TEXAS A&M UNIVERSITY
Mays Business School
Lecture1: Stats camp
Marco Rossi, PhD
Finance Department
TEXAS A&M UNIVERSITY
Mays Business School
Finance
Stats Camp Outline
•Measuring risk and returns
•Measuring the co-movement of
returns (correlations)
•Portfolio statistics
2
TEXAS A&M UNIVERSITY
Mays Business School
Finance
Risk and Return
• We typically assume that investors are risk
averse:
• Given the same level of risk, investors prefer a security with a
higher return.
• Given the same return, investors prefer a security with a lower
risk.
• First, we need to understand what we mean
by expected return and risk.
• How do we measure expected return?
• How do we measure risk?
TEXAS A&M UNIVERSITY
Mays Business School
Finance
Scenario Analysis
• Need to compute return in various states of the world
and the likelihood of each future state.
• Consider the following forecast of the expected future
return on equities:
State
Probability
Requities
High Economic Growth
30%
21%
Stable Economic Growth
60%
10%
Recession
10%
-3%
Estimating Expected Returns
• Given the forecasted return in each different state of the world Rs
and the probability of each state occurring Probs the expected
return is calculated as follows:
S
E ( R) =  (Probs  Rs )
s =1
• For our example forecast, this gives:
S
E ( R) =  Probs (Rs ) = .30(.21) + .60(.10) + .10(−.03) = 12.0%
s =1
Estimating Risk (continued)
• Variance: weighted average of squared deviations from expected value
2
S
 =  Probs (Rs − E ( R) )
2
s =1
• The standard deviation is equal to the square root of the variance.
= 
2
Estimating Risk (continued)
• Reconsider the return forecasts given previously:
State
Probability
Requities
High Economic Growth
30%
21%
Stable Economic Growth
60%
10%
Recession
10%
-3%
• Based on these forecasts, the variance σ2 and standard deviation σ are
calculated as follows:
 2 = .30(.21 − .12) 2 + .60(.10 − .12) 2 + .10(−.03 − .12) 2 = 0.00492
 = 0.00492 = 7.01%
Estimation with historical data
• If we were to use past data to estimate expectations and risk then:
• Type equation here.
R= σ𝑻𝒕=1
𝑹𝒕
𝑻
 (R − R )
T
2 =
t =1
2
t
 = 2
T
• where Rt = return during period t, T is the number of periods, and the mean return is
defined as before.
• Example: Consider a stock that earned returns of 5%, 7%, -4%, and 12% over the
past four years (E(R)=5%). The variance and standard deviation would then
equal:
(
)
 2 = (.05 − .05)2 + (.07 − .05)2 + (− .04 − .05)2 + (.12 − .05)2 / 4 = 0.00335
 = .00335 = 5.7879%
TEXAS A&M UNIVERSITY
Mays Business School
Finance
Stats Camp Outline
•Measuring risk and returns
•Measuring the co-movement of
returns (correlations)
•Portfolio statistics
10
TEXAS A&M UNIVERSITY
Mays Business School
Finance
Measures of Association
• We may also wish to know the relationship between the
returns on two investments or how they co-vary with one
another.
• Measures of co-variation are key factors in portfolio theory
because they affect diversification.
• As we will see in next lecture, co-variation is also key for
asset pricing (price of a security is typically determined by
its co-variation with risk factors).
• We will calculate two measures of association:
Covariance and Correlation.
Measures of Association: covariance
• The Covariance between the returns on stock i and the
returns on stock j is defined as follows:
 (R
T
Covariance(Ri ,R j ) =
t =1
i ,t
− Ri )(R j ,t − R j )
T
• If the returns on two stocks tend to move in the same
(opposite) direction, they will have a positive
(negative) covariance.
• If they move independently, they will have a
covariance of zero.
Measures of Association: correlation
• Standardized Covariance (independent of units of
measurement). The Correlation between the returns on stock i
and the returns on stock j is defined as follows:
Correlation(Ri ,R j ) = i , j =
Covariance(Ri ,R j )
 i  j
• Correlation ranges from -1 to +1.
• If the returns on two stocks tend to move by the same
amounts in the same (opposite) direction, the correlation
will be close to +1 (-1).
• If they move independently, the correlation will be zero.
Example: Covariance vs. Correlation
• Suppose you are given the following data:
• Std(A)=45%
• Std(B)=30%
• Std(C)=2%
• Cov(A,B)=0.0135
• Cov(B,C)=0.0054
• Calculate the correlations of security B with securities A and C.
 B, A =
Cov( B, A)
0.0135
=
= 0.1
 A  B
0.45  0.30
 B ,C =
Cov( B, C )
0.0054
=
= 0.9
 B  C
0.30  0.02
TEXAS A&M UNIVERSITY
Mays Business School
Finance
Lecture 1 - Outline
•Measuring risk and returns
•Measuring the co-movement of
returns (correlations)
•Portfolio statistics
16
TEXAS A&M UNIVERSITY
Mays Business School
Finance
What is the expected return and variance of a portfolio?
• Consider the following two risky assets:
Security A
Security B
E(RA) = 8%
σA = 13%
E(RB) = 16% σB = 20%
• If we place weight wA in security A and weight wB in security B
(where wB=1–wA), the expected return and risk of the resulting
portfolio are defined as follows:
Rule 1 : E ( RP ) = wA  E ( RA ) + wB  E ( RB )
*
Rule 2 :  = w  + w  + 2wA wBCov( RA , RB )
*
2
P
2
A
2
A
2
B
2
B
Example (Two Risky Assets)
• What are the expected return and standard deviation of a portfolio
that invests 60% in Security A and 40% in Security B? Assume a
covariance of 0.0052 (or a correlation of +0.2).
Rule 1* : E ( RP ) = (0.6)(.08) + (0.4)(.16) = 11.2%
Rule 2* :  P2 = (0.6 2 )(.132 ) + (0.4 2 )(.202 ) + 2(0.6)(0.4)(.0052) = 0.01498
 P = 0.01498 = 12.24%
• How would the results change if the correlation were -0.2?
Rule 2* :  P2 = (0.62 )(.132 ) + (0.42 )(.202 ) − 2(0.6)(0.4)(.0052) = 0.00998
 P = 0.00998 = 8.19%
Combining Two Risky Assets (Corr. = +1)
• Consider the following examples of portfolio expected returns
and risk:
• Case A: =1
E(RP)
20.0%
16.0
12.0
8.0
4.0
P
23.5%
20.0
16.5
13.0
9.5
20.0%
B
15.0%
E(R)
wA
-0.5
0.0
0.5
1.0
1.5
25.0%
10.0%
A
5.0%
0.0%
0.0%
5.0%
10.0%
15.0%
Std. Dev.
20.0%
25.0%
30.0%
Combining Two Risky Assets (Corr. = -1)
25.0%
E(RP)
20.0%
16.0
12.0
11.15
8.0
4.0
P
36.5%
20.0
3.5
0.0
13.0
29.5
20.0%
B
15.0%
E(R)
wA
-0.5
0.0
0.5
0.6061
1.0
1.5
10.0%
A
5.0%
0.0%
0.0%
5.0%
10.0%
• Note: If correlation () =-1, we can create a perfect hedge
(zero risk) portfolio that includes positive weights in both
securities.
15.0%
Std. Dev.
20.0%
25.0%
30.0%
Combining Two Risky Assets (corr. = 0)
25.0%
E(RP)
20.0%
16.0
12.0
10.9
8.0
4.0
P
30.7%
20.0
11.9
10.38
13.0
21.9
20.0%
B
15.0%
E(R)
wA
-0.5
0.0
0.5
0.703
1.0
1.5
10.0%
A
5.0%
0.0%
0.0%
5.0%
10.0%
15.0%
Std. Dev.
20.0%
25.0%
30.0%
Combining Two Risky Assets
25.0%
20.0%
= -1
For most real-world
securities,
0<correlation<1.
B
E(R)
15.0%
10.0%
=+1
A
5.0%
0.0%
0.0%
5.0%
10.0%
15.0%
Std. Dev.
20.0%
25.0%
30.0%
When correlation is
negative, you can think
of one asset as being
insurance for the other.
The Minimum Variance Portfolio
Minimum Variance Portfolio
• The
is defined as the
portfolio combination that gives the lowest possible level of risk. What
is the standard deviation of this portfolio in the case of correlations
=1, =-1, =0 ?
25.0%
=-1
=0
20.0%
B
E(R)
15.0%
10.0%
A
5.0%
=1
0.0%
0.0%
5.0%
10.0%
15.0%
Std. Dev.
20.0%
25.0%
30.0%
The Minimum Variance Portfolio
• Weight on MVP:
mvp
1
w
 22 − Cov1, 2
= 2
 1 +  22 − 2Cov1, 2
• If correlation = -1, this simplifies to:
w1mvp =
2
1 +  2
• If correlation = 0, this simplifies to:
mvp
1
w
 22
= 2
 1 +  22
TEXAS A&M UNIVERSITY
Mays Business School
Finance
Risk and Return with Many Assets
• General formulas for portfolio expected returns and variance:
N
E ( R P ) =  wi E ( Ri )
i =1
 P2
N
=
N
 wi w j Cov( Ri , R j )
i =1 j =1
• How does this simplify for N=3?
 P2 = w1w1Cov( R1 , R1 ) + w1w2Cov ( R1 , R2 ) + w1w3Cov ( R1 , R3 )
+ w2 w1Cov ( R2 , R1 ) + w2 w2Cov ( R2 , R2 ) + w2 w3Cov ( R2 , R3 )
+ w3 w1Cov ( R3 , R1 ) + w3 w2Cov ( R3 , R2 ) + w3 w3Cov ( R3 , R3 )
 P2 = w12Var ( R1 , R1 ) + w2 2Var ( R2 , R2 ) + w32Var ( R3 , R3 )
+2 w1w2Cov( R1 , R2 ) + 2 w1w3Cov( R1 , R3 ) + 2 w2 w3Cov( R2 , R3 )