Mathematical Finance II (6CCM338A) Winter 2026
Homework Problems 6CCM338A- Week 8
On the Keats page “Tutorial 8: 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. You have 3 stocks A, B, C and some cash where you invested 1000, 1500, 500, 1000
pounds respectively. The corresponding β’s are βA = 1.5, βB = 2, βC = 1. What is the risk-free
interest rate if the expected return of the market portfolio is x% and the expected return of your
portfolio is y%. Is there any condition on x, y?
Example solution. The β of my portfolio is:
Cov(KV , KM ) Cov(wA KA + wB KB + wC KC + wrisk−f ree Krisk−f ree , KM )
βV = M
=
σ σM
wA Cov(KA , KM ) + wB Cov(KB , KM ) + wC Cov(KC , KM ) + wrisk−f ree Cov(Krisk−f ree , KM )
= 2
σM
1 3 1
= wA βA + wB βB + wC βC + wrisk−f ree βrisk−f ree = 1.5 + 2 + 1 + 0
4 8 8
From the CAPM model,
µV − βV µM y − βV x
R= = .
1 − βV 1 − βV
y−βV x
Conditions: we want 0 ≤ 1−βV
.
Problem 2. Using the properties of the capital market line (CML) and the security market line
(SML), determine which of the following scenarios are consistent or inconsistent with the CAPM.
Justify your answers in details. Let A and B denote arbitrary securities while F and M represent
the riskfree asset and the market portfolio respectively.
Security E[R] β Security E[KV ] σ(KV )
(a) Scenario I: A 25% 0.8 (b) Scenario II: A 25% 30%
B 15% 1.2 M 15% 30%
Security E[KV ] σ(KV ) Security E[KV ] β
A 25% 55% A 20% 1.5
(c) Scenario III: (d) Scenario IV:
F 5% 0% F 5% 0
M 15% 30% M 15% 1
, Security E[KV ] β
(e) Scenario V: A 35% 2.0
M 15% 1.0
Example solution. (a) Inconsistent. Higher beta requires higher expected return.
(b) Inconsistent. Portfolio A lies above the CML. This would suggest that the market portfolio is
inefficient.
(c) Inconsistent. Portfolio A lies above the CML. This would suggest that the market portfolio is
inefficient.
(d) Consistent. Portfolio A does lie on the SML.
(e) Inconsistent. The implied risk-free rate would be negative if A lies on the SML.
Problem 3. The risk-free rate is 5%, the expected return on the market portfolio is 14%, and the
standard deviation of the return on the market portfolio is 25%. Consider a portfolio with expected
return of 16% and assume that it is on the efficient frontier.
(a) What is the beta of this portfolio?
(b) What is the standard deviation of its return?
(c) What is its correlation with the market return?
Example solution. (a) CAPM tells us:
µV = R + βV (µM − R)
V −R
Thus, βV = µµM −R
0.16−0.05
= 0.14−0.05 = 1.22.
(b) Portfolio P is on the efficient frontier. As we discussed in class, in presence of a risk-free asset
and under the CAPM assumption, the efficient frontier is the half-line starting from the risk-free
point and going through the market portfolio point. So the efficient frontier will satisfy:
.14 − .05 .25
µV = 0.05 + σV ⇐⇒ σV = µV − .05
.25 − 0.0 .09
So, σA = .11×.25
.09
≈ 30.5%.
(c) Obviously since A is in the efficient frontier and in this case this is a straight line with a positive
slope, the ρ has to be equal to 1. But we can prove this formally as well. From CML we have:
.11
.16 = .05 + βA (.14 − .05) ⇐⇒ βA = .
.09
Now, from the definition of βA we have
Cov(KA , KM ) ρA,M σA σM σM .11 .25
βA = 2
= 2
⇐⇒ ρA,M = βA = ( ) .11×.25 = 1.
σM σM σA .09 .09
Problem 4. Are the following statements true or false? Give brief but precise explanations for
your answers.
Homework Problems 6CCM338A- Week 8
On the Keats page “Tutorial 8: 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. You have 3 stocks A, B, C and some cash where you invested 1000, 1500, 500, 1000
pounds respectively. The corresponding β’s are βA = 1.5, βB = 2, βC = 1. What is the risk-free
interest rate if the expected return of the market portfolio is x% and the expected return of your
portfolio is y%. Is there any condition on x, y?
Example solution. The β of my portfolio is:
Cov(KV , KM ) Cov(wA KA + wB KB + wC KC + wrisk−f ree Krisk−f ree , KM )
βV = M
=
σ σM
wA Cov(KA , KM ) + wB Cov(KB , KM ) + wC Cov(KC , KM ) + wrisk−f ree Cov(Krisk−f ree , KM )
= 2
σM
1 3 1
= wA βA + wB βB + wC βC + wrisk−f ree βrisk−f ree = 1.5 + 2 + 1 + 0
4 8 8
From the CAPM model,
µV − βV µM y − βV x
R= = .
1 − βV 1 − βV
y−βV x
Conditions: we want 0 ≤ 1−βV
.
Problem 2. Using the properties of the capital market line (CML) and the security market line
(SML), determine which of the following scenarios are consistent or inconsistent with the CAPM.
Justify your answers in details. Let A and B denote arbitrary securities while F and M represent
the riskfree asset and the market portfolio respectively.
Security E[R] β Security E[KV ] σ(KV )
(a) Scenario I: A 25% 0.8 (b) Scenario II: A 25% 30%
B 15% 1.2 M 15% 30%
Security E[KV ] σ(KV ) Security E[KV ] β
A 25% 55% A 20% 1.5
(c) Scenario III: (d) Scenario IV:
F 5% 0% F 5% 0
M 15% 30% M 15% 1
, Security E[KV ] β
(e) Scenario V: A 35% 2.0
M 15% 1.0
Example solution. (a) Inconsistent. Higher beta requires higher expected return.
(b) Inconsistent. Portfolio A lies above the CML. This would suggest that the market portfolio is
inefficient.
(c) Inconsistent. Portfolio A lies above the CML. This would suggest that the market portfolio is
inefficient.
(d) Consistent. Portfolio A does lie on the SML.
(e) Inconsistent. The implied risk-free rate would be negative if A lies on the SML.
Problem 3. The risk-free rate is 5%, the expected return on the market portfolio is 14%, and the
standard deviation of the return on the market portfolio is 25%. Consider a portfolio with expected
return of 16% and assume that it is on the efficient frontier.
(a) What is the beta of this portfolio?
(b) What is the standard deviation of its return?
(c) What is its correlation with the market return?
Example solution. (a) CAPM tells us:
µV = R + βV (µM − R)
V −R
Thus, βV = µµM −R
0.16−0.05
= 0.14−0.05 = 1.22.
(b) Portfolio P is on the efficient frontier. As we discussed in class, in presence of a risk-free asset
and under the CAPM assumption, the efficient frontier is the half-line starting from the risk-free
point and going through the market portfolio point. So the efficient frontier will satisfy:
.14 − .05 .25
µV = 0.05 + σV ⇐⇒ σV = µV − .05
.25 − 0.0 .09
So, σA = .11×.25
.09
≈ 30.5%.
(c) Obviously since A is in the efficient frontier and in this case this is a straight line with a positive
slope, the ρ has to be equal to 1. But we can prove this formally as well. From CML we have:
.11
.16 = .05 + βA (.14 − .05) ⇐⇒ βA = .
.09
Now, from the definition of βA we have
Cov(KA , KM ) ρA,M σA σM σM .11 .25
βA = 2
= 2
⇐⇒ ρA,M = βA = ( ) .11×.25 = 1.
σM σM σA .09 .09
Problem 4. Are the following statements true or false? Give brief but precise explanations for
your answers.
