7. 6-7
Covariance and Correlation
Portfolio risk depends on the correlation
between the returns of the assets in the
portfolio
Covariance and the correlation coefficient
provide a measure of the returns on two
assets to vary
8. 6-8
Two Asset Portfolio
Return â Stock and Bond
Return
Stock
ht
Stock Weig
Return
Bond
Weight
Bond
Return
Portfolio
ď˝
ď˝
ď˝
ď˝
ď˝
ďŤ
ď˝
S
S
B
B
P
S
S
B
B
p
r
w
r
w
r
r r
w
r
w
9. 6-9
Covariance and Correlation Coefficient
Covariance:
Correlation
Coefficient:
1
( , ) ( ) ( ) ( )
S
S B S S B B
i
Cov r r p i r i r r i r
ď˝
ďŠ ďš ďŠ ďš
ď˝ ď ď
ďŤ ďť ďŤ ďť
ďĽ
( , )
S B
SB
S B
Cov r r
ď˛
ďł ďł
ď˝
10. 6-10
Correlation Coefficients:
Possible Values
If ď˛ = 1.0, the securities would be
perfectly positively correlated
If ď˛ = - 1.0, the securities would be
perfectly negatively correlated
Range of values for ď˛ 1,2
-1.0 < ď˛ < 1.0
11. 6-11
Two Asset Portfolio St Dev â
Stock and Bond
Deviation
Standard
Portfolio
Variance
Portfolio
2
2
,
2
2
2
2
2
2
ď˝
ď˝
ďŤ
ďŤ
ď˝
ďł
ďł
ď˛
ďł
ďł
ďł
ďł
ďł
p
p
S
B
B
S
S
B
S
S
B
B
p w
w
w
w
12. 6-12
rp = Weighted average of the
n securities
ďłp
2 = (Consider all pair-wise
covariance measures)
In General, For an n-Security Portfolio:
13. 6-13
Three Rules of Two-Risky-Asset Portfolios
Rate of return on the portfolio:
Expected rate of return on the portfolio:
P B B S S
r w r w r
ď˝ ďŤ
( ) ( ) ( )
P B B S S
E r w E r w E r
ď˝ ďŤ
14. 6-14
Three Rules of Two-Risky-Asset Portfolios
Variance of the rate of return on the portfolio:
2 2 2
( ) ( ) 2( )( )
P B B S S B B S S BS
w w w w
ďł ďł ďł ďł ďł ď˛
ď˝ ďŤ ďŤ
15. 6-15
Numerical Text Example: Bond and Stock
Returns (Page 169)
Returns
Bond = 6% Stock = 10%
Standard Deviation
Bond = 12% Stock = 25%
Weights
Bond = .5 Stock = .5
Correlation Coefficient
(Bonds and Stock) = 0
16. 6-16
Numerical Text Example: Bond and Stock
Returns (Page 169)
Return = 8%
.5(6) + .5 (10)
Standard Deviation = 13.87%
[(.5)2 (12)2 + (.5)2 (25)2 + âŚ
2 (.5) (.5) (12) (25) (0)] ½
[192.25] ½ = 13.87
22. 6-22
Dominant CAL with a Risk-Free
Investment (F)
CAL(O) dominates other lines -- it has the best
risk/return or the largest slope
Slope = ( )
A f
A
E r r
ďł
ď
23. 6-23
Dominant CAL with a Risk-Free
Investment (F)
Regardless of risk preferences, combinations of
O & F dominate
( ) ( )
P f A f
P A
E r r E r r
ďł ďł
ď ď
ďž
28. 6-28
Extending Concepts to All Securities
The optimal combinations result in lowest
level of risk for a given return
The optimal trade-off is described as the
efficient frontier
These portfolios are dominant
32. 6-32
Single Factor Model
βi = index of a securitiesâ particular return to the
factor
M = unanticipated movement commonly related to
security returns
Ei = unexpected event relevant only to this
security
Assumption: a broad market index like the
S&P500 is the common factor
( )
i i i i
R E R M e
ď˘
ď˝ ďŤ ďŤ
33. 6-33
Specification of a Single-Index Model of
Security Returns
Use the S&P 500 as a market proxy
Excess return can now be stated as:
â This specifies the both market and firm risk
i i M
R R e
ďĄ ď˘
ď˝ ďŤ ďŤ
36. 6-36
Components of Risk
Market or systematic risk: risk related to the
macro economic factor or market index
Unsystematic or firm specific risk: risk not
related to the macro factor or market index
Total risk = Systematic + Unsystematic
41. 6-41
Are Stock Returns Less Risky in the Long
Run?
Consider a 2-year investment
Variance of the 2-year return is double of that of the
one-year return and Ď is higher by a multiple of the
square root of 2
1 2
1 2 1 2
2 2
2
Var (2-year total return) = (
( ) ( ) 2 ( , )
0
2 and standard deviation of the return is 2
Var r r
Var r Var r Cov r r
ďł ďł
ďł ďł
ďŤ
ď˝ ďŤ ďŤ
ď˝ ďŤ ďŤ
ď˝
42. 6-42
Are Stock Returns Less Risky in the Long
Run?
Generalizing to an investment horizon of n
years and then annualizing:
2
Var(n-year total return) =
Standard deviation ( -year total return) = n
1
(annualized for an - year investment) =
n
n
n n
n n
ďł
ďł
ďł
ďł ďł ď˝
43. 6-43
The Fly in the âTime Diversificationâ
Ointment
Annualized standard deviation is only appropriate
for short-term portfolios
Variance grows linearly with the number of years
Standard deviation grows in proportion to n
44. 6-44
The Fly in the âTime Diversificationâ
Ointment
To compare investments in two different
time periods:
â Risk of the total (end of horizon) rate of return
â Accounts for magnitudes and probabilities