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Financial Economics (ECON 369)
- Read IS 6.1-6.4 Portfolio. In general, the portfolio has return rp and its expected value as well as variance:
rp = α 1 r 1 + α 2 r 2 + · · · + αnrn, E(rp) = E(α 1 r 1 + α 2 r 2 + · · · + αnrn) = α 1 E(r 1 ) + α 2 E(r 2 ) + · · · + αnE(rn) Var(rp) = Var(α 1 r 1 + α 2 r 2 + · · · + αnrn) = Var(α 1 r 1 ) + Var(α 2 r 2 ) + · · · + Var(αnrn) + Cov(α 1 r 1 , α 2 r 2 ) + Cov(α 1 r 1 , α 3 r 3 ) + · · · + Cov(α 1 r 1 , αnrn) + Cov(α 2 r 2 , α 1 r 1 ) + Cov(α 2 r 2 , α 3 r 3 ) + · · · + Cov(α 2 r 2 , αnrn) + · · · + Cov(αnrn, α 1 r 1 ) + Cov(αnrn, α 2 r 2 ) + · · · + Cov(αnrn, αn− 1 rn− 1 ) = α^21 Var(r 1 ) + α^22 Var(r 2 ) + · · · + α n^2 Var(rn) + α 1 α 2 Cov(r 1 , r 2 ) + α 1 α 3 Cov(r 1 , r 3 ) + · · · + α 1 αnCov(r 1 , rn) + α 2 α 1 Cov(r 2 , r 1 ) + α 2 α 3 Cov(r 2 , r 3 ) + · · · + α 2 αnCov(r 2 , rn) + · · · + αnα 1 Cov(rn, r 1 ) + αnα 2 Cov(rn, r 2 ) + · · · + αnαn− 1 Cov(rn, rn− 1 ) = α 1 α 1 Cov(r 1 , r 1 ) + α 1 α 2 Cov(r 1 , r 2 ) + · · · + α 1 αnCov(r 1 , rn) + α 2 α 1 Cov(r 2 , r 1 ) + α 2 α 2 Cov(r 2 , r 2 ) + · · · + α 2 αnCov(r 2 , rn) + · · · + αnα 1 Cov(rn, r 1 ) + αnα 2 Cov(rn, r 2 ) + · · · + αnαnCov(rn, rn).
Or we can use the matrix/vector notations. Notice that we can summarize the variance and covariance among the n assets in the following table (or matrix):
Cov(r 1 , r 1 ) Cov(r 1 , r 2 ) · · · Cov(r 1 , rn) Cov(r 2 , r 1 ) Cov(r 2 , r 2 ) · · · Cov(r 2 , rn) · · · · · · · · · · · · Cov(rn, r 1 ) Cov(rn, r 2 ) · · · Cov(rn, rn)
we use matrix (vectors) notation:
E(rp) =
α 1 α 2 · · · αn
E(r 1 ) E(r 2 ) · · · E(rn)
Var(rp) =
α 1 α 2 · · · αn
Cov(r 1 , r 1 ) Cov(r 1 , r 2 ) · · · Cov(r 1 , rn) Cov(r 2 , r 1 ) Cov(r 2 , r 2 ) · · · Cov(r 2 , rn) · · · · · · · · · · · · Cov(rn, r 1 ) Cov(rn, r 2 ) · · · Cov(rn, rn)
α 1 α 2 · · · αn
a) You are managing $1bn hedge fund, you plan to invest in one thousand different assets, and $1 million to each instrument. Assuming each asset has historical ¯rn = 7%, standard deviation 15%. What’s the portfolio’s expected return? What’s the portfolio variance and standard deviation when there is NO correlation between each pair of assets?
Answer: In this case, n = 1000 and α 1 = α 2 = · · · = α 1000 = 10001. Thus we have:
rp =
r 1 +
r 2 + · · · +
rn
=
(r 1 + r 2 + · · · + rn) ,
E(rp) =
E(r 1 + r 2 + · · · + rn)
=
1000¯rn
= 7%.
And since there is NO correlation between each pair of assets, the variance of the portfolio should be:
Var(rp) = α^21 Var(r 1 ) + α^22 Var(r 2 ) + · · · + α^2 nVar(rn) = 1000Var(ri) 10002
20%^2
b) You are planning to invest in n different assets with equal weights ( (^1) n ). Assuming each asset has historical ¯rn = ¯r, standard deviation σ. What’s the portfolio expected return? What’s the portfolio variance and standard deviation when there is NO correlation between each pair of asset?
Answer: Notice that we can summarize the variance and covariance among the n assets in the following table (or matrix):
Cov(r 1 , r 1 ) Cov(r 1 , r 2 ) · · · Cov(r 1 , rn) Cov(r 2 , r 1 ) Cov(r 2 , r 2 ) · · · Cov(r 2 , rn) · · · · · · · · · · · · Cov(rn, r 1 ) Cov(rn, r 2 ) · · · Cov(rn, rn)
σ^2 0 · · · 0 0 σ^2 · · · 0 · · · · · · · · · · · · 0 0 0 σ^2
and there is NO correlation among different assets:
E(rp) = α 1 E(r 1 ) + α 2 E(r 2 ) + · · · + αnE(rn) =
n nr¯ = ¯r.
Var(rp) =
n
[Var(r 1 ) + Var(r 2 ) + · · · + Var(rn)] =
nVar(ri) n^2
σ^2 n
To find out the optimal α, we need to solve the above equation:
2 ασ A^2 − 2(1 − α)σ^2 B + 2(1 − 2 α)σAσB ρAB = 0.
Thus we need to solve:
ασ^2 A − (1 − α)σ B^2 + (1 − 2 α)σAσB ρAB = 0,
or
α
σ^2 A + σ^2 B − 2 σAσB ρAB
= σ B^2 − σAσB ρAB.
Hence the optimal share in assets A and B are:
α∗^ =
σ^2 B − σAσB ρAB σ^2 A + σ B^2 − 2 σAσB ρAB
1 − α∗^ = σ^2 A − σAσB ρAB σ^2 A + σ B^2 − 2 σAσB ρAB
Insert the number, and we have α∗^ ≈ 0 .7617, and 1 − α∗^ ≈ 0 .2383. b) What is the value of this minimum standard deviation?
The minimized variance is:
L(α∗) = (α∗)^2 σ A^2 + (1 − α∗)^2 σ B^2 + 2α∗(1 − α∗)σAσB ρAB ≈ 0. 00918.
Thus the minimized standard deviation is about 0.0958 or 9.58%. c) What is the expected return of this portfolio. Answer:
E(rp) = α∗E(rA) + (1 − α∗)E(rB ) ≈ 15 .95%.
- The covariance between assets A and B is denoted by σAB , and other notations are given below.
Table 2: Two Correlated Assets
Asset ¯r σ A ¯rA σA B r¯B σB
Notice that we can also present the variance-covariance information using the help of the following table (matrix): ( Cov(rA, rA) Cov(rA, rB ) Cov(rB , rA) Cov(rB , rB )
σ^2 A σAB σBA σ B^2
a) Find the proportions α of A and (1 − α) of B that define a portfolio of A and B having minimum standard deviation.
Answer: Define the variance of the portfolio as a function of α:
L(α) = Var(rp) = α^2 Var(rA) + (1 − α)^2 Var(rB ) + 2α(1 − α)Cov(rA, rB )
= α^2 σ^2 A + (1 − α)^2 σ^2 B + 2α(1 − α)σAB.
The optimization problem can be written parsimoniously as:
min α L(α).
The optimal condition to this problem is:
L′(α) = 0.
To find out the optimal α, we need to solve the above equation:
2 ασ A^2 − 2(1 − α)σ^2 B + 2(1 − 2 α)σAB = 0.
Thus we need to solve:
ασ^2 A − (1 − α)σ^2 B + (1 − 2 α)σAB = 0,
or
α
σ^2 A + σ B^2 − 2 σAB
= σ B^2 − σAB.
Hence the optimal share in assets A and B are:
α∗^ =
σ^2 B − σAB σ A^2 + σ^2 B − 2 σAB
1 − α∗^ =
σ^2 A − σAB σ A^2 + σ^2 B − 2 σAB
b) What is the expected return of this portfolio with the smallest possible standard deviation?
Answer:
E(rp) = α∗E(rA) + (1 − α∗)E(rB )
= σ^2 B − σAB σ^2 A + σ B^2 − 2 σAB
r ¯A + σ^2 A − σAB σ^2 A + σ B^2 − 2 σAB
r ¯B.
isolate the α from the other terms:
α∗^ =
r¯ 1 − ¯r 2 A[σ^21 + σ 22 − 2 σ 12 ]
σ 22 − σ 12 σ 12 + σ 22 − 2 σ 12
=
r¯ 1 − r¯ 2 A[σ^21 + σ 22 − 2 ρ 12 σ 1 σ 2 ]
σ^22 − ρ 12 σ 1 σ 2 σ 12 + σ^22 − 2 ρ 12 σ 1 σ 2
1 − α∗^ =
r¯ 2 − ¯r 1 A[σ^21 + σ 22 − 2 σ 12 ]
σ 12 − σ 12 σ 12 + σ 22 − 2 ρ 12 σ 1 σ 2
= r¯ 2 − r¯ 1 A[σ^21 + σ 22 − 2 ρ 12 σ 1 σ 2 ]
σ^21 − σ 12 σ 12 + σ^22 − 2 ρ 12 σ 1 σ 2
- a) Two securities are available. The covariance between securities 1 and 2 is denoted by σ 12 , variance being σ^21 and σ^22 , respectively. We can also present the variance-covariance information using the following table (matrix): ( Cov(r 1 , r 1 ) Cov(r 1 , r 2 ) Cov(r 2 , r 1 ) Cov(r 2 , r 2 )
σ 12 σ 12 σ 21 σ^22
Notice that σ 12 = σ 21. The expected rates of return are ¯r 1 and ¯r 2. What percentages of total investment should be invested in each of the two securities to maximize the mean-variance utility of the following?
U (rp, σ^2 p ) = rp −
Aσ^2 p.
Answer: The expected return and variance of the portfolio are calculated below, when α is the weight invested in security 1 (and hence 1 − α in security 2):
rp = E(rp) = αE(r 1 ) + (1 − α)E(r 2 ) = α¯r 1 + (1 − α)¯r 2 ; σ^2 p = Var(rp) = α^2 Var(r 1 ) + (1 − α)^2 Var(r 2 ) + 2α(1 − α)Cov(r 1 , r 2 ) = α^2 σ^21 + (1 − α)^2 σ 22 + 2α(1 − α)σ 12.
The utility function is thus a function of the only endogenous variable, α:
J(α) = U (rp, σ^2 p ) = rp −
Aσ^2 p
= [α¯r 1 + (1 − α)¯r 2 ] −
A
[
α^2 σ 12 + (1 − α)^2 σ^22 + 2α(1 − α)σ 12
]
And the utility maximization problem can be presented succinctly as:
max α J(α).
The optimal condition to this problem is:
J′(α) = 0.
More specifically, we have the optimal condition as:
¯r 1 − r¯ 2 − A
[
ασ^21 − (1 − α)σ^22 + (1 − 2 α)σ 12
]
To find the optimal share invested security 1 and 2, α∗ and 1 − α∗, we isolate the α from the other terms:
α∗^ = r¯ 1 − r¯ 2 A(σ 12 + σ^22 − 2 σ 12 )
σ 22 − σ 12 σ 12 + σ^22 − 2 σ 12
1 − α∗^ =
¯r 2 − ¯r 1 A(σ^21 + σ^22 − 2 σ 12 )
σ^21 − σ 12 σ^21 + σ^22 − 2 σ 12
b) Suppose σ 1 = 0.15, σ 2 = 0.35, σ 12 = 0.02625, ¯r 1 = 15% and ¯r 2 = 25%. The risk aversion coefficient, A, is equal to 4. What percentages of total investment should be invested in security 1 and 2? Notice that the variance-covariance table/matrix can be reported as following:
( σ^21 σ 12 σ 21 σ^22
Answer: To apply the results from part a):
α∗^ =
r¯ 1 − ¯r 2 A(σ^21 + σ 22 − 2 σ 12 )
σ 22 − σ 12 σ^21 + σ 22 − 2 σ 12
=
1 − α∗^ =
r¯ 2 − ¯r 1 A(σ^21 + σ 22 − 2 σ 12 )
σ 12 − σ 12 σ^21 + σ 22 − 2 σ 12
=
