LECTURE 5 :
PORTFOLIO THEORY
(Asset Pricing and Portfolio Theory)
Contents


Principal of diversification
Introduction to portfolio theory (the Markowitz
approach) – mean-variance approach
– Combining risky assets – the efficient frontier
– Combining (a bundle of) risky assets and the risk free rate
– transformation line
– Capital market line (best transformation line)
– Security market line

Alternative (mathematical) way to obtain the MV
results
– Two fund theorem
– One fund theorem
Introduction


How should we divide our wealth ? – say £100
Two questions :



Between different risky assets (s’s > 0)
Adding the risk free rate (s = 0)
Principle of insurance is based on concept of
‘diversification’
 pooling of uncorrelated events
 insurance premium relative small proportion of
the value of the items (i.e. cars, building)
Assumption : MeanVariance Model

Investors :
prefer a higher expected return to lower returns

ERA ≥ ERB
Dislike risk

var(RA) ≤ var(RB) or SD(RA) ≤ SD(RB)
Covariance and correlation :
Cov(RA, RB) = r SD(RA) SD(RB)
Portfolio : Expected
Return and Variance
Formulas (2 asset case) :
Expected portfolio return :
ERp = wA ERA + wb ERB
Variance of portfolio return :
var(Rp) = wA2 var(RA) + wB2 var(RB) + 2wAwBCov(RA,RB)
Matrix notation :
Expected portfolio return :
Variance of portfolio return :
where
ERp = w’ERi
var(Rp) = w’Sw
w is (nx1) vector of weights
ERi is (nx1) vector of expected returns of individual assets
S is (nxn) variance covariance matrix
Minimum Variance
‘Efficient’ Portfolio




2 asset case : wA + wB = 1 or wB = 1 – wA
var(Rp) = wA2 sA2 + wB2 sB2 + 2wA wB rsAsB
var(Rp) = wA2 sA2 + (1-wA)2 sB2 + 2wA (1-wA) rsAsB
To minimise the portfolio variance : Differentiating
with respect to wA
∂sp2/∂wA = 2wAsA2 – 2(1-wA)sB2 + 2(1-2wA)rsAsB =
0
Solving the equation :
wA = [sB2 – rsAsB] / [sA2 + sB2 – 2rsAsB]
= (sB2 – sAB) / (sA2 + sB2 – 2sAB)
Power of Diversification


As the number of assets (n) in the portfolio
increases, the SD (total riskiness) falls
Assumption :
– All assets have the same variance : si2 = s2
– All assets have the same covariance : sij = rs2
– Invest equally in each asset (i.e. 1/n)
Power of Diversification
(Cont.)

General formula for calculating the portfolio
variance
s2p = S wi2 si2 + SS wiwj sij

Formula with assumptions imposed
s2p = (1/n) s2 + ((n-1)/n) rs2
If n is large (1/n) is small and ((n-1)/n) is close to 1.
Hence : s2p  rs2
Portfolio risk is ‘covariance risk’.
Standard deviation
Random Selection of
Stocks
Diversifiable /
idiosyncratic risk
C
Market / non-diversifiable risk
0
1
2 ...
20
40
No. of shares in portfolio
Example : 2 Risky Assets
Equity 1
Equity 2
Mean
8.75%
21.25%
SD
10.83%
19.80%
Correlation
-0.9549
Covariance
-204.688
Example : Portfolio Risk
and Return
Share of wealth
in
w1
w2
Portfolio
ERp
sp
1
1
0
8.75%
10.83%
2
0.75
0.25
11.88%
3.70%
3
0.5
0.5
15%
5%
4
0
1
21.25%
19.80%
Example : Efficient
Frontier
25
0, 1
Expected return (%)
20
0.5, 0.5
15
1, 0
10
0.75, 0.25
5
0
0
5
10
15
Standard deviation
20
25
Efficient and Inefficient
Portfolios
ERp
A
U
mp* = 10%
x
L
mp** = 9%
B
x
x
sp**
x
x
x
P1
x
x
x
x
sp*
x
x
x
P1
x
x
x
x
C
sp
Risk Reduction Through
Diversification
Y
Expected return
r = -0.5
r = -1
r = +1
B
A
r = 0.5
Z
C
r=0
X
Std. dev.
Introducing Borrowing and
Lending : Risk Free Asset

Stage 2 of the investment process :
– You are now allowed to borrow and lend at the
risk free rate r while still investing in any SINGLE
‘risky bundle’ on the efficient frontier.
– For each SINGLE risky bundle, this gives a new
set of risk return combination known as the
‘transformation line’.
– Rather remarkably the risk-return combination
you are faced with is a straight line (for each
single risky bundle) - transformation line.
– You can be anywhere you like on this line.
Example : 1 ‘Bundle’ of Risky
Assets + Risk Free Rate
Returns
T-Bill
(safe)
Equity
(Risky)
Mean
10%
22.5%
SD
0%
24.87%
‘Portfolio’ of Risky Assets
and the Risk Free Asset

Expected return
ERN = (1 – x)rf + xERp

Riskiness
s2N = x2s2p
or
sN = xsp
where
x = proportion invested in the portfolio of risky assets
ERp = expected return on the portfolio containing only risky assets
sp = standard deviation of the portfolio of risky assets
ERN = expected return of new portfolio (including the risk free asset)
sN = standard deviation of new portfolio
Example : New Portfolio
With Risk Free Asset
Share of wealth
in
(1-x)
x
Portfolio
ERN
sN
1
1
0
10%
0%
2
0.5
0.5
16.25%
12.44%
3
0
1
22.5%
24.87%
4
-0.5
1.5
28.75%
37.31%
Example : Transformation
Line
35
30
0.5 lending +
0.5 in 1 risky bundle
25
20
No borrowing/
no lending
15
10
5
-0.5 borrowing +
1.5 in 1 risky bundle
All lending
0
0
5
10
15
20
25
Standard deviation (Risk)
30
35
40
Transformation Line
Expected Return, mN
Borrowing/
leverage
Z
Lending
r
Q
L
X
all wealth in risky asset
all wealth in
risk-free asset
sX
Standard Deviation, sN
The CML – Best
Transformation Line
Transformation line 3
– best possible one
ERp
Portfolio M
Transformation line 2
Transformation line 1
rf
Portfolio A
sp
The Capital Market Line
(CML)
Expected return
CML
Market Portfolio
Risk Premium / Equity Premium
(ERm – rf)
rf
20
Std. dev., si
The Security Market Line
(SML)
Expected return
SML
Market Portfolio
Risk Premium / Equity Premium
(ERi – rf)
rf
0.5
1
1.2
The larger is bi, the larger is ERi
Beta, bi
Risk Adjusted Rate of
Return Measures

Sharpe Ratio :
Treynor Ratio :

Jensen’s alpha :

SRi = (ERi – rf ) / si
TRi = (ERi – rf ) / bi
(ERi – rf)t = ai + bi(ERm – rf)t + eit
Objective :
Maximise Sharpe ratio (or Treynor ratio, or
Jensen’s alpha)
Portfolio Choice
IB
ER
Z’
Capital Market Line
K
IA
ERm
A
Y
M
Q
ERm - r
r
a
L
sm
s
Math Approach
Solving Markowitz Using
Lagrange Multipliers

Problem : min ½(Swiwjsij)
Subject to
SwiERi = k (constant)
Swi = 1

Lagrange multiplier l and m
L = ½ Swiwjsij – l(SwiERi – k) – m(Swi – 1)
Solving Markowitz Using
Lagrange Multiplier (Cont.)


Differentiating L with respect to the
weights (i.e. w1 and w2) and setting the
equation equal to zero
For 2 variable case
s12w1 + s12w2 – lk1 – m = 0
s21w1 + s22w2 – lk2 – m = 0
The two equations can now be solved for the two
unknowns l and m.
 Together with the constraints we can now solve
for the weights.

The Two-Fund Theorem

Suppose we have two sets of weight : w1 and w2
(obtained from solving the Lagrangian), then
aw1 + (1-a)w2

for -∞< a < ∞ are also solutions and map out the
whole efficient frontier
Two fund theorem :
If there are two efficient portfolios, then any other efficient
portfolio can be constructed using those two.
One Fund Theorem


With risk free lending and borrowing is
introduced, the efficient set consists of a
single line.
One fund theorem :
There is a single fund M of risky assets, so that
any efficient portfolio can be constructed as a
combination of this fund and the risk free rate.
Mean = arf + (1-a)m
SD = asrf + (1-a)s
References


Cuthbertson, K. and Nitzsche, D. (2004)
‘Quantitative Financial Economics’,
Chapter 5
Cuthbertson, K. and Nitzsche, D. (2001)
‘Investments : Spot and Derivatives
Markets’, Chapters 10 and 18
END OF LECTURE