The Investment Advisor Body of Knowledge + Test Bank. IMCA

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      If we hold R_Annual constant as n increases, R_Nominal gets smaller, but at a decreasing rate. Though the proof is omitted here, using L'Hôpital's rule, we can prove that, at the limit, as n approaches infinity, R_Nominal converges to the log rate. As n approaches infinity, it is as if the instrument is making infinitesimal payments on a continuous basis. Because of this, when used to define interest rates the log rate is often referred to as the continuously compounded rate, or simply the continuous rate. We can also compare two financial products with different payment periods by comparing their continuous rates.

      Sample Problem

      Question:

      You are presented with two bonds. The first has a nominal rate of 20 percent paid on a semiannual basis. The second has a nominal rate of 19 percent paid on a monthly basis. Calculate the equivalent continuously compounded rate for each bond. Assuming both bonds have the same credit quality and are the same in all other respects, which is the better investment?

      Answer:

      First we compute the annual yield for both bonds:

      Next we convert these annualized returns into continuously compounded returns:

      All other things being equal, the first bond is a better investment. We could base this on a comparison of either the annual or the continuously compounded rates.

      Discount Factors

      Most people have a preference for present income over future income. They would rather have a dollar today than a dollar one year from now. This is why banks charge interest on loans, and why investors expect positive returns on their investments. Even in the absence of inflation, a rational person should prefer a dollar today to a dollar tomorrow. Looked at another way, we should require more than one dollar in the future to replace one dollar today.

      In finance we often talk of discounting cash flows or future values. If we are discounting at a fixed rate, R, then the present value and future value are related as follows:

      (3.7)

      where V_t is the value of the asset at time t and V_t+n is the value of the asset at time t + n. Because R is positive, V_t will necessarily be less than V_t+n. All else being equal, a higher discount rate will lead to a lower present value. Similarly, if the cash flow is further in the future – that is, n is greater – then the present value will also be lower.

      Rather than work with the discount rate, R, it is sometimes easier to work with a discount factor. In order to obtain the present value, we simply multiply the future value by the discount factor:

      (3.8)

      Because δ is less than one, V_t will necessarily be less than V_t+n. Different authors refer to δ or δⁿ as the discount factor. The concept is the same, and which convention to use should be clear from the context.

      Geometric Series

      In the following two subsections we introduce geometric series. We start with series of infinite length. It may seem counterintuitive, but it is often easier to work with series of infinite length. With results in hand, we then move on to series of finite length in the second subsection.

      INFINITE SERIES

      The ancient Greek philosopher Zeno, in one of his famous paradoxes, tried to prove that motion was an illusion. He reasoned that, in order to get anywhere, you first had to travel half the distance to your ultimate destination. Once you made it to the halfway point, though, you would still have to travel half the remaining distance. No matter how many of these half journeys you completed, there would always be another half journey left. You could never possibly reach your destination.

      While Zeno's reasoning turned out to be wrong, he was wrong in a very profound way. The infinitely decreasing distances that Zeno struggled with foreshadowed calculus, with its concept of change on an infinitesimal scale. Also, an infinite series of a variety of types turn up in any number of fields. In finance, we are often faced with series that can be treated as infinite. Even when the series is long, but clearly finite, the same basic tools that we develop to handle infinite series can be deployed.

      In the case of the original paradox, we are basically trying to calculate the following summation:

      (3.9)

      What is S equal to? If we tried the brute force approach, adding up all the terms, we would literally be working on the problem forever. Luckily, there is an easier way. The trick is to notice that multiplying both sides of the equation by

has the exact same effect as subtracting from both sides:

      The right-hand sides of the final line of both equations are the same, so the left-hand sides of both equations must be equal. Taking the left-hand sides of both equations, and solving:

      (3.10)

      The fact that the infinite series adds up to one tells us that Zeno was wrong. If we keep covering half the distance, but do it an infinite number of times, eventually we will cover the entire distance. The sum of all the half trips equals one full trip.

      To generalize Zeno's paradox, assume we have the following series:

(3.11)

      In Zeno's case, δ was

. Because the members of the series are all powers of the same constant, we refer to these types of series as geometric series. As long as |δ| is less than one, the sum will be finite and we can employ the same basic strategy as before, this time multiplying both sides by δ.

(3.12)

      Substituting

for δ, we see that the general equation agrees with our previously obtained result for Zeno's paradox.

Before deriving Equation 3.12, we stipulated that |δ| had to be less than one. The reason that |δ| has to be less than one may not be obvious. If δ is equal to one, we are simply adding together an infinite number of ones, and the sum is infinite. In this case, even though it requires us to divide by zero, Equation 3.12 will produce the correct answer.

      If δ is greater than one, the sum is also infinite, but Equation 3.12 will give you the wrong answer. The reason is subtle. If δ is less than one, then δ^∞ СКАЧАТЬ