2. Lecture
Financial and Investment
Mathematics
Dr. Eva Cipovova
Bonds
• Bonds are a form of financing the operations
of a company. Bonds are a debt security
because the principle must eventually be
returned to the bondholders
Bond Terms
• Every bond will feature a set of terms which
include:
• Principle Amount
• Number of Years Until Maturity
• Contract Interest Rate
• Market Interest Rate
• Interest Payment Schedule
Bond Principle
• The bond principle represents the investment
made by the bondholder
• Eventually this principle must be paid back
with interest
• Bond principle can be used by the company to
finance operations
• Known as Face value - FV
Years Until Maturity
• Every bond will feature a maturity date
• The maturity date is the date that the bond
principle must be paid back
• Usually there will be a period of several years
between the issue date and the maturity date
Contract Interest Rate
• Every bond will feature a contract rate
• This is the rate of interest that must be paid
every year to the bondholders
• The interest paid out represents the cost of
financing for the company
• The interest received by the bondholder
represents a return on the investment
• It is called also coupon rate (can be divided
on fixed and variable coupon rate)
Market Interest Rate
• On the open market, there is always a
prevailing interest rate
• Most of the time this rate will be different than
the contract rate on any given bond
• The relationship between the market rate and
contract rate will help to determine the issue of
the bond
• Know as yield of the bond
Interest Payment Schedule
• The company must pay each bondholder the
contract rate of interest each year. The
payment can be scheduled a variety of ways
including:
• Annual (once per year)
• Semi-annual (twice per year)
• Quarterly (four time each year)
• Monthly (twelve times each year)
Interest Rate Relationships
• Contract Rate = Market Rate:
The bond sells at par
• Contract Rate < Market Rate:
The bond will sell at a discount price
• Contract Rate > Market Rate:
The bond will sell at a premium price
Bond Issued at Par
• A company sells a $200,000 bond with a 5 %
contract rate which matures in 10 years. The
interest payments are to be made semiannually and the market rate for a similar bond
is 5 %. The journal entry required to record the
sale is:
Cash $200,000
Bond Payable $200,000
Interest Payment
• The journal entry to record the first semiannual interest payment follows:
Bond Interest Expense $ 5,000
Cash $5,000
• Ultimately, this journal entry will be recorded
20times over 10 years
Bond Maturity
• Once the bond matures in 10 years, the
following entry will be made:
Bond Payable $200,000
Cash $200,000
Bond Sold at Discount
• A company sells a $100,000 bond with a 8 %
contract rate which matures in 2 years. The
interest payments are to be made semiannually and the market rate for a similar bond
is 10 %. The journal entry to sell the bond is:
• Cash $96,454
• Discount $3,546
Bond Payable $100,000
Bond Interest Payment
• The journal entry to record the first semi-annual
interest payment and amortize the discount
follows:
Bond Interest Expense $4,886.50
Discount $883,5
Cash $4,000
Bond Maturity
• The journal entry to retire the bond at maturity
is:
Bond Payable $100,000
Cash $100,000
Bond Sold at Premium
• A company sells a $100,000 bond with a 12 %
contract rate which matures in 2 years. The
interest payments are to be made semiannually and the market rate for a similar bond
is 10 %. The journal entry to sell the bond is:
Cash $103,546
Premium $3,546
Bond Payable $100,000
Bond Interest Payment
• This journal entry is required for the first semiannual interest payment and amortization of
the premium:
Bond Interest Expense $5,113.50
Premium $886.50
Cash $6,000
Secured/Unsecured Bonds
• Secured bonds are backed up by an asset
which serves as collateral
• Unsecured bonds are only backed up by the
reputation of the company
• Typically, the interest rate is higher on an
unsecured bond
Registered/Bearer Bonds
• A registered bond is registered in the name of
the specific bondholder for security purposes
• A bearer bond is not registered and could be
redeemed by anyone
• In the United States, bearer bonds are no
longer issued
Risk/Reward
• A relationship exist between risk and reward
• The greater the risk, the greater the potential
reward
• The lower the risk, the lower the potential
reward
Debt Investment Classification
• No we will purchasing our investment, not selling
anymore
• Debt investments (bonds) can be classified in the
following categories:
– Held to maturity (HTM) – this classification
signals the intent of the company to hold the
bond as an investment until it has fully matured
– Trading – signals the intent to actively trade
the bond as soon as a profit could be made
– Available for Sales (AFS) – signals that the
company does not intend to hold the
investment to maturity or actively trade it. In
other words, this is a catch-all category
Classification of bonds
• Government Bonds
• In general, fixed-income securities are classified according to the
length of time before maturity. These are the three main categories:
1. Treasury Bills. T-bills have maturities of up to 12 months. They
are zero coupon bonds, so the only cash flow is the face value
received at maturity.
2. Treasury Notes. Notes have maturities between one year and ten
years. They are straight bonds and pay coupons twice per year,
with the principal paid in full at maturity.
3. Treasury Bonds. T-Bonds may be issued with any maturity, but
usually have maturities of ten years or more. They are straight
bonds and pay coupons twice per year, with the principal paid in
full at maturity.
Classification of bonds
• Corporate Bonds
• We will consider three major types of corporate bonds:
• Mortgage Bonds. These bonds are secured by real property such as
real estate or buildings. In the event of default, the property can be
sold and the bondholders repaid.
• Debentures. These are the normal types of bonds. It is unsecured
debt, backed only by the name and goodwill of the corporation. In
the event of the liquidation of the corporation, holders of debentures
are repaid before stockholders, but after holders of mortgage bonds.
• Convertible Bonds. These are bonds that can be exchanged for
stock in the corporation.
• In the United States, most corporate bonds pay two coupon
payments per year until the bond matures, when the principal
payment is made with the last coupon payment.
Classification of bonds
•
•
•
•
Bonds according of length of maturity
Short-term bonds (up to 1 year)
Medium-term bonds (up to 10 years)
Long-term bonds
• Domestic bonds, foreign bonds, eurobonds….
Classification of bonds
• Bonds with the possibility of early repayment
• Callable bonds - A bond that can be redeemed by the issuer prior to its
maturity. Usually a premium is paid to the bond owner when the bond is
called. In other words, on the call date(s), the issuer has the right, but not
the obligation, to buy back the bonds from the bond holders at a defined
call price. Technically speaking, the bonds are not really bought and held
by the issuer but are instead cancelled immediately.
• Also known as a "redeemable bond.“
• Puttable bonds - (put bond, putable or retractable bond) is a bonds with an
embedded put option. The holder of the puttable bond has the right, but not
the obligation, to demand early repayment of the principal. The put
option is exercisable on one or more specified dates.
• Convertible bonds
Classification of bonds
• Bonds according to the repayment of bonds
1. Perpetuity bonds (Consol bonds)
2. Zero-coupon bonds (discount bonds)
3. Coupon bonds
4.
See attached .pdf file with formulas
Summary
• Rule 1:
If the yield y is equal to the coupon rate c, then the bond
price P is equal to face value FV, if yield y is higher, resp.
less than the coupon rate c, then the bond price P is smaller,
resp. greater than the face value FV.
• Rule 2:
If the price of the bond increases, resp. decreases, this
results in a decrease, resp. increase of the yield of the bond.
Reverse: decrease, resp. rise in interest rates (yields) results
in an increase, resp. decrease in bond prices.
• Rule 3:
• If the bond comes closer to its maturity, then the bond price
comes closer to the face value of bond.
• Rule 4:
• The closer is the bond to its maturity the higher is the
velocity of approaching the face value by the price of the
bond.
Relationship of the
bond price and time
to maturity of the
bond
• Rule 5:
The decrease in a bond yield leads to an increase in bond
price by an amount higher than is the amount
corresponding to the decrease (in absolute value) in the
price of the bond if the yield increases by same percentage
as previously decreased.
Example:
Assume a 5-year bond with a face value FV = 1.000 CZK,
coupon rate c = 10 % and yield y = 14 %.
Yield
Price
Price change
12 %
13 %
14 %
15 %
16 %
927,90 894,48 862,68 832,39 803,54
65,22
31,80
0
-30,29
-59,14
Examples
1. Zero-coupon bond with a principal $1,000 has a maturity
10 years and yield 8 %. Calculate bond’s price
P= 1000/1,08^10 = $463.2
2. Coupon bond with the principal $1,000 has a maturity 10
years, coupon rate = 14.9 % and yield = 8%. Calculate bond’s
price
P= $1,463
Bond pricing
• The aliquot interest yield is a part of the nominal yield
which pertaining to the bond holder for the period from
the bond issue or from the last coupon payment
calculated until the transaction settlement date. Aliquot
interest yield is added automatically to the bond price
and can be positive or negative. For a client buying a
bond, a positive aliquot interest yield means
compensation that he has to pay when s/he becomes
entitled to the whole coupon without holding the bond
all the time. For a client buying a bond, a negative
aliquot interest yield represents compensation for the
period when the bond is held without entitlement to the
coupon.
• A + B = 360
• Example:
Assume a 5-year bond with a face value FV =
10.000 CZK issued at 6. 2. 1998 with maturity
6. 2. 2003 and with coupon rate c = 14.85%.
The yield of this bond was y = 7% on 9. 11.
1999. Calculate the clean price PCL of the
bond.
Results: