Lecture 0

Statistics Review
Yexiao Xu
School of Management The University of Texas at Dallas
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Outline
   

Probability Random variables Distributions Characterizing a random variable (ex-ante)
• Expected returns • variances, covariances • Correlation

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Statistics (ex-post)
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Random Events
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Many events occurs with uncertainty
• Working condition of hard drive in your laptop
─ Working properly versus hard drive crash

• Driving home
─ Safely get to home versus accident

• Stock price
─ Going up versus going down
 

We know the possible outcomes of a random event, but we do not know which outcome will actually occur An outcome of a event could be
• Discreet: roll of a dice {one of the six faces} • Continuous: plane arrival time {sometime today} • Multivariate: weight and height of a person
─ { (short and heavy), (short and light), (tall and heavy), (tall and light)}
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What Is Probability?


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The likelihood that each outcome will occur are usually different Probability is a numerical measure of the likelihood of an event to occur
• It can only take a number between 0 and 1

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Interpreting the probability scale •
0 Event never will occur 0.5 Event and “not event” are equal likely

1 Event will Always occur

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Random Variables
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A Random Variable: maps from outcomes of an event to numerical values
• It assigns numerical values to each possible outcome
─ Instead of using descriptive language to describe outcomes, we use numerical values for different outcomes

• In the case of toss a coin, we can assign “1” to head, “-1” to tail
─ That is, let X denote a random variable of tossing a coin ─ x = 1, if the head is up ─ x = -1, if the tail is up

• This is useful since numbers can be manipulated
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Each value of a random variable is associated with a probability
• The same probability as that of the corresponding outcome of an event • In the case of toss a coin, the probability associated with each outcome is 0.5, that is 50%
─ Prob(x = 1) = 0.5, Prob(x = -1) = 0.5
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Example of A Random Variable
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Roll a dice:
• Let Z be a random variable of rolling a dice
─ z = 1, if the first face is at the top ─ z = 2, if the second face is at the top, … ─ z = 6, if the sixth face is at the top

• The probability associated with each outcome is 1/6,
─ Prob(z = 1) = … = Prob(z = 6) = 1/6
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Car accident:
• Let X denote a random variable of possible loss in driving a car
─ x = 10,000, if an accident occurs ─ x = 0, if safely reach a destination

• The probability of an accident is 1%
─ Prob(x = 1) = 0.01, Prob(x = 0) = 0.99
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Distribution
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Distribution: a collection of probabilities that each outcome may occur
• In other words, the probability number as a function of a random event • Example of discreet outcomes
─ Fair coin toss: 50% head, 50% tail

• Example of continuous outcomes
─ Stock returns: Normal distribution (see next slide)
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The distribution of a random variable
• A probability function with respect to a random variable • Example: the likelihood of observing a particular face in rolling a dice f(x) 1/6

x 1 2 3 4 5 6
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Normal Distribution (Bell Curve)

The area in the green region represents the likelihood that the random variable will have the value between -0.9 and -0.5
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Returns as Random Variables
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Definition of return (R) on a stock: D P P  R  • P P t t t 1 t

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We can treat a stock return as a random variable
─ Due to different expectations, and ─ Uncertain future firm performance ─ Many possible returns

• Return = Dividend yield + Capital gain

─ D: dividend and ─ P: stock price

t 1

t 1

• There are more than one possible future returns (outcome) • We don’t know ex-ante (before investing) what exact return we will get • We can only find out the exact return ex-post (after investing)
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An Example
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Example: Suppose a stock, BULL, is currently at $20, and the following 4 scenarios are possible in a year’s time. What are the returns in each case?
Scenario 1 2 3 4 Prob(%) 10% 20% 40% 30% Price 0.00 10.00 20.00 30.00 Dividend 0.00 0.00 0.50 0.50 Return (R) -100.0% -50.0% 2.5% 52.5%

• e.g. R(4) = (0.50/20) + [(30.00 – 20.00)/20.00] = 52.50%
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Distribution:
20% 10%

f(x)

40% 30%

x
-100 -50 2.5 52.5 10

Statistical Characteristics of Returns
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Since the return random variable can take many possible values ex-ante, we need some measures to characterize it
• E[R] : Expected (Mean) Return • 2 : Variance
─ Measures the central tendency of the return random variable Ri ─ Measures the average squared deviation from the mean ─ Measures the average deviation from the mean

•  : Standard Deviation (SD) • Cov : Covariances • Corr (or ): Correlations
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─ Measures the co-movement of the two stock returns, i and j ─ Measures the tightness of the co-movement of the two stock returns

Assume there are S number of possible returns that might occur next period:
• For a particular return s, (s could be 1, 2, …, S), denote
─ R(s) as the s-th possible return ─ p(s) as probability s-th return might occur
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Expected Return
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The Expected Return can be calculated Sas
{p(s)}
s 1

E(R )  p(1) R(1)  p( 2) R( 2)   p( S ) R( S )   p( s ) R( s )

• It is a probability weighted average return of all the possible returns {R(s)} in the future • The expected value is the average outcome if the event was repeated infinitely often
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Example:
Scenario 1 p(1)=0.2 First Stock: R1 Second Stock : R2 Third Stock : R3 -0.01 0.20 0.10 Scenario 2 p(2)=0.3 0.30 0.15 -0.1 Scenario 3 p(3)=0.4 0.10 -0.03 0.35 Scenario 4 p(4)=0.1 -0.06 0.07 0.20
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Example


The expected return of the fist stock return:
• E (R1) = 0.2×(-0.01) + 0.3×0.30 + 0.4×0.10 + 0.1×(-0.06) = 0.122 • E (R1) = 12.2% • E (R2) = 0.2×0.20 + 0.3×0.15 + 0.4×(-0.03) + 0.1×0.07 = 0.08 • E (R2) = 8.0%

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The expected return of the second stock return:

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The expected return of the third stock return:
• E (R3) = 0.2×0.10 + 0.3×(-0.10) + 0.4×0.35 + 0.1×0.20 = 0.15 • E (R3) = 15.0%
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Variance and Standard Deviation
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The variance can be computed as
 2  Var ( R )  p (1) [ R (1)  E ( R )]2  p ( 2 ) [ R ( 2)  E ( R )]2    p ( S ) [ R ( S )  E ( R )]2
  p ( s ) [ R ( s )  E ( R )]2 s 1 S

• Variance is the average of the squared deviation from the mean E[R]
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Standard Deviation (SD) is just the square-root of Variance
  2
• It measures the average deviation from the mean • It is usually referred to as volatility
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Example (Cont’d)
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Continue from the last example
Scenario 1 p(1)=0.2 First Stock: R1 Second Stock : R2 Third Stock : R3 -0.01 0.20 0.10 Scenario 2 p(2)=0.3 0.30 0.15 -0.1 Scenario 3 p(3)=0.4 0.10 -0.03 0.35 Scenario 4 p(4)=0.1 -0.06 0.07 0.20

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Variance of the first stock:

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Variance of the second stock: 22 = 0.0092 Variance of the third stock: 32 = 0.0355
• Standard deviation  3 = 18.8% • Standard deviation  2 = 9.6%

• 12 = 0.2×[-0.01 - 0.122]2 + 0.3×[0.30 - 0.122]2 + 0.4×[0.10 - 0.122]2 + 0.1×[-0.06 - 0.122]2 = 0.0164 • Standard deviation  1 = 12.8%

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Covariance
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Covariance measures whether deviations of two stock’s returns from their means are related,
Covij  Cov( Ri , R j )  p(1) [ Ri(1)  E( Ri )][R(j1)  E( R j )]  p(2) [ Ri( 2)  E( Ri )][R(j 2)  E( R j )]    p( S ) [ Ri( S )  E( Ri )][R(j S )  E( R j )]   p( s ) [ Ri( s )  E( Ri )][ R(j s )  E( R j )] s 1 S



Two stock returns move
• in the same direction if Cov > 0 • in the opposite direction when Cov < 0

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Note that we can say nothing about how tight the two stock returns are related
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Example (Cont’d)


Continue from the last example
Scenario 1 p(1)=0.2 First Stock: R1 Second Stock : R2 Third Stock : R3 -0.01 0.20 0.10 Scenario 2 p(2)=0.3 0.30 0.15 -0.1 Scenario 3 p(3)=0.4 0.10 -0.03 0.35 Scenario 4 p(4)=0.1 -0.06 0.07 0.20



The covariance between the first and the second stock:
• Cov12 = 0.2×[-0.01 - 0.122]×[0.20 - 0.08] + 0.3×[0.30 - 0.122]×[0.15 - 0.08] +0.4×[0.10 - 0.122]× [-0.03 - 0.08]+0.1×[-0.06 - 0.122]×[0.07 - 0.08] = 0.0017

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The covariance between the first and the third stock: The covariance between the second and the third stock:
• Cov23 = -0.0153
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• Cov13 = -0.0147

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Correlation


A correlation is a scaled covariance by the standard deviations,
 i j The correlation coefficient between stock i and stock j’s returns should always be between -1 and 1 • ij close to 1  the two stock returns are tightly move together • ij close to 0  the moves in the two stock returns are unrelated • ij close to -1  returns are tightly move in the opposite direction Continue from the above example:
• The correlation between the first and the second stock returns:
12 
0 .0017  0 .139 0 .096  0 .128

 ij 

Cov ij

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• The correlation between the first and the third stock returns:
─ 13 = -0.607

• The correlation between the second and the third stock returns:
─ 23 = -0.846
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Measures for Continuous Distributions
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All the examples we have studied so far are for the case of discrete outcomes
• Fixed number of scenarios

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But returns are continuous
• Can have many many many outcomes • The corresponding distribution looks like a normal distribution

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When we increase the number of scenarios:
• we start to do infinite sum • which becomes integration rather than summation

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For simplicity, we focus on the discrete case in this class, but you get the idea …
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Introducing Statistics
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In most cases, we do not know the distribution of the return random variable
• How can we calculate the characteristics of stock returns?

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Solution
• Learning from past returns • That is to collect actual returns from the past • Apply statistical formulas to figure out these characteristics

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Suppose, there is a sample (a collection) of T realized returns r1, r2, … rT
• Since each of the T returns has occurred, we can think as if each outcome is equal likely • The previous formulas for mean, variance, covariance, and correlation carry over if we image that the probability is (1/T) for each actual return

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Basic Statistics
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

The most commonly used statistics are mean, variance, and covariance Mean estimator – the arithmetic average
1 1 1 1 rA  T  rA,1  T  rA, 2    T  rA,T  T ( rA,1  rA, 2    rA,T )



Variance estimator: average squared deviation from mean
2  A  T1 1 [( rA,1  rA ) 2  ( rA, 2  rA ) 2    ( rA,T  rA ) 2 ] 

• Note that we actually use (T-1) instead of T for mathematical reason


Covariance estimator: average comovement between stock A’s returns and stock B’s returns cov AB  T1 1 [(rA,1  rA )(rB,1  rB )  (rA,2  rA )(rB,2  rB )   (rA,T  rA )(rB,T  rB )] 
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Example
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Microsoft stock’s monthly returns in the past six months are: 3.17%, -2.06%, -4.72%, 4.90%, 3.16%, 3.84%,
• Average Return = (1/6)× 3.17%+ (1/6)×(-2.06%)+ (1/6)×(-4.72%)+ (1/6)×4.90%+ (1/6)×3.16%+ (1/6)×3.84% = 1.38% • Variance estimate = (1/5)× (3.17% - 1.38%)2 + (1/5)×(-2.06% - 1.38%)2 + (1/5)×(-4.72% - 1.38%)2 + (1/5)×(4.90% - 1.38%)2 + (1/5)×(3.16% - 1.38%)2 + (1/5)×(3.84% - 1.38%)2 = 0.001476
─ Standard Deviation = √0.001476 = 3.84%
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Example—High Correlation
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Weekly returns on Amazon.com (y axis) versus Ebay (x axis) in the period October 98-January 99
70% 50% 30% 10% 10% -10% -30%
80% 70% 60% 50% 40% 30% 20% 10% 0% -10% -20% -30%
Ebay Amazon.com

Weakly Return

-40%

60%

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Example—Low Correlation
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Weekly returns on GM (y axis) and Ebay (x axis) in the period October 98-January 99
12% 8% 4%
80% 70% 60% 50% 40% 30% 20% 10% 0% -10% -20% -30%
Ebay GM

-30%

-10%

10% 0% -4%

30%

50%

70%

Weakly Return

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