The Investment Advisor Body of Knowledge + Test Bank. IMCA

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      FINITE SERIES

      In many financial scenarios – including perpetuities and discount models for stocks and real estate – it is often convenient to treat an extremely long series of payments as if it were infinite. In other circumstances we are faced with very long but clearly finite series. In these circumstances the infinite series solution might give us a good approximation, but ultimately we will want a more precise answer.

      The basic technique for summing a long but finite geometric series is the same as for an infinite geometric series. The only difference is that the terminal terms no longer converge to zero.

(3.14)

We can see that for |δ| less than 1, as n approaches infinity δⁿ goes to zero and Equation 3.14 converges to Equation 3.12.

      In finance, we will mostly be interested in situations where |δ| is less than one, but Equation 3.14, unlike Equation 3.12, is still valid for values of |δ| greater than one (check this for yourself). We did not need to rely on the final term converging to zero this time. If δ is greater than one, and we substitute infinity for n, we get:

      (3.15)

      For the last step, we rely on the fact that (1 – δ) is negative for δ greater than one. As promised in the preceding subsection, for δ greater than one, the sum of the infinite geometric series is indeed infinite.

      We start by evaluating the summation, using a discount factor of δ =

≈ 0.96:

      Part II Probabilities

      The concept of probability is central to risk management. Many concepts associated with probability are deceptively simple. The basics are easy, but there are many potential pitfalls.

      In this chapter, we will be working with both discrete and continuous random variables. Discrete random variables can take on only a countable number of values – for example, a coin, which can only be heads or tails, or a bond, which can only have one of several letter ratings (AAA, AA, A, BBB, etc.). Assume we have a discrete random variable X, which can take various values, x_i. Further assume that the probability of any given x_i occurring is p_i. We write:

      (3.16)

      where P[ ·] is our probability operator.

      An important property of a random variable is that the sum of all the probabilities must equal one. In other words, the probability of any event occurring must equal one. Something has to happen. Using our current notation, we have:

      (3.17)

      In contrast to a discrete random variable, a continuous random variable can take on any value within a given range. A good example of a continuous random variable is the return of a stock index. If the level of the index can be any real number between zero and infinity, then the return of the index can be any real number greater than –1.

      Even if the range that the continuous variable occupies is finite, the number of values that it can take is infinite. For this reason, for a continuous variable, the probability of any specific value occurring is zero.

      Even though we cannot talk about the probability of a specific value occurring, we can talk about the probability of a variable being within a certain range. Take, for example, the return on a stock market index over the next year. We can talk about the probability of the index return being between 6 percent and 7 percent, but talking about the probability of the return being exactly 6.001 percent or exactly 6.002 percent СКАЧАТЬ