Problem Set: Asset Pricing and Portfolio Allocation - ForEcoStudies

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Problem Set: Asset Pricing and
Portfolio Allocation
Author: Michael Donadelli
E-Mail: michael.donadelli@gmail.com
Date of Creation: 15/12/2011
Date of Update: 02/03/2012
Field: Finance
Keywords: CCAPM, Mean-Variance Ecient
Frontier
Additional Info: MoSec: Problem Set (Financial
Economics Courses)

Problem Set III
Exercise 2.3
Consider a two period economy with I agents and J risky assets, and one risk-free asset. Agent’s
wealth is defined as follows:
⎛ ⎞
Ỹ i =
⎝Y i
0 − ∑ j
x i j
⎠ (1 + r f ) + ∑ j
x i j (1 + ˜r j )
Y0 i = agent i’ initial wealth
r f = riskfree interest rate
˜r j = random rate of return on the risky asset j
x i j = amount invested in asset j by agent i
.
The individual’s choice problem is,
max x i
j
E[U i (Ỹi)]
Let’s now substitute Ỹi in the utility function. We obtain:
⎡ ⎛⎛
⎞
max x i
j
E ⎣U i
⎝⎝Y 0 i − ∑ x i ⎠
j (1 + r f ) + ∑
j
j
⎞⎤
x i j (1 + ˜r j ) ⎠⎦
or
⎡ ⎛
max x i
j
E ⎣U i
⎝Y 0 i + Y0 i r f − ∑ j
x i j − ∑ j
x i jr f + ∑ j
x i j + ∑ j
⎞⎤
x i j ˜r j
⎠⎦
First Order Condition:
∂E[U i (Ỹi)]
∂x i j
= 0
⇒ E[U ′ (Ỹi)(−1 − r f + 1 + ˜r j )] = 0
⇒ E[U ′ (Ỹi)(˜r j − r f )] = 0
.
Re-arranging and using the properties of the covariance (E (XY ) = E (X) E (Y ) + cov (X, Y ))
we get:
E[U ′ (Ỹi)(˜r j − r f )] = E[U ′ (Ỹi)]E[(˜r j − r f )] + cov(U ′ (Ỹi), ˜r j )
⇒ E[U ′ (Ỹi)]E[(˜r j − r f )] + cov(U ′ (Ỹi), ˜r j ) = 0
⇒ E[U ′ (Ỹi)]E[(˜r j − r f )] = −cov(U ′ (Ỹi), ˜r j )
1

Applying Stein’s theorem cov[g(x), y] = E[g ′ (x), y)]cov[x, y] we can re-write the latter as follows:
cov(U ′ (Ỹi), ˜r j ) = E[U ′′ (Ỹi)] cov(Ỹi, ˜r j )
or
Let’s define RA i = −
⇒ E[U ′ (Ỹi)]E[(˜r j − r f )] = −E[U ′′ (Ỹi)] cov(Ỹi, ˜r j )
′′
E[U (Ỹi)]
E[U
E[(˜r j − r f )] = − E[U ′′ (Ỹi)]
E[U ′ (Ỹi)] cov(Ỹi, ˜r j )
′ (Ỹi)]
, then equation above can be written in the following way:
E[U ′ (Ỹi)]E[(˜r j − r f )] = RA i cov(Ỹi, ˜r j )
∀ i
.
Aggregating across the individuals we can show that E[U ′ (Ỹi)]E[(˜r j − r f )] = RA i cov(Ỹi, ˜r j )
can be rewritten as follows: 1
E[(˜r j − r f )] = ∑ i
RA i cov(Ỹi, ˜r j )
⇒ E[(˜r j − r f )] = ∑ i
cov(Ỹi, ˜r j RA i )
⇒ E[(˜r j − r f )] = cov( ∑ i
∑
Ỹ i , ˜r j RA i )
i
where Y MO (1 + ˜r M ), hence
∑
⇒ E[(˜r j − r f )] = cov(Y MO (1 + ˜r M ), ˜r j RA i )
∑
⇒ E[(˜r j − r f )] = Y MO cov((˜r M , ˜r j RA i )
∑
⇒ E[(˜r j − r f )] = Y MO RA i cov(˜r M , ˜r j )
i
i
i
1 Remark: cov(aX, bY ) = ab cov(X, Y ).
⇒ E[(˜r j − r f )] =
⇒ E[(˜r j − r f )] =
Y MO
∑ 1
i RAi cov(˜r M , ˜r j )
∑
i Ỹ i
(1+˜r M )
∑ 1
i RAi
2

Suppose ˜r M = ˜r j :
⇒ E[(˜r j − r f )] =
Y MO
∑ 1 cov(˜r M , ˜r M )
i RAi
⇒ E[(˜r M − r f )] =
Derive the Capital Asset Pricing Model: 2
• E[(˜r j − r f )] =
• E[(˜r M − r f )] =
Y MO
∑ 1
i RA i
Y MO
∑ 1
i RA i
cov(˜r M , ˜r j )
var(˜r M )
Y MO
∑ 1
i RAi var(˜r M )
and dividing the first equation over the second we end up with the following result:
E[(˜r j − r f )] =
E[(˜r M − r f )] =
Y MO
∑ 1
i RA i
Y MO
∑ 1
i RA i
cov(˜r M , ˜r j )
var(˜r M )
⇒
⇒ E[(˜r j − r f )]
E[(˜r M − r f )] = 1 · cov(˜r M , ˜r j )
var(˜r M )
.
E[(˜r j − r f )] = cov(˜r M , ˜r j )
var(˜r M ) E[(˜r M − r f )]
Exercise 2.4
E[mR i ] = 1 where R i is the gross return on asset i and m the stochastic discount factor.
• m t+1 ≡ δ U ′ (c t+1)
U ′ (c t)
→ stochastic discount factor.
• (1 + R i,t+1 ) → gross return on asset i.
Note that this fundamental equation holds also for the gross return on the risk-free asset and
on the market portfolio R M .
[
E[mR i ] ⇒ 1 = E t δ U ′ ]
(c t+1 )
U ′ (c t ) (1 + R i,t+1)
but this should also holds for R M ,
[
E[mR M ] ⇒ 1 = E t δ U ′ ]
(c t+1 )
U ′ (c t ) (1 + R M,t+1)
.
2 CAPM: E[(˜r j − r f )] = β E[(˜r M − r f )] where β = cov(˜r M ,˜r j )
var(˜r M ) .
3

.
.
Then, using the property of the covariance E(XY ) = E(X)E(Y ) + cov(X, Y ) we have: 3
[
1 = E t δ U ′ ] [
(c t+1 )
U ′ E t [(1 + R M,t+1 )] + cov t δ U ′ ]
(c t+1 )
(c t )
U ′ (c t ) , R M,t+1
Remark: from discount bond price (p f,t )
[
1
(1 + R f,t+1 ) = p f,t = E t δ U ′ ]
(c t+1 )
U ′ · 1
(c t )
[
Then substituting 1 = E t
[
]
p f,t = E t δ U ′ (c t+1)
U ′ (c t)
· 1
E t
[
δ U ′ (c t+1)
U ′ (c t)
δ U ′ (c t+1)
U ′ (c t)
]
E t [(1 + R M,t+1 )] + cov t
[
we obtain the following result:
]
E t [(1 + R M,t+1 )] + cov t
[
(1 + R f,t+1 )
δ U ′ (c t+1)
U ′ (c t)
, R M,t+1
Now, some algebra manipulations are required:
[
E t δ U ′ ] [
(c t+1 )
U ′ E t [(1 + R M,t+1 )] + cov t δ U ′ (c t+1 )
(c t )
U ′ (c t ) , R M,t+1
[ ]
and dividing both sides by E t δ U ′ (c t+1)
Remark: E t
[
U ′ (c t)
[
cov t
E t [(1 + R M,t+1 )] + [
E t
δ U ′ (c t+1)
U ′ (c t)
δ U ′ (c t+1)
U ′ (c t)
δ U ′ (c t+1)
U ′ (c t)
]
]
= E t
[
, R M,t+1
]
[
= E t δ U ′ ]
(c t+1 )
U ′ · 1
(c t )
δ U ′ (c t+1 )
U ′ (c t )
]
, R M,t+1
] = (1 + R f,t+1 )
[
cov t
⇒ E t [(1 + R M,t+1 )] − (1 + R f,t+1 ) = − [
E t
δ U ′ (c t+1)
U ′ (c t)
]
=
1
(1+R f,t+1 )
[
cov t
⇒ E t [(1 + R M,t+1 )] − (1 + R f,t+1 ) = −
δ U ′ (c t+1)
U ′ (c t)
δ U ′ (c t+1)
U ′ (c t)
δ U ′ (c t+1)
U ′ (c t)
1
(1+R f,t+1 )
, R M,t+1
]
, R M,t+1
[
⇒ E t [(R M,t+1 )] − R f,t+1 = −(1 + R f,t+1 )cov t δ U ′ ]
(c t+1 )
U ′ (c t ) , R M,t+1
]
]
in
1
(1+R f,t+1 ) =
]
(1 + R f,t+1 )
Exercise 2.5
To compute the weights, expected returns and standard deviation of the tangency portfolio we
use a dataset composed by 16 risky assets and 1 riskless security (1 month interbank interest rate).
Risky expected returns, riskfree returns, 4 risky indicators (Variance and Standard Deviation) and
a performance index for each asset are defined in table 1:
[ ]
3 In our example: E(X) ≡ E δ U ′ (c t+1 )
and E(Y ) ≡ E[(1 + R M,t+1 )].
U ′ (c t )
4 ITALY INTERBANK 1 MONTH - OFFERED RATE. Since the time series provides data on annual basis we
computed the equivalent monthly rate by applying the following formula: r 12 = (1 + r) 1 12 − 1. The Expected Return
is obtained as the mean of the equivalent monthly rates
4

Ep Var Sd Sharpe Ratio
RISKFREE 0.4% 0.0% 0.0%
ALLEANZA 0.5% 0.5% 6.9% 0.02
ANTONVENETA 1.0% 0.4% 6.6% 0.08
AUTOSTRADE 2.4% 0.4% 6.7% 0.30
ENI 1.3% 0.2% 4.3% 0.20
GENERALI 0.5% 0.3% 5.7% 0.01
INTESA 1.7% 1.2% 11.1% 0.12
MEDIASET 1.4% 0.9% 9.2% 0.11
MEDIOBANCA 1.0% 0.8% 8.8% 0.07
MEDIOLANUM 1.8% 1.4% 11.6% 0.12
POPNOVAR 0.3% 0.3% 5.6% -0.01
RAS 1.0% 0.5% 6.7% 0.09
SAIPEM 1.8% 0.5% 7.1% 0.19
SANPAOLO 1.0% 0.6% 7.7% 0.08
SNAM 1.0% 0.1% 2.6% 0.25
TELECOM 0.9% 1.6% 12.5% 0.04
UNICREDITO 1.6% 0.6% 7.6% 0.16
Table 1: Risky Assets and Riskfree Rate
Figure 1: Efficient Frontier (16 Risky assets)
Global Min-Var Portfolio
Expected Return 0.92%
Standard Deviation 1.58%
Table 2: Global Minumun Variance
5

First we compute the Efficient Frontier solving the usual contrained optimization problem under
the hypothesis to live in an economy composed by only risky assets. Figure 1 depicts the result
(efficient frontier) of such optimization problem.
Remark: minw w’Vw subject to w’1 = 1 and w’e = Ep.
Note also that Figure 1 shows the position of the global minimun variance portfolio (portfolio with
the smallest variance among all the portfolios on the efficient frontier).
Global Minimun Variance Portfolio Features:
Global Min-Var Portfolio
Stock
Weights
ALLEANZA -18%
ANTONVENETA 7%
AUTOSTRADE -8%
ENI 17%
GENERALI 35%
INTESA -6%
MEDIASET 1%
MEDIOBANCA -26%
MEDIOLANUM -7%
POPNOVAR 7%
RAS 6%
SAIPEM -5%
SANPAOLO -9%
SNAM 70%
TELECOM -2%
UNICREDITO 39%
Table 3: Global Minumun Variance Portfolio Weights
Note that our optimization problem (N risky assets economy) suggests to buy 70% of SNAM,
which is in line to what is illustrated in table 1 where SNAM (on historical basis) is the asset with
the smallest standard deviation (2.6%).
We next develop the same general optimization problem introducing one riskless asset.
Formally the problem: minw w’Vw subject to w’e + (1 − w’1)r f = Ep. Such a problem provides
us a new efficient frontier (straight line ≡ Capital Market Line) with intercept in Ep = r f
5 and tangent to the efficient frontier obtained by solving the problem for an economy composed
by only risky assets. The tangency point correspond to a portfolio composed by only risky assets
(riskfree = 0%), which belongs also to the efficient section of the frontier of the risky portfolios.
An important property of the Tangency Portfolio is that its Sharpe Ratio, (E T −r f )
σ T
, correspond to
the slope of the Capital Market Line.
Efficient Frontier in Figure 2 and results in Table 4 and Table 5 are obtained by solving the following
equations:
16 Risky Assets + 1 Riskless Security Portfolio
W p = V −1 (e − 1r f ) (E p − r f )
H
σ 2 p = 1 H (E p − r f ) 2
5 Basically in this point we are assuming to hold a portfolio with no risk.
6

E p = r f ± √ Hσ p
Tangency Portfolio
E[ r˜
T ] = r f +
H
1 ′ V −1 (e − 1r f )
√
H
σ T =
1 ′ V −1 (e − 1r f )
where H = (e − 1r f ) ′ V −1 (e − 1r f ).
W T = V−1 (e − 1r f )
1 ′ V −1 (e − 1r f )
Figure 2: Efficient Frontier (16 Risky Assets + 1 Riskless Security)
The tangency portfolio is the portfolio that maximises the Sharpe Ratio, (E T −r f )
σ T
our problem:
= √ H. In
or
4.31% − 0.4%
4.38%
= 0.89
√
H =
√
0.7984 = 0.89
.
The optimizer suggests to invest 96% in AUTOSTRADE and to underweighted ALLEANZA (-
144%) and POPNOVARA (-63%). Note that AUTOSTRADE is the asset with the highest Sharpe
Ratio (i.e. 0.3) and ALLEANZA, POPNOVARA are those assets with the lowest one (respectively
0.02 and -0.01). Basicallly the optimizer prefers those asset with the highest sharpe ratio, more
7

Figure 3: Portfolio Assets
Tangency Portfolio
Expected Return 4.31%
Standard Deviation 4.38%
Table 4: Tangency Portfolio
TANGENCY PORTFOLIO
STOCK
WEIGHTS
ALLEANZA -144%
ANTONVENETA 29%
AUTOSTRADE 96%
ENI 36%
GENERALI 26%
INTESA 26%
MEDIASET -14%
MEDIOBANCA -3%
MEDIOLANUM 41%
POPNOVAR -63%
RAS 70%
SAIPEM 21%
SANPAOLO -54%
SNAM -20%
TELECOM -1%
UNICREDITO 52%
RISKFREE 0%
Table 5: Tangency Portfolio Weights
8

Figure 4: Tangency Portfolio Weights (16 Risky Assets)
precisly those assets having sufficiently high returns and sufficiently low standard deviation (see
Figure 3). Note also that if we multiply our tangency portfolio weights by their expected return
we obtaint the expected return of the tangency portfolio. Formally: W ′ T · e = E[˜r T ].
Suppose one does not want to run optimization and decides to invest in an equally weighted
portfolio composed as follows: then, what is going to happen to our portfolio return ?
Stock
Weights
ALLEANZA 5.9%
ANTONVENETA 5.9%
AUTOSTRADE 5.9%
ENI 5.9%
GENERALI 5.9%
INTESA 5.9%
MEDIASET 5.9%
MEDIOBANCA 5.9%
MEDIOLANUM 5.9%
POPNOVAR 5.9%
RAS 5.9%
SAIPEM 5.9%
SANPAOLO 5.9%
SNAM 5.9%
TELECOM 5.9%
UNICREDITO 5.9%
RISKFREE 5.9%
100.0%
Table 6: Equally Weighted Portfolio
The return of the equally weighted portfolio is going to be equal to 1.16% [W ′ ewp · e = E(˜r ewp )],
far below the one obtained by the tangency portfolio. We would like to stress the fact that these
9

esults are obtained by the “past”. Then, in the future, assuming no re-allocation 6 it could be that
the second portfolio 7 may perform better than the one obtained through the optimization 8
6 Investors will not change the composition of the portfolio.
7 In our example the equally weighted portfolio.
8 In our example the tangency portfolio.
10