The Investment Advisor Body of Knowledge + Test Bank. IMCA

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СКАЧАТЬ substituting the equation into itself, we can see how the equation evolves over multiple periods:

      (3.102)

      At time t, X is simply the sum of its initial value, x₀, plus a series of random steps. Using this formula, it is easy to calculate the conditional mean and variance of x_t:

      (3.103)

      If the variance increases proportionally with t, then the standard deviation increases with the square root of t. This is our familiar square root rule for independent and identically distributed (i.i.d.) variables. For a random walk, our best guess for the future value of the variable is simply the current value, but the probability of finding it near the current value becomes increasingly small.

      Though the proof is omitted here, it is not difficult to show that, for a random walk, skewness is proportional to t^−0.5 and kurtosis is proportional to t⁻¹. In other words, while the mean, variance, and standard deviation increase over longer time spans, skewness and kurtosis become smaller.

      The simple random walk is not a great model for equities, where we expect prices to increase over time, or for interest rates, which cannot be negative. With some rather trivial modification, though, we can accommodate both of these requirements.

      Variance and Autocorrelation

      Autocorrelation has a very important impact on variance as we look at longer and longer time periods. For our random walk, as we look at longer and longer periods, the variance grows in proportion to the length of time.

      Assume returns follow a random walk:

      (3.104)

      where ϵ_t is an i.i.d. disturbance term. Now define y_n,t as an n period return; that is:

      (3.105)

      As stated before, the variance of y_n,t is proportional to n:

      (3.106)

      and the standard deviation of y_n,t is proportional to the square root of n. In other words, if the daily standard deviation of an equity index is 1 percent and the returns of the index follow a random walk, then the standard deviation of 25-day returns will be 5 percent, and the standard deviation of 100-day returns will be 10 percent.

      When we introduce autocorrelation, this square root rule no longer holds. If instead of a random walk we start with an AR(1) series:

      (3.107)

      Now define a two-period return:

      (3.108)

      With just two periods, the introduction of autocorrelation has already made the description of our multiperiod return noticeably more complicated. The variance of this series is now:

      (3.109)

      If λ is zero, then our time series is equivalent to a random walk and our new variance formula gives the correct answer: that the variance is still proportional to the length of our multiperiod return. If λ is greater than zero, and serial correlation is positive, then the two-period variance will be more than twice as great as the single-period variance. If λ is less than zero, and the serial correlation is negative, then the two-period variance will be less than twice the single-period variance. This makes sense. For series with negative serial correlation, a large positive return will tend to be followed by a negative return, pulling the series back toward its mean, thereby reducing the multiperiod volatility. The opposite is true for series with positive serial correlation.

      Time series with slightly positive or negative serial correlation abound in finance. It is a common mistake to assume that variance is linear in time, when in fact it is not. Assuming no serial correlation when it does exist can lead to a serious overestimation or underestimation of risk.

      CHAPTER 4

      Applied Finance and Economics

      This chapter is laid out in two sections: applied finance (the time value of money) and economics. The readings that follow offer a thorough review of time value of money (TVM) concepts and an overview of macroeconomic principles and analysis.

      Section I: Time Value of Money

      Investment advisors and consultants must understand the mathematics behind the concepts in the area of time value of money. They should be able to perform calculations by hand or with the help of a spreadsheet, computer program, or on their financial calculator. Simply understanding the intuition is not enough…the ability to calculate and apply are essential. The readings in this chapter explain each concept and provide numerous mathematical examples. Exercises and questions will help readers apply these formulae to solve practical problems in finance.

      Part I Foundations and Applications of the Time Value of Money: The Basics of the Time Value of Money

      Learning Objectives

      ■ Explain the concept of the time value of money; and describe compounding, discounting, present value, and future value.

      ■ Express present value and future value in an equation, and calculate each when given a fact pattern or problem to solve.

      ■ Discuss frequency of compounding and explain the annual percentage rate (APR).

      ■ Solve problems converting and comparing interest rates based on monthly, quarterly, semiannual, and annual interest payments.

      Part II Foundations and Applications of the Time Value of Money: Don't Discount Discounting

      Learning Objectives

      ■ Describe discounting and express through a formula solving for present value (PV).

      ■ Solve for present value using various inputs for future value, time, and discount rate.

      ■ Solve for present value using different compounding periods.

      Part III Foundations and Applications of the Time Value of Money: Cash Happens

      Learning Objectives

      ■ Calculate the value of a stream of future cash flows.

      ■ Define and calculate the value of a perpetuity.

      ■ Differentiate between an ordinary annuity, an annuity due, and a deferred annuity.

      ■ Calculate the value of an ordinary annuity and an annuity due.

      Part СКАЧАТЬ