# Prices Example T Bill Invoice Price Quote Date 12 14 1990 33221 Use the value function in excel to convert a date into a serial number Bid d

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```					Prices:
Example: T-Bill: Invoice Price
Quote Date:                 12/14/1990      33221 (Use the value function in excel to convert a date into a serial number.)
Bid (d)                         0.0678 -- By convention, T-Bill prices are usually quoted as discount rates. (If you do not u
Ask (d)                         0.0676 -- Oddly these are based on a 360-day year convention.
Maturity                     5/23/1991      33381
160 (We see this instrument has 160 days to maturity.)
Use the Excel FunctionTBILLPRICE to compute the invoice price:

96.98666667 at the bid quote

Pricing Example T-Bond:
Coupon:                          0.08
Maturity:                 11/15/2021
Today's Date:             12/17/1991
Bid:                          102.29 -- By convention, T-Note and T-Bond quotes are expressed in 32nds.

When we buy a Note or bond, we must pay accrued interest to the last owner.
Definitions:
The flat price is the price quoted.
The invoice price is the price the buyer must pay (and the seller receives) this inclu
By convention, interest payments are paid every 6 months following the issue date.
Here, then interest payments are made on Nov. 15 and May 15 until maturity.
Days between Nov and May pmts:                                            30
Elapsed days since last pmt:                                              32
Proportion of next interest payment due to seller:
Interest payment:                                              4
Accrued interest:                                  0.703296703
Invoice Price:
Dollars required for \$1 million principal:

Yields:
Refer to the T-Bill example above. What is the yield-to-maturity on that instrument?
Bond Equivalent Yield (Since the life is less than 182 days):
BEY = (100-P)/P x (365/n)
3.0133333
0.0310696
0.0708774 BEY at Bid

Using Excel Function
TBILEQ:                   0.0708774

For a discount instrument with more than 180 days until maturity, this is made a bit more complicated by th
every 6 months, and by assumption, these intermediate payments are reinvested at t
y    y      365      y
The BEY is y in the following:                   P(1 + ) +   (n -     )(1 + ) P = 100
2 365        2       2

Ex:
T-Bill                                         This first part is the first 180 days.
Settle Date                  4/18/1991
Bid d                           0.0594                       This is the next period. Note that interest is ea
Mat                           4/9/1992
TBILEQ:                     0.0630239 BEY using bid discount
TBILEQ:                     0.0628018 BEY using ask discount
Price                       93.772157 Use function TBILLPRICE to compute the price, using the ask discount

T-Bonds:
Ex:
Settle:                     1/9/1992
Mat:                     11/15/2021
Coupon:                          0.08
Price:                            107
Use excel yield function to compute the yield to maturity (By convention the Treasury Bond market uses actu
Yield to Maturity:        0.0740868

In this example, the 7.41% is the (market-determined) yield on this 29-year T-Bond.
1) The bond is selling at a premium. Thus it is likely that rates have fallen since this b
2) --Since most bonds are "priced" to originate at par. When this bond was issued, th
3) The yield is the bond's internal rate-of-return as such it assumes re-investment at t
4) Remember the quoted price does not include accrued interest.
Issue date:                 11/15/1991
1st int pmt:                 5/15/1992
settle:                        1/9/1992
rate:                               0.08
par                            1000000
freq:                                                 2
basis:                                                1

ACCRINT:                    12087.912 (If you buy \$1 million of these T-Bonds, you will also have to pay \$12,08
purchase.)

As discussed, the yields on such an instrument are market-determined: the price and the yield move in conce
We use the Excel Function PRICE to return the bond's flat price for a variety of yields:

Yield                      Price
0.12     67.670616
0.115     70.622875
0.11     73.820628
0.105      77.29135
0.1    81.066104
0.095     85.180061
0.09     89.673086
0.085     94.590428
0.08   99.983502
0.075    105.9108          Here, I start with the yield and infer the price, a
0.07   112.43894
0.065   119.64389
0.06   127.61236
0.055   136.44343
0.05   146.25045
0.045   157.16321
0.04   169.33049
0.035     182.923
0.03   198.13678

250

200
Bond (Market) Price

150

100

50

0
0                       0.02             0.04
Notes:
1) Since this is a Treasury Bond, the bond's value (market price) is strictly a function of the yield on this type o
2) Note that when the yield equal the bond's coupon rate, the bond sells at par (the few cents difference is du
3) Note that the plot of price against yield is convex. The effect on price of a 50 basis point drop is much larg

Price Risk:
A measure of price risk is the change in the bond's price, given a change in its yield. Notation:

This is the slope of the above graph at the current P, y combination.
The convexity implies that this risk decreases with y.
Like any derivative, this is a local measure--relevant for small changes in y only.
This measure of price risk is most naturally calculated with duration.

Duration:
Duration is a powerful tool in fixed income analysis.
It is a summary measure of the timing of the cash flows. So for example, the duration
duration decreases.
Example of the relationship between duration and coupon for bonds with the s
Bond 1             Bond 2              Bond 3
Settle Date               12/15/2002                12/15/2002 12/15/2002
Mat Date                    6/15/2032                 6/15/2032     6/15/2032
Coupon                               0                      0.01          0.02
Price                      23.296568              38.63725421 53.9779407
Yield                             0.05                      0.05          0.05
(Term:
Duration:                        29.5           22.36916388 19.2915362
Here I use excel to compute the bonds' prices given their terms and the market yield.

It should be clear that for bonds with the same coupon, the duration will increase with term.
Also note that duration changes get smaller as the coupon gets larger. Note the big effect on duration as the
coupon increases from 7 to 10%.
This standard measure of duration is also known as Macaulay's duration. Its relationship to price risk is as fo

Dividing both sides of Macaulay's duration by 1+y provides a more direct link to price risk. This is called Mod

From the above bonds, we can compute modified durations using excel mduration.:

Mod. Duration:            28.780488                21.82357452     18.8210109
My suggestion is to use excel's modified duration function to compute the derivative measure of price risk. It is common to use
Fixed income pros often use the measure: price value of a basis point (PVBP).
PVBP is also called DV01 (The dollar value of .01%). (By convention the sign is switched.)
In the case of more complex securities, it is often efficacious to calculate DV01 numerically.

For bonds, obtain the delta-p by multiplying MD by (P/100)

-(DELTA-p / DELTA-y)    670.48658                  843.2029964 1015.91941
DV01                    0.0670487                     0.0843203 0.10159194
Numerical DV01          0.0669506                  0.084207368       0.1014641
This shows that the riskiest of the 30-year bonds in this example is the zero-coupon b
Why?
If interest rates rise today then we can reinvest the next coupon paymen
life. So the bigger is the next coupon, the more
This arises from the fact that the bond's term fixes the "planning horizon." Early paym
This also suggests the importance of aligning the planning horizon with the duration t
Note that the Numerical DV01 is slightly different from the DV01 compute using the first derivative only. The

Portfolio duration is simply value-weighted average of the securities in the portfolio.
Here is an example of portfolio duration and DV01.

Settle                         9/10/1992     9/10/1992                9/10/1992       9/10/1992
Mat                            8/31/1994     8/15/1995                8/31/1997       7/15/1999
coupon                            0.0425       0.04625                  0.05625         0.06375
yield                            0.03801       0.04269                  0.05188         0.05753
(Flat) Price                100.8444604     100.96763              101.8919662      103.469051
Accrd Intrst:               0.117403315     0.3288494               0.15538674      0.98743207
Invoice Price               100.9618638     101.29648              102.0473529      104.456483
Mod Duration:               1.875470538     2.7080139              4.294736356      5.46750773
delta-p                     189.1308145     2.7342175              4.375991316      5.65717836
PVBP                        0.018932827     0.0274261               0.04381544      0.05709261
Portfolio AMT               100961863.8              0                        0               0
Portfolio %ge               0.333427311              0                        0               0

% * Mod Dur                  0.625333099             0                          0             0
Portfolio Modified Duration:
Portfolio PVBP:
Par Amount:               100,000,000.00
Par Units                    1,000,000.00         0.00                      0.00           0.00
\$ Investment              100,961,863.76          0.00                      0.00           0.00
PVBP Effect:                    18,932.83         0.00                      0.00           0.00
Notes and warnings:
1) Note that the figure shows the relationship between a bond's price and its
It is overly simplistic to treat the bond price as a simple func
2) Duration as a measure of risk does rely on this simplification.
Duration only characterizes the riskiness of bonds (precisely
3) Nevertheless duration is always used by fixed income managers and analy
The most important aspect of duration is the intuition that it
effects of (the standard) interest-rate risk (or discounting ris
4) Note that duration and DV01 are both measures of a security's riskiness.
Duration gives us a percentage change in price to a change in interest rate
DV01 gives a dollar change to a change in interest rates.
excel to convert a date into a serial number.)
lly quoted as discount rates. (If you do not use %ge points, you'll get an error.) This is affirmed by the "(d)" notation.
y year convention.
In particular, the discount yield is defined as [(100-P)/100 x (360/n)]
as 160 days to maturity.)

Using the definition directly:
d / (360/n) = (100-P)/100             0.030133333
100*(d / (360/n) )-100 = -P          -96.98666667
P=      96.98666667 Note that this confirms that the Excel function TBILL

uotes are expressed in 32nds.

must pay (and the seller receives) this includes the accrued interest.
wing the issue date.
until maturity.
31                 31              29                 31            30          182

0.175824176

102.96875      102.96875 The Excel function DOLLARDE can do the conversion from 32nds directly.
103.6720467 -- By convention bond prices are quoted as percentage of par.
\$1,036,720.47

Note that there is no compounding.
3.004444444
0.030975073

0.070661886

rity, this is made a bit more complicated by the notion that bonds make interest payments
se intermediate payments are reinvested at the bond's yield to maturity.
365      y
(n -       )(1 + ) P = 100                               This formula looks complicated. But what it does is adjust the gross return (100/P) for
2       2                                                       1) The fact the the holding period is between 181 and 365 days,
2) The semi-annual compounding assumption of bonds.

first 180 days.

his is the next period. Note that interest is earned on the interest from the first 180 days.

E to compute the price, using the ask discount quote.

nvention the Treasury Bond market uses actual days basis, (Choice 1 in the Function):
(For corporates we would use the option 0 (or omit) which is a 30/360 basis).

his 29-year T-Bond.
us it is likely that rates have fallen since this bond was issued.
iginate at par. When this bond was issued, the ytm on 30-years was probably very close to 8%.
f-return as such it assumes re-investment at that same rate.
include accrued interest.                                We can obtain the accrued interest using the EXCEL function ACCRINT:

ese T-Bonds, you will also have to pay \$12,087.91 to the seller at the time of

rmined: the price and the yield move in concert (in opposite directions).
for a variety of yields:
ere, I start with the yield and infer the price, and do not round the price to a 32nd (i.e,. These prices are in US\$).

Bond Price and Yield

0.04                              0.06                              0.08                                0.1
(Market) Yield
(Market) Yield

) is strictly a function of the yield on this type of instrument (as a function of coupon and term).
ond sells at par (the few cents difference is due to the timing of the interest pmts).
on price of a 50 basis point drop is much larger for low rates than it is for high rates.

a change in its yield. Notation:
¶P
¶y
eases with y.
ure--relevant for small changes in y only.
ally calculated with duration.

f the cash flows. So for example, the duration of a zero-coupon bond is equal to its term. As the coupon gets larger, (holding the ytm consta

duration and coupon for bonds with the same term:
Bond 4             Bond 5         Bond 6                     Bond 7       Bond 8      Bond 9    Bond 10
12/15/2002       12/15/2002   12/15/2002                   12/15/2002 12/15/2002 12/15/2002 12/15/2002
6/15/2032        6/15/2032    6/15/2032                    6/15/2032 6/15/2032 6/15/2032 6/15/2032
0.03             0.04         0.05                         0.06       0.07       0.08       0.09
69.3186271      84.65931355            100                115.3406864 130.68137 146.02206 161.36275
0.05             0.05         0.05                         0.05       0.05       0.05       0.05

17.57610857      16.48236892 15.72420361            15.1677152 14.741879 14.405517 14.133111
prices given their terms and the market yield. Then I use the excel function duration to compute duration.

tion will increase with term.
larger. Note the big effect on duration as the coupon increases from 0 to 1%, and the small effect as the

duration. Its relationship to price risk is as follows:                              ¶P 1 + y
D= -
¶y P
ore direct link to price risk. This is called Modified Duration:

¶P                   1
MD              = -
¶y                   P
ng excel mduration.:

17.147423        16.08035992 15.34068645            14.79777093      14.382321 14.054163 13.788401
ve measure of price risk. It is common to use change in price for a percentage change in yield:
sis point (PVBP).
ention the sign is switched.)                                                           ¶P
o calculate DV01 numerically.                                      DV 01= -                /(10 , 000 )
¶y
1188.635821        1361.352233 1534.068645          1706.785057      1879.5015 2052.2179 2224.9343
0.118863582        0.136135223 0.153406864          0.170678506      0.1879501 0.2052218 0.2224934
0.118720833        0.135977565 0.153234297          0.170491029      0.1877478 0.2050045 0.2222612
ar bonds in this example is the zero-coupon bond.

hen we can reinvest the next coupon payment at that higher rate and earn that higher rate over the bond's remaining
e. So the bigger is the next coupon, the more this reinvestment effect can offset the direct discount effect.
term fixes the "planning horizon." Early payments in the planning horizon are subject to re-investment risk.
gning the planning horizon with the duration to mitigate risk.
compute using the first derivative only. The numerical computation is exact and takes into account the second derivative.

issued:         8/31/1992
8/31/1993
9/10/1992         9/10/1992                     Portfolio                          33857
8/15/2002         8/15/2022                                                        33847
0.06375            0.0725                                                           10
0.06289           0.07216                                                        34028
100.6220749       100.4069507                                                           181
0.450407609       0.512228261                                                    0.0552486
101.0724825        100.919179                                                        2.8125
7.254597382       12.12816498                                                    0.1553867
7.299726413       12.17752063                                                         34212
0.073290581        0.12227515                                                           184
0        201838358                      302800221.7
0      0.666572689                         1

0     8.084303538
8.709636637                     Note that duration is \$ weighted

200,000,000.00                  300,000,000.00
0.00     2,000,000.00
0.00   201,838,357.98
0.00       244,550.30                      263,483.13                      Note that PVBP is Par-amount weight
The portfolio will gain (lose) this amount if the "yield" falls (rises) one ba
p between a bond's price and its yield to maturity.
t the bond price as a simple function of the yield-to-maturity.
n this simplification.
the riskiness of bonds (precisely) with respect to a parallel shift in a flat yield curve.
xed income managers and analysts to characterize the riskiness of the portfolio.
of duration is the intuition that it engenders with respect to the (generally) offsetting
erest-rate risk (or discounting risk) and re-investment risk.
asures of a security's riskiness.
price to a change in interest rates, while
interest rates.
hat the Excel function TBILLPRICE uses the definition.

om 32nds directly.
the gross return (100/P) for two factors:
en 181 and 365 days,
ion of bonds.

ion ACCRINT:
0.12   0.14
rger, (holding the ytm constant),

Bond 11
12/15/2002
Duration and Coupon Rates
6/15/2032
0.1                             35
176.70343
0.05
30
13.908004
25
Duration (Years)

20

15

10

5

0
0   0.02   0.04      0.06       0.08     0.1
Coupon Rate

13.568784
2397.6507
0.2397651
0.239518

2/28/1993

duration is \$ weighted

PVBP is Par-amount weighted
he "yield" falls (rises) one basis point.
Series1

0.1   0.12
In this example, a trader wants to bet on a steepening of the yield curve. By buying a 2-year note and selling a 30-year
curve--if she equates the PVBP of the long and short positions.

Data:
Long-term Bond:
Maturity:        5/15/2020
Cpn:                0.0875
First Int:      11/15/1991                  12/2/1991        108.8405               108.903
Last Int:       11/15/1991                  12/3/1991        109.0806              109.1431
Next Int:        5/15/1992                  12/4/1991        109.8076              109.8701

Short-term Note:
Maturity:      11/30/1993
Cpn:                0.055
First Int:      5/30/1992                  12/2/1991        100.3735               100.436
Last Int:      11/30/1991                  12/3/1991        100.4664              100.5289
Next Int:       5/30/1992                  12/4/1991        100.5967              100.6592

n2 x PVBP2 = n30 x PVBP30                (For small changes in yields the position neutralizes interest rate

So suppose that n30 = 100 million position in the 30-Year Bond.                100

n2 = n30 x (PVBP30 / PVBP2)               =            638.5847981

Position:    Long             638.5848 million in the 2-Yr Note
Short                 100 million in the 30-Yr Bond

Transactions:
12/2/1991
Borrow cash and purchase the 2-Yr T-Note
Post 2-Yr Note as Collateral                                -641,562,006.79

Borrow 30-Yr T-Bond and (short) sell
Post Cash as Collateral                                       109,249,153.85

Net Investment                                               -532,312,852.94
12/4/1991
Sell 2-Yr Note                                                 642,781,191.48

Repay the amount borrowed +
repo interest                                               -641,697,447.66

Collect Cash + Interest                                        109,271,003.68
Buy the 30-Yr Bond to Cover:                         -110,326,830.77

Profits:                                                   27,916.73

(% of net position:)                                    0.086718834

Note:
We computed DV01 (or PVBP) using the modified duration formula. So equating the PVBPs of the position
If we had computed DV01 using the numerical method, we would not equate the durations exactly.
note and selling a 30-year bond, the trader is not exposed to a level shift in the yield

Yield        Accrd Int.   PVBP
0.0796    0.408654     0.119725             On December 2, put on the position.
0.079348     0.432692     0.120179             Evaluate the position over the
0.078749     0.456731     0.121476              next 2 days.

Yield      Accrd Int.     PVBP
0.052667    0.03022       0.018748             On December 2, put on the position.
0.0525   0.04533       0.018741             Evaluate the position over the
0.0518   0.06044       0.018745              next 2 days.

on neutralizes interest rate risk.)

This translates into the dollar amount:                         108,840,500.00

This translates into the dollar amount:                         641,369,027.87

Verify that this portfolio is "duration neutral:"
Long          11.00002
Short          1.86671
Portfolio

Repo Interest Paid on loan:                                  0.038

Repo Interest Earned on Collatr:                                0.036
Note that only the transactions on 12/4/91 affect our profits.

he PVBPs of the positions neutralizes duration.
urations exactly.
Date Values:
33557      33574
33557      33575
33739      33576

Date Values:
33754      33574
33572      33575
33754      33576

1197247520
1197247520
0
Example:
Today:      9/9/1992
Strips:    Mat        Price     Yld        Mod Dur PVBP
1 8/15/1997      76.86 0.05408 4.802213 0.036900049
2 8/15/2002      50.99 0.068978 9.60094    0.04893052
3 8/15/2012       21.6 0.078382 19.18037 0.041388894

Cx is the convexity. This is
Recall that DV01 (or PVBP)
So here we have calculated

Here we consider a position that is sometimes (and erroneously) referred to as arbitrage.
We go long a barbell position and short a bullet position. The desire is to have no exposure to a parallel shift in the yield curve

In this example, we also have a "self-financing" (zero-net investment) portfolio.
(Contrast to Bloomberg's BBA portfolio which instead of this condition, sets the risk on both sides of the "barbell" to

n2 P2 = n1 P + n3 P3 = V p
1
2 equations in 2 unknowns. Requires a numeraire: in this case n2

n1 P       n P
MD2 =          1
MD1 + 3 3 MD3
Vp          Vp

Set n2 = 100, substitute the values from above                                             Portfolio Duration:
n1            44.19987                                                                     Portfolio Convexity:
n2                 100
n3            78.78693
Vp                5099

Eqn 1                  0 Sqr'd:                0
Eqn 2                  0 Sqr'd:                0
0 Invoke Solver to minimize this cell by changing n1 and n3.
(May need to modify Solver options to converge, or use algebra to s

Future Scenarios:
Assume parallel shifts in the yield curve:
y2          P2          V Bullet     y1    p1                      y3           p3          V Barbell
0.05 61.23218 6123.218 0.035103       84.229361             0.0594044    31.13592    6176.031
0.051 60.64186 6064.186 0.036103 83.82218939                  0.0604044    30.53916    6111.017
0.052 60.05752 6005.752 0.037103 83.41718455                  0.0614044    29.95412    6047.022
0.053 59.47908 5947.908 0.038103 83.01433391                  0.0624044    29.38056    5984.027
0.054 58.9065 5890.65 0.039103 82.61362496                    0.0634044    28.81825    5922.013
0.055 58.3397 5833.97 0.040103 82.21504527                    0.0644044    28.26697    5860.962
0.056 57.77863 5777.863 0.041103 81.81858249                  0.0654044    27.72649    5800.856
0.057 57.22322 5722.322 0.042103 81.42422434                  0.0664044    27.19661    5741.678
0.058   56.67341                             5667.341   0.043103     81.03195864    0.0674044    26.6771   5683.409
0.059   56.12916                             5612.916   0.044103     80.64177326    0.0684044   26.16775   5626.033
0.06   55.59039                             5559.039   0.045103     80.25365617    0.0694044   25.66837   5569.534
0.061   55.05705                             5505.705   0.046103     79.86759539    0.0704044   25.17876   5513.895
0.062   54.52908                             5452.908   0.047103     79.48357903    0.0714044   24.69871     5459.1
0.063   54.00642                             5400.642   0.048103     79.10159528    0.0724044   24.22804   5405.134
0.064   53.48903                             5348.903   0.049103     78.72163239    0.0734044   23.76656   5351.981
0.065   52.97684                             5297.684   0.050103      78.3436787    0.0744044   23.31409   5299.626
0.066    52.4698                              5246.98   0.051103     77.96772261    0.0754044   22.87044   5248.055
0.067   51.96785                             5196.785   0.052103     77.59375258    0.0764044   22.43544   5197.254
0.068   51.47094                             5147.094   0.053103     77.22175718    0.0774044   22.00893   5147.208
0.069   50.97903                             5097.903   0.054103     76.85172502    0.0784044   21.59071   5097.903
0.07   50.49204                             5049.204   0.055103     76.48364479    0.0794044   21.18064   5049.325
0.071   50.00994                             5000.994   0.056103     76.11750525    0.0804044   20.77855   5001.463
0.072   49.53268                             4953.268   0.057103     75.75329523    0.0814044   20.38429   4954.301
0.073   49.06019                             4906.019   0.058103     75.39100364    0.0824044   19.99768   4907.829
0.074   48.59244                             4859.244   0.059103     75.03061944    0.0834044   19.61859   4862.032
0.075   48.12937                             4812.937   0.060103     74.67213168    0.0844044   19.24686     4816.9
0.076   47.67093                             4767.093   0.061103     74.31552946    0.0854044   18.88235    4772.42

Value of Convexity

60

50
Long Barbell / Short Bullet

40

30

20

10

0
0.04              0.045           0.05                     0.06
0.055 on intermediate strip
Yield
Delta y:        0.0001
New y       New P          MD        PVBP
0.0541802      76.82309995 4.801979 0.036880539
0.0690776      50.94106948 9.600476 0.048881204
0.0784819      21.55861111 19.17945 0.041307601

Cx is the convexity. This is defined as the second derivative of the price, with respect to yield, divided by the price.
Recall that DV01 (or PVBP) is the first derivative divided by 10,000.
So here we have calculated Cx numerically by evaluating DV01 at two contiguous rates, and evaluating its rate of change.

parallel shift in the yield curve, but be long in convexity.

both sides of the "barbell" to be the same.)

s a numeraire: in this case n2.

Portfolio Duration:           0.0000
Portfolio Convexity:        22.90274

changing n1 and n3.
converge, or use algebra to solve.)

Value of Long Barbell, Short Bullet
52.8127782
46.8307828
41.270397              The Barbell has more convexity because while duration
36.118832               increases linearly with maturity, convexity increases with the square of maturity.
31.3636142
26.9925783
22.9938601
19.3558893
16.0673831
13.1173394
10.4950303
8.1899959        Note that for any rate change the barbell portfolio
6.19203801         will have a higher value than the bullet portfolio, although their values are
4.4912141         identical at the current rate. (Note that this is ony true for parallel shifts in
3.07783144         the yield curve.)
1.9424413
1.07583331
0.46902994
0.11328111
5.8898E-05
0.12105241
0.46816274
1.03349802
1.80936864
2.78828251
3.96294051
5.32623193

ue of Convexity

0.06         0.065
d on intermediate strip          0.07             0.075              0.08
Chg in first-derivative       Cx
0.195104477            12.6921986
0.493156086            48.3581179
0.812929665            188.178163

, divided by the price.

evaluating its rate of change.

the square of maturity.
h their values are
r parallel shifts in
Bloomberg has a useful tool to build butterfly positions: BBA.
Bloomberg uses the following 2 criteria (equations): 1) Portfolio has 0 Duration (a
2) PVBP of both ends of t
(and therefore equal to
Therefore, the position does not have 0-net investment.
The reason for the 2nd equation is an attempt also to minimize the effect of chang

To use this you must identify 3 US Treasury securities. You can buy convexity by buying a short- and long-term security, and
selling an intermediate security.

To identify, one way is to go into the active Governments screen: <GOVT> BBT <GO>
In this screen, you should note that the on-the-runs are
highlighted in white.
placing the cursor over that security and right-clicking the mouse.
This will bring up the description screen in the alternative panel:

In this case, we see the on-the-run 10-Year T-Note.
For the BBA analysis, we need its CUSIP, so we note that this is: 912828CT5.

Go the the BBA screen by entering <GOVT> BBA <GO>

Enter the settlement date for the 3 securities and under the security description fileds enter the CUSIP <GOVT>
The screen will fill in all other data for you:

One potentially confusing aspect of the BBA screen is the shift table. Here is Bloomberg's clarification of the cells in that table:

The second screen in BBA is a picture of how well the position has done over the past several days:
o has 0 Duration (and PVBP)
BP of both ends of the barbell position are equal to each other
therefore equal to 1/2 of the PVBP of the bullet position).

the effect of changes in the slope of the yield curve on the position.

d long-term security, and

on of the cells in that table:
Extension of Convexity analysis to coupon-paying bonds and notes.
While computation of individual securities' is done using the flat price (or clean pri
Portfolio weights therefore should be based on the invoice price.

Here's an example using actual T-Securities from Bloomberg.
Today:     10/18/2004
Mat             Price     Yld         Mod Dur                         PVBP     Accrued Interest
1 11/15/2008       106.3438 0.03081326                       3.640018189 0.039434 2.013587
2 2/15/2014        99.70313 0.0403793                        7.664644464 0.076917 0.695652
3 11/15/2024       134.1875 0.04822401                       11.41486464 0.156676 3.179348

Set n2 = 100, substitute the values from above
n1            44.69254 *price =                         4842.764856
n2                  100 *price =                        10039.87802
n3            37.83382 *price =                         5197.113161
Vp            10039.88                                  10039.87802

Eqn 1                  0 Sqr'd:                                 0
Eqn 2                    Sqr'd:               0.0000000000000000
5.8556E-18

Note even though n2 is 100, because we have to account
for accrued interest, we must be careful to define portfolio
duration accordingly. (This is done in cell K15).

Scenarios:
Assume parallel shifts in the yield curve:
y2           P2          V Bullet                   y1                p1          y3          p3
0.0213793    115.6662     11636.183           0.011813254             114.1599    0.029224    169.1486
0.0223793     114.755    11545.0668           0.012813254              113.732    0.030224    167.0206
0.0233793    113.8522    11454.7842           0.013813254             113.3059    0.031224     164.928
0.0243793    112.9576    11365.3271           0.014813254             112.8818    0.032224    162.8702
0.0253793    112.0712    11276.6873           0.015813254             112.4595    0.033224    160.8466
0.0263793    111.1929    11188.8567           0.016813254             112.0391    0.034224    158.8564
0.0273793    110.3226    11101.8273           0.017813254             111.6205    0.035224    156.8992
0.0283793    109.4603    11015.5913           0.018813254             111.2038    0.036224    154.9742
0.0293793    108.6058    10930.1409           0.019813254             110.7889    0.037224     153.081
0.0303793     107.759    10845.4683           0.020813254             110.3759    0.038224    151.2188
0.0313793      106.92    10761.5658           0.021813254             109.9646    0.039224    149.3872
0.0323793    106.0886     10678.426           0.022813254             109.5552    0.040224    147.5855
0.0333793    105.2648    10596.0412           0.023813254             109.1476    0.041224    145.8133
0.0343793    104.4484    10514.4042           0.024813254             108.7417    0.042224    144.0699
0.0353793    103.6394    10433.5075           0.025813254             108.3377    0.043224    142.3549
0.0363793    102.8378     10353.344           0.026813254             107.9354    0.044224    140.6678
0.0373793    102.0434    10273.9063           0.027813254             107.5348    0.045224     139.008
0.0383793    101.2562    10195.1875           0.028813254             107.1361    0.046224     137.375
0.0393793                             100.4762   10117.1805   0.029813254       106.739   0.047224   135.7683
0.0403793                             99.70313   10039.8783   0.030813254      106.3438   0.048224   134.1875
0.0413793                             98.93709   9963.27399   0.031813254      105.9502   0.049224   132.6321
0.0423793                             98.17796   9887.36084   0.032813254      105.5584   0.050224   131.1016
0.0433793                             97.42567   9812.13205   0.033813254      105.1682   0.051224   129.5956
0.0443793                             96.68016   9737.58096   0.034813254      104.7798   0.052224   128.1137
0.0453793                             95.94136   9663.70095   0.035813254      104.3931   0.053224   126.6553
0.0463793                              95.2092   9590.48548   0.036813254       104.008   0.054224   125.2201
0.0473793                             94.48363   9517.92809   0.037813254      103.6247   0.055224   123.8077
0.0483793                             93.76457   9446.02237   0.038813254       103.243   0.056224   122.4176
0.0493793                             93.05197   9374.76198   0.039813254       102.863   0.057224   121.0495
0.0503793                             92.34575   9304.14064   0.040813254      102.4846   0.058224   119.7029
0.0513793                             91.64587   9234.15215   0.041813254      102.1078   0.059224   118.3775
0.0523793                             90.95225   9164.79036   0.042813254      101.7328   0.060224   117.0728
0.0533793                             90.26484   9096.04919   0.043813254      101.3593   0.061224   115.7886
0.0543793                             89.58357   9027.92262   0.044813254      100.9875   0.062224   114.5244
0.0553793                             88.90839   8960.40469   0.045813254      100.6172   0.063224     113.28
0.0563793                             88.23924   8893.48951   0.046813254      100.2486   0.064224   112.0548
0.0573793                             87.57606   8827.17125   0.047813254       99.8816   0.065224   110.8487
0.0583793                             86.91879   8761.44412   0.048813254      99.51617   0.066224   109.6612
0.0593793                             86.26737   8696.30242   0.049813254      99.15232   0.067224    108.492
0.0603793                             85.62175   8631.74048   0.050813254      98.79005   0.068224   107.3409

Value of Convexity

80
lue of Long Barbell - Short Bullet

70
60
50
40
30
20
10
0
Value of Long Ba
0
-102.14%   2.64%   3.14%   3.64%   4.14%
Yield on Intermediate Trea
price (or clean price), cash flows depend on accrued interest as well.

Change in y:    0.0001
Invoice Price                             New y        New P       MD          PVBP        Chg in first-derivative
108.3573                                  0.030913     106.3043    3.639747    0.039416   0.172822
100.3988                                  0.040479     99.62621    7.663568    0.076847   0.697092
137.3668                                  0.048324     134.0308    11.40941    0.156422   2.534415

Portfolio Duration:      0.0000
Portfolio Convexity:   16.88331

V Barbell   Value of Long Barbell, Short Bullet
11711.91                         75.73064
11612.28                          67.2105
11514.07                         59.28227
11417.26                          51.9289
11321.82                         45.13371
11227.74                         38.88044
11134.98                          33.1532
11043.53                         27.93651
10953.36                         23.21524
10864.44                         18.97461
10776.77                         15.20024
10690.3                         11.87804
10605.04                         8.994301
10520.94                          6.53563
10438                          4.48896
10356.19                         2.841538
10275.49                         1.580924
10195.88                         0.694977
10117.35   0.171853
10039.88    -1.5E-06
9963.442     0.16814
9888.026   0.665281
9813.613   1.480694
9740.185   2.603916
9667.726   4.024741
9596.219   5.733215
9525.648     7.71963
9455.997   9.974518
9387.251   12.48865
9319.394   15.25301
9252.411   18.25884
9186.288   21.49757
9121.01   24.96085
9056.563   28.64055
8992.933   32.52874
8930.107   36.61769
8868.071   40.89987
8806.812   45.36792
8746.317   50.01469
8686.574   54.83321

nvexity
4.14%   4.64%   5.14%   5.64%
rmediate Treasury
Cx
8.125643
34.95839
94.43559

```
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