculate the dividend yield, but this index is not normally published. There is a problem with constructing a satisfactory dividend index as well as a satisfactory price index. When stocks enter or leave a share index, or change their nominal share issue, the index can be “chain-linked” so that there is no discontinuity in the price index. But normally there is a discontinuity in the corresponding dividend index. When we have changed from one index to another as in 1928, 1929 and 1962, we have had to adjust the yield to give us a continuous dividend index, but we can make no adjustments to the ongoing FTSE-Actuaries All-Share Index, used since 1962. However, we assume that the movements of stocks in and out of the index are not so very large as to affect the whole index significantly, especially since the index contains many stocks, none overwhelmingly large (the same is not true of individual sectors within the indices, but we do not use these). We show in Figure 5 values of the dividend index, D(t) (multiplied by 2.5 to fit the scale better) from December 1922 to December 2014, and in Figure 6 K(t), the annual change in lnD(t), over the same period.

Figure 5

Values of 2.5×D(t), the dividend index, monthly, from December 1922 to December 2014.

Figure 6

Values of K(t), monthly, from December 1922 to December 2014.

3.2.2. Like the dividend yield index the dividend index was linearly interpolated between year-end values in 1923, so we start our analysis in June 1924 and carry it through to June 2014. In Figure 7, we show the monthly changes in DL(t) from 1923 to the end of 2014. One can see a few constant values at the start of the series, which indicate the linear interpolation. Otherwise the scatter does not indicate any problems with the source data. However, we can identify some rather large decreases and rather fewer large increases.

Figure 7

Monthly changes in lnD(t) from 1923 to 2014.

3.2.3. We assume BBs rather than OU bridges throughout. We start by comparing the estimated values of σ _m for DL and for DL2. That for the former is 0.01337 and for the latter is much bigger, at 0.03936. Allowing explicitly for DL1 does not help us at all. This is also not surprising. Dividends are declared by companies often twice a year, as interim and final, but in either order within the calendar year, because they have different year-ends. Many companies have accounting year-ends on 31 December, so declare a final dividend in about March of the following year, with an interim in the autumn. But many other have their year-ends on other quarters, and quite a number of companies pay dividends quarterly, which is common practice in the United States. Then some companies make losses and perhaps pay no dividends at times. A few, as a matter of policy, pay no dividends during a period when they are growing rapidly and have opportunities to use the shareholders money effectively in the business. Yet, others feel they have sufficient capital and return money to shareholders by share buybacks, capital distributions, or special dividends. In principle special dividends are not included in the FTSE index dividend yield, though they are allowed for in the “xd adjustment”, which shows actual cash attributed on the xd date for each stock. The yield index should allow for dividends as they are announced. They may go “ex dividend” some weeks later, and be actually paid some weeks later than that, but these last two steps should not affect the index.

3.2.4. All these practical aspects of the dividend index mean that it may alter in rather a lumpy way, as bigger companies increase or cut their dividends. But in detail it is not connected with the retail or consumer price indices. Consumer prices have their effect slowly and indirectly. If there is high general inflation, companies may be able to increase their own prices, and may make higher profits and pay higher dividends, all with quite a long time lag. This is all reflected in the annual model, and we would not expect any simultaneous correlation.

3.2.5. In Table 6 we show statistics for the forwards deviations, first unadjusted, and then standardised by dividing each value of Df _j by the estimated value of σ _m for the same year calculated from the values of Df _j in that year. We consider first the unadjusted values. These show quite large positive deviations in February, March and May, the period when we might expect companies to announce increases to final dividends, with quite large negative deviations in October, December and January, periods when announcements are scarcer. However, the kurtosis in every month is very large, going up to 36.03 for November, and often the skewness is large and negative, but never large and positive. In general this may reflect the actions of companies, whose directors may well increase dividends cautiously from year to year, but when dividends must be reduced, feel that it is better to do that in a big cut, which may allow gentle increases from a lower level in subsequent years. But very specifically this identifies the months when there were isolated big changes. The biggest negative values of Dfj occur in November 2008, −0.1303; August 2009, −0.0850; and June 2010, −0.0866, all within the most recent crisis period. The large drops in 1931 and 1932, visible in Figure 5, do not provide such large values of Df _j , because they occurred in years when the overall drop was large, and Df _j measures movements relative to the mean for the June to June year, so the drop in March 1931 gives a Df _j value of −0.0556, and that in June 1930, which was a smaller decrease in DL(t), gives a Df _j of −0.0714. When we note that the overall value of σ _m is 0.01337, we see how large these drops are. The data are very fat-tailed and far from normally distributed.

Table 6

Statistics for forwards deviations, Df _j , of DL, from June 1924 to June 2014; unadjusted and standardised.

3.2.6. The effect of the irregular monthly pattern in the forwards deviations follows through to the sideways deviations, which are mostly significantly negative; but again the kurtosis is so large that conclusions based on significance tests based on normality are suspect. We can take account of some of this irregular seasonal pattern by including a correlation between the same months of successive years, as we did for QL and as we shall investigate, but we shall not make any other adjustment for it.

3.2.7. We next consider the variation in the values of σ _m year by year. They are very variable and as before vary with the change in DL over the year, that is, DLD(t)=K(t+1). We show the correlation coefficients of the regressions in Table 7. The quadratic and absolute models centred on DSC show high correlation coefficients with the best being that for σ _m and the absolute value (formula 5):

with DSA=0.005893, DSB=0.075657 and DSC=0.052764. The T-ratios of A and B are very large (6.93 and 8.49, respectively). We show this regression in Figure 8 as formula 5. There is a big bunch of apparently uncorrelated points in the middle, but the extreme values indicate that the regression is useful.

Figure 8

Standard deviations, and estimates of σ _m in each year, related to DLD(t) in each year, from 1924 to 2014.

Table 7

Correlation coefficients from regressions.

3.2.8. We now look at the correlation between the standardised forwards deviations in the same months of successive years. For DL the correlations are weaker than for either QL or WL, with an average correlation coefficient of 0.1782, though this varies very much from month to month. We might have expected a higher value, but the high variation in average forwards deviation from month to month may be caused more by exceptionally large values than by a consistent seasonal variation, in spite of the plausibility of there being correlation because of the annual cycle of company announcements.

3.2.9. There is, however, another correlation well worth taking into account. There is simultaneous correlation between the standardised deviations between those for YL and those for DL, with a coefficient of 0.2580. This is not huge, but still very significant. It is, further, not surprising. Since the dividend yield is calculated by dividing the dividend index by the share price index, a jump in the dividend index automatically causes a corresponding jump in the yield, unless the price also changes correspondingly. We are looking at changes over a month, and a lot of other things may influence prices and hence yields in that time, so the correlation is reasonably a lot less than unity. But it is included in our model. We find no other larger cross-correlations with earlier series. As with correlation coefficients in successive years, provided the values of the correlation coefficients in corresponding months are the same, the Zs and the Z*s have the same correlation coefficient.

3.3. Simulations

3.3.1. We now have a model for the dividend index. We use a BB for DL, with σ _m (t) varying according to the formula in section 3.2.7 and with correlation of the Z _j values with those of YL _j . We show one simulation for 50 years for K in Figure 9 and for D in Figure 10. We make the vertical scale of Figure 9 the same as that of Figure 6. The annual values do not fluctuate as much in the simulations as in the actual data, but the bridging appears to us to be comparable. In so far as one can observe anything from these figures, Figure 10 seems locally similar to Figure 5.

Figure 9

Stochastic yearly and monthly simulation for 50 years of K(t), with initial conditions as at June 2014.

Figure 10

Stochastic yearly and monthly simulation for 50 years of D(t), with initial conditions as at June 2014.

3.4. The share price index

3.4.1. Part of the basic source data for the share indices is the prices index, P(t), from which the dividend index is derived as D(t)=P(t)×Y(t). In Figure 11 we show the actual values of P(t), along with the dividend index, D(t), multiplied by 25 so as to get comparable scales, and in Figure 12 we show the annual changes in lnP(t), which we denote as PLD(t).

Figure 11

Values of the share prices index, P(t), monthly, along with 25×D(t), from December 1922 to December 2014.

Figure 12

Values of annual changes in logarithm of share prices index, PLD(t), from December 1922 to December 2014.

3.4.2. Now that we have models for simulating YL and DL we can construct simulations for the price index, PL, since PL(t)=DL(t)−YL(t) and P(t)=exp(PL(t)). We show in Figure 13 a simulation of P(t) for 50 years, based on the simulated values for Y and D in Figures 4 and 10, on which we include the simulated values of D(t) multiplied by 25. We can compare this with the graph of the actual values of P(t) and 25D(t) in Figure 11. To complete the picture we show in Figure 14 the simulated values of PLD(t), on the same scale as Figure 12, which we can compare with the actual values in Figure 12. The results look plausible, though the extreme values seen in Figure 12 do not appear in the simulated series. This may be because we have used normally distributed random innovations in the simulations, whereas we know that the actual data are distributed in a more fat-tailed way. We hope to return to this in later part of this series.

Figure 13

Stochastic yearly and monthly simulation for 50 years of P(t), along with 25 times simulated D(t), with initial conditions as at June 2014.

Figure 14

Stochastic yearly and monthly simulation for 50 years of PLD(t), with initial conditions as at June 2014.

4.1. The long-term interest rate (“Consols”) model

4.1.1. The model for long-term interest rates, which we still describe as the “Consols” yield although for many years we have not based our data on the yield on the UK government’s $2{\scale 75%{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}\%\,$ Consolidated Stock (which was wholly redeemed in July 2015), is, in principle:

with parameters suggested in Part 1 (Wilkie et al., Reference Wilkie and Şahin2011):

Two different constraints are then introduced, one of which is applied to past data, where we know the values of C(t) but wish to ensure that the values of CR(t) are positive. For this we put

so that we may reduce CM(t) to ensure that CR(t)≥CMIN.

The other is applied to future simulations, in which we simulate CRL(t), ensuring that CR(t) is positive, but then put

so that we may increase CM(t) to ensure that C(t)≥CMIN.

We could use different values of CMIN in these two situations, but in practice put $$CMIN\, {\equals}\, {\rm 0}{\rm .005}\, {\equals}\, {\scale 75%{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}{\rm \%\,}$$ in both cases.

4.1.2. C(t) too can be decomposed into two parts, which we have already denoted CM(t) and CR(t). However, CM(t) is not observable in the market; we have constructed it as a proxy for “the market’s” estimate of future inflation, for which, as noted in Part 2, “implied inflation”, the difference between the yields on conventional and index-linked stocks, might be an alternative for more recent years. Thus, although the annual values of CM(t) are given by the model, we can define CM(t) for intervening months in different ways, and we investigate some of these. We could then simulate bridges for CRL(t), calculate CR(t), add CM(t) and apply the second constraint to ensure that C(t)≥CMIN. An alternative is to simulate bridging for CL(t) directly, thus ensuring that C(t)>0. For both methods an OU bridge might be natural, but, as we show, there is very little difference between an OU bridge and a BB for these, especially since the value of the autoregressive parameter CA is large.

4.2. The “Consols” yield data

4.2.1. We have available data at monthly intervals from 1923 to 2014 for this series, first from the yield on $2{\scale 75%{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}\%\,$ Consols, then the yield on irredeemables from the Financial Times Actuaries Fixed Interest Indices (now the FTSE Actuaries UK gilts Indices), and most recently the 45-year yield from those indices. Apart from the possibility that in the early years, up to about 1930, the “monthly” figures might be “monthly averages” (in practice probably the mid-point of the range of daily closing yields) rather than end month yields, there are no problems with these figures. We plot them in Figure 15.

Figure 15

Values of long-term interest rates, C(t), monthly from December 1922 to December 2014.

4.2.2. Our formulae for CM(t) for annual intervals is

subject to the constraint

An obvious way to calculate CM(t) for intervening months is to use the same formula, using values of I(t) ending in the same month in previous years. This is what we have quoted in Table 8 of Part 2, as being appropriate for annual simulations starting in the given months. But the values, which contain values of I(t) for past years with exponentially decaying weights, are perhaps more erratic than they should be if the purpose is to indicate “the market’s” estimate of future inflation. We denote these monthly values as CM1.

Table 8

Values of C%, CM1, CM2 and CM3, and CR1, CR2 and CR3, from June 2013 to June 2014.

4.2.3. A second way of calculating monthly values would be to slide from CM(t) to CM(t+1), a year later, gradually. CM(t+1)=CD.I(t+1)+(1−CD).CM(t), and we can separate this into two components denoted, CM2A and CM2B. For year t, month m, we define

and

Then we put CM2(t, j)=CM2A(t, j)+CM2B(t, j) and we reach CM2(t, 12)=CM(t+1).

4.2.4. A third way is by simple linear interpolation

4.2.5. In each of these cases we need to consider the constraint. For CM1(t, j) we can apply the constraint as for annual values, carrying forwards the adjusted value of CM1(t, j) to the next year. For CM2(t, j) we can apply the constraint to that single value, letting the two components CM2A and CM2B continue unaltered. For CM3(t, j) we can apply the constraint, and then interpolate linearly between the adjusted value of CM3(t, j) and the final “target” value, CM(t+1), over the remaining months. In some years we may have several such adjustments. In Table 8 we give an example of these different values of CM, over the year from June 2013 to June 2014, along with C% and the corresponding values of CR, denoted CR1, CR2 and CR3. In no case does the constraint apply.

4.2.6. Calculations show that, for this short period, the standard deviation of the values for the 11 intervening months, is, for CM1 0.0067, for CM2 0.0008 and for CM3 0.0001, so (rather obviously) CM3 is the smoothest. The standard deviation for CR1, CR2 and CR3 are 0.00083, 0.00047 and 0.00046, respectively. Since the standard deviation for C alone is 0.00046, the effect of CM1 is to add to the variability of CR1 rather than to reduce it, whereas the other two formulations leave the variability almost unchanged.

4.2.7. In Figure 16 we show the values of C (as a fraction, not a percentage), and CM1, CM2 and CM3. C is shown as a dashed line at the top of the group, CM3 is shown by a solid line; it is mainly short straight lines between June values. CM2 is shown by a dashed line, which is almost indistinguishable from CM3. CM1 is shown by a dotted line, also mostly indistinguishable from CM3. But in December 1998 all three move down to equal an amount CMIN=0.005 below the reduced values of C; the sequence from October 1998 in percentage terms is 5.14, 4.72, 4.55. 4.44. 4.76. 4.72, so C is low again in January 1999, but goes back up a bit in February 1999 and thereafter. CM2 and CM3 are reduced temporarily, but the reduced value of CM1 is carried forward into subsequent years, decaying only slowly, so that the “hiccup” that can be seen in the graph continues for several years. This is not very satisfactory. It is an instructive example of how one can see from a graph features that are not easily observed in a table of numbers.

Figure 16

The Consols yield, C, CM1, CM2 and CM3, from 1922 to 2014.

4.2.8. Rather than work directly with C and CR, we analyse the bridging properties of CL(t)=ln(C(t)) and CRL(t)=ln(CR(t)), both as BBs and as OU bridges. The initial estimates of σ _m for each of what are now eight possibilities are shown in Table 9. These numbers are not strictly comparable as between CL and CRL because of the non-linear relationship between them, but we see that the values of σ _m for CL are much lower than for CRL. Of the three versions of CRL, CRL2 gives rather lower values than the others. There is very little difference between the values of σ _m for the BB and OU bridge.

Table 9

Estimates of σ _m for CL(t), CRL1(t), CRL2(t) and CRL3(t), 1923–2014.

4.2.9. An important influence on the value of σ _m for the CRL bridges is the value of CMIN. In the 1,093 months from June 1923 to June 2014 there were 63 months in which CR1(t)=0.005, 64 months for CR2(t) and 40 months for CR3(t); in the case of CR1(t) these included every month in the year from June 1998 to June 1999. The lowest value of CRL(t) is ln(0.005)=−5.2983. Since our purpose is solely to keep CR(t)>0 we could use a much smaller value but if we change to CMIN=0.001 with ln(0.001)=−6.9077, then we get slightly fewer values at the minimum, but very greatly increased values of σ _m , around 0.22–0.25. This is uncomfortable.

4.2.10. Experiments with simulations, using CMIN=0.005 (but in none of the simulated years did it apply), and using the same innovations, showed very little difference between the simulated values of C(t) with any of the eight methods. The furthest out were those using CM1, both BB and OU. Excluding these two, the largest difference, in only one simulation over only 10 years, was 0.096%, less than 10 basis points, and that was between CRL2 with an OU bridge and CL with a BB. The difference between corresponding OU bridges and BBs was in no case greater then 0.0016, less than 2 basis points. This inclines us towards recommending simulating with the simplest model, with CL and a BB. Simulating bridging for CL keeps the value of C(t) necessarily positive and saves us having to choose between different versions of CM, and different values of CMIN. The strong effect of inflation on interest rates is taken into account in the annual model, and does not need to be considered separately for the bridging model.

4.3. Further analysis

4.3.1. We now consider the analysis of CL(t) more closely. We assume a BB. Statistics for the forwards deviations (not standardised) are shown in Table 10, and for the sideways ones in Table 11. Among the forwards deviations only October shows a significantly low mean (just), and June a significantly high one. But the first few months show negative means, and this results in the means for the sideways deviations being all negative. In both cases the kurtosis is high in many months. There are particularly large forwards deviations in January 2009 and September 1931 and a particularly large negative one in September 1977, but apart from the first of these we cannot identify any particular economic reasons for these.

Table 10

Statistics for forwards deviations, Df _j , of CL, from June 1923 to June 2014.

Table 11

Statistics for sideways deviations, Ds _j , of CL, from June 1923 to June 2014.

4.3.2. The means of the 55 upwards deviations are all negative, as are those of the 32 downwards ones and the 4 medium ones. This does not suggest that an OU bridge is any more appropriate. Indeed when we analyse the data as an OU bridge with α _y =0.92, the statistics are quite similar. But with such a large α _y this is not surprising. As with so many of our series, the residuals are fatter tailed than normal, and tests based on an assumption of normality are unreliable.

4.3.3. We investigate the variation of the monthly estimate of σ _m in each year with the overall change in CL(t) in the year. We put CLD(t)=CL(t+1)−CL(t) and consider the same eight possible regressions as before. The constant CSC is the calculated mean value of CLD(t) which is quite small at −0.0025. The long-term expected value of this is 0, so the first four regressions are also relevant. The correlation coefficients are shown in Table 12. The coefficients for σ _m are larger than those for σ _m ²; those for CSC=0 are a little larger (but very little larger) than those for CSC=−0.0025, and those for the squared function are a little larger than those for the absolute function. We therefore use formula (3) and put

with CSA=0.028392 and CSB=0.165597. We note that the T-ratio of CSB is 2.36, so it is reasonably, but not extremely, significant.

Table 12

Correlation coefficients from regressions.

4.3.4. Correlation of the standardised forwards deviations in corresponding months in successive years is very small, the overall correlation coefficients being 0.0573, and the largest, that for December, being only 0.16. When we look at the forwards deviations as a single series there is some autocorrelation, just significant, with negative partial autocorrelation coefficients at all lags up to about 18 months. But we showed in section 2.2.6 that we expect an autocorrelation within the year of −1/11=−0.09 so this result is only to be expected.

4.3.5. The one cross-correlation of interest is that between the residuals of CL(t) and YL(t) in the same month, which is 0.2558 for the standardised and 0.2610 for the unstandardised residuals. This is not surprising. To some extent the equity and gilt markets can move together in the short term, and this is reflected in our model by changes in the yields. We therefore take this into account, using a value of 0.2558. There is, however, no apparent significant correlation monthly between changes in CL and changes in QL, WL or DL.

4.4. Simulations

4.4.1. We end up therefore with a similar model to those for many of our series. We use a BB for CL(t), with a σ _m varying according to formula (3) with

and with CSA=0.028392 and CSB=0.165597; with a further correlation coefficient with the monthly innovations of YL(t) of 0.2558. One simulation on this basis for 50 years is shown in Figure 17, with the same vertical scale as Figure 15. We can compare this with the actual values of C(t) % shown in Figure 15. The rates do not rise so high as they did in 1974, nor are they as low as they have fallen since June 2014, but these, if anything, are criticisms of our annual model, and we consider that the monthly variation looks reasonable.

Figure 17

Stochastic yearly and monthly simulation for 50 years of C(t)%, with initial conditions as at June 2014.

5.1. The short-term (“Base Rate”) interest rate model

5.1.1. The model for short-term interest rates, for which we use the Bank of England’s Base Rate and its predecessors, is

In 2009, the change in B(t) down to ${\scale 75%{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}\%\,$ was so large, and the corresponding value of BD(t) was so large, that we used, for that year only, an “intervention” term, BInt(t) putting

But this was exceptional and is not part of the normal model. Parameters suggested in Part 1 (Wilkie et al., Reference Wilkie and Şahin2011) are as follows:

5.2. The “Base Rate” data

5.2.1. Base Rate changes on particular days, and remains constant between changes. Although we have available complete daily data, we have also recorded it at the end of each month, and we use these end-of-month values. Up to 1972 the values were always multiples of ${\scale 75%{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}\%\,$ , but in October of that year the rate was $ }7 {\scale 75%{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$4$}}}\%\,$ for a few days, and in December it was $7{\scale 75%{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 4}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$4$}}}\%\,$ for a few days; the first ${\scale 75% {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$4$}}}\%\,$ we have recorded is for January 1973. In recent years changes have occurred or not occurred at monthly intervals, but in earlier years there might have been several changes in one month. For long periods it has remained unchanged; from June 1932 to November 1951 it was at 2%, except for a few weeks in August to October 1939, and from March 2009 to August 2016 it was at ${\scale 75%{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}\%\,$ . It is the sort of series that would lend itself to being analysed as a Poisson process with discrete jumps, but we have in our annual model pretended that it has a continuous distribution, so we keep to that here.

5.2.2. The value clearly goes up and down along with long-term interest rates. In Figure 18 we plot both C(t) and B(t) at monthly intervals from December 1922 to December 2014. Note that B(t) always jumps discretely at one or more days in the month, so the slightly sloping lines joining the dots should be vertical. We express the relation between B(t) and C(t) through BD(t), the difference between their logarithms, or equivalently the logarithm of their ratio, and we plot the values of BD(t) in Figure 19. Up to 2009 BD(t) clearly is mean reverting, wandering up and down around some middle position, but in 2009 it jumped to a different level altogether.

Figure 18

Values of the long-term interest rate, C(t), and bank Base Rate B(t), monthly from December 1922 to December 2014.

Figure 19

Values of BD(t)=ln(C(t)/B(t)), monthly from December 1922 to December 2014.

5.2.3. We have the choice of modelling the monthly bridges either on the logarithm of B(t), which we can call BL(t), or on BD(t), deriving monthly values of B(t) from those of C(t) and BD(t). We use BL(t) because that ensures that the values of B(t) remains positive, but this is perhaps not now a necessary constraint, since some other countries have introduced negative bank rates (Switzerland and Denmark, for example). We could, if we wished, round the simulated future values of B(t) to ${\scale 75%{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$4$}}}\%\,$ units. We could use BBs or OU bridges in either case. However, a problem with analysing BL(t) is that in many years there is no change in its value at all, and the estimated value of σ _m is therefore 0. This occurs in 27 of the 91 tears. When there are changes, they are in discrete jumps, slightly concealed because of taking logarithms. This very much distorts the figures, which are nothing like normally distributed. On the other hand, in these same periods, changes in BD(t) are caused entirely by changes in C(t) so we may seem to be repeating our analysis of that variable.

5.2.4. In Table 13 we show statistics for the forwards deviations for both BL and BD assuming BBs. Because of their construction, deviations for BL and BD have opposite signs, so we expect the means to have sign reversed, and this occurs for 9 out of the 12 months. There are large T-ratios for three months, but the skewness and kurtosis are so large for every month that any significance test would be invalid. The statistics for the forwards deviations for an OU model are very similar, and we do not show them.

Table 13

Statistics for forwards deviations, Df _j , of BL and BD, from June 1923 to June 2014.

5.2.5. The sideways deviations show even larger kurtosis, and we can learn little from them. We determine whether a deviation is “upwards”, “downwards” or “medium” by the relation with the assumed mean if there were an OU bridge. For BL, the 52 years with upwards deviations show positive means in every month; but for downwards deviations so do 8 out of the 11 months, and so does every month for the medium deviations. For BD the numbers are almost the same, but the signs are reversed, negative means prevailing.

5.2.6. There is nothing from the available statistics that allows us to express any firm preference for bridging on BL or BD, or on using a Brownian or an OU bridge. Since the simulated results of using OU bridges are very close to those with BBs, as is usual with our series, we continue to consider only BBs. The simulated values for BL and BD are quite similar (for BD we reverse the signs of the innovations used for BL), but either shows reasonable looking results.

5.2.7. We consider the regressions of σ _m and σ _m ² on the usual functions of the annual changes, defined as BLD(t)=BL(t+1)−BL(t) and BDD(t)=BD(t+1)−BD(t). The values are shown in Table 14. The larger correlation coefficients are certainly worth taking into account. For BL the largest is for σ _m on the absolute difference from the mean BSC=−0.01969, but this is only marginally higher than that when BSC=0; for BD it is for is σ _m ² on the squared difference from the mean, a different BSC=0.01714, with again a very small difference when BSC=0. Since the long-term expected values of both BLD and BDD are zero we prefer these forms.

Table 14

Correlation coefficients from regressions for BL and for BD, 1923–2014.

5.2.8. In Figure 20 we show the actual values of σ _m for BL estimated from the data in each year along with estimates on two regression formulae: (the formula numbers are as in section 8.5 of Part 3A)

where BSA=0.029183 and BSB=0.112781, and

where BSA=0.001616 and BSB=0.021683.

Figure 20

Standard deviations, and two estimates of σ _m in each year, related to BLD(t) in each year, 1923–2014.

In Figure 21 we show corresponding values for BD along with estimates on two regression formulae:

where BSA=0.004317 and BSB=0.010021, and

where BSA=0.001852 and BSB=0.019851.

Figure 21

Standard deviations, and two estimates of σ _m in each year, related to BDD(t) in each year, 1923–2014.

5.2.9. We can see how both are dominated by the extreme value for 2009, on the left for BL and on the right for BD. Nevertheless, there is a sort of V shape in the position of the other points, which must be partly caused by the discrete nature of the data. If the change in B in the year is small, and changes are restricted to ${\scale 75%{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$4$}}}\%\,$ or ${\scale 75%{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}\%\,$ points, they cannot be evenly spaced across the months, so the variance cannot be very small. On the other hand, in the graph for BL there are 27 points piled up on (0, 0).

5.2.10. We look also at possible auto- and cross-correlations. For neither variable is there any significant correlation from year to year in corresponding months. But simultaneous cross-correlations can be observed. For BL the correlation between standardised residuals, the values of DfZ _j , is moderate for both YL, at 0.1796, and CL at 0.1476, and negligible elsewhere. For BD the correlation with YL is small, but that with CL is large at 0.5286. This is not surprising, since CL is one of the constituents of BD, and it would be appropriate to take it into account.

5.3. Simulations

5.3.1. We now have two possible models, one based on bridging for BL, the other on bridging for BD. BBs are appropriate for both; there is no strong evidence in favour of an OU bridge in either case. Both show significant variation in σ _m from year to year depending on the change in BL or BD over the year according to the formulae in section 5.2.8; we prefer formulae (1) and (4), respectively. Neither shows correlation over the residuals from year to year, but BD shows a strong simultaneous correlation with the residuals of CL.

5.3.2. We show one simulation for 10 years, with values of B(t) simulated with bridging both BL and BD and then rounded to the nearest ${\scale 75%{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$4$}}}\%\,$ . We use the same innovations for both sets of simulations, but with the sign reversed, since BD and BL are negatively related. We also show values of C(t) for the corresponding simulation.

5.3.3. Visually, the results seem quite similar, with many months showing no changes in B(t), and many months showing the same values for the two sets of simulations. However, simulations using BD seem to “jiggle” up and down a little more than those using BL. We can compare the results numerically. In the 10 years there are 110 intermediate months (excluding the annual values which are necessarily equal), and in 14 of these the values of B(t) using BL and BD show different values. There are 120 months when a change could occur, and there are changes in 29 of these using BL, and 43 using BD. The sum of the absolute value of the monthly changes is 7.5% using BL and 11.5% using BD. In both cases most changes are of ${\scale 75%{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$4$}}}\%\,$ , but using BL there is one change of ${\scale 75%{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}\%\,$ and using BD there are three. Figures for simulations over 50 years, which we do not show fully, are comparable. Neither bridging method is incompatible with the real data, but neither shows the long periods of stability that can be seen in Figure 22. On balance we prefer to use bridging over BL.

Figure 22

Stochastic yearly and monthly simulation for 10 years of B(t)%, bridging both with BL and with BD, and showing also values of C(t)%, with initial conditions as at June 2014.

5.3.4. In Figure 23 we show the simulations continued for a total of 50 years using bridging over BL. The overall pattern is similar. We note that nowhere in this simulation do the values rise to anything like the levels actually seen in the 1970s and 1980s, nor does C(t) drop as far as it actually has since June 2014. But for all our simulations we are using QMU=0.043, the value found from the data in 2009, rather than the lower value of say 0.025 that we suggested then might be more appropriate. Further, in both the annual and the monthly models we use normally distributed innovations, rather than the fatter-tailed ones that would correspond better with the experiences. But that is for another part of our investigations.

Figure 23

Stochastic yearly and monthly simulation for 50 years of B(t)%, bridging with BL, and showing also values of C(t)%, with initial conditions as at June 2014.

6.1. The model for “real” yields on index-linked stocks

6.1.1. The model for “real” yields that we suggested in 2009, altered from that suggested in Wilkie (1995), is

with parameters suggested in Part 1 (Wilkie et al., Reference Wilkie and Şahin2011):

6.1.2. This is a simple AR(1) model for R(t), with the only dependence on other variables being a simultaneous correlation with the innovations for C(t). An OU bridge would be the obvious choice, but the suggested value of RA of 0.95 is so high that an OU bridge and a BB would be almost indistinguishable.

6.2. The real yield data

6.2.1. We have used throughout the yields from the FTSE-Actuaries index-linked indices for stocks over 5 years, with an assumption of 5% inflation, which are available from May 1981 when the first index-linked stocks in the United Kingdom were issued. In Figure 24 we plot the values of R(t) at monthly intervals from May 1981 to December 2014. The values rose and fell again in the first years, as if centred somewhere around 3%, but from 2001 there has been a steady decline, with the rates going negative in 2011 and staying below zero most months since then. But it is economically most implausible that they will continue downwards indefinitely. There are specific institutional arrangements in the United Kingdom at present which almost force pension funds and insurance companies to bid for the limited supply of government index-linked stock, but these may ease sometime. In any case, our current problem is not with the overall annual model, which does suggest reversal to a mean of 3%, but with stochastic bridging within that model.

Figure 24

Values of real yield, R(t)%, monthly from May 1981 to December 2014.

6.2.2. We have only 33 full years to analyse, from June 1981 to June 2014. We calculate forwards and sideways deviations in the usual way, assuming both BBs, and OU bridges. The numbers are very similar. We show statistics for forwards and sideways deviations for BBs in Table 15. We multiply the means by 100 to improve the presentation. The values of the kurtosis statistic for most months are very high, though less high than for some of our series. This makes the T-ratios unreliable as a test of whether the means are equal to zero. The mean is apparently well below zero for August, just significantly above zero for April, but not unusual for any other months. The sideways deviations consequently show means well below zero for December through to April.

Table 15

Statistics for forwards and sideways deviations, Df _j and Ds _j , of R(t), from June 1981 to June 2014.

6.2.3. The overall estimate of σ _m is 0.001569, which corresponds with an annual σ _y of 0.0054. This is much larger than the value of the estimated annual standard deviation RSD, of 0.003.

6.2.4. There are 16 years with upwards deviations, and 9 out of 11 months show negative means for these. There are 15 years with downwards deviations, and 10 out of 11 months show negative means. There are only two with medium deviations. This does not give strong evidence in favour of OU bridges.

6.2.5. We calculate the usual eight regressions of the values of σ _m and σ _m ² on the annual changes of R(t), which we denote RD(t)=R(t+1)−R(t). We show the correlation coefficients for BBs in Table 16. The overall observed mean value of RD(t) is −0.00848, whereas the long-term expected value assuming that an annual AR(1) is correct would be 0. All the correlation coefficients are negative; none is significant at all. The largest value of the T-ratio of RSB is only 0.7, so we assume a constant value of σ _m of 0.001569.

Table 16

Correlation coefficients from regressions for R with Brownian bridges, 1981−2014.

6.2.6. We look also at correlations. The correlation coefficients of the standardised forwards deviations with those of the corresponding month in the previous year are all quite small, so we can ignore these. The cross-correlations with other series, however, are not negligible. The standardised residuals show correlation coefficients of 0.2398 with YL, 0.6367 with CL and a much smaller 0.1116 with BL. Over the same period of 33 years the cross-correlation coefficient of the standardised residuals of YL and CL is 0.1607 and over the whole period is 0.26. The extra information by taking into account the correlation between R and YL is small and we can ignore it, leaving us the large coefficient of 0.6367 between R and CL.

6.3 Simulations

6.3.1. We now have quite a simple model for bridging R, a BB, with a constant value of σ _m and a simultaneous cross-correlation with the innovations of CL. We show in Figure 25 one simulation for 50 years. Because we retain our assumed value of RMU of 3% the annual values creep up, rather slowly, towards that level again. Whether this is reasonable or not is not discussed here. Here, the important point is that, given the annual values, the monthly values seem to behave reasonably similarly to the actual values shown in Figure 24.

Figure 25

Stochastic yearly and monthly simulation for 50 years of R(t)%, with initial conditions as at June 2014.