Economics 134a

1
Economics 134a
2
Chapter 10

• Return and Risk
• Im not lecturing on Ch. 9, but you should read
  it

3
Big Ideas in Finance

• NPV
• Portfolio theory and equilibrium under
  uncertainty
• Efficient capital markets
• Capital Structure
• Derivatives

4
Up to now

• We have been ignoring risk in doing NPV
  exercises.
• Recognizing uncertainty means
• discounting expected payoffs, not actual payoffs
• Expected payoffs means treating payoffs,
  explicitly or implicitly, as random variables.
• (defined later in this lecture)
• Discounting at riskless rate of interest doesnt
  make sense if the payoff is risky
• discount rate makes allowance for risk. How do
  you do this?

5
Today review of probability
6
Random Variables

• What exactly is a random variable?
• Random variable a variable that takes on
  different values (or maybe the same value) with
  different probabilities
• Special case -- all possible values are the same
  -- no uncertainty
• So we are generalizing the deterministic case

7
Strictly speaking

• A RV is a function on a probability space
• AKA sample space
• probability space set of possible outcomes with
  assigned probabilities
• The probability assigned to the whole sample
  space is 1
• The probability assigned to any subset is
  positive or zero
• The probability assigned to the union of 2
  disjoint events is the sum of the probabilities
  of the 2 events

8
Example

• Suppose I give you 1 if a coin toss comes up
  heads, and you give me 2 if the coin comes up
  tails
• The payoff to me is a random variable

9
Probability distributions

• In the example, the distribution of the payoff
  is
• -1 with probability 1/2 and 2 with probability
  1/2
• As you see, the probability space and the
  definition of the random variable implies its
  distribution
• Sometimes you work with the random variable,
  sometimes with its distribution.
• You need to understand the difference.

10
Definition of a Random variable
11
Definition of its distribution

• -1 with probability ½
• payoff to me
• 2 with probability
  1/2

12
More examples

• You should make up more examples of random
  variables
• Number of heads in 3 coin tosses
• x 0 with probability 1/8
• x 1 with probability 3/8
• x 2 with probability 3/8
• x 3 with probability 1/8
• Number of hearts in a (5-card) poker hand
• Number of students over 6 feet tall in a sample
  of 10 from this class

13
Uncertainty about the future

• treat future-dated asset prices, returns, etc. as
  random variables investors dont know the
  actual values, but they have some idea about how
  likely various returns are.
• We model that by assuming future returns are
  random variables.
• Note this means were assuming investors (act
  as if they) know the possible values the random
  variable takes on, and also the probabilities

14
How do investors know the probabilities?

• Not clear. Past history?
• But assuming they know the probabilities is more
  realistic than ignoring uncertainty by assuming
  they know the actual future outcomes, which is
  what weve done so far.
• Finally, were usually assuming investors agree
  about the distributions of random variables.

15
Expectation

• of random variables
• A measure of the AVERAGE VALUE a random variable
  takes on.
• multiply the values the RV takes on by the
  relevant probabilities, add up.
• Heads-tails case above expected payoff to me
• 1/2 (-1) 1/2 (2) 1/2
• Other terms for expectation mean, average

16
Population vs. Sample

• You need to distinguish between the population
  mean of a random variable
• vs.
• the sample mean in a random sample.
• The sample mean is an estimator of the population
  mean
• The sample mean of the height of 10 students is a
  random variable the population mean is a number.

17

• The expectation might be of return,
  return-squared, return-minus-expected return,
  etc.
• these are all random variables
• So expectation has a more specific meaning in
  probability theory than in general usage.

18
Example returns of 2 firms
19
Expected return (in the population)

• Assume each state is equally likely (prob. 1/4)
• Supertech (-20 10 3050)/4
• 17.5
• Slowpoke (-5 20 - 12 9)/4 5.5.
• So Slowpoke has a much lower (average return/
  mean return/ expected return) than Supertech

20
Variance

• A measure of volatility or dispersion -- on
  average, how far is the RV from its mean?
• Variance the expectation of the squared
  deviation from mean.
• (not the only possible measure another measure
  would be average absolute value of the deviation
  from the mean)

21
(No Transcript)
22

• So variance of Supertech 668.75
• Variance of Slowpoke 132.25

23
Standard deviation

• SD square root of variance
• Supertechs SD is 25.9
• Slowpokes SD is 11.5
• SD as a measure of risk. Risk is a bad.
• SD has natural units -- same as returns (variance
  doesnt).

24
Note well

• Variance, SD count the upper tail as well as the
  lower tail.
• Why? Suppose we compare 2 payoffs with the same
  mean but different SD. The one with the higher
  SD attaches higher probability to payoffs in both
  the upper tail and the lower tail (otherwise the
  mean wouldnt be the same!).

25
Risk aversion

• People prefer low standard deviation, holding
  constant expected return
• Risk-return tradeoff.
• Need to be compensated for risk
• You expect that risky assets will have higher
  return on average
• And they do.

26
Special case -- binomial returns

• Binomial
• means 2 state (return can take on only 2 values)
• usually equal probabilities (Ill assume that
  probs are equal henceforth)
• Under equal probabilities, the mean is the
  average of the 2 payoffs
• SD is the upper payoff minus the mean
• (or the mean minus the lower payoff)
• Check this from the SD formula just presented
• Example if a stock has returns equal to 20 and
  - 10 with equal probability, its expected
  return is 5 and its SD is 15.

27
If probs not equal

• calculation of expected payoff and SD of payoff
  is a little more complicated.
• You should still be able to do it.

28
Covariance

• Between 2 random variables
• A measure of their association
• Positive covariance when one goes up the other
  (usually) goes up.
• Negative covariance when one goes up the other
  (usually) goes down.

29
Covariance defined

• expectation of the product of (one RV minus its
  mean) and (the other minus its mean)
• Example with Supertech and Slowpoke, p. 246.
• So the variance of a RV equals its covariance
  WITH ITSELF

30
Covariance of Supertech and Slowpoke
31
Correlation

• covariance divided by the product of the SDs of
  the 2 random variables.
• Varies from -1 when theres a perfect linear
  relation with negative coefficient
• To zero when theres no association
• To 1 when theres a perfect linear relation with
  positive coefficient

32
Supertech and Slowpoke

• Correlation is -0.16.
• So these returns are negatively associated (you
  already knew this from the fact that the
  covariance was negative).

33
In the binomial case

• (Namely, when both returns are binomial)
• There are only 3 possibilities
• correlation 1 if the 2 outcomes are high in
  the same state
• 0 if one (or both) of the outcomes is (are)
  constant
• - 1 if one is high when the other is low.

34
Binomial example
35
Example, continued

• SD of Supertech is 10
• SD of Slowpoke is 2
• Covariance is 20
• So correlation is 1.

36
Example, continued

• Repeat the calculations in this case, verify that
  correlation is minus-1

37
Example, continued

• If either of the firms has 0 standard deviation,
  correlation is undefined.