Post on 21-Dec-2015
FRMhttp://pluto.mscc.huji.ac.il/
~mswiener/zvi.html
Zvi Wiener02-588-3049
mswiener@mscc.huji.ac.il
Introduction to Financial Markets
Zvi WienerIntroduction to Financial
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Call Option
European Call
X Underlying
premium
Zvi WienerIntroduction to Financial
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Put Option
European Put
X Underlying
premium
X
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Collar
• Firm B has shares of firm C of value $100
• They do not want to sell the shares, but need
money.
• Moreover they would like to decrease the
exposure to financial risk.
• How to get it done?
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Collar
1. Buy a protective Put option (3y to maturity,
strike = 90% of spot).
2. Sell an out-the-money Call option (3y to
maturity, strike above spot).
3. Take a “cheap” loan at 90% of the current
value.
Zvi WienerIntroduction to Financial
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Collar payoff
payoff
90 100 K stock
90
K
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Inverse Floater
Today -100
1 yr 7.5%
2 yr 9% - LIBOR
3 yr 10% - LIBOR
4 yr 11% - LIBOR
5 yr 12% - LIBOR + 100
Callable!
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Inverse Floater
Today
1 yr
2 yr
3 yr
4 yr
5 yr
A
-100
L
L
L
L
L+100
B
-100
5
5
5
5
105
C
-100
5
4
5
6
105
D
-call option
0
0
0
0
2
B + C + D - A
Zvi WienerIntroduction to Financial
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Yield Enhancement
Today you have 100 NIS invested in shekels for 1 year and 100 NIS invested in dollars for one year.
Yields are 4.5% NIS, 2% USD.
You can create a deposit that offers 7% NIS or 4.5% USD (the linkage is chosen by the bank!).
Zvi WienerIntroduction to Financial
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Yield Enhancement
USD
1.07
1.045
Payoff at the year end
Sell some amount of USD Put options,the money received invest in SHEKELaccount!
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Combined CPI deal
• You are underexposed to CPI
• You have TA25 exposure
• One can sell an out-of-the-money call on TA25
• Buy a Call on CPI
FRMhttp://pluto.mscc.huji.ac.il/
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Zvi Wiener02-588-3049
mswiener@mscc.huji.ac.il
Example of Risk Management
Zvi WienerIntroduction to Financial
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Investment Decision
$1000M bonds
capital capital
$900M bonds
$100M stocks5%13%
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Investment Decision
You manage $1B (OPM) and consider a
decision to transfer $100M to a more risky
investment (stocks).
Your trader claims that on average he can
earn 13% on the risky portfolio instead of 5%
that you have now.
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Investment Decision
Your stockholders have required rate of return on
capital 15%.
1. Calculate VaR before the transaction VaRo=15.
2. Calculate VaR after the transaction VaR1=24.
3. The difference is an additional capital that will be
used to back this transaction:
additional capital = (VaR1- VaR0)*3 = 27M
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Investment Decision
required additional net profit is
Additional Capital * Required rate of return
$27M * 15% = $4.05M
required additional profit before tax is
$4.05M/(1-tax) = $7.4M
this profit should be earned by an extra return on
the risky investment.
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Investment Decision
Thus the required return on the stock portfolio is
$7.4M = (x%-5%)*100M
x = 12.4%
You should accept the proposed transaction.
Zvi WienerIntroduction to Financial
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Tax in Financial Sector
%299.45%)361(17.1
17.0%36
Zvi WienerIntroduction to Financial
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Options in Hi Tech
Many firms give options as a part of
compensation.
There is a vesting period and then there is a
longer time to expiration.
Most employees exercise the options at
vesting with same-day-sale (because of tax).
How this can be improved?
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Long term options
payoff
k K stock
50
K
Sell a call
Your option
Result
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ExampleYou have 10,000 vested options for 10 years
with strike $5, while the stock is traded at $10.
An immediate exercise will give you $50,000
before tax.
Selling a (covered) call with strike $15 will
give you $60,000 now (assuming interest rate
6% and 50% volatility) and additional profit at
the end of the period!
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Example
payoff
10 15 26
50
K
Your option
Result
60
exercise
FRMhttp://pluto.mscc.huji.ac.il/
~mswiener/zvi.html
Bond Market Bootcamp:Handouts
Session One
FRMhttp://pluto.mscc.huji.ac.il/
~mswiener/zvi.html
Bond Market Bootcamp
2001 FRM Certification Review
Session One
Zvi WienerIntroduction to Financial
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Fixed Income Securities
•Definition has evolved to include any security that obligates specific payments at specified dates.
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Overview of Bond Markets
•Bond
•Note
•Money Market Securities
•Sovereign, Agency,Corporate Debentures
•Handout A-1 & A-2, Street Software Inc
Zvi WienerIntroduction to Financial
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Fixed Income Securities
•Overview of major bond markets
•Types of instruments & day counts
•Repo and Securities Lending
•Basic tools of analysis
•Mortgage Backed Securities
•Forward Rate Pricing
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Types of Fixed Income Securities
•Corporate bonds
•Foreign bonds
•Eurobonds
•Mortgage Backed Securities (pass throughs)
•ABS
•Brady Bonds
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World Bond Markets
•Particular focus on differences in nomenclature and conventions; expanded section of FRM in recognition of significant increase in candidates from emerging markets
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UK Government Bonds Gilts• straights = bullet bonds (some callable)
• convertibles (option to holder to convert to longer gilts)
• index linked low coupon 2-2.5%
• irredeemable (perpetual)
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Brady Bonds
Argentina, Brazil, Costa Rica, Dominican Republic, Ecuador, Mexico, Uruguay, Venezuela, Bulgaria, Jordan, Nigeria, Philippines, Poland.
Partially collateralized by US government securities
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Types of Securities MBS & ABS
• Mortgage Loans
• Mortgage Pass-Through Securities
• CMO and Stripped MBS
• ABS
• Bonds with Embedded Options
• Analysis of MBS
• Analysis of Convertible Bonds
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Arbitrage Motivations of ABS
•Direct descendant of zero coupon bonds, replacing rate risk with credit risk
•Necessity for investors to comprehend motivation of arb desk maintaining syndication book of primary issue
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Fixed income Analysis
• Pricing of Bonds
• Yield Conventions
• Bond Price Volatility
• Factors Affecting Yields and the Term Structure of IR
• Treasury and Agency Securities Markets
• Corporates & Municipals
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Types of Fixed Income Securities
• Government securities (sovereign)– Bills (discount)
– Notes
– Bonds (including new index linked)
• Government agency and guaranteed securities– GNMA, SLMA, FNMA
• Municipal Securities– State and local obligations
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Securities Sectors
• Treasury sector: bills, notes, bonds
• Agency sector: debentures (no collateral)
• Municipal sector: tax exempt
• Corporate sector: US and Yankee issues– bonds, notes, structured notes, CP
– investment grade and non-investment grade
• Asset-backed securities sector
• MBS sector
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Fixed Income Universe
•Fixed coupon securities–6.75% UST 3/05
•Floating Rate notes–WB 3/05 T+15
•Zero Coupon Bonds–0% USP 3/05 (or USC)
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Fixed Income Universe
•Perpetual notes (consols in UK)
•Structured notes
•Inverse floaters
•Callable bonds
•Puttable bonds
•Convertible notes
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Characteristics of a Bond
• Issuer
• Time to maturity
• Coupon rate, type and frequency
• Linkage
• Embedded options
• Indentures
• Guarantees or collateral
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Basic security structures
• Coupon, discount and premium bonds• Zero coupon bonds• Floating rate bonds• Inverse floaters• Perpetual notes• Convertible bonds•Interest Only, Principal Only notes•ABS & Structured Products
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Applications
• Active Bond Portfolio Management
• Indexation
• Liability Funding Strategies
• Bond Performance Measurements (AIMR)
• Interest Rate Futures & Options
• Interest Rate Swaps, Caps, Floors
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Analytic Tools to be Reviewed
•Time Value of $
•Yield Conventions
•Pricing Factors for Specific Securities
•Converting Yield Measurements
•Yield Curve Analysis
•Day Counts
•Repo
Zvi WienerIntroduction to Financial
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Analytic Tools to be Reviewed (cont’d)
•Price volatility for option free bonds
•Duration
•Convexity
•Embedded options & their applications
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FRM Cheat Sheet
•The answers are (virtually always):
•Negative convexity
•Effective duration
•SMM
•Double the BEY big figure when quoting Europeans
•Know your current duration ratios by heart
Zvi WienerIntroduction to Financial
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Basic NomenclatureCoupon securities are quoted in terms of price expressed in dollars.
Clean price excludes accrued interest.
Accrued interest =
next coupon*fraction of time that passed.
Bills are quoted in terms of discount rate as % of face value. Assuming 360 days in a year, i.e. multiplied by 360 and divided by the actual number of days remaining to maturity.
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UST Nomenclature
•Clean v. Dirty Pricing
•6.25% UST 5/30 104-12
•Actual/Actual Day Count
•AI=Coupon x actual days since last coupon
actual days in current coupon period
Price 20mm bonds for settlement April 12
Zvi WienerIntroduction to Financial
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UST Pricing Example 1
•8.75 UST 11/08
•Security was purchased 06 Jun @ 110-31
•Security was sold 06 Sep @ 109-27+
•Calculate the loss
Zvi WienerIntroduction to Financial
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UST Pricing Example 1
•Bought at 110-31 11,151,562,50
•Sold at 109-27 11,257,812.50
•Net “loss” is a profit of $106,350.00
•See Handouts 1-1 and 1-2
Zvi WienerIntroduction to Financial
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UST Pricing Example 2
•$3.125 (semi-annual coupon)
•$3.125 x 163 = 2.798763
•($20mm/100) x [(104 + 12/32) + 2.798763 = $21,434,753]
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Discount Nomenclature (T Bills)
•DR = (Face-Price)/Face x(360/t)
•$P = Face x [1-DR x (t/360)]
•$P = $100 x [1-5.19% x (91/360) = $98.6881
•YTM = F/P = (1+y x t/365), or 5.33% for the above 5.19%
Zvi WienerIntroduction to Financial
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Price quotes for T-Bills
360
FtYD d
3601
tYFDFprice d
Zvi WienerIntroduction to Financial
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Price quotes for T-Bills
%75.8100
360
100
569.97100
dY
100 days to maturityprice = $97,569 will be quoted at 8.75%
360
1000875.01000,100$569,97$
Zvi WienerIntroduction to Financial
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FRM 98:13T Bill Calculation
•$100,000 USB 100 days out, 97.569 should be quoted on a bank discount basis at:
•A) 8.75%
•B) 8.87%
•C) 8.97%
•D) 9.09%
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FRM 98:13
• A US T-Bill selling for $97,569 with 100 days to maturity and a face value of $100,000 should be quoted on a bank discount basis at:
• A) 8.75%
• B) 8.87%
• C) 8.97%
• D) 9.09%
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FRM 98:13Bank Discount Rate Question
•DR= (Face-Price)/Face x (360/t)
•($100,000-$97,569)/$100,000 x (360/100)=
•8.75%
•VERY IMPORTANT: NOTE THAT THE YIELD IS 9.09%, WHICH IS HIGHER
Zvi WienerIntroduction to Financial
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Price quotes for T-Bills
The quoted yield is based on the face value and not on the
actual amount invested.
The yield is annualized on 360 days basis.
Bond equivalent yield = CD equivalent yield
d
d
Yt
YyielequivCD
360
360.
%97.8%75.8100360
%75.8360.
yielequivCD
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TIPS
•Index linked government securities
•Pricing key is the “compression factor”, which relates its spread to normal government securities of comparable maturity
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Comparing Yields
bond equivalent yield of Eurodollar bond
= 2[(1+yield to maturity)0.5-1]
for example: A Eurodollar bond with 10% yield has the bond equivalent yield of
2[1.100.5-1] = 9.762%
Eurobond equivalent yield is always greater than UST
Zvi WienerIntroduction to Financial
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Annualizing Yield
Effective annual yield = (1+periodic rate)m-1 examples
Effective annual yield = 1.042-1=8.16%
Effective annual yield = 1.024-1=8.24%
price
ratecouponannualyieldcurrent
Zvi WienerIntroduction to Financial
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The Yield to Maturity
The yield to maturity of a fixed coupon bond y is given by
n
i
ytTi
iectp1
)()(
Zvi WienerIntroduction to Financial
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Embedded Options
•Calls, Puts
•Repricing Features (Inverse Floaters)
•Prepayment Features
•Credit Features
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Callable bond
The buyer of a callable bond has written an option to the issuer to call the bond back.
Rationally this should be done when …
Interest rate fall and the debt issuer can refinance at a lower rate.
Zvi WienerIntroduction to Financial
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Callable Bond
•Long callable bond = long bond + (call)
•Therefore, px of callable bond need be the price of the straight bond – straight call option px (adjusted for credit spread where applicable)
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Puttable bond
The buyer of a such a bond can request the loan to be returned.
The rational strategy is to exercise this option when interest rates are high enough to provide an interesting alternative.
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Putable Bond
Long Bond + Put
PX = Straight Bond + Put Option (adjusted for credit spread as appropriate)
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FRM 00:09Callable Bonds
• An investment in a callable bond can be decomposed into a:
• A) long position in a non-callable bond and short a put
• B) short position in a non-callable bond and long a call
• C) long position in a non-callable bond and long a call
• D) long position in a non-calable bond and short a call
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FRM 00:74Derivatives v. Cash Bonds
• In a market crash, the following are usually true:
• I) fixed income portfolios hedged with short UST and futures lose less than those hedged with interest rate swaps given equivalent durations
• II) bid offer spreads widen due to less liquidity
• III) spread between off the runs and benchmarks widen
• A) all of the above B) II & III
• C) I & III D) None of the above
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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 WienerIntroduction to Financial
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Repurchase Agreements
Borrowing and lending using Treasuries and other debt as collateral.
Repo (loan). You sell a security to counterparty and agree to repurchase the same security at a specified price at a later date (often next day).
Reverse Repo - you agree to purchase a security and sell it back at a specified price later.
Zvi WienerIntroduction to Financial
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Repurchase Agreements
Most repos are general-collateral repo rate.
Some securities are special (for example on-the-run).
Specialness peaks around next auction, then declines sharply.
NY FED operates a securities lending for primary dealers using FED’s portfolio while posting other Treasury security as collateral.
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Repo ExampleYou 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 WienerIntroduction to Financial
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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 WienerIntroduction to Financial
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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 WienerIntroduction to Financial
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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 WienerIntroduction to Financial
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Yield Curve Analysis
•Normal Curve
•Inverted Curve
•Twister
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Yield Curve Analysis
•Par curve
weighted avg of spot ratesweighted avg of spot rates
•Spot Curve
currently priced zero curveurrently priced zero curve
Forward Curve
commence at future datecommence at future date
Zvi WienerIntroduction to Financial
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Handouts 2 & 3
•Illustrations of current swap yield curves for US, UK, Germany and Japan as of 06 Sep 01
•Note inversion
•Note normality
•Note “twister”
•All three types exhibited in Big Four
Zvi WienerIntroduction to Financial
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Forward RatesBuy a two years bond
Buy a one year bond and then use the money to buy another bond (the price can be fixed today).
)1+r2)=(1+r1)(1+f12(
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Forward Rates
(1+r3)=(1+r1)(1+f13)= (1+r1)(1+f12)(1+f13)
Term structure of instantaneous forward rates.
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Time Value of Money
•Future Value
•Discounted Present Value (DPV)
•Internal Rate of Return
•Implications of curve structure on pricing
•Conventional Yield Measurements
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Time Value of Money
• present value PV = CFt/(1+r)t
• Future value FV = CFt(1+r)t
• Net present value NPV = sum of all PV-PV5555105
5
4
1 )1(
105
)1(
5
rrPV
tt
Zvi WienerIntroduction to Financial
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Determinants of the Term Structure
Expectation theory
Market segmentation theory
Liquidity theory
Mathematical models: Ho-Lee, Vasichek,
Hull-White, HJM, etc.
Zvi WienerIntroduction to Financial
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TT
T
tt
t
r
C
r
CPV
)1()1(1
Term structure of interest rates
TT
TT
tt
t
t
r
C
r
CPV
)1()1(1
Yield = IRR
TT
T
tt
t
y
C
y
Cice
)1()1(Pr
1
How do we know that there is a solution?
Zvi WienerIntroduction to Financial
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Parallel shift
T
r
Current UST
Downward move
upward move
Zvi WienerIntroduction to Financial
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Twist
T
r
steepening
flattening
Zvi WienerIntroduction to Financial
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Butterfly
T
r
Zvi WienerIntroduction to Financial
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Do not use yield curve to price bonds
Period A B
1-9 $6 $1
10 $106 $101
They can not be priced by discounting cashflow with the same yield because of different structure of CF.
Use spot rates (yield on zero-coupon Treasuries) instead!
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Hedge Ratios for On the Run Treasuries
•See Handout 4
•Note discrepancies between employing hedge ratios and risk factors -- convexity rearing its ugly head
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Position Duration Management
•See Handouts 5-1 through 5-3
•Hedging the current UST 30 with UST10: The NOB Spread
•Why is this trade “a perfect arbitrage” in any direction of interest rates?
•Why did I note employ the Bond future contract cheapest to deliver?
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FRM 00:95Curve Risk
• Which statement about historic UST yield curve changes is TRUE?
• A) changes in long term yields tend to be larger than short term yields
• B) changes in long term yields tend to approximate those of short term yields
• C) the same size yield change in both long term and short term rates tends to produce a larger price change in short term instruments when securities are trading near par
• The largest part of total return variability of spot rates is due to parallel changes with a smaller portion due to slope changes and the residual due to curvature changes.
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FRM 98:39Yield Curve Analysis
• Which of the following statements about yield curve arbitrage are true?
• A) no arb conditions require that the zero curve is either upward sloping or downward sloping
• B) it is a violation of the no-arb condition if the USB 1 yr rate is 10% or more, higher than the UST 10.
• C) as long as all discounted factors are less than one but greater than zero, the curve is arb free
• D) the no-arb condition requires all forward rates to be non-negative.
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FRM 98:39
•D) discount factors need be below one, as interest rates need be positive (JGB’s ???), but in addition forward rates also need be positive.
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FRM 97:1Yield Curve Arbitrage
• Suppose a risk manager made the mistake of valuing a zero coupon bond using a swap (par) curve rather than a zero curve. Assume the par curve is normal. The risk manager is therefore:
• A) indiffernt to the rate used
• B) over-estimating the value of the security
• C) under-estimating the value of the security
• D) does not have enough information
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FRM 97:1
•B) In a normal interest rate environment, the par curve need always be below the spot curve. As a result, the selected par curve is too law, over-estimating the value of the security.
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FRM 99:1Yield Curve Analysis
• Assume a normal yield curve. Which statement is TRUE?
• A) the forward rate curve is above the zero curve, which is above the coupon-bearing bond curve
• B) the forward rate curve is above the par curve, which is above the zero coupon yield curve
• C) the coupon bearing curve is above the zero coupon curve, which is above the forward rate curve
• D) coupon bearing curve is above the forward curve, which is above the zero curve.
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FRM 99:1
•A) In a normal (upwardly sloping) yield curve, the coupon curve (which is the avg of the spot or zero curve) lies below the zero curve. The forward curve can be interpolated as the spot curve plus the slope of the spot curve, so must be above the spot curve.
FRMhttp://pluto.mscc.huji.ac.il/
~mswiener/zvi.html
Bond Market Bootcamp
2001 FRM Certification Review
Session Two
Zvi WienerIntroduction to Financial
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YTM and Reinvestment Risk
• YTM assumes that all coupon (and
amortizing) payments will be invested at the
same yield.
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YTM and Reinvestment Risk
• An investor has a 5 years horizon
Bond Coupon Maturity YTM
A 5% 3 9.0%
B 6% 20 8.6%
C 11% 15 9.2%
D 8% 5 8.0%
What is the best choice?
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Bond selling at Relationship
Par Coupon rate=current yield=YTM
Discount Coupon rate<current yield<YTM
Premium Coupon rate>current yield>YTM
Yield to call uses the first call as cashflow.
Yield of a portfolio is calculated with the total cashflow.
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FRM 00:6Dollar value of an 01
• A Eurodollar futures contract has a constant PVBP of $25.00 per million. The bank bill contract in Sydney trades on a discount basis and the PVBP is therefore different at each yield level. Assuming positive yields, the PVBP :for the Sydney contract will be
• A) always less than the Eurodollar contract
• B) always greater than the Eurodollar contract
• C) dependent upon market yield
• D) A$27.00 per million
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FRM 99:53Dollar Value of an 01
• Consider a 9% annual coupon 20 year bond trading at 6% with a price of 134.41.When rates rise 10bp, price reduces to 132.99, and when rates drop 10bp, price rises to 135.85.
• What is the modified duration?
• A) 11.25
• B) 10.63
• C) 10.50
• D) 10.73
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FRM 99:53Dollar Value of an 01
•(135.85-132.99)/134.41/[0.001*2] = 10.63
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Volatility and Bond Valuation
•Volatility plays a critical role in theoretical value of bonds with embedded options. Not readily comprehended, this is the concept behind OAS
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Inverse Floater
Is usually created from a fixed rate security.
Floater coupon = LIBOR + 1%
Inverse Floater coupon = 10% - LIBOR
Note that the sum is a fixed rate security.
If LIBOR>10% there is typically a floor.
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FRM 98:3The price of an inverse floater:
•A) increases as interest rates increase
•B) decreases as rates increase
•C) remains constant as rates change
•D) behaves like none of the above
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FRM 98:3Inverse Floater Question
•(B) decreases as rates increase
•As rates increase, the coupon decreases. Additionally, the discount factor increases. Hence the value of the note need decrease even more than a regular fixed income security.
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FRM 98: 3
•Answer is DR = 8.75%
•Yield is 9.09%
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Duration and IR sensitivity
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Understanding of Duration/Convexity
What happens with duration when a coupon is paid?
How does convexity of a callable bond depend on interest rate?
How does convexity of a puttable bond depend on interest rate?
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FRM 98:31Duration
• A 10 year zero coupon bond is callable annually at par, commencing at the beginning of year six. Asssume a flat yield curve of 10%. What is the bond’s duration?
• A) 5 years
• B) 7.5 years
• C) 10 years
• D) cannot be determined from given data
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FRM 98:31Zero Coupon Question
•Trick Question (both Zvi and I got it wrong on first read thru)
•It’s a zero, the bond will never be called because it will never trade above par prior to maturity
•C) regular 10 year duration for a zero
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Duration
nn y
M
y
C
y
C
y
CP
)1()1()1()1( 2
nn y
nM
y
nC
y
C
y
C
P
DurationMacaulay
)1()1()1(
2
)1(
112
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Duration
y
DurationMacaulayDurationModified
1
DurationModifiedPdy
dP 1
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Meaning of Duration
Dpecdy
d
dy
dp n
i
yTi
i
1
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Macaulay Duration
Definition of duration, assuming t=0.
p
ecTD
n
i
yTii
i
1
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Macaulay Duration
T
tt
tT
tt y
CFt
iceBondwtD
11 )1(Pr
1
A weighted sum of times to maturities of each coupon.
What is the duration of a zero coupon bond?
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KEY MISCONCEPTION OF DURATION
Do not think of duration as a measure of time!
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FRM 98:32IO’s and PO’s
• A 10 yr reverse floater pays seminannual coupon of 8% less 6 month LIBOR.Assume the yield curve is 8% flat, the current UST 10 yr has a duration of 7 yrs, and interest on the note was reset today. What is the note’s duration?
• A) 6 mos B) shorter than 7 yrs
• C) longer than 7 yrs D) 7 years
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FRM 00:73Duration
• What assumptions does a duration-based hedging scheme make about interest rate movement?
• A) all interest rates change by the same amount
• B) a small parallel shift in the yield curve
• C) parallel shift in the term structure
• D) rate movements are highly correlated
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Example
Portfolio consists of $1M of a bond with duration of 1 year and $1M worth of a bond with duration of 20 years.
What is the duration of the portfolio?
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Rough calculation
Duration of the first bond is 1 year, of the second bond is 20 years.
This means that when IR go 1% up we will lose 1% of the first bond and 20% of the second.
All together we will lose 10.5% of the portfolio.
The duration is (roughly) 10.5 years.
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FRM 97:49FRN Duration
• A money markets desk holds a floating rate note with an 8 year maturity. The interest rate is floating at 3 mo LIBOR, reset quarterly. The next reset is in one week. What is the security’s duration?
• A) 8 yrs
• B) 4 yrs
• C) 3 months
• D) 1 week
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FRM 97:49Floating Rate Note Question
•(d) duration is not related to maturity when coupons are not fixed for the life of the security. The duration or price risk is only related to the time to the next reset, which is one week.
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Duration
8212.907.1
20
05.1
1
2
1
8212.9
1
07.1
1
05.1
2
1
0
2020
20
1
xxrxrdx
d
dr
dA
ADA
1
BA
BD
BA
ADD BA
BA
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Convexity
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Convexity
368
1
07.1
1
05.1
0
2020
20
12
2
x
xrxrdx
d
2
2
dr
AdCA
For a simple bond portfolio it does not help much!
It is much more important to consider 2 risk factors!
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-0.03 -0.02 -0.01 0.01 0.02 0.03
1.6
1.8
2.2
2.4
2.6
2.8
Value
2
07.01
07.1
05.01
05.1
1
07.1
1
05.120
20
2020
20
1
rr
Parallel shift
value
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FRM 99:40Effective Duration &
Convexity• Which attribute of a bond is NOT a reason for
using effective duration rather than modified duration?
• A) its life may be uncertain
• B) its cash flow may be uncertain
• C) its price volatility tends to decline as maturity approaches
• D) it may include changes in adjustable rate coupons with caps or floors
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FRM 99:40
•C) all attributes are reasons for using effective convexity, except that the price risk decreases as maturity approaches since this would hold for a regular security as well.
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Negative Convexity and Duration
•MBS and particularly I/Os have negative convexity, the result of contraction risk and extension risk
•Accordingly, effective duration and effective convexity need always be computed
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Negative Convexity
•Negative convexity means that the PX appreciation will be less than the price depreciation for a large change in yields:
•BP +C -C
•+100 more than x% less than Y%
•-100x% Y%
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Negative Convexity
–Also address topic of price compression, lack of linearity in PX
–Limited appreciation as yields decline, which is why probability on straight bonds skews towards par in options trading; allusion to necessity to employ yield volatility rather than price volatility (session four)
FRMhttp://pluto.mscc.huji.ac.il/
~mswiener/zvi.html
Bond Market Bootcamp
2001 FRM Certification Review
Session Four
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Bootcamp Warmups
•Closing outstanding value of contracts, in US$ trillions, of OTC contracts as measured by BIS last year: 65, 16, 2
•Which is FX
•Which is equity
•Which is interest rate
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Bootcamp Warmups
•Which exchanges merged to form Euronext?
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Bootcamp Warmups
•What risks would a Euro denominated fund take when investing in the Euronext index?
•Interest rate
•Foreign exchange
•Equity price
•Dividend risk
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Bootcamp Warmups
•Which market is mean-reverting?
•California energy futures
•TA-25 index
•HM T-Bills
•Notes/Bond spread
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Emerging Markets Risk Warmups
• Majority of EM is Latin (over 40% of total EM market)
• Mexican IPC & Brazilian BOVESPA are most important
• Argentine MERVAL is most volatile, and the tail that wags the dog from BOVESPA to Chilean IPSA.
• Now major, liquid contracts in local mkts
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Emerging Markets Warmup: EVT
• Extreme Value Theory adds two magnitudes of risk not otherwise calculated in major markets (until WTC/Pentagon):
• Magnitude of an “X” year return (the norm in Buenos Aires) and
• Excess loss given Value-at-Risk
• It is NOT a scenario analysis per se in the manner of VAR
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Bonds 102
•A quick return and review (Promises, Promises) to duration, convexity and yield curve analysis
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Basis
•Any expression of negative convexity in the relationship of two securities
•Cash/futures (most common, but not exclusive)
•On-the-run/off-the-run
•Deliverables v. Cheapest-to-Deliver
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Duration Revisited
• Influences on duration, in order of importance:
• Coupon
• Frequency of coupon payment
• YTM
• Life at issue (what is the difference in duration between a 20/30 and a 30/40? This is a key concept)
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KEY MISCONCEPTION OF DURATION
Do not think of duration as a measure of time!
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Duration Reconsidered
•Duration is NOT an approximation, it is a first-order derivative
•It’s APPLICATION is an approximation
•This makes it a particularly seductive error
•Duration can reagularly exceed remaining life of a security (inverse floaters, I/Os)
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Duration Reconsidered
•At low yields, prices rise at an increasing rate as yields fall
•At high yields, prices rise at a decreasing rate as yields rise
•Why?? Coupon effect takes over in importance
FRMhttp://pluto.mscc.huji.ac.il/
~mswiener/zvi.html
Bond Market Bootcamp
2001 FRM Certification Review
Session Five
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NPV Warmups
•Arb Desk buys DM 100,000 of new issue John Fairfax HY 9/30/16 step-up note, priced at par, coupons are +2% (annual 30/360).
•Internally assigned cost of capital for HY Eurobonds, 7-15 yrs is 6%.
•Calculate the NPV
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NPV Warmups
•Arb Desk buys 1mm 10% World Bank 10/16 at par
•Internal cost of capital discount rate is 4% ANNUAL for supranationals from 7-15 yrs.
•What is the DPV?
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Duration Warmups
•USC of the 9.375% UST 10/04
•YTM 4.30%
•Calculate the modified duration
•(trick question, there will be one or two of these on each exam testing nomenclature)
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Modified Duration
•3/ (1+.043/2) =
•3/1.0215 =
•2.9368
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Duration Warmups
•5% UST 9/03
•Purchased today @ YTM 4.33%
•Calculate the duration
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Duration Review: Gilts
•Calculate the modified duration of a UKT with a McCauley duration of 7.865 years. Assume rates are 4.75%
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Duration on Gilts
•Don’t panic, the US used to be a colony even if Ben Franklin insisted we switch traffic flows to the French side of the road in 1776 as a sign of independence.
•7.865/1+.0475/2=
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Convexity Reconsidered
•Most critical (potentially flawed) assumption when calculating convexity is its reliance on YTM and therefore a flat yield curve
•Tattoo this onto the top of your Bloomberg before performing quick-and-dirty hedges
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Implications for Yield Curve Analysis
•Forward rate curve requires that all yield curve interpolation be done in a “steps” manner rather than simple linear curve smoothing
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FRM 98:50Leverage Factors
•Hedge fund invests $100mm by a factor of 3 in HY bonds yielding 14% at an average borrowing cost of 8%. What is its yearend return on capital?
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FRM 98:50
•Fund borrows 200mm and invests 300m.
•300 x 0.14 = $42mm
•200 x 0.08 = $16mm
•Net profits are $26mm on $100mm, or 26%.
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BONDS WITH EMBEDDED OPTIONS
•Convertibles
•Mortgage Backeds (first generation)
•I/Os and P/Os
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Convertibles
•Q in class last week: why are there calls?
•A: forces conversion, as convertibles are generally highly advantageously priced for the issuing entity, not the option holder
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Convertibles
•Bond is convertible at 40, redemption call at 106
•Bond trades at 115, stock is at 45
•A) sell the bond
•B) convert & sell equity
•C) await the call at 106
•D) do nothing and earn the coupon
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Convertibles
•$1000 (face value is always $1000)/40= 25 shares
•25 shares @45=$1,125
•Bond may be sold at higher price than convertible value
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Convertibles
•ABS issue: we locate a convertible priced very attractively post WTC/Pentagon, but cannot maintain the credit name on a term basis
•Hedge the risk
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Convertible Hedge
•Requires an asset swap to maintain investment structure yet modify underlying credit to an acceptable name and tenure.
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FRM 98:34
•A 3 yr convertible paying 4% p.a. priced at par, right to conversion ratio of 10 @ $75, forced conversion at maturity. Convexity relative to underlying equity is:
•a) zero b) always positive
•c) always negative d) none of the above
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FRM 98:34
•B) as the convertible includes a warrant ( a call option on the underlying stock) its convexity must trade positive relative to the underlying equity. This is what the purchaser paid premium to receive.
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FRM 97:52Convertible Risk
•Trader purchases convertible with call provision. Assuming a 50% conversion risk, which combination of stock price and interest rates would constitute “a perfect storm”?
•a) lower rates, lower equity prices
•b) lower rates, higher equity prices
•c) higher rates, lower equity prices
•d) higher rates, higher equity prices
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Convertible RiskFRM 97:52
•C) value of the fixed rate bond will fall as rates increase, value of the embedded warrant will fall as equity prices decline.
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FRM 98:9Equity Indeces
•To prevent arbitrage, the theoretical price of a stock index need be fully determined via:
•I) cash price II) financing cost
•III) inflation IV) dividend yield
•
•a) I & II b) II & III
•c) I, II & IV d) all of the above
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Equity IndecesFRM 98:9
•While embedded in the underlying nominal interest rate of the futures, inflationis not a direct calculation of any futures index.
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IO/PO Key Concepts
•I/O+ P/O must equal the MBS
•IO’s are bullish securities with negative duration.
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Duration and I/O’s P/O’s
•Five year note dollar duration is:
•$50m x DF + $50m x D1F = $100m x D
•Duration of inverse floater must be:
•D1F = ($100m/50m) x D = 2 x D
•Or twice that of the original note
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FRM 99:79IO’s and PO’s
• Suppose the coupon and modified duration of a 10 yr note priced to par is 6% and 7.5, respectively. What is the approximate modified duration of a 10 yr inverse floater priced to par with a coupon of (18%-2x 1m LIBOR)?
• A) 7.5 B) 15.0
• C) 22.5 D) 0.0
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FRM 99:79
•C) following the same reasoning, we must divide the fixed rate bonds into 2/3 FRN and 1/3 inverse floater. This will ensure that the inverse floater payment is related to twice LIBOR. As a result, the duration of the inverse floater must be 3x the bond.
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Mortgage Backed Securities: Conceptual Review
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Fixed Rate Mortgage
A series of equal payments with PV=loan.
Example: 100,000 for 20 years with 6% and equal monthly payments.
20*12
1
1206.0
1
000,100i
i
x
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Adjustable-Rate Mortgage (ARM)
The contract rate is reset periodically, based on a short term interest rate.
Adjustment from one month to several years.
Spread is fixed, some have caps or floors.
Market based rates.
Rates based on cost of funds for thrifts.
Initially low rate is often offered = teaser rate.
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Balloon Mortgage
One payment at the end.
Sometimes they have renegotiation points.
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Prepayments
Prevailing mortgage rate relative to original.
Path of mortgage rates.
Level of mortgage rates.
Seasonal factors (home buying is high in
spring summer and low in fall, winter).
General economic activity.
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Prepayments
Prepayment speed, conditional prepayment rate CPR (prepayment rate assumed for a pool).
Single-Monthly mortality rate SMM.
SMM = 1 - (1-CPR)1/12
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PSA prepayment benchmarkThe Public Securities Association benchmark is
expressed as monthly series of annual prepayment
rates.
Low prepayment rates of new loans and higher for
old ones.
Assumes CPR increasing 0.2% to 6% with life of a
loan.
Actual rate is expressed as % of PSA.
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PSA standard default assumptions
0 30 60 120Age in months
Annual default rate (SDA) in%
0.3
0.6Month 1 - 0.02%
increases by 0.02% till 30mstable at 0.6% 30-60m
declines by 0.01% 61-120mremains at 0.03% after 120m
0.02
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100 PSA
030Age in months
Annual CPR in%
0.2
6
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Prepayments
A general model should be based on a dynamic transition matrix, very similar to credit migration.
But note the difference of a pool of not completely rational customers and a single firm.
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Example of prepayments
Example: let CPR=6%, then
SMM = 1-(1-0.06)1/12 = 0.005143.
An SMM of 0.5143% means that approximately 0.5% of the mortgage balance will be prepaid this month.
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Example of prepayments
If the balance at the beginning of a month is $290M, SMM = 0.5143% and the scheduled principal payment is $3M, then the estimated repayment for this month is
0.005143 (290,000,000-3,000,000)=$1,476,041
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FRM 99:44Prepayment Risk
• The following are reasons why a prepayment model will not accurately predict future mortgage prepayments. Which of these will have the greatest effect on convexity of mortgage pass throughs?
• A) refinancing incentive
• B) seasoning
• C) refinancing burnout
• D) seasonality
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FRM 99:44MBS Prepayments
•A) the factor influencing most the decision to repay early (the embedded option) is, from this list, refinancing incentives
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Prepayment Risk and Convexity
Negative convexity - if interest rates go up
the price of a pass through security will
decline more than a government bond due to
lower prepayment rate.
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FRM 99:51CPR to SMM Conversion
• Suppose the annual prepayment rate CPR for a mortgage backed security is 6%. What is its corresponding single-monthly mortality (SMM) rate?
• A) 0.514%
• B) 0.334%
• C) 0.5%
• D) 1.355%
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FRM 99:51Convert CPR to SMM
•(A) 0.51%
•(1-6%)=(1-SMM) 12
•SMM = 0.51%
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MBS Bond Equivalent YieldBond equivalent yield = 2[ (1+yM)6 - 1]
Yield is based on prepayment assumptions and must
be checked!
PSA benchmark = Public Securities Association.
Assumes low prepayment rates for new mortgages,
and higher rates for seasoned loans.
FRMhttp://pluto.mscc.huji.ac.il/
~mswiener/zvi.html
Bond Market Bootcamp
2001 FRM Certification Review
Session Six
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Options 102
•Review of Basic Concepts & Their Applications
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Options 101
•Never forget the fact that lognormal distributions are positively skewed
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Options 101
•Delta
•Gamma
•Vega
•Theta
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Compound Option Risks
• Option risk compounds with each layer of additional risk embedded in position. Therefore, while all recognize the risk of a short gamma position, for example, consider the additional incremental risk involved in whether this was established at a delta neutral or short/long delta price. The delta risk can easily exceed the originally accepted embedded (and priced) gamma risk if not properly hedged.
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Compound Option Example
•Arb desk owns 1mm shares of GE at 50
•Writes 3mm ATM calls exp 12/01 at 40% volatility post-WTC/Pentagon madness to capitalize on spike in volatility
•Calculate the delta hedge
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FRM 97:49
• An option strategy exhibits unfavorable sensitivity to increases in implied volatility while experiencing significant daily time decay. The portfolio may be hedged by:
• A) selling short-dated options & buying longer-term options
• B) buying short-dated options & selling longer-term options
• C) selling both periods D) buying both periods
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Options 102
•Continuously rebalancing an options portfolio to small change in delta is called dynamic hedging
•Because of its transaction costs, it is virtually never profitable long term from the short side
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Options 102
•Butterfly strategies are employed in very stable markets precisely as means of capitalizing on dynamic hedging from the long side
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Options 102
•From the short side, a condor would accomplish a similar objective, except it would be expressed as vega positive while remaining delta neutral
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Options 102
•European v. American
•Model implications
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American Options
•May be exercised at any time to maturity
•Accordingly, on equity options, early exercise of an American option on a non-dividend paying stock can never be optimal strategy.
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Core Concept: Options
•An American call option on a non-dividend paying stock (or asset with no income) should never be exercised early. If the asset pays income, there is a possibility of early exercise, which increases with the size of the income payments.
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Options 102
•Discrete time models will make a stochastic path into steps, thereby eliminating intraperiod volatility – this will ALWAYS make them value volatility (and therefore options) cheaper than continuous time models
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Put Call Parity (and other myths of mathematics)
•Highly problematic when applied to american options
•Very problematic when applied to volatility smiles (let alone emerging market “smirks”)
•Even in European options not necessarily valid
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FRM 99:35Put Call Parity
• According to put call parity, writing a put is equivalent to:
• a) buying a call,buying stovk and lending on repo
• b) writing a call, buying stock and borrowing
• c) writing a call, buying stock and lending
• d) writing a call, selling stock and borrowing.
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FRM 99:35Put Call Parity Theory
•B) a short put position is equivalent to a long asset position plus a short call. To finance the purchase, we need borrow, as the value of the options is minute relative to the value of the underlying asset.
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Convexity Adjustment
•Because interest rate futures are more highly correlated with the underlying reinvestment rate, profits would be reinvested at a different rate than calculated in a static forward price. To offset this advantage, futures are priced above the forward price. This becomes increasingly significant in longer maturity contracts.
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Equity Options
•Key concept: focus upon whether option model requires inclusion or exclusion on dividend
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Dividend Paying Stocks
•In EUROPEAN options, the stock price need be recalculated in pricing the option by first deducting the discounted dividend. The fact that the dividend comes later need be accounted for in the option price. This is a critical and frequent error in calculation.
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Options 102:Nomenclature Question
•Buy 1 43P @ $6
•Sell 2 37P @ $4
•Buy 1 32P @ $1
•Stock expires at $19
•Calculate P/L
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Options 102: Nomenclature
•(43-19) + (2 *-37+19)+(32-19) + (-6+4+4-1) =
•2 per share
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Options 102
•180 day Call option, strike price @ 50
•Current price 55, option price 5
•What underlying instrument are we pricing?
•A) Eurodollar futures
•B) DAX equity index
•C) JGBs
•D) KOSPI 200 Index
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Options 102:Bonds
•When related to fixed income, a model must accommodate mean-reversion in calculating stochastic behavior, a concept rarely considered in equity options
•This is the BS model demise in fixed income
•Recall the warmup question
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Options 102:“Myron, the Damned Model
Doesn’t Work”
• BS does not account for transaction costs, occassionally the most significant factor in emerging markets, less liquid markets, or three standard deviation events like WTC/Pentagon
• At an extreme, the BS assumes the option is a tradable instrument (how does FIBI price 83% of open interest in index options, in but only one EM example…cost TF Bank the ranch in identical trade in 1998, and Ulusal/Demir this year
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Options 102:“Myron, you still deserve that
damned Nobel”
•prices embedded options very simply and accurately
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BS Assumptions
•Price of the underlying asset moves in a continuous fashion
•interest rates are known and constant
•variance of returns is constant
•perfect liquidity and transaction capabilities (not simply liquidity, also short sales, taxes, etc)
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Stupid Dog Tricks You Need Comprehend but Not Calculate
•Martingales are the quant soup-du-jour solution, as they represent a zer-drift stochastic process
•Beware bespeckled thirtysomethings bearing Martingale solutions
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Backwardation/Negative Price Options
•Any option on an underlying instrument that can go/regularly goes negative in price (long terms WTI Crude, JGBs, waste and environment) MUST employ an arithmetic Brownian motion
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Options on Index Securities
•Options relate to the INDEX, not the underlying intent – think of Israeli mortgages or TIPS. The options need relate to the INDEX, not the true inflation rate.
•Difference is but another example of basis, and the risk would be another example of negative convexity.
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Options 102
•Rank delta, gamma, vega, theta, rho as risks for the following options:
•Deep ITM 5 days to expiration
•ATM 180 days to expiration
•Slightly OTM LEAPs
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Options 102
•We own a swaption on 10 year Yen LIBOR to 3% annual swap
•Trader hedged by shorting JGB 83s (couldn’t resist a JGB example in Tel Aviv as payback for the Shachars and Gilboas)
•What risks are hedged, what risks remain?
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Options 102
•Volatility risk remains as primary risk
•Basis risk remains, and has added another demension
•Interest rate reduced, but not curve risk
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Options 102:Bermuda Triangles
•Any option with a discontinuous payoff function necessitates an exceedingly high gamma near the strike price
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Options 102
•Therefore, any such option: Asiatic, Bermudan will require a specific model loosely based upon but VERY different than a traditional BS
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FRM 99:34
•What is the lower pricing bound for an ITM European call option, strike at 80, current price 90, expiration one year? GB 12m is 5%.
•A) 14.61 B) 13.90
•C) 10.00 D) 5.90
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FRM 99:34
•$90-[$80 (-0.05 x 1)] = 90-76.10=$13.90
•pricing a simple European option, and likely one American to comprehend the difference, is a guaranteed set of questions on any FRM exam. Like reading a bond quote, one cannot walk around with 3 letters after their name without this capability mastered.
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FRM 99:52American option pricing question• Price of an American equity call option equals an
otherwise equivalent European option at time t when:
• I) stock pays continuous dividends from t to option expiration T.
• II) interest rates are mean reverting from t to T.
• III) stock pays no dividends from t to T.
• IV) interest rates are non-stochastic between t and T.
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FRM 99:52
•B) an American call option will not be exercised early when there is no income payment on the underlying asset.
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FRM 98:58Options on Futures
•Which statement is true regarding options on futures?
•A) an American call equals a European call
•B) an American put equals a European put
•C) put/call parity holds for both European & American options
•D) none of the above
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FRM 98:58
•D) futures have an implied income stream equal to the risk free rate. As a result, both sets of calls may be exercised early as distinct from “normal” American call options. Similarly, American puts would certainly be likely exercised early, dismissing laws of put/call parity.
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Options Exotica
•Binary & digital options
•Barriers (knock-in, knock-out)
•Down & Out, Down & In, Up & Out, Up & In
•Asian options, or average rate options
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Options ExoticaFRM 98:4
•A knock-in barrier option is harder to hedge when it is:
•a) ITM
•b) OTM
•c) at the barrier and near maturity
•d) at the barrier and at trade inception
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FRM 98:4
•Discontinuous are harder to price at barrier with little time remaining.
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FRM 97:10
•Knock out options are often employed rather than regular options because:
•a) they have lower volatility
•b) they have lower premium
•c) they have a shorter average maturity
•d) they have a smaller gamma
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FRM 97:10
•Knockouts are no different from regular options in terms of maturity or underlying volatility, but are much cheaper than equivalent European options since they involve a much lower probability of exercise.
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Swaps, FX, Caps & Collars
•Review of Nomenclature and Core Concepts
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Core Concepts: Swaps
•A position receiving a fixed rate swap is eqivalent to a long position in bond with similar coupon and maturity characteristics offset by a short position in an FRN. Its duration is equivalent to the fixed rate note, adjusted for the near coupon of the floater.
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FRM 99:42Swaps
•Client may either issue a fixed rate bond or an FRN with an interest rate swap. To achieve this, client should:
•a) issue FRN of same maturity and enter IRS paying fixed/receiving float
•b) issue FRN and enter IRS paying float/receiving fixed
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FRM 99:42Swaps
•A) receiving float on the swap will offset payments on the FRN and leave a net fixed income obligation, presumably at a lower cost to issuer.
•Why would this make sense for Israeli issuers in general?
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FRM 99:59Swap Convexity
•If an interest rate swap is priced off the Eurodollar futures strip curve without correcting the rates for convexity, the resulting arbitrage may be exploited by an:
•a) receive fixed swap + short ED position
•b) pay fixed + short ED position
•c) receive fixed + long ED position
•d) pay fixed + long ED position
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FRM 99:59
•A) futures rate need be corrected downward to forward rate; otherwise too high a fixed rate is implied. The arb would be closed by shorting ED futures and “rolling the thunder” until futures and forwards price consistently closer to maturity.
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FRA & Forward Pricing
•6x9 FRA $10mm 4.25% LIBOR 30/360
•settles at 4.85%
•calculate P/L
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FRA Pricing Example One
•(1mm x 0.425) = 42,500 x 90/360 = 10,625
•(1mm x 0.485) = 48,500 x .25 = 12,125
•12,125-10,625 = 1,527.00
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FRA Pricing Example One
•$1,527 was, of course, wrong
•We neglected to discount for the forward rate
•1527/[1+(.0485 * .25)] = 1,508.71
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FRA Pricing Example Two
•6x9 FRA 4.25% LIBOR, 30/360 daycount
•settles at 3.95% on $10mm
•calculate the P/L
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FRA Pricing Example Two
•$10mm @ 4.25 * .25= $106,250
•$10mm @ 3.95 * .25= $ 98,750
•$7500/ 1.009875 = $7,426.66
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FX Swaps
•$25mm 4% fixed for GBP 17mm 5% fixed
•18 months tenor
•1 GBP = $1.4775
•present rates are US LIBOR 3% for 180, 3.5% for 365 days
•present rates are UK LIBOR 4.0% for 180, 4.5% for 365 days
•Calculate the swap
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US $ cash flows
•25mm 4% each coupon is 1mm
•6m DPV@3% = 985,000
•12mDPV@3.5%= 965,000
•18mDPV@4%= 940,000
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GBP cash flows
•GBP17mm 5% coupon = GBP 850,000
•6m DPV @ 4% = 828,750
•12mDPV@4.5% = 811,750
•18mDPV@5% = 786,250
•convert cumulative GBP cash flows @ 1.4775
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FX Swaps Example
•6m cash flow: $985,000-951,197.81 =$33,802.19
•12m c/f: $965,000-931,686.86 =$33,313.14
•18m c/f: $940,000-902,418.43 =37,581.57
•total p/l adjustment = $ 104,696.90
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FX Swaps Example Two
•Estimate the forward rate for 6 month Eur/$.
•US$ LIBOR is 3%, Eur is 4%.
•Eur/$ is 0.9100 spot
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•0.9100 ( -.01/2) = -0.00455
•forward FX rate of 0.90545
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Caps & Floors
•Caps are simply call options on interest rates, usually written to FRN issuers to provide a maximum cost of borrowing
•Floors are simply put options on rates.
•Collars are a combination of caps/floors locking in a predefined range of potential interest rates.
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FRM 99:54
•Cap/Floor parity can be stated as:
•a) short cap + long floor = fixed rate bond
•b) long cap + short floor = fixed swap
•c) long cap + short floor = FRN
•d) short cap + short floor = collar
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FRM 99:54
•A) with same strike price, a short cap/long floor loses money if rates increase which is equivalent to a fixed rate bond position
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FRM 99:60Cap Risk
• For a 5 yr ATM cap on LIBOR, what can be said about the individual caplets in a downward sloping term structure?
• A) short mturity caplets are ITM,longer are OTM
• b) longer maturities are ITM, longer are OTM
• c) all are ATM
• d) The moneyness of the caplets is also dependent upon the volatility term structure.
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SwaptionsFRM 97:18
•The price of an option to receive fixed on a swap will decrease as:
•a) time to expiry of the option increases
•b) time to expiry of the swap increases
•c) swap rate rises
•d) volatility increases
•(think, this is a bit of a trick question)
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FRM 97:18Swaptions
•C) value of the call increases with the maturity of the call and underlying asset value. In contrast, the value of the right to receive an asset at K decreases as K increases.
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FRM 99:60
•A) in an inverted interest rate environment, forwards are higher for short rates. As caplets involve the right to buy a series of FRN options stuck at the same fixed strike price, short dates will be ITM and longer dates OTM for an ATM cap.
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FX 101Core Concept
• receiving an FX swap is equivalent to a long position in a foreign pay bond and a short position in a dollar pay bond
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FX 101
•$:JPY 120
•12 month Libor is 6%
•12 month Yen rates are 1%
•Calculate the forward
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FX Options 102
•Therefore, in FX options the BS model need be ammended to include consideration of the all-important foreign interest rate
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Garman Kohlhagen Model
•Applied BS to FX, employing the foreign interest rate as the yield in the original BS model. In short, any income payment is automatically reinvested in the asset.
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Physical & Precious Commodities
•Beans in the Teens, or why there are still no Nice Jewish Boys in the Pork Bellies pit
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Commodities
•If a commodity is more expensive for immediate delivery than for future delivery, the commodity curve is said to be in backwardation. As distinct from interest rate curves where interest accrues with the normal passage of time, backwardation is the “normal” term structure for almost all physical storage and delivery commodities.
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Backwardation Aspect
•GSCI is useful in that it provides an accurate market value of the major commodity indeces adjusted for their storage costs at present market conditions
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“Beans in the Teens”
•agricultural futures: grains (corns, wheat, soybeans) and food & fiber (cocoa,coffee,sugar, OJ…think Eddie Murphy)
•livestock: cattle, hogs, pork bellies
•trade with inflation rather than against inflation (such as financial assets)
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Metals
•Base metals: aluminum, copper, nickel, zinc
•Precious: gold, silver, platinum
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Convenience Yield in Physicals
•Forward prices are only at a discount v. spot prices in backwardation markets.
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FRM 99:32
•Spot April Corn is 207/bushel. CNU1 is 241.50. What statement is true about the expected spot price in September?
•A) higher than 207 B) lower than 207
•C) higher than 241.50 D) lower than 241.50
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Gold Pricing
•Spot gold $288/ounce
•12m LIBOR (continuou compounded) is 5.73%
•storage costs are $2/ounce p.a.
•Price the one year forward
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Gold Pricing
•290 (1.0573) = 307.10
•price rose rather than dropped, as forward purchaser need not incur the storage costs for the year
•a reverse dividend, conceptually
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Energy Products
•Natural gas, heating oil, WTI unleaded gasoline, crude oil
•electricity (California/Oregon Border)
•Weather & Waste
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FRM 98:27Metallgesellscaft AG
• MG Trading’s losses were the direct result of employing an interest rate type strategy of “stack-and-roll”. This hedge involved:
• a) buying short dates futures to hedge long term exposures, expecting long term oil to rise
• b) buying short dated futures to hedge long term exposure, expecting long term oil prices to decline
• c) selling short dates futures, expecting short term prices to rise
• d) selling short term futures, expecting short term prices to decline
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FRM 98:27MG Trading AG
• A) MG hedged sales of oil forward by buying short term oil contracts on a “duration ratio basis”. In theory, price declines in one should have offset the other. Compound Hedging Error: They got convexity wrong as well. In futures markets, however, losses were realized immediately, which led to significant liquidity problems even if the long term price of oil had remained constant (which, of course, it did not).