A STOCHASTIC MODEL FOR RETURN ON INVESTED ASSETS
...l for the movement of prices of ordinary shares based on dividends and dividend yields in 1980 (Ford et al, 1980). Such models are also described as “investment models”, but perhaps “economic models” would be a better term, since they often include models for inflation and wages. These models have many similar features, but of course differ in a number of ways. 2.3 The models that we consider are: (1) the Wilkie model, as described in Wilkie (1995); (2) the ARCH variation of the Wilkie model, also described in Wilkie (1995); (3) the random walk model with a Wilkie-style inflation model and normally distributed innovations, described in Smith (1996) and there attributed by him to Kemp; (4) as (3), but with -stable distributions for innovations, also as described by Smith (1996); (5) Smith’s jump diffusion model, as described in Smith (1996), and further explained by Huber (1998); (6) the TY model, as described in Yakoubov, Teeger & Duval(l999); (7) the Cairns model, as described in Cairns (1999a and 1999b); (8) the Whitten & Thomas model, as described in Whitten & Thomas (2000). 2.4 There are several features common to all the actuarial models. The first conspicuous one is that they are all designed to be used in Monte Carlo simulation exercises, and so in general can not be treated analytically, except in limited circumstances. 2.5 The next common feature is that they are mostly defined in terms of annual steps, rather than more frequently. Such models are therefore not suitable for, and were not designed for, the pricing of derivatives. Exceptions are the Smith and Cairns models, which are defined as continuous time models, although they can be simulated with any desired frequency. Some of the actuarial models are consistent with a continuous time model, or at least partially so, or with simulation over any shorter intervals, such as monthly. 2.6 Each model, except the Smith and Cairns models is defined in terms of variables which take annual values, and are connected from one time step to the next by a series of formulae, introducing for each basic variable at each time step some random innovation or driver. The Cairns model is defined in terms of stochastic differential equations, with driving Brownian motions instead of discrete innovations. The Smith model is defined in terms of continuous time equations, with driving compound Poisson diffusion processes. However, for both of these models, when they are simulated by Monte Carlo methods it is necessary to introduce random innovations for them too. 2.7 The first important feature that distinguishes the models when one wishes to use them is the set of output variables that they produce. Many of the variables come in necessary pairs, such as a price index and an annual rate of change of that index. Using the notation of the Wilkie model, we have: The retail price index at time is denoted by . The force (continuously compounded rate) of inflation over the year to is denoted by , which is calculated, when analysing past data, as: , and for future simulation we have: . We treat and , the retail prices index and the annual force of change in that index as one variable, rather than two. We could alternatively treat the force of inflation, as a basic variable, since it is simulated directly, and the prices index, , which is derived from , as a derived variable. However there are more complicated cases where we introduce derived variables, and it gives less clutter if we treat such pairs as one. 2.8 All the models include price inflation, bonds, or interest rates on bonds, and shares, at least to some extent. The RWV (random walk variant) and Smith models, however, only give total returns on shares, and not dividend yields. They also omit wages, though it would not be difficult to add them using Cairns’s or Wilkie’s approach to these. All, except Whitten & Thomas, include index-linked bonds. The Smith and Cairns models provide full yield curves, both for conventional and for index-linked bonds while the other models don’t. Only the RWV, Wilkie and Smith models cover property (income and values in the case of the Wilkie model and total return in the case of RWV and Smith). The TY model includes earnings on shares as well as dividends, and also covers overseas assets in sterling terms. The Wilkie model is the only one to cover exchange rates, though these could easily be incorporated into the other models in a similar way. 2.9 The Smith model is symmetric as between all asset classes, and may in theory be used to model “dividend yield” curves, analogously to those for fixed income and inflation-linked bonds, but the details of how to do this are not given by Smith (1996) or Huber (1998). 2.10 The Whitten & Thomas model is very similar in overall structure to the Wilkie model, and could easily be extended to cover index-linked, property and exchange rates by using the Wilkie model versions for these. 2.11 The Wilkie model explicitly models other countries on the same lines as the UK, and provides an exchange rate for each country modelled. In this respect it is wider than any of the other published models, though as mentioned above they too could presumably be applied to other countries, along with a model for exchange rates. 2.12 Within each model each variable has a certain range of values that it may attain. A typical range for an index type of variable is 0 to , and for the annual force of change of that variable is to . It is most unlikely that such a variable will even approach either of these extremes, but no value within the ranges is inherently impossible, so a simulated variable that allows such a range is acceptable. We make no comment on such variables. 2.13 The situation is different for interest rates. It is in principle unacceptable for the value of a variable that represents nominal interest rates to be negative. A model that allows such a possibility may need modification to exclude it. The Wilkie model would allow the nominal “consols” yield, to become negative if inflation were sufficiently negative and the “real yield” part were not large enough to counter this. In practical application of the Wilkie model one inserts a minimum value for , denoted , as one of the parameters, with a typical value of 0.5%. This value is used in place of the simulated value in any year that it applies. Such a barrier may either be applied without affecting future simulated values, or it may affect the carried forward value of the relevant variable as used in the simulation for next year. If such minima need to be applied it is not always clear what authors of models would recommend. 2.14 In the case of real interest rates, such as might apply to index-linked bonds, it is less obvious that negative yields are impossible in practice. An index-linked zero- coupon bond might well sell “above par”, and hence at a negative real yield, if circumstances were thought to favour this. However, it is implausible that an index- linked bond with a negative coupon would be issued. The practical difficulty for the borrower in collecting the negative interest payments would not worry a simulation exercise, but the market value of such a bond could become negative if real yields rose to be sufficiently positive at some later point of the simulation, and this could be undesirable. It might therefore be desirable to restrict investment in index-linked bonds either to zero-coupons at all times or to zero-coupons bonds if the simulated real yields were negative. This does not involve a discontinuity in procedure: as real yields move down to zero the coupon approaches zero, and at zero a coupon bond and a zero-coupon bond are the same; but it does imply an asymmetry of treatment when real bond yields are on either side of zero. 2.15 Each model is driven by a series of innovations. These are always treated as independent from year to year and from simulation to simulation. If they were to be treated as not independent then the dependence should enter the “skeleton” of the model, that is, the deterministic part before the innovations are added. In many of the models the innovations are modelled as a series of independent identically distributed normal variates. However, Wilkie’s ARCH model allows for a varying standard deviation in the inflation model, and Whitten & Thomas allow for different standard deviations in the two states of their model. Their innovations are still normally distributed. However, Smith’s innovations in effect consist of the differences between two sets of independent gamma-distributed variables and the RWV -stable model uses independent -stable variates (also known as Levy-stable or stable Paretian variates). 2.16 Every stochastic economic model needs some initial conditions, i.e. values of the state space at time . However, in some cases the initial conditions may be arbitrary and may have no effect on the effective development of the model. In some models there may be no unique neutral initial condition. Consider a model where the system may be in one of two or more states, with a system of transition probabilities determining how moves between states happen. Each simulation must start in some specific state, but no state is neutral in our sense. Instead it may be best to allow the initial state to be selected at random for each simulation, with specified probabilities. If the chosen probabilities are those that are the long-term probabilities of the states themselves, then one can speak of a neutral probability distribution for the initial state. But one might well wish to simulate conditional on knowing what the initial state is, so one may wish to use some non-neutral probability structure. 2.17 One can take this idea further, and start each simulation in a random state such that the long-run probability distribution of the state variables is replicated. It may be complicated to calculate the joint probability distribution of all the state variables analytically and then simulate from this joint distribution, and an alternative way of reproducing this idea might be to run each simulation for say 100 years from some fixed starting state and then start with the simulated position at time , recording only the changes from that point onwards. However, this method relies on the model being ergodic. Most of the models are ergodic, though Smith’s jump diffusion model appears not to be in the long run, as noted by Huber. 2.18 In Whitten & Thomas’s model it is not obvious what either the median or mean values of the variables are. To calculate them analytically seems complicated, and the distribution is not an obvious one. There are two sets of values that could be thought of as neutral, one for each state, in the sense that, if started with these values, and with zero innovations, the values of the variables would remain the same. We have started the simulations using these neutral values for the lower state. 2.19 To derive initial conditions suitable for simulation from a set of conditions observable in the market may not be trivially easy. Neither Smith nor Cairns explain in their published papers how this can be done for their models, and we have not considered it in detail. It is a matter of finding a set of state variables within the simulation model which reproduce a given yield curve in some way. It may not be possible to do this exactly, and there is then a choice as to which error function one wishes to adopt. It would be made easier if an unambiguous zero coupon yield curve for the market were immediately available, but the construction of such a yield curve remains a matter of choice at present. 2.20 Where the models include foreign (overseas) investments, three different approaches have been adopted. Wilkie models investments in each currency separately, though with allowance for simultaneous correlation in the contemporaneous ...