Mathematical Finance II (6CCM338A) winter 2026
Homework Problems 6CCM338A- Week 7
On the Keats page “Tutorial 7: Suggestion” section, please indicate two questions from below which
you would like to be covered in the next tutorial (please finish this by Friday night, the week before).
Problem 1. Let us assume two portfolios V and V ′ (both consisting of two assets A and B) are on
the minimum variance line (MVL). We further know that the weight vector for portfolios V and V ′
are respectively wV = ( 21 , 12 ) and wV ′ = ( 34 , 14 ). Show that for all the portfolios below either belongs
to MVL or does not belong to MVL:
1. w = ( 31 , 23 )
2. w = ( 31 , 43 )
3. w = ( 87 , 38 )
4. w = ( 85 , 38 )
Hint: Use the two-fund theorem. We will learn this the first thing in the next class but since you
have the quiz in the following week, and the two fund theorem is a very important topic I am adding
it to this weeks HW.
Example solution. Firstly observe that for case 2 and case 3 the weights do not sum up to 1. So
they are not valid portfolios. For the other two case: w = ( 58 , 83 ) and w = ( 13 , 23 ) belongs to MVL. I
will do the calculation for w = ( 13 , 23 ).
What we are looking for is if there exists α ∈ R such that
1 2 = αwV + (1 − α)wV ′
3 3
⇐⇒ 1 2 =α 1 1 + (1 − α) 3 1
3 3 2 2 4 4
(
1
= α 12 + (1 − α) 34
⇐⇒ 32
3
= α 12 + (1 − α) 14
This leads to a unique solution for α. So indeed this belongs to MVL.
Problem 2. Under the quadratic performance criteria; justify why the following pictures can not
be pictures of the indifference curves.
Example solution. a. There are two portfolio on the indifference curve which has the same ex-
pected return but one has a strictly lower risk than the other. So one portfolio is clearly
preferable to the other, and hence both can not belong to the same indifference curve.
b. For c1 < c2 , the indifference curve µV − γ2 σV2 = c1 represents portfolios which has lower perfor-
mance criteria than the indifference curve µV − γ2 σV2 = c2 . The picture shows they intersect.
So the portfolio corresponding to the intersection point (call it P ), has both performance c1
and c2 , which is a contradiction.
, c. The indifference curve in σ − µ plane can not be decreasing, as this will imply we will have
two portfolios A and B in the same indifference curve which dominates each other. This is a
contradiction to the definition of the indifference curve.
d. If you draw the different indifference curves changing the values of γ (keeping everything else
to be fixed), the curves are monotone wrt gamma in the σ − µ plane, which is not the case in
the picture.
Problem 3. (difficult) Which of the following portfolios can not be on the Markowitz efficient
frontier? Explain in details.
Portfolio Expected Return (µ = E(K)) sd (σ)
Q 10% 15%
R 10.5% 16.5%
S 11.5% 18.5%
Example solution. The portfolio R can not be on the Markowitz efficient frontier. It is dominated
by a weighted portfolio V of Q and S, with weights 23 on Q and 13 on S. Indeed:
2 1
E[KV ] = E[KQ ] + E[KS ] = 10.5% = E[KR ]
3 3
Homework Problems 6CCM338A- Week 7
On the Keats page “Tutorial 7: Suggestion” section, please indicate two questions from below which
you would like to be covered in the next tutorial (please finish this by Friday night, the week before).
Problem 1. Let us assume two portfolios V and V ′ (both consisting of two assets A and B) are on
the minimum variance line (MVL). We further know that the weight vector for portfolios V and V ′
are respectively wV = ( 21 , 12 ) and wV ′ = ( 34 , 14 ). Show that for all the portfolios below either belongs
to MVL or does not belong to MVL:
1. w = ( 31 , 23 )
2. w = ( 31 , 43 )
3. w = ( 87 , 38 )
4. w = ( 85 , 38 )
Hint: Use the two-fund theorem. We will learn this the first thing in the next class but since you
have the quiz in the following week, and the two fund theorem is a very important topic I am adding
it to this weeks HW.
Example solution. Firstly observe that for case 2 and case 3 the weights do not sum up to 1. So
they are not valid portfolios. For the other two case: w = ( 58 , 83 ) and w = ( 13 , 23 ) belongs to MVL. I
will do the calculation for w = ( 13 , 23 ).
What we are looking for is if there exists α ∈ R such that
1 2 = αwV + (1 − α)wV ′
3 3
⇐⇒ 1 2 =α 1 1 + (1 − α) 3 1
3 3 2 2 4 4
(
1
= α 12 + (1 − α) 34
⇐⇒ 32
3
= α 12 + (1 − α) 14
This leads to a unique solution for α. So indeed this belongs to MVL.
Problem 2. Under the quadratic performance criteria; justify why the following pictures can not
be pictures of the indifference curves.
Example solution. a. There are two portfolio on the indifference curve which has the same ex-
pected return but one has a strictly lower risk than the other. So one portfolio is clearly
preferable to the other, and hence both can not belong to the same indifference curve.
b. For c1 < c2 , the indifference curve µV − γ2 σV2 = c1 represents portfolios which has lower perfor-
mance criteria than the indifference curve µV − γ2 σV2 = c2 . The picture shows they intersect.
So the portfolio corresponding to the intersection point (call it P ), has both performance c1
and c2 , which is a contradiction.
, c. The indifference curve in σ − µ plane can not be decreasing, as this will imply we will have
two portfolios A and B in the same indifference curve which dominates each other. This is a
contradiction to the definition of the indifference curve.
d. If you draw the different indifference curves changing the values of γ (keeping everything else
to be fixed), the curves are monotone wrt gamma in the σ − µ plane, which is not the case in
the picture.
Problem 3. (difficult) Which of the following portfolios can not be on the Markowitz efficient
frontier? Explain in details.
Portfolio Expected Return (µ = E(K)) sd (σ)
Q 10% 15%
R 10.5% 16.5%
S 11.5% 18.5%
Example solution. The portfolio R can not be on the Markowitz efficient frontier. It is dominated
by a weighted portfolio V of Q and S, with weights 23 on Q and 13 on S. Indeed:
2 1
E[KV ] = E[KQ ] + E[KS ] = 10.5% = E[KR ]
3 3
