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Note: Read theappendage to this
document about thedifference between zero
rates and forward rates
e ore rev ew ng t e
lecture material about
forward rates This is
difficult and conceptually
complex material.
Chapter 4 from Hull
(c)2006-2013, Gary R. Evans. This material may not be used by others without permission of the author.
Exam resultsExam results
13
12
14
185 A 13
175 A 7
170 B+ 4
7
6
8
10A
B+
B
B
C+
C
165 B 4
160 B 4
150 C+ 4
140 C 2
120 C 2
Below ? 0
2 2
00
2
4
1
C
?
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Where we are going in this lecture Review of interest rate formulas
Calculating "zero rates"
Calculating "forward rates"
Calculating "duration"
Justif in ield s reads and ield curves
Discussing yield arbitrage (intro)
What we are skipping from Hull chapter 4What we are skipping from Hull chapter 4
Section 4.7 Forward Rate Agreements
requires knowledge of the LIBOR/swap zero curve, which isn't
covered until chapter 7.
Section 4.9 Convexity
not very important and I don't want to throw too much at you all at
once.
On duration, I am using discrete examples compared to
' out there in the real world you are more likely to see discrete andthe two approaches are similar anyway.
also in chapter 5 I am using discrete rather than continuous upper
and lower limit calculations.
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Types of Risk Embodied in the Yields of YBFAsTypes of Risk Embodied in the Yields of YBFAs
Default risk reflected in corporate and municipal YBFAs
Market (maturity) risk
due to capital gains and losses when interest rates fluctuate
the longer the maturity, the higher the risk
common to all YBFAs
Economic risk
ue to n at on or s m ar econom c or po t ca s oc s
the longer the maturity, the higher the risk(the more likely you
will be holding this asset during the "event"). Treasuries may experience "flight to safety" during political
shocks.
Yield SpreadsYield Spreads
YBFAs with long maturities have a higher probability of
gain or loss when interest rates fall or rise. They are also
subject to a higher level of economic risk.
Therefore, according to our notions of risk, longer term
YBFA maturities are seen as having higher risk, therefore
their yields have arisk premium.
Therefore, typically when we map the yields of a full
maturity range for a class of YBFA securities, like
Treasuries, from short-term to long-term, that mapping will
r se see next s e). These are calledyield spreads oryield curves and the
mapping is called theterm structure of interest rates.
Sometimes we see an atypicalflat or even invertedterm
structure of interest rates.
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The yield curveThe yield curve today: March 2013today: March 2013
The configuration is normal.
but the s read between lon
and short is atypically narrow
because of QE2/QE3.
Note this configuration
compared to 2012 a carbon
copy, which shows the
stability of QE3.
Flat and Inverted YieldFlat and Inverted Yield Spreads: 1990Spreads: 199010.00
Jan 1990 yield curve flat
4.00
6.00
.
3 mo 6 mo 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr
longposition
shortposition
This tends to happen during periods of high inflation or when
the FRS is aggressively raising short term rates to combat
inflation. Inverted (1980) at end of hyper-inflation period.
Presents an ideal spread arb situation.
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Treasury YieldTreasury Yield Spreads: 2004Spreads: 2004--0505
6.00
This slide was from the 2005 lecture.
2.00
3.00
4.00
5.00
Mar-05
Jan-04
FRS raises
FFR targets
As can be seen, this Fed action
is influencing only short-term
rates.
0.00
1.00
1 mo 3 mo 6 mo 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr
"Short on short term" means that you are betting that 3 mo prices will fall and yields will rise.
TheThe 20062006 yield spreadyield spread
6.00
March 28, 2006
4.774.754.744.724.724.734.50 4.59
2.00
3.00
4.00
5.00
A flat yield curve after the 2-year ... largely due to
overseas purchases of Treasury Securities (especially
China and OPEC favorin the much thinner lon end
loaded
to riseFRS raising
0.00
1.00
3 Mon 6 Mon 2 Yr 3 Yr 5 Yr 10 Yr 20 Yr 30 Yr
of the spectrum. This is tied very strongly to our
Merchandise Trade Deficit.
rates this end
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Yield curve in March 2007Yield curve in March 2007
March 6, 2007: Yield curve is
inverted from 2 Year Note out.
Rates were slowly rising at this
time.
Yield Curve in March 2008Yield Curve in March 2008
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The simple bond valuation formulaThe simple bond valuation formula
C Par n
r r
n
i 1 11where
MV = market valuepresently of the bond
n = number of years to maturity
C = cou on a ment ar times the cou on interest rater =present yield of this bond (market determined)
This formula assumes that there is only one interest paymentper year and that this bond was priced on the day after the
most recent interest payment was made.
The elementary bond formula (reduced form)The elementary bond formula (reduced form)
r n
11
1 1
r r n
1
whereMV: the present market value of the bondCR: cou on rate ori inal ield of the bond
Note: This formula cannot be used
to value an actual bond because it
assumes only one interest payment
r: current market rate on equivalent bondsn: number of remaining years to maturity
per year and that the bond is beingbought on the day of the coupon
payment. For actual bond pricing a
more complicated version of this is
shown at the end of this lecture.The original formula is a geometric series and
this is a reduced-form equation. To see its
justification and derivation, read the appendix in
the reading assigned for this lecture.
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The formula when interest is paidThe formula when interest is paid
semisemi--annuallyannuallym
MV
r ri m
i
1 2 1 21
where
C = coupon interest payment (par X r/2)
r = present market yieldm = number of coupon payments remaining (years X 2)
TheThe final formula ask yield (final formula ask yield (ytmytm))
aCParC
MVapn
n
ai 1
1
This term represents an adjustment for accrued interest.
MV= market value, the quoted askprice of the bond,
C= the annual coupon payment, equal to the coupon rate times par,
r
m
r m1
ar=r = the prevailing annual market yield, expressed asask yieldoryield to maturity,
m = the number of coupon payments per year,
n = the number of remaining coupon payments,
p = the number of days in this coupon period (between 181-184, use 182 if unknown),
a = the number of days between the last coupon payment and the settlement day.
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an examplean example
Purchase date: February 8, 2009 (146 days since last coupon payment, 36 to next),Next coupon date: March 15, 2009 (the first of 42 coupons remaining),
Redemption date: September 15, 2029 (for par and last coupon),
You are buying a 30-yr bond that was issued on September 15, 1999:
Coupon rate/amount: 8% yielding $4per coupon payment,Present market rate (askyield): 10%. 8022.0
182
146
p
a
8022.02
8
10.1
100
2
10.1
1
2
8099.05.20
42
18022.0
ii
MV
31.8321.3
10.1
100
05.1
1460.20
42
18022.0
i
iMV
ReducedReduced--form Version of the Complex Formulaform Version of the Complex Formula
aCr
Cn
11
11
Solving the same problem from the last slide (rounding error explains the difference):
pmr
ar
mr
mrm
apm
n
pa
3651
/
1
1
31.838022.0410.1
1100
05.1
05.0
05.111
4099.05.20
8022.0
42
MV
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The continuous TThe continuous T--bond yield formulabond yield formula
P Ce eby i
i
n y n
21
2100
This formula above is the generalized continuous bond yield
formula shown in Hull (he only shows an example, below). This
formula is solved fory using an iterative technique. P is the price
of the bond, C is the semi-annual coupon payment (equal to the
coupon rate times 100 divided by 2, and n is the number of
3 3 3 103 98 390 5 1 0 1 5 2e e e ey y y y . . . .
coupon payments to e ma e ou e t e num er o years o
maturity). Compare this to the discrete bond formula.
The Market Value of a CMO or CDOThe Market Value of a CMO or CDO
MV PI ri
, 1 12360
A Collateralized Mortgage Obligation (called a CDO if made up of other
classes of debt) is a pass-through debt obligation where the principle and
interest payments (PI) made by mortgage holders is passed through to the
owner of the CMO.
A CMO differs from bonds in three respects: (1) The numerator is not a
known value and must be treated as a random variable, 2 the a ment is
monthly, and (3) there is no principal value at the end.
Because mortgages get paid off or refinanced, the nominal front-end value
of PIis considerably higher than at the tail.
Pieces of this can be sold off as Cash Pass-Thru Certificates and therefore
must be priced.
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A very important question ...A very important question ...
Regarding the conventional bond formula or any similar formula ...
nnn rPar
r
C
r
C
r
C
r
C
r
CMV
111111
132
... is this piece, the third payment, considered by itself, being priced
properly, given that you are discounting it at a 20 year rate when the
yield on a 3-year asset is not the same as a 20-year asset.
Why is this relevant? Suppose you want to sell off this piece (atranche).
This issue is at the root of the importance of the difficult material
ahead, the derivation of thezero rate andforward rates.
Hull's "zero rate"Hull's "zero rate"Hull's "zero rate," more typically called "zero coupon rate," is the continuous
version of the traditional formula used to value a zero-coupon bond (like a US
Treasur STRIP . This is a lon -term bond that makes no interest a ment a s
no "coupon") but simply returns the par value at maturity.
This is the formula, where r is the current market
rate, and t is the time in years to maturity.
P
r t
100
1For example, a 30-year zero-coupon bond with22 years and three months remaining to
maturity with current yields of equivalents at 100
5.25% is worth: 1 0 0525 22 25. . .
Hull's continuous
rate equivalent is: P e ert 100 100 31090 0525 22 25. . .
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Determining "Treasury Zero Rates"Determining "Treasury Zero Rates"
"Treasury Zero Rates," which are difficult to calculate (conceptually ...
canned software exists to do it for you), are used to give a correct value to
each individual payment of the cashflow stream. It consists of continuous
discounting to the present each payment from the known cashflow of
Treasury securities. Remember that a Treasury security is almost never
purchased at par and that yields on these securities are expressed as simple
interest compounded quarterly or semi-annually. Therefore to make the
conversion to continuous, the we go back to a conversion formula that we
saw in an earlier lecture:
r m r
mcd
ln 1
Note: This zero-rate conversion can be used for other types of securities as well.
Where we are going in the next fewWhere we are going in the next fewslidesslides
We are trying to define and calculate the rates for this zero rate
table. Compare this to Tables 4.3 and 4.4 in Hull. This step isnecessary for the calculation of forward rates, which are shown in
Table 4.5 in Hull.
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First row in the table: 3-month TBill
The 3-mo TBill pays no coupon and is always sold at a discount
(97.50 in this example). Therefore the interest earned in one
quarter is 100-price (2.50 in this example). First we calculate the
simple quarterly interest:
r
P
Pq
4 100 4 2 50
97 5 10 256%
.
. .
e t en convert t s rate to cont nuous us ng t e convers on
formula:
r mr
c
q
ln ln
..1
4 4 1
010256
4 10127%
2nd row in the table: 62nd row in the table: 6--month TBillmonth TBill
This TBill also pays no coupon. Here we assume you paid
94.90 and because it has a maturity of 6 months it is paying
semi-annual interest:
r
P
Psa
2 100 2 510
94 90 10 748%
.
. .
which is converted to the continuous annual rate:
r mr
c
sa
ln ln
..1
2 2 1
010748
2 10 469%
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3rd row in table: 13rd row in table: 1--year TBillyear TBill
This is easy. Here the conversion is direct from annual to
continuous. Assuming the TBill is worth for 90.00:
ra
100 90
90 1111%.
Once we get to TBonds, however, we have to take into
10536.1111.1ln1ln ac rr
-
payment equal to Par (100) times the initial coupon rate,
then pay a final payment equal to the Par value of thebond. For each six months we have to use an iterative
process where we take the semi-annual and annual rates
that we have already calculated and ...
4th row in the table: TBond with 1.5 years4th row in the table: TBond with 1.5 years
... discount the first coupon payment at the 6-month continuous rate, discount
the second coupon payment at the 1-year continuous rate, then solve for the
. .
In this example, we assume a coupon payment of $4 (8% annually) and the
current secondary market value of a 1.5 year bondof 96 (which implies that
the current yield is higher than 8% because this bond is at discount):
This is the value of the
remaining piece.
4 4 104 960 10469 0 5 0 10536 1 5e e e R
. . . .
0 949 0 90 26 241 5. . . e R
R1 5085196
15 10 681%.
ln( .
. .
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5th row in the table:5th row in the table: TBondTBond with 2.0 yearswith 2.0 years
... discount the first coupon payment at the 6-month continuous rate, discountthe second coupon payment at the 1-year continuous rate, discount the third
coupon payment at the 1.5-year continuous rate, then solve for the 2 year
- - .
In this example, we assume a coupon payment of $6 (12% annually) and the
current secondary market value of a 2- year bondof 101.60 (which implies
that the current yield is lower than 12% because this bond is at premium):
60.101106666 25.110681.010536.05.010469.0 Reeee
9333.16667.178520.0900.0949.02
R
e
10808.0
2
80561.0ln2
R
... and later maturities... and later maturities
This same iterative process, which is called the "bootstrap
method," continues out (typically quarterly or semi-annually)
un e en o e ma ur y spec rum say years .
Clearly this would be rather easy to program in C++ or
equivalent.
For data, you simply use the coupon rates and current prices
for Treasuries of these various maturities quoted in The Wall
.
Note how this is different from the traditional bond formula ...
here each payment is discounted at the rate that is relevant
for its maturity, rather than at a single "market" discount rate.
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... giving us the zero... giving us the zero--rate tablerate table
-... .
The implicit formula that we haveThe implicit formula that we havecreated ...created ...
i n
P Ce eb ii n
2
1
2100
Now each cashflow component of this bond is correctly valued, which is essential
if you plan to sell off pieces of this bond as cashflow certificates or even to
calculate the true effective ield that ou will be earnin if ou sell this bond
before maturity.
Note: You should realize that the Pb this bond should not and will not be different
than the equivalent in the original discrete formula ... it can't be, given the
technique we used. Each piece is priced differently than the original formula.
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Review: What did we just do?Review: What did we just do?
In effect, we pretended that a ten-year bond is not a ten-year
on . nstea t s a co ect on o zero-coupon notes one or
each coupon payment and one much larger one for par) that are
each continuously discounted back to the present at the
appropriate interest rate for that maturity.
For example, the coupon payment that will be paid in two years is
discounted back to the present at the rate that is appropriate for 2-
year notes.
We determine that rate by taking the implied continuous marketrate for a 2-year note discounting the coupon payment back to the
present
A simple application of the zeroA simple application of the zero--couponcouponconcept (to motivate its acceptance)concept (to motivate its acceptance)
Many exotic but commonly used financial instruments, especially debt
obligations, like Collateralized Debt Obligations (CDOs ... often
mortgage pools) or Carry Trade paper (borrowing from one large lender
at a low rate to re-loan to many different smaller borrowers at a higher
rate), are often chopped up and sold off in pieces. These pieces are
commonly called "tranches."
Suppose you have a debt contract promising to pay you $100 per year
or t e next t ree years. uppose you ec e to se t e t r payment asa tranche to a third party? How would you price it?
If the 3-year zero rate is 5%, the tranche would be sold at $86.07:
P e 100 86 070 05 3. .
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Step 2: Determining Forward RatesStep 2: Determining Forward Rates(warning: conceptually very difficult to grasp)(warning: conceptually very difficult to grasp)
As Hull points out in section 4.6 and 4.7, forward rates are used to price forward
rate agreements. To calculate forward rates, we have to have access to (or to
- - .
Here is an example: Suppose you have a debt instrument that pays you interest
for five years according to a zero coupon schedule (with zero rates) that goes out
for five years. [Note important to understand: This may be a traditional debt
instrument for which you have made a zero rate schedule as we did in the
previous section]. If you ask the question, "What is the interest yield that I will
earn in the 5th year alone?" you are asking "What is the 5-year forward rate?"
How does this differ from the five-year zero rate? There we are asking the
question, "What is my present interest yield on the coupon payment that will
made to me in the 5th year."
Stop and read the document entitled Justification of the technique used to
calculate forward rates from zero rates by example.
Calculating the Forward Rate fromCalculating the Forward Rate fromZero RatesZero Rates
Time Zero Rate Forward Rate
1 2.000
2 2.500 3.000
3 3.000 4.000
4 3.500 5.000
5 4.000 6.000
Forward Rates for 5 Year Investment
.
to calculate forward rates in this
table is easy:
RF RZ t RZ tt t t 1 1
For example:
RF4 350 4 3 00 3 5% . ( . )
Why so simple?
See also Hull Table
4.5
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Why so simple??Why so simple??
The value of the investment over all four years, equal to
principal value compounded at the four year zero rate, will
equal the principal value compounded at the three year zero
rate compounded again at the 4th year forward rate (because
that is the rate earned for only that year):
100 1000 030 3 0 035 44e e e
FR. .
Taking the natural logs and solving for FR4 leaves us with
the equation on the previous page. Normally thedenominator is 1 as in this example, but we do allow for
multi-period forward rates.
Macaulay DurationMacaulay Duration"Duration is a weighted average of the times that interest payments and the final
return of principal are received. The weights are the amounts of the payments
discounted by the yield-to-maturity of the bond."* Duration is a common measure
t at a ows ana ysts to compare on s w t erent matur t es an coupons. e
discrete form of the equation assuming annual interest payments is:
D
t c y
P
t c y
c y
i i
t
i
n
i i
t
i
n
i
tn
i i
i
1 1
1
1 1
whereD is duration,y is
current yield, c is the cash
payment (coupon or coupon
plus redemption when t=n),
P is the price of the bond,
*Source: Subject: Bonds - Duration Measure,by Rich Carreiro, 1998, at ht tp://invest-
faq.com/articles/bonds-duration.html. This article provides a good introduction to duration. The
numerical example above and the example on the next page is from that source.
i 1
Note: Not in Hull
and n is the number ofyears maturity.
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Calculating durationCalculating duration
Like most of these summation formulas, duration is easy to
calculate in an Excel workbook or even easier et in a C++ or
V Basic program. For example, consider a 5-year annual note
with a 7% coupon and a current yield of 5%. Duration would
be equal to:
D years
6 677 2 6 349 3 6 047 4 5 759 5 83838
6 6 77 6 3 49 6 0 47 5 759 83 838 4 41
. ( . ) ( . ) ( . ) ( . )
. . . . . .
Memo Note: When payments arecompounded semi-annually or quarterly,there is a simple conversion formula that
you can apply (see Hull p. 91):
DD
y ma 1
Calculating Duration with ExcelCalculating Duration with Excel
Maturity value: 100
Calculating Duration fo r a 5-year Annual Note The weight of each
a ment relativeoupon rate: .
Current yield: 0.05
Years to maturity: 5
Year Cash payment Present Value Numerator Weight
1 7 6.667 6.667 0.061
2 7 6.349 12.698 0.058
3 7 6.047 18.141 0.056
4 7 5.759 23.036 0.053
5 107 83.837 419.186 0 .772
Totals Note value (denom): 108.659 479.728 1.000
total note value. It
must sum to 1.0
This number has
no inherent
meaning ... it is to.
Refer to this slide when doing your homework (after exam?).
Also compare it to Table 4.6 in Hull (where he is using a
continuous rather than discrete formula).
be compared toanother bond.
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How is duration used?How is duration used?When comparing the duration of two bonds, a bond with a higher duration
carries more riskand is likely to have a higher price volatility.
Why do we need to calculate duration when we know that a long-term bond
as more r s an vo at ty t an a s orter-term note on t we nee to
know only the time to maturity?
No, because the yield relative to
the coupon also affects risk and
volatility. For example, a 10-year
bond with a low coupon rate and a
low yield will have a higher
volatility and risk to cashflow than
a 10-year bond with a high coupon
and high yield because the latterreceives payment proportions at a
faster rate.
Image source: investopedia.com, section on advanced bonds, duration.
Hull's Duration FormulaHull's Duration Formula
Hull's duration formula (4.12), which is continuous
is clearly a similar formula to our own, which is more commonly
used:
D
t c e
P
i i
yt
i
n
i
1
D
t c y
P
i it
i
i
11
All resulting theory is the same, of course.
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