Lecture 11
■
Staticstics review and portfolio theory
■
Readings:
– Reader, lecture 13
– BM Chapter 7
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Where are we?
■
We know, in principle, how to make investment decisions:
– PV/NPV
– Discount cash flows using an appropriate discount rate
■
Where does the discount rate come from?
■
We need the Capital Asset Pricing Model (CAPM)
– Gives quantitative tradeoff between risk and expected return.
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What we need to know
■
■
■
How to calculate the expected return on a stock.
How to calculate the variance of stock returns.
How to apply the above to a portfolio of stocks:
– How to calculate the expected return on a portfolio.
– How to calculate the covariance between the return on
two stocks.
– How to calculate the variance of a portfolio.
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A Single Stock:
Notation and Definitions
■
Notation:
–
~r is the (random) return on stock i.
i
– rix is one possible return for stock i, which
occurs with probability px
– ri is the expected return on stock i.
Expected Return = E(~r ) = r = p r
i
i
x ix
x
2
2
~
~
Variance = Var( ri ) = σi = E( ri − ri ) =
x
Standard Deviation = σi = Variance
© M. Spiegel and R. Stanton, 2000
p x (rix − ri )
2
4
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A Single Stock:
Expected Return
Expected Return = E(~ri ) = ri =
p x rix
x
rix
pX
-5
0
5
12
© M. Spiegel and R. Stanton, 2000
pxrix
-1.25
0
1.25
3
3
0.25
0.25
0.25
0.25
ri=
5
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A Single Stock:
Variance
2
~
~
Stock Variance = Var( ri ) = E( ri − ri ) =
p x (rix − ri )
2
x
rix
-5
0
5
12
© M. Spiegel and R. Stanton, 2000
Px
0.25
0.25
0.25
0.25
(rix - ri)2
px(rix - ri)2
64
16.00
9
2.25
4
1.00
81
20.25
Var =
39.50
6
U.C. Berkeley
Portfolio Weights
■
Take a portfolio worth $200 that contains
– $40 worth of stock A,
– $60 worth of stock B,
– $100 worth of stock C.
■
Then the portfolio has investment weights of:
– wA = 40/200 = .20 in stock A
– wB = 60/200 = .30 in stock B
– wC = 100/200 = .50 in stock C
■
The weights on any portfolio must sum to 1
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Portfolios: Expected Return
■
Formula for the (expected) return on a portfolio:
~r = w ~r + w ~r (actual returns)
p
i i
j j
rp = w i ri + w jrj (expected returns)
■
Notation:
–
–
–
–
■
the expected return on stock i is ri,
the expected return on stock j is rj,
the weight for stock i is wi,
the weight for stock j is wj.
Example: ri = 3, rj = 5, wi = .4, and wj = .6. Then:
rp = (.4)(3) + (.6)(5) = 4.2.
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CD and NBF Sample Problem:
Finding Portfolio Weights
■
Invest in Corporate Disasters (CD) at 4%.
■
Or, invest in Nevada beach front property (NBF):
Investment will have an expected return of 10%.
■
You want a portfolio with expected return 6%.
■
What portfolio fraction should you invest in the
Nevada property?
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Finding Portfolio Weights:
Solution
■
Use formula for expected return on a portfolio
6 = wCD(4) + wNBF(10)
Since wCD + wNBF = 1, wCD = 1 − wNBF.
■
So we can write 6 = (1 − wNBF)(4) + wNBF(10)
■
We find wNBF = 1/3
■
■
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Covariance: The relationship
between two returns
■
■
■
A zero covariance implies no relationship.
A positive covariance implies that when stock i
has an exceptionally high return, so (usually) does
stock j.
A negative covariance implies that when the return
on stock i is unusually high, the return on stock j
tends to be unusually low.
Cov(~ri , ~rj ) = E(~ri − ri )(~rj − rj ) = σij =
© M. Spiegel and R. Stanton, 2000
x
y
11
p xy (rix − ri )(rjy − rj ).
U.C. Berkeley
Covariance vs. Correlation
■
■
■
People often talk about the correlation between
stock i and j (ρij) instead of their covariance (σij).
The two measures are very closely related.
We can easily convert from one to the other:
~
~
σij
Cov( ri , rj )
ρij =
=
.
~
~
SD( r )SD( r ) σ σ
i
© M. Spiegel and R. Stanton, 2000
j
i
j
12
U.C. Berkeley
Example: Computing covariance
between two stocks
■
Probabilities for returns on each stock
Joint Probabilities for each pair of
returns on stocks i and j
r ix Returns on i
r jy Returns on j
2
7
8
0
0.1
0
0
6
0
0.2
0.1
12
0
0.2
0.4
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Example 1: Computation Matrix
for Variance and Covariance
rix
pxy
r jy
pxy r ix
pxy r jy
r ix - r i
pxy ( r ix - r i )( r jy - r j )
r jy - r j
0
2
0.1
0.0
0.2
-9
-5
4.5
6
7
0.2
1.2
1.4
-3
0
0.0
12
7
0.2
2.4
1.4
3
0
0.0
6
8
0.1
0.6
0.8
-3
1
-0.3
12
8
0.4
4.8
3.2
3
1
1.2
ri =
rj =
Cov ( ~r i , ~r j )=
9.0
7.0
5.4
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Rules Governing Portfolio
Covariance and Variance
■
Rules for covariance:
cov(w i ~ri , w j~rj ) = w i w jcov(~ri , ~rj )
cov(~r , ~r + ~r ) = cov(~r , ~r ) + cov(~r , ~r )
i
j
k
var (~ri ) = cov(~ri , ~ri )
■
i
j
i
k
The variance of a portfolio with two stocks:
(
)
()
(
var w i ~ri + w j~rj = w i2 var(~ri ) + w 2j var ~rj + 2w i w jcov ~ri , ~rj
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15
)
U.C. Berkeley
Example 1: Computing Portfolio
Expected Return and Variance
■
■
■
■
In previous example, we found ri = 9.0, rj = 7.0.
If we form a portfolio with wi = .4, wj = .6, then
rp = E (.4~ri + .6~rj ) = .4(9 ) + .6(7 ) = 7.8.
~
~
(
)
(
var
r
=
16.2,
and
var
rj ) = 3.
For homework, show that
i
We also know that cov(~ri , ~rj ) = 5.4, so
var(~r ) = var (.4~r + .6~r )
p
i
j
= .4 2 (16.2 ) + .6 2 (3) + 2(.4 )(.6 )(5.4 )
= 6.264.
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Multiple Stock Portfolios:
Expected Return and Variance
■
Expected return on portfolio, rP:
rp = w1r1 + w 2 r2 + .... + w n rn
Just plug in values and add terms together.
Variance on portfolio, VarP:
Varp = Var (w1~r1 + w 2 ~r2 + .... + w n ~rn )
■
Use “box method” to compute portfolio variance.
Notation : σi2 = variancei ; σi = SD i ; σ ij = covarianceij
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Multiple Stock Portfolios:
Computing Variance
■
Add up all boxes in the weighted variance-covariance matrix:
1
Stock
Stock
2
…
w1w 2σ12 …
n
1
w12σ12
2
w1w 2σ12
w 22σ 22
w 2 w n σ 2n
·
n
·
·
·
w2nσ2n
w1w n σ1n w 2 w n σ 2n
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w1w n σ1n
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Multiple Stock Portfolios:
Example with 2 Stocks
Var(w1~r1 + w 2 ~r2 ) = w 12 σ12 + w 22 σ 22 + 2w 1w 2 σ12
= w 12 Var (~r1 ) + w 22 Var (~r2 ) + 2w 1w 2 Cov(~r1 , ~r2 )
STOCK
Stock
© M. Spiegel and R. Stanton, 2000
1
2
1
w 12 σ12
w 1w 2 σ12
2
w 1w 2 σ12
w 22 σ 22
19
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Numerical Example 1
■
What is the variance of a portfolio with:
w 1 = .2, w 2 = .8, σ12 = 10, σ 22 = 20, and σ12 = 5?
STOCK
Stock
( )
1
2
1
(.22)10
(.2)(.8)5
2
(.2)(.8)5
(.82)20
( )
σ 2p = .2 2 10 + .82 20 + 2(.2 )(.8)5 = 14.8
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Numerical Example 2
■
You have a portfolio of 15 stocks:
–
–
–
–
■
The first 5 stocks have portfolio weights 0.1 (10%);
The remaining stocks have portfolio weight .05 (5%);
Each stock has a variance of 10,
All stocks have a covariance of 2 with each other.
What is the variance of your portfolio?
– To solve this problem, set up the matrix, and use the
patterns in the matrix to simplify the box addition.
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Numerical Example 2, Continued
1
…
5
6
2
(.1 )(2)
(.1 )(2)
2
(.1 )(10)
6
…
(.1)(.05)(2)
(.1)(.05)(2) (.05 )(10)
15
(.1)(.05)(2)
(.1)(.05)(2) (.05 )(2)
1
…
(.1 )(10)
5
…
15
2
(.1)(.05)(2)
(.1)(.05)(2)
2
(.1)(.05)(2)
(.1)(.05)(2)
2
(.05 )(2)
2
2
(.05 )(10)
2
Top left hand group of 25 boxes (stocks 1-5): The 5 diagonals are (.12)(10).
The remaining 20 entries are (.12)(2). Total = 5x.12x10 + 20x.12x2 = .9.
■
Bottom right hand 100 boxes (stocks 6-15): The 10 diagonals = (.052)(10).
The other 90 entries are .052x2. Total = 10x.052x10 + 90x.052x2 =.7.
■
Top right and bottom left group for total of 100 boxes. Every box has the
entry (.1)(.05)(2). Total = 100x.1x.05x2 = 1.
■ Total portfolio variance = .9 + .7 +1 = 2.6.
■
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Example: Eliminating “firm
specific risk” via Diversification
■
Consider a special portfolio, in which wi = 1/n.
STOCK
1
2
…
n
1
σ12 /n 2
σ12 /n 2
…
σ1n /n 2
2
σ12 /n 2
σ 22 /n 2
σ 2n /n 2
·
·
·
·
·
·
·
·
n
σ1n /n 2
σ 2n /n 2
σ 2n /n 2
Stock
■
For portfolio variance, add up the boxes.
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Computing Variance of the Fully
Diversified Portfolio
■
Write the sum of the diagonal elements as
n
i =1
σ i2 1
=
2
n
n
n
i =1
σ i2 1
= (avg. variance).
n n
Now add up off-diagonal elements. There are n x n boxes
in total of which n are diagonals leaving n x n - n off
diagonal boxes. Add up the off-diagonal boxes to get
■
σij
σij
n2 − n
æ 1ö
=
= ç1 − (avg. cov.)
2
2
2
n off −diagonal boxes n − n è n
off − diagonal boxes n
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Variance of the Fully Diversified
Portfolio: Conclusion
■
Therefore the variance of the portfolio equals
1~ö 1
æ1~ 1~
æ 1ö
Varç r1 + r2 + ... + rn ÷ = x Avg.Var + ç1 − x Avg. Cov.
n
n
n
èn
è n
■
So what is the variance of a “perfectly” diversified
portfolio? This occurs as n goes to infinity. In
that case:
Var (
1~
r
n 1
+
© M. Spiegel and R. Stanton, 2000
1~
r
n 2
+ ... +
1~
r
n n
) = Avg.Cov.
25
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Variance of a Fully Diversified
Portfolio: Comment
■
■
Note that diversification eliminates all risk except
the average covariance of the stocks.
Since people can do this for themselves, we’ll see
that
– The market only rewards people for holding “market
risk” (covariance between all stocks)
– No reward for “firm specific risk” (variance of a single
stock).
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