Fixed Income Instruments 5
Zvi Wiener
02-588-3049
mswiener@mscc.huji.ac.il
Fall-01
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
FIBI
Fixed Income 5
• Mortgage loans
• Pass-through securities
• Prepayments
• Agencies
• MBS
• CMO
• ABS
Zvi Wiener
FIFIBI - 5
slide 2
Bonds with Embedded Options (14)
Traditional yield analysis compares yields of
bonds with yield of on-the-run similar
Treasuries.
The static spread is a measure of the spread
that should be added to the zero curve
(Treasuries) to get the market value of a bond.
Zvi Wiener
FIFIBI - 5
slide 3
Active Bond Portfolio Management (17)
Basic steps of investment management
Active versus passive strategies
Market consensus
Different types of active strategies
Bullet, barbell and ladder strategies
Limitations of duration and convexity
How to use leveraging and repo market
Zvi Wiener
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slide 4
Investment Management
• Setting goals, idea of ALM or benchmark
• GAAP, FAS 133, AIMR - reporting
standards
• passive or active strategy - views, not
transactions
• available indexes
• mixed strategies
Zvi Wiener
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slide 5
Major risk factors
• level of interest rates
• shape of the yield curve
• changes in spreads
• changes in OAS
• performance of a specific sector/asset
• currency/linkage
Zvi Wiener
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slide 6
r
Parallel shift
upward move
Current TS
Downward move
T
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Twist
r
flattening
T
steepening
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slide 8
r
Butterfly
T
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slide 9
Yield curve strategies
Bullet strategy: Maturities of securities are
concentrated at some point on the yield curve.
Barbel strategy: Maturities of securities are
concentrated at two extreme maturities.
Ladder strategy: Maturities of securities are
distributed uniformly on the yield curve.
Zvi Wiener
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slide 10
Example
bond
coupon
maturity
yield duration
convex.
A
B
C
8.5%
9.5%
9.25%
5
20
10
8.5 4.005
9.5 8.882
9.25 6.434
19.81
124.17
55.45
Bullet portfolio: 100% bond C
Barbell portfolio: 50.2% bond A, 49.8% bond B
Zvi Wiener
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slide 11
Dollar duration of barbell portfolio =
0.502*4.005 + 0.498*8.882 = 6.434
it has the same duration as bullet portfolio.
Dollar convexity of barbell portfolio =
0.502*19.81 + 0.498*124.17 = 71.78
the convexity here is higher!
Is this an arbitrage?
Zvi Wiener
FIFIBI - 5
slide 12
The yield of the bullet portfolio is 9.25%
The yield of the barbell portfolio is 8.998%
This is the cost of convexity!
Zvi Wiener
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Leverage
Risk is not proportional to investment!
This can be achieved in many ways: futures,
options, repos (loans), etc.
Duration of a levered portfolio is different
form the average time of cashflow!
Use of dollar duration!
Zvi Wiener
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Repo Market
Repurachase agreement - a sale of a security
with a commitment to buy the security back at
a specified price at a specified date.
Overnight repo (1 day) , term repo (longer).
Zvi Wiener
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Repo Example
You are a dealer and you need $10M to purchase
some security.
Your customer has $10M in his account with no
use. You can offer your customer to buy the
security for you and you will repurchase the
security from him tomorrow. Repo rate 6.5%
Then your customer will pay $9,998,195 for the
security and you will return him $10M tomorrow.
Zvi Wiener
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slide 16
Repo Example
$9,998,195 0.065/360 = $1,805
This is the profit of your customer for offering the
loan.
Note that there is almost no risk in the loan since
you get a safe security in exchange.
Zvi Wiener
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slide 17
Reverse Repo
You can buy a security with an attached
agreement to sell them back after some time at a
fixed price.
Repo margin - an additional collateral.
The repo rate varies among transactions and may
be high for some hot (special) securities.
Zvi Wiener
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Example
You manage $1M of your client. You wish to
buy for her account an adjustable rate
passthrough security backed by Fannie Mae.
The coupon rate is reset every month according
to LIBOR1M + 80 bp with a cap 9%.
A repo rate is LIBOR + 10 bp and 5% margin is
required. Then you can essentially borrow
$19M and get 70 bp *19M.
Is this risky?
Zvi Wiener
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Indexing
The idea of a benchmark (liabilities, actuarial
or artificial).
Cellular approach, immunization, dynamic
approach
Tracking error
Performance measurement, and attribution
Optimization
Risk measurement
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slide 20
r
Flattener
Current TS
Sell, Buy
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T
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slide 21
Example of a flattener
• sell short, say 1 year
• buy long, say 5 years
• what amounts?
In order to be duration neutral you
have to buy 20% of the amount sold
and invest the proceedings into
money market.
• Sell 5M, buy 1M and invest 4M into MM.
Zvi Wiener
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Use of futures to take position
Assume that you would like to be longer
then your benchmark.
This means that you expect that interest
rates in the future will move down more
than predicted by the forward rates.
One possible way of doing this is by taking
a future position.
How to do this?
Zvi Wiener
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slide 23
Use of futures to take position
Your benchmark is 3 years, your current
portfolio has duration of 3 years as well and
value of $1M. You would like to have
duration of 3.5 years since your expectation
regarding 3 year interest rates for the next 2
months are different from the market.
Each future contract will allow you to buy
5 years T-notes in 2 months for a fixed
price.
Zvi Wiener
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slide 24
Use of futures to take position
Each future contract will allow you to buy 5
years T-notes in 2 months for a fixed price.
If you are right and the IR will go down
(relative to forward rates) then the value of the
bonds that you will receive will be higher then
the price that you will have to pay and your
portfolio will earn more than the benchmark.
Zvi Wiener
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slide 25
Use of futures to take position
0
2M
3Y
5Y
-x(1+r2M/6)
(1+r3Y)3
x(1+r5Y)5
One should chose x such that the resulting
duration will be 3.5 years.
Zvi Wiener
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slide 26
Bond Risk Management
Zvi Wiener
02-588-3049
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
Fall-01
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
FIBI
Duration
N
tCt
1
Macauley duration DMacauley  
B t 1 (1  y ) t
Modified duration
Effective duration
Dollar duration
Zvi Wiener
D* 
DMacauley
1 y
1 dB
D
B dy
DB
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slide 28
Fixed Income Risk
Arises from potential movements in the level
and volatility of bond yields.
Factors affecting yields
– inflationary expectations
– term spread
– higher volatility of the low end of TS
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slide 29
Volatilities of IR/bond prices
Price volatility in %
Euro 30d
Euro 180d
Euro 360d
Swap 2Y
Swap 5Y
Swap 10Y
Zero 2Y
Zero 5Y
Zero 10Y
Zero 30Y
Zvi Wiener
End 99
0.22
0.30
0.52
1.57
4.23
8.47
1.55
4.07
7.76
20.75
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End 96
0.05
0.19
0.58
1.57
4.70
9.82
1.64
4.67
9.31
23.53
slide 30
Duration approximation
 P 

  D *  (y )
 P 
What duration makes bond as volatile as FX?
What duration makes bond as volatile as stocks?
A 10 year bond has yearly price volatility of 8%
which is similar to major FX.
30-year bonds have volatility similar to equities
(20%).
Zvi Wiener
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slide 31
Models of IR
Normal model (y) is normally distributed.
Lognormal model (y/y) is normally distributed.
Note that:
Zvi Wiener
 y 
 (y )  y    
 y 
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slide 32
Principal component analysis
• level risk factor 94% of changes
• slope risk factor (twist) 4% of changes
• curvature (bend or butterfly)
See book by Golub and Tilman.
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Forwards and Futures
Ft e
 rt
 St e
 yt
The forward or futures price on a stock.
e-rt the present value in the base currency.
e-yt the cost of carry (dividend rate).
For a discrete dividend (individual stock) we
can write the right hand side as St- D, where D
is the PV of the dividend.
Zvi Wiener
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slide 34
Hedging Linear Risk
Following Jorion 2001, Chapter 14
Financial Risk Manager Handbook
Fall-01
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
FIBI
Hedging
Taking positions that lower the risk profile of
the portfolio.
• Static hedging
• Dynamic hedging
Zvi Wiener
FIFIBI - 5
slide 36
Unit Hedging with Currencies
A US exporter will receive Y125M in 7
months.
The perfect hedge is to enter a 7-months
forward contract.
Such a contract is OTC and illiquid.
Instead one can use traded futures.
CME lists yen contract with face value
Y12.5M and 9 months to maturity.
Sell 10 contracts andFIFIBI
revert
in 7 months.
-5
Zvi Wiener
slide 37
Market data
0
7m
time to maturity
9
2
US interest rate
6%
6%
Yen interest rate
5%
2%
Spot Y/$
125.00
150.00
Futures Y/$
124.07
149.00
1 
 1
Y 125M  

  $166,667
 150 125 
P&L
1 
 1
 10  Y 12.5M  

  $168,621
 149 124.07 
Zvi Wiener
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slide 38
Stacked hedge - to use a longer horizon and
to revert the position at maturity.
Strip hedge - rolling over short hedge.
Zvi Wiener
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slide 39
Basis Risk
Basis risk arises when the characteristics of
the futures contract differ from those of the
underlying.
For example quality of agricultural product,
types of oil, Cheapest to Deliver bond, etc.
Basis = Spot - Future
Zvi Wiener
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slide 40
Cross hedging
Hedging with a correlated (but different) asset.
In order to hedge an exposure to Norwegian
Krone one can use Euro futures.
Hedging a portfolio of stocks with index future.
Zvi Wiener
FIFIBI - 5
slide 41
The Optimal Hedge Ratio
S - change in $ value of the inventory
F - change in $ value of the one futures
N - number of futures you buy/sell
V  S  N  F

2
V

2
S
N 
2
2
F
 2 N S ,F

2
 2 N F  2 S ,F
N
2
V
Zvi Wiener
FIFIBI - 5
slide 42
The Optimal Hedge Ratio

2
 2 N F  2 S ,F
N
2
V
N opt
 S ,F
 S
  2    S ,F
 F
 F
Minimum variance hedge ratio
Zvi Wiener
FIFIBI - 5
slide 43
Hedge Ratio as Regression Coefficient
The optimal amount can also be derived as the
slope coefficient of a regression s/s on f/f:
s
f
    sf

s
f
 sf
s
 sf  2   sf
f
f
Zvi Wiener
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slide 44
Optimal Hedge
One can measure the quality of the optimal
hedge ratio in terms of the amount by which
we have decreased the variance of the original
portfolio.
2
2
R 
2
( s   V * )

2
s
V*   s 1 R

2
sf
2
If R is low the hedge is not effective!
Zvi Wiener
FIFIBI - 5
slide 45
Optimal Hedge
At the optimum the variance is

Zvi Wiener
2
V*

 

2
S
FIFIBI - 5
2
SF
2
F
slide 46
Example
Airline company needs to purchase 10,000
tons of jet fuel in 3 months. One can use
heating oil futures traded on NYMEX.
Notional for each contract is 42,000 gallons.
We need to check whether this hedge can be
efficient.
Zvi Wiener
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slide 47
Example
Spot price of jet fuel $277/ton.
Futures price of heating oil $0.6903/gallon.
The standard deviation of jet fuel price rate of
changes over 3 months is 21.17%, that of
futures 18.59%, and the correlation is 0.8243.
Zvi Wiener
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slide 48
Compute
• The notional and standard deviation f the
unhedged fuel cost in $.
• The optimal number of futures contracts to
buy/sell, rounded to the closest integer.
• The standard deviation of the hedged fuel
cost in dollars.
Zvi Wiener
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slide 49
Solution
The notional is Qs=$2,770,000, the SD in $ is
(s/s)sQs=0.2117$277 10,000 = $586,409
the SD of one futures contract is
(f/f)fQf=0.1859$0.690342,000 = $5,390
with a futures notional
fQf = $0.690342,000 = $28,993.
Zvi Wiener
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slide 50
Solution
The cash position corresponds to a liability
(payment), hence we have to buy futures as a
protection.
sf= 0.8243  0.2117/0.1859 = 0.9387
sf = 0.8243  0.2117  0.1859 = 0.03244
The optimal hedge ratio is
N* = sf Qss/Qff = 89.7, or 90 contracts.
Zvi Wiener
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slide 51
Solution
2unhedged = ($586,409)2 = 343,875,515,281
- 2SF/ 2F = -(2,605,268,452/5,390)2
hedged = $331,997
The hedge has reduced the SD from $586,409
to $331,997.
R2 = 67.95%
Zvi Wiener
(= 0.82432)
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slide 52
Duration Hedging
dP   D * P  dy
Dollar duration
S   DS*  S  y
F   DF*  F  y
  D  S   
2
S
2
*
S
2
y
  D  F   
2
F
*
F
2
2
y
 SF  D  F D  S  
*
F
Zvi Wiener
*
S
FIFIBI - 5
2
y
slide 53
Duration Hedging
 SF
D S
N*   2  
F
D F
*
S
*
F
If we have a target duration DV* we can get it by using
D V  D  S
N
*
DF  F
*
V
Zvi Wiener
*
S
FIFIBI - 5
slide 54
Example 1
A portfolio manager has a bond portfolio worth
$10M with a modified duration of 6.8 years, to
be hedged for 3 months. The current futures
prices is 93-02, with a notional of $100,000.
We assume that the duration can be measured
by CTD, which is 9.2 years.
Compute:
a. The notional of the futures contract
b.The number of contracts to by/sell for optimal
protection.
Zvi Wiener
FIFIBI - 5
slide 55
Example 1
The notional is:
(93+2/32)/100$100,000 =$93,062.5
The optimal number to sell is:
D S
6.8  $10,000,000
N*  

 79.4
D F
9.2  $93,062.5
*
S
*
F
Note that DVBP of the futures is 9.2$93,0620.01%=$85
Zvi Wiener
FIFIBI - 5
slide 56
Example 2
On February 2, a corporate treasurer wants to
hedge a July 17 issue of $5M of CP with a maturity
of 180 days, leading to anticipated proceeds of
$4.52M. The September Eurodollar futures trades
at 92, and has a notional amount of $1M.
Compute
a. The current dollar value of the futures contract.
b. The number of futures to buy/sell for optimal
hedge.
Zvi Wiener
FIFIBI - 5
slide 57
Example 2
The current dollar value is given by
$10,000(100-0.25(100-92)) = $980,000
Note that duration of futures is 3 months,
since this contract refers to 3-month LIBOR.
Zvi Wiener
FIFIBI - 5
slide 58
Example 2
If Rates increase, the cost of borrowing will
be higher. We need to offset this by a gain, or
a short position in the futures. The optimal
number of contracts is:
D S
180  $4,520,000
N*  

 9.2
D F
90  $980,000
*
S
*
F
Note that DVBP of the futures is 0.25$1,000,0000.01%=$25
Zvi Wiener
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slide 59
Beta Hedging
Rit   i   i Rmt   it
 represents the systematic risk,  - the
intercept (not a source of risk) and  - residual.
S
M

S
M
A stock index futures contract
Zvi Wiener
FIFIBI - 5
F
M
1
F
M
slide 60
Beta Hedging
M
M
V  S  NF  S
 NF
M
M
S
The optimal N is N *  
F
The optimal hedge with a stock index futures
is given by beta of the cash position times its
value divided by the notional of the futures
contract.
Zvi Wiener
FIFIBI - 5
slide 61
Example
A portfolio manager holds a stock portfolio
worth $10M, with a beta of 1.5 relative to
S&P500. The current S&P index futures price
is 1400, with a multiplier of $250.
Compute:
a. The notional of the futures contract
b. The optimal number of contracts for hedge.
Zvi Wiener
FIFIBI - 5
slide 62
Example
The notional of the futures contract is
$2501,400 = $350,000
The optimal number of contracts for hedge is
N*  
 S
F
1.5  $10,000,000

 42.9
1 $350,000
The quality of the hedge will depend on the
size of the residual risk in the portfolio.
Zvi Wiener
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slide 63
A typical US stock has correlation of 50% with S&P.
Using the regression effectiveness we find that the
volatility of the hedged portfolio is still about
(1-0.52)0.5 = 87% of the unhedged volatility for a
typical stock.
If we wish to hedge an industry index with S&P
futures, the correlation is about 75% and the
unhedged volatility is 66% of its original level.
The lower number shows that stock market hedging
is more effective for diversified
portfolios.
Zvi Wiener
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slide 64
FRM-GARP type question
Zvi Wiener
Fall-01
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
FIBI
FRM-GARP 98:18
A portfolio consists of two positions: One
position is long $100M of a two year bond
priced at 101 with duration of 1.7, the other
position is short $50M of a five year bond
priced at 99 with a duration of 4.1. What is
the duration of the portfolio?
Zvi Wiener
FIFIBI - 5
slide 66
FRM-GARP 98:18
The dollar duration is sum of dollar durations, so
$100M 101/100 1.7 = $171.7M
-$50M 99/100 4.1 = -$202.95M
total dollar duration is -$31.25M, portfolio’s
value is $50M, thus its duration is -0.61.
Zvi Wiener
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slide 67
Cap and Floor
Cap:
Max[iT-iC, 0]
Floor:
Max[iF-iT, 0]
What is Long Cap and Short Floor position?
Cap - Floor =
Max[iT-iC, 0] - Max[iC-iT, 0] = iT-iC
pay fixed swap
Zvi Wiener
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slide 68
FRM-GARP 98:50
A hedge fund leverages its 100M of investor
capital by a factor of 3 and invests it into a
portfolio of junk bonds yielding 14%. If its
borrowing costs are 8%, what is the yield on
investor capital?
Zvi Wiener
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slide 69
FRM-GARP 98:50
300M invested at 14% yield 42M, borrowing
costs are 200 at 8% or 16M, the difference of
26M provides 26% yield on equity of 100M.
Zvi Wiener
FIFIBI - 5
slide 70
FRM-GARP 98:51
A portfolio consists of two long assets $100
each. The probability of default over the next
year is 10% for the first asset, 20% for the
second asset, and the joint probability of
default is 3%. What is the expected loss on
this portfolio due to credit risk over the next
year assuming 40% recovery rate for both
assets.
Zvi Wiener
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slide 71
FRM-GARP 98:51
0.1(1-0.2) - default probability of A
0.2(1-0.1) - default probability of B
0.03 - default probability of both
Expected losses are
0.1(1-0.2)100(1-0.4)
0.2(1-0.1)100(1-0.4)
0.03200(1-0.4)
4.8 + 10.8 + 3.6 = 19.2M
Zvi Wiener
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slide 72
Example
Assume a 1-year US Treasury yield is 5.5%
and a Eurodollar deposit rate is 6%. What is
the probability of the Eurodollar deposit to
default (assuming zero recovery rate)?
1
1

1.06 1.055
  0.5%
Zvi Wiener
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slide 73
FRM-GARP 97:24
Assume the 1-year US Treasury yield is 5.5%
and a default probability of a one year
Commercial Paper is 1%. What should be the
yield on the CP assuming 50% recovery ratio?
1
1
0.5


1  x 1.055 1.055
x  6%
Zvi Wiener
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slide 74
FRM-GARP 00:47
Which one of the following deals has the
largest credit exposure for a $1,000,000 deal
size. Assume that the counterparty in each
deal is a AAA-rated bank and there is no
settlement risk.
A. Pay fixed in an interest rate swap for 1 year
B. Sell USD against DEM in a 1 year forward
contract.
C. Sell a 1-year DEM Cap
D. Purchase a 1-year Certificate of Deposit
Zvi Wiener
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slide 75
FRM-GARP 00:47
Which one of the following deals has the
largest credit exposure for a $1,000,000 deal
size. Assume that the counterparty in each
deal is a AAA-rated bank and there is no
settlement risk.
A. Pay fixed in an interest rate swap for 1 year
B. Sell USD against DEM in a 1 year forward
contract.
C. Sell a 1-year DEM Cap
D. Purchase a 1-year Certificate of Deposit
Zvi Wiener
FIFIBI - 5
slide 76
FRM-GARP 98
A step-up coupon bond pays LIBOR for 2
years, 2LIBOR for the next two years and
3LIBOR for the last two years. The
principal amount is paid at the end of year 6.
Prices of zero coupon bonds maturing in 2, 4,
and 6 years are Z2, Z4, Z6. What is the price
of the step-up bond?
Zvi Wiener
FIFIBI - 5
slide 77
FRM-GARP 98
0
1
2
3
4
5
6
?
L
L
2L
2L
3L
3L+100
Zvi Wiener
FIFIBI - 5
slide 78
0
1
2
3
4
5
6
?
L
L
2L
2L
3L
3L+100
100 L
L
L
L
L
L+100
300 3L
3L
3L
3L
3L
3L+300
100 L
L
L
L+100
100 L
L+100
Z2
100
Z4
100
2Z6
200
? = 300 - 100 - 100 + Z2 + Z4- 2Z6
Zvi Wiener
FIFIBI - 5
slide 79