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Bonds are debt instruments that represent cash flows payable during a specified time period.

They are essentially

loans. The cash flows they represent are the interest payments on the loan and the loan redemption. Unlike
commercial bank loans, however, bonds are tradable in a secondary market.
Bonds are commonly referred to as fixed-income instruments. This term goes back to a time when bonds paid
fixed coupons each year. That is no longer necessarily the case. Asset-backed bonds, for instance, are issued in
a number of tranchesrelated securities from the same issuereach of which pays a different fixed or floating
coupon. Nevertheless, this is still commonly referred to as the fixed-income market.
Bond prices are expressed per 100 nominalthat is, as a percentage of the bonds face value. (The convention
in certain markets is to quote a price per 1,000 nominal, but this is rare.) For example, if the price of a U.S.
dollardenominated bond is quoted as 98.00, this means that for every $100 of the bonds face value, a buyer
would pay $98.
The principles of pricing in the bond market are the same as those in other financial markets: the price of a
financial instrument is equal to the sum of the present values of all the future cash flows from the instrument.
The interest rate used to derive the present value of the cash flows, known as the discount rate, is key, since it
reflects where the bond is trading and how its return is perceived by the market.
All the factors that identify the bondincluding the nature of the issuer, the maturity date, the coupon, and
the currency in which it was issuedinfluence the bonds discount rate.
If compounding takes place m times per year, then at the end of n years, m*n interest payments will have been
made, and the future value of the principal is computed using the formula

The present value of $100 to be received at the end of five years, assuming an interest rate of 5 percent, with
quarterly compounding is
=100/ (1+0.05/4) ^ (4*5) =78.00085483
As compounding becomes continuous and m approaches infinity,

continuous compounding is
=100/EXP (0.05*5) = 77.88007831
Discount Factors: An n-period discount factor is the present value of one unit of currency that is payable at the
end of period n.

For instance, the five-year discount factor for a rate of 6 percent compounded annually is

Q: Describe the basic features of a fixed-income security.

The basic features of a fixed-income security include the specification of:
Issuer: It is the entity that has issued the bond. The types of issuers include supranational organizations,
Sovereign governments, non-sovereign governments, Quasi-government entities and companies.
Maturity: It is the date on which the last payment is made for a bond. Based on maturity, bonds can be
classified into money market security, capital market security and perpetual bonds.
Par value: It is the amount an issuer agrees to pay the investor or bondholder on maturity date. A bond is
premium if sold above par, discount if sold below par and at par if sold at face value.
Coupon rate and frequency: Coupon rate is the interest rate paid on a bond every year by the issuer until its
maturity date. Bonds that have only one payment at maturity are called zero-coupon bonds.
Currency denomination: A bond can be issued in any currency: the local currency, or in a widely traded one
like the yen, euro or U.S. dollar. Bonds that pay coupon in one currency and principal in another currency are
called dual currency bonds.

Q. Describe functions of a bond indenture.

Bond indenture is a legal agreement between the bond issuer and the investor. It includes all the terms of a
bond issue such as the type of bond, its features such principal value and coupon rate, maturity date. It also
includes the dollar amount of issue, issuers obligations, bondholders rights, when the coupon payments will
be made, if the bonds are secured or not, covenants and contingency provisions.

Q. Describe how cash flows of fixed-income securities are structured.

The cash flows to investors can be divided into principal repayment and coupon payments. Principal can be
repaid in three ways:
Bullet bond: Principal is paid all at once at maturity.
Fully amortized: In this method the principal is paid little by little, in equal payments over the bonds life, so
that it is repaid in full by the maturity date.
Partially amortized: Only a part of the principal is repaid over the bonds life. The remaining big part of the
principal is paid at maturity, making it a balloon payment. This is a hybrid between the bullet and the fullyamortized bond.
The sinking fund arrangement allows for full or partial amortization of a bond prior to its maturity. Three
sinking fund arrangements are standard, accelerated and call provision. The different types of coupon payments
are listed below:
Fixed periodic coupon: A fixed interest is paid either semi-annually or annually.
Floating rate notes: The coupon payments are not fixed; instead they are linked to a benchmark reference
rate such as Libor, short-term Treasury bills etc.
Step-up bond: May be a fixed-rate or floating rate bond. The coupon rate increases over time.
Credit-linked Coupon Bonds: Coupon changes when the bonds credit rating changes.
Payment-in-kind Coupon Bonds: The issuer pays interest by issuing additional bonds.
Deferred Coupon Bonds: Bonds that do not pay a coupon in the initial years, but compensate by paying a
higher coupon in the later years.
Index-Linked Bonds: Coupon and/or principal payments are linked to an index.
TIPS: Treasury-inflation protected securities issued by the U.S. government, is linked to the U.S.CPI.

Q. Describe contingency provisions affecting the timing and/or nature of the cash flows of fixed-income securities
and identify whether such provisions benefit the borrower or the lender.
A contingency provision is a clause in a legal document that allows for some action if the event or circumstance
does occur.
Callable bond: It gives the issuer the right to redeem all or part of the bond before the specified maturity date.
Investors face reinvestment risk with callable bonds, as it is not possible to reinvest the proceeds at the previous
higher interest rates. When a bond is redeemed early, the issuer has to make the lump-sum payment equal to
the present value of future coupon payments and principal. The three types of callable bonds are American call,
European call and Bermuda style call.
Putable bond: It gives the bondholder the right to sell the bond back to the issuer at a predetermined price on
specified dates. Putable bonds offer a lower yield and sell at a higher price relative to otherwise non-putable
bonds. Putable bonds are beneficial to the bondholder. The three types of putable bonds are American put,
European put and Bermuda-style put.
Convertible bond: It allows the bondholder the right to exchange the bond for a fixed number of common
shares of the issuing company any time before maturity.
Advantages of Convertible Bonds

Warrant: It is an attached option, not an embedded option. It gives the bondholder the right to buy the
underlying common shares at a fixed price called the exercise price any time before the expiration date.
Contingent Convertible Bonds: These are bonds with contingent write-down provisions. The bonds can be
converted into equity contingent to a specific condition.

Q. Describe the use of interbank offered rates as reference rates in floating-rate debt.
Interbank offered rates are the average interest rates at which banks may borrow unsecured funds from other
banks. The rates differ for different periods ranging from overnight to one year. Examples of interbank offered
rates include Libor, Euribor (Euro interbank offered rate), Mibor (Mumbai interbank offered rate) etc. In a
floating rate bond, the coupon payment is linked to a floating rate which is usually a reference rate plus a
spread. The reference rate contributes to most of the coupon rate and is usually an interbank offered rate.

Q. Describe securities issued by sovereign governments, non-sovereign governments, government agencies, and
supranational agencies.
Sovereign bonds are issued by national governments, primarily for fiscal reasons. Recently issued sovereign securities are
called on-the-run. Off-the-run refers to securities that were issued some time ago. Sovereign bonds are not backed by
collateral. Instead, they depend on the taxing authority to repay the debt. The types of sovereign bonds include fixed rate
bonds, floating rate bonds and inflation linked bonds.
Non-sovereign bonds are issued by local government instead of national government. They have a higher credit risk than
sovereign bonds and therefore demand a higher yield. The yield of a non-sovereign bond is higher than a comparable
sovereign bond to compensate for the credit risk.
Quasi-government bond or agency bonds are issued by quasi-government entities. These are not government entities, but
they are usually backed by the government. The credit risk is low. They fund specific projects and cash flows from the
project are used to service the debt.
Supranational bonds are issued by international organizations such as the World Bank, IMF, EIB, ADB etc. They are usually
plain-vanilla bonds. Sometimes callable or floaters are also issued.

Q. Describe types of debt issued by corporations.

Debt issued by companies includes the following types:
Bank loans and Syndicated Paper: A bilateral loan is a loan from a single lender to a single borrower. A syndicated
loan is a loan from a group of lenders, called the syndicate, to a single borrower.
Commercial paper: It is a short-term, unsecured promissory note issued in the public market or via a private
placement that represents a debt obligation of the issuer.
Corporate notes and bonds: Corporate bonds can be categorized as short term, medium term and long term
security. Coupon payments for corporate notes and bonds vary based on the type of bond. Principal repayment
can be based on serial maturity structure, term maturity structure and sinking fund arrangement. Corporate
bonds have varying amount of risk so they are backed by collateral to protect the investors. These bonds can
have a call provision, put provision or can be convertible bonds.

Q. Describe the short term funding alternatives available to banks.

Retail Deposits: One of the primary sources of funds for a bank is the money deposited by retail investors in
their accounts. The three types of retail accounts are demand deposits, saving accounts and money market
Central bank funds: When a bank receives deposits from customers, a certain percentage of this money must
be kept as a reserve with the national central bank. The funds stashed in the central bank by all banks are
collectively known as central bank funds market.
Interbank Funds: Banks lend to and borrow from each other in the interbank market. It is an unsecured system
of lending and the term may vary from overnight to one year.
Certificate of deposit: It is a savings instrument with a maturity date, fixed interest rate and can be issued in
any denomination. The investor or bearer of the certificate receives an interest at the end of the deposit period.
There are two forms of CD: negotiable and non-negotiable CD.

Q. Describe repurchase agreements (repos) and their importance to investors who borrow short term.
A repurchase agreement or repo is a sale and repurchase agreement. It is an agreement between two parties
where the seller sells a security with a commitment to buy the same security back from the purchaser at an
agreed-on price at a future date. The interest rate negotiated between both the parties is called the repo rate.
A Haircut or repo margin is the difference between market value of security (collateral) and amount lent to
dealer. Repurchase agreements are a common source of funding for dealer firms and are also used to borrow
securities to implement short positions. If you look at the agreement from lenders perspective, it is a reverse
repo agreement.

Q. Suppose the coupon rate on a bond is 4% and the payment is made once a year. The time-to maturity is five
years and the market discount rate is 6%. What is the bond price per 100 of par value?
A bonds price is the present value of all future cash flows at the market discount rate, which is the rate of
return required by investors given the risk of investment in the bond.

Q. Define spot rates. Calculate the price of a bond using spot rates.
Spot rates are yields-to-maturity on zero coupon bonds maturing at the date of each cash flow. Since zero
coupon bonds have no intermediate cash flows, the actual yield on a zero-coupon bond is used as the discount
rate for a cash flow occurring at the same maturity date.

Case 1. A bond that matures in 2 years, that has a coupon rate of 6%, and pays coupon semi-annually. The spot
rates are 3.9% for 6 months, 4% for 1 year, 4.15% for 1.5 years, and 4.3% for 2 years.
The cash flows from this bond are $30, $30, $30, and $1030.
The value of the bond will be calculated as follows:
Bond value = $30/ (1+3.9%/2) ^1+$30/ (1+4%/2) ^2+$30/ (1+4.15%/2) ^3+$1030/ (1+4.3%/2) ^4
= 29.42618931+ 28.83506344+ 28.20740206+ 945.9838908
Bond value = $1032.45

Case 2. Calculate the value of a $1000 par, 5% annual coupon, 5 year maturity bond. Spot rates for the next 5

The bond will have the following cash flows.

Year 1: $50
Year 2: $50
Year 3: $50
Year 4: $50
Year 5: $1000 + $50
The value of the bond can be calculated by discounting these cash flows by their respective spot rate.
Bond Value = 50/(1.04)^1+50/(1.0430)^2+50/(1.0451)^3+50/(1.047)^4+1050/(1.048)^5
Bond Value = $1010.033

Q. Describe and calculate the flat price, accrued interest, and the full price of a bond.
When a bond is between coupon dates, its price has two parts: flat price and accrued interest. The sum of the
two parts is called the full price or dirty price.
Full price or dirty price = flat price + accrued interest
Flat price = full price accrued interest
Accrued Interest = Interest* (t/T)
T= Number of Days between Payments
t=Number of Days since Last Payment

Q. Define and compare the spot curve, yield curve on coupon bonds, par curve and forward curve.
Spot rate curve is also called the zero or strip curve. The spot rate curve plots different maturities on the x-axis and
corresponding spot rates on the y-axis.
Yield curve plots yields of bonds on the y-axis versus maturity on the x-axis. The main difference between a yield curve
and a spot rate curve is that the yield curve considers the coupon payments as well.
Par curve plots yields-to-maturity for different maturities, but the bonds are assumed to be priced at par.
Forward curve plots the forward rates, which is an estimation of what investors expect the short-term interest rates to

Q. Define forward rates and calculate spot rates from forward rates, forward rates from spot rates, and the price
of a bond using forward rates.
A forward rate is an interest rate in the future. The one year interest rate after one year is an example of a
forward rate. The implied forward rate can be calculated using spot rates.
Consider a scenario where the one-year spot rate is 2%, and the two-year spot rate is 3%. This is illustrated

The forward rate f2 is the rate over the second year. This can be calculated if we assume the following: $1
invested for two years at the 2-year spot rate of 3% should give the same result at $1 invested for 1 year at the
one year spot rate of 2% and then again at the forward rate, f2. Mathematically, this can be expressed as:

Elucidation: Take a two-year example where the spot rate over the first year is 8 percent and the spot rate
over the two years is 10 percent. Here, an individual investing $1 in a two-year zero coupon bond would have
$1* (1.10) ^2 in two years.

Above equation tells us something important about the relationship between one- and two-year rates. When
an individual invests in a two-year zero coupon bond yielding 10 percent, his wealth at the end of two years is
the same as if he received an 8 percent return over the first year and a 12.04 percent return over the second
year. This hypothetical rate over the second year, 12.04 percent, is called the forward rate.
More generally, if we are given spot rates r1 and r2, we can always determine the forward rate, f2, such that:

Forward rates can be calculated over later years as well. The general formula is:

Where fn is the forward rate over the nth year, rn is the n-year spot rate, and rn-1 is the spot rate for n-1

the forward rate over the first year is, by definition, equal to the one-year spot rate.

Q. Given the spot rates r1 equals 8 percent and r2 equals 10 percent, what should a 5 percent coupon, two-year
bond cost? The cash flows C1 and C2 are illustrated in the following time chart:

: Yield to maturity is not a simple average of r1 and r2. Rather, financial

economists speak of it as a time-weighted Average of spot rates r1 and r2.


Q. Explain the relationship between price and yield

There is a direct relationship between the price of a bond and its yield. The price is the amount the investor
will pay for the future cash flows; the yield is a measure of return on those future cash flows. Hence price will
change in the opposite direction to the change in the required yield.

Q. Explain the term structure of interest rates.

The term structure of interest rates, also known as the yield curve, is a very common bond valuation method.
Constructed by graphing the yield to maturities and the respective maturity dates of benchmark fixed-income
securities, the yield curve is a measure of the market's expectations of future interest rates given the current
market conditions.
There are three main patterns created by the term structure of interest rates:

1) NORMAL YIELD CURVE: This is the yield curve shape that forms during normal market conditions,
wherein investors generally believe that there will be no significant changes in the economy, such as in
inflation rates, and that the economy will continue to grow at a normal rate. During such conditions, market
expects long-term fixed income securities to offer higher yields than short-term fixed income securities. This
is a normal expectation of the market because short-term instruments generally hold less risk than long-term
instruments. To invest in one instrument for a longer period of time, an investor needs to be compensated
for undertaking the additional risk.
2) FLAT YIELD CURVE: These curves indicate that the market environment is sending mixed signals to
investors, who are interpreting interest rate movements in various ways. During such an environment, it is
difficult for the market to determine whether interest rates will move significantly in either direction farther
into the future. A flat yield curve usually occurs when the market is making a transition that emits different
but simultaneous indications of what interest rates will do. In other words, there may be some signals that

short-term interest rates will rise and other signals that long-term interest rates will fall. This condition will
create a curve that is flatter than its normal positive slope. When the yield curve is flat, investors can
maximize their risk/return trade-off by choosing fixed-income securities with the least risk, or highest credit
quality. In the rare instances wherein long-term interest rates decline, a flat curve can sometimes lead to an
inverted curve.
3) INVERTED YIELD CURVE: These yield curves are rare, and they form during extraordinary market
conditions wherein the expectations of investors are completely the inverse of those demonstrated by the
normal yield curve. In such abnormal market environments, bonds with maturity dates further into the future
are expected to offer lower yields than bonds with shorter maturities. The inverted yield curve indicates that
the market currently expects interest rates to decline as time moves farther into the future, which in turn
means the market expects yields of long-term bonds to decline. Remember, also, that as interest rates
decrease, bond prices increase and yields decline.
Proposition 1- If the yield curve is not flat, then bonds with the same maturity but different coupons will have
different yields.
Proposition 2- If the yield curve is upward-sloping, then for any given maturity, higher coupon bonds will
have lower yields.
Proposition 3-If the yield curve is downward-sloping, then for any given maturity, higher coupon bonds will
have higher yields.

Hypothesis on Interest Rates

What determines the term structure of interest rates?
Expected future spot rates.
Risk of long bonds.

Expectations Hypothesis: Forward rates predict future spot rates, ft = E [r1 (t)].
It states that long-term interest rates hold a forecast for short-term interest rates in the future. The theory
postulates that an investor earns the same amount of interest by investing in a one-year bond in the present
and rolling the investment into a different one-year bond after one year as compared to purchasing a two-year
bond in the present.

Implications: The slope of the term structure reflects the markets expectations of future short-term interest rates.
Liquidity Preference Hypothesis: Investors regard long bonds as riskier than short bonds.
The liquidity preference theory suggests that an investor demands a higher interest rate, or premium, on
securities with long-term maturities, which carry greater risk, because all other factors being equal, investors
prefer cash or other highly liquid holdings. According to this theory, interest rates on short-term securities are
lower because investors are sacrificing less liquidity than they do by investing in medium-term or long-term

Long bonds on average receive higher returns than short bonds.
Forward rate on average over-predict future short-term rates.
Term structure reflects (a) expectations of future interest rates, and (b) risk premium demanded by
investors in long bonds.

Q. Explain bond pricing and yield.

Bonds can be priced at a premium, discount, or at par. The term At Par refers to the face value of the bond,
that is, the value at which the issuer will redeem the bond at maturity.
If the bond's price is higher than its par value, it will sell at a premium because its interest rate (Coupon) is
higher than current prevailing rates. If the bond's price is lower than its par value, the bond will sell at a
discount because its interest rate is lower than current prevailing interest rates.

When you calculate the price of a bond, you are calculating the maximum price you would want to pay for the
bond, given the bond's coupon rate in comparison to the average rate most investors are currently receiving in
the bond market. Required yield or required rate of return is the interest rate that a security needs to offer in
order to encourage investors to purchase it. Usually the required yield on a bond is equal to or greater than
the current prevailing interest rates.
Fundamentally, however, the price of a bond is the sum of the present values of all expected coupon payments
plus the present value of the par value at maturity.
Calculating bond price is simple: all we are doing is discounting the known future cash flows. Remember that
to calculate present value (PV) - which is based on the assumption that each payment is re-invested at some
interest rate once it is received--we have to know the interest rate that would earn us a known future value.
For bond pricing, this interest rate is the required yield.

By incorporating the annuity model into the bond pricing formula, which requires us to also include the
present value of the par value received at maturity, we arrive at the following formula:

Example: Calculate the price of a bond with a par value of $1,000 to be paid in ten years, a coupon rate
(semi-annually) of 10%, and a required yield of 12%.
Step 1. Determine the Number of Coupon Payments 20 coupon payments.
Step 2. Determine the Value of Each Coupon Payment Each semi-annual coupon payment will be $50
($1,000 X 0.05).
Step 3. Determine the Semi-Annual Yield required semi-annual yield is 6% (0.12/2).

Step 4. Plug the Amounts into the Formula

The CURRENT YIELD calculates the percentage return that the annual coupon payment provides the investor.

So, if you purchased a bond with a par value of $100 for $95.92 and it paid a coupon rate of 5%, this is how
you'd calculate its current yield:

If we were considering a zero-coupon bond that has a future value of $1,000 that matures in two years and
can be currently purchased for $925, we would calculate its current yield with the following formula:

YIELD TO MATURITY: YTM is the resulting interest rate the investor receives if he or she invests all of his
or her cash flows (coupons payments) at a constant interest rate until the bond matures. YTM is the return
the investor will receive from his or her entire investment.

Q. Explain duration and convexity of a bond

Duration is a way of measuring how much bond prices are likely to change if and when interest rates move. In
more technical terms, duration is measurement of interest rate risk.
The bigger the duration, the greater the interest-rate risk or reward for bond prices.
For each of the two basic types of bonds the duration is the following:
1. Zero-Coupon Bond Duration is equal to its time to maturity.
2. Vanilla Bond - Duration will always be less than its time to maturity.
How Duration Affects the Price of Your Bonds?
As a general rule, for every 1% increase or decrease in interest rates, a bond's price will change approximately
1% in the opposite direction for every year of duration.
For example, if a bond has a duration of five years and interest rates increase by 1%, the bond's price will decline
by approximately 5%. Conversely, if a bond has a duration of five years and interest rates fall by 1%, the bond's
price will increase by approximately 5%.

MACAULAY DURATION is calculated by adding the results of multiplying the present value of each cash flow
by the time it is received and dividing by the total price of the security.

n = number of cash flows

t = time to maturity
C = cash flow
i = required yield
M = maturity (par) value

P = bond price =
MODIFIED DURATION is a modified version of the Macaulay model that accounts for changing interest rates.
Because they affect yield, fluctuating interest rates will affect duration, so this modified formula shows how
much the duration changes for each percentage change in yield.

Example: X holds a five-year bond with a par value of $1,000 and coupon rate of 5%. Assume that the coupon
is paid annually and that interest rates are 5%.

= 4.55 years

= 4.33 years
If the bond's yield changed from 5% to 6%, the duration of the bond will decline to 4.33 years.

As maturity increases, duration increases and the bonds price becomes more sensitive to interest
rate changes.
As the bond coupon increases, its duration decreases and the bond becomes less sensitive to
interest rate changes.
As interest rates increase, duration decreases and the bond becomes less sensitive to further rate
If term to maturity and a bond's initial price remain constant, the higher the coupon, the lower the
volatility, and the lower the coupon, the higher the volatility.
If the coupon rate and the bond's initial price are constant, the bond with a longer term to maturity
will display higher price volatility and a bond with a shorter term to maturity will display lower price
volatility. Therefore, if you would like to invest in a bond with minimal interest rate risk, a bond with
high coupon payments and a short term to maturity would be optimal.
A bond with a high yield to maturity will display lower price volatility than a bond with a lower yield to
maturity, but a similar coupon rate and term to maturity.

Yield to maturity is affected by the bond's credit rating, so bonds with poor credit ratings will have higher
yields than bonds with excellent credit ratings. Therefore, bonds with poor credit ratings typically display lower
price volatility than bonds with excellent credit ratings

All three factors affect the degree to which bond price will change in the face of a change in prevailing interest
rates. So, if a bond has both a short term to maturity and a low coupon rate, its characteristics have opposite
effects on its volatility: the low coupon raises volatility and the short term to maturity lowers volatility. The
bond's volatility would then be an average of these two opposite effects.
LIMITATION OF DURATION: It assumes a linear relationship between interest rates and bond price. In reality,
the relationship is likely to be curvilinear.

Convexity is the rate that the duration changes along the price-yield curve, and, thus, is the 1st derivative to the equation
for the duration and the 2nd derivative to the equation for the price-yield function.
A bond with greater convexity is less affected by interest rates than a bond with less convexity.
Also, bonds with greater convexity will have a higher price than bonds with a lower convexity, regardless of whether
interest rates rise or fall.

Bond A has greater convexity than Bond B, but they both have the same price and convexity when price equals *P and
yield equals *Y. If interest rates change from this point by a very small amount, then both bonds would have approximately
the same price, regardless of the convexity. When yield increases by a large amount, however, the prices of both Bond A
and Bond B decrease, but Bond B's price decreases more than Bond A's.
A bond is positively convex when the price increases from falling yields is greater than the price decrease caused by rising
yields. Bonds without any embedded options, such as call features, will always have positive convexity. Callable bonds at
par, including most mortgage-backed securities, have negative convexity.


In estimating price changes, duration treats the price-yield function as though it were along the line tangent to the actual
price-yield curve. Because duration provides a linear or first-order approximation of price changes, is often referred to as
the first derivative of the price-yield function. Since the actual price-yield relationship is not linear, we can get a better
price approximation if we use a higher-order derivative. Convexity represents the second derivative of the price-yield

Price Change = (- Duration x Price Yield) + (0.5 x Convexity x (Yield Change) ^2))

Consider an 8% 10-year note at a price of 100 and a modified duration of 6.80. If the yield increases
by 10%, duration estimates the price change as follows:
Price change = - [Duration] x [Yield Change]
Price change = - [6.80] x [2] = -13.60
The price change is negative to reflect the fact that prices decline as yields rise. The duration estimated price is 86.40 (100
13.60). Similarly, for a 2% decrease in yields, the estimated price is 113.60 (100 + 13.60).
The actual prices should be 87.54 and 114.88, and therefore, duration tends to underestimate the price during both rising
and falling rates.
Using our previous example, if the 8% 10-year note has a 0.60 convexity, the new estimated price change is calculated as
Price Change = (-6.80 x 2) + (0.5 x 0.60 x 4)
Price Change = -12.40

The estimated price using convexity is now 87.60 (100 12.40). The convexity estimate of 87.60 is much closer
to the actual price of 87.54 than the duration estimate.

Example. Consider a 4-year T-note with face value $100 and 7% coupon, selling at $103.50, yielding 6%.
For T-notes, coupons are paid semi-annually. Using 6-month intervals, the coupon rate is 3.5% and the yield is 3%.

Duration (in 1/2 year units) is D = (738.28)/103.50 = 7.13.

Modified duration (volatility) is MD = D/ (1 + y) = 7.13/1.03 = 6.92.
If the semi-annual yield moves up by 0.1%, the bond price decreases roughly by 0.692%.
For small yield changes, pricing by MD is accurate.
For large yield changes, pricing by MD is inaccurate.

Bond price is not a linear function of the yield. For large yield changes, the effect of curvature (i.e.,
nonlinearity) becomes important.

Q. Estimate the effect of a 100 basis point increase and decrease on a 10-year, 5%, option-free bond currently
trading at par, using the duration/convexity approach. The bond has a duration of 7 and a convexity of 90.
Using the duration/convexity approach: percentage bond price change ~ duration effect + convexity effect

Q. Describe the relationships among a bonds holding period return, its duration, and the investment horizon.
The impact of a sudden change in yield on the price of a bond is of particular concern to short-term investors
(price risk). Long term investors will also be concerned about the impact of a change in yield on the reinvestment
income (reinvestment risk). An investor who plans to hold the bond to maturity will only be concerned about
reinvestment risk. Macaulay duration indicates the investment horizon for which coupon reinvestment risk and
market price risk offset each other.
Duration gap = Macaulay duration Investment horizon
If Macaulay duration < investment horizon, the duration gap is negative: coupon reinvestment risk dominates.
If investment horizon = Macaulay duration, the duration gap is zero: coupon reinvestment risk offsets market
price risk.
If Macaulay duration > investment horizon, the duration gap is positive: market price risk dominates.

Q. Identify the relationships among a bonds price, coupon rate, maturity, and market discount rate (yield-tomaturity).

Managing Interest Rate Risks

Given a fixed-income position, we can take another fixed-income position to offset the interest rate risk of the original
position. Thus, the interest rate risk of the total position (the portfolio) is reduced. Such a strategy is called hedging.
Assume a flat term structure. Consider a bond portfolio consisting of nA units of bond A and nB units of bond B.

Example. Suppose that you are long in 4-year bonds and you want to use 3-year bonds to hedge

the interest rate risk. To hedge the long position in 4-year bond, we need to sell 3-year
bond. How much to sell?

For each dollar worth of 4-year bond, short dollars worth of 3-year bond such that the total portfolio
has zero volatility: ( is called the hedge ratio)
MD4 + MD3 = 0

Default Premium and Risk Premium

Example. Suppose all bonds have par value $1,000 and
10-year Treasury strip is selling at $463.19, yielding 8%
10-year zero issued by XYZ Inc. is selling at $321.97
Expected payoff from XYZs 10-year zero is $762.22

Promised YTM is the yield if default does not occur.

Expected YTM is the probability-weighted average of all possible yields.
Default premium is the difference between promised yield and expected yield.
Bond risk premium is the difference between the expected yield on a risky bond and the yield on a
risk-free bond of similar maturity and coupon rate.
Q. Compute the price of a $100 face value, 2-year, and 4% semi-annual coupon bond using the annualized spot rates.
M at uri t y (yrs)
1 .5

S po t R at e (%)

Bond Price =2*EXP (-0.025/2) + 2*EXP ((-0.026/2)*2) + 2*EXP ((-0.027/2)*3) +102*EXP ((-0.029/2)*4)
= 102.0967387 = 102.10

Q. If the 1-year rate is 2.136% and the 2-year rate is 2.915%, what is the 1-year forward rate one year from now?
Ri = the spot rate corresponding with Ti periods
RForward = the forward rate between T1 and T2
RForward = (R2T2-R1T1)/ (T2-T1) = R2 + (R2-R1)* (T1/ (T2-T1))
RForward =0.02915+ (0.02915-0.02136)*(1/ (2-1)) = 0.03694=3.694%

Q. Explain forward rate agreement, cash flow in FRA and its value.
A forward rate agreement (FRA) is a forward contract obligating two parties to agree that a certain interest
rate will apply to a principal amount during a specified future time. Obviously, forward rates play a crucial
role in the valuation of FRAs. The T2 cash flow of an FRA that promises the receipt or payment of RK is:

Q. Suppose an investor has entered into an FRA where he has contracted to pay a fixed rate of 3% on $1 million
based on the quarterly rate in three months. Assume that rates are compounded quarterly. Compute the payoff of
the FRA if the quarterly rate is 1% in three months.
For this FRA, the payoff will take place in six months. The net payoff will be the difference between
the fixed-rate payment and the floating rate receipt.
If the floating rate is 1% in three months, the payoff at the end of the sixth month will be:
$1,000,000 (0.01 -0.03X0.25) =-$5,000
Q. Suppose the 3-month and 6-month LIBOR spot rates are 4% and 5%, respectively (continuously compounded
rates). An investor enters into an FRA in which she will receive 8% (assuming quarterly compounding) on a
principal of $5,000,000 between months 3 and 6. Calculate the value of the FRA.
Rforward = 0.05+ (0.05-0.04)*(1/ (2-1)) = 0.06= 6%
Rforward (With Quarterly compounding) = 4* (e0.06/4 1) = 0.060452 = 6.05%
Value of FRA = $5,000,000 * (0.0800-.0605) * (0.5-0.25) * e(0.05)*(0.5) =$23,773

Q. A $100 face value, 1-year, 4% semi-annual bond is priced at 99.806128. If the annualized 6-month spot rate
(z1) is 4.1%, what is the 1-year spot rate (z2)? (Both spots are continuously compounded rates.)

Q. What is the bond price of a $100 face value, 2.5-year, 3% semi-annual coupon bond using the following annual
continuously compounded spot rates: z1= 3%, z2 = 3.1%, z3 = 3.2%, z4 = 3.3%, and z5 = 3.4%?

Q. Explain yield spread.

Despite the imperfections of the Treasury yield curve as a benchmark for the minimum interest rate that an
investor requires for investing in a non-Treasury security, it is commonplace to refer to the additional yield
over the benchmark Treasury issue of the same maturity as the yield spread.
In general, the yield spread between any two bond issues, bond X and bond Y, is computed as follows:
Yield spread = yield on bond X - yield on bond Y
When a yield spread is computed in this manner it is referred to as an absolute yield spread and it is measured
in basis points.
For example, on February 8, 2002, the yield on the 10-year on-the-run Treasury issue was 4.88% and the yield
on a single A rated 10-year industrial bond was 6.24%. If bond X is the 10-year industrial bond and bond Y is
the 10-year on-the-run Treasury issue,
The absolute yield spread was: yield spread = 6.24% - 4.88% = 1.36% or 136 basis points.
Yield spreads can also be measured on a relative basis by taking the ratio of the yield spread to the yield of the
reference bond. This is called a relative yield spread.

Q. Explain Credit Spreads.

The yield spread between non-Treasury securities and Treasury securities that are identical in all respects except
for credit rating is referred to as a credit spread or quality spread. Identical in all respects except credit rating
means that the maturities are the same and that there are no embedded options.

Q. What is price value of a basis point?

It is a measure of the price volatility of a bond to quantify interest rate riskthe price value of a basis point
(PVBP). This measure, also called the dollar value of a 01 (DV01), is the absolute value of the change in the
price of a bond for a 1 basis point change in yield.
PVBP = |initial price - price if yield is changed by 1 basis point|

Q. What is Securitization?
Securitization is the process in which certain types of assets are pooled so that they can be repackaged into
interest-bearing securities. The interest and principal payments from the assets are passed through to the
purchasers of the securities.
The benefits to investors are:
Securitization converts an illiquid asset into a liquid security.
It gives investors direct access to the payment streams of the underlying mortgage loans that would otherwise
be unattainable.
There are higher risk-adjusted returns to investors: pooling loans results in diversification and lower risk for
It gives investors anywhere an opportunity to buy a small part of homebuyers mortgage in the form of a
security issued by the SPV.
Exposure to the market, real estate in this example, without directly investing in it.
The benefits to the bank or loan originator are:
It enables banks to increase loan origination, monitoring, and collections.
It reduces the role of the intermediaries like the bank.
Banks have the ability to lend more money if the demand for ABS and MBS is high relative to if the money
was self-financed.
There is greater efficiency and profitability for the banking sector.
The borrowers of the loan benefit from securitization in the following ways:
It lowers the risk as the pooled loans offer a diversification benefit. The lower risk, in turn, decreases the cost
of borrowing for homeowners.

Q. Describe the securitization process, including the parties to the process, the roles they play, and the legal
structures involved.
The parties involved in securitization process include:
SPV (Special Purpose Vehicle): It is also called the trust or the issuer.
Seller of pool of securities: It is also known as the originator or depositor.
Servicer: Servicing involves collecting payments from borrowers, notifying borrowers who may be delinquent, and if
necessary, seizing equipment from borrowers who do make payments on time.
Other parties: Independent accountants, underwriters, trustees, rating agencies, and guarantors.
The motivation for the creation of different types of structures is to redistribute prepayment risk and credit risk efficiently
among different bond classes in the securitization.
Prepayment risk is the uncertainty that the actual cash flows will be different from the scheduled cash flows.
Time trenching is the creation of bond classes to distribute the prepayment risk.
Subordination is another layering structure in securitization. The bond classes differ in their exposure to credit risk.


A mortgage is a loan that is collateralized with a specific piece of real property. Before the 1970s, mortgages existed solely
in the primary market where banks that issued the mortgage loans collected all interest and principal payments from the
borrower. Within the past few decades, it is more common for mortgage lenders to sell the loans in the secondary market
through a process known as securitization. The secondary market has allowed more banks to issue mortgage loans.
In the secondary market, mortgages are pooled together and packaged to investors in the form of a mortgage backed
security (MBS). The payments of an MBS can follow a pass-through structure where the interest and principal collected
from the borrower pass through the banks and ultimately end up with the MBS investor. Because default risk is present
in mortgage lending, banks will often guarantee the borrowers payments when mortgages are securitized .
LIEN STATUS: Whether the mortgage is a first lien, a second lien, or a subsequent lien will greatly impact the lenders
ability to recover the balance owed in the event of default. For example, a first lien would give the lender the first right to
receive proceeds on liquidation, so from a seniority perspective, a first lien is more desirable than a second lien.

Original Loan Term: Mortgage terms of 10 to 30 years are common, with the most popular being 30 years (long
term). However, medium terms in the 10- to 20-year range are starting to become more common, given the
desire of many individuals to pay off their mortgages as soon as possible.
Classifying loans between prime and subprime is determined mainly by credit score (i.e., Fair Isaac Corporation
or FICO model).
Prime (A-grade) loans constitute most of the outstanding loans. They have low rates of delinquency and default as a result
of low loan-to-value (LTV) ratios (i.e., far less than 95%), borrowers with stable and sufficient income (i.e., front income
ratio of no more than 28% of monthly income to service payments relating to the home and back income ratio of no more
than 36% for those payments plus other debt payments), and a strong history of repayments (e.g., FICO score of 660 or
greater). Home payments include interest, principal, property taxes, and homeowners insurance.
Subprime (B-grade) loans have higher rates of delinquency and default compared to prime loans. They could be associated
with high LTV ratios (i.e., 95% or above), borrowers with lower income levels, and borrowers with marginal or poor credit
histories (e.g., FICO score below 660). High LTV ratios suggest a higher risk of default. Upon issuance, subprime loans are
carefully scrutinized by the servicer to ensure timely payments.
Alternative-A loans are the loans in between prime and subprime. Although they are essentially prime loans, certain
characteristics of Alternative-A loans make them riskier than prime loans. For example, the loan value may be unusually
high, the LTV ratio may be high, or there may be less documentation available (e.g., income verification, down payment


Fixed-rate mortgages have a set rate of interest for the term of the mortgage. Payments are constant for the
term and consist of blended amounts of interest and principal.
Adjustable-rate mortgages (ARMs) have rate changes throughout the term of the mortgage. The rate is usually
based on a base rate (e.g., prime rate, LIBOR) plus a spread. Rates can usually change on a monthly, semiannual, or annual basis. The risk of default is high, especially if there are large rate increases after the first year,
thereby significantly increasing the total payment amount (due to the increase in interest).

The ability to create mortgage-backed securities requires loans that have credit guarantees.
Government loans are those that are backed by federal government agencies (e.g., Government National
Mortgage Association or GNMA).
Conventional loans could be securitized by either government-sponsored enterprises (GSEs): Federal Home
Loan Mortgage Corporation (FHLMC) or Federal National Mortgage Association (FNMA). For a guarantee fee,
these GSEs will guarantee payment of principal and interest to the investors.
Agency (or conforming) MBSs are those that are guaranteed by any of three government sponsored entities
(GSEs): GNMA, FNMA, and FHLMC. Most of the MBSs are issued by these GSEs.
Also known as private label securitizations, non-agency (or non-conforming) MBSs grew along with U.S. home
prices over time up to the 2007 credit crisis. The GSEs have restrictions on what mortgages they can
guarantee/securitize [e.g., dollar value limit, loan to-value (LTV) ratio limit], which opened up the private label
market for those participants willing to take on the risks inherent in nonconventional loansjumbo loans
(mortgage principal balance over the limit) and/or loans with high LTVs. The rising prices of the underlying
homes held as collateral provided some risk mitigation. Unfortunately, the falling prices of homes and the credit
crisis beginning in 2007 caused a significant drop in MBS issuances in the non-agency segment because they
did not have government guarantees.
With agency MBSs, the investor bears no credit risk because the GSEs have been paid a fee to guarantee the
underlying mortgages. If there is a default with a mortgage, the GSE will pay the outstanding balance to the
investors. With a non-agency MBS, there is some credit risk but that is mitigated through the process of


A conventional mortgage is the most common residential mortgage. The loan is based on the creditworthiness
of the borrower and is collateralized by the residential real estate that it is used to purchase.
If a borrowers credit quality is questionable or the borrower is lacking a sufficient down payment, the mortgage
lender may require mortgage insurance to guarantee the loan. Mortgage insurance is made available by both
government agencies and private insurers. The cost of the insurance is borne by the borrower and effectively
raises the interest rate on the mortgage loan.
There are a wide variety of mortgage designs that specify the rates, terms, amortization, and repayment methods.
All of the concepts associated with risk analysis and valuation, however, can be understood through an
examination of fixed-rate, level payment, fully amortized mortgage loans. This common type of mortgage loan
requires equal payments (usually monthly) over the life of the mortgage. Each of these payments consists of an
interest component and a principal component.
There are four important features of fixed-rate, level payment, fully amortized mortgage loans to remember
when we move on to mortgage-backed securities (MBS):
1. The amount of the principal payment increases as time passes.
2. The amount of interest decreases as time passes.
3. The servicing fee also declines as time passes.
4. The ability of the borrower to repay results in prepayment risk. Prepayments and curtailments reduce the
amount of interest the lender receives over the life of the mortgage and cause the principal to be repaid sooner.

Notice that the monthly interest charge is based on the beginning-of-period outstanding principal. As time
passes, the proportion of the monthly payment that represents interest decreases, and, because the payment is
level, the proportion that goes toward the repayment of principal increases. This process continues until the
outstanding principal reaches zero and the loan is paid in full.

The collection of payments and all of the other administrative activities associated with mortgage loans are paid
for via a servicing fee, also known as the servicing spread, because it is usually built into the mortgage rate. For
example, if the mortgage rate is 10.5% and the servicing fee is 35 basis points, the provider of the mortgage
funds will receive 10.15%. This amount is called the net interest or net coupon. The dollar amount of the
servicing fee is based on the outstanding loan balance; thus, it declines as the mortgage is amortized.

Allocation between Principal and Interest

Fully amortizing fixed-rate mortgage:
The mortgage payment consists primarily of interest in the early years.
Interest is calculated on a declining principal balance so the interest payable will gradually decrease over time.
As a result, more of the fixed mortgage payment will be applied toward reducing the principal amount.
The crossover point is the point in the mortgage where principal and interest allocation amounts are the
same. After that point, relatively more amounts will be allocated to principal.
Mortgages with shorter amortization periods result in less interest paid and more of the payment applied
toward reducing the principal balance sooner. In other words, equity build-up occurs at a quicker rate when
the amortization period is shorter.

As indicated previously, the reason principal payments will increase over time is due to the reduction in the
outstanding loan balance.
Figure 2 illustrates the relationship between loan balance and time for a $100,000 loan.

In the previous example, it was assumed that the borrower paid the exact amount of the monthly payment, and
the interest and principal followed the amortization schedule. However, it is possible for a borrower to pay an
amount in excess of the required payment or even to pay off the loan entirely. The option to prepay a mortgage
is essentially a call option for the borrower. The borrower is in a position that is very similar to the issuer of
a callable bond. A prepayment will effectively free the borrower of the mortgage obligation.

Mortgage Prepayment Option

Mortgage prepayments come in two forms: (1) increasing the frequency or amount of payments (where
permitted) and (2) repaying/refinancing the entire outstanding balance.
Prepayments are much more likely to occur when market interest rates fall and borrowers wish to
refinance their existing mortgages at a new and lower rate.
For the lender, prepayments represent a loss for two reasons:
(1) They stop receiving interest income at the high rate and
(2) They have to reinvest the proceeds received from prepayment at the prevailing lower market rates.
Therefore, the pricing of the initial mortgage rate should be somewhat higher to take into account the possibility
of prepayment. With agency MBSs, prepayments and defaults have the same impact on investors. Prepayments
result in the investors actually receiving cash from the borrowers, whereas with defaults, the borrower does not
pay the outstanding mortgage balances, but the GSE does, thereby causing a prepayment.
Other Factors That Influence Prepayments
Seasonality. The summertime is a popular time for individuals to move (and mortgages must be paid out prior
to the sale of a home), so it is the period of time with the greatest prepayment risk. Given some time lags, the
prepayments often start to appear in the late summer and early fall.

Age of mortgage pool. Refinancing often involves penalties and administrative charges, so borrowers tend not
to do so until several years into the mortgage. Also, it takes some time for borrowers to build up equity and
savings to make prepayments and/or attempt to refinance. As a result, the lower the age of the mortgage pool,
the less likely the risk of prepayment.

Personal. Marital breakdown, loss of employment, family emergencies, and destruction of property are
commonly cited reasons for prepayments based on personal reasons. It is difficult to assess this type of
prepayment risk.

Housing prices. Property value increases may spur an increase in prepayments caused by borrowers wanting
to take out some of the increased equity for personal use. Property value decreases reduce the value of collateral,
reduce the ability to refinance, and, therefore, decrease the risk of prepayment. The increasingly popular use of
home equity lines of credit where the mortgage balance is revolving (i.e., mortgage balance can be drawn up
to a certain limit and paid down to zero at any time) reduces refinancing and prepayment risk due to the
nature of the loan.

Refinancing burnout. To the extent that there has been a significant amount of prepayment or refinancing
activity in the mortgage pool in the past, the risk of prepayment in the future decreases. That is because
presumably the only borrowers remaining in the pool are those who were unable to refinance earlier (e.g., due
to poor credit history or insufficient property value), and those who did refinance have been removed from the
pool already. Also, those who made only large prepayments (instead of folly refinancing) in the past would have
exhausted their savings to make the prepayment and would require quite some time to do so again in the

To reduce the risk from holding a potentially undiversified portfolio of mortgage loans, a number of financial
institutions (i.e., originators) will work together to pool residential mortgage loans with similar characteristics
into a more diversified portfolio. They will then sell the loans to a separate entity, called a special purpose
vehicle (SPV), in exchange for cash.
An issuer will purchase those mortgage assets in the SPV and then use the SPV to issue MBSs to investors; the
securities are backed by the mortgage loans as collateral.
As of now, the securitization process has become a crucial part of the U.S. credit system. Financial institutions
expect to originate mortgage loans and sell them through securitization. The lack of a securitization market for
mortgages would lead to the downfall of mortgage lending because financial institutions would not want to
retain the risks.

A mortgage pass-through security represents a claim against a pool of mortgages. Any number of mortgages
may be used to form the pool and any mortgage included in the pool is referred to as a securitized mortgage.
The mortgages in the pool have different maturities and different mortgage rates. The weighted average maturity
(WAM) of the pool is equal to the weighted average of all mortgage ages in the pool, each weighted by the
relative outstanding mortgage balance to the value of the entire pool. The weighted average coupon (WAC) of
the pool is the weighted average of the mortgage rates in the pool. The investment characteristics of a mortgage
pass-through are a function of its cash flow features and the strength of its government guarantee.
As illustrated in Figure 3, pass-through security investors receive the monthly cash flows generated by the
underlying pool of mortgages, less any servicing and guarantee/insurance fees. The fees account for the fact
that pass-through rates (i.e., the coupon rate on the pass-through) are less than the average coupon rate of the
underlying mortgages in the pool.

Because pass-through securities may be traded in the secondary market, they effectively convert illiquid
mortgages into liquid securities (as mentioned, this process is called securitization). More than one class of
pass-through securities may be issued against a single mortgage pool.
The timing of the cash flows to pass-through security holders does not exactly coincide with the cash flows
generated by the pool. This is due to the delay between the time the mortgage service provider receives the
mortgage payments and the time the cash flows are passed through to the security holders.
The most important characteristic of pass-through securities is their prepayment risk; because the mortgages
used as collateral for the pass-through can be prepaid, the pass-through themselves have significant prepayment

Collateralized Mortgage Obligations

All investors have varying degrees of concern about exposure to prepayment risk. Some are primarily concerned
with extension risk (the increase in the expected life of a mortgage pool due to rising interest rates and lower
prepayment rates), while others want to minimize exposure to contraction risk (the decrease in the expected
life of a mortgage pool due to falling interest rates and higher prepayment rates). Fortunately, all of the passthrough securities issued on a pool of mortgages do not have to be the same. The ability to partition and
distribute the cash flows generated by a mortgage pool into different risk packages has led to the creation of
collateralized mortgage obligations (CMOs).
CMOs are securities issued against pass-through securities (securities secured by other securities) for which
the cash flows have been reallocated to different bond classes called tranches. Each tranche has a different
claim against the cash flows of the mortgage pass-through or pool from which it was derived. Each CMO
tranche represents a different mixture of contraction and extension risk. Hence, CMO securities can be more
closely matched to the unique asset/liability needs of institutional investors and investment managers.

Planned Amortization Class Tranches

The most common type of CMO today is the planned amortization class (PAC). A PAC is a tranche that is
amortized based on a sinking fund schedule that is established within a range of prepayment speeds called the
initial PAC collar or initial PAC bond. What makes a PAC bond work is that it is packaged with a support, or
companion, tranche created from the original mortgage pool. Support tranches are included in a structure with
PAC tranches specifically to provide prepayment protection for the PAC tranches (each tranche is, of course,
priced according to the timing risk of the cash flows). If prepayment rates are faster than the upper repayment
rate, the PAC tranche receives principal according to the PAC schedule, and the support tranche absorbs (i.e.,
receives) the excess. If prepayment speeds are below the lower repayment rate, the funds needed to keep the
PAC on schedule come from the cash flows scheduled for the support tranche(s). It should be pointed out that
the extent of prepayment risk protection provided by a support tranche increases as its par value increases
relative to its associated PAC tranche.
There is an inverse relationship between the prepayment risk of PAC tranches and the prepayment risk
associated with the support tranches. In other words, the certainty of PAC bond cash flow comes at the expense

of increased risk to the support tranches.

To understand the relatively high prepayment risk for support tranches, consider the situation in which
prepayments are slower than planned. Because the PAC tranches have priority claim against the cash flows,
principal payments to the support tranches must be deferred until the PAC repayment schedule is satisfied.
Thus, the average life of the support tranche is extended. Similarly, when actual prepayments come at a rate
that is faster than expected, the support tranches must absorb the amount that is in excess of that required to
maintain the repayment schedule for the PAC. In this case, the average life of the support tranche is contracted.
If these excesses continue to occur, the support tranches will eventually be paid off and the principal will then
go to the PAC holders. When this happens, the PAC is referred to as a broken or busted PAC, and any further
prepayments go directly to the PAC tranche. Essentially, the PAC tranche becomes an ordinary sequential pay
Notice that the prepayment risk protection provided by the support tranches causes their average lives to
extend and contract. This relationship is such that as the prepayment risk protection for a PAC tranche
increases, its average life variability decreases, and the average life variability of the support tranche increases.

A distinguishing characteristic of a traditional pass-through security is that the interest and principal payments
generated by the underlying mortgage pool are allocated to the bondholders on a pro rata basis. This means
that each pass-through certificate holder receives the same amount of interest and the same amount of principal.
Stripped MBS differ in that principal and interest are not allocated on a pro rata basis. The unequal allocation
of principal and interest results in a price/yield relationship that is different from that of the
underlying pass-through.
The two most common types of stripped MBSs are principal-only strips (PO strips) and interest-only strips (IO
strips). PO strips are a class of securities that receive only the principal payment portion of each mortgage
payment, while IO strips are a class that receive only the interest component of each payment.

PO strips are sold at a considerable discount to par. The PO cash flow stream starts out small and increases
with the passage of time as the principal component of the mortgage payments grows. The investment
performance of a PO is extremely sensitive to prepayment rates. Higher prepayment rates result in a fasterthan-expected return of principal and, thus, a higher yield. Since prepayment rates increase as mortgage rates
decline, PO prices increase when interest rates fall. The entire par value of a PO is ultimately paid to the PO
investor. The only question is whether realized prepayment rates will cause it to be paid sooner or later than
In contrast to PO strips, an IO strip cash flow starts out big and gets smaller over time. Thus, IOs have shorter
effective lives than POs.
The major risk associated with IO strips is that the value of the cash flow investors receive over the life of the
mortgage pool may be less than initially expected and possibly less than the amount originally invested. Why?
The amount of interest produced by the pool depends on its beginning-of-month balance. If market rates fall,
the mortgage pool will be paid off sooner than expected, leaving IO investors with no interest cash flow.
Therefore, IO investors want prepayments to be slow.
An interesting property of an IO is that its price has a tendency to move in the same direction as market rates.
When market rates decline below the contract rate and prepayment rates increase, the diminished cash flow
usually causes the IO price to decline, despite the fact that the cash flows are discounted at a lower rate. As
interest rates rise above the contract rate, the expected cash flows improve. Even though the higher rate must
be used to discount these improved cash flows, there is usually a range above the contract rate for which the
price increases.
Both IOs and POs exhibit greater price volatility than the pass-through from which they were derived. This
occurs because IO and PO returns are negatively correlated (their prices respond in opposite directions to
changes in interest rates), but the combined price volatility of the two strips equals the price volatility of the

The price/yield relationships for IO and PO securities are shown in Figure. Notice the following:
The underlying pass-through security exhibits significant negative convexity.
The PO exhibits some negative convexity at low rates.
The IO price is positively related to mortgage rates at low current rates.
The PO and IO prices are more volatile than the underlying pass-through.


A collateralized debt obligation (CDO) is a security backed by a diversified pool of one or more of the following
types of debt obligations:
U.S. domestic high-yield corporate bonds
structured financial products (i.e., mortgage-backed and asset-backed securities)
emerging market bonds
bank loans
special situation loans and distressed debt
When the underlying pool of debt obligations are bond-type instruments (high-yield corporate, structured
financial products, and emerging market bonds), a CDO is referred to as a collateralized bond obligation (CBO).
When the underlying pool of debt obligations are bank loans, a CDO is referred to as a collateralized loan
obligation (CLO).
In a CDO structure, there is an asset manager responsible for managing the portfolio of debt obligations. There
are restrictions imposed (i.e., restrictive covenants) as to what the asset manager may do and certain tests that
must be satisfied for the tranches in the CDO to maintain the credit rating assigned at the time of issuance
and determine how and when tranches are repaid principal.
The funds to purchase the underlying assets (i.e., the bonds and loans) are obtained from the issuance of debt
obligations (i.e., tranches) and include one or more senior tranches, one or more mezzanine tranches, and a
subordinate/equity tranche. There will be a rating sought for all but the subordinate/equity tranche. For the
senior tranches, at least an A rating is typically sought. For the mezzanine tranches, a rating of BBB but no less
than B is sought. As explained below, since the subordinate/equity tranche receives the residual cash flow, no
rating is sought for this tranche.
The ability of the asset manager to make the interest payments to the tranches and payoff the tranches as they
mature depends on the performance of the underlying assets. The proceeds to meet the obligations to the CDO
tranches (interest and principal repayment) can come from (1) coupon interest payments of the underlying
assets, (2) maturing assets in the underlying pool, and (3) sale of assets in the underlying pool.

In a typical structure, one or more of the tranches is a floating-rate security. With the exception of deals backed
by bank loans which pay a floating rate, the asset manager invests in fixed-rate bonds. Now that presents a
problempaying tranche investors a floating rate and investing in assets with a fixed rate. To deal with this
problem, the asset manager uses derivative instruments to be able to convert fixed-rate payments from the
assets into floating-rate payments. In particular, interest rate swaps are used. This derivative instrument
allows a market participant to swap fixed-rate payments for floating-rate payments or vice versa. Because of
the mismatch between the natures of the cash flows of the debt obligations in which the asset manager invests
and the floating-rate liability of any of the tranches, the asset manager must use an interest rate swap. A rating
agency will require the use of swaps to eliminate this mismatch.

The first breakdown in the CDO family is between cash CDOs and synthetic CDOs. A cash CDO is backed by
a pool of cash market debt instruments. These were the original types of CDOs issued. A synthetic CDO is a
CDO where the investor has the economic exposure to a pool of debt instrument but this exposure is realized
via a credit derivative instrument rather than the purchase of the cash market instruments.
Both a cash CDO and a synthetic CDO are further divided based on the motivation of the sponsor. The
motivation leads to balance sheet and arbitrage CDOs. In a balance sheet CDO, the motivation of the sponsor
is to remove assets from its balance sheet. In an arbitrage CDO, the motivation of the sponsor is to capture a
spread between the return that it is possible to realize on the collateral backing the CDO and the cost of
borrowing funds to purchase the collateral (i.e., the interest rate paid on the obligations issued).
Cash CDOs that are arbitrage transactions are further divided in cash flow and market value CDOs depending
on the primary source of the proceeds from the underlying asset used to satisfy the obligation to the tranches.
In a cash flow CDO, the primary source is the interest and maturing principal from the underlying assets. In a
market value CDO, the proceeds to meet the obligations depends heavily on the total return generated from
the portfolio. While cash CDOs that are balance sheet motivated transactions can also be cash flow or market
value CDOs, only cash flow CDOs have been issued.
CREDIT DEFAULT SWAP: The type of credit derivative used in a synthetic CDO is a credit default
swap. A credit default swap is conceptually similar to an insurance policy. There is a protection buyer who
purchases protection against credit risk on the reference asset. In a synthetic CDO, the insurance buyer is the
asset manager. The protection buyer (the asset manager in a synthetic CDO) pays a periodic fee (like an
insurance premium) and receives, in return, payment from the protection seller in the event of a credit event
affecting any asset included in the reference asset. Who is the seller of the protection seller? It is the SPV on
behalf of the junior note holders.
The junior note holders are getting payments that come from two sources:
(1) The income from the high quality securities purchased with the funds from the issuance of the junior debt
(2) The insurance premium (premium from the credit default swap) paid by the asset manager to the SPV
Effectively, the junior note holders are receiving the return on a portfolio of high-quality assets subsidized by
the insurance premium (i.e., the payment from the credit default swap).
In credit derivatives, a payoff by the protection seller occurs when there is a credit event. Credit events are
defined in the credit derivative documentation. On a debt instrument a credit event generally includes:
bankruptcy, failure to pay when due, cross default/cross acceleration, repudiation, and restructuring. This credit
event applies to any of the assets within the reference asset. For example, if a high-yield corporate bond index
is the reference asset and Company X is in the index, a credit event with respect to Company X results in a
payment to the protection buyer. If a designated portfolio of bank loans to corporations is the reference asset
and Corporation Ys loan is included, then a credit event with respect to Corporation Y results in a payment to
the protection buyer.