Problem Set 1
Asset Pricing 1
Stefanie Schraeder
31 October 2022
Notes for the solution - No sample solution!
Important Notes:
• The solutions to the problem set have to be HANDWRITTEN and READABLE.
• Solutions have to be uploaded on Moodle before Monday 7 November 2022
8am.
• Solutions have to be developed by each student individually!
• Formulas, which were not derived in the lecture, cannot be used without
derivation.
• If you are stuck at some point, please make a reasonable assumption and
continue with the rest of the exercise.
• Round your results to four digits after the decimal point, percentages to
two digits after the decimal point.
Exercise:
Consider
thecase of three assets (A,B, and C). The corresponding
expected returns are


0.07
1 0.5 0.4


1 0.5 
µ=
 0.10  and the correlation matrix is Π =  0.5
.
0.06
0.4 0.5
1
The return variances of the asset returns are σA2 = 0.04, σB2 = 0.09, σC2 = 0.04. The risk-free
rate is rf = 0.02.
a) Determine the covariance matrix.
0.04 0.03 0.016


Σ =  0.03 0.09 0.03 
0.016 0.03 0.04


b) For a required expected return µ̄ = 0.09, determine the portfolio return volatility as
1
a square root of a polynomial function of λC . (For this subquestion, do not use matrix
notation)
λA = 1 − λB − λC
0.09 = (1 − λB − λC ) · 0.07 + λB · 0.10 + λC · 0.06
0.02 = 0.03λB − 0.01λC
λB = 32 + 31 λC
λA = 1 − λB − λC = 1 − 23 − 31 λC − λC = 13 − 43 λC
σP2 F = 0.04 ·
1
3
− 43 λC
2
+ 0.09 ·
2
3
+ 31 λC
2
+ 0.04 · λ2C + 2 · 0.03 ·
1
3
− 43 λC ·
2
3
+ 13 λC
+2 · 0.016 · 13 − 34 λC λC + 2 · 0.03 · 32 + 13 λC · λC
[4 · (1 − 8λC + 16λ2C ) + 9 · (4 + 4λC + λ2C ) + 36 · λ2C + 6 · (2 − 8λC + λC − 4λ2C )
= 0.01
9
+3.2 · (3λC − 12λ2C ) + 6 · (6λC + 3λ2C )]
= 0.01
[(4 + 36 + 12) + (−32 + 36 − 48 + 6 + 9.6 + 36)λC + (64 + 9 + 36 − 24 − 38.4 + 18)λ2C ]
9
0.52
= 9 + 0.076
λC + 0.646
λ2C
9
9
2
= 0.0578 + 0.0084λC + 0.0718λ
qC
Thus, the volatility is σP F = 0.0578 + 0.0084λC + 0.0718λ2C
c) For a required expected return µ̄ = 0.09, determine the portfolio weights and the volatility
of the portfolio with the lowest return volatility. (For this subquestion, do not use matrix
notation)
Minimizing the volatility with respect to λC leads to the first order condition:
∂σP F
1
· (0.0084 + 0.14369λC ) = 0
= 0.5 √
2
∂λC
0.0578+0.0084λC +0.0718λC
λC = −0.0084
= −0.0585
0.14369
4
1
λA = 3 − 3 λC = 0.4113
λB = 23 + 31 λC = 0.6472
√
σP F = 0.0578 − 0.0084 · 0.0585 + 0.0718 · 0.05852 = 0.2399
d) Generalize your finding in c) for a general required expected return µ̄ and use it obtain
the volatility of the overall minimum variance portfolio. (For this subquestion, do not use
matrix notation)
λA = 1 − λB − λC
µ̄ = (1 − λB − λC ) · 0.07 + λB · 0.10 + λC · 0.06
µ̄ − 0.07 = 0.03 · λB − 0.01λC
λB = µ̄−0.07
+ 13 λC
0.03
µ̄−0.07
λA = 1 − 0.03 − 31 λC − λC = 0.10−µ̄
− 43 λC
0.03
σP2 F = 0.04 ·
+2 · 0.03 ·
0.10−µ̄
0.03
0.10−µ̄
−
0.03
µ̄−0.07
+
0.03
− 43 λC
2
4
λ
3 C
1
λ
3 C
2
·
+ 0.09 ·
µ̄−0.07
0.03
µ̄−0.07
0.03
+ 13 λC
2
+ 0.04 · λ2C
+ 13 λC + 2 · 0.016 ·
0.10−µ̄
0.03
− 43 λC λC
+2 · 0.03 ·
· λC
0.01
= 9 (400 + 40000µ̄ + 64λ2C − 8000µ̄ − 320λC + 3200µ̄λC + 90000µ̄2 + 441 + 9λ2C − 12600µ̄
+1800µ̄λC − 126λC + 36λ2C + 6000µ̄ − 420 + 60λC − 60000µ̄2 + 4200µ̄ − 600µ̄λC − 2400µ̄λC
+168λC − 24λ2C + 96λC − 960µ̄λC − 38.4λ2C + 1800µ̄λC − 126λC + 18λ2C )
2
= 0.01
· (421 + 70000µ̄2 + 64.6λ2C − 10400µ̄ − 248λC + 2840µ̄λC )
9
= 0.0718λ2C + λC · (−0.2756 + 3.1556µ̄) + 0.4678 + 77.7778µ̄2 − 11.5556µ̄
We determine the optimal portfolio weights, by differentiating with respect to λC . This leads
to the first order condition:
0.1436λC − 0.2756 + 3.1556µ̄ = 0
λC = 1.9192 − 21.9749µ̄
+ 13 (1.9192 − 21.9749µ̄) = −1.6936 + 26.0084µ̄
λB = µ̄−0.07
0.03
λA = 1 −B −λC = 1 − 1.9192 + 21.9749µ̄ + 1.6936 − 26.0084µ̄ = 0.7744 − 4.0335µ̄
The
resulting minimum volatility as a function of the required expected return is σP F =
h
0.04 · (0.7744 − 4.0335µ̄)2 + 0.09 · (−1.6936 + 26.0084µ̄)2 + 0.04 · (1.9192 − 21.9749)2
+0.06·(0.7744 − 4.0335µ̄)·(−1.6936 + 26.0084µ̄)+0.032·(0.7744 − 4.0335µ̄) (1.9192 − 21.9749µ̄)
+0.06 · (−1.6936 + 26.0084µ̄) · (1.9192 − 21.9749µ̄)]0.5
We want the overall minimum volatility portfolio. Thus, we differentiate with respect to µ̄
to find the minimum:
∂σP F
= 0.5 · σP1F (−0.2499 + 1.3015µ̄ − 7.9286 + 121.7586µ̄ − 3.3739 + 38.6317µ̄
∂ µ̄
+1.2085 + 0.4099 − 12.5886µ̄ − 0.5446 − 0.2477 + 5.6727µ̄ + 2.9949 + 2.2330
−68.5838µ̄ = 0
5.4984 = 86.1921µ̄
µ̄ = 0.0638
The corresponding portfolio weights are:
λC = 1.9192 − 21.9749 · 0.0638 = 0.5172
λB = −1.6936 + 26.0084 · 0.0638 = −0.0344
λA = 0.7744 − 4.0335 · 0.0638 = 0.5172
The corresponding volatility of the minimum variance portfolio is:
σM V P = (0.04 · 0.51722 − 0.06 · 0.5172 · 0.0344 + 0.032 · 0.51722
0.5
+0.09 · 0.03442 − 0.06 · 0.5172 · 0.0344 + 0.04 · 0.51722 ) = 0.1671
e) Agent 1 has the preference µP F − 0.5 · σP2 F . Calculate the optimal investment in the risky
assets.
maxλ (µ − 0.02)T λ − 21 λT Σλ
FOC: (µ − 0.02)T − λT Σ = 0
−1
λ1 = Σ
(µ − 0.02)

34.7222 −9.2593 −6.9444


Σ−1 =  −9.2593 17.2840 −9.2593 
−6.9444 −9.2593 34.7222


0.05


(µ − 0.02) =  0.08 
0.04

 



34.7222 −9.2593 −6.9444
0.05
0.7176
 



λ1 = 
 −9.2593 17.2840 −9.2593  ·  0.08  =  0.5494 
−6.9444 −9.2593 34.7222
0.04
0.3009
f) Determine agent 1’s optimal investment in the risk-free asset and determine the portfolio
weights of the tangency portfolio.
3
λ0,1 = 1 − 0.7176 − 0.5494
− 0.3009
=
−0.5679



0.7176
0.4577




1
1
λT = 1.5679 λ1 = 1.5679  0.5494  =  0.3504 
0.3009
0.1919
g) Provide the formula for the capital market line.
µT = λTT · µ = 0.0786
T
σT2 = λ
qT ΣλT = 0.0374
σT = σT2 = 0.1934
µ −r
µ = rf + TσT f σ = 0.02 +
0.0586
σ
0.1934
= 0.02 + 0.3030σ
h) Determine the risk-aversion coefficient b2 of the agent 2, who holds the tangency portfolio
exactly.
λ2 = λT =
1
λ.
b2 1
From f) we know λT =
1
λ.
1.5679 1
Thus, b2 = 1.5679
i) The price vector of the assets is p = (50, 100, 200)T . How many units of each asset does
investor 1 hold, if his/her total wealth is 1000 EUR.
λA,1 ·1000
= 717.6
= 14.3520
pA
50
λB,1 ·1000
549.4
θB,1 = pB
= 100 = 5.4940
λc,1 ·1000
θc,1 = pc
= 300.9
= 1.5045
200
θA,1 =
k) In total there are 4 investors in the economy, each having 1000 EUR wealth. The net
investment in the risk-free asset is 1000 EUR. Calculate the market capitalization of each of
the three risky assets.
The total investment in the risky assets is 4000 EUR - 1000 EUR = 3000 EUR.
CapA = 3000 · 0.4577 = 1373.0000
CapB = 3000 · 0.3504 = 1051.2000
CapC = 3000 · 0.1919 = 575.8000
4