Homework #3: Practice Problems

Homework #3: Practice Problems
We will use a minimum of 3 decimal places for both prices and yields.
Question 1
Recall that the yield is a single rate at which we discount all cash flows of a bond so that
the sum of their present values equals the price of the bond. We could also find a specific
rate r1 to discount every cash flow that happens in one period, rate r2 to discount every cash
flow that happens in two periods, etc. These rates are called “spot rates” or “zero-coupon
rates”, and your job in this question is to calculate and use these rates.
Consider three default-free bonds that pay coupons annually, at the end of the year and are
priced using annual compounding (to simplify the calculations). Their times to maturity,
coupon rates and yields to maturity based on current market prices are as follows:
Bond Maturity Coupon rate Yield
A 1 year 2.0% 2.5%
B 2 years 5.0% 4.0%
C 3 years 8.0% 7.0%
a) What should be the fair prices of these three bonds at the given yield? Complete the
table by adding a column labeled “Price”.
b) Starting with Bond A and then Bond B and then Bond C, extract the current spot
rates. This is a puzzle where you will need to “backsolve”. Focus on the cash flows
you receive if you own each coupon bond above, and how each separate cash flow
should be valued using these spot rates. Hint: write out the equation for the price of
each bond, and see which elements are unknown. You will have three equations (one
for each bond) and three unknowns (the spot rates you’re looking for). For example,
the equation to compute the price of bond B is:
𝑃𝐵 =
5
(1 + 𝑟1)
+
105
(1 + 𝑟2)
2
You already calculated the price 𝑃𝐵 in part a) above. Where would r1 come from?
Write down the price equation for Bond A and you’ll see. With a bit of algebra you
can solve for r2. Next, write down the pricing equation for Bond C. Solve for r3.
c) Now that you have the spot rates figured out, you are ready to price any other defaultfree bond that has cash flows in the first three years. To do that, you simply need to
discount the cash flows of that bond at the spot rates you just calculated. If you were
not able to solve part b), assume that the one-, two-, and three-year spot rates are
2.5%, 4.5%, and 6.5%, respectively, and use these spot rates to solve the question that
follows. If you did solve b), use the rates you got.
Assume that the spot rates you calculated are correct. Suppose now that a quote for a
fourth bond, Bond D, flashes across your trading screen. This bond matures in three
years, has a 4% coupon rate and its quoted yield is 5.5%. Calculate the price implied
by this yield and compare it to the bond price implied by the spot rates. Is Bond D
correctly priced? How do you know?
Question 2
The last page of this pdf contains information on prices and yields of select U.S. Treasury
coupon-paying bonds and strips. Assume you can buy (at the ask) or sell (at the bid) any
amount (even fractions) of any of those bonds/strips at the prices shown. For the purposes of
this exercise, assume that all trades settle on January 31, 2021.
The key point of this question is to “replicate” the cash flows generated from one treasury
security by buying a series of other treasury securities, like we discussed in Class 2.
Sometimes the costs of buying or selling these cash flows as a package (e.g., as a couponpaying bond) versus individually (e.g., as a portfolio of zero-coupon bonds) are different. It
is like buying single goods that exactly make up an assortment pack and possibly paying a
different price. In situations like this, it may be possible to earn arbitrage profit from such
potential mispricing.
We are going to see if we can profit by trading the 1.75% bond maturing on June 15, 2022
and a portfolio of strips (zero-coupon bonds) that exactly replicates the cash flows of this
bond.
a) What is the total amount of money you would have to spend on buying $10,000,000
in face value of the 1.75% bond maturing on June 15, 2022? Note that the prices quoted
on the page are clean prices and you should add to them the accrued interest to arrive
at the total price at which you would buy or sell. You can verify that the accrued
interest in this case is $0.22596 per $100 of face value (using the formula from the
Class 2 slides or Excel’s ACCRINT function).
b) What is the total amount of money you would receive from short selling $10,000,000
in face value of the 1.75% bond maturing on June 15, 2022? As in part a), you have to
add the accrued interest of $0.22596 per $100 of face value to arrive at the invoice
(dirty) price.
c) You can create a portfolio of Treasury Strips which would replicate the cash flows of
the Treasury Bond referred to above. What is the total amount of money you would
have to spend to buy such a replicating portfolio? What is the total amount of money
you would receive from short selling such a replicating portfolio?
d) Would the replicating portfolio be any more or less “risky” than the actual couponpaying Treasury Bond? Why is this important to know when assessing whether there
is an arbitrage opportunity?
e) Does an arbitrage strategy involving buying or short selling the replicating portfolio
and the actual Treasury Bond exist given the prices provided? If you believe it does,
show all cash flows involved. Would you make a profit if you traded on this strategy?
If there were an arbitrage opportunity, do you think it would persist over time, and
why?
Bid and ask prices of select U.S. government coupon-paying bonds (source: Wall
Street Journal, Jan 25, 2021)
MATURITY COUPON BID ASK
6/15/2021 2.625 100.3120 100.3160
6/30/2021 1.125 100.1420 100.1460
6/30/2021 1.625 100.2100 100.2140
6/30/2021 2.125 100.2800 100.2840
12/15/2021 2.625 102.0740 102.0800
12/31/2021 1.625 101.1320 101.1360
12/31/2021 2.000 101.2420 101.2460
12/31/2021 2.125 101.2800 101.2840
6/15/2022 1.750 102.0840 102.0900
6/30/2022 0.125 100.0060 100.0120
6/30/2022 1.750 102.1100 102.1140
6/30/2022 2.125 102.2800 102.2840
Bid and ask prices of select U.S. government zero-coupon bonds
MATURITY COUPON BID ASK
6/15/2021 0 99.914 99.918
12/15/2021 0 99.829 99.834
6/15/2022 0 99.697 99.702

Research Paper 101
Calculate your paper price
Pages (550 words)
Approximate price: -

Reasons to trust Research Paper 101

On Time Delivery

We pride ourselves in meeting the deadlines of our customers. We take your order, assign a writer but allow some more time for ourselves to edit the paper before delivering to you. You are guaranteed a flawless paper on a timely manner...

24x7 Customer Live Support

Our team at Research Paper 101 is committed to handling your paper according to the specfications and are available 24*7 for communication. Whenever you need a quick help, you can talk to our writers via the system messaging or contact support via live chat and we will deliver your message instantly.

Experienced Subject Experts

Online Experts from Research Paper 101 are qualified both academically and in their experiences. Many are Masters and Phd holders and therefore, are qualified to handle complex assignments that require critical thinking and analyses...

Customer Satisfaction

We offer dissertation papers as per students’ wishes. We also help craft out the best topics and design concept papers. By ordering with us, you are guaranteed of defending and making through those hard faced professors in the defense panel!

100% Plagiarism Free

We at Research Paper 101 take plagiarism as a serious offence. From the start, we train our writers to write all their papers from scratch. We also check if the papers have been cited appropriately. Our website also has a tool designed to check for plagiarism that has been made erroniusly. In essense, the paper you get will be 100% legit...

Affordable Prices

We understand that being a student is very challenging, some students balance between work and studies in order to survive. We therefore offer pocket friendly rates that are very competitive in the market.

Try it now!

Calculate the price of your order

Total price:
$0.00

How it works?

Follow these simple steps to get your paper done

Place your order

Fill in the order form and provide all details of your assignment.

Proceed with the payment

Choose the payment system that suits you most.

Receive the final file

Once your paper is ready, we will email it to you.

Our Services

No need to work on your paper at night. Sleep tight, we will cover your back. We offer all kinds of writing services.

Essays

Essay Writing Service

No matter what kind of academic paper you need and how urgent you need it, you are welcome to choose your academic level and the type of your paper at an affordable price. We take care of all your paper needs and give a 24/7 customer care support system.

Admissions

Admission Essays & Business Writing Help

An admission essay is an essay or other written statement by a candidate, often a potential student enrolling in a college, university, or graduate school. You can be rest assurred that through our service we will write the best admission essay for you.

Reviews

Editing Support

Our academic writers and editors make the necessary changes to your paper so that it is polished. We also format your document by correctly quoting the sources and creating reference lists in the formats APA, Harvard, MLA, Chicago / Turabian.

Reviews

Revision Support

If you think your paper could be improved, you can request a review. In this case, your paper will be checked by the writer or assigned to an editor. You can use this option as many times as you see fit. This is free because we want you to be completely satisfied with the service offered.

error: