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					Econ 136: Financial Economics                   Section Notes – I                                               Aug 29, 2008
Rachita Gullapalli


I. Return

Define:     Pt =   the price of a security at time t
            Dt+1 = the dividend/coupon paid by the security on date t+1
            t+1 = the inflation rate between date t and t+1

         Gross simple return between date t and t+1: how much $1 invested in the security at time t is worth at time t+1
          (1+Rt+1) = Pt+1 + Dt+1
                        Pt
         Net simple return, how much $1 invested in the security at time t has changed in value at time t+1
          (Rt+1) = Pt+1 + Dt+1 – 1
                       Pt
          Note that net simple return is sum of the investment’s capital gain = Pt+1 – Pt and its dividend yield = Dt+1
                                                                                     Pt                                 Pt
         Log return (rt+1) = log (1 + Rt+1), the natural logarithm of the gross simple return and it is  Rt+1 for small Rt+1

         Gross compound return (1 + Rt,t+n) between date t and t+n = (1+Rt+1)(1+Rt+2)…(1+Rt+n)
          Notational Note: When writing a return that is not between consecutive periods we need two subscripts representing
          the starting date (t) and ending date of our calculation (t+n).
          Also, note how working with log returns makes this calculation easier: r t,t+n = rt+1 + rt+2 + …. + rt+n

         Real gross simple return = (1 + Rt+1) , how much $1 invested in the security at time t is worth in purchasing power at
          time t+1.                  (1 + t+1)

                   (Pt+1 + Dt+1)/CPIt+1 = (1 + Rt+1)
                             Pt/ CPIt     (1+ t+1)


II. Some Examples of Return Calculations

#1 (from ps1 ’05) Calculate the net simple return of High Tech Disaster, Inc. for 2001 given that its stock price equaled $120 on
January 1, 2001 and dropped down to $23.47 by December 31 st, 2001. The company paid $.53 in dividends on November 30,
2001. Also calculate the two components of the net simple return: the capital gain and the dividend yield.

net simple return = (23.47 +.53)/120 – 1 = -.8 or -80%

capital gain = (23.47 – 120)/120 = -.8044 or -80.44%

dividend yield = .53/120 = .0044 or .44%


#2 (from ps1 ’05) (a) Calculate the nominal net simple return for Boney M Co. in 1980 given: stock price on January 1, 1980
was $100; stock price on December 31, 1980 was $107.50; the company paid a dividend of $.50 per share on December 30,
1980.

nominal net simple return = (107.50+.50)/100 – 1 = .08 or 8%

note that no adjustment for inflation is necessary here to get the nominal return.


(b) Calculate the real net simple return given that the inflation rate in 1980 was 14% per year.

real net simple return = 1.08/1.14 – 1 =-.0526 or –5.26%


(c) Which of these would you use to compare the 1980 performance of Boney M to that of other companies? Which would you
use to compare it to the company’s performance in some other year, say 1990?
If you are comparing the 1980 performance of Boney M to that of other companies in 1980 in the same country (where the
same inflation rate applies), then comparing nominal returns would give you an accurate comparison of the companies’
performances. You could also compare their real returns.

If you are comparing the 1980 performance to performance in some other year (when inflation would likely be different), you
can only accurately compare the company’s performance using real returns. Notice that in the example above the nominal
return for the company 8%, which looks pretty good until you consider that inflation was 14% so the stock in the company went
down in terms of its purchasing power. Note that if in 1990 inflation was 0% and the stock’s nominal return was 5%, then by
comparing nominal returns, you would inaccurately think that the company performed better in 1980 than 1990.


III. Statistics Review:

        Random Variable: When outcomes of a decision can be quantified, the decision outcome is called a random variable.
         Ex. Google’s stock price was $473. The price in one year (August 28, 2009) is expected to be $550 with probability 0.6
         and $500 with probability 0.4. As the future price can be assigned a numerical value, we can quantify the return on
         investing today  one year return on Google stock is a random variable.

         Return:           x1 = $27 with probability 0.4
                           x2 = $77 with probability 0.6

         Discrete random variables are variables whose values are countable. Ex. if a coin is tossed twice and random
         variable x is the number of heads. Then only the following are possible values of x: 0, 1 or 2.

         If a random variable has values that are not countable i.e. the data can take infinitely number of possible values, we
         call these continuous random variables. Ex. height of students in this class. Any positive value is possible (with an
         upper bound).


        Expected Return:
         The expected value of a random variable is basically its average or central value. It can be thought of as a probability-
         weighted average of the possible values that the random variable can assume (the ``center of mass'' of the probability
         distribution function). For a discrete random variable

         E(x) = Prob1* x1 + Prob2* x2

         E(x) = 27*0.4 + 77*0.6 = 57

				
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