## Pricing

Let us go back to our very first example of an investment situation, the first offer from your uncle, but now let us turn it around. Imagine that there is an investment opportunity that will pay exactly $110 at the end of one year We ask: How much is this investment worth today? In other words, what is the appropriate price of this investment, given the overall financial environment?

If the current interest rate for one-year investments is 10%, then this investment should have a price of exactly $100. In that case, the $J 10 paid at the end of the year would correspond to a rate of return of 10% If the current interest rate for one-year investments is less than 10%, then the price of this investment would be somewhat greater than $100 In general, if the interest rate is r (expressed as a decimal, such as /■■ = 10), then the price of an investment that pays X after one year should be X/( \ -f r).

We determined the price by a simple application of the comparison principle. This investment can be directly compared with one of investing money in a one-year certificate of deposit (or one-year Treasury bill), and hence it must bear the same effective interest rate.

This interest rate example is a simple example of the general pricing problem: Given an investment with known payoff characteristics (which may be random), what is the reasonable price; or, equivalently, what price is consistent with the other securities that are available? We shall encounter this problem in many contexts. For example, early in our study we shall determine the appropriate price of a bond, Later we shall compute the appropriate price of a share of stock with random return characteristics Still later we shall compute suitable prices of more complicated securities, such as futures and options Indeed, the pricing problem is one of" the basic problems of modern investment science and has obvious practical applications.

As in the simple interest rate example, the pricing problem is usually solved by use of the comparison principle In most instances, however, the application of that principle is not as simple and obvious as in this example. Clever arguments have been devised to show how a complex investment can be separated into parts, each of which can be compared with other investments whose prices are known. Nevertheless, whether by a simple or a complex argument, comparison is the basis for the solution of many pricing problems.

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