Introduction to investments

1
Introduction to investments

• Expected returns versus required returns
• The expected return on a security is the return
  an investor expects to receive over some future
  time period.
• The expected return is estimated by determining
  an expected future price and income for a
  security and measuring the return in a manner
  similar to the actual return calculation.

2
Introduction to investments

• The required rate of return
• Recall that an investment is the current
  commitment of dollars for a period of time in
  order to derive future payments that will
  compensate the investor for
• a. The time the funds are committed
• b. The expected rate of inflation and
• c. The uncertainty of future payments.
• These three factors determine an investors
  required rate of return

3
Introduction to investments

• Differences between the required rate of return
  and the expected rate of return
• The expected return is merely the return that we
  expect from our investment in the future.
• The development of these expectations can be
  based upon historical returns, fundamental
  analysis, market analysis, etc.
• Thus, the expected return and the required return
  do not have to be the same.

4
Introduction to investments

• Comparing the required rate of return and the
  expected rate of return
• a. ER gt RR investment is undervalued, buy
• b. ER RR investment is fairly valued,
  indifferent
• c. ER lt RR investment is undervalued, sell
• In an efficient market, ER RR on average
• Choosing securities that have ER gt RR is essence
  of any active investment technique.

5
Example

• Suppose we have two assets, A and B, that are
  both expected to provide a return of 15 and have
  a price of 100. (thus, both stocks currently
  have an expected return of 15)
• Suppose that stock A is riskier than stock B.
• What would happen?
• What if the price of A fell to 75 and B rose to
  150?

6
Introduction to investments

• Expected returns
• Mathematical definition of the expected return
• ER ((Pt1-Pt) incomet1)/Pt

7
Introduction to investments

• Expected returns
• We can also measure the expected return by
  estimating several potential outcomes and
  assigning a probability to each outcome. The
  expected return in this case would be equal to
• ? PiERi

8
Examples

• 1. Assume we have estimated that the price of
  General Motors will be 90 one year from now and
  GM will pay 2.00 in dividends during that time.
  If the current price of GM is 85 what is the
  expected annual return for GM?
• ER ((90-85) 2)/85 0.0824 8.24

9
Examples

• 2. Assume we have estimated GMs expected return
  over four equally likely states of nature.
• ERi Pi
• 5 .25
• 8 .25
• 12 .25
• 25 .25
• ER (.255) (.258) (.2512) (.2525)
  12.5

10
Example Expected Returns

• 4. Suppose you have predicted the following
  returns for stocks C and T in three possible
  states of nature. What are the expected returns?
• State Probability C T
• Boom 0.3 0.15 0.25
• Normal 0.5 0.10 0.20
• Recession 0.2 0.02 0.01
• RC .3(.15) .5(.10) .2(.02) .099 9.99
• RT .3(.25) .5(.20) .2(.01) .177 17.7

11
Variance and Standard Deviation of Expected
Returns

• Variance and standard deviation still measure the
  volatility of returns
• Using unequal probabilities for the entire range
  of possibilities
• Weighted average of squared deviations

12
Example

• Consider the previous example. What are the
  variance and standard deviation for each stock?
• Stock C
• ?2 .3(.15-.099)2 .5(.1-.099)2 .2(.02-.099)2
  .002029
• ? .045
• Stock T
• ?2 .3(.25-.177)2 .5(.2-.177)2 .2(.01-.177)2
  .007441
• ? .0863

13
Another Example

• Consider the following information
• State Probability ABC, Inc.
• Boom .25 .15
• Normal .50 .08
• Slowdown .15 .04
• Recession .10 -.03
• What is the expected return?
• What is the variance?
• What is the standard deviation?

14
Introduction to investments

• Actual return calculations
• 1. Percentage return on an investment
• ((Pt-Pt-1) incomet)/Pt-1

15
Example

• Example We buy a Widget for 15 and sell it in
  a year for 18. The stock pays a dividend of
  1.00 during the year. What is our percentage
  return on the stock?
• return ((18-15)1)/15
• 26.67
• The book defines this calculation as the holding
  period yield (HPY).

16
Introduction to investments

• 2. What if we dont have the beginning and
  ending values of the investment?
• In this case the rate of return over the life of
  the investment is
• Rate of return for entire time period
  (1 R1)(1 R2)(1 R3)...(1
  Rn)-1
• Ri return in intermediate time period i,
• n the number of intermediate time periods.

17
Example

• Suppose that we know that Widgets stock earned
  10, 5, -5, 6, and 12 over the last five
  years. What would the rate of return on Widgets
  stock been if we had held the stock over the last
  five years?
• (1.10)(1.05)(1-.05)(1.06)(1.12) 1
  1.3027 1 .3027 30.27

18
Introduction to investments

• C. Finding the return for subperiods of the
  entire life of the investment
• Sometimes you will want to take a rate of return
  and split it into an equivalent compounded return
  for a smaller time period.

19
Example 1

• From the example in section B, suppose that we
  wanted to find the annual rate of return that,
  when compounded over five years, will yield the
  total rate of return over the five year life of
  the investment.
• (1.10)(1.05)(1-.05)(1.06)(1.12)1/5 1
  (1.3027)1/5 1 1.0543 1 5.43
• Thus, if we earned 5.43 compounded each year for
  five years, we would have earned the same return,
  30.27, as we earned on Widgets stock over five
  years.

20
Example 2

• From the example in section A, suppose that we
  wanted to find the weekly rate of return that,
  when compounded will yield the total rate of
  return over last year for Widgets stock.
• 1 ((453) 40/40)1/52 1 (1 .20)1/52
  1 1.0035 1 .0035 .35.
• Thus, if we had earned .35 per week compounded
  over the last year we would have earned the same
  rate of return (20) as Widgets stock.

21
Introduction to investments

• Arithmetic and geometric average returns

22
Example

• AM 1 (-0.5) /2 0.25 25
• GM 20.51/2 -1 0
• The question of which average return measure is
  superior (geometric or arithmetic) becomes a
  question of whether or not the holding period of
  the investment is important.

23
Introduction to investments

• The standard deviation of historical returns

24
Introduction to Investments

• Coefficient of variation

25
Introduction to Investments

• Compound returns and continuous compounding
• Realized annual return
• (1 (R/N))NT -1
• N Number of compounding periods per year
• T Number of years, generally this will be one
• What happens if we want continuous compounding?

26
Introduction to Investments

• Compound returns and continuous compounding
• With continuous compounding we have

27
Example of the effect of compounding