Quick Review of Some Statistical Concepts
Expected Return of a Security
Variance of the Return of a Security
Standard Deviation of the Return of a Security
Covariance Between the Returns of Two Securities
Correlation Between the Returns of Two Securities
Expected Return and Variance of a Portfolio of Two
Securities
Consider a world in which N possible states of the world
can occur. A state S, S=1...N, has probability of
occurrence P(S).
Let there exist two securities in this world, securities A
and B. The returns on securities A and B in state S are
denoted rA(S) and rB(S) respectively.
Expected Return:
E (rA ) ≡ P (1)rA (1) + ... + P (N )rA (N )
the average over all states of the return.
Example
State
1
2
Probability
0.5
0.5
rA
0.30
0.10
rB
-0.05
0.10
E (rA ) = 0.5 × 0.30 + 0.5 × 0.1 = 0.20
E (rB ) = 0.5 × (-0.05) + 0.5 × 0.1 = 0.025
Variance of the Return:
var(rA ) ≡ E (rA 2 ) − [E (rA ) ]2 ≡ σ 2A
a measure of dispersion around the average return.
Standard Deviation of the Return:
sd (rA ) ≡
var(rA ) ≡ σ A
the square root of the variance.
Example
State
1
2
Probability
0.5
0.5
rA
0.30
0.10
rB
-0.05
0.10
2
2
2
var(rA) =0.5—0.3 +0.5—0.1 -0.2 = 0.01
2
2
2
var(rB) =0.5—(-0.05) +0.5—0.1 -0.025 =0.005625
VA =
0.01 = 0.1
VB =
0.005625 = 0.075
Covariance between two Returns:
cov(rA , rB) = E(rA —rB) - E(rA) —E(rB)
a measure of the co-variation between the two
returns.
Correlation between two returns:
ρ A, B
cov(rA , rB )
≡
σ a ×σ B
a normalized measure of covariance.
Example
State
1
2
Probability
0.5
0.5
rA
0.30
0.10
rB
-0.05
0.10
cov(rA , rB) = 0.5— 0.3 — (-0.05)
+ 0.5 — 0.1 —— ρ A, B
− 0.0075
=
= -1
0.1 × 0.075
Expected Return and Variance of a Portfolio:
Consider an investor who chooses to invest a fraction D
of his wealth in security A and the remaining fraction (1 D) in security B. Let the resulting portfolio be called AB.
We have
E(rAB) = D — E(rA) + (1 - D) — E(rB)
The expected return on a portfolio is the weighted
average of the returns on the securities in the portfolio.
Similarly, the variance of a portfolio is a weighted
average of the variance of each security and the
covariance between securities
2
var(rAB) = D — var(rA) + (1 - D) — var(rB)
+ 2 — D — (1 - D) — cov(rA , rB)
Example
Suppose that E(rA)= 0.16, E(rB)= 0.10, VA= 0.30, VB=
0.16 and •A,B= -0.5.
If the investor divides his wealth equally between
securities A and B (D = 0.5), the expected return on the
resulting portfolio AB is
E(rAB) = 0.5 — 0.16 + 0.5 — 0.1 = 0.13
The covariance between the two securities is
cov(rA , rB)= UA,B —VA —VB = -0.5—— The standard deviation of the portfolio is therefore
VAB=(0.52————)1/2=0.13
The standard deviation of the portfolio is lower than that
of either security A or security B.
This is not always the case, however. The extent to
which one risk can offset another depends on the
correlation between them.