Bond Valuation
Case Study 3
Fundamentals of financial management
BEAM047
Alexander Pereverzev
Yana Shkrebenkova
Verónica Deambrosi
Dina Sharipova
Bond
Bond - a long-term contract under which a borrower agrees to
make payments of interest and principal on specific dates to the
holders of the bond.
Grouping based
on the issuer
Treasury bonds
Corporate
bonds
Municipal bonds
Foreign bonds
A
Bond’s Key features
• Par Value – the face value of a bond
• Coupon Interest Rate - the stated annual interest rate on a bond:
• Fixed-Rate Bond - a bond whose interest rate is fixed for its entire life
• Floating-Rate Bond - a bond whose interest rate fluctuates with shifts in
the general level of interest rates
• Zero Coupon Bond – a bond that pays no annual interest but is sold at
a discount below par, thus compensating investors in the form of capital
appreciation.
• Original Issue Discount (OID) Bond - any bond originally offered at a
price below its par value.
• Maturity Date - a specified date on which the par value of a bond
must be repaid. Original Maturity - the number of years to maturity
at the time a bond is issued.
B
Provisions and sinking fund provisions
Call Provision – a provision in a bond contract
that gives the issuer the right to redeem the bonds
under specified terms prior to the normal maturity date:
• immediately callable bonds
• deferred call provision bond
Refunding operation: valuable to the firm but
detrimental to long-term investors
Sinking Fund Provision – a provision in a bond contract
that requires the orderly retirement of the issue.
Sinking funds are
considered to protect
investors as bonds are
going to be retired in an
orderly fashion
These funds work to the
detriment of bondholders if
the bond’s coupon rate > the
current market rate
On balance bonds with a sinking fund are regarded as
being safer than those without such a provision → so at
the time they are issued, coupon rate of sinking fund
bonds < similar bonds without sinking funds.
Types of Bonds
• Convertible Bond – a bond that can be exchanged into shares of
common stock at a fixed price at the option of the bondholder.
• Warrant - a long-term option to buy a stated number of shares of
common stock at a stated price.
• Putable Bond - a bond with a provision that gives the right to its
investors to require the company to buy it prior to maturity at a
prearranged price.
• Income Bond – a bond that pays interest only if the issuer has
enough earnings to pay the interest.
• Indexed (Purchasing Power) Bond - a bond that has interest
payments is related to an inflation index so as to protect the holder
from inflation.
Valuation
The present value: PV =
N
Bond value =
t=1
FV
(1+𝑟𝑑 )t
C
F
+
t
(1 + rd )
(1 + rd )N
Price of callable bond=
C
N
t=1 (1+r )t
d
𝐶
1
Bond value =
1−
𝑟
1+𝑟
𝑇
+
Call price
(1+rd )N
𝐹
+
(1 + 𝑟)𝑇
Bond value = present value of the coupons + present value of the
face amount
D
How is a bond’s value determined? What is the value of a 10-year,
$1,000 par value bond with a 10% annual coupon if its required return is
10%?
• The general way of bond valuation:
Calculate the present value of a bond’s cash flows and
then add these up to yield a fair price of the bond
• 3 STEPS:
• Estimate the bond’s cash flows = the coupon payments and
the return of the principal
• Determine the appropriate discount rate based on the risk of
the cash flows
• Calculate the present value of the cash flows, and their sum
D
How is a bond’s value determined? What is the value of a 10-year,
$1,000 par value bond with a 10% annual coupon if its required
return is 10%?
Period
CF
PVF 10%
PV
Y0
(?)
1
(?)
Y1-Y10
100
6.145*
6145
Y10
1000
0.386
386
Total
The value of
the debt =
6145+386=
=(1000)
0
IF bond has coupon rate = the interest rate=> price of the bond=par value
Hence Bond offers current yield = YTM => zero capital gain
E1
What is the value of a 13% coupon bond that is otherwise identical
to the bond described in Part D? Would we now have a discount or
a premium bond?
Period
CF
PVF @ 10%
PV
Y0
?
1
?
Y1-Y10
130
6.145
798.85
Y10
1000
0.386
386
Total
0
Premium Bond sells more (1184.34) than its face value (1000)
B=798.85+386
B=(1184.34)
E2
What is the value of a 7% coupon bond with these
characteristics? Would we now have a discount or a premium
bond?
Period
CF
PVF @ 10%
PV
Y0
?
1
?
Y1-Y10
70
6.145
430.15
Y10
1000
0.386
386
Total
B= 430.15+386
B=(816.15)
0
Discount bond sells for less (816.15) than its par value (1000)
E3
What would happen to the values of the 7%, 10%, and 13% coupon bonds
over time if the required return remained at 10%? (Hint: With a financial
calculator, enter PMT, I/YR, FV, and N; then change (override) N to see
what happens to the PV as it approaches maturity.)
13%
10
%
7%
F. (1)
What is the yield to maturity on a 10-year, 9% annual coupon,
$ 1,000 par vale bond that sells for $ 887.00?
• That sells for $ 1,134.20?
• YTM = 7.08% (Excel function)
What does the fact that it sells at a discount or at a
premium tell you about the relationship between rd and
the coupon rate?
• Whenever the bond sells at a discount it
means that the going rate of interest is
above the coupon rate, i.e. bond sells for $
887.00, coupon rate is 9% while rd is
10.91%.
• On the other hand, whenever the bond sells
at a premium the going rate of interest is
below the coupon rate, i.e. bond sells for $
1,134.20, coupon rate is 9% while rd is
7.08%.
F. (2)
What are the total return, the current yield, and the capital gains yield for
the discount bond? Assume that it is held to maturity and the company
does not default on it.
• Current yield = Coupon interest divided by the bond’s price.
Current yield = 90/887 = 10.15%
• Capital gains = Difference between the purchase price and the selling
price divided by the current price.
Capital Gains Yield = (1,000 – 887)/887 = 12.74%
• Total Return = Current Yield + Capital Gains Yield.
Total Return = 10.15% + 12.74% = 22.89%
G.
What is the interest rate (or price) risk?
•
The risk of a decline in a bond’s price due to an increase in
interest rates.
• For example, 15-year bond, par value of $ 1,000, 10% annual
coupon. Bond Price at moment 0 is $ 1,000.
• If the going interest rate rises to 15%, by using the formula
•
•
•
•
C = coupon payment
n = number of payments
i = interest rate, or required yield
M = value at maturity, or par value
we calculate the PV of the bond at this time to be $ 707.63.
Which has more interest rate risk, an annual payment 1year bond or a 10-year bond? Why?
• Interest rate risk is higher on bonds that
have long maturities than on bonds that will
mature in the near future. Therefore, a 10year bond has more interest rate risk than a
1-year bond.
• This is because the longer the maturity, the
longer before the bond will be paid off and
the bondholder can replace it with another
bond with a higher coupon.
H.
What is reinvestment rate risk?
• The risk that a decline in interest rates will lead
to a decline in income from a bond portfolio.
Which has more reinvestment rate risk, a 1-year bond or a 10-year
bond?
• Reinvestment risk is higher in a 1-year bond.
This is because the shorter the bond’s maturity,
the fewer the years before the relatively high
old-coupon bonds will be replaced with the new
low-coupon issues.
I.
How does the equation for valuing a bond change if semiannual
payments are made? Find the value of a 10-year, semiannual payment,
10% coupon bond if nominal rd = 13%.
Equation for valuing bond with annual payments:
𝑃𝑟𝑖𝑐𝑒𝑏𝑜𝑛𝑑 =
𝐶𝑡
𝑛
𝑡=1 (1+𝑟 )𝑛
𝑑
+
𝑁
(1+𝑟𝑑 )𝑛
Equation for valuing bond with semi-annual payments:
𝑃𝑟𝑖𝑐𝑒𝑏𝑜𝑛𝑑 =
𝐶𝑡 /2
2𝑛
𝑡=1 (1+𝑟 /2)𝑡
𝑑
𝑁
+
(1+𝑟𝑑 /2)2𝑛
=> Divide the interest rate and coupons by 2, increase
the number of periods by 2. (NB! Annual coupon rate is
assumed)
Ex.:
n=10; YTM=13%; q=10%; N=100
𝑃𝑟𝑖𝑐𝑒𝑏𝑜𝑛𝑑 =
10/2
2∗10
𝑡=1 (1+13%/2)𝑡
+
100
(1+13%/2)10∗2
= 83.47
J (1).
Suppose for $1,000 you could buy a 10%, 10-year, annual payment
bond or a 10%, 10-year, semiannual payment bond. They are
equally risky. Which would you prefer?
Let’s assume N=900
1000 =
90
10
𝑡=1 (1+𝑌𝑇𝑀)𝑡
1000 =
90/2
2∗10
𝑡=1 (1+𝑌𝑇𝑀/2)𝑡
+
900
(1+𝑌𝑇𝑀)10
+
=> YTM=8.32% annually
900
(1+𝑌𝑇𝑀/2)10∗2
=> YTM=8.51% annually
Thus, we will prefer semiannual payments.
In fact, the effective rate of semiannual payments will always be
higher than effective rate of annually paid CFs.
E.g.:
If a bond’s annual YTM=10%, the effective YTM rate of that bond
paid semiannually = (1+10%/2)2-1=10.25%.
Remember about the opportunity of received coupon payments’
reinvesting!
J (2).
If $1,000 is the proper price for the semiannual bond, what is the
equilibrium price for the annual payment bond?
Let’s assume semiannual bond with following
characteristics: q=10%; YTM=13%; N=1200; n=10;
Price=1000. Annual bond’s Price will be:
𝑃𝑟𝑖𝑐𝑒𝑏𝑜𝑛𝑑 =
100
10
𝑡=1 (1+13%)𝑡
1000
+
(1+13%)10
= 1003
=> The annual bond’s price must be higher at this YTM
rate.
K.
Suppose a 10-year, 10%, semiannual coupon bond with a par value of $1,000 is
currently selling for $1,135.90, producing a nominal yield to maturity of 8%.
However, it can be called after 4 years for $1,050. What is the bond’s nominal
yield to call (YTC)? If you bought this bond, would you be more likely to earn the
YTM or the YTC? Why?
The yield to call is the yield to maturity assuming that the bond is
called on its first callable date for the callable value.
1
2
3
4
5
6
7
8
CF
50
50
50
50
50
50
50
1100
Total
PV
48,28
46,61
45,01
43,46
41,96
40,51
39,12
830,95
1135,90
Thus, YTC for this bond will equal to
7.14%, which is lower than nominal YTM.
=> It is cheaper for the issuer to call the
bond and issue another one with lower
nominal YTM afterwards. So, we should
expect YTC rather than YTM.
Does the yield to maturity represent the promised or
expected return on the bond? Explain.
The yield to maturity can be viewed as the bond’s promised
rate of return, which is the return that investors will receive if
all the promised payments are made.
The yield to maturity equals the expected rate of return only
if:
• The probability of default is zero
• The bond cannot be called.
These bonds were rated AA- by S&P. Would you
consider them investment-grade or junk bonds?
What factors determine a company’s bond rating?
1.Various ratios
9.Antitrust
2.Mortgage provisions
10.Overseas operations
3.Subordination provisions 11.Environmental factors
4.Guarantee provisions
12.Product liability
5.Sinking fund
13.Pension liabilities
6.Maturity
14.Labor unrest
7.Stability
15.Accounting policies
8.Regulation
If this firm were to default on the bonds, would the
company be immediately liquidated? Would the
bondholders be assured of receiving all of their promised
payments?
• Liquidation or reorganization
• The reorganization plan call for restructuring
=> reduce the financial charges
• Liquidation – if company worth more dead than
alive => assets are auctioned off and the cash
obtained is distributed
Suppose a firm will need $100,000 20 years from now to replace some
equipment. It plans to make 20 equal payments, starting today, into an
investment fund. It can buy bonds that mature in 20 years or bonds
that mature in 1 year. Both types of bonds currently sell to yield 10%,
i.e., rd = YTM = 10%. The company’s best estimate of future interest
rates is that they will stay at current levels, i.e., they may rise or they
may fall, but the expected rd is the current rd. How much should the
firm plan to invest each year?
• 100,000 =
𝑋
20
𝑡=1 (1+10%)𝑡
=>
• To obtain $100,000 at the end of every year, the company
will have to invest the sum equal to $11,745.96
If the company decides to invest enough right now to
produce the future $100,000, how much must it put up?
• Let’s assume a zero-coupon bond. Thus, to obtain
$100,000 at the end of the 20 year period, the
company has to invest the sum equal to
100,000/(1+0,1)20=$14,864.36 right away.
Can you think of any other type of bond that
might be useful for this company’s purposes?
• The best type of bonds to be invested in is
putable one. That is the bond with embedded
option to be redeemed at a date chosen with a
pre-specified price by a bond holder.
THANK YOU FOR YOUR ATTENTION
Any Questions?