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B8306 Fall 2023
Professor Xuelin Li
Due: Sunday, October 15
Problem Set 2
PROBLEM SET INSTRUCTIONS
Please hand in your answers using an Excel spreadsheet.
Add the names of all group members at
the top of the first worksheet and use a separate worksheet for each question.
Hand in one
assignment per group.
Please turn in the homework by midnight on the due date.
Question 1. Understanding bond price volatility
Download 6-month and 10-
year Treasury yield data from the St. Louis Fed’s FRED website
.
These
are called,
respectively, the “6
-
month Treasury constant maturity rate” and the “10
-year
Treasury constant maturity rate.”
Start your download from September 1981, which is when the
6-month series begins.
Download the series at a weekly frequency, taking the yield level on
Friday, and using “End of Period” as the aggregation method (these options are available under
the “Edit Graph” tab of the FRED charting tool).
a)
Plot the two time series.
b)
Compute weekly changes in yields (i.e. the yield in week
t
minus the yield in week
t-1
) and
compute the annualized standard deviation of the yield changes. To do this, compute the
standard deviation of the weekly % changes and multiply by
√52
to account for the
number of weeks in a year. (Assuming weekly returns are uncorrelated,
𝐴𝑛𝑛?𝑎?𝑖??? ?????? = √52
× 𝑊????? ??????
).
c)
Compute zero coupon bond prices for both maturities, assuming the yields you
downloaded are actually spot rates.
1
Remember to use semi-annual compounding, and
make sure to convert each yield to decimals, e.g. 0.07 and not 7.
d)
Compute weekly percentage changes in each bond
’s
prices (i.e. returns) and compute the
annualized standard deviation of bond returns (again, multiply by
√52
).
2
e)
Are 6-month or 10-year bond yields more volatile? Are 6-month or 10-year bond prices
more volatile? Provide one or two sentences of explanation.
Question 2. Hedging interest rate risk of a corporate bond
After extensive fundamental research, you decide to go long the bonds of SuperRocket Inc. (SRI)
which is a start-up company interested in space exploration.
The company has one bond: a 12-
year unsecured bond, with an 8% annual coupon paid semi-annually.
This bond currently trades
at $95.
You decide to invest $10mm into the SRI bond (i.e., to buy bonds having a market value
of $10 million).
While you think the SRI bond is a good investment, you are concerned that the
1
The “10
-
year yield” here is actually the
yield on the 10-year Treasury bond, not on the 10-year STRIPS.
Unfortunately, we don’t have easy access to STRIPS prices/yields going back to 1982. However, the STRIPS and
Treasury yields should be close.
2
Please note that in this step again we are makin
g an approximation: the “return” we are calculating here isn’t really
a return. It is the ratio of the price of the 10-year bond in a week to the price of the 10-year bond today, minus 1.
The problem with this is that, in a week, the 10-year bond today is really a 9-year, 51-week bond. However, since
these yields are usually very close, this approximation error should be small.
B8306 Fall 2023
Professor Xuelin Li
Due: Sunday, October 15
current Fed tightening cycle will result in an increase in yields on all bonds, corporate and
Treasury bonds included.
To hedge out the risk that interest rates rise, you decide to go short a
10-year Treasury note with a 3% coupon that trades at par, also with semi-annual coupons.
a)
What is the yield and modified duration of the SRI bond?
b)
What is the yield and modified duration of the 10-year Treasury?
c)
To duration hedge the SRI bond with the Treasury, how much market value of the
Treasury should you short against the SRI bond?
d)
If the yield of both the SRI bond and the Treasury rise by 1%, what will be the change in
the value of your portfolio?
Calculate this exactly using the actual new dollar prices of the
bonds.
How does this compare to the forecasted value change from a duration
approximation?
Why are they not exactly the same?
e)
What if the SRI bond yield increases by 1% but the Treasury yield increases by 1.5%?
Would you make or lose money?
Explain why this happens.
Question 3. Bond pricing with default risk
Consider a risky zero-coupon bond and risk-free STRIP as follows:
Risky bonds
STRIP
Maturity
5 years
5 years
YTM
5%
4%
Probability of default
4%
Expected loss given default
60%
Both bonds promise to pay $100 at maturity.
Assume that default, if it occurs, will occur at the
risky
bond’s maturity
, and not before.
Use annual compounding in all calculations.
a)
What is the risky bond’s yield spread relative to the STRIP?
b)
Find the price of the risky zero-coupon bond.
c)
Find the bond’s expected cash flow, and from this determine its expected return
?[?]
.
d)
What is
the risky bond’s risk premium, i.e.
?[?] − 𝑌??
???𝐼𝑃
, relative to the STRIP?
e)
Why is the risk premium different
from the bond’s spread in part (a)?
Question 4. Default risk and CDS pricing
In this problem, you will price and analyze a simplified version of a credit default swap (CDS).
Revisit the default risk example from class, in which there are two 1-year 0% coupon bonds (Safe
and Risky) and two cash flow scenarios for each bond. To this scenario diagram, we now add a
CDS based on the reference risky bond. This CDS pays the full par value of the risky bond ($100)
in exchange for the bond if and only if a default occurs. The only scenario in which default
happens is the one where the risky bond pays $50, meaning that the CDS pays a net amount of
$50 in this scenario and $0 otherwise. The CDS and bond cash flows and probabilities are as
follows:
B8306 Fall 2023
Professor Xuelin Li
Due: Sunday, October 15
a)
Consider a trading strategy in which you short sell the risk-free (safe) bond and use the
$95 proceeds to buy the risky bond and the CDS to protect against possible default. What
are the net cash flows of this strategy in each of the two scenarios? Throughout this
problem, ignore short selling constraints and margin requirements.
b)
If the CDS price were $1.50, what would be the cost of the trading strategy in part (a)?
c)
According to the no-arbitrage principle, what must be the price of the CDS?
d)
What is the 1-yr expected return (using annual compounding) for an investor who buys
the CDS at the price in part (c)?
NOTE: Assume the investor buys only the CDS, and not
the risky bond, i.e. what is the return on a $100 investment only into the CDS contract?
e)
Given your answer in (d), why would a risk-averse investor be willing to invest in a CDS?
Provide one or two sentences of explanation.
(CF
Safe
, CF
Risky
, CF
CDS
)
(-95, -93,
?
)
(100, 100, 0)
(100, 50, 50)
0.02
0.98
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