Chapter 2: Price-yield convention
2.1.1 Future values
When you invest in debt securities, you receive period coupon, which must be reinvested. Long-term
wealth of investors depends at what rate they are able to invest these periodic coupons. Likewise, if
they want to sell the security at time t+1, the price at which they sell will depend on the interest rates
at time t+1.
There are two methods of interest calculation: simple inter- est and compound interest. Compoundinterest calculations vary with respect to the number of compounding intervals used in a period.
When they invest in debt securities, investors receive periodic coupons, which must be reinvested.The
wealth accumulated by investors will then depend on the rates at which they are able to invest their
coupon incomes. Likewise, the price at which they may be able to sell the security in the future will
depend on the prevailing interest rates in the markets at the time of sale. We examine these issues
now.
In addition, the rate at which money placed in coupon-bearing bond portfolio grows depends on the
method used for computing the interest payments and rein- vestment assumptions. There are two
methods of interest calculation: simple inter- est and compound interest. Compound-interest
calculations vary with respect to the number of compounding intervals used in any given period.
Simple interest rate calculation 
and If 365 days, then  becomes P=(1+y). What if 3 years? Sum year 1
+ 2 + 3 = final value
 Compounded interest, as it is to the power of N
Compounding interest calculation 
If the interest is compounded semiannually, that is, we are paid interest at the end of half a year and
we earn interest on that interest, what will be the future value of our capital? At the end of half a year,
we will have
And if we reinvest this half for another half a year (To have 1 full year) we end up
with
So, with semiannual compounding, the terminal value after one year is
Proceeding this way, the future value of setting aside P today for N years (with m compounding
intervals per year) is
However, with continuous compounding (Tending to infinity) 
Converting interest rates quoted under one convention (say, annual compounding) to another
convention
Generally, we can convert from one method of compounding to another. If we are given the annually
compounded interest rate y*, we can convert it to the semian- nually compounded interest rate y
using
 y* corresponds to annual simple interest and normal y correspond
to the compounding interest. So basically, we can convert annual y* into semiannual compounding
interest rate y!
If y* = 8%, meaning that under ANNUAL compounding we would get 8%, then solving for y, we get
SEMIANNUAL compounded interest of 7.846%
2.1.2 Annuities
A security that pays C (in dollars) per period for N periods is known as an annuity.
We can determine the future value of an annuity that pays C for two years as follows: The first year’s
payment can be reinvested for one more year at a rate y to get, at the end of Year 2,an amount C(1 +
y).
We can determine the future value of an annuity that pays C for two years as follows:
In this way we can determine the future value of an annuity C for N years with an annual interest
payment as follows:
2.1.3 Present values
Similarly, the PV of $1 received after N years with m interest compounding inter- vals per year is
With continuous compounding (as m approaches infinity) we get the present value corresponding to
continuous compounding as:
2.2 YIELD TO MATURITY OR INTERNAL RATE OF RETURN
With annual compounding 
The internal rate of return (IRR) of a bond, denoted by y and sometimes referred to as the yield to
maturity, is the rate of discount at which the present value of the promised future cash flows equals
the price of the security.
With annual compounding, the price P of a bond that pays annual dollar coupons of C for N years, per
$100 of face value, is
Assuming that there is no default, the price of a bond will equal the present value of promised future
cash flows when discounted at its yield to maturity.
Eq 2.6
2.2.1 Semiannual compounding
The U.S.Treasury market uses semiannual compounding.The price of a default-free bond, which has a
round number of N coupons remaining, trading at a semiannual yield y, is given by the expression in
Equation 
2.8
The first term, with the summation sign, is the present value of all future semiannual coupons, and
the second term is the present value of the balloon payment.
N= Number of coupons remaining
As before, we can set the dollar coupon C = 100c, where c is the coupon rate in decimals formula for
pricing a bond with semiannual coupon payments (Simplification of 2.8) 
Eq 2.9
Using Equation 2.9 we can determine the price, P, of a bond, given its yield to maturity y. Alternatively,
we can determine the yield, y, of a bond, given its price P. Simple case when N goes to infinity 
Perpetuity formula (which pays every six months a dollar coupon of C) From Equation 2.9 we solve for
the price of perpetuity as follows:
2.3 PRICES IN PRACTICE
It is customary in fixed-income markets to quote values in terms of yields and/or prices. As we noted
in the previous section, given a yield to maturity, we can com- pute the price, and vice versa.
2.4 PRICES AND YIELDS OF T-BILLS
We start with the concept of invoice price (or dirty price) of a security. Invoice price is the price that
the buyer of a security has to pay.
Treasury bills  Invoice price
Treasury bills, which are discount instruments; they do not pay any coupons and pay a fixed sum of
money (say, $100) at a stated maturity date. Conceptually T-bills are zero-coupon bonds.
Treasury bills are quoted on a discount yield basis.The procedure for obtaining the invoice price from
discount quotes is illustrated in the next example,
The BEY of T-bills is a better measure of the actual return that investors will get by buying the T-bill
and holding it until its maturity date.
Simple relation between the discount yield d that traders quote, and the BEY as follows:
The difference between BEY and d increases with time to maturity.
2.4.2 Yield of a T-Bill with n > 182 Days
When a T-bill has more than six months to maturity, the calculation must reflect the fact that a T-bill
does not pay interest whereas a T-note or T-bond will pay a semiannual as we saw. The industry
convention is to assume that an interest y is paid after six months and that it is possible to reinvest
this interest
Solving for y gives the BEY for T-bills with a maturity of more than 182 days as
With excel  by applying the =TBILLPRICE function, we can determine the price of a T-bill, knowing
(a) the settlement date, (b) the maturity date, and (c) the quoted dis- count yield. Likewise, by applying
the =TBILLEQ function, we can determine the BEY of the T-bill.
2.5 PRICES AND YIELDS OF T-NOTES AND T-BONDS
For T-bills, the invoice or dirty price IS the quoted price cause they don’t pay interests. For T-notes and
T-bonds, the quoted price (also referred to as the clean or flat price)  The quoted price or flat price
is not the invoice or dirty price. To get to invoice or dirty price  We do the quoted price + we add
accrued interests. This accrues to the seller of the security and must be paid by the buyer to get the
full
Dollar coupon on the next coupon date.
1)
Compute accrued interest  determine the last coupon date (LCD), or the dated date (when
the first coupon starts to accrue), which in this case is May 15, 2008. This is also referred to as
the previous coupon date (PCD). The next coupon date (NCD) is November 15, 2008. So, the
number of days between the NCD and the LCD is 184 days (6 Months).
2)
The number of days between the last coupon date (15 Mai) and the settlement date, June 27,
2008, is 43 days. The accrued interest AI is
AI = (Last coupon date) / (NCD - LCD) x (Coupon rate)/2
 Need more explanation, dates are wrong, 15 May 2008???? OR 15 May 2018????
There is an Excel function that can directly compute the accrued interest but requires as an input the
issue date. The U.S. Treasury Website provides the issue date of all outstanding Treasury issues.
 This is the appropriate price yield relation equation.
To apply it  See how to setup excel page at page 39 – 41 of the book.
In a similar manner, we can compute the price of a Treasury debt security given its yield  page 4142
2.6 PRICE-YIELD RELATION IS CONVEX
As we increase the yield to maturity, the price will decline in general.This is due to the fact that the
dollar coupon on debt securities is fixed, and therefore, to com- pensate for the increases in yields in
the market, the price must fall.
Note that the prices fall as yields increase, but the rate of drop in price actu- ally goes down at higher
yields
Important to understand how the value of debt securities changes as market moves interest rates
A drop of 400 basis points from 4% to 0% produces a price increase from 98.98 to 138.30, which is an
increase of 39.72%. On the other hand, an increase of 400 basis points from 4% to 8% pro- duces a
price loss from 98.98 to 72.18, which is a decrease of 27.08% x
In deriving this relationship, we have assumed that the T-note is a “bullet” security with no call feature.
I guess you are now wondering what is a callable bond, don’t you motherfucker? Well, A callable bond,
also known as a redeemable bond, is a bond that the issuer may redeem before it reaches the stated
maturity date. A callable bond allows the issuing company to pay off their debt early.
This is due to the fact that an investor holding a callable bond faces the risk that the bond may be
called away by the issuer when interest rates go down or when the issuer’s credit rating improves, or
both. These circumstances will warrant the issuer to refinance the old (callable debt) with cheaper
new debt. As a result, a callable bond will not show the price increase in a regime of falling interest
rates. Instead, a callable bond price will approach the call price as interest rates fall.
Chapter 7:
Fixed income securities display varying price sensitivities to changing interest rates. The purpose of
this chapter is to develop certain quantitative measures of interest rate risk.
7.1 DV01/PVBP OR PRICE RISK
Dollar value of an 01 (DV01) or price value of a basis point (PVPB) measures risks of fixed income
securities.
Risk of a bond  Changes in price due to changes in rates
DV01 or PVBP measures this price change in debt securities for 1 basis point change in rates (0.01%
change). If P is the price of the bond and y is its yield, a measure of the bond’s risk is the change in its
price for a change in its yield.
Consider a bond with one year to maturity. It pays a 4% coupon semiannually on a par value of $100
and has a YTM of 6%. The price of the bond is  REMEMBER, semiannual coupon hence formula
should be this 
Hence 
 Price of the bond
Now what if the yields change by a small amount, increasing from 6% to 6.1%?
How can we isolate the risk?  Do p – p’  101.91347 – 101.903764 = 0.009705
To express it in millions  0.009705 x 10,000 = 97,053 OU 0.009705 x (1,000,000/100) = 97,053
We are estimating the slope of the tangent to the price-yield relationship at 6%.
Using the =PRICE function of Excel it is fairly easy to compute PVBP or DV01 of debt securities as shown
in the following example.  See page 107-108 of book to compute PVBP or DV01 of a debt security
1) Compute the clean price of each security using =Price function. Next, we recomputed the
clean price at a yield, one basis point different (1 basis point higher) than the prevailing yield.
2) We take the difference of those prices and multiply by #10.000 to get the PVPB per million.
Note that the 2-year T-note changes by $187.68 per million-dollar par when its yield changes by one
basis point. On the other hand, a 30-year bond changes by $1,665.26 for a basis point change in its
yield. This implies the following: If the 2-year yield and 30-year bond yield were to move down by
exactly one basis point, the 30-year bond will appreciate by $1,666.26, whereas the 2-year bond will
appreciate by only 187.68. We can therefore say that the 30-year bond is 1,666.26/187.68 = 8.87 times
more risky than a 2-year T-note under these hypothesized assumptions.
We can compute DV01 by first calculating the price at half a basis point below the prevailing yield
and then computing the price at half a basis point above the prevail- ing yield. We can then compute
the difference between these two resulting prices and evaluate the DV01. The results of such an
approach are shown in Table 7.3.
7.2 DURATION
Another concept widely used to measure risk is duration. Macaulay duration of a debt security is its
discounted-cash-flow-weighted time to receipt of all its promised cash flows divided by the price of
security. In this sense, the duration measures the average time taken by the security, on a discounted
basis, to pay back the original investment:The longer the duration, the greater the risk.
We can think of Bond X as having duration of six years and Bond Y as having duration of three years.
We can then interpret Bond X being more risky than Bond Y. Macaulay duration can also be
interpreted as the price elasticity, which is the percentage change in price for a percentage change in
yield; in this sense, the greater the duration of a security, the greater the risk of the security.
Macaulay duration can also be interpreted as the price elasticity, which is the percentage change in
price for a percentage change in yield; in this sense, the greater the duration of a security, the
greater the risk of the security.
Interpretation 7.1
Returning to coupon bonds, we can define Macaulay duration in more general terms as follows:
In the general definition of Macaulay duration, in Equation 7.1, N is the number of years until maturity,
y is the yield to maturity, and the cash flow at period i is denoted by Ci .
One of the applications of this concept is in bond portfolio immunization: If we can fund liabilities
with assets in such a way that their Macaulay durations are the same, such a portfolio is immune from
interest rate fluctuations. This is because the price elasticity of assets is the same as the price elasticity
of liabilities. Hence their fluctuations cancel each other out.
Interpretation 7.2
Duration is the price elasticity of interest rates; duration is also the price elasticity, which is the
percentage change in price for a percentage change in yield.
The negative sign is just a reminder that prices and yields move in opposite directions.
Dy/(1+y)  Percentage change in the yield
With semiannual compounding, sometimes the following convention is also used:
In this way of thinking about risk, we say that duration measures the elasticity of the bond price to
interest rates: the percentage change in bond price for a percent- age change in interest rates. Note
that we are not measuring the change in interest rates but the percentage change in interest rates.
Another related measure is modified duration (MD). Modified duration is the percentage change in
price for a change in yield. Modified duration is denoted as
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 Rearange to get the following MD 
Price at a yield of 4.995% is 100.0136175 (rounded to seven decimals in Table 7.5). Price at a yield of
5.005% is 99.986385 (rounded to seven decimals in Table 7.5). Note that the price, P, in Equation 7.5
is the dirty price.
Modified duration is simply DV01 divided by the price, which leads to MD = 2.72. What is the economic
intuition behind modified duration? 
Modified duration is always smaller than Macaulay duration and is more exten- sively used in practice.
The PVBP is approximately $734 for $1 million par. The T-bond has a PVBP of 1,667.16. Hence on a par
value basis strip is much less risky than the 30-year T-bond. This is because with same par values,the
dollar exposure of strip is far less than that of the 30-year T-bond. The main message is the
following:When comparing investments with unequal dollar amounts, PVBP gives more transparent
answer. In using duration, one must exercise care to reflect the difference is dollar amounts.
7.2.2 Properties of duration and PVBP
The finding that the duration of a zero coupon bond is equal to its maturity means that the zero
coupon bond is the most interest-rate-elastic security for a given matu- rity class. It is easy to verify
that duration is generally increasing in maturity and decreasing in coupons and yield to maturity, as
shown earlier. The duration of coupon bonds will be less than their maturity. Clearly, as time passes,
duration will change. This requires some attention in portfolios of assets and liabilities for which the
durations are held the same.
7.2.3 PVBP and duration of portfolios
The PVBP of a portfolio is simply the par value-weighted PVBP of individual securi- ties in the portfolio.
For a portfolio with two securities, the PVBP can be computed as shown here:
7.3 TRADING AND HEDGING
A trader is evaluating the shape of the yield curve for settlement on September 12, 2007. (Refer to
Table 7.7 for information.) The yield spread between the 10- year T-note and the 2-year T-note stood
at 43.10 basis points ([4.364% 3.933%] 10,000 43.10) on September 12, 2007. The trader is expecting
this spread to sig- nificantly increase in a few days; in other words, the trader is expecting the yield
curve to get steeper. This expectation may be motivated by many considerations, some of which we
get into later in this section.
The trader wants to set up a trade that will break even if the spread stays at 43.10 basis points and
will make money if the spread widens. Of course, the trader must be willing to accept the risk that
there will be a loss if the yield curve were to flatten; that is, if the spread decreases and moves against
his or her beliefs. How can the trader implement the trade reflecting his or her view about the yield
curve? The over- all yields may either go down or go up, but it is the spread that the trader is betting
on.
First, the trader recognizes that for the spreads to increase in a bullish market (when all rates are
expected to fall), the 2-year yields must drop by much more than the 10-year yields (upward slopping
yield curve, short term cost of capital is lower than longer term return on investment). Similarly, in a
bearish market (when all rates are expected to go up), the 2-year yields must increase by much less
than the 10-year yields. This calls for a long position in the 2-year T-note and a short position in the
10-year T-note.
Second, the trader must determine the amount of the 2-year T-note to buy and the amount of the 10year T-note to short. This is where the concepts that we have developed come in handy. The trader
will want to set up the trade such that the total PVBP is zero.
7.4 CONVEXITY
The slope of price-yield relationship changes with yield levels. Furthermore, the slope of the tangent
becomes steeper as the interest rates (yields) fall. This leads to what is known as convexity of the
price-yield curve. Convexity measures the rate at which DV01 changes as yields change.
Note that the slope of the price-yield function (given by Equation 7.12) and plot- ted in column E of
the spreadsheet in the table decreases (ignoring the negative sign) as yields increase. This change in
slope is measured by the second derivative (given by Equation 7.13), which is plotted in column F.
Note that the second deriva- tive is high at low yields and small at high yields.
The price change of a debt security, according to Equation 7.16, consists of two terms. The first term
is the duration effect, and it is negative. As the yields increase, prices decline. Note that the convexity
effect on price change is positive as seen from the sign of the second term. This is referred to as the
gain from convexity. We can explicitly compute gain from convexity using the PVBP estimates as
shown in Table 7.10.
We find that the convexity contributes favorably to the price change. Holding maturity and yield to
maturity fixed, the convexity decreases as the coupon increases. Convexity increases with duration.
We find that the convexity contributes favorably to the price change. Holding maturity and yield to
maturity fixed, the convexity decreases as the coupon increases. Convexity increases with duration.
7.4.1 Bullet versus barbell securities (butterfly trade)
Let’s consider Table 7.10 and examine the following trading strategy. Is it possible to replace a 5-year
T-note with a portfolio of a 2-year T-note and a 10-year T-note such that (a) there is no cash outlay
and (b) the PVBP remains the same? If so, what is the difference between these two positions?
A long position in 5-year T-note is a bullet position. A long position in a portfolio of a 2-year T-note and
a 10-year T-note is a barbell position, reflecting the two balloon payments.
Let n2 be the par value of the 2-year T-note and let n10 be the par value of the 10- year T-note needed
to replace $100 million par value of a 5-year T-note.We require that the cash proceeds from the sale
of a 5-year T-note to be sufficient to buy the requisite numbers of 2-year and 10-year T-notes. This is
the self-financing condition.
We further require that the DV01 of the 5-year T-note that is sold is equal to the PVBP of the portfolio
that is purchased.
The portfolio we have created is very similar but not identical to the 5-year T-note that we sold. To
see why this is the case, we need to analyze the effects of changes in yields on the 5-year T-note in
the portfolio we have created.
By construction, at the prevailing market yields (underlined in Table 7.11), the market value of a 5year T-note and its PVBP are exactly matched by those of the barbell portfolio. When there is a parallel
shift in the yields, the value of the barbell portfolio dominates the value of the bullet security.
Consider what happens to the portfolio when the yields drop. The PVBP of the barbell portfolio, given
in the last column, exceeds the PVBP of strip 2. This indi- cates that the barbell portfolio will benefit
more from the reduction in yields. On the other hand, as the yields go up, the PVBP of the barbell
portfolio is always lower than that of the 5-year T-note. As a consequence, the barbell portfolio will
lose less value compared to the bullet position.
Trades of this sort, in which an intermediate maturity security is sold (bought) and two securities
whose maturities straddle the intermediate maturity are bought (sold), are known as butterfly trades.
To get a better perspective, we have plotted in Figure 7.3 the amount by which the value of the barbell
portfolio exceeds the value of the 5-year T-note at different levels of yield.
To get a better perspective, we have plotted in Figure 7.3 the amount by which the value of the
barbell portfolio exceeds the value of the 5-year T-note at different levels of yield.
Note that the convexity effect really kicks in only at very high or very low yield levels. In fact, for a
100 basis point change in yields, the effect of convexity is hardly evident. A critical assumption we
have maintained throughout this discussion is that the shift in yields is parallel.
7.5 EFFECTIVE DURATION AND EFFECTIVE CONVEXITY
In our analysis of interest rate risk, we have maintained an assumption that cash flows of debt
securities are unaffected by changes in market interest rates. Thus in computing DV01, duration, and
modified duration, we have assumed that cash flows do not change when interest rates change. In a
number of circumstances, the cash flows of debt securities may depend on interest rate. Callable
bonds and MBS are two obvious examples. In such situations, we need to use a concept that reflects
the fact that cash flows might change when interest rates change. One such measure is known as
effective duration.
We illustrate the idea of effective duration with a simple example of a callable bond with a stated
maturity of three years but callable at any time at 100. Let’s assume that the annual coupon is 6%.
Clearly, the bond will be called if the issuer can issue a similar bond at a lower coupon rate. Thus, the
cash flows of this callable bond are sensitive to interest rates.
1. We project possible interest rate scenarios into the future, covering the life of the debt security,
the effective duration of which we want to estimate. For example, to compute the effective duration
of a 30-year MBS, we will project the interest rates out to a horizon of 30 years.
A simple motivation for us to determine interest rates of different matu- rities can be given using an
annual coupon-paying callable bond as an example. In valuing such a callable bond, we need to
know at each node the one-year inter- est rate, to perform discounting of cash flows. In addition, we
need to know at each node the interest rates of noncallable bonds with the same stated maturity
as the callable bond. This latter information will be used in determining whether the bond should be
called or not.
2. Next we select a random path of interest rates.
3. Next we estimate the cash flows along that path. For a callable bond, when interest rates
go down, the bond may be called. For a mortgage, when refi- nancing rates go down,
mortgages may be prepaid. So, at each node, cash flows will reflect the optimal behavior
of bond issuers (in the case of call) or investors (in the case of mortgages). The result will
be a set of cash flows at each node, as shown in Figure 7.6.
4.
Next we estimate the present value of cash flows along each simulated path.
5. Next we compute option-adjusted spreads (OASs). Once we have the present values of all
simulated paths, we average all the present values. If the average present value across all
simulated paths is exactly equal to the market price, we define the OAS as zero. If the
average of present values is higher than the mar- ket price, we add a constant spread z to
the discount rate at each node until the average is equal to the market price. This spread z
is defined as the OAS.
Article 1: Understanding duration and volatility
Paper 1  Understanding duration and volatility
Duration is one of the most used tools by fixed income managers  One use is immunization 
Portfolio of assets selected such that the duration of asset matches the duration of liabilities to hedge
against interest rates risk  Good estimate of the volatility or sensitivity of the market value of my
bond portfolio to changes in rates. Measure of “risk”
II. Macaulay Duration
In the bond market, securities are commonly referred to by their maturities. While this is a useful
benchmark, it is deficient, because it measures only when the final cash flow is paid and ignores all of
the interim flows.
He described a measure he called duration, which measures the weighted average time until cash
flow payment. The weights are the present values of the cash flows themselves.
Duration and Maturity for Nonzero-Coupon Bonds
As the maturity is lengthened further, the bond begins to look more like a perpetual annuity, the
duration of which is given by
This duration versus maturity pattern is shown in Figure 3; however, this applies only to par and
premium bonds.
The duration pattern for discount bonds is more complex. Very low coupon bonds have a duration
pattern that lies close to the zero-coupon pattern (D=m) up to a reasonably long maturity. For very
long maturities, however, even a low- coupon bond begins to resemble a perpetual annuity.
For an extreme example, consider a 1/2% coupon bond. If the maturity was 20 years, the duration
would be approximately 17 years at 10% yield, because the coupons are relatively insignificant,
compared with the redemption payment. If the maturity was 100 years, however, the bond would act
like an annuity, because the redemption payment would be so distant. The duration would be 10.6
years, very close to the perpetual annuity duration of 10.5 years. Thus, the duration can actually
decrease with increasing maturity, approaching the duration of a perpetual annuity from above. This
pattern is shown in Figure 4.
Therefore, we will refer to Figure 3 as the basic duration versus maturity pattern, even though long
maturity discount bonds can behave differently. For comparison, the patterns of premium, par and
discount bonds are shown in Figure 5.
The Effect of the Yield Level
It is natural to expect that the yield level will affect duration. Using Figure 2 as a guide, consider how
the figure would change as the yield increased. All of the present values would decline, but the cash
flows that are the farthest away would show the largest proportional decrease.9 As a result, the early
cash flows would have a greater weight relative to the later cash flows, and the fulcrum would have
to be moved to the left (shorter duration) to keep the system in balance. Yield increase  duration
decrease
It may help to remember that duration changes in the same direction as price when yield changes.
The Effect of the Coupon Frequency
How does the coupon payment frequency affect the duration? Referring to the seesaw diagrams,
imagine that every coupon was divided into two parts and that one of the parts was paid one-half
period earlier than the other. On the diagram, this represents a shift of weight to the left, as part of
each coupon is paid earlier. This shift to the left moves the balance point to the left; thus, increasing
the coupon frequency shortens the duration. Decreasing the frequency lengthens the duration.
Duration as Time Elapses (and Maturity Approaches)
Consider the duration of a par bond as time elapses and the bond's maturity decreases (holding yield
constant). Using Figure 3 as a guide, we can see that duration will initially decline slowly, and then at
a more rapid pace as the bond approaches maturity.
Duration Between Coupon Dates
What happens to duration as time elapses between coupon payments (with no change in yield).
Consider what happens as one day elapses. Each cash flow and the original duration fulcrum are now
one "day" closer to the investor. If the position of the fulcrum does not change relative to the cash
flows, then the duration (the time from the "investor" to the fulcrum) will have decreased by one day.
As we will show, this is the case: The fulcrum's position will not have changed relative to the cash
flows.
+1 day, hence  All of the cash flows will increase in present value as the discounted period is
shortened. As time elapses between coupon dates (or any cash flow dates), duration shortens by the
same amount of time that elapsed. After each day, the duration will be one day shorter.
We now have to consider what happens as the coupon date approaches. As before, each day that
elapses brings the fulcrum one day closer (that is, duration shortens by one day). What happens to
duration after a coupon payment? The coupon is no longer part of the bond's cash flows and thus, is
not a factor in its duration.10 To bring the system into balance after the coupon is paid, the fulcrum
must be moved to the right (as shown in Figure 10) and the duration increases. Except for the extreme
maturity range of Figure 4, this increase is less than the time between coupon payments. Thus, the
duration shows a slight decline from one coupon date to the next
With this new information, the duration versus time pattern can now be redrawn more accurately
(see Figure 11 for an example of a semiannual bond). The downward sloping straight-line segments
represent the duration decreasing between coupon dates, with the upward jumps occurring on the
coupon dates. This sawtooth duration pattern normally evokes a question about the volatility of the
bond, namely whether the volatility follows a similar pattern.11 As described more fully in Section IV,
the answer is no: Volatility follows a smooth path as time elapses (at constant yield).
Duration of a Portfolio
Portfolio duration is a weighted average of the durations of the individual security durations. The
weights are the present values (full prices) of the securities divided by the full price of the entire
portfolio, and the duration that results is often referred to as a "market-weighted" duration. This
approach is actually very similar to the determination of duration of a single bond, in which the bond
is considered a portfolio of zero-coupon instruments. The duration of each payment (time to maturity)
is multiplied by its present value divided by the value of the whole bond.
It is quite likely, however, that a portfolio may exhibit more convexity than individual securities.
III. Modified Duration
While Macaulay duration is appropriate for use in immunization, another measure - modified
duration - is better as a volatility measure.
As mentioned above, modified duration, not Macaulay duration, is appropriate for volatility
measurements. Second, modified duration provides a measure of percentage price volatility, not
absolute dollar volatility. Third, the percentage volatility applies to the full price of the security
(including accrued interest), not the quoted (flat) price.
IV. Volatility Weighting for Hedging, Bond Swaps and Arbitrage
The motivations for entering hedging, bond swaps and arbitrage transactions are usually quite
different. The hedger is usually attempting to minimize a risk that cannot otherwise be conveniently
eliminated. The bond swapper is attempting to increase return by swapping into a security that is
expected to outperform (even in the absence of a general market move) the original position over
some specified time horizon. The arbitrageur is creating an entirely new position to capitalize on an
expected realignment of yield spreads. Despite these differing motivations, however, volatility
weighting is similarly used in all three cases.
Hedging attempts to offset price changes in one security with equal changes in another. Because most
securities in the debt market are positively correlated with one another in terms of price movement,
a short position normally offsets a long position.
Volatility weighting is occasionally and unfortunately referred to as "duration weighting," leading
some to assume - incorrectly - that the ratio of the durations is the proper hedge ratio. Duration can
be used to weight trades, but it is more complex than the simple ratio of the durations. In the sections
that follow, we determine the correct weighting for one bond versus another. This ratio is appropriate
for hedging one bond with the other, swapping from one to another (except for rate-anticipation
swaps) or establishing an arbitrage position.
The Hedge Ratio
The objective of weighting the position is to equalize the total changes in value of the two offsetting
positions.
Method 1: Price Value of a Basis Point
The measure known as the price value of a basis point (PVBP) or, alternatively, as the dollar value of
a 0.01 (DV.01), is simply the change in price for a bond that corresponds to a change in yield to
maturity of one basis point (0.01 %).
Figure 12 shows the price-yield curve pattern that is common to noncallable bonds and expands this
to present the graphic interpretation of PVBP, which is a direct measure of price volatility relative to
yield change. Figure 12 also demonstrates that a given bond has a greater price sensitivity to a given
yield change when rates are low.
We can use the PVBP to determine the appropriate volatility weighting for trades. For example,
assume that the price of one bond would change by 0.08 (from 98.60 to 98.68, for example) if its yield
moved by one basis point, and we wished to hedge that change by taking a position in a security that
would change by 0.06 per basis point. If we assume that both securities will change by the same
number of basis points, then it is obvious that we need 1.3333 units of the hedge vehicle per unit of
target security. Thus, if yields changed by ten basis points, we would expect that the target security
would change by approximately 0.80, and 1.3333 hedge vehicle units, changing by 0.60 each for ten
basis points, would also change by 0.80 (that is, 1.3333 x 0.60). How does this intuitive approach
compare with equation [8] above?
Method 2: Yield Value of 1/32
Many Treasury bond traders use yield value of 1/32 for weighting trades
Figure 13 presents the yield value of 1/32 and shows that this is an inverse measure of volatility. A
high value indicates low price volatility (and vice versa), because it means that a large yield change is
necessary to produce a 1/32 price change.
Using Duration in Volatility Weighting
V. Convexity
The basic price-yield pattern of a straight bond
Because of its shape, it is referred to as convex: The degree of curvature is loosely referred to as the
convexity. Convexity is the reason that estimates of price changes using duration or price value of a
basis point increase in error as the yield change increases.
Let us compare the price (and duration) changes that occur to three different investments for equal
yield moves. The three securities are structured as follows: The first pays a single payment of 162.89
in five years, the second pays 67.00 in three years and 98.99 in seven years, and the third pays 55.13
in one year and 120.33 in nine years. While these securities seem to be quite different, they have at
least two characteristics in common - a present value at 10% (discounted semiannually) of 100 and a
Macaulay duration of 5.0.
The duration of Security 1 is obviously five years, because it is a zero-coupon bond with a maturity of
five years. The duration of Security 2 is also five years, because the present values of each cash flow
are the same (50) and are equal distances from the five-year point, so the balance point is five years.
A similar argument shows that the duration of Security 3 is also five years.
Figures 20 and 21 demonstrate the convexity patterns of the three securities, but they do not explain
why convexity occurs. For this, we will return to the seesaw diagrams. We begin by examining what
happens to security I (the least convex), compared with security 3 (the most convex).
V1. Duration for Other Securities
Money Market Instruments: duration = maturity (like zero coupon bonds)
Securities with Embedded Options: Duration calculations are made difficult by the uncertainty of the
cash flows associated with the bonds, in addition to the problem of determining the appropriate yield
(to call, maturity or first put date) to use. Securities with embedded options must be analyzed with a
model of price behavior to properly determine duration.
Mortgages: The duration of mortgage securities is even more difficult to determine than that of
callable bonds. First, the cash flows of the mortgage are even more uncertain than those of a callable
bond, reflecting the unknown pattern of prepayments that may prevail. Second, the call feature of
mortgages is not as totally yield driven as it is for callable bonds, and many prepayments occur even
when the option is out of the money.
it is more likely that the search for duration is driven by the need to estimate the sensitivity of the
mortgage to changes in the market level of rates. One method is to attempt to model the price or
yield behavior of the mortgage versus a benchmark for the market, for example, ten-year Treasury
notes.
Futures Contracts: The duration of a futures contract cannot be determined using the standard
calculation. There are no definable cash flows associated with a futures contract. We can view a
contract as pure volatility, and because there is no cash outflow (price) paid to enter the contract, its
percentage volatility (and, thus, its modified duration) is infinite. A long position adds volatility, and a
short position reduces volatility in a portfolio.
Floaters: Floating-rate securities defy attempts at the standard duration calculation because of the
unknown level of the future cash flows. If a sensitivity to rate changes is the objective, however, then
a duration can be inferred. If a particular floater is reset every quarter to the then-prevailing threemonth rate based on some index, the primary volatility of the floater will be the same as that of a
three-month instrument. As time elapses and the coupon payment approaches, the implied primary
duration will approach zero and will reset to three months on the coupon date.
Interest Rate Swaps: An interest rate swap can be analyzed as essentially an exchange of two
securities, usually involving a fixed-rate and a floating-rate component. In a manner similar to futures
contracts, an interest rate swap contract adds (or subtracts) volatility without involving a purchase
"price." As a result, it is impossible to determine duration as a percentage volatility measure, but it is
possible to estimate the volatility characteristics of the swap and how entering into the swap affects
the duration of an existing portfolio.
The reason that we multiply the price change by 5 is simple:The price dif- ference is over a 20-basispoint overall change. So, multiplying by 5 gives an estimate of the price change for a 100-basis-point
change. The effective dura- tion takes into account the effect of changes in interest rates on cash
flows.
Paper 2: Understanding Convexity
Bond’s price-yield curve  Indicates the bond’s market value at different yields. Yet, the rate of
change in value is important as yield increase or decrease. A bond that loses value quickly when yield
rises, but gains little value when the yields decrease, is not ideal.
For example, the price-yield curve of a bond free from options or sinking funds – a bullet issue – is
convex. That is, the price is always falling at a slower rate as the yield increases. For a given increase
in rates, the curve falls much faster at point A than at Point B.
High convexity is a desirable property for a bond or portfolio of bonds. If two bonds have tangent
price-yield curves (as in figure 2), then when rates move, the price of the more convex bond (bond
A) will fall more slowly and rise more quickly than the price of the less convex bonds.
If you want to increase portfolio convexity, while leaving the modified duration on your bonds’
fixed, you would need options (hence premium paying) or accept a lower yield.
The advantages of increased convexity is often outweighed by the costs and the risk. Using
barbells strategy instead of bullet issued with the same duration generally does not increase
returns. In the other hand, using options can be highly beneficial in terms of the risk/reward ratio
given by increased convexity. Paying for more convexity entails that we are taking a position on
interest rate volatility and the reshaping of the yield curve.
Understanding price yield curve is useful for two reasons 
-
It is easy to translate a percentage price change into an absolute
price change
-
One can easily compare the price yield curves of two investments
opportunity
Modified duration:
Modified duration approximates the percentage price change of a bond when rates change by 1%.
Formula:
For example, suppose we have a 12% 30-year bonds. We find a modified duration of 8.08. If we want
to approximate the price after yield changes by 200 Basis point, we can just move down the tangent
line and meet point B.
CHAPTER 8
Yield curve is a term used to describe the plot of yield to maturity against time to maturity or against
a risk measure, such as the modified duration of debt securities in a certain market segment (such as
Treasury or corporate bonds). It is therefore natural to speak of “Treasury yield curve” or “corporate
AAA yield curve.” By incor- porating the expectations of diverse participants in the marketplace, the
shape of the yield curve succinctly captures and summarizes the cost of credit for various maturities
of different issuers.
Recall that the concepts of duration and convexity are strictly valid only when the movements in the
yield curve are parallel. Parallel shift? All the rates move up by 1% for example.
Recall that the concepts of duration and convexity are strictly valid only when the movements in the
yield curve are parallel. What is the meaning of a parallel shift in yield curve? Consider Table 8.1, which
records yield to maturity of selected benchmarks in the Treasury market. The yield to maturities of
the benchmarks on November 8, 1979, are shown.These are plotted in Figure 8.2. Suppose all the
yields moved up by 0.50%.
Note the sharp differences in (a) levels of interest rates, (b) slope of the yield curve, and (c) the overall
shape of the yield curve. On November 8, 1979, inter- est rates were rather high and short-term
interest rates were higher than long- term interest rates, producing an inverted yield curve.
By October 9, 1992, the levels had fallen significantly: Two-year yields stood at 4% and 30-year yields
at 7.52%.The spread between 30-year yields and 2-year yields was 352 basis points, resulting in an
upward-sloping yield curve, as illustrated in Figure 8.3.
Figures 8.2–8.4 vividly illustrate three different types of risk: (a) levels of interest rates, (b) slope of the
yield curve, and (c) shape of the yield curve. These three variables (level, slope, and curvature) have
been found extremely useful in explaining the variations in yield curve.
8.1.1 Principal components analysis of yield curves
The PCA approach recognizes that the interest rates of various maturities (zero yields) are very likely
correlated with a relatively small number of underlying economic variables that cannot be observed
directly. Such unobserved economic variables are known as latent variables. For example, zero yields
of var- ious maturities are all correlated with each other. Economic intuition would then suggest that
perhaps a few (three or four) common latent variables drive them all.
PCA identifies independent factors that explain the maximum amount of observed correlation
among yields.
Litterman and Scheinkman (1991) performed a PCA on gov- ernment yield curve and concluded that
there are three major principal compo- nents that can help explain the comovements of yields of
various maturities. They associated these three latent variables with (a) level of interest rates, (b)
steepness of yield curve, and (c) the convexity of the yield curve or its curvature.
They conclude that considering explicitly these three variables in hedging interest rate risk is likely to
produce much better outcomes than simply holding a zero duration position.
8.1.2 Volatility of short and long rates
Volatility measures the variability of interest rates relative to their expected average levels. Loosely
speaking, volatility measures the degree of variation of any variable around its mean.
The levels of interest rates and their volatility might sys- tematically incorporate the changes in the
factors that affect them. As a consequence, the time series of interest rates might exhibit a systematic
clustering effect.
Short terms rates are more volatile than long term rates  Prices can’t deviate too much from par
value in the short term. Yield however will move much more based on macro environment and
economic response. In the longer term, they care less about that noise.
8.1.3 Price-based versus yield-based volatility
It is useful to distinguish between price-based and yield-based volatilities in the context of fixedincome securities markets. The prices of fixed income securities tend to par as their maturity dates
approach. The price-yield relation that we developed earlier can be used to derive one volatility, given
the other. For example, we showed that the modified duration is given by.
Though short-term yields are generally more volatile than long-term yields, long-term bond prices are
much more volatile than short-term debt securities, which pay par value at maturity and hence cannot
trade too far away from par. In pricing options on bonds, bond price volatility is more important; on
the other hand, in pricing options on yields, it is the volatility of yields that is more relevant.
If I care about return on investment, I will be more interested in price volatility. If I’m a hedge fund
yield volatility will be more important. Return = prices
8.1.4 Economic news announcements and volatility
Economic news arrives in a lumpy fashion in capital markets. In equity markets, compa- nies announce
their earnings every quarter. Market participants form expectations about earnings, and actual
earnings may produce either a positive or negative surprise relative to expectations. Such surprises
may result in positive or negative jumps in stock prices and can result in a jump in volatilities.
They found that surprises in news announcements (as measured by deviations from forecasts) can
explain a substantial fraction of price volatility that follows announcements. Markets are generally
very fast in adjusting to news. Volatility and trading volume increase significantly after the
announcements. To sum up, bond yields and their volatilities depend on many factors, including (a)
maturity, (b) business cycles, and (c) surprises in economic news announcements.
8.1.6 Coupon and vintage effects
In each duration range, there are clusters of Treasury securities of varying vintages, coupons, and
contractual provisions, as illustrated in Table 8.4.
Note, for example, in the duration bucket 10 to 17 years there are debt securities, which are more
than 13 years old, and there are newly issued securities. This “vin- tage” problem is even more severe
in the duration buckets 5 to 10 and 2 to 5. Many callable 30-year bonds with fairly high coupons, which
were issued 25 years ago, are still outstanding.The range of yields in these duration buckets is very
large. The dif- ferences in yields can be very substantial, even for securities maturing on exactly the
same date
Table 8.5 gives anecdotal evidence to the effect that high- coupon securities tend to trade at a slightly
higher yield. One possible explanation is that many institutional investors with long-dated liabilities
might prefer low-coupon securities, which tend to have a longer duration.
As noted in Chapter 7, higher dura- tion means a higher sensitivity to interest rates. Such assets may
be ideal to match the interest rate sensitivity of long-dated liabilities.
The important message to take away from the evidence is that the yield curve is populated by many
debt issues with varying vintage, coupons, and contractual features. These debt issues may differ in
liquidity. Some of them may trade at a discount and some at a premium above par. For pricing
purposes, we need a good benchmark at each maturity date clearly indi- cating the yield at that
maturity date. Ideally, we would like to price a zero coupon bond at each maturity. With so many
variations in vintages and coupons, it is very difficult to estimate the correct yield for a zero coupon
bond at any given maturity date.
In addition, there are some maturity dates for which no issues are present in the market. For example,
as noted earlier,Treasury suspended the auctions of 30-year T-bonds during 2001–2006.This action
resulted in maturity sectors for which we have no yields. To develop estimates of yields at those
maturities, we need a theoreti- cal benchmark as well. This leads us to the concept of term structure.
8.2 TERM STRUCTURE
To develop a sharp intuition about the shape of the yield curve and the factors that underlie the levels
and the shape, we need a more parsimonious representation of the yield curve than Figure 8.1. It is in
this context that we define the term structure of interest rates.
Term structure of interest rates refers to the relationship between the yield to maturity of defaultfree zero-coupon securities and their maturities.
Often the yield to maturity on a default-free zero coupon (pure discount) bond is termed the spot rate
of interest. The relationship between the spot rate of a pure discount bond and its maturity is referred
to as the spot curve. To get a better handle on the pricing of zero-coupon bonds, we first develop
the pricing principles for a default-free pure discount bond.
Term structure vs yield curve  For the yield curve, we use all the YTM to constitute on y (one rate of
discount for the bond). In the term structure, they discount each cash flow by their respective discount
rate.
8.2.1 Implied zeroes
The concept of term structure is best developed in terms of pure discount bonds or zero-coupon
bonds. We define the notation for the price of a zero, suggested earlier, more formally as zj , the price
of a pure discount bond today that pays $1 in j periods from now. If we set j = 2, then z2 will represent
the price of a two-year zero coupon bond. We assume that the discount bonds are free from default
risk.
Examples of default-free zero coupon securities are T-bills
8.2.2 Bootstrapping procedure
Spot rates are associated with specific maturities.Thus, the spot rate yj for a pure dis- count bond
maturing j years from now may be defined as the discount rate at which the present value of the
promised terminal cash flow of the pure discount bond is equal to its price. Recall that zj is the price
today of a pure discount bond paying $100 in j periods; then
Often, we are confronted with situations in which the prices of coupon bonds are readily available,
but zero coupon prices are difficult to get. So, it is necessary to try to estimate zero coupon bond
prices based on the prices of coupon bond prices. A procedure known as bootstrapping is used for this
purpose. This procedure is illustrated in Example 8.1.
Before proceeding to address these important estimation problems, we briefly state the general
relationship between coupon bond prices and spot rates of interest. In Example 8.1, we used the
information on coupon bond prices as input to derive the zero coupon bond prices. Such estimates of
zero coupon bond prices are known as implied zeroes, since they are implied by coupon bond prices.
Let’s denote the cash-flow information about coupon bonds as Matrix A:
See how to do this and solve it in excel at page 146
8.2.3 Par bond yield curve
A concept that is used in the industry is the par bond yield curve. It is the relation- ship between the
yield to maturity and time to maturity of bonds that sell at their par value.
Useful if i’m trying to issue a bond to finance my investment and I want to determine a good coupon
rate that is reasonable to charge. Good guide for the issuer on where to set the coupon rate.
8.3 FORWARD RATES OF INTEREST
Formally, a forward rate between two future dates j and k, where k = j is a currently agreed-upon rate
at which one may borrow or lend on date j for a loan maturing on date k. How do we determine the
forward rate on date t, so that we can lock the rate in for a loan that begins on date j and matures on
date k (naturally, k >/ j >/ t)?
The prices of pure discount bonds are provided in Table 8.17 for four maturities. Using this as the basic
data, compute all the applicable forward rates as well as the par bond yields.
Note that the forward rates at Year 0 may be computed for the future periods 1 to 2, 1 to 3, 1 to 4, 2
to 3, 2 to 4, and 3 to 4. In effect, there are six forward rates that we can compute at Year 0.
`
Our analysis, however, does illustrate that the coupon bond prices contain infor- mation about the
prices of pure discount bonds or discount factors. If the prices of coupon bonds are out of alignment
with those of pure discount bonds, after account- ing for liquidity and coupon effects, there will be
profitable trading opportunities with- out any risk. Since there are coupon and vintage effects in these
securities, we should use those coupon securities that sell close to or at par in constructing the implied
zero coupon bond prices. We illustrate this idea later in the chapter. More generally, this analysis
indicates that the schedule of coupon bond prices and pure discount bond prices must stay in
alignment to preclude profitable trading opportunities.
8.4 STRIPS MARKETS
Reinvestment risk  We assume that each coupon payment can be reinvested at the same rate.
However, if rates go to 0, you won’t get any coupon payment anymore.
These are called STRIPS or strips, short for Separate Trading of Registered Interest and Principal
Securities.
It should be stressed that strips are not implied zeroes. Strips are traded secu- rities directly subject
to demand and supply. Implied zeroes are estimated pure discount functions derived from the prices
of coupon-paying Treasury securities. Their prices are influenced by the demand supply forces in the
coupon securities markets. Yet, as expected, implied zeroes provide a natural benchmark for
assessing the relative richness or cheapness of Treasury securities compared to strips.
Why would investors want to hold zero coupon securities such as strips? Several motivating factors
are at work here
the duration of Treasury coupon securities change with interest rates. As a result, investors who buy
Treasury coupon securities to hedge against their liabilities (by matching the duration of assets with
liabilities) may have to frequently rebalance their positions. When a 30-year zero coupon bond is
purchased, its duration is always its time to maturity, irrespec- tive of interest rates. This may
significantly reduce the need to rebalance positions.
There may be many institutional investors holding liabilities with a duration of 20 or more years, and
for these investors, strips may be the only real- istic alternative. A 30-year strip has a duration of 30
years and may thus be preferred by investors with long-dated liabilities for hedging purposes. If
there are many such inves- tors, the strong demand for such securities may drive up the prices of
long-dated strips and bring down their yields.
Principal-only strips tend to be generally less liquid compared with strips made from coupons.
Why? First, there are far fewer principal-only strips.
Second, interest-based strips are more uniformly distributed across the whole range of maturity,
whereas the distribution of principal-based strips is skewed in favor of longer maturity sectors.
If this condition were not to hold, then one can sell a longer maturity zero (sell- ing at the same or
higher price) and buy a shorter maturity zero without any cash outlay.
8.5 EXTRACTING ZEROES IN PRACTICE
Recall from our bootstrapping procedure that, given a set of bond coupons and maturities, it is
possible for us to extract the spot rates of interest.
Once we know the spot rates of interest, we can compute the relevant implied forward rates of
interest.
See book section if needed
`
To explain variation in returns  Essential to distinguish between systematic from specific risks
(Influencing each security individually without putting at arm the entire portfolio). We can use
duration analysis to estimate how much a change in interest rate will impact the prices of my fixed
income securities.
The analysis of the author of the paper suggests that most of the variation in returns on fixed income
securities can be explained by the three “factors” or attributes of the yield curve.  Level, steepness
and curvature.
Investors, by considering those factors, can achieve a better hedge than they would get simply by
holding 0 duration bonds.
The implied zero curve:
Ideally, what we need are ‘zero coupon obligations” priced in such a way that they can be used to
obtain correct prices for coupon bonds.
Duration hedging
Duration  Price sensitivity of a bond to a parallel shift of the yield curve. However, in reality yields
do not move equally together. Hence, such a parallel shift only explains part of the price changes.
`
9.4 INTEREST RATE DERIVATIVES
ENTIRE CHAPTER 9 IS TRASH.
In one simple and versatile model of interest rates, all security prices and rates depend on only one
factor?the short rate. The current structure of long rates and their estimated volatilities are used to
construct a tree of possible future short rates. This tree can then be used to value interest-ratesensitive securities.
The model has three key features.
1. Its fundamental variable is the short rate? the annualized one-period interest rate. The short rate is
the one factor of the model; its changes drive all security prices.
2. The model takes as inputs an array of longrates (yields on zero-coupon Treasury bonds) for various
maturities and an array of yield volatilities for the same bonds. We call the first array the yield curve
and the second the volatility curve. Together these curves form the term structure.
3. The model varies an array of means and an array of volatilities for the future short rate to match
the inputs. As the future volatility changes, the future mean reversion changes.
We examine how the model works in an imaginary world in which changes in all bond yields are
perfectly correlated; expected returns on all securities over one period are equal; short rates at any
time are lognormally distributed; and there are no taxes or trading costs.
Chapter 11 
A fixed-income portfolio manager may manage funds against a bond market index or against the
client’s liabilities. In the former approach, the chief concern is performance relative to the selected
bond index; in the latter, it is performance in funding the payment of liabilities.
2. A FRAMEWORK FOR FIXED-INCOME PORTFOLIO MANAGEMENT
To make our discussion easier to follow, let us revisit the four activities in the investment management
process:
1. Setting the investment objectives (with related constraints);
2. developing and implementing a portfolio strategy;
3. monitoring the portfolio
4. adjusting the portfolio.
There are two types of investors based on investment objectives.
A)
Investor with no liability matching as a specific objective. ex: a bond mutual fund has a great deal of freedom in how to invest its funds (No liabilities
that require specific cash flow stream). Fund gets money from investors and
invests it for them. No guanratee rate of return. No liability matching ? -
More likely to select a bond market index as benchmark for the portfolio.
The investor taking this approach will evaluate of bonds holding in relation
to the benchmark index and in relation to the overall risk of portolio.
B) The second type of investor has a liability (or set of liabilities) that needs to
be met. For ex- ample, some investors create a liability by borrowing money
at a stated rate of interest, thereby leveraging the portfolio. Success =
Enough cash flow to pay the liabilities interest payments. Liabilities =
benchmark
2. MANAGING FUNDS AGAINST A BOND MARKET INDEX
A passive management strategy assumes that the market’s expectations are essentially cor- rect or,
more precisely, that investors cannot add value by second-guessing these expectations.
By setting the portfolio’s risk profile (e.g., interest rate sensitivity and credit quality) identical to the
benchmark index’s risk profile and pursuing a passive strategy, the investor accepts an average risk
level (as defined by the index’s and port- folio’s risk profile) and an average rate of return. No need to
make independent forecasts and the portfolio should track the index.
An active management strategy essentially relies on the manager’s forecasting ability. The portfolio’s
return should increase if the manager’s forecasts of the future path of the factors that influence fixedincome returns (e.g., changes in interest rates or credit spreads) are more accurate than those
reflected in the current prices of fixed-income securities
3.1. Classification of Strategies
1) Pure bond indexing (or full replication approach)  The goal here is to produce a portfolio
that is a perfect match to the index. The pure bond indexing approach attempts to dupli- cate
the index by owning all the bonds in the index in the same percentage as the index.
2) Enhanced indexing by matching primary risk factors  They only focus on a sample of the
risks, eliminate some noise rather than using the full replication sample. Primary risk factors
are typically major influences on the pricing of bonds, such as changes in the level of interest
rates, twists in the yield curve, and changes in the spread between Treasuries and nonTreasuries.
A) By investing in a sample of bonds rather than the whole index, the manager reduces the
construction and maintenance costs of the portfolio. Although a sampling ap- proach will usually track
the index less closely than full replication, this disadvantage is expected to be more than offset by the
lower expenses.
B) By matching the primary risk factors, the portfolio is affected by broad market-moving events (e.g.,
changing interest rate levels, twists in the yield curve, spread changes) to the same degree as the
index.
3) Enhanced indexing by small risk factor mismatches  While matching duration (interest rate
sensitivity), this style allows the manager to tilt the portfolio in favor of any of the other risk
factors. The mismatches are small and are intended to simply enhance the portfolio’s return
and/or risk profile enough to overcome the difference in administrative costs between the
portfolio and the index.
 Small mismatches, administrative costs covered. Matches duration.
4) Active management by larger risk factor mismatche . The difference between this style
and enhanced indexing is one of degree. This style involves the readiness to make deliberately
larger mismatches on the primary risk factors than in Type 3—definitely active management. The portfolio manager is now actively pursuing opportunities in the market to increase
the return. The manager may overweight A rated bonds relative to AA/Aaa rated bonds,
overweight corporates versus Treasuries, position the portfolio to take advantage of an
anticipated twist in the yield curve, or adjust the portfolio’s duration slightly away from the
index’s duration to take advantage of a perceived opportunity. The objective of the manager
is to produce sufficient returns to overcome this style’s additional transaction costs while
controlling risk.
5) Full-blown active management. Full-blown active management involves the possibility of
aggressive mismatches on duration, sector weights, and other factors. Often, the fund
manager is seeking to construct a portfolio with superior return and risk characteristics,
without much day-to-day consideration of the underlying index composition.
“Why should an investor consider investing in an indexed portfolio?” 
-
Lower fees than actively managed accounts, outperforming being a
difficult task, some people don’t even try & provides excellent
diversification
You should choose as a benchmark the index containing characteristics that match closely with the
desired characteristics of your portfolio. The choice depends heavily on four factors:
1) Market value risk  Deried market value risk of portfolio and index should be
comparable. Ex: Upward slopping yield curve, yield increase with maturity. It does not
mean higher returns than securities with shorter marutiries. A long duration portfolio
is more sensitive to changes in rates, it will likely fall more in prices during rate hikes
than a short duration bond. Duration and maturity increase? Market risk increases
2) Income risk  The chosen index should produce an income stream that is desired for
the portfolio. Some investors prefer portfolio generating high level of income while
conserving capital. If stability and dependability of income is important for investor,
then the long portfolio is the least risky and the short one the more risky.
Index portfolio  Lower fees than actively managed accounts
Broadly based index portfolio provide excellent diversification
3) Credit risk  Average credit risk of the index should be appropriate for the portfolio’s
role in the overall portfolio of the investor and satisfy constraints place on credit
quality and policy statements.
4) Liability framework risk  This risk should be minimized. Prudent to match
investment characteristics (duration) of assets and liabilities. Investors with long term
liabilities should select a long index! This means that bond investors with no liabilities
have much more room in choosing the appropriate investment vehicle
From a macro perspective, the bond market may be separated into sectors, in which sim- ilar securities
are grouped together.
-
Divided by issuer (bond market)  a) ABS and MBS, corporate bonds,
government bonds …
-
Or bond can separate by credit risk  b) Low to high credit risk
(AAA/Aaa, AA/Aa…)
-
Bonds can also be separated by other features, such as maturity, fixed
versus floating coupon rates…
3.2.2. Bond Index Investability and Use as Benchmarks
The creation of a bond index requires many decisions and choices. The typical criteria used to
construct a bond index concern:
-
country, credit risk, liquidity, maturity, currency, and sector
classification. Furthermore, compared with equities, there are more
issuers of bonds.
Compared with equities  More issuers of bonds, most bonds have less liquid secondary markets,
less data than equity markets data.
Secondly, owing to the heterogeneity of bonds, bond indices that appear similar can often have very
different composition and performance.
A third potential challenge is that the index composition tends to change frequently. Al- though
equity indices are often reconstituted or rebalanced quarterly or annually, bond indices are usually
recreated monthly.
A fourth issue is what Siegel (2003) referred to as the “bums” problem, which arises be- cause
capitalization-weighted bond indices give more weight to issuers that borrow the most (the “bums”).
The bums in an index may be more likely to be downgraded in the future and experience lower returns.
The bums problem is applicable to corporate as well as government issuers. With global bond indices,
the countries that go the most into debt have the most weight. An index heavily weighted by bums
will likely have increased risk compared with an equally weighted index.
Because of the infrequent trading of bond issues, their heterogeneity, and often limited size, one issue
with bond index investability is that a passive manager faces challenges in tracking a broad index.
A fifth issue is that investors may not be able to find a bond index with risk characteristics that match
their portfolio’s exposure. Because bonds differ in terms of credit rating, duration, prepayment risk,
and other characteristics, a bond index will have a unique exposure that is unlikely to exactly match
that desired for the portfolio.
In sum, Due to small size and heterogeneity of bond issues, their infrequent trading, and other issues,
many bonds indices will not be easily replicated or investable. These indices are often recommended
as benchmarks for manager performance analysis, manager selection and retention, and for other
performance measurement purposes. However, if bond indices are not investable, it is unrealistic and
unfair to expect a manager to match its performance. As such, bond indices often do not serve as valid
benchmarks.
Identification and measurement of risk factors play a role both in index selection and in portfolio
construction  Major risk of bonds  Related to yield curve risks !  Parallel shift in yield curve,
twist of yield curve (movement in contrary directions of rates at two maturities) and other curvature
changes of the yield curve. Yield curve shifts are expected to account for 90% of the noise. The
manager must also examine each index’s risk profile separately and compare it to the overall risk of
his portfolio. He needs to ask himself, how sensitive are my return to changes in rates (interest rates
risks), changes in the shape of yield curve (yield curve risk)? Or changes in the spread between treasury
and non-treasury security (spread risk)
How can the manager use the index risk profile to construct an effecrtive indexed portfolio? 
-
Cell matching technique: (Stratified sampling), divides index into cells
that represent qualities designed to reflect risk factors of the index. If
the A rated bonds make up 4% of the entire index, then A rated bonds
will be sampled and added until it makes up 4% of manager’s portfolio
-
Multifactor model technique – Makes use of a set of factors
accounting for most bond returns. Focus on the most important
primary factors.

Primary risk factors measures are:
1. Duration,  An index’s effective duration measures the sensitivity of the index’s price to a
relatively small parallel shift in interest rates (i.e., interest rate risk). (For large parallel
changes in interest rates, a convexity adjustment is used to improve the accuracy of the
index’s estimated price change. A convexity adjustment is an estimate of the change in price
that is not explained by duration.) The manager’s indexed portfolio will attempt to match
the duration of the index as a way of ensuring that the exposure is the same in both
portfolios.
2. Key rate duration and present value distribution of cash flows. Nonparallel shifts in the yield
curve (i.e., yield curve risk), such as an increase in slope or a twist in the curve, can be captured
by two separate measures. Key rate duration is one established method for measuring the
effect of shifts in key points along the yield curve.
3. Sector and quality percent. To ensure that the bond market index’s yield is replicated by the
portfolio, the manager will match the percentage weight in the various sectors and qualities
ofthe index.
4. Sector duration contribution. A portfolio’s return is obviously affected by the duration of
each sector’s bonds in the portfolio. For an indexed portfolio, the portfolio must achieve the
same duration exposure to each sector as the index.
5. Quality spread duration contribution
6. Sector/coupon/maturity cell weights. Because duration only captures the effect of small interest rate changes on an index’s value, convexity is often used to improve the accuracy of
the estimated price change, particularly where the change in rates is large.
7. Issuer exposure. Event risk for a single issuer is the final risk that needs to be controlled. If a
manager attempts to replicate the index with too few securities, issuer event risk takes on
greater importance.
3.2.4. Tracking Risk
Tracking risk (also known as tracking error) is a measure of the variability with which a portfolio’s return tracks the return of a benchmark index. More specifically, tracking risk is defined as
the standard deviation of the portfolio’s active return, where the active return for each period is
defined as
Therefore,
Active return = Portfolio’s return – Benchmark index’s return
Tracking risk = Standard deviation of the active returns
Tracking risk arises primarily from mismatches between a portfolio’s risk profile and the
benchmark’s risk profile.
Any change to the portfolio that increases a mismatch for any of these seven items will potentially
increase the tracking risk. Examples (using the first five of the seven) would include mismatches in
the following:
Although there are expenses and transaction costs associated with constructing and rebalanc- ing an
indexed portfolio, there are no similar costs for the index itself. Therefore, it is reasonable to expect
that a perfectly indexed portfolio will underperform the index by the amount of these costs. For this
reason, the bond manager may choose to recover these costs by seeking to enhance the portfolio’s
return.
1) Lower cost enhancement: Managers can increase the portfolio’s net return by
simply main- taining tight controls on trading costs and management fees.
2) Issue selection enhancements. The manager may identify and select securities that
are under- valued in the marketplace, relative to a valuation model’s theoretical
value.
3) Yield curve positioning. Some maturities along the yield curve tend to remain
consistently overvalued or undervalued.
4) Sector and quality positioning. This return enhancement technique takes two
forms:
-
Maintaining a yield tilt toward short duration corporates. Experience
has shown that the best yield spread per unit of duration risk is
usually available in corporate securities with less than five years to
maturity (i.e., short corporates). A manager can increase the return
on the portfolio without a commensurate increase in risk by tilting
the portfolio toward these securities.
-
Periodic over- or underweighting of sectors (e.g., Treasuries vs.
corporates) or qualities. Conducted on a small scale, the manager
may overweight Treasuries when spreads are expected to widen
5) Call exposure positioning. A drop in interest rates will inevitably lead to some
callable bonds being retired early. As rates drop, the investor must determine the
probability that the bond will be called.  crossover point at which the average
investor is uncer- tain as to whether the bond is likely to be called. Near this point,
the actual performance of a bond may be significantly different than would be
expected, given the bond’s effec- tive duration20 (duration adjusted to account for
embedded options).
Active management strategies?  This is the view of Petit, I believe we did the same in
earlier chapters or later chapters? Compare both
Second
chapter
of
PETIT
book

5.1. Combination Strategies
1) Active + passive management combination  allocates a core component of the portfolio to a
passive strategy and the balance to an active component. A large pension fund might have a large
allocation to a core strategy, consisting of an indexed portfolio, with additional active strategies
chosen on the margin to enhance overall portfolio returns.
2) Active + immunization management combination  also consists of two component portfolios:
The immunized portfolio provides an assured return over the planning horizon while the second
portfolio uses an active high-return/high-risk strategy. An example of an active immunization strategy
is a surplus protection strategy for a fully funded pension plan in which the liabilities are immu- nized
and the portion of assets equal to the surplus is actively managed.
5.2. Leverage
Leverage = Enhance portfolio rate of return.
For example, if a manager can borrow €100 million at 4 percent (i.e., €4 million interest per year) and
invest the funds to earn 5 percent (i.e., €5 million return per year), the difference of 1 percent (or €1
million) represents a profit that increases the rate of return on the entire portfolio. When a manager
leverages a bond portfolio, however, the interest rate sensitivity of the equity in the portfolio usually
increases
They call that leverage lmao. Long 75x at resistance?
Skip part 5 and 7, hence juming to part 6 of chapter directly
6. INTERNATIONAL BOND INVESTING
International bond investing = foreign investment in bonds
Remember currency risk increases
Motivation?  Risk reduction (Low correlation with domestic bonds) and return enhancement.
Highest correlation  European bond market denominated in euro currency  Increased liquidity
and integrated bond market.
Overall, local currency correlations tend to be higher than their US dollar equivalent correlations.
Such deviations are attributed to currency volatility, which tends to reduce the correlation among
international bond indices when measured in US dollars.
6.1. Active versus Passive Management
The opportunities for active management are created by inef- ficiencies that may be attributed to
differences in tax treatment, local regulations, coverage by fixed-income analysts, and many
arbitrage opportunities
Active management  Add value through the following means:

Bond market selection.

Currency selection

Duration management/yield curve management

Sector selection

Credit analysis of issuer

Investing in markets outside the benchmark
Relative to duration management, the relationship between duration of a foreign bond and the
duration of the investor’s portfolio including domestic and foreign bonds deserves further comment.
Earlier defined duration concept  Percentage change in value of a bond for 100 bps change in
rates (0.1%). It makes sense in the context of domestic bonds investment but not for international
bonds. Why? Interest rates from different countries issuing bonds won’t change by 100 bps at the
same time (mismatch). They are not perfectly correlated. of international bond portfolio duration
would not be meaningful.
 The duration measure of a portfolio that includes domestic and foreign bonds must rec- ognize
the correlation between the movements in interest rates in the home country and each
nondomestic market.
6.2. Currency Risk
Foreign currency depreciates against home currency? Currency loss occurs
Investor protection? Diversification  Invest in many foreign bonds, in the hope that one
depreciation is offset by an appreciation.
 a multi-currency portfolio has less currency risk than a portfolio denominated in a single currency.
Standard measure of currency risk?
-
Forward discount (If negative)/premium (If positive): (F1 – S0)/S0
-
Unexpected effect  Unexpected movement of foreign currency
relative to the forward rate already locked in!
Currency hedge instruments such as call/put options, future, forward etc…
The bond investor should be aware of a basic result in economics concerning the forward
discount/premium called covered interest rate parity as it suggests an approach to comparing (fully)
hedged returns across international bond markets.
6.2.1. Interest Rate Parity
IRP states that forward foreign exchange rate (discount or premium) over a fixed period should be
equal to the risk-free rate difference between two countries to prevent arbitrage opportunity.
The currency quotation convention used—domestic currency/foreign currency—called direct
quotation, means that from the perspective of an investor in a foreign asset an increase in the spot
exchange rate is associated with a currency gain from holding the foreign asset.
According to IRP
For example  Domestic rate = 3%, foreign rate = 4.5%
Difference is -1.5% and suppose the spot rate is 0.800.  Forward rate is (0.7880 – 0.8)/0.8 = -1.5%
indeed
If the Eurozone investor makes a US dollar bank deposit, the higher interest earned is offset by a
currency loss.
6.2.2. Hedging Currency Risk
The three main methods of currency hedging are:
1) Forward hedging  Forward contract between two currencies is used.
Example: Investor holds a position an USD$ 5M maturing in 9 months. He locks it forward at a
rate of 1.2 euro per Canadian dollar. He is locking a payoff of 5M$ X 1.2 €  6 million euro. If the
exchange rate is lower than this, he is happy, if not, he is mad. He would just not exercise the
forward rate and lose the premium.
2) Proxy hedging  Forward contract between home currency and a currency highly correlated
with the bond currency.
3) Cross hedging  refers to hedging using two currencies other than the home currency and is
a technique used to convert the currency risk of the bond into a different exposure that has
less risk for the investor.
Currency exposures associated with investments with variable cash flows, such as variable coupon
bonds or collateralized debt obligations, cannot be hedged completely because forward contracts
only cover the expected cash flows.
A first, basic fact is that a foreign bond return stated in terms of the investor’s home currency, the
unhedged return (R), is approximately equal to the foreign bond return in local currency terms, rl,
plus the currency return, e, which is the percentage change in the spot exchange rate stated in
terms of home currency per unit of foreign currency (direct quotation, as before):
If the investor can hedge fully with forward contracts, what return will the investor earn? The (fully)
hedged return, HR, is equal to the sum of rl plus the forward discount (premium) f, which is the price
the investor pays (receives) to hedge the currency risk of the foreign bond. That is,
6.3. Breakeven Spread Analysis
Breakeven spread analysis can be used to quantify the amount of spread widening required to
diminish a foreign yield advantage. Breakeven spread analysis does not account for exchange rate
risk, but the information it provides can be helpful in assessing the risk in seeking higher yields.
Furthermore, even a constant yield spread across markets may produce different returns. One
reason is that prices of securities that vary in coupon and maturity respond differently to changes in
yield: Duration plays an important role in breakeven spread analysis. Also, the yield advantage of
investing in a foreign country may disappear if domestic yields increase and foreign yields decline.
6.4. Emerging Market Debt
Emerging market debt (EMD) includes sovereign bonds (bonds issued by a national government) as
well as debt securities issued by public and private companies in those countries.
Because of its low correlation with do- mestic debt portfolios, EMD offers favorable diversification
properties to a fixed-income port- folio. EMD has played an important role in core-plus fixed-income
portfolios. Core-plus is a label for fixed-income mandates that permit the portfolio manager to add
instruments with relatively high return potential, such as EMD and high-yield debt, to core holdings
of investment-grade debt.
6.4.2. Risk and Return Characteristics
- May be a source of consistent and above average market returns
- Sovereign emerging market governments possess several advantages over private corporations.
They can react quickly to negative economic events by cutting spending and raising taxes and interest
rates (actions that may make it more difficult for private corpora- tions in these countries to service
their own debt). They also have access to lenders on the world stage, such as the International
Monetary Fund and the World Bank. Many emerging market nations also possess large foreign
currency reserves, providing a shock absorber for bumps in their economic road. Using these
resources, any adverse situation can be rapidly addressed and reversed.
Risks  Volatility in the EMD market is high. EMD returns are also frequently characterized by
significant negative skewness (Unlimited downside). Negative skewness is the po- tential for
occasional very large negative returns without offsetting potential on the upside.
Emerging market countries frequently do not offer the degree of transparency, court-tested laws,
and clear regulations that developed market countries do. The legal system may be less developed
and offer less protection from interference by the executive branch than in developed countries.
Also, developing countries have tended to over borrow, which can damage the position of existing
debt.
6.4.3. Analysis of Emerging Market Debt
Default risk remains a critical consideration in evaluating EMD, and investors should not rely solely on
bond ratings. Political risk and currency risk are significant sources of uncertainty in portfolios with
international exposures, including the risk of war, government collapse, political instability,
expropriation, confiscation, adverse changes in taxation, and limitations on converting foreign
currency holdings. Sovereign EMD bears greater credit risk than developed market sovereign debt due
to developing countries' less-developed banking and financial market infrastructure, lower
transparency, and higher political risk. The article concludes by mentioning the importance of
selecting a fixed-income portfolio manager.
13.1 OVERVIEW OF INFLATION-INDEXED DEBT
Though it is certainly true that nominal securities issued by the U.S. Treasury are liquid, default-free,
and carry certain tax advantages (exemption from local and state taxes), they are still subject to the risk
of inflation.
he United States began to sell indexed Treasury securities in January 1997. These securities have little
inflation risk, and they enjoy the same advantages of the normal treasury bonds. These securities are
known as Treasury inflation-protected securities, or simply TIPS.
Yields are lower for TIPS as they account for inflation. They compensate investors for inflation risk. If
realized inflation is higher than expected inflation  Investors are compensated.
The spread at the five-year benchmark is (3.56% - 1.25%) = 231 basis points. This is referred to as the
break-even inflation (BEI) in the five-year maturity. The reasoning is based on the following economic
intuition: If the actual inflation rate is below the BEI, TIPS will underperform nominal debt, and vice
versa.
The interest in issuing and investing in indexed debt securities is obviously tied to the perception of the
risk of inflation. Note that the underlying indices have not varied much: Most countries have used the
consumer prices as the index, but some have used wholesale prices, gold prices, wage index, and the
like.
Many Latin American countries (and a few others) tended to issue indexed bonds when the rate of
inflation ran very high.
13.2 ROLE OF INDEXED DEBT
- Protect the welfare of investors against downside risk of inflation
- Government issuing those bonds give itself good credit in fighting inflation In the long term  This
incentive can also be encouraged by forcing the issuance of nominal securities in
the shorter end of the yield curve. This obliges the Treasury to refinance every time the short-term debt
matures. Unless the inflation rate is kept low, such refinancing costs can be potentially high.
The basic idea behind these arguments is the follow- ing: Either short-term nominal debt (requiring
frequent refinancing) or longer-term inflation-indexed bonds will reduce the government’s incentive to
inflate.
inflation-indexed bond issuance provides a more cost-effective approach in the sense that we substitute
by the one-time issu- ance cost of indexed bonds the multiple issuances associated with short-term debt
and frequent refinancing. Any inflationary adjustment simply increases the principal of the indexed
security and is thus a forced savings rather than an outright cash payout. Thus, the issuance of inflationindexed bonds simultaneously eliminates the moral hazard problem associated with the issuance of a
long-term nominal debt and reduces the need to roll over and refund short-term nominal debt.
Furthemore, indexed securities might give precious information on the expected future rate of inflation.
It is however difficult in practice because  First, most indexed bonds have lags in indexing. Second,
the tax treatment of TIPS will definitely influence the pricing and yields of TIPS. Third, investors will
typically require a risk premium associated with the inflation risk. This risk premium has to be estimated
and the manner in which it affects the expected inflation rate has to be determined.
The Treasury has also argued that the issuance of indexed bonds might reduce the cost of public debt.
The reasoning is as follows: By offering securities that are indexed to inflation,theTreasury is able to
attract investors who are very averse to inflation risk. Such investors will be willing to pay a higher
price to buy securities that are default- free and that are indexed to inflation.
Likewise, institutions such as pension funds might find TIPS attractive as their liabilities are generally
highly correlated with inflation. It makes sense for them to be willing to invest more in TIPS and other
assets indexed to inflation. With the introduction of TIPS, first-time investors have a reliable financial
security to hedge in the long term against the risks of inflation.
13.3 DESIGN OF TIPS
Roll (1996), in his insightful analysis of TIPS, identifies the following key features in their design: (a)
choice of index, (b) indexation lag, (c) maturity composition, (d) strippability, (e) tax treatment, and (f)
cash-flow structure.
13.3.1 Choice of index
The key question in the design of TIPS is the choice of the index. Several candidates exist in the market:
the Consumer Price Index (CPI), wage indexes, indexes related to the costs of industrial. production, and
indexes that are related to other important items in the household’s expenses. the index should be
maintained and updated in a scrupulous manner so that it reflects the true cost of a representative
consumption basket.
13.3.2 Indexation lag
With the existing technology it is nearly impossible to adjust the cou- pon payment to reflect the inflation
rate up to the last minute. This is due to the fact that the inflation numbers have to be computed by the
Bureau of Labor Statistics, and the process takes time. Thus, there is an unavoidable delay between the
time the infla- tion is measured and the time the cash flows are indexed to the measured inflation rate.
This makes the indexed bond have some residual exposure to the inflation risk.
13.3.3 Maturity composition of TIPS
The deci- sion to issueTIPS in the long-term maturity sectors is a very strong credible signal by the Treasury
that it intends to keep the inflation rate low. In addition to issuing TIPS in long maturity sectors, the
Treasury also has allowed stripping of securities, which implies that long-dated strips that are indexed to
inflation will be available to investors. In the Canadian Treasury bond market, already inflation-indexed
bonds have been stripped. Now indexed strips have maturities ranging from a few months to more than
25 years. Such index-linked zeroes offering “real returns” may be quite valuable to institutions that have
indexed liabilities with long maturities. The decision to offer TIPS in a broad maturity spectrum will clearly
improve the TIPS products that will be offered by the dealers. In TIPS, under normal inflationary
conditions, the nature of inflation indexing “backloads”the cash flows. With longer maturities,this effect
will be even stronger.
13.3.4 Strippability of TIPS
all the interest-only strips from Treasury inflation-indexed securities with the same maturity date to be
interchangeable (i.e., fungible). This is likely to promote liquid markets for stripped interest-only inflationindexed securities and could potentially increase the overall demand for the underlying TIPS. Note that
investors can now buy long-dated real strips to hedge against inflation. This should prove very attrac- tive
to tax-sheltered retirement accounts
Fund retirement benefits indexed to inflation  Jackpot for retiring
13.3.5 Tax treatment
Taxation of inflation-indexed bonds poses a special issue: Should the appreciation in the principal amount
due to inflation and the resulting increase in coupon be taxed? The U.S. Treasury says they must be taxed.
This produces a “phantom income” that is subject to taxation. At high enough inflation rates, tax- able
investors may experience negative cash flows from TIPS
Roll (1996) argues that taxing inflation accruals may, in fact, be necessary to improve the liquidity of the
TIPS market. Absent taxes on inflation accruals,TIPS will trade at very low yields, and tax-exempt
institutions will prefer nominal securities, which are likely to have higher pre-tax yields. Since tax- exempt
institutions represent a significant pool of investment capital, this will lead to an illiquid market for TIPS.
13.4 CASH-FLOW STRUCTURE
Here we briefly outline the various structures and review their relative merits. To simplify presentation, we
will think of CPI as the index
13.4.1 Indexed zero coupon structure
This structure is the simplest and perhaps the most elementary unit of a real bond. As we saw earlier,
stripping produces a security of this type except that the indexation lags will make the strip from TIP
different from the pure zero coupon structure that we described previously. Note that the zero coupon
structure presents no reinvestment risk and presents the best protection from the risk of inflation. Such
countries as Canada, the United States, and Sweden have indexed zeroes, either through stripping or by
outright issuance. From the perspective of forecasting, expected inflation rates as well as the zero coupon
structure are probably the best. Pension funds and insurance companies should find this ideal in putting
together indexed annuities without the risk of reinvestment. Unfortunately, the tax treatment in many
countries would generate negative cash flows to taxable investors who must recognize the accrual of
interest as well as inflation in this structure. This could be one reason that we do not see this structure
widely used in indexed bond markets.
13.4.2 Principal-indexed structure
13.4.3 Interest-indexed structure
13.5 REAL YIELDS, NOMINAL YIELDS, AND BREAK-EVEN INFLATION
As the expected inflation rates change, so does the inflation risk premium. This is readily seen by noting
that the changes in the nominal rates of interest arise from changes in the expected infla- tion rates and
changes in the inflation risk premium.
Not surprisingly, the yields on nominal securities are much more volatile than the yields on TIPS. This
is because much of the inflation risk is already reflected in the prin- cipal value of TIPS, whereas the
nominal securities have a fixed principal and coupon. The yields of TIPS also reflect the poor liquidity
of the market. DuPont and Sack (1999) report that 50% of the largest price changes in TIPS took place
around auctions of TIPS. The liquidity of TIPS is lower than nominal treasury securities. As a
consequence, inves- tors prefer the more liquid securities in periods of financial distress.
13.6 CASH FLOWS, PRICES, YIELDS, AND RISKS OF TIPS
Then the index ratio IRt is defined as follows:
Example 13.1
A TIP was issued on April 15, 1996, with a coupon rate of 3.5%. The first interest payment date
for this TIP was October 15, 1996. The reference CPI number for the issue date of April 15,
1996, was 120.00. The reference CPI number for October 15, 1996, was l35.00. Then, for a par
value of $1 million, what was the coupon income on October 15, 1996?
`
13.7 INVESTOR’S PERSPECTIVE
From the investor’s perspective,TIPS offer protection from unanticipated increases in inflation by the
process of indexation.
The price sensitivity of TIPS to inflation rate is low due to the fact that coupons and prin- cipal
automatically change to reflect market inflation. In addition,TIPS have a fairly low to negative correlation
with stock market indexes  Good to include in a broad diversified portfolio
If the TIPS are perfectly indexed, they will carry no inflation risk and, therefore, will have zero duration
with respect to the inflation rate
When real rates go up, both prices will fall, and vice versa.
When inflation goes up, TIPS appreciate
This implies that surprises in expected inflation, which result in higher than anticipated inflation, will
cause the yields on TIPS to fall and the yields on nominals to increase, result- ing in widening spreads.
Chapter 11: Fixed income management
11.1 OVERVIEW OF MORTGAGE CONTRACTS
Mortgage  Home ownership  Secured loan (Collateral)
Origination fee  Fee obtained by the bank to give you the loan
11.1.1 Lenders’ risks
5) Default risk (Banker will assess the appropriate LTV ratio)
6) Prepayments: Lenders also face the prospect that borrowers may choose to refinance their
previ- ously taken loans. They will take a mortgage loan at a lower interest rate to repay a
loan they took at a higher interest rate. Lenders can protect from this to some extent by
charging a higher interest rate on the mortgage loan or by hedging.
7) Interest rate risk: When the interest rates go up, the key risk is not the risk of prepayments
but the fact that the loan portfolio is sensitive to changes in interest rates. For example, a
fixed-rate loan portfolio will lose value when interest rates go up.
11.2 TYPES OF MORTGAGES
11.2.1 Fixed-rate mortgages (FRMs)
FRMs are offered in two different maturities: 15 years and 30 years. FRMs result in constant monthly payments for the borrower. The borrower has no uncertainty about his or her mortgage obligations, the interest
rate specified in an FRM does not change during the life of the contract.
11.2.2 Adjustable-rate mortgages (ARMs)
The interest rate on an ARM changes over the life of the contract. The rates are linked to certain indexes
of borrowing rates. Two indexes are widely used for setting interest rates on ARMs. They are (1) the
Eleventh Federal Home Loan Bank Board District Cost of Funds Index (COFI) and (2) the National
Cost of Funds Index. Indices such as LIBOR are also used.
ARMs are frequently designed by lenders to have the following features: First, during the first few years
of the mortgage (ranging from one to five years), the interest rate is kept at a fixed level. Second, ARMs
carry a lifetime cap on interest rates, above which the borrower will never be charged. In addition, ARMs
also carry a year-to- year cap, which ensures that the borrower’s interest cannot exceed the previous year’s
interest by more than a certain percentage point.
It should be noted that ARMs are typically indexed to short-term interest rates. This would therefore imply
that the future monthly mortgage obligations may be much more volatile to a home- owner with an ARM.
ARMs have a lower prepayment risk, since the rates are indexed to market conditions.This allows the
lenders to worry less about prepayment risk and focus on managing credit risk.
Mortgage loans are also classified along other dimensions, such as (a) the size of the initial loan taken, (b)
LTV ratios, (c) credit scores, (d) the level of documenta- tion, (e) the ability of lenders to sell their
mortgage loans to federal agencies, and (f) underwriting standards employed at the time the loan was
extended.
11.2.3 Agency mortgages
Agency mortgages are mortgage loans that must conform to the standards set forth by federal agencies.
Standards depending on point a to f in the upper paragraph in bold.
A mortgage is typically considered “conventional” or “conforming” if the LTV ratio is small (80% or
lower).
11.2.4 Jumbo mortgages
Jumbo mortgages cannot be sold by lenders to federal agencies. These are relatively
large loans, and the average credit quality of the borrowers tends to be high.
11.2.5 Alt-A mortgages
Alt-A mortgages are mortgages that generally conform with agency standards in terms of loan size and
borrower credit score. On the other hand, these mortgages can have other unattractive features, such as
low documentation.
11.2.6 Subprime mortgages
Subprime mortgages tend to have much lower FICO scores relative to agency stan- dards.They also attract
borrowers who are relatively more heavily levered. Also + low documentation as well.
11.3 MORTGAGE CASH FLOWS AND YIELDS
Bonds pay interest semiannually. Bond yields are there- fore quoted in nominal annualized terms,
assuming semiannual compounding. This is known as bond-equivalent yield (BEY). On an annualized basis,
semiannual BEY can be reported as follows:
Mortgages differ from bonds in several respects. First, they pay monthly cash flows. Second, these
monthly cash flows include both interest payments and (amor- tizing) principal payments. Hence
mortgage yields are quoted in annualized terms, assuming monthly compounding. This is called mortgageequivalent yield, or MEY.
We can compute the monthly cash flows of an FRM :
Excel example for case
From the borrower’s perspective, the stated rate in the mortgage contract is the most important cost, but
it’s not the only one. The borrower might have to perform an appraisal of the property, which will result
in an appraisal fee. There may be costs associated with insuring the property. Other costs can include
broker fees, title fees, and taxes. Once all the relevant costs are included, the effective interest rate for the
borrower will exceed the stated interest rate on the mortgage. Federal regulations require that lenders
disclose to borrowers the effective rate charged rather than sim- ply the stated interest on the mortgage.
The effective interest rate is referred to as the annual percentage rate, or APR.
11.4 FEDERAL AGENCIES
The mortgage market has two segments. One is the primary mortgage market, where borrowers get their
loans from lenders.
Mortgages that were previously originated are bought and sold in the secondary mortgage markets.
Government-sponsored enterprises (GSEs) were created to promote the availabil- ity of credit to housing.
1. Credit guarantees. Fannie Mae and Freddie Mac purchase mortgages and issue mortgage-backed securities
on which they guarantee the timely payment of principal and interest.
2. Mortgage investments. GSEs purchase whole mortgages, mortgage-backed securities, and other mortgagerelated securities in the capital market. The GSEs take on three forms of risk with these investments: credit
risk, interest rate risk, and liquidity risk. Interest rate risk arises from prepayments, as noted ear- lier.
Liquidity risk tends to be more significant for GSEs because they hold a very significant fraction of the
secondary mortgage market; it would be very hard for them to sell parts of their portfolio without an adverse
price reac- tion. If GSEs package all the mortgages that they bought and immediately sell them as mortgagebacked securities, they do not face prepayment risk. But as it stands, GSEs do own a very significant amount
of mortgages. One motiva- tion for owning mortgages is the spread between the income from mortgages
and the costs of issuing debt to finance the purchase of mortgages.
3. Advances. The FHLB makes secured loans, called advances, to the approxi- mately 8000 banks and thrifts
that are system members. These subsidized funds are frequently used by members to make further
mortgage loans, but they are also used for nonhousing purposes.
12.1 OVERVIEW OF MORTGAGE-BACKED SECURITIES
Lenders sell their originated mortgage loan port- folios to institutional buyers.This occurs through a process
known as securitization, which leads to the creation of mortgage-backed securities (MBSs). MBSs are bonds
that are secured or backed by a portfolio of underlying mortgage loans. Mortgage owners pay monthly
payment. Such payments of individual borrowers are aggregated and are used to make interest and principal
payments on MBSs.This process of creating an MBS from individual mortgage loans is called securitization.
12.1.1 Securitization
The process of securitization, which transforms illiquid, individual mortgages into liquid mortgage-backed
securities, involves several players and steps. In essence, securitization involves three important steps:
1. Pooling of individual residential mortgage loans. Individual loans originated by lending institutions such as
banks and thrifts are pooled into a portfolio of sufficient size.This pooling is necessary to create a
sufficiently large size that would be of interest to institutional investors such as asset management firms,
pension funds, insurance companies, and the like.
2. Provision of credible guarantees. This is to ensure that payments promised by the MBS will materialize in a
timely fashion.
3. Issuance of the MBS. Relying on the strength of the underlying mortgage loans and guarantees, the MBS
is issued with the help of financial intermediar- ies such as dealers and investment banks.These securities
are then purchased by institutional investors.
Figure 12.1 explains the process of securitization in greater detail. Lenders (originators) assemble a portfolio
of loans and create a pool of mortgages with some homogeneity (Typically only FRMs or ARMs). The rate
prevailing on the FRMs will typically not differ by more than 1% to 2% between securities so that the
weighted average coupon of the pool is a reasonable measure of the coupons received by each loans. They
tend to group loans from the same geographical area together. All these considerations allow investors to
forecast prepayments with greater accuracy.The goal of originators is to move the loan portfolio out of
their balance sheet and into a special-purpose vehicle (SPV), which will issue the MBS. Once the process
of securitization is completed, the originators simply become servicers of the loans.
As servicers, financial institutions provide a number of functions, including:
1. Maintaining the status of individual loans (principal outstanding,
prepayments…)
2. Collecting interests, principal payments and prepayments
3. Handling defaults and foreclosures…
This servicing fee is subtracted from the WAC of the underlying mortgage loans, and the remaining
amount is “passed through” to the purchasers of these mortgage loans.As a consequence, these securities
are referred to as mortgage pass-through securities.
12.1.2 Guarantees and credit enhancement
For a fee, such a standardized portfolio of pooled mortgages is then guaranteed by a federal agency (or a
private entity of sufficiently high credit reputation) against default. Such defaults may occur at the level of
individual units in the pool (such as a given homeowner in a pool of mortgages) or at the level of issuers
Since September 2008, Fannie Mae and Freddie Mac have been brought under “conservatorship” of the
federal government and their debt obligations may now be regarded as having the direct backing of the
federal government.
The key idea here is to put some distance between the originators and the pool of assets.“Bankruptcy
remoteness” is the goal. In other words, the SPV is structured such that the bankruptcy of the originator(s)
does not affect the pool of financial assets held by the SPV.This separation is criti- cal to obtaining the
necessary credit enhancements, which usually lead to a high credit rating.The structure of the collateral may
itself provide some credit enhance- ment; for example, there may be “overcollateralization” of mortgage
loans that back the pass-through securities.
12.1.4 Cash flows and market conventions
Cash flows are passed back to the owner of the MBS for the exception of the fees. Servicing fees and
guarantee fees will be subtracted from the cash flows generated by the loan portfolio that is backing the
mortgage-backed security.
The prices quoted refer to percentages of the outstanding principal balance in the underlying pool.This
requires the calculation of the outstanding balance, which in turn requires compilation of the scheduled
interest and principal payments as well as any prepay- ments. For these computations, the servicing
institutions calculate a pool factor. The term pool factor pf (t ) is defined as follows:
Note that the accrued interest calculations differ from Treasuries in important ways. First, interest accrues
from the first day of the month; in Treasury markets, the last coupon date is the relevant date from which
interest accrues. In the case of GNMA, the convention is actually over 360, as the example illustrates.
Note that this means that an investor buying a GNMA in April for settlement in the middle of April (say,
April 15) is buying a pro rata share in the outstanding principal balance of a mort- gage pool as of the end
of March.This investor will expect to receive on May 15 the interest on the balance, computed as of the
end of March, plus any prepayments during the month of April. When agency pass-through securities are
traded, they are identified with some key characteristics of the underlying pool. A pool number is assigned
that enables investors to learn about the features of the underlying pool, such as whether the pool is fixed
or adjustable, the issuer, and the weighted-average coupon.
12.2 RISKS: PREPAYMENTS
Mortgages in the United States permit the homeowners to prepay their loans.This prepayment provision
introduces timing uncertainty into the originating bank’s cash flows from its loan portfolio. For example,
if the bank originates a pool of mortgages with a weighted-average rate of 8% and six months later the
mortgage rates drop significantly below 8%, say to 7%, then the loan portfolio is certain to experience
significant prepayments as borrowers rush to refinance their mortgages with less costly loans.The lender
has a long position in the mortgage loan that entitles him or her to monthly scheduled payments, but the
lender has also sold an option to the homeowners that gives them the right to prepay the loan when the
circumstances demand it.This means that the bank cannot predict with certainty the future cash flows
from its loan portfolio. Clearly, the option to prepay will be priced into the loan by the bank, and the
borrower will pay a higher interest rate on the loan as a consequence.
12.2.1 Measuring prepayments
12.2.1.1 Twelve-year retirement
This is perhaps the simplest and the least important measure of prepayment. It assumes that the mortgage
is prepaid exactly after 12 years. If this assumption is made, at the end of 12 years we can add the
prepayments to the scheduled pay- ments.The cash flows of the mortgage loan in the absence of default
can then be determined for all future months.This measure is clearly inconsistent with what we know
about the factors that determine prepayments.
12.2.1.2 Constant monthly mortality
This measure assumes that there is a constant probability that the mortgage will be prepaid following the
next month’s scheduled payments. For instance, consider the assumption that there is a 0.50% probability
that the mortgage will be prepaid fol- lowing the first month.This 0.50% probability is referred to as the
single monthly mortality, or SMM, rate. Using the SMM, we can compute the probability that the
12.3 FACTORS AFFECTING PREPAYMENTS
12.3.1 Refinancing incentive
Perhaps the most important reason for prepayments is the refinancing incentive. If the market rates for
mortgage loans drop significantly below the rate that a borrower is paying, the borrower has a very strong
reason to prepay as long as the borrower is able to qualify for a new loan.This incentive means that the
prepayments accelerate in periods of falling interest rates
12.3.2 Seasonality factor
Families typically do not move during the school year. Things remaining equal, families typically move
during the period from the middle of June through the first week of September; this results in increased
prepayments during this part of the year. This can be thought of as the seasonality factor
12.3.3 Age of the mortgage
During the early part of the mortgage loan, interest payments far exceed the prin- cipal component. This,
in part, means that the interest savings associated with refi- nancing are greater during the earlier part of
the mortgage loan.
12.3.4 Family circumstances
A number of factors pertaining to mortgage holders’ family circumstances lead to prepayments. These
factors include marital status (divorce decisions often lead to prepayment) and job switching. Sometimes a
household’s inability to make the monthly payment (due to job loss or disability) leads to default; under
some circum- stances, this can precipitate a prepayment.
12.3.5 Housing prices
The price of a home is yet another factor in prepayments.The housing price affects the LTV ratio, which
in turn affects a household’s ability to qualify for refinancing. When housing prices increase, the LTV
decreases. This enhances the homeowner’s ability to refinance if the going interest rates and family
circumstances warrant refinancing. On the other hand, when the housing prices drop, the LTV ratio
increases; this diminishes the homeowner’s ability to qualify for refinancing, even if other factors favor
refinancing.
12.3.6 Mortgage status (premium burnout)
The relationship between the contractual interest rate in a mortgage loan and the going mortgage interest
rates is a major determinant of the value of the loan. If the contractual interest rate r is greater than the
going interest rate R, the loan is a prime
candidate for prepayment. Such mortgages are referred to as premium mortgages. If r R, the mortgage is said
to be a discount mortgage.
12.3.7 Mortgage term
12.4 VALUATION FRAMEWORK
The basic insight into the valuation of mortgage-backed securities is to recognize that default-free
assumable mortgage-backed securities consist of an annuity and a call option that gives the homeowners
the right to buy the annuity at a strike price equal to the remaining par amount at any time prior to
maturity (from 15 to 30 years).Thus, the factors that determine the value of a fixed-rate mortgage are the
fol- lowing: (a) coupon, (b) time to maturity, (c) amortization schedule, and (d) interest rates on
comparable mortgages at the time of valuation.
In the valuation of mortgage-backed securities, we have thus far treated interest rates as the only variable
affecting the value of the security and have assumed that the mortgage-backed security is default-free. In
reality, the fact that some homeowners might default affects the pricing of mortgage-backed securities. If
the mortgage- backed security is fully insured and assumable (such as GNMAs), then upon default the
guarantor will pay off the mortgage.
When rates are high, the mortgage-backed securities sell below par, but default produces a cash flow equal
to par, leading to a windfall gain. It is also useful to recognize the incen- tives to voluntary default that the
homeowner might have. If the value of the house is relatively high compared to the value of the
mortgage, the homeowner might not want to default. If the value of the house is well below the value of
the mortgage, the incentive to default is high.This may be thought of as a put option or a walk-away option.
Our arguments suggest that housing prices affect the valuation in two dis- tinct ways:
12.5.1 Empirical behavior of an OAS
The OAS is a guide to determine which securities are rich and which are cheap. If the market price of an
MBS is 97 and the model price is 98, the OAS is positive. We say that the security is “cheap.” This does
not necessarily imply that we should buy it. It will be useful to compare the OAS with the history of OAS
and then interpret the current OAS in light of current economic circumstances. In a similar manner, if the
market price of an MBS is 97 and the model price is 95,the OAS is negative.We say that the security is
“rich.”
Note that the OAS of the GNMA is smaller than that of a Fannie Mae 15-year pass-through, which is in
turn smaller than a Fannie Mae 30-year pass-through as of December 2005. By the criteria developed
earlier, we would regard GNMA to be expensive (as its OAS 0) as of that date, but over time, GNMA
became cheaper relative to the model used in producing the OAS.Then investors would look at the
history of OAS to determine whether a particular pass-through is attractive as invest- ment or not.
12.6 REMICS
REMICs are real estate mortgage investment conduits
CMOs were issued as debt obligations of the issuer; thus, such issues appeared in a balance sheet as a
liability. REMICs, on the other hand, are a legal framework within which mortgage-backed securities are
treated as asset sales for tax purposes.
REMICs can be structured in a senior-subordinated format.This allows for credit enhancements to
mortgage-backed securities with multiple tranches. REMICs rep- resent an innovative way to redistribute
the cash flows from a pool of mortgages or mortgage-backed securities to various investor classes. Recall
that investors in mortgage-backed pass-through securities get a pro rata share of the cash flows of the
security, including prepayments. REMICs, through careful structuring, can offer varying levels of
protection against prepayment risk.The REMIC issuance is backed by pools of residential mortgages or
mortgage-backed securities, such as GNMAs, which serve as the collateral.
12.6.1 REMIC structure
REMIC securities tend to be rated AAA or Aaa by the rating agencies.The key to this high credit
reputation is the basic requirement that the cash flows generated by the underlying mortgages or the
agency securities are more than sufficient to meet the obligations of all tranches, even under the most
extreme prepayment assumptions.
The REMIC securities are typically overcollateralized.The purpose of this overcollateralization is to create
an insurance cushion that helps offset any cash-flow shortages that may result due to a fall in
reinvestment income from the underlying monthly cash flows.The cash flows from the collateral are
divided and allocated to several classes or tranches of bonds. There are usually two basic REMIC structures:
(1) a sequential structure and (2) a planned amortization class (PAC) structure.
12.6.2 Sequential structure
A typical generic REMIC sequential structure has four tranches. Specific rules dic- tate how the cash
flows (including prepayments) from the collateral are allocated to each tranche.
The last (here the fourth) tranche is called the Z bond and receives no cash flows until all earlier tranches
are fully retired.The face amount, however, accrues at the stated coupon. After all tranches have been
retired, the Z bond receives the coupon on its current face amount plus all the prepayments. Trustees
ensure that the remaining collateral is large enough at all times so that all tranches get their promised cash
flows.
12.6.3 Planned amortization class structure
In a PAC structure, the tranches are created to provide varying levels of protection from prepayment. In
this structure, the collateral’s principal is divided into two cat- egories.The first category is designated
PAC bonds,and the second category is the companion group.The amortization schedule for the PAC
bonds remains fixed over a range of prepayment rates measured by a range of PSAs.The more stable
amortiza- tion schedule of the PAC group is at the expense of the companion group.The struc- ture,
therefore, allows for many PAC bonds with stable average lives.The companion bonds, on the other
hand, have much less stable lives than otherwise similar sequen- tial bonds. PAC bonds, because of their
more stable amortization schedules, tend to be priced tightly to respective Treasuries. By the same token,
bonds in the compan- ion group are priced at much wider spreads relative to Treasuries.
Another type of REMIC, known as the targeted amortization class (TAC) CMO, are very similar to PAC
CMOs; they also enjoy a specified redemption schedule backed by support tranches in the CMO
structure. Unlike PACs,TACs have a longer average maturity when interest rates fall and the prepayments
are slower than expected.