## Normal Price Distribution

it is instructive to solve explicitly for a few of the prices from (11.2). By direct substitution we have

= a2S(0) + mi(0) + ¡'(I) By simple induction it can be seen that for general k,

S(k) = akS(0) + ak-lu(0)+ak-2u(\)+ + u(k - 1) (11.3)

Hence S(k) is akS(0) plus the sum of k random variables,

Frequently we assume that the random variables u(k), k = 0, 1, 2, ... , N — 1, are independent normal random variables with a common variance a2. Then, since a linear combination of normal random variables is also normal (see Appendix A), it follows from (11 3) that S(k) is itself a normal random variable.

If the expected values of all the u(kY& are zero, then the expected value of S(k) is

When a > 1, this model has the property that the expected value of the price increases geometrically (that is, according to ak). Indeed, the constant a is the growth rate factor of the model.

The additive model is structurally simple and easy to work with The expected value of price grows geometrically, and all prices are normal random variables. However, the model is seriously flawed because it lacks realism. Normal random variables can take on negative values, which means that the prices in this model might be negative as well; but real stock prices are never negative. Furthermore, if a stock were to begin at a price of, say, $1 with a a of, say, $ 50 and then drift upward to a price of $100, it seems very unlikely that the a would remain at $.50 It is more likely that the standard deviation would be proportional to the price. For these reasons the additive model is not a good general model of asset dynamics. The model is useful for localized analyses, over short periods of time (perhaps up to a few months for common stocks), and it is a useful building block for other models, but it cannot be used alone as an ongoing model representing long- or intermediate-term fluctuations. For this reason we must consider a better alternative, which is the multiplicative model (However, our understanding of the additive model will be important for that more advanced model.)

11.3 THE MULTIPLICATIVE MODEL

The multiplicative model has the form

for k — 0, I,. . , N - I. Here again the quantities u(k), k — 0, 1, 2, , N - I, are mutually independent random variables The variable u{k) defines the relative change in price between times k and k 4-1. This relative change is S{k -f \ )/S(k), which is independent of the overall magnitude of S(k). It is also independent of the units of price, For example, if we change units from U.S. dollars to German marks, the relative price change is still u(k)

The multiplicative model takes a familiar form if we take the natural logarithm of both sides of the equation This yields

for k = 0, 1,2,. ..., N — 1, Hence in this form the model is of the additive type with respect to the logarithm of the price, rather than the price itself. Therefore we can use our knowledge of the additive model to analyze the multiplicative model.

It is now natural to specify the random disturbances directly in terms of the ln«(A)'s. In particular we let w{k) = In u{k)

for k = 0, 1,2, .. , N - I, and we specify that these w(ky$ be normal random variables. We assume that they are mutually independent and that each has expected value w(k) ~ u and variance a2.

We can express the original multiplicative disturbances as u(k)^ewik) (11.6)

for k — 0, 1, 2, . , N — 1. Each of the variables u(k) is said to be a lognormal random variable since its logarithm is in fact a normal random variable

Notice that now there is no problem with negative values. Although the normal variable w(k) may be negative, the corresponding u(k) given by (11.6) is always positive. Since the random factor by which a price is multiplied is u(k), it follows that prices remain positive in this model.

## Retirement Planning For The Golden Years

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