Fixed Income Investments - Bond Features
Purpose of a Bond Indenture
A bond indenture is the contract between a bondholder and the issuer. It is a legal document that states
what the issuer can and cannot do, and states the bondholders rights. Since there tends to be a ton of
legalese involved, the contract is managed by the corporate trustee who polices the actions of the issuer to
ensure the rights of the bondholder are upheld.
Within the indenture, there are affirmative and negative covenants:
1. Affirmative Covenants
Affirmative covenants are what the issuer promises to do for the investor. These promises include
things such as paying interest and principle in a timely manner; paying taxes and other expenses
when due; maintaining the assets backing the bond and issuing reports to the trustee to ensure
compliance.
2. Negative Covenants
Negative convents are the restraints put on a borrower. These restraints include issuing additional
securities or taking on additional debt that may harm the current bondholders. This is generally
done without meeting certain tests and/or ratios or receiving permission from the current
bondholders.
Basic Features of Bonds
In order to better understand more complicated topics, the CFA Institute requires CFA candidates to have
the ability to describe the basic features of a bond. These features include:
1. Maturity
Maturity is the time at which the bond matures and the holder receives the final payment of principal and
interest. The "term to maturity" is the amount of time until the bond actually matures. There are 3 basic
classes of maturity:

A. Short-Term Maturity - One to five years in length

Intermediate-Term Maturity - Five to twelve years in length

C. Long-Term Maturity - Twelve years or more in length
Maturity is important because:

It indicates the length of time in which an investor will receive interest as well as when he or she
will receive principal payments.

It affects the yield on the bond; longer maturities tend to yield higher rates.

The price volatility of a bond is a function of its maturity. A longer maturity typically indicates
higher volatility or, in Wall Street lingo, simply the "vol".
2. Par Value
Par value is the dollar amount the holder will receive at the bond's maturity. It can be any amount but is
typically $1,000 per bond. Par value is also known as principle, face, maturity or redemption value. Bond
prices are quoted as a percentage of par.
Example: Premiums and Discounts
Imagine that par for ABC Corp. is $1000, which would =100. If the ABC Corp. bonds trade at 85 what
would the dollar value of the bond be? What if ABC Corp. bonds at 102?
Answer:
At 85, the ABC Corp. bonds would trade at a discount to par at $850. If ABC Corp. bonds at 102, the
bonds would trade at a premium of $1,020.
3. Coupon Rate
A coupon rate states the interest rate the bond will pay the holders each year. To find the coupon's dollar
value, simply multiply the coupon rate by the par value. The rate is for one year and payments are usually
made on a semi-annual basis. Some asset-backed securities pay monthly, while many international
securities pay only annually. The coupon rate also affects a bond's price. Typically, the higher the rate, the
less price sensitivity for the bond price because of interest rate movements.
4. Currency Denomination
Currency denomination indicates what currency the interest and principle will be paid in. There are two
main types:

Dollar Denominated - refers to bonds with payment in USD.

Nondollar-Denominated - denotes bonds in which the payments are in another currency besides
USD.
Other currency denomination structures can use various types of currencies to make payments.
Because the provisions for redeeming bonds and options that are granted to the issuer or investor are more
complicated topics, we will discuss them later in this LOS section.
Example: Bond Table
Let's take a look at an example of a bond with the features we've discussed so far, within a bond table
format you'd see in a paper.
ABC Corp 7.00% 6/1/10 at 90.
The issuer is ABC Corp.
The maturity is 2010 with a term to maturity of roughly 5 years.
Par value is 1,000 per bond or 100
Coupon rate is 7%.
Coupon Payment is $70 per year (coupon=coupon rate* par value = .07 *$1,000 = $70
Trading Price in dollars in $900.00 (par price * .90)
ABC Corp is a U.S. company and all payments of interest ant principle are in USD.
Fixed Income Investments - Basic Coupon Structures
Make sure that you understand the various coupon rate structures that can be developed in the bond
market. Let's discuss each in detail:

Zero-Coupon Bonds - These instruments pay no interest to the holder and are issued at a deep
discount. As the bond nears maturity, its price increases to reach par value. At maturity, the
bondholder will receive the par price. The interest earned is the difference between the purchase
price of the bond and what the holders receives at maturity.

Step-up Notes - The interest rate of these bonds increases or "steps-ups" at a stated date(s). The
rate may remain at this level until maturity and in this case would be considered a "single step-up
note". Step-up notes can also have a series of rate increases and are then referred to as "multiple
step-up notes".

Deferred Coupon Bonds - A structure that essentially incorporates features of both a zero
coupon bond and a coupon paying bond. These bonds typical do not pay interest for the first
couple of years. After this period the cash interest accrues at a stated rate and is usually paid
semi-annually to the bondholders. The coupon rate is typically higher than other issues in order to
entice investors to purchase these issues. Companies in the high yield arena typically issue these
bonds to conserve their cash flows in the earlier years of their business life.

Floating-Rate Bonds - These bonds have coupon rates that reset at predetermined times. The rate
is usually based on an index or benchmark with some sort of spread added or subtracted to the
benchmark.
Structure of Floating-Rate Bonds
To find the coupon rate of floating rate bonds, all one has to find out is what the benchmark or reference
rate is trading at and add or subtract the stated amount of basis points or other influencing variable. Let's
take a closer look at the above example of the Federal Fund Floater:
Example: Floating Rate Security: Federal Funds
Assume the coupon rate of a floating-rate bond is based on the Federal Funds rate plus 25 basis points at
three-month intervals. If the Federal Funds are at 3%, what would the coupon rate for this bond be?
Formula 14.1
Coupon rate = Reference Rate + influencing variable.
Answer:
Coupon rate = 3% (Fed Funds) + 25 basis points.
Coupon rate = 3.25%
The coupon rate for this bond would be 3.25% until the next reset date. Floating- rate securities come in
many forms. Other forms of floating-rate securities involve caps and floors; these are discussed in detail
below.
Caps and Floors
Some floating-rate securities have restrictions placed on how high or how low the coupon rate can
become.

Caps - state how high the coupon rate can go. Once it hits that level, there can be no further
increase in the rate. Caps are less advantageous for investors because the rate can only keep pace
with market rates up to a point. On the other hand, they protect the issuer by keeping the cost of
borrowing below a certain level.
Example: Caps
Referring back to our Federal Funds example, let's add a cap of 3.90 % and assume that Fed
Funds are trading at 3.75%. What is the coupon rate?
Answer:
Coupon rate = 3.75%(Fed Funds) + 25 basis points.
Coupon rate = 4.00%
Even though the formula states a 4% coupon should be paid this period, the cap holds the coupon at
3.90%.

Floor - states how low a coupon rate can go. Once the coupon rate hits the floor, it can no longer
decline beyond that point. Floors are more advantageous to investors because as rates continue to
decrease, the investor is protected from that decrease at a stated point.
Example: Floors
Now lets add a floor of 2% and assume that Fed Funds are trading at 1.50%
Answer: Coupon rate = 1.50% (Fed Funds) + 25 basis points
Coupon rate = 1.75%
Even though the formula states a 1.75% coupon should be paid, there is a 2% floor in place, which means
that the investor will receive 2% instead of the 1.75% derived from the formula.
Accrued Interest and Price Terminology

Accrued interest - the amount of interest that builds up in between coupon payments that will be
received by the buyer of the bond when a sale occurs between these coupon payments, even
though the seller of the bonds earned it.

Full Price - is sometimes referred to as a bond's dirty price, which is the amount the buyer will
pay the seller. It equals the negotiated price of the bond plus the accrued interest.

Clean Price - is simply the price of the bond without the accrued interest.
Fixed Income Investments - Early Retirement
A bond generally have two main maturity structures:

Bullet Maturity - Most corporate and government bonds use this structure, which requires the
borrower to pay the investor one lump sum of the principle on the stated maturity date.

Amortizing Securities - Asset-Backed Securities (ABSs) along with Mortgage Backed Securities
(MBSs) have structures that pay the principal back at certain intervals during the bond's life. For
example, a mortgage payment includes part principle and part interest. They are called amortizing
securities because the principle amount shrinks as the security matures, so that the last payment
made to the investors closes out the issuer's responsibility concerning this bond.
Fixed Income Investments - Provisions for Redeeming Bonds
The provisions for redeeming bonds are found in the indenture. They can be:
1.Called
2.Refunded
3.Have Prepayment Options and/or
4.Sinking Fund Provisions
1. Call Redemption
By adding a call feature in the indenture, a bond becomes a callable bond. A callable bond gives the
issuer the right to redeem the bonds on a stated date or a schedule of dates before the stated maturity date
for the bonds arrives.
Let's look at callable bonds in a little more detail. First, some terminology:

Call Price - This is the price that the issuer will pay the bondholder; also know as the redemption
price.

Call Date - This is the date or dates that the issuer can call the bond from the holders.

Deferred Call - When a callable bond is originally issued, it is said to have a deferred call of so
many years up to the first call date, which is the first day the bond can be called by the issuer.
Redemption Pricing

Regular or General Redemption Prices - These price tend to be above par until the first par call
date. The price is typically known before the redemption occurs.

Special Prices - These occur because of certain events such as sinking funds, repossessions,
forced sales, and eminent domain. These usually occur at par value but could be less, depending
on the collateral backing the bonds.
Calling Bonds
When callable bonds are called, it can be for the entire issue or for just a part of it. A partial call can be
done on a random basis, like picking numbers out of a hat, or on a pro rata basis. A pro rata call allows all
holders to redeem a certain percentage of their holdings while with a random, partial call it could be
anyone's guess as to which bonds will be called by the issuer.
Price can be determined as a fixed price, regardless of dates, based on a predetermined schedule of dates
in which price decreases as it nears the bond's maturity date, as well as through a make whole call.
Example: Call Redemption
Let's use the Stone and Co 12's of 20, or Stone and Co 12% bonds of 2020 to illustrate a scheduled call.
Fixed price regardless of date:
This call provision allows the bond to be called at par plus interest at any date past Jan.1, 2010.
Price based on Schedule.
This call provision bases its price on stated dates with the price decreasing as the bond nears maturity.
Jan 1. 2010 Price = 103 or $1,030 based on a par value of 100
Jan 1. 2012 Price = 102 or $1,020 based on a par value of 100
Jan 1. 2015 Price = 101.5 or $1,015 based on a par value of 100
Price based on a Make-Whole Premium
This structure incorporates various formulas that can be structured to develop the price. The formula is
structured to protect the yield the investor had been receiving on his bond.
Fixed Income Investments - Refunding
2. Refunding
The refunding of an issue is the replacement of a current high coupon rate bond. This is done by issuing
newer bonds at a lower coupon rate. With regards to a callable issue, refunding offers little protection to a
holder. At least with a callable bond the holder has a date on which the call will occur. The refunding
could occur as soon as it becomes advantageous to the issuer to replace older, higher rate bonds.
3. Prepayments
This form of redemption occurs in ABS and MBS securities. In this instance the investor could receive
additional principal payments before the maturity date. For example, a homeowner with a mortgage
payment of $500 a month could pay more than that amount, say $700 a month. This additional $200
would constitute a prepayment of principal. If this were to happen in the payment of a bond, the bond
would be redeemed before maturity.
4. Sinking Fund Provisions
This helps redeem and retire bonds. It requires an issuer to retire or pay for the retirement of a specific
portion of the issue at certain times. This helps reduce credit risk by having something in the "kitty" each
year as a protection against a default. It can be structured to retire the entire issue at its maturity date or
only a portion of the balance of the issue. If provision is only for a balance of the issue, the final payment
is paid by a balloon payment.
Fixed Income Investments - The Importance of Embedded Options
Embedded options grant the issuer or bondholder certain rights in order to dispose of or redeem a bond.
Because these options carry some sort of value, they can have a dramatic effect on the price of a security's
cash flow as well as its total return.
Options that Benefit the Issuer:

Call options - allows the issuer to call the bonds prior to maturity if prevailing rates decrease
enough to make it economically feasible for the issuer to replace the existing issue (consisting of
higher rate coupons) with lower coupon bonds.

Prepayments - the benefit of prepayment is the same as a call option, in that it allows a company
to end a contract early.

Caps - Acap puts a lid on the amount that an issuer has to pay in the face of rising interest rates,
when their bonds coupon is based on a floating interest rate.
Options that Benefit the Holder

Puts - This option is the exact opposite of a call. It allows the bondholder to sell the bond or "put"
the bond back to the issuer at a certain price and date(s) before its maturity. As rates rise, this
helps the bondholders dump their holdings and reinvest their proceeds at a higher rate.

Floor - As mentioned earlier, a floor enables a firm to set a limit on how low the payment of their
coupon can be. This benefits the holder because as rates decrease, it holds their interest payment
at a certain level even as market rates decline below the floor level.

Conversion Privilege - This allows the bondholders to exchange their current bond with equity
in the same firm using convertible bonds. They may also receive equity or fixed income securities
in another firm by the use of exchangeable bonds. This benefits the holder because if the equity or
other securities of the firm is outperforming the bonds, the bonds can be converted, allowing the
holder to realize a higher return.
Fixed Income Investments - Institutional Investors and Financing Purchases
Institutional investors tend to finance their purchases in two ways instead of purchasing securities
outright. They are:

Buying on Margin - In this form of financing the buyer borrows funds from a broker/dealer who
in turn gets the cash from a bank. The institution is charged an interest rate plus some additional
charges for using this method. Regulations T and U, as well as the Securities Act of 1934, limit
the degree to which the margin can be extended to the buyer.

Repurchase Agreements - These are collateralized loans in which the institution sells a security
with the commitment to purchase the same security at a later date. The length of time could be as
short as overnight or extend all the way to the maturity of the security. The price that is agreed
upon is the repurchase price and the institution is charged a repo rate. The repo rate is an
implied interest rate, which is the cost that the institution incurs for funding the position. Its
benefit is that it is a very cheap way to finance a position because the repo rate tends to be set
around the Federal Funds rate.
Fixed Income Investments - Interest Rate Risk
Interest rate risk is concerned with a decline in the price of a bond or a portfolio of bonds due to an
increase in market rates. As rates increase, bond prices decline and vice versa.
The features of a bond affect the bond's interest rate in the following ways:

Maturity - The longer the maturity of the bond, the more sensitive it is to interest rate
movements. The reason for this effect is that more cash flows will be affected over a longer
period of time.

Coupon Rates - Lower coupon rates are more sensitive to interest rates. Why? If your bond is
paying 4% and rates are in an upward swing, the difference in the market yield and your yield
will continue to widen, which will push your bond values down.

Embedded Options - As interest rates decline the value of the option becomes more valuable to
the issuing company. The price will increase as rates decline but will be held at the call or
redemption price. This is because as rates decline it will become more likely that the issuer will
call the bonds and that the holder will only receive the call price and not a true market price.
Likewise, when rates rise the price will not drop as much because the option will maintain some
sort of value when compared to a bond with no options.
Look Out!
Recognize that the various bond features work with or
against each other in determining a bond's price
volatility.
As market yield volatility increases, the interest rate risk increases. This is because there is a greater
chance of rates breaking out of their current ranges, either by rising or declining. Typically, as rates
increase there is a greater chance of this risk occurring because market prices of bonds will decline as
interest rates rise.
Fixed Income Investments - Call and Prepayment Risk
Call and prepayment risk is concerned with the holders having their bonds paid off earlier than the
maturity date. This is due to decreasing marker rates, which cause the issuer to call the bonds. It can also
occur when the borrowers in a MBS or ABS refinance or pay off their debt earlier than the stated maturity
date.
Disadvantages of Investing in Bonds that are Callable or Prepayable
1. It is difficult to develop and forecast the cash flows for the security because of the possible early
redemption of the bond.
2. The reinvestment risk is another disadvantage. As rates decrease and bond are called or prepaid,
investors will not be able to invest their proceeds at the old rates and will have to use new, lower market
rates to put their cash to work.
3. Price compression is the final disadvantage for investors. When rates decline there is a greater chance
that the issuer will call the bonds. This compresses or holds the bond at its call price while bonds without
this option will continue to increase in market value as rates continue to decrease.
Fixed Income Investments - Reinvestment Risk
Reinvestment risk is the risk that the proceeds from the payment of principal and interest, which have to
be reinvested at a lower rate than the original investment.

Call features affect an investor's reinvestment risk because corporations typically call their bonds
in a declining interest rate environment. This allows them to finance their operations at a cheaper
rate, but the investor also has to take those proceeds and invest them at lower rates.

Amortizing securities have even greater reinvestment risk. Since the investor is receiving both
interest and principle payments each month, they have to continue to invest a greater amount of
proceeds than with a regular type of bond. Also, the investor is exposed to prepayment risk as
rates decline because mortgage and credit card holders can refinance their debt at lower levels.

Zero coupon bonds have no reinvestment risk because there are no coupon payments made to the
investor. Because of the lower coupon rate, however, zeros expose the investor to a higher
interest rate risk.
Because amortizing securities pay part of their principal with regular interest payments, investors receive
the maturity value in installments in a current period. Non-amortizing securities make their payment at a
later maturity date. If prevailing interest rates are declining, holders of amortizing securities have to
reinvest their coupons and portion of principal at a lower rate. Meanwhile, the non-amortizing securities
offer the holder some protection by holding the principal payment off until a later date. This enables the
non-amortizing holder to maintain a higher rate at a longer period of time and reduces the reinvestment
risk for a portion of his or her holdings
Fixed Income Investments - Yield Curve Risk
The yield curve risk is how your portfolio will react with different exposures based on how the yield
curve shifts.

Investors portfolios tend to move in a parallel shift on the yield curve. This happens when all
maturities on the yield curve move by an equal amount. The yield curve encompasses all
maturities in the fixed income market. In a parallel shift, the investor would see an increase or
decrease in all maturities by, for example, 25 basis points.

In the real world, however, the yield curve does not move this way. The curve tends to steepen, or
yield an increase in the long end of the curve compared to shorter issues. The curve can also
flatten,with yield on the long end decreasing at a faster rate than at the short end of the curve.

Because any measure of interest-rate risk assumes an equal amount of basis point moves on the
yield curve, anything will be an approximation of how an investor's portfolio will react. The risk
involved here is the degree to which this approximation will not match of the actual yield curve
movement.
Fixed Income Investments - Credit Risk
1. Default Risk - Default risk is the risk that the issuer will go belly up and not be able to pay its
obligations of interest and principle. To help measure this risk, an investor can look at default rates. A
default rate is the percentage of a population of bonds that are expected to default. Another ratio that an
investor can look at is the recovery rate. This rate indicates how much and investor can expect to get back
if a default occurs.
2. Credit Spread Risk - This second type of credit risk deals with how the spread of an issue over the
treasury curve will react. For example, Ford five-year bonds may trade at 50 basis points above the
current five-year treasury. If the five-year bond is trading at 3.5%, then the Ford bonds are trading at a
yield of 4%. If this spread of 50 bps widens out compared to other bond issues, it would mean that the
Ford bonds are not performing as well as the other bonds in the marketplace. Spreads tend to widen in
poor performing economies.
3. Downgrade Risk - The third type of credit risk deals with the rating agencies. These agencies, such as
Moody's, S&P and Fitch, give an issuer a rating or grade that indicates the possibility of default. On the
more secure side, the ratings range from AAA, which is the best rating to AA, A, BBB. These are the
ratings for investment-grade bonds. Once bonds dip into the BB, B, CCC ranges they become junk bonds
or, in politically correct language, high yield securities. If one of these rating agencies downgrades a
company's rating, it may be harder for the corporation to pay. This will typically cause its marker value to
decrease. That is what this risk is all about.
Fixed Income Investments - Liquidity Risk
Liquidity risk is concerned with an investor having to sell a bond below its indicated value, the
indication having come from a recent transaction.

Liquidity refers to how deep or liquid the market is for a particular security. If the market is deep,
an investor can purchase or sell a security at current prices. If the market is not liquid, it is harder
to sell or buy a security at the last market price.

Liquidity is typically measured by the bid/ask spread. If the spread is wide, the market is illiquid.
If the spread is narrow, the market is more liquid.

Liquidity risk is important because it tells you how easily you can get rid of a position if you need
to close it near the last market price.

This is even more important if you plan to hold a security to maturity because of the marking to
market of your positions. In an illiquid market, it may be hard to obtain quotes, and when you
revalue the security it could be well below market prices, affecting the reports you send to clients
and management.

This risk also changes over time, so a manager has to keep abreast of this risk. This is especially
true when looking to invest in new complex bond structures.
Fixed Income Investments - Exchange-Rate Risk
Exchange-rate risk is the risk of receiving less in domestic currency when investing in a bond that is in a
different currency denomination than in the investor's home country.
When investors purchase a bond that is designated in another currency other than their home countries,
investors are opened up to exchange risk. This is because the payment of interest and principal will be in a
foreign currency. When investors receive that currency, they have to go into the foreign currency markets
and sell it to purchase their home currency. The risk is that their foreign currency will be devalued
compared to the currency of their home countries and that they will receive less money than they
expected to receive.
As an example, suppose that a U.S. investor purchases a Euro denominated bond. When the interest
payment comes due and if the Euro has declined in value compared to USD, the investor will receive less
in USD than expected when he or she transacts in the foreign currency markets. In short, the investor will
receive fewer Euros to purchase USD.
Fixed Income Investments - Volatility Risk
Volatility risk involves bonds with embedded options. Expected volatility or "vol" effects the option price
within a callable or puttable bond. Greater expected yield vol yields a greater increase in the value of the
option
1. Callable bonds- Let's look at the components of a callable bond:
Formula 14.2
Price of Callable Bond = Price of option-free bond - Price of Embedded option
As volatility increases, with all other factors holding the same, the price of the option will increase. This
decreases the price of the bond.
The risk here is that volatility increases.
2. Puttable bonds - Let's looks at the components of a puttable bonds.
Formula 14.3
Price of a Puttable Bond = Price of option-free bond + Price of embedded option
As volatility decreases with all other factors holding the same, the price of the option will decrease, which
depresses the price of the bond.
The risk here is that volatility decreases.
Fixed Income Investments - Inflation Risk
Inflation Risk is also known as Purchasing Power Risk, this risk arises from the decline in value of
securities cash flow due to inflation, which is measured in terms of purchasing power.
Here's how it works:
You buy a bond with a coupon rate of 4%. The inflation rate is at 2%. Even though you are earning 4% on
your money, inflation is chipping 2% of it away only leaving you with 2% of your money or purchase
power, which you can use when you receive your payments.
Only Inflation Protection Bonds such as TIPS offer protection against this risk. Floaters help reduce this
risk because of the resetting of the interest rates. All other bonds expose the investor to this risk because
the interest rate is fixed for the life of the bond.
Fixed Income Investments - Event Risk
Event risk is the risk that an issuer will not being able to make a payment because of dramatic and
unexpected events. Such risks can affect a single issuer or an entire sector depending on the type of risk.
Event risk falls into three categories:
a) Natural Disasters or Industrial Accidents that hamper their ability to make payments causing a
downgrade in their credit rating.

Things like floods, earthquakes, or a plant catching fire.

For example, if a hurricane hits the FloridaCoast, all of the insurers that have underwritten
policies there may see their bonds take a hit if the damage is large and they have to make big
payments to their policy holders.
b) Corporate Takeovers/Restructuring

Happens when a company is taken over or restructured, causing the firm to take on new or
additional debt that may be too heavy for them to make their payments of interest and principal.

The company may also have to issue this new or additional debt at higher yields, which will also
increase the debt burden.

This could also cause the rating agencies to get spooked and downgrade the issue.
c) Regulatory Risk

Comes in various forms and extends across several industries such as investment companies,
insurance companies and depository institutions.

If a change occurs that causes an entity to divest itself form certain forms of investments, the
sudden flood of the divesture of those investments could depress the market and the price of those
securities and similar types of investments.
Fixed Income Investments - Pricing Bonds
How bond coupon rates and market rates affect bond price

If a bond's coupon rate is above the yield required by the market, the bond will trade above its par
value or at a premium. This will occur because investors will be willing to pay up on the bonds
price to achieve the additional yield. As investors continue to buy the bond, the yield will
decrease until it reaches market equilibrium. Remember that as yield decrease, bond prices
rise.

If a bond's coupon rate is below the yield required by the market, the bond will trade below its par
value or at a discount. This happens because investors will not buy this bond at par when other
issues are offering higher coupon rates, so yields will have to increase, which means the bond
price will drop in order to induce investors to purchase these bonds. Remember that as yields
increase, bond prices fall.
The relationship between the price of a callable bond, an option free bond, and the price of the
embedded call option
The price of a callable bond will always be truncated when compared to an option- free bond. This is
because a bond with a call option will have a set price at which the bond can be called. An option-free
bond does not, and it could trade into higher prices than the stated call price of the embedded option bond.
Example:
You have two bonds that are alike except one of them can be called this year at a price of $102. If market
rates decline, the price of both bonds will increase. The option free bond could trade up to $105, however.
This is because of the possibility that the callable bond will be called by the issuer and, therefore, it will
tend to hold the dollar price at the call price of $102. Why would you buy this bond at $105 when you
will only receive $102 if the issuer calls the bond?
The price of the embedded call option will rise when interest rates decrease. This is because as rates
decrease, the possibility that the bonds will be called increases, which adds value to the call option. As the
price of the call option increases, the dollar price of the callable bond will decrease or be maintained at
the call price.
When market rates increase, both issues will tend to decline in value lock step. This is because there will
be less of a chance that the bonds will be called; this will make the callable bond act like an option free
bond. The price of the option will also decrease, which will also make it trade and be valued more like an
option free bond.
Interest Rate Risk of a Floating-Rate Security
Although a floating rate helps to reduce interest-rate risk because of the reset of the rate at periodic times;
the price can fluctuate for three reasons:
1. The longer the reset period, the greater the potential price movements will be. The basis is the same as
it is for a fixed-rate bond. The longer the time to maturity or the longer to reset, the more market events
can affect the price of the bond.
2. The required margin could change. For example, let's say that when an investor originally purchases a
bond, the spread over the index that was required by the market was 15 basis points. Because of market
events, however, the spread that is required by the market is now 45 basis points. This would cause the
floater's price to decline. The opposite would occur if the spread came in to 5 basis points.
3. Floaters usually come with caps. Once this cap value is breached, the reset interest rate can no longer
go above a certain level. This will cause it to act much like a fixed-rate security in a rising interest rate
environment, meaning the value of the floater will decrease because it can't keep up with current market
rates.
Fixed Income Investments - Duration
Duration is an estimated measure of the price sensitivity of a bond to a change in interest rates. It can be
stated as a percentage or in dollar amounts. It can be helpful to "shock" or analyze what will happen to a
bond when market rates increase or decrease.
Example:
Let's assume that the calculation yields a duration of 6.14, this means that if interest rates change, the
value of the bond will change by 6.14%. If there is a 50 basis point change, the value will change by
3.07% and for a 25 basis point change would equal a 1.53% change.
Calculating Duration
The easiest way to calculate duration or the percentage price change is to average the percentage price
change coming from an equal increase and decrease in interest rates measured in basis points. To
compute duration one needs to develop a valuation model to determine the new prices. The
duration value is only as good as the valuation model.
Example:
Stone & Co. bonds are selling at 95, yielding 5.25%
Let's assume that yields increase by 25bps, causing the price to decline to 93.
Therefore, the price changes by 2.1%.
Then, take 2.1% and divide it by 25bps equaling a .084% change. This represents a 1 basis points move.
Now let's assume that yields decrease by 25bps, causing the price to increaseto 98. The price change is
now 3.06%.
Then, take 3.06% and divide it by 25bps equaling a .1224% change. This represents a 1 basis points
move.
As a final step, just average the two percentage price changes for a 1 basis points move in rates.
Answer:
Here's how the calculation should look:
Duration = (.084 + .1224)/2 = .1032 = price change of 10.32% (.1032*100).
Formula 15.4
Duration = Price if yield decline - Price if yield increase / 2 * (initial
price) *change in yield in decimals
As such:
98-93/ 2*95*.0025 = 10.52
Approximate Percentage Price Change of a Bond Given a Change in Duration
Let's continue with the above duration of 10.52. This would equal a percentage price change of 10.52 %
for a change of 100 basis points in either direction. If the basis points change were 50, then the percentage
price change would be 5.26% (10.52/2). If it were a 25bps change, the value would be 2.63% (10.52 / 4).
Approximate New Price of a Bond Given the Duration and New Yield Level
Let's return once again to working with a duration of 10.52. This time, we'll add a total market value of
the Stone & Co bonds of $10,000,000.
Assume that the rates change by 100 bps. This would cause the value of the bonds to change by $
1,052,000 ($10,000.000 *.1052). This is also known as dollar duration.The price will then range from
$11,052,000 to $8,948,000.
If rates increase by 50 basis points, however, the dollar change would be $526,000 giving the bonds a
price range of $ 10,526,000 to $ 9,474,000.
Duration and Yield-Curve Risk for a Portfolio of Bonds
Portfolios have different exposures to how the yield curve shifts. These differences represent yield-curve
risk. Because a portfolio tends to have different maturities, if there is not a parallel movement of rates or
an equal amount of change in the yield curve across all maturities of the yield, the durations for the
different maturities will not react in the same manner. Therefore, the simple procedure discussed above
concerning duration will not be able to be applied to the entire portfolio, but will have to be applied over
the different maturities as well as to the amount of rate changes in those maturities.
Fixed Income Investments - International Bonds
In general, a country has two bond markets: An Internal Market and an External Bond Market.
The internal market is the mechanism for trading securities within the country in which the issuers are
based. This includes the country's domestic and foreign markets.
There are several types of international bonds traded on the External Market:

Foreign Bonds - Bonds by issuers who do not reside in the country where they are issued and
traded. An example of a foreign would be a bond that is issued by a non-U.S. entity but then
trades in the U.S. market. Such bonds can be issued in any currency and can have colorful
nicknames such as "Yankee Bonds", which are foreign bonds issued in the U.S., or "Bulldog
Bonds", which are sterling-denominated bonds traded in the U.K. foreign bond mark. One last
type of a foreign bond is a Supranational. These bonds are issued when two or more central
governments issue foreign bonds to promote economic development for the member countries.
These include bonds issued by the International Bank for Reconstruction and Development, or
World Bank, and the International American Development Bank.

Eurobonds - Bonds are bonds issued in a different currency denomination than that of the
country in which the bond is issued. Eurobonds are considered the external market for a country,
or its international bond market. They are classified by their currency denomination. For example,
Eurodollars are denominated in USD while a Euroyen bond would be denominated in Japanese
Yen. All Eurobonds have four features:
1. Underwritten by an international syndicate
2. When issued, offered simultaneously to investors in a number of countries
3. Issued outside the jurisdiction of any single country
4. They are in unregistered form.

Global Bonds - A bond that is issued and traded in the foreign bond market of one or more
countries as well as in the Eurobond market

Sovereign Bonds - Bonds issued by a country's central government. Sovereign bonds tend to be
the largest sector of a bond market in any country. They can be issued in their home country, the
Eurobond market or the foreign sector of another country. They are typically denominated in the
home country's currency, however, they are not required to be. They also have two different
ratings:
1. Local Currency Debt Rating
2. Foreign Currency Debt Rating
Why two different ratings? The defaults of the bonds tend to differ based on the currency denomination.
In general, there are greater defaults on the foreign currency denominated debt. The reason for this is that
a government can raise taxes and can control its own financial system. When dealing with a foreign
currency, this element of control is lost because the foreign currencies are purchased in the open market.
Therefore, if the local currency has depreciated in the markets as compared to the foreign currency, it will
be that much harder for an issuer to pay off his obligation.
Fixed Income Investments - Government Bonds
The U.S. Government issues four types of securities:
1.Treasury Bills - Treasury bills have a maturity of less than 12 months, no coupon rate, are issued at a
discount to par value, mature at par value and pay no coupon interest. The return the investor receives is
the difference between the purchase price and the maturity price.
2.Treasury Notes - Treasury notes have a maturity of one to ten years. They have a coupon rate set by
the market place at issue. They are issued approximately at par value and mature at par value.
3.Treasury Bonds - Treasury bonds are the same as treasury notes except that they have maturities that
are greater than ten years. The U.S. Government has not issued this type of bond for a while, but there are
still some issues that are outstanding.
4.Treasury Inflation Protected Securities (TIPS) - TIPS are issued as notes or bonds and help to
protect the investor against inflation risk. The example below illustrates how these securities work.
Example:

The coupon rate on an issue is set at a fixed rate, which is determined when the bonds are
auctioned.

This is considered the "real rate" because it is what the investor will earn over the inflation rate.

The inflation index the government uses is the non-seasonally adjusted U.S. City Average All
Item Consumer Price Index for All Urban Consumers (CPI-U)

These indexes work in deflationary environments as well, but the government has structured them
so that the investor receives the higher inflation-adjusted amount, or par value, when redeemed at
a later date.
Now on to the number crunching part concerning TIPS:
Coupon Rate = 4%
Annual Inflation Rate = 2%
Investor buys 1,000,000.00 USD of TIPS
At the end of the first six months the investor's coupon she will receive:
Inflation rate (2%/2) =1%
Coupon rate (4%/2) = 2%
Answer:
Inflation adjusted principle amount = (par value x 1+ semi-annual inflation rate) = 1,000,000 x 1.01 =
1,010,000.00
Coupon Payment = (Inflation adjusted principle amount x semi-annual coupon rate) = 1,010,000 x .02 =
20,200.00
Now let's move ahead another six months:
Coupon rate= 4% or 2% in semi-annual terms
Inflation rate = 3 % or 1.5 % in semi-annual terms
Inflation adjusted principle amount = (new par value x 1 + semi-annual inflation rate) = 1,010,000 x
1.015 = 1,025,150
Coupon Payment = (Inflation adjusted principle amount x semi-annual coupon rate) = 1,025,250 x .02 =
20,503.00
On-the-run vs. Off-the-run Government Securities
On-the-run Securities
On-the-run securities are the most current security issued by the U.S.Treasury Department. These issues
tend to be more liquid in the marketplace.
Off-the-run Securities
Off-the-run securities are the securities that are replaced by the on-the-run securities. These issues tend to
be less liquid in the marketplace.
How Stripped Government Securities, & Coupon and Principal Strips Are Created
The U.S. government does not issue zero coupon notes and bonds and there is a strong demand for an
instrument with no credit risk and a maturity of greater than one year. Based on consumer demand,
therefore, the private sector created securities with these features.

Let's look at a treasury bond that has five years to maturity with a coupon rate of 7 %. This
constitutes ten interest payments of US$70 based on $1,000 par value and one principal payment
of $1,000 for a total of 11 payments.

You now can discount these 11 single payments and create zero coupon instruments with
maturity dates that correspond with the payment dates of the Treasury securities.

These are issued through the Treasury's Separate Trading and Registered Interest and Principal
Securities (STRIPS) program to facilitate the stripping process. The zero-coupon securities
created are the obligations of the U.S. Government.
These securities come in two different forms:
1. Coupon Strips
Coupon strips come from the coupon payment part of the security.
2. Principal Strips
Principal strips come from the principal payment.
The difference between coupon strips and principal strips, besides maturity dates and amount received,
has to do with taxes. Coupon strips accrue interest and are taxed each year even though interest is not paid
until maturity. This causes negative cash flows for a taxable entity. Foreign investors often like principal
strips because of the preferred tax treatments they can receive in their home countries.
Fixed Income Investments - Mortgage-Backed Securities (MBS)
An investment instrument that represents ownership of an undivided interest in a group of mortgages.
Principal and interest from the individual mortgages are used to pay investors' principal and interest on
the MBS.
When you invest in a mortgage-backed security you are lending money to a homebuyer or business. An
MBS is a way for a smaller regional bank to lend mortgages to its customers without having to worry if
the customers have the assets to cover the loan. Instead, the bank acts as a middleman between the
homebuyer and the investment markets.
A mortgage-backed security (MBS) is secured by the collateral of mortgages on real estate for which the
borrower has agreed to make a predetermined series of payments. The mortgage gives the lender the right
to take a property in case the borrower fails to make the payments on his loan, thus ensuring that the debt
is paid off. These securities are amortizing, meaning they will decrease to zero as the payments are made.
The cash flows consist of a principal payment and an interest payment that can be paid in full at anytime
by the borrower. The investor in an MBS does not receive the full payment made by the borrower because
the issuer charges servicing fees for doing the administrative work and prepayments.
Example:
Let's look at a $150,000 mortgage with a mortgage rate of 6%, a monthly payment of $1,000 and a term
of 30 years or 360 months.
Beginning
Month
Month
Mortgage
Balance
Mortgage
Payment
Interest
End of
Scheduled
Month
Principle
Mortgage
Repayment
Balance
1
$1,000
$750
$250
$150,000
$149,750
2
$149,750
$1,000
$748.75
$251
$149,498.75
3
$149,498.75 $1,000
$747.49
$252.51
$149,246.62
This process continues until the mortgage balance reaches zero, either by the scheduled payments or
through any sort of prepayment. As you can see the interest decreases through the term of the loan as the
mortgage balance decreases. This also means that that as the loan matures, more of the scheduled
mortgage payment is applied to the mortgage balance.
Prepayment
Prepayment occurs when a bond's payments to its holders incorporates both interest and principal.
Typically, in asset-backed securities (ABS) and Mortgage-backed securities (MBS) there is always a
chance for a prepayment. To go back to an old example, a homeowner may only have to pay $500 a
month on his mortgage, but decide to pay $700 a month. This additional amount is an example of
prepayment. It can occur in chunks like this or it may be paid off in one lump sum.
Risk of Prepayment
The risk of prepayment is that they typically occur in declining rate environments. When this happens,
individuals tend to refinance their mortgages or credit cards at lower rates, causing the securities that were
made of these obligations to be prepaid before their stated maturity date. This causes the investors in these
securities to have to reinvest their proceeds at a lower market rate.
Fixed Income Investments - Federal Issues
Central governments can develop entities that issue bonds. These securities are referred to as semigovernment bonds or government agency bonds. In the U.S. they are referred to as federal agency
securities.
The agency bond market can be further broken down into two categories:
1. Federally Related Institutions
Federally related institutions are arms of the federal government. They include Export-Import, Tennessee
Valley Authority (TVA), Government National Mortgage Association (Ginnie Mae). With exception to
TVA and the Private Export Funding Corp., these securities are backed by the full faith and credit of the
U.S. Government.
2. Government Sponsored Enterprises (GSEs)
Government sponsored enterprises are privately owned, publicly chartered entities that were developed to
help lower the cost of funding in certain sectors of the marketplace that the government feels are
important enough to warrant assistance.
They include the more familiar names such as:

Federal National Mortgage Association (Fannie Mae) - provides credit for the residential
housing sector.

Federal Home Loan Mortgage Corporation (Freddie Mac) - provides credit for the residential
housing sector.

Federal Home Loan Bank - provides credit for the residential housing sector.

Federal Agriculture Mortgage Corporation - provide credit for farm proprieties

Federal Farm Credit System - provide credit for agricultural part of the economy

Student Loan Marketing Association (Sallie Mae) -provides support for higher education.
GSEs issue two forms of debt: debentures are notes or bonds with typical maturities of one to 20 years,
while discount notes are short-term papers with maturities ranging from overnight to 360 days.
Fannie Mae and Freddie Mac, as noted above, provide credit and support for the housing sector. In doing
so, they issue securities that are backed by the mortgage loans that they purchase.
The loans act as collateral for the bonds and they come in three forms:
1. Mortgage Pass through Securities: Mortgage pass through securities are created when one or
more bondholders form a pool (or collection) of mortgages and sell shares or certificates in the
pool. The cash flows depend on the payments of the mortgage and opens the investor to
prepayment risk. The monthly cash flows include net interest, scheduled principal payments and
any principal prepayments.
2. Collateralized Mortgage Obligations (CMOs): CMOs are a derivative securities. They help an
investor pick the type of cash flows he wants to be exposed to based on how the pool of
mortgages are sliced up into tranches. The tranches offer investors different payment rules and
par values. For example Tranche A might receive all principal payments until the balance is zero
then the payments would flow to Tranche B and so on.
3. Stripped Mortgage-Backed Securities: For CFA exam purposes, just know that the name exists
in case they want three examples of Freddie or Fannie.
Motivation Behind CMO Creation
The motivation behind creating a CMO is to spread the risk of prepayment among different classes of
bonds. A CMO has several tranches that splits the mortgage pool into different layers. These layers have
different par values and prepayment speeds. This aids investors in helping them manage the risk
exposures they want in this arena.
Fixed Income Investments - Bondholder's Rights
The Bankruptcy Process
Bankruptcy - this is the dirty little word that most debt holders hate to hear. It occurs when an entity can
no longer make the payments to its creditors. Bankruptcy grants the entity protection from creditors. The
entity can then decide whether to liquidate the company by selling everything and leaving no surviving
entity or to have a reorganization. With a reorganization, a new entity will emerge after the bankruptcy. A
bankruptcy can occur in two ways:

Voluntary - the company decides to pull the plug by itself.

Involuntary
Fixed Income Investments - Other Types of Bonds
Corporate Bonds vs. Medium-term Notes
The main difference between medium-term notes and corporate bonds is the way they are issued in the
marketplace. MTNs can be offered to investors by the issuer's agent instead of being underwritten by
investment banks and then sold to the public in one shot.

This helps to cover the funding gap between commercial paper and long-term bonds, hence the
"medium-term" designation.

When a firm what to issue this type of debt they have to file a "shelf registration" that lists the
details of the offering, such as rates, maturities and the investment banks acting as their agent.

It is also important to know that MTNs can also come in different structures instead of mirroring
a corporate bond. These include: step up notes, inverse floaters, deleveraged floaters, range notes
and index amortizing notes.
What is a Structured Note?
A synthetic medium-term debt obligation with embedded components and characteristics that adjust the
risk/return profile of the security. A structured note is a hybrid security that attempts to change its profile
by including additional modifying structures. A simple example would be a 5 year bond tied together with
an option contract for increasing the returns. A motivation for their issuance is the fact that they allow
investors to realize a profit from favorable price movements.
What is Commercial Paper?
Commercial paper is a short term unsecured promissory note that is fewer than 270 days to maturity and
is issued as a zero-coupon security. Companies continue to "roll over" or pay off the holders by issuing
new commercial paper in the market. The risk to investors is that the issuing company will not be able to
place the new commercial paper to pay off their older debt.
Commercial Paper is issued in two ways:
1. Directly Placed: The issuing company sells the paper directly to the investing public without the
help of an agent or intermediary. An example would include GE Capital.
2. Dealer-Placed: The issuing company uses an agent to help sell its paper in the marketplace.
Commercial paper has its own credit rating and can de divided into financial and non-financial
companies.
Bank Obligations
Negotiable CDs - A savings certificate entitling the bearer to receive interest. A CD bears a maturity
date, a specified fixed interest rate and can be issued in any denomination. CDs are generally issued by
commercial banks and are insured by the Federal Deposit Insurance Corporation (FDIC). The term of a
CD generally ranges from one month to five years.
A certificate of deposit is a promissory note issued by a bank. It is a time deposit that restricts holders
from withdrawing funds on demand. Although it is still possible to withdraw the money, this action will
often incur a penalty.
For example, let's say that you purchase a $10,000 CD with an interest rate of 5% compounded annually
and a term of one year. At year's end, the CD will have grown to $10,500 ($10,000 * 1.05).
CDs of less than $100,000 are called "small CDs"; CDs for more than $100,000 are called "large CDs" or
"jumbo CDs". Almost all large CDs, as well as some small CDs, are negotiable.
Bankers Acceptances - A short-term credit investment created by a non-financial firm and guaranteed by
a bank. Acceptances are traded at a discount from face value on the secondary market. Banker's
acceptances are very similar to T-bills and are often used in money market funds.
Fixed Income Investments - Asset-Backed Securities (ABS)
An asset-backed security is a security that is backed by a pool of loans or receivables. These include: auto
loans, consumer loans, commercial assets (planes, receivables), credit cards, home equity loans, and
manufactured housing loans.
ABSs are essentially the same thing as a mortgage-backed security except that the security is backs assets
such as loans, leases, credit card debt, a company's receivables, royalties, etc and not mortgage based
securities.
Special Purpose Vehicles and Their Effect on Asset-backed Securities
SPVs are also referred to as a "bankruptcy-remote entity" whose operations are limited to the acquisition
and financing of specific assets. The SPV is usually a subsidiary company with an asset/liability structure
and legal status that makes its obligations secure even if the parent company goes bankrupt.
Thanks to Enron, SPVs/SPEs are household words. These entities aren't all bad though. They were
originally (and still are) used to isolate financial risk.
A corporation can use such a vehicle to finance a large project without putting the entire firm at risk.
Problem is, due to accounting loopholes, these vehicles became a way for CFOs to hide debt. Essentially,
it looks like the company doesn't have a liability when they really do. As we saw with the Enron
bankruptcy, if things go wrong, the results can be devastating.
Why Issue Asset-Backed Securities?
The primary motive for issuing asset-backed securities is to take an asset, such as a receivable, a loan or
some other form of illiquid asset, and move it off the balance sheet. This helps the parent to clean up its
balance sheet and monetize those receivables rather than waiting for the payments to come in. It can also
help protect those assets in case the parent defaults. This is possible because the SPV that was created is a
separate entity.
Types of Credit Enhancements
Credit enhancement is designed to reduce risk. It comes in two forms:
1. Internal Enhancements: Internal enhancements come in the form of reserve funds over
collateralization and senior/subordinated structures. These will be covered in more detail in the
CFA Level II exam.
2. External Enhancements: External enhancements come in three forms of third-party guarantees.
These include: corporate guarantee, letter of credit and bond insurance. The enhancement can
come from the parent company or from the newly created company that holds the assets. One
problem with external enhancements is that one not only has to analyze the assets and the
company that owns them but also the company that is "wrapping" or insuring the debt.
Collateralized Debt Obligations
An investment-grade security backed by a pool of bonds, loans and other assets. CDOs do not specialize
in one type of debt but are often non-mortgage loans or bonds.
Similar in structure to a collateralized mortgage obligation (CMO) or collateralized bond obligation
(CBO), CDOs are unique in that they represent different types of debt and credit risk. In the case of
CDOs, these different types of debt are often referred to as 'tranches' or 'slices'. Each slice has a different
maturity and risk associated with it. The higher the risk, the more the CDO pays.
Fixed Income Investments - Yield Curves
The U.S Federal Reserve (the Fed) has four tools it uses to directly influence short-term and, indirectly,
long-term rate as well. They are:
1. Open Market Operations: The Fed buys Treasuries or adds funds to the system; this reduces
short-term rates. The Fed also sells Treasuries or takes funds out of the system to increase shortterm rates.
2. The Discount Rate: This is the rate at which banks can borrow on a collateralized basis at the
Fed's discount window. If the Fed raises rates, they makes it more costly for the banks to do
business, which drains cash from the system. If the Fed eases this rate, banks will find it cheaper
to borrow additional funds, which will add cash to the system.
3. Bank Reserve Requirements: This is hardly used these days. If the Fed raises these
requirements, money is kept out of the economy. If they lower the rate, additional money will hit
the economy.
4. Verbal persuasion to influence how bankers supply credit to businesses and consumers:
This simple method requires no additional explanation.
What is a Yield Curve?
A yield curve represents the relationship between maturity and yields. As an example:
1 Month
3 Month
6 Month
1 Year
2 Year
5 Year
10 Year
30 Year
1.00%
1.25%
1.50%
1.75%
2.00%
2.35%
2.68%
3.00%
If you were to graph this data you would see the yield curve develop. This date is only good for one single
point in time because rates are constantly moving. If you are searching for a point on the yield curve that
does not have a maturity represented by an actual "on the run security", that point will only be an
approximation.
Yield Curve Shapes
Yield Curves come in three shapes:
1. Upward or Normal Yield Curve: This curve occurs when short-term rates are lower than longterm rates, as noted in the above example.
2. Inverted Yield Curve: This curve is formed when short-term rates are higher than the longer part
of the curve.
3. Flat Yield Curve: This curve occurs when there is little or no change between short-term and
long-term rates.
Fixed Income Investments - The Term Structure of Interest Rates
There are three main theories that try to describe the future yield curve:
1. Pure Expectation Theory: Pure expectation is the simplest and most direct of the three theories.
The theory explains the yield curve in terms of expected short-term rates. It is based on the idea
that the two-year yield is equal to a one-year bond today plus the expected return on a one-year
bond purchased one year from today. The one weakness of this theory is that it assumes that
investors have no preference when it comes to different maturities and the risks associated with
them.
2. Liquidity Preference Theory: This theory states that investors want to be compensated for
interest rate risk that is associated with long-term issues. Because of the longer maturity, there is a
greater price volatility associated with these securities. The structure is determined by the future
expectations of rates and the yield premium for interest-rate risk. Because interest-rate risk
increases with maturity, the yield premium will also increase with maturity. Also know as the
Biased Expectations Theory.
3. Market Segmentation Theory: This theory deals with the supply and demand in a certain
maturity sector, which determines the interest rates for that sector. It can be used to explain just
about every type of yield curve an investor can came across in the market. An offshoot to this
theory is that if an investor wants to go out of his sector, he'll want to be compensated for taking
on that additional risk. This is known as the Preferred Habitat Theory.
Implications of the Yield Curve for the Yield-Curve Theories
1. Pure Expectation Theory
According to this theory, a rising term structure of rates means the market is expecting short-term rates to
increase. So if the two-year rate is higher than the one-year rate, rates should rise. If the curve is flat, the
market is expecting that short-term rates will remain low or hold constant in the future. A declining rateterm structure indicates the market believes that rates will continue to decline.
2. Liquidity Preference Theory
Under this theory, the curve starts to get a little bit more bent. With an upward sloping yield curve, this
theory really has no opinion as to where the yield curve is headed. It could continue to be upward sloping,
flat, or declining, but the yield premium will increase fast enough to continue to produce an upward curve
with no concerns about short-term interest rates. When it comes to a flat or declining term structure of
rates, this suggests that rates will continue to decline in the short end of the curve given the theory's
prediction that the yield premium will continue to increase with maturity.
3. Market Segmentation Theory
Under this theory, any type of yield curve can occur, ranging from a positive slope to an inverted one, as
well as a humped curve. A humped curve is where the yields in the middle of the curve are higher than
the short and long ends of the curve. The future shape of the curve is going to be based on where the
investors are most comfortable and not where the market expects yields to go in the future.
Fixed Income Investments - Types of Yield Measures
faEven though the way most investors discuss spreads is based on a Treasury security with the same
maturity as the one it is being compared to, an investor can also talk about spreads between any two
bonds with the following measures:
1. Absolute Yield Spread
This is the way most spreads are measured in the market. This spread measures the difference in spread
between two bonds in terms of basis points.
The equation is: Yield Spread = Yield on Bond A - Yield on Bond B
2. Relative Yield Spread
This ratio measures the yield spread relative to the reference bond.
This equation is: Relative Yield Spread = Yield on bond A - Yield on Bond B/ Yield on Bond B
3. Yield Ratio
This is just the ratio of the yields between the two bonds.
The equation is: Yield Ratio = Yield on Bond A / Yield on Bond B
Market convention is to use the on-the-run government security as the reference yield or bond. So in the
above equations, one would replace Bond B with the comparable government security.
Example: Yield Ratios
We want to compare an IBM five-year bond with a yield of 4.5 % and the on- the-run government fiveyear with a yield of 3.75%
Answer:
Absolute Yield Spread = 4.5% - 3.75% = .75% or 75 basis points
Relative Yield Spread = 4.5% - 3.75% / 3.75% = .20 = 20%
Yield Ratio = 4.5% / 3.75% = 1.20
Why Relative Spreads Are Better
Investors may find relative spreads a better measure because they measure the magnitude of the yield
spread and the way it is affected by interest-rate levels. While absolute spread may be maintained as rates
change, relative spreads will move in or out depending on the level of rates.
Example:
Use the IBM and Treasury bond from the previous example, except now assume that yields have
increased.
Absolute Yield Spread 5.75% - 5.00% = .75% or 75 basis points. Even though yields have increased the
spread is the same. However, the Relative Spread has changed too:
Answer:
5.75% - 5.00% / 5.00% = .15 or 15%.
This example shows that the relative spread can give an investor a better reading of how spreads are
actually moving relative to the generic yield spread.
Fixed Income Investments - Intermarket vs. Intramarket Sector Spreads
The bond market is carved into different sectors based on the issuer. Typically, these sectors are:
1. U.S. Government Securities
2. U.S. Government Agency Securities
3. Municipal Securities
4. Corporate Bonds
5. Mortgage Backed Bond
6. Asset Backed Bonds
7. Foreign Bonds
These sectors also can be broken down even further. For example, in the Corporate Sector, issuers can fall
into one and sometimes more categories such as industrial, utilities, financials and bank.
Spreads tend to be wider the farther one goes out the curve. Spreads can be based on individual sectors or
crossed between them.
Intermarket Sector Spreads
Intermarket sector spreads deal with the yield spreads between two bonds in different sectors of the
market. The most popular of these is a non-treasury security as opposed to a comparable treasury security.
A comparable treasury security would be one with the same maturity.
Intramarket Sector Spreads
Intramarket sector spreads deal with the yield spread between two bonds in the same market sector. This
can be done by developing a yield curve that is similar to the treasury yield curve but instead using the
issuers' securities to develop the curve.
Some other factors that affect spreads between bonds besides maturity are credit risk, any options that the
bonds may have, the liquidity of the issuers and the tax bracket of investors who receive interest
payments.
Credit Spreads and Their Relationship to Economic Activity
A Credit Spread is the yield spread between non-treasury and treasury securities. These are equal in all
respects except their individual credit ratings. This means that their maturities are the same and that there
are no options thrown into the equation.
Look Out!
It is important to note that spreads increase with maturity
and lower credit ratings.
Spreads interact with economic growth or decline in two key ways:
1. Spreads narrow or tighten - When the economy is growing, cash flows are increasing. Therefore, a
corporation should have an easier time paying off its debt. Individuals will purchase more non-treasury
securities than treasury securities because the increased economic activity reduces the default risk,
causing spreads to tighten.
2. Spreads widen - When economy is faltering or slowing down, spreads widen. When this happens, the
possibility of defaults increases because cash flows are declining. Individuals will sell or dump nontreasury securities for government securities because there is less of a chance that the government will
default on their debt when compared to a corporation. This is also known as a flight to safety.
Fixed Income Investments - Options and their Benefits
For options that benefit the issuer, such as calls, investors will want yield spreads that are greater than
bonds and that do not have options embedded in them. Because there is a risk that the bonds will be
called, investors want a higher yield to compensate for that risk, causing the spread to widen over the
treasury security when compared to bonds without options. The longer the call period, the less spread
widening investors will be needed because of a longer protection period against the call.
Options That Benefit the Holder
For options that benefit the holder, such as puts, investor will require a smaller yield spread than bonds
that do not have embedded options in them, such as treasury bonds. There is even the possibility that the
coupon rate could be lower than the treasury coupon rate, depending on how favorable the option is to the
investors.
Spreads and Liquidity
When issues are less liquid, yield spreads tend to widen because there are fewer bonds to buy or it is
harder to find a buyer. When issues are more liquid, such as on-the-run treasuries, yield spreads are
tighter or narrower because there are plenty of buyers and sellers.
The larger the issue size, the more liquidity compared to a smaller issues in the market leads to tighter or
narrower spreads and vice versa
Fixed Income Investments - After Tax Yield of a Taxable Security
The after-tax yield is the yield on a taxable bond after federal income taxes are paid. It is computed with
the following formula.
Formula 14.5
After-tax yield = pre-tax yield x (1- marginal rate)
The marginal rate will vary depending on the tax bracket the investor is at that given time.
Example: Taxable Bond Yield
Taxable bond yield is 7.5%
The Marginal tax rate for this investor is 31%
Answer:
After-tax yield = .075 x (1-.31)
= .05175
= 5.175%
Tax-Equivalent Yield
The tax-equivalent yield is the yield that must be offered on a taxable bond issue to give the same aftertax yield as a tax-exempt issue. It is computed with the following formula.
Formula 14.6
Taxable-equivalent yield = tax-exempt yield / (1- marginal tax rate)
Example: Tax-Exempt Yield
Tax exempt yield = 5.00%
Marginal Tax Rate = 31%
Answer:
Taxable-equivalent yield = .05 / (1-.31)
= .05 / .69
= .072464
= 7.2464 %
This means that a taxable issue must yield more than 7.25 % for the investor at the 31% tax bracket in
order to beat the 5% yield offer in the tax-exempt bond.
Look Out!
Notice that the higher the marginal tax rate, the higher the
taxable equivalent yield would be needed in the taxable
bond market.
Fixed Income Investments - London Interbank Offer Rate (LIBOR)
An interest rate at which banks can borrow funds, in marketable size, from other banks in the London
interbank market. The LIBOR is fixed on a daily basis by the British Bankers' Association. The LIBOR is
derived from a filtered average of the world's most creditworthy banks' interbank deposit rates for larger
loans with maturities between overnight and one full year.
The LIBOR is the world's most widely used benchmark for short-term interest rates. It's important
because it is the rate at which the world's most preferred borrowers are able to borrow money. It is also
the rate upon which rates for less preferred borrowers are based. For example, a multinational corporation
with a very good credit rating may be able to borrow money for one year at LIBOR plus 4 or 5 points.
Countries that rely on the LIBOR for a reference rate include the United States, Canada, Switzerland and,
of course, England
Fixed Income Investments - Bond Valuation Basics
The fundamental principle of valuation is that the value is equal to the present value of its expected cash
flows. The valuation process involves the following three steps:
1. Estimate the expected cash flows.
2. Determine the appropriate interest rate or interest rates that should be used to discount the cash flows.
3. Calculate the present value of the expected cash flows found in step one by using the interest rate or
interest rates determined in step two.
Fixed Income Investments - Cash Flow
Bonds With Difficult Expected Cash Flow Estimation
The bonds for which it is difficult to estimate expected cash flows fall into three categories:
1. Bonds for which the issuer or investor has an option or right to change the contract due date for
the payment of the principal. These include callable bonds, puttable bonds, MBSs and ABSs.
2. Bonds for which coupon payment rate is reset occasionally based on a formula with values that
change, such as reference rates, prices or exchange rates. A floating-rate bond would be an
example of this type of category.
3. Bonds for which investor has the option to convert or exchange the security for common stock.
The problems when estimating the cash flows of these types of bonds include:
1. In the case of bonds for which the issuer or investor has the option/right to change the contract
due date for the payment of principal, the bonds can be affect by future interest rates. If rates
decline, a corporation may issue new bonds at a lower cost and call the older bonds. The same
thing happens with MBSs and ABSs. As rates decline, borrowers have the right to refinance their
loans at cheaper rates. This causes the bond to be paid off earlier than the stated maturity date.
2. When rates increase, a puttable bond will be sold back to the issuing corporation at the put price
once the increase in rates drives the price of the security below the put price.
3. For bonds in which the coupon payment rate is reset occasionally based on a formula with
changing values, because the rate is always changing based on other variables it is hard to
estimate the cash flows. Also, for bonds that give the investor the option to convert or exchange
the security for common stock, the cash flows will stop altogether once the investor decides that it
would be more profitable to exchange the fixed income security for equity. The investor will have
no certain idea as to when this may occur, making it difficult to value the cash flows until the
maturity of the bond.
4. Because the value of the bond rests on the performance of the securities that back the bond, it is
hard to determine whether the bonds may be converted into those securities.
Determining Appropriate Interest Rates
The minimum interest rate that an investor should accept is the yield that is available in the market place
for a risk-free bond, or the Treasury market for a U.S. investor. The Treasury security that is most often
used is the on-the-run issues because they reflect the latest yields and are the most liquid securities.
For non-treasury bonds such as corporate bonds, the rate or yield that would be required would be the onthe-run government security plus a premium that takes up the additional risks that come with non-treasury
bonds.
As for the maturity, an investor could just use the final maturity date of the issue compared to the
Treasury security. However, because each cash flow is unique in its timing, it would be better to use the
maturity that matches each of the individual cash flows.
Computing a Bond's Value
First of all, we need to find the present value (PV) of the future cash flows in order to value the bond. The
present value is the amount that would be needed to be invested today to generate that future cash flow.
PV is dependent on the timing of the cash flow and the interest rate used to calculate the present value. To
figure out the value the PV of each individual cash flow must be found. Then, just add the figures
together to determine the bonds price.
Formula 14.7
PV at time T = expected cash flows in period T / (1 + I) to the
T power
After you develop the expected cash flows, you will need to add the individual cash flows:
Formula 14.8
Value = present value @ T1 + present value @ T2 + present
value @Tn
Let's throw some numbers around to further illustrate this concept.
Example: The Value of a Bond
Bond GHJ matures in five years with a coupon rate of 7% and a maturity value of $1,000. For simplicity's
sake, the bond pays annually and the discount rate is 5%.
Answer:
The cash flow for each of the years is:
Year one = $70 Year Two = $70 Year Three = $70, Year Four is $70 and Year Five is $1,070.
PV of the cash flows is: Year one = 70 / (1.05) to the 1st power = $66.67
Year two = 70 / (1.05) to the 2nd power = $ 63.49
Year three = 70 / (1.05) to the 3rd power = $ 60.47
Year four = 70 / (1.05) to the 4th power = $ 57.59
Year five = 1070 / (1.05) to the 5th power = $ 838.37
Now to find the value of the bond:
Value = 66.67 + 63.49 + 60.47 + 57.59 + 838.37
Value = 1, 086.59
Fixed Income Investments - Bond Value and Price
How Does the Value of a Bond Change?
As rates increase or decrease, the discount rate that is used also changes appropriately. Let's change the
discount rate in the above example to 10% to see how it affects the value of the bond.
Example: The Value of a Bond when Discount Rates Change
PV of the cash flows is: Year one = 70 / (1.10) to the 1st power = $ 63.63
Year two = 70 / (1.10) to the 2nd power = $ 57.85
Year three = 70 / (1.10) to the 3rd power = $ 52.63
Year four = 70 / (1.10) to the 4th power = $ 47.81
Year five = 1070 / (1.10) to the 5th power = $ 664.60
Answer:
Value = 63.63 + 57.85 + 52.63 + 47.81 + 664.60 = $ 886.52

As we can see from the above examples, an important property of PV is that for a given discount
rate, the older a cash flow value is, the lower its present value.

We can also compute the change in value from an increase in the discount rate used in our
example. The change = 1,086.59 - 886.52 = 200.07.

Another property of PV is that the higher the discount rate, the lower the value of a bond and the
lower the discount rate the higher the value of the bond.
Look Out!
If the discount rate is higher than the coupon rate the PV will
be less than par. If the discount rate is lower than the coupon
rate, the PV will be higher than par value.
How Does a Bond's Price Change as it Approaches its Maturity Date?
As a bond moves closer to its maturity date, its price will move closer to par. The break down on the three
scenarios is as follows:
1. If a bond is at a premium, the price will decline over time towards its par value.
2. If a bond is at a discount, the price will increase over time towards its par value
3. If a bond is at par, its price will remain the same.
To show how this works lets use our original example of the 7% bond, but now let's assume a year has
passed and a discount rate remains the same at 5%.
Example: Price Changes Over Time
Let's compute the new value to see how the price moves closer to par. You should also be able to see how
the amount by which the bond price changes is attributed to it being closer to it's maturity date.
PV of the cash flows is: Year one = 70 / (1.05) to the 1st power = $66.67
Year two = 70 / (1.05) to the 2nd power = $ 63.49
Year three = 70 / (1.05) to the 3rd power = $ 60.47
Year four = 1070 / (1.05) to the 4th power = $880.29
Answer:
Value = 66.67 + 63.49 + 60.47 + 880.29 = 1,070.92
As the price of the bond decreases, it moves closer to its par value. Theamount of change attributed to the
year's difference is 15.67.
An individual can also decompose the change that results when a bond approaches its maturity date and
the discount rate changes. This is accomplished by first taking the net change in the price that reflects the
change in maturity, and then adding it to the change in the discount rate. The two figures should equal the
overall change in the bond's price.
Computing the Value of a Zero-coupon Bond
This may be the easiest of securities to value because there is only one cash flow - the maturity value.
Value of a zero coupon bond that matures N years from now is:
Formula 14.9
Zero coupon bond value = Maturity value / (1 + I) to the
power of the number of years x 2
Where I is the semi-annual discount rate.
Example: The Value of a Zero-Coupon Bond
For illustration purposes, let's look at a zero coupon with a maturity of three years and a maturity value of
$1,000 discounted at 7%
Answer:
I = 0.035 (.07 / 2)
N=3
Value of a Zero = 1,000 / (1.035) to the 6th power (3 x 2)
= 1,000 / 1.229255
= 813.50
Fixed Income Investments - Arbitrage-free Valuation Approach
Under a traditional approach to valuing a bond, it is typical to view the security as a single package of
cash flows, discounting the entire issue with one discount rate. Under the arbitrage-free valuation
approach, the issue is viewed, instead, as various zero-coupon bonds that should be valued individually
and added together to determine value. The reason this is the correct way to value a bond is that it does
not allow a risk-free profit to be generated by "stripping" the security and selling the parts at a higher
price than purchasing the security in the market.
As an example, a five-year bond that pays semi-annual interest would have 11 separate cash flows, and be
valued using the appropriate yield on the curve that matches its maturity. So the markets implement this
approach by determining the theoretical rate the U.S. Treasury would have to pay on a zero-coupon
treasury for each maturity. The investor then determines the value of all the different payments using the
theoretical rate and adds them together. This zero-coupon rate is the treasury spot rate. The value of the
bond based on the spot rates is the arbitrage-free value.
Determining Whether a Bond is Under or Over Valued
What you need to be able to do is value a bond like we have done before using the more traditional
method of applying one discount rate to the security. The twist here, however, is that instead of using one
rate, you will use whatever rate the spot curve has that coordinates with the proper maturity. You will
then add the values up as you did previously to get the value of the bond. You will then be given a market
price to compare to the value that you derived from your work. If the market price is above your figure,
then the bond is undervalued and you should buy the issue. If the market price is below your price, then
the bond is overvalued and you should sell the issue.
How Does a Dealer Generate Arbitrage Profits?
A dealer has the ability to strip a security or to take apart the cash flows that make up the bond. These
Treasury strips can be sold to investors. So if the market price of a Treasury security is less than the value
using the arbitrage-free valuation, a dealer will buy the security, strip the bond and then sell the Treasury
strips at a higher amount than the purchase price for the whole bond.
On the other hand, if the market price is more than the value using the arbitrage-free valuation, the dealer
will buy the strips, make the bond "whole" and sell it at a higher price than that of the purchased strips.
Fixed Income Investments - Typical Yield Measures
There are three sources of return an investor can expect to receive by investing in bonds:
1. The coupon payment made by the issuer.
2. Any capital gain or loss (negative dollar return) when the bond matures, is called or is sold. This is the
difference between the purchase price and the price when the bond is no longer owned by you.
3. Income from the reinvestment of interim cash flows such as interest payments and any prepayments of
principal prior to the final or stated maturity date. You take the interim payment and invest it in another
fixed income security to earn additional returns. This is also known as interest on interest.
This section is all about formulas and bond math; some of the questions you see on your CFA Level 1
exam will almost certainly come out of this section.
1. Current Yield
Current yield relates the annual dollar coupon interest to the bond's market price:
Formula 14.10
Current Yield = annual dollar coupon interest / price
Example: Current Yield
IBM ten-year bond with a rate of 5% and market price of 98.
Answer:
Step1 - Figure out the annual dollar coupon interest= .05 x $100 = 5$
Current Yield = $5 / 98 = .05102= 5.1%
Current yield is greater when bond is selling at a discount. The opposite is true for a premium bond. If a
bond is selling at par, the current yield will equal the coupon rate.
The drawback using current yield is that it only considers the coupon interest and nothing else.
2. Yield to Maturity (YTM)
Yield to maturity is the most popular measure of yield in the market. It isthe rate that will make the
present value of a bond's cash flows equal toits market price plus accrued interest. To find YTM, one has
to develop the cash flows and then, through trial and error, find the interest rate thatmakes the present
value of cash flow equal to the market price plus accrued interest.
This is basically a special type of internal rate of return (IRR).
Example: Yield to Maturity
An example using the above IBM bond the cash flows will consist of 20 payments of $2.50 every six
months and a payment, twenty six-month periods from now, of $100. The present values, using various
semi-annual discounts, are as follows:
Semi-annual Present
interest Rate Value
2.50%
100
2.60%
99.5
2.70%
99
2.80%
98.5
2.90%
98
When the rate of 2.9% is used the present value of the cash flows is equal to a price of $98.00. Hence the
semi-annual yield to maturity is 2.9%. Now that we have found this we must make it into a market
convention rate or the bond-equivalent yield. To get this yield, just double the semi-annual rate. In this
example, it would be 5.8% yield to maturity.
Bond Price, Coupon Rate, Current Yield and Yield to Maturity

For a bond selling at par:
Coupon Rate = Current Yield = Yield to Maturity

For a bond selling at a discount:
Coupon Rate < Current Yield < Yield to Maturity

For a bond selling at a premium:
Coupon Rate > Current Yield > Yield to Maturity
The limitations of the yield to maturity measure are that it assumes that thecoupon rate will be reinvested
at an interest rate equal to the YTM. Besides that it does take into considerationthe coupon income and
capital gains orloss as well as the timing of the cash flows.
3. Yield to First Call
Yield to first call is computed for a callable bond that is not currently callable. The actual calculation is
the same as the Yield to Maturity with the only difference being that instead of using a par value and the
stated maturity, the analyst will use the call price and the first call date in calculating the yield.
4. Yield to First Par Call
Again, yield to first par call is the same procedure as above, with the difference being that the maturity
date that will be used instead of the stated maturity date is the first time the issuer can call the bonds at par
value.
5. Yield to Refunding
Yield to refunding is used when the bonds are currently callable but there are certain restrictions on the
source of funds used to buy back the debt when a call is exercised. The refunding date is the first date the
bond can be called using a lower-cost debt. The calculation is the same as YTM.
6. Yield to Put
Yield to put is the yield to the first put date. It is calculated the same way as YTM but instead of the stated
maturity of the bond, one uses the first put date.
7. Yield to Worst
Yield to worst is the yield occurs when one calculates every possible call and put date that has the lowest
possible yield. For example if you calculate all the call dates and the yield comes out as follows 5.6%,
7.6%, 8.2% and 7.5%, the yield to worst would be 5.6%. This measure means little to the potential return;
it is supposed to measures the worst possible return the investor will receive if the bond is called or put.
8. Cash Flow Yield
Cash flow yield deals with mortgage-backed and asset-backed securities. The cash flows of these
securities are interest and principal payments. What makes this complicated is that the borrowers who
make up the mortgage or asset pool can prepay their loans in whole or in part prior to the scheduled
principal payment. Because of this, the cash flows have to be estimated and an assumption must be made
as to when these principle prepayments may occur. The rate that exists when the prepayments occurs is
called the prepayment rate or prepayment speed.Once this rate is estimated, a yield can be calculated. The
yield is the interest rate that will make the present value of the estimated cash flows equal the price plus
accrued interest.
Example: Cash Flow Yield
Because cash flows for these securities are usually monthly, a bond-equivalent yield must be developed.
The math here is a little different than in the above examples:
Step 1 - the effective semi-annual yield must be computed from the monthly yield by compounding it for
six months.
Effective semi-annual yield = (1 + monthly yield) to the 6th power -1
Step 2 - Double the effective semi-annual yield to get the annual cash flow on a bond equivalent basis.
Cash flow yield = 2 x {(1 + monthly yield) to the sixth power-1}
Answer:
So if the monthly yield is .8% then:
Cash flow yield = 2*{ 1.008) to the sixth power -1}
= 2 x .04897
= 9.79%
Fixed Income Investments - Assumptions Underlying Traditional Yield Curve Measures
The main underlying assumptions used concerning the traditional yield measures are:
1. The bond will be held to maturity.
2. Coupons can be reinvested at the yield to maturity
Limitations:
1. Current yield- Current yield only considers the coupon interest and no other sources for an investors
return. It does not take into consideration the capital gain when a bond is purchased at a discount or the
capital loss when the bond is purchased at a premium. Also, reinvestment income is not taken into
consideration.
2. Yield to Maturity - Yield to maturity measures assume that the coupon payments will be reinvested at
the coupon rate
3. Yield to Call - Yield to call assumes investor will hold the bond to the assumed call price and that the
issuer will call the bond on that date which both are unrealistic. Also, the comparison of different yields to
call with the YTM are meaningless because the cash flows stop once the issuer calls the bond.
4. Yield to Put - This assumes that coupon payments will be reinvested at the calculated yield and that
the bonds will be put on the first date.
5. Yield to Worst - This measure does not identify the potential return over some time horizon and fails
to take into account that the calculation for a YTW has different exposures to reinvestment risk.
6. Cash Flow Yield - Cash flow yield assumes that the coupons will be reinvested at the coupon rate and
that the bond will be held to maturity. However, because cash flow yield tend to be used for MBSs or
ABSs there is a risk that the bonds will be prepaid and the measure of cash flow yield will be thrown out
the window.
Fixed Income Investments - Importance of Reinvestment Income and Reinvestment Risk
Reinvestment income can make up a large portion of the return for a bond. Before beginning with
calculations, it is important to understand the difference between total future dollars, which is equal to all
the dollars an investor expects to receive and the total dollar return, which is equal to the dollars the
investor will realize from the three sources of income for a bond (coupon payment, capital gain/loss, and
reinvestment income)
Example: Reinvestment Income
Let's look at an investor that has $96 to invest in a certificate of deposit (CD) that will mature in five
years. The bank will pay 3% every six months, which equals a bond equivalent basis of 6%. The total
future value of this investment today would be:
Answer:
96 x (1.03) to the tenth power = $129.02
So the investment of $96 for five years at 6% on a BEY will generate $129.02
To further break it down:
Total Future Dollars = 129.02
Return of Principle = 96.00
Total interest = 33.02
Now let's turn to a bond that has a price of $96, five-year maturity and with a coupon of 5% and YTM of
6%. As shown above an investor must generate $129.02 to provide a yield of 6% or the total dollar return
must be $33.02.
So with this bond, the sources of return are a capital gain of: $4 ($100 - $96) and coupon interest of $2.50
for ten periods or $25. That equals $29 without the reinvestment of the coupon payments. As we can see,
this leads to a shortfall of $4.02 when compared to the CD example above. This $4.02 can be generated if
the coupon payments are invested at a 3% semi-annual rate at the time it is paid.
For the first payment the reinvestment income earned is:
$2.50 x (1.03) to 10 - 1 power - 2.50 = $2.50 x (1.03) to 9th power = $0.76.
If you were to continue this effort, which is unlikely to be required on the exam, you would find the
reinvestment income would equal $4.02.
To continue this with the three sources of income would produce the following:
Capital Gain of $4
Coupon Interest of $25
Reinvestment Income or $4.02
The total would be $33.02.
Therefore, reinvestment income accounts for 12% of the total return, illustrating how important
reinvestment income can be for an investor.
Factors That Affect Reinvestment Risk
There are two characteristics that affect reinvestment risk:
1. For a given yield to maturity and a given non-zero coupon rate, the longer the maturity, the more the
bond's total return depends on reinvestment revenue to realize the yield to maturity at purchase time.
Longer maturity = greater reinvestment risk.
2. For a given coupon-paying bond with a given maturity and yield to maturity, the higher the coupon
rate, the more the total dollar return depends on the reinvestment of the coupon payments. This must
occur in order to produce the yield to maturity at the time of purchase.
Fixed Income Investments - Spot Rates and Bond Valuation
On some occasions, such as with non-U.S. government bonds which pay annual interest compared to
semi-annual interest in the U.S., an adjustment needs to be made in order to compare their yields.
The computation is as follows:
Formula 14.11
Bond-equivalent yield of an annual-pay bond = 2[(1 + yield
on annual-pay bond) to the .5 power - 1]
Example:
Assume that the YTM on an annual-pay bond is 8%.
Answer:
Bond-equivalent yield = 2 [(1 + .08) to the .5 power - 1]
= 2 [.03923]
= .078461 = 7.95%
Look Out!
The bond equivalent yield will always be less than the
annual-yield.
Example:
Now if you want to convert the bond equivalent yield of a U.S. bond into an annual-pay bond the
calculations are as follows:
Formula 14.12
Yield on annual-pay basis = [(1 + yield on bond-equivalent
basis/2)2-1
Example:
The yield of a U.S. bond quoted on a bond-equivalent basis of 8%:
Answer:
Yield on annual-pay basis = [(1 + 8/2 to the 2nd power) -1]
= [(1.04) to the 2nd power - 1]
= .0816 = 8.16%
Look Out!
The yield on an annual-pay basis is always greater than the
yield on a bond-equivalent basis. This is because of
compounding.
Example: Computing the Value of a Bond Using Spot Rates
Suppose you have a bond that matures in 1.5 years that has a coupon rate of 8% and the spot curve is 5%
for six months, 5.25% for 1 year and 5.50% for 1.5 years.
Answer:
Bond price = 40/ (1.05) + 40 / (1.0525) to the second power + 1040 / (1.055) to the third power.
Bond Price = 38.09 + 36.12 + 931.06
Bond Price = 1005.27
This can be applied to any maturity; all you need to do is to continue theformula out to that maturity to
discover the price of the bond.
Example: Compute the Theoretical Treasury Spot Rate Curve Using Bootstrapping
Again let's look at an example to get through this LOS. We have a six month annualized yield of 4% and
similarly of the 1 year Treasury Security the rate is 4.40%. Given these two rates we can compute the 1.5
year theoretical spot rate of a zero coupon bond. For our example let's use a coupon of 6% with them
selling at par.
Answer:
First let's get the cash flows:
0.5 year = .06 x $100 x .5 = 3.00
1.0 year = .06 x $100 x .5= 3.00
1.5 year = .06 x $100 x .5 = 3.00 +100(par value) = 103
On to the next step:
3.00/ 1.02 + 3 / (1.02) to the second power + 103 / (1 +x3) to the third power = 100
2.94+ 2.88 + 103 / (1 + x3 ) to the third power = 100
103/ (1 +x3) to the third power = 94.18
(1 + x3) to the third power = 103 /94.18
Limitations of the Nominal Spread
As we discussed earlier, a nominal spread is the spread between a non-treasury bond's yield and the yield
to maturity on the comparable Treasury security in terms of maturity. For example, if an IBM is trading at
a YTM or 6.25% and the comparable Treasury is at 5%, then the nominal spread is 125 basis points. This
spread measure takes into consideration the extra credit risk, option risk and any liquidity risk that may be
associated with the non-treasury security.
Even though this is a quick and dirty way to describe the yield difference, it has two drawbacks. They are:
1. For bond bonds, the yield does not take into consideration the term structure of spot rates.
2. In the case of callable/puttable bonds, expected interest-rate volatility may change the cash flows of the
non-treasury security.
Fixed Income Investments - Differentiating Between Spreads
The nominal spread is simply the difference in basis points between the Treasury and non-treasury
security. For example, if the Stone & Co. bonds have a yield of 5.5% and the comparable Treasury
security has a yield of 4.5% the nominal spread is 100 bps. (5.5% - 4.5%).

Zero-Volatility Spread or Z-spread
This measures the spread the investor would capture over the entire Treasury spot- rate curve if
the bond was held to maturity. The Z-spread is calculated as the spread that will make the present
value of cash flows from the non-treasury security when they are discounted at the Treasury spot
rates plus the Z-spread equal to the non-Treasury securities price. This is done by trial and error.
This is different than the nominal spread because the nominal spread just uses one point on the
curve.
For example, take the spot curve and add 50 basis points to each rate on the curve. If the two year
spot rate is 3%, the rate you would use to find the present value of that cash flow would be
3.50%. After you have calculated all of the present values for the cash flows, add them up and see
whether they equal the bonds price. If they do, then you have found the Z-spread, if not, you have
to go back to the drawing board and use a new spread until the present value of those cash flows
equals the bonds price.

Option-Adjusted Spread (OAS)
This takes the dollar difference between the fair price and the market price and converts it into a
yield measure. The OAS helps reconcile the value to market price by finding a spread that will
equate the two. This is also done on a trial-and-error basis and is very model dependent.
Remember:
1. Interest rate volatility is critical. The higher the volatility, the lower the OAS. Check this
assumption when making comparisons.
2. The OAS is a spread over the Treasury spot-rate curve or benchmark that is used in the analysis.
3. As the name implies, the security's embedded option can change the cash flows and the value of
the security should take this change in account. The difference between the OAS And the Zspread is that the Z-spread doesn't take this into consideration.

Option Cost
This cost can be derived by calculating the difference between the OAS at the assumed interest
rate or yield volatile and the Z-spread.
Z-spread = OAS + option cost
Therefore,
Option Cost = Z-spread - OAS
The option cost is measured in this way because if rates do not change, the investor would earn
the Z-spread. When future rates are uncertain, the speed tends to be different because of the
embedded option. The option cost for a callable bond and most MBS and ABS securities are
positive. This is because the issuer's ability to alter the bond's cash flows will result in an OAS
that is less than the Z-spread. For puttable bonds the option cost is negative because of the
investor's ability to alter the cash flows
Fixed Income Investments - What are Forward Rates?
Forward rates can be defined as the way the market is feeling about the future movements of interest
rates. They do this by extrapolating from the risk-free theoretical spot rate. For example, it is possible to
calculate the one-year forward rate one year from now. Forward rates are also known as implied forward
rates.
To compute a bond's value using forward rates, you must first calculate this rate. After you have
calculated this value, you just plug it into the formula for the prices of a bond where the interest rate or
yield would be inserted.
Example:
An investor can purchase a one-year Treasury bill or buy a six-month bill and roll it into another sixmonth bill once it matures. The investor will be indifferent if they both produce the same result. An
investor will know the spot rate for the six-month bill and the one-year bond, but he or she will not know
the value of a six-month bill that is purchased six months from now. Given these two rates though, the
forward rate on a six-month bill will be the rate that equalizes the dollar return between the two types of
investments mentioned earlier.
Answer:
An investor buys a six-month bill for $x. At the end of six months, the value would equal:
x(1 + z1)
where z1 = one half of the bond equivalent yield on the six month spot rate.
F= one half the forward rate (expressed as a BEY) of a six-month rate six months from now. If he bought
the six-month bill and reinvested the proceeds for another six months the dollar return would be
calculated like this:
X(1 +z1) (1 + F)
For the one year investment the future dollars would be x(1 +z)2
So F = (1 + z2)2/ (1 + z1) - 1
Then double F to get the BEY.
Here are some numbers to try in this formula:
Six-month spot rate is 0.05 = 0.025 = z1
1-year spot rate is 0.055 = 0.0275= z2
F = ( 1.0275)2/ (1.025) -1
F = .030 or .06 or 6% BEY
To confirm this:
X(1.025)(1.03) = 1.05575
X(1.02575)2 = 1.05575
Once you have developed the future rate curve, you can continue to run and gun in the basic bond
equation using the forward rates instead of the discount rate to value the bond.
Fixed Income Investments - Forward Rates vs Spot Rates
Let's say an investor buys a two-year zero-coupon bond. The proceeds will equal:
X (1 + z6)6.
The investor could also buy a six-month Treasury bill and reinvest the proceeds every six months for two
years. In this case, the value would be:
X (1 + z1)(1+ future rate at time 1)(1 + future rate at time 2)(1+ future rate at time 3) (1 + future rate at
time 4)
Because these two investments must be equal this tells us that:
X (1 + z6)6 = X (1 + z1)(1+ future rate at time 1)(1 + future rate at time 2)(1+ future rate at time 3)
So Z6 = [(1 + z1)(1+ future rate at time 1)(1 + future rate at time 2)(1+ future rate at time 3)] ¼ - 1
This equation states that the two-year spot rate depends on the current six-month rate and the following
three six-month spot rates.
As we can see, short-term forward rates must equal spot rates or else an arbitrage opportunity can exist in
the market place.
Compute Spot Rates if Given Forward Rates, and Forward Rates if Given Spot Rates
Computing a forward rate by using spot rates is covered above. Using spot rates, an investor can develop
any forward rate.
There are two elements to the forward rate. The first is when the future rate begins. The second is the
length of time for that rate. The notation is length of time of the forward rate f when the forward rate
began. For example, a 2 f 8 would be the 1-year (two six-month periods) forward rate beginning four
years (eight six-month periods) from now.
To solve for tFm use the following equation:
Formula 15.13
tFm =[ (1 + Zm+t)m+t / (1 + Zm)m] 1/t - 1
So for a 3f5 it would equal an equation of: [(1 + z8)8/ (1 + z5)5]1/3 -1
Example:
Z3(the 1.5 year spot rate) = 3.5%/2 = .0175
Z5 (the 2.5 year spot rate) = 4.25%/2 = .02125
Answer:
So 3f5 =[(1.02125)/ (1.0175)5]1/3 -1
S3f5 = .027916
Doubling this rate gives you a rate of 5.58%
Fixed Income Investments - Measuring Interest Rate Risk
The Full Valuation Approach
The full valuation approach to measuring the interest rate risk is to re-value the bond or portfolio for a
given interest-rate change scenario. This rate change can be parallel or non-parallel. It is also referred to
as a scenario analysis because it involves the way in which your exposure will change as a result of
certain interest rate scenarios. For example, an investor may evaluate the portfolio based on an increase in
rates of 50, 100 and 200 basis points. Each bond is valued and then the total value of the portfolio is
computed under the various scenarios.
The Duration/Convexity Approach
In contrast, the duration/convexity approach just looks at one time parallel move in interest rates using the
properties of price volatility.
Because the full valuation approach uses various outcomes to measure the risk of the bond or portfolio, as
compared to a one time move for the duration/convexity approach, it bears that the full valuation
approach is better suited to measuring interest-rate risk even though it can be very time consuming.
Example: Compute the Interest-Rate Risk Exposure
Let's take an option-free bond with an 8% coupon, ten-year bond with a price of 125. Yield to maturity is
7%
Answer:
Scenario 1 is an increase of 50bps that drives the price down to 120 (this is just an estimate). To see the
percentage change you take the new price after the yield change and subtract it from the initial price after
the change divided by the initial price.
120 - 125 / 125 = -.04 = a 4 % decrease in the price of the bond due to a 50 bps change
Scenario 2 is an increase of 100 bps that drives the price down to 114.
114 - 125 / 125 = - .088 = an 8.8% decrease in price due to a 100 bps change.
You can use this for any type of scenario concerning a change in yields.
Fixed Income Investments - Price Volatility
Price Volatility for Option-free Bonds
The fundamental change in price is that which causes yields to increase as price decreases and vice versa.
This relationship is not linear, however, it is convex.
There are four properties concerning the price volatility of an option free bond:
1. Price moves in the opposite direction of a change in yields, but the percentage change is not the same
for all bonds.
2. For small changes in yields, the percentage change is roughly the same no matter what direction rates
move.
3. For large changes in yields, the percentage price change is not the same for an increase in yield as it is
for a decrease in yield
4. For a given large change in yield, the percentage price increase is greater than the percentage price
decrease.
Although the above properties apply to percentage change they still apply to dollar changes.
Properties three and four involve the convex shape of the price yield relationship. Property four states that
with a long a bond, the price increase when rates decline will be greater than the price decrease when
rates rise.
Positive Convexity
Positive convexity is what market participants refer to the yield/price relationship of option free bonds.
Price-Volatility Characteristics of Callable and Prepayable Securities
The price of a callable bond will react in the same way as an option-free bond when market rates are high.
This is because there is less of a chance of the bonds being called by the issuer because they probably will
not be able to refinance the bonds at a lower interest rate. When rates decline, on the other hand, the price
increase of a callable bond will be held to that call price because of the increased chance of the bonds
being called by the issuer. Meanwhile, option-free bonds will continue to see an increase in price as rates
fall. Would you pay $105, for example, for a bond that could be called at $101 as rates drop? In essence,
you give $4 dollars away. It is for this same reason that bonds are unlikely to be called by issuer when the
rates are low.
These bonds contain negative convexity. That is, the price appreciation is less than its price decline when
rates change by a large amount. The bonds will not always exhibit negative convexity; at higher rates they
will exhibit positive convexity just like an option-free bond.
Price Volatility Characteristics of Putable Bonds
A bondholder can redeem puttable bonds on certain dates and at certain prices. The advantage of these
bonds to an investor is that if market yields rise and the value of the bond falls below the put price, the
investor can exercise the put option and stem his losses to the put price. This can not be done with an
option-free bond.
Value of puttable bond = value of option free bond + the option.
The price of a puttable bond will react same way as an option-free bond at low yield levels. As rates rise,
the puttable bond's price will decrease at the same rate as an option-free bond, but the decline will be
lessened because of the value of the put option.
Fixed Income Investments - Effective, Modified, and Macaulay Duration
Effective Duration
Duration is the approximate percentage change in price for a 100 basis point change in rates. To compute
duration, you can apply the following equation that was presented earlier in the guide.
Price if yield decline - price if yield rise / 2(initial price)(change in yield in decimal)
Let's make:
∆y = change in yield in decimal (∆ = "delta")
V1 = initial price
V2 = price if yields decline by ∆y
V3 = price if yields increase by ∆y
Duration = V2 - V3 / 2(V1)(? y)
Example:
Stone & Co 9% of 10 are option free and selling at 106 to yield 8.5%. Let's change rates by 50 bps. The
new price for the increase in 50 bps would be 104 and the new price for a decrease in rates would be 109.
Then:
Answer:
Duration = 109 - 104 / 2 *(106) * (.005)
Duration = 5 / 1.06
Duration = 4.717
This means that for a 100 basis point change, the approximate change would be 4.717%
Price Change Given the Effective Duration and Change in Yield
Once you have computed the effective duration of a bond it is easy to find the approximate price change
given at change in yield.
Formula 14.13
Approximate Percent Price change = - duration x change in
yield x 100
Example:
Using the duration for 4.717% obtained from the previous example, let's see the approximate change for a
small movement in rates such as a 20 bps increase.
Percentage Price Change = - 4.717 x (+0.0020) x 100 = -.943%
And for a large change, a 250 bps increase:
Percentage Price Change = -4.717. (+0.0250) x 100 = -11.79%
As noted before, these changes are estimates. For small changes in rates, the estimate will be almost dead
on. For larger movements in rates, the estimate will be close but will underestimate the new price of the
bond regardless of whether the movement in rates is up or down.
Modified Duration
Modified duration is the approximate percentage change in a bond's price for a 100 basis points change in
yield, assuming that the bond's expected cash flow does not change when the yield changes. This works
for option-free bonds such as Treasuries but not with option-embedded bonds because the cash flows may
change due to a call or prepayment.
Effective Duration
Effective duration takes into account the way in which changes in yield will affect the expected cash
flows. It takes into account both the discounting that occurs at different interest rates as well as changes in
cash flows. This is a more appropriate measure for any bond with an option embedded in it.
Macaulay Duration
In order to better understand Macaulay duration, let's first turn to the modified duration equation:
Formula 14.14
modified duration = 1/(1+yield/k)[1 x pvcf1 + 2 x pvcf2
+...+n x pvcfn / k x Price
Where:
k = the number of periods: two for semi-annual, 12 for monthly and so on.
n = the number of periods to maturity
yield = YTM of the bond
pvcf = the present value of cash flows discounted at the yield to maturity.
The bracket part of the equation was developed by Frederick Macaulay in 1938 and is referred to as
Macaulay Duration.
So Modified duration = Macaulay's Duration/ (1 + yield/k)
Macaulay's duration gives the analysis a short cut to measure modified duration. But because modified
duration is flawed by not incorporating the change in cash flows due to an embedded option, so are
Macaulay durations.
When is Effective Duration a Better Measure?
When a bond has an embedded option, the cash flows can change when interest rates change because of
prepayments and the exercise of calls and puts. Effective duration takes into consideration the changes in
cash flows and values that can occur from these embedded options.
Why is duration the best interpretation of a measure of the sensitivity of a bond or portfolio to
changing interest rates?
As expressed throughout this guide, duration gives an approximate percentage change for a 100 basis
point change in rates. Once you understand duration, it is a quick way to calculate the change in a bond's
value. It also allows an investor to get a "feel" for the price change. For example, you can tell a client that
the duration of measure of 7 for their portfolio would equal roughly a 7% change in their portfolio's value
if rates change, plus or minus 100 basis points. It also allows a manager or investor a way to compare
bonds regarding the interest rate risk under certain assumptions.
A portfolio's duration is equal to the weighted average of the durations of the bonds in the portfolio. The
weight is proportional to how much of the portfolio consists of a certain bond.
Formula 14.15
Portfolio Duration = w1D1 + w2D2 ...+ wkDk
Example:
Let's take 3 bonds:
$6,000,000 market value of Stone & Co 7% of 10 with duration of 5.5
$3,400,000 market value of Zack Stores 5% or 15 with duration of 7.8
$1,535,000 market value of Yankee Corp. 9% or 20 with duration of 12
Total market valve of $10,935,000
Answer:
First let's find the weighted average of each bond
Stone & Co. weighted average is 6,000,000 / 10,935,000 = .548
Zack Stores weighted average is 3,400,000 / 10,935,000 = .311
Yankee Corp. weighted average is 1,535,000 / 10935,000 = .14
So the portfolio duration = .548(5.5) + .311(7.8) + .14 (12)
= 7.119
This means that if rates change by 100 bps the portfolio's value will change by approximately by 7.119%.
Keep in mind that the individual bonds will not change by this much because each will have their own
duration.
You can also use this to figure out the dollar amount of the change. This is done by using the dollar
duration equation and adding up the change for all of the bonds in the portfolio.
Going back to our example of those three bonds and a 50 bps yield change.
Percentage price change = -duration x change in yield x market value
Stone & Co = -5.5 x .005 x 6,000,000 = 165,000
Zack Stores = -7.8 x .005 x 3,400,000 = 132,600
Yankee Corp = -12 x .005 x 1,535,000 = 92,100
So the dollar change for a 50 bp change would be equal to approximately $389,700
Limitations of the Portfolio Duration Measure
The primary limitation of this measure is that each of the bonds in the portfolio must change by the 100 or
50bps, or there must be a parallel shift in the yield curve for the duration measure to be useful.
Fixed Income Investments - Convexity
Convexity helps to approximate the change in price that is not explained by duration. If you go back to
the third property of a bond's price volatility you will see that when there is a large change in rates, the
duration measure can be way off because of the convex nature of the yield curve.
To calculate convexity the formula is:
Formula 14.16
Convexity adjustment to the percentage price change = C x
change in yield squared x 100
To find the C in the equation, use this equation that has the same notation as duration:
C = V3 +V2 - 2(V1) / 2V1(change in yield) squared
Estimate a Bond's Price Given Duration, Convexity and Change in Yield
This is done by simply adding the convexity adjustment and the percentage price change due to duration
equations to achieve an estimate that is closer than just a duration measure.
Formula 14.17
Total Price change = (-duration x change in yield x 100) + (C x
change in yield squared x 100)
Example: Total Price Change
Using the Stone & Co. bonds that had duration of 5.5, let's add a convexity of 93 and an increase of 150
bps in yield.
Answer: Price Increase
Total Price Change = (-5.5 x .0150 x 100) + (93 x .0150 squared x 100)
= -8.25 + 2.0925
= 6.157
So if rates increase by 150 bps, the price will decrease by 6.157%
Now let's look at a decrease of 150 bps in yield.
Answer: Price Decrease
Total Price Change = (-5.5 x -.0150 x 100) + (93 x -.0150 squared x 100)
= 8.25 + 2.0925
= 10.34
So if rates decrease by 150 bps, the price will increase by 10.34 %
Again, if you refer to the properties of price volatility, you can see that as rates decrease, the price
increase will be greater than the decrease in price when rates rise.
Modified Convexity vs. Effective Convexity
With modified convexity the cash flows do not change due to a change in interest rates.
Effective Convexity, on the other hand, assumes that cash flow does change due to a change in interest
rates.
When bonds have options, it is best to use effective convexity just like you should use effective duration.
For option-free bonds, either convexity measure will be a positive value, whereas when it comes to bonds
with options, the effective convexity could be negative even if the modified convexity is positive.
Fixed Income Investments - Price Value of a Basis Point (PVBP)
This measure is the absolute value of the change in price of a bond for a one basis point change in yield. It
is another way to measure interest-rate risk. It does not matter if it is an increase or decrease in rates,
because such a small move in rates will be about the same in either direction according to the second
property of a bond's price . This is also know as Dollar Value of an 01 (DV01).
Formula 14.18
PVBP = initial price - price if yield is changed by 1 basis point
Example: Price Value of a Basis Point
Assume that the initial price is 98 and the new price is 97.75. Because of a 1 bps increase in rates the
PVBP would be .25 (98 - 97.75).
DV01 is related to duration. It is just a special case of dollar duration. Instead of using a 100 basis point
change you are simply using a 1 basis point change.
An example using the Stone & Co. bonds with duration of 5.5
5.5 x 0.0001 x 100 = .055% change
If the price was 98 the dollar price change would be:
.055% x 98 = $ 0.53