IF3200 Mock Exam paper (Indicative answers)

Question 1
(25 marks)

Assume that interest rates are compounded annually. Ignore transaction cost, credit risk and
accrued interest. No other bonds are actively traded in the market except for those mentioned
below.

A term structure ranging from 1 year to 10 years (i.e., d1, d2, …, d10) is estimated using 1-year,
3-year, 5-year, 7-year, 9-year strips. Suppose now we have a 3-year bond that pays cash flows
X1, X2, and X3 at the end of each year respectively. This 3-year bond is quoted in the market
with a price (P) and yield to maturity (y).

Based on the above information, we construct a variable V:

V = X1d1 + X2d2 + X3d3,

Explain whether and why the following statements are true or false.

a) Both the bootstrapping approach and parametric approach can be applied to the estimation
of the term structure.
(10 marks)
Key points include:
• Bootstrapping approach is not feasible as the maturities from 1 year to 10 years
are not fully identified by the 5 traded bonds.
• Parametric approach such as the Nelson-Siegal model is appropriate given the
information. The 5 given bonds is sufficient to identify the 4 parameters in the NS
model.

b) It must be that V=P, otherwise there would exist arbitrage opportunities.
(15 marks)
Key points include:
• The term structure of discount factors can only be derived from parametric models
such as NS and therefore permits pricing errors.
• V is the theoretical value consistent with the term structure estimated in this way.
• However, it does not have to be exactly equal to the actual price observed in the
market.
• When P not equal to V, one cannot construct an arbitrage stategy using only the
bonds available (to hedge all relevant payment dates: 1-year, 2-year, and 3-year).
Note to the marker: please mark based on the overall quality of the student’s answer taking
into account factors such as whether the answer is concise and clear. There are no credits
for simply stating T/F per se.

Question 2
(25 marks)

Compute the dollar duration (DV01) for the following fixed-income instruments and explain

a) A perpetuity that pays £1,000/year. Suppose the term structure is flat at 1.00%, and interest
rates are annually compounded.
(10 marks)
P = C/r, where C=£1,000, r=1%, so P=£100,000
dP/dr = - C/r2
DV01 = - 0.01% x (dP/dr) = 0.01% x 1000 / (1%)2 = £1,000
Intuition: The present value of the perpetuity drops by £1,000 if the interest rate
increases by 1 basis point.

Note to the marker: Students may or may not have a minus sign in the definition of DV01.
This shouldn’t affect the marking as long as the overall calculation makes sense, and the
intuition is clearly and correctly phrased.

b) An investment of \$1m cash in a bank account for 1 year earning the 1-year LIBOR of
1.00%.
(15 marks)
T=0 deposit PV=\$1m cash, obtain \$1m x (1+1.00%) = \$1.01m at T=1

If 1 sec later, the interest rate increases by 1bp to 1.01%, the PV becomes:
\$1.01m / (1+1.01%) = \$1.01m / 1.0101 = \$999,901

Hence, the present value of the bank account drops by \$1,010,000 - \$999,901 = \$99,
in response to a 1bp interest rate increase.

Question 3
(25 marks)

Dynamic term structure models are useful tools for pricing interest rate derivatives. Answer the
following questions regarding the pricing of interest rate derivatives and dynamic term structure
models:

a) Suppose the 1-year LIBOR is governed by the risk-neutral binomial tree process (time t is
measured in years, and at any time t, LIBOR can move up or down with equal probability):

t=0 t=1 t=2 t=3

8%
7%
6% 6%
5% 5%
4% 4%
3%
2%

Based on the above information, calculate the present value (at time t=0) of a 3-year floorlet
on 1-year LIBOR at the strike of 5.5%. Assume notional amount of \$100.
(10 marks)

Denote F(rt , t) the value of the floorlet at time t when the 1-year LIBOR from t to t+1 is rt.

t=3:
F(8%,3) = [5.5% - 8%]+ = 0
F(6%,3) = [5.5% - 6%]+ = 0
F(4%,3) = [5.5% - 4%]+ = 1.5%
F(2%,3) = [5.5% - 2%]+ = 3.5%

t=2:
F(7%,2) = [ (1/2)xF(8%,3) + (1/2)xF(6%,3) ] / 1.07 = 0
F(5%,2) = [ (1/2)xF(6%,3) + (1/2)xF(4%,3) ] / 1.05 = 0.71%
F(3%,2) = [ (1/2)xF(4%,3) + (1/2)xF(2%,3) ] / 1.05 = 2.43%

t=1:
F(6%,1) = [ (1/2)xF(7%,2) + (1/2)xF(5%,2) ] / 1.06 = 0.33%
F(4%,1) = [ (1/2)xF(5%,2) + (1/2)xF(3%,2) ] / 1.06 = 1.51%

t=0:
F(5%,0) = [ (1/2)xF(6%,1) + (1/2)xF(4%,1) ] / 1.05 = 0.88%

Since the notional amount is \$100, the value of the floorlet is \$100 x 0.88% = \$0.88.

b) Discuss the weakness and limitations of using one-factor term structure models.
(15 marks)
The key points include:
• 1-factor models assumes that the term structure of interest rate is perfectly driven
by the dynamics of the short rate.
• Empirically, at least 3 factors (level, slope, curvature) are needed to account for the
yield curve dynamics.
• For example, short rate can also be driven by the slope of the yield curve.
• Theoretical price depends on the dynamic term structure model. Therefore, models
that are too simple to capture the important dynamic properties of the term structure
tend to fail in pricing.
Note to the marker: please mark based on the overall quality of the student’s answer
taking into account factors such as whether the answer is concise and clear.

Question 4
(25 marks)

Fixed income securities are in general not free from credit risk.

a) Discuss the main credit spread determinants.
(10 marks)

Key points include:
• firm-specific default cost
• systematic risk
• (il)liquidity of corporate bonds relative to treasury bonds
Note to the marker: Students should provide their own explanations for the above
components. Please mark based on the overall quality of the student’s answer taking into
account factors such as whether the answer is concise and clear.

b) “One of the disadvantages of securtization is that it increases the risk exposure of financial
institutions through the creation of a pool of underlying assets, leading to an inefficient risk
sharing mechanism.” Do you agree with this statement? Explain.
(15 marks)
Key points include:
• the portfolio of underlying assets improves diversification and is supposed to
reduce idiosyncratic risk.
• It is information asymmetry and incentive problems that lead to inefficient risk
sharing.
Note to the marker: Students should provide their own explanations based on the benefits
and costs of securitization. Please mark based on the overall quality of the student’s
answer taking into account factors such as whether the answer is concise and clear.

IF3200 Mock Exam paper (Indicative answers)

Question 1
(25 marks)

Assume that interest rates are compounded annually. Ignore transaction cost, credit risk and
accrued interest. No other bonds are actively traded in the market except for those mentioned
below.

A term structure ranging from 1 year to 10 years (i.e., d1, d2, …, d10) is estimated using 1-year,
3-year, 5-year, 7-year, 9-year strips. Suppose now we have a 3-year bond that pays cash flows
X1, X2, and X3 at the end of each year respectively. This 3-year bond is quoted in the market
with a price (P) and yield to maturity (y).

Based on the above information, we construct a variable V:

V = X1d1 + X2d2 + X3d3,

Explain whether and why the following statements are true or false.

a) Both the bootstrapping approach and parametric approach can be applied to the estimation
of the term structure.
(10 marks)
Key points include:
• Bootstrapping approach is not feasible as the maturities from 1 year to 10 years
are not fully identified by the 5 traded bonds.
• Parametric approach such as the Nelson-Siegal model is appropriate given the
information. The 5 given bonds is sufficient to identify the 4 parameters in the NS
model.

b) It must be that V=P, otherwise there would exist arbitrage opportunities.
(15 marks)
Key points include:
• The term structure of discount factors can only be derived from parametric models
such as NS and therefore permits pricing errors.
• V is the theoretical value consistent with the term structure estimated in this way.
• However, it does not have to be exactly equal to the actual price observed in the
market.
• When P not equal to V, one cannot construct an arbitrage stategy using only the
bonds available (to hedge all relevant payment dates: 1-year, 2-year, and 3-year).
Note to the marker: please mark based on the overall quality of the student’s answer taking
into account factors such as whether the answer is concise and clear. There are no credits
for simply stating T/F per se.

Question 2
(25 marks)

Compute the dollar duration (DV01) for the following fixed-income instruments and explain

a) A perpetuity that pays £1,000/year. Suppose the term structure is flat at 1.00%, and interest
rates are annually compounded.
(10 marks)
P = C/r, where C=£1,000, r=1%, so P=£100,000
dP/dr = - C/r2
DV01 = - 0.01% x (dP/dr) = 0.01% x 1000 / (1%)2 = £1,000
Intuition: The present value of the perpetuity drops by £1,000 if the interest rate
increases by 1 basis point.

Note to the marker: Students may or may not have a minus sign in the definition of DV01.
This shouldn’t affect the marking as long as the overall calculation makes sense, and the
intuition is clearly and correctly phrased.

b) An investment of \$1m cash in a bank account for 1 year earning the 1-year LIBOR of
1.00%.
(15 marks)
T=0 deposit PV=\$1m cash, obtain \$1m x (1+1.00%) = \$1.01m at T=1

If 1 sec later, the interest rate increases by 1bp to 1.01%, the PV becomes:
\$1.01m / (1+1.01%) = \$1.01m / 1.0101 = \$999,901

Hence, the present value of the bank account drops by \$1,010,000 - \$999,901 = \$99,
in response to a 1bp interest rate increase.

Question 3
(25 marks)

Dynamic term structure models are useful tools for pricing interest rate derivatives. Answer the
following questions regarding the pricing of interest rate derivatives and dynamic term structure
models:

a) Suppose the 1-year LIBOR is governed by the risk-neutral binomial tree process (time t is
measured in years, and at any time t, LIBOR can move up or down with equal probability):

t=0 t=1 t=2 t=3

8%
7%
6% 6%
5% 5%
4% 4%
3%
2%

Based on the above information, calculate the present value (at time t=0) of a 3-year floorlet
on 1-year LIBOR at the strike of 5.5%. Assume notional amount of \$100.
(10 marks)

Denote F(rt , t) the value of the floorlet at time t when the 1-year LIBOR from t to t+1 is rt.

t=3:
F(8%,3) = [5.5% - 8%]+ = 0
F(6%,3) = [5.5% - 6%]+ = 0
F(4%,3) = [5.5% - 4%]+ = 1.5%
F(2%,3) = [5.5% - 2%]+ = 3.5%

t=2:
F(7%,2) = [ (1/2)xF(8%,3) + (1/2)xF(6%,3) ] / 1.07 = 0
F(5%,2) = [ (1/2)xF(6%,3) + (1/2)xF(4%,3) ] / 1.05 = 0.71%
F(3%,2) = [ (1/2)xF(4%,3) + (1/2)xF(2%,3) ] / 1.05 = 2.43%

t=1:
F(6%,1) = [ (1/2)xF(7%,2) + (1/2)xF(5%,2) ] / 1.06 = 0.33%
F(4%,1) = [ (1/2)xF(5%,2) + (1/2)xF(3%,2) ] / 1.06 = 1.51%

t=0:
F(5%,0) = [ (1/2)xF(6%,1) + (1/2)xF(4%,1) ] / 1.05 = 0.88%

Since the notional amount is \$100, the value of the floorlet is \$100 x 0.88% = \$0.88.

b) Discuss the weakness and limitations of using one-factor term structure models.
(15 marks)
The key points include:
• 1-factor models assumes that the term structure of interest rate is perfectly driven
by the dynamics of the short rate.
• Empirically, at least 3 factors (level, slope, curvature) are needed to account for the
yield curve dynamics.
• For example, short rate can also be driven by the slope of the yield curve.
• Theoretical price depends on the dynamic term structure model. Therefore, models
that are too simple to capture the important dynamic properties of the term structure
tend to fail in pricing.
Note to the marker: please mark based on the overall quality of the student’s answer
taking into account factors such as whether the answer is concise and clear.

Question 4
(25 marks)

Fixed income securities are in general not free from credit risk.

a) Discuss the main credit spread determinants.
(10 marks)

Key points include:
• firm-specific default cost
• systematic risk
• (il)liquidity of corporate bonds relative to treasury bonds
Note to the marker: Students should provide their own explanations for the above
components. Please mark based on the overall quality of the student’s answer taking into
account factors such as whether the answer is concise and clear.

b) “One of the disadvantages of securtization is that it increases the risk exposure of financial
institutions through the creation of a pool of underlying assets, leading to an inefficient risk
sharing mechanism.” Do you agree with this statement? Explain.
(15 marks)
Key points include:
• the portfolio of underlying assets improves diversification and is supposed to
reduce idiosyncratic risk.
• It is information asymmetry and incentive problems that lead to inefficient risk
sharing.
Note to the marker: Students should provide their own explanations based on the benefits
and costs of securitization. Please mark based on the overall quality of the student’s
answer taking into account factors such as whether the answer is concise and clear.