PROBLEM SETS

1. Define the following types of bonds:

a. Catastrophe bond.
b. Eurobond.
c. Zero-coupon bond.
d. Samurai bond.
e. Junk bond.
f. Convertible bond.
g. Serial bond.
h. Equipment obligation bond.
i. Original issue discount bond.
j. Indexed bond.
k. Callable bond.
l. Puttable bond.
Definition of Zero-coupon Bond
These are bonds that pay no coupons but pay at par value on maturity.

Definition of Samurai Bond

These bonds are yen-dominated bonds which are sold in Japan by non-Japanese issuers
called Samurai bonds.

Definition of Junk Bond

These are those speculative bonds which are rated low and hence are called junk bonds

Definition of Convertible Bond

These are those bonds that may be exchanged at bond holder’s discretion for a specific
number of shares of stock. This option is availed by the ‘bondholder’ by paying and
accepting a lower coupon rate on the security.

Definition of Serial Bond

It is an issue in which the firm sells bonds with staggered maturity dates. As bonds
mature, the principal repayment burden of the firm is spread over time. It is more
like sinking funds and but do not include call provisions.

Definition of Equipment Obligation Bond

It is a bond that is issued with specific equipment pledged as collateral against the
bond.

Definition of Original – issue discount Bond

These are bonds which are issued intentionally with low coupon rates that cause the
bond to sell at a discount from par value. These bonds are less common than coupon
bonds issued at par.
 Definition of Indexed Bond
These are bonds that make payments tied to a general price index or the price of a
particular commodity

2. Two bonds have identical times to maturity and coupon rates. One is callable at 105, the
other at
110. Which should have the higher yield to maturity? Why?

The issuer of the callable bond can redeem the bond before the date of maturity.
Explanation on the higher yield
As the call provision is more valuable to the firm, the bond callable at 105 should sell
at a lower price. Therefore, its yield to maturity will be higher.

3. The stated yield to maturity and realized compound yield to maturity

of a (default-free) zero- coupon bond will always be equal. Why?
The bonds which do not pay interests but trade at a deep discount from
their face value is known as zero coupon bond.
Explanation on compound yield to maturity on zero coupon bond
Zero coupon bonds provide no coupons for reinvestment. Therefore
the final value of the coupon comes from the principal of the bond and
is independent of the rate at which this could be reinvested. There is no
reinvestment rate uncertainty with zeros.
4. Why do bond prices go down when interest rates go up? Don’t lenders like high
interest rates?
Bond prices and interest rates are inversely proportional. So, an increase in interest
rates decreases the bond prices.
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Explanation on price and interest rates relationship

Changes of the market rates don’t affect the bond’s coupon interest payment or principal
repayment. Therefore on the increase of market rates, the bond investors in the
secondary market are not willing to pay as much for a claim on bond’s fixed interest.
Explanation on the reason for the price decline
If the interest rates are lower, investors would not want to invest in such bonds that
would lead to its decline.
This inverse relationship between interest rate and present value can be noted from the
decrease in present value of future cash flows with increase in discount rate.
A bond with an annual coupon rate of 4.8% sells for $970. What is the bond’s current
yield?
Annual coupon rate = 4.8%
Par value = $1000
Bond Price = $970
Calculation of Current yield
Annual coupon = Annual coupon rate x Par value
= 4.8% x $1000
= $48
Current yield = Annual Coupon / Bond Price
= $48 / $970
= 4.95%

5. Which security has a higher effective annual interest rate?

a. A 3-month T-bill selling at $97,645 with par value $100,000.
b. A coupon bond selling at par and paying a 10% coupon semiannually.
Calculation of effective price yield of T- Bill
The formula for calculating effective rate EAR=(1+PV−P/P)365/t

Effective rate of interest on a 3 month T-bill = (1 + $100,000 - $97645 /

$97645)365/90

= ($100,000 / $97645)4 – 1

= (1.02412)4 -1

= 1.1048 - 1

= 0.10148

= 10.148%

Calculation of effective annual rate of bond

The formula for calculating effective annual rate of interest of coupon bond paying
10% semi-annually EAR = (1 + r /m)m -1

= (1 + 10/2)2 - 1

= (1+0.05)2 – 1
 =0.1025

= 10.25%

Therefore the coupon bond has higher effective annual interest rate.
6. Treasury bonds paying an 8% coupon rate with semiannual payments
currently sell at par value. What coupon rate would they have to pay in
order to sell at par if they paid their coupons annu- ally? (Hint: What is
the effective annual yield on the bond?)
The formula for calculating effective annual rate of interest of coupon
bond paying 8% semi-annually EAR = (1 + r /m)m -1

= (1 + 8/2)2 - 1

= (1+0.04)2 – 1

= 1.8016 - 1

= 8.16%

Therefore the effective annual yield on the semi-annual coupon bond =

8.16%

If the annual coupon bonds are to sell at par, they must offer the same
yield i.e. an annual coupon of 8.16%
7. Consider a bond with a 10% coupon and with yield to maturity 5 8%.
If the bond’s yield to maturity remains constant, then in 1 year, will the
bond price be higher, lower, or unchanged? Why?
The bond price will be lower than it was at the beginning of the year.
‫شرحها‬
The total anticipated return on a bond if it is held till maturity is
known as Yield to Maturity or YTM.
Explanation on the higher yield
With the passage of time, the bond value will decrease when yield to
maturity is lower than the coupon rate
Consider an 8% coupon bond selling for $953.10 with 3 years until maturity
making annual cou- pon payments. The interest rates in the next 3 years
will be, with certainty, r1 5 8%, r2 5 10%, and r3 5 12%. Calculate the yield
to maturity and realized compound yield of the bond.
To calculate the realized compound yield of the bond, we need to find the
average annual compound rate of return that takes into account the
different interest rates over the next three years. We can use the following
formula:
Realized compound yield = (FV/PV)^(1/n) - 1
where FV is the face value or principal repayment at maturity, PV is the
market price of the bond, n is the number of years to maturity.
To calculate the realized compound yield, we need to first find the future
value of the bond's cash flows at each interest rate. We can use the
following formula to calculate the future value of a single cash flow:
FV = C x (1 + r)^n
where C is the cash flow, r is the interest rate, n is the number of years.
Using this formula, we get the following future values for the cash flows
of the bond:
 Cash flow at year 1: FV1 = 80 x (1 + 0.08)^1 = $86.40
 Cash flow at year 2: FV2 = 80 x (1 + 0.10)^2 = $97.60
 Cash flow at year 3: FV3 = (80 + 1000) x (1 + 0.12)^3 = $1293.04
Now we can calculate the realized compound yield of the bond as follows:
Realized compound yield = (FV/PV)^(1/n) - 1
Realized compound yield = ((86.40 + 97.60 + 1293.04)/953.10)^(1/3) - 1
Realized compound yield = 0.099 or 9.9%
Therefore, the yield to maturity of the bond is 8.39% and the realized
compound yield is 9.9%.

8. Assume you have a 1-year investment horizon and are trying to choose
among three bonds. All have the same degree of default risk and mature
in 10 years. The first is a zero-coupon bond that pays $1,000 at
maturity. The second has an 8% coupon rate and pays the $80 coupon
once per year. The third has a 10% coupon rate and pays the $100
coupon once per year.
a. If all three bonds are now priced to yield 8% to maturity, what are their prices?
b. If you expect their yields to maturity to be 8% at the beginning of
next year, what will their prices be then? What is your before-tax
holding-period return on each bond? If your tax bracket is 30% on
ordinary income and 20% on capital gains income, what will your
after- tax rate of return be on each?
c. Recalculate your answer to (b) under the assumption that you expect
the yields to maturity on each bond to be 7% at the beginning of
next year.
 SOLUTIONS

1. The callable bond will sell at the lower price. Investors will not be willing to pay as much if they
know that the firm retains a valuable option to reclaim the bond for the call price if interest rates
fall.
2. At a semiannual interest rate of 3%, the bond is worth $40 3 Annuity factor (3%, 60) 1
$1,000 3 PV factor(3%, 60) 5 $1,276.76, which results in a capital gain of $276.76. This
exceeds the capital loss of $189.29 (i.e., $1,000 2 $810.71) when the semiannual interest rate
increased to 5%.
3. Yield to maturity exceeds current yield, which exceeds coupon rate. Take as an example the 8%
coupon bond with a yield to maturity of 10% per year (5% per half year). Its price is $810.71, and
therefore its current yield is 80/810.71 5 .0987, or 9.87%, which is higher than the coupon rate
but lower than the yield to maturity.
4. The bond with the 6% coupon rate currently sells for 30 3 Annuity factor (3.5%, 20) 1 1,000 3 PV
factor(3.5%, 20) 5 $928.94. If the interest rate immediately drops to 6% (3% per half-year), the
bond price will rise to $1,000, for a capital gain of $71.06, or 7.65%. The 8% coupon bond
currently sells for $1,071.06. If the interest rate falls to 6%, the present value of the scheduled
payments increases to $1,148.77. However, the bond will be called at $1,100, for a capital gain of
only $28.94, or 2.70%.
5. The current price of the bond can be derived from its yield to maturity. Using your calculator,
set: n 5 40 (semiannual periods); payment 5 $45 per period; future value 5 $1,000; interest
rate 5 4% per semiannual period. Calculate present value as $1,098.96. Now we can calculate
yield to call. The time to call is 5 years, or 10 semiannual periods. The price at which the bond will
be called is $1,050. To find yield to call, we set: n 5 10 (semiannual periods); payment 5 $45 per
period; future value 5 $1,050; present value 5 $1,098.96. Calculate yield to call as 3.72%.
6. Price 5 $70 3 Annuity factor(8%, 1) 1 $1,000 3 PV factor(8%, 1) 5 $990.74

$70 1 ($990.74 2 $982.17)

Rate of return to investor 5 5 .080 5 8%
$982.17
7. By year-end, remaining maturity is 29 years. If the yield to maturity were still 8%, the bond
would still sell at par and the holding-period return would be 8%. At a higher yield, price
and return will be lower. Suppose, for example, that the yield to maturity rises to 8.5%.
With annual payments of $80 and a face value of $1,000, the price of the bond will be
$946.70 [n 5 29; i 5 8.5%; PMT 5 $80; FV 5 $1,000]. The bond initially sold at $1,000 when
issued at the start of the year. The holding-period return is

80 1 (946.70 2 1,000)
HPR 5 5 .0267 5 2.67%
1,000

which is less than the initial yield to maturity of 8%.

 8. At the lower yield, the bond price will be $631.67 [n 5 29, i 5 7%, FV 5 $1,000, PMT 5 $40].
Therefore, total after-tax income is

Coupon $40 3 (1 2.38) 5 $24.80

Imputed interest ($553.66 2 $549.69) 3 (1 2 .38) 5 2.46
Capital gains ($631.67 2 $553.66) 3 (1 2 .20) 5 62.41
Total income after taxes $89.67
Rate of return 5 89.67/549.69 5 .163 5 16.3%.

9. It should receive a negative coefficient. A high ratio of liabilities to assets is a poor omen for a
firm that should lower its credit rating.
10. The coupon payment is $45. There are 20 semiannual periods. The final payment is assumed
to be $500. The present value of expected cash flows is $650. The expected yield to maturity is
6.317% semiannual or annualized, 12.63%, bond equivalent yield.