Capital Asset Pricing Model: CAPM

The study of chaotic dynamics was made popular by the computer experiments of Robert May and Mitchell Feigenbaum on a mapping known as the logistic map. The remarkable feature of the logistic map is the contrast between the simplicity of its form (it is a polynomial mapping of degree two) and the complexity of its dynamics.

The logistic map is the simplest model in population theory that incor- porates the effects of both birth and death rates. We imagine an experiment where a census is taken each year of the population of rabbits in an isolated region in South Australia. The basic question facing population dynamicists is to find a model that accurately predicts the future rabbit population, given that the present rabbit population is known. This seemingly innocent question is much too difficult to answer with present knowledge. A more modest approach is to construct a model of the rabbit population that gives some insights into what may actually happen. In other words, we try to create models that give insight into how population dynamics works. It turns out that the logistic map is the simplest model we can construct that incorporates a realistic mechanism for birth and death. In spite of the rudimentary character of the model, the logistic map displays an astonishing range of complex dynamical behavior. Indeed, the simplicity of this model provides compelling evidence that the real dynamics of rabbit populations, whatever they are, are likely to be at least as complex as those of the logistic map.

At this stage, it is helpful to introduce some economical notation to describe the future population of rabbits. To this end, we shall let pndenote the population of rabbits after n years and, in particular, let p0 denote the present rabbit population.

We begin by deciding what factors might influence pn. We shall start by making the reasonable assumption that the population in any given year depends only on the previous year’s population: in symbols this means that there is a rule

gsuch that

pn+1= g(pn).

Otherwise said, we have a formula ‘g’ that gives the population of rabbits after

n+ 1 years in terms of the population after n years. Next we discuss what form

the rule g should have. A natural assumption to make is that the rule g depends only on the birth and death rates of the rabbit population. Moreover, we shall assume that the birth and death rates do not depend on the year (though we shall allow the possibility that they depend on the size of the population). Finally, in order to determine g, we must specify how the birth and death rates actually enter the formula for g.

If there are no deaths and the birth rate is constant and equal to b, then the simplest dependence of the rabbit population on the birth rate would be

pn+1 = pn+ bpn.

That is, this formula states that the rabbit population after n+ 1 years is the sum of the rabbit population after n years (that is, pn) with the number of new rabbits

born in year n (that is, bpn). For example, if b = 1 (a rather low birth rate for

rabbits), we would have

pn+1 = pn+ pn= 2pn,

and so the model for the population of rabbits would obey the doubling rule of Chapter 1. In any case, whenever the birth rate is greater than zero, it is easy to see that in this model the population grows without bound, leading to the absurd conclusion that after a few years the rabbit population would cover the land of South Australia to a depth of several hundred feet.

In short, any reasonable model must incorporate a mechanism for death, presuming that the population of rabbits is to remain bounded. One way of incorporating death into our model is to assume that a constant proportion of the population dies each year and that new baby rabbits are born only to that portion of the population that survives. For example, if the population at the start of the year were N rabbits, we might assume that over the year only sN rabbits reproduce and the rest die (think of s as the survival rate). Consequently, at the end of the year there would be (1+ b)sN rabbits leading to the model

pn+1 = (1 + b)spn.

Unfortunately, this model has the same difficulty as the previous one. If the

bound. There is a new feature that appears in this model. If the survival rate is so small that the growth rate is less than unity, then the model predicts that eventually all of the rabbits will die (this should be compared with the halving map of Chapter 1). Of course, in the unlikely event that the growth rate is exactly 1, the population will remain constant for all time. Since rabbits have a way of surviving, even though their numbers do not grow without bound, our model is unrealistic and cannot be correct.

From a mathematical point of view, the difficulty with both of the previous models is that they are linear: next year’s population is just a constant multiple of this year’s population (though the exact interpretation of the constant differs in the two models). Linear models lead naturally either to exponential growth or to exponential extinction, neither of which is a satisfactory conclusion. It follows that a model must be nonlinear if it is to yield realistic predictions of bounded and positive population size.

From a modeling point of view, one way to address the difficulty of unbounded population growth is to assume that there is a maximum number of rabbits P that can be supported on the land, where the constant P depends on such features as the availability of food and the existence of predators. In such a model, it is reasonable that the survival rate is proportional to the proximity of the population to P . In other words, we might suppose that if the population at the beginning of the year was N , then the survival rate would be s(P − N) and

so s(P − N)N rabbits would survive to the end of the year and reproduce. We

assume here that s is a constant which does not depend on the population size (or year). This reasoning leads to the new model

pn+1= (1 + b)[s(P − pn)]pn.

Finally, we can simplify this formula by setting xn= pn/P, the propor-

tion of the maximum population present at year n, and λ = (1 + b)sP. The number xnis called the relative population; it is the ratio of the actual population

to the maximum possible population and is therefore a number between 0 and 1. In terms of xn, this model simplifies to

xn+1 = g(xn)≡ λxn(1− xn),

where g is called the logistic mapping and λ is called the effective growth rate. There is a natural limit on the size of λ for this model. Since we obtain each year’s relative population by applying the mapping g to the previous year’s relative population, we must assume the result of applying g to a number between 0 and 1 is also a number between 0 and 1. Thus, in order for this formula to make sense

as a model of population growth, the function g can never achieve a value greater than unity when 0 < xn ≤ 1. Since the maximum value of g is g(1₂) = 1₄λ,

we must assume that 0≤ λ ≤ 4. Also note that, as promised, the mapping g is nonlinear—there is a quadratic term x2

nin its definition. Moreover, the mapping

ghas the simplest form possible for a nonlinear mapping.