Econ 331 Financial Economics Tutorial 5 answers
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Econ 331 Financial Economics
Spring 2021
Tutorial 5 answers
Question 1
Consider the problem of portfolio choice with one risky asset (A) and one riskless asset (RF). Recall that in class we have shown that the expected return and standard deviation of the portfolio which allocated proportion wA to the risky asset are given by
rP = rRF + ç ÷sP
P A A (2)
Suppose that investor’s utility function is given by U(rP ,sP(2)) = rP - sP(2) , where higher value of g correspond to larger aversion to risk.
(a) Explain why equation (1) represents the constraint on the investor’s portfolio choice. Illustrate your answer on a graph.
This equation represents all possible portfolios one can construct with one risky and one riskless asset.
(b) Substitute constraint (1) into the utility function and solve the utility maximization problem by finding the value of the standard deviation, sP(*) , which maximizes investor’s utility. Then use
this solution to find the corresponding expected return, rP(*) .
After substituting for rP using equation (1) we have:
U(rP ,sP(2)) = rP - sP(2) = rRF + sP - sP(2)
The investor’s utility function is now a function of sP only. Take the derivative with respect to this variable we get
¶ U æ r - r ö
¶sP = çè A sA RF ø÷ -gsP = 0
* rA - rRF
s =
gsA
(c) Use (2) to find the optimal weight of asset A which corresponds to the portfolio found in (b).
Using equation (2) we get
*
* sP rA - rRF
w = =
sA gsA
The share of risky assets decreases if the investor is more risk averse, that is if g is larger, or if the risky asset has a return with larger variance, sA(2) . On the other hand, it increases when the risk premium on the risky asset, rA - rRF , increases.
Question 2
A fund manager can invest in any combination of the three assets with the following expected returns and standard deviations:
Asset 1: r1 = 4%, s1 = 4%
Asset 2: r2 = 8%, s2 = 8%
Asset 3: r3 = 6%, s3 = 6%
All three assets returns are uncorrelated: p12 = p13 = p23 = 0 . Suppose the manager was instructed by investors to construct a portfolio with an expected return of 6% and the lowest risk possible.
Initially he considers investing all his funds in asset 3, which has an expected return of 6% and a standard deviation of 6%. But, having studied portfolio theory, he realizes that he could reduce his risk by combining all three assets in a single portfolio while still ensuring a 6% expected return. In particular he sets out to solve the following problem:
Minimize portfolio variance
sP(2) = å i(3)=1 wi(2)si2 + åi(3)=1 åj(3)¹i wi wj cov(ri , rj )
subject to the following constraints
(i) Portfolio weights should sum to one: w1 + w2 + w3 = 1
(ii) The expected return on the portfolio should be equal to 6: wr + w r + w r = 6
Show how one can solve this problem by completing the following steps:
(a) Simplify the formula for the portfolio variance using the fact that all returns are uncorrelated;
Because the returns are uncorrected, all covariance terms are equal to zero:
sP(2) = åi(3)=1 2wisi2 = w1(2)s1(2) + w2(2)s2(2) + w3(2)s3(2)
s2 = 16w2 + 64w2 + 36w2
(b) Combine constraints (i) and (ii) into one constraint using the fact that w3 = 1 - w1 - w2 ;
wr + w r + w r = 6
w1r1 + w2r2 +(1 - w1 - w2 )r3 = 6
4w1 + 8w2 +6(1 - w1 - w2 ) = 6
2w - 2w = 0
(c) Set up and solve the constrained optimization problem of choosing portfolio weights to minimize the variance found in (a) subject to the constraint derived in (b) and compute the resulting portfolio variance;
Minimize sP(2) = 16w1(2) + 64w2(2) + 36(1 - w1 - w2 )2
Subject to 2w1 - 2w2 = 0
Construct the Lagrangian function:
L(w1 , w2 , 入) = 16w1(2) + 64w2(2) + 36(1 - w1 - w2 )2 + 入(0 - (2w1 - 2w2 ))
The first order conditions for minimization are given by
(i) 32w1 - 72(1 - w1 - w2 ) - 2入= 0
(ii) 128w2 - 72(1 - w1 - w2 )+ 2入= 0
(iii) 2w1 - 2w2 = 0
Now note that (iii) implies w1(*) = w2(*) . Substituting this into (i) and (ii) we get
(i) 32w1(*) - 72(1 - 2w1(*)) = 2入
(ii) 128w1(*) - 72(1 - 2w1(*)) = -2入
(i) 16w1(*) - 36(1 - 2w1(*)) = 入
(ii) -64w1(*) + 36(1 - 2w1(*)) = 入
Combining these two equations we get
16w1(*) - 36(1 - 2w1(*)) = -64w1(*) + 36(1 - 2w1(*))
80w1(*) = 72(1 - 2w1(*))
80w* = 72 - 144w*
224w* = 72
w* = w* = 0.32
w* = 1 - w* - w* = 0.36
(d) Find the standard deviation of the optimal portfolio and verify that it is indeed lower than the standard deviation of the return on Asset 3. What is the gain from diversification in this case?
sP = 16(0.32)2 +64(0.32)2 +36(0.36)2 = 3.6
Holding asset 3 alone provides an expected return of 6% with a standard deviation of 6%. By combining all three assets into an optimally chosen portfolio we can obtain the same return of 6% with a much lower risk, given by the standard deviation of 3.6%.
2022-09-05