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 =
(1 + 𝑟1)
(1 + 𝑟2)
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
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
Bid and ask prices of select U.S. government coupon-paying bonds (source: Wall
Street Journal, Jan 25, 2021)
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
6/15/2021 0 99.914 99.918
12/15/2021 0 99.829 99.834
6/15/2022 0 99.697 99.702

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