Derivatives: A Primer on Bonds

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Derivatives A Primer on Bonds

• First Part Fixed Income Securities
• Bond Prices and Yields
• Term Structure of Interest Rates
• Second Part TSOIR
• Term Structure of Interest Rates
• Interest Rate Risk Bond Portfolio Management

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Bond Prices and Yields

• Time value of money and bond pricing
• Time to maturity and risk
• Yield to maturity
• vs. yield to call
• vs. realized compound yield
• Determinants of YTM
• risk, maturity, holding period, etc.

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Bond Pricing

• Equation
• P PV(annuity) PV(final payment)
• Example Ct 40 Par 1,000 disc. rate 4
  T60

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Prices vs. Yields

• P ? ? yield ?
• intuition
• convexity
• BKM6 Fig. 14.3 BKM4 Fig. 14.6
• intuition yield ? ? P ? ? price impact ?

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Measuring Rates of Return on Bonds

• Standard measure YTM
• Problems
• callable bonds YTM vs. yield to call
• default risk YTM vs. yield to expected default
• reinvestment rate of coupons
• YTM vs. realized compound yield
• Determinants of the YTM
• risk, maturity, holding period, etc.

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Measuring Rates of Return on Bonds 2

• Yield To Maturity
• definition
• discount rate such that NPV0
• interpretation
• (geometric) average return to maturity
• Example Ct 40 Par 1,000 T60 sells at
  par

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Measuring Rates of Return on Bonds 3

• Yield To Call
• definition
• discount rate s.t. NPV0, with TC earliest call
  date
• deep discount bonds vs. premium bonds
• BKM6 Fig. 14.4 BKM4 Fig. 14.7
• Example Ct 40, semi Par 900 T60 P
  1,025 callable in 10 years
  (TC20), call price 1,000

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Measuring Rates of Return on Bonds 4

• Yield To Default
• definition
• discount rate s.t. NPV0, with TD expected
  default date
• default premium and business cycle
• economic difficulties and flight to quality
• Example Ct 50, semi Par 1,000 T10 P
  200 expected to default in 2 years
  (TC4), recover 150

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Measuring Rates of Return on Bonds 5

• Coupon reinvestment rate
• YTM assumption average
• problem not often true
• solution realized compound yield
• forecast future reinvestment rates
• compute future value (BKM6 Fig.14.5 BKM4
  Fig.14.9)
• compute the yield (rcy) such that NPV 0
• practical?
• need to forecast reinvestment rates

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Bond Prices over Time

• Discount bonds vs. premium bonds
• coupon rate lt market interest rates
• ? built-in capital gain (discount bond)
• coupon rate gt market interest rates
• ? built-in capital loss (premium bond)
• Behavior of prices over time
• BKM6 Fig. 14.6 BKM4 Fig. 14.10
• Tax treatment
• capital gains vs. interest income

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Discount Bonds

• OID vs. par bonds
• original issue discount (OID) bonds
• less common
• coupon need not be 0
• par bonds
• most common
• Zeroes
• what? mostly Treasury strips
• how? certificates of accrual, growth
  receipts, ...
• annual price increase 1-year disc. factor
  (BKM6 Fig. 14.7 BKM4 Fig. 14.11)

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OID tax treatment -- Discount Bonds 2

• Idea for zeroes
• built-in appreciation implicit interest
  schedule
• tax the schedule as interest, yearly
• tax the remaining price change as capital gain or
  loss
• Other OID bonds
• same idea
• taxable interest coupon computed schedule

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OID tax treatment -- Discount Bonds 3

• Example
• 30-year zero issued at 57.31 Par 1,000
• compute YTM
• 1st year taxable interest

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OID tax treatment -- Discount Bonds 4

• Example (continued)
• interests on 30-year bonds fall to 9.9
• capital gain
• tax treatment taxable interest 5.73 capital
  gain

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Term Structure of Interest Rates

• Basic question
• link between YTM and maturity
• Bootstrapping short rates from strips
• forward rates and expected future short rates
• Recovering short rates from coupon bonds
• Interpreting the term structure
• does the term structure contain information?
• certainty vs. uncertainty

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Terminology

• Term structure yield curve (BKM6 Fig. 15.1)
• plot of the YTM as a function of bond maturity
• plot of the spot rate by time-to-maturity
• Short rate vs. spot rate
• 1-period rate vs. multi-period yield
• spot rate current rate appropriate to
  discount a cash-flow of a given maturity
• BKM6 Figure 15.3 BKM4 Figure 14.3

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Extracting Info reShort Interest Rates

• From zeroes
• non-linear regression analysis
• bootstrapping
• From coupon bonds
• system of equations
• regression analysis (no measurement errors)
• Certainty vs. uncertainty
• forward rate vs. expected future (spot) short rate

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Bootstrapping Fwd Rates from Zeroes

• Forward rate
• break-even rate BKM Fig. 15.4
• equates the payoffs of roll-over and LT
  strategies
• Uncertainty
• no guarantee that forward expected future spot
• General formula
• f1 YTM1 and

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Bootstrapping Fwd from Zeroes 2

• Data
• BKM Table 15.2 Fig. 15.1
• 4 bonds, all zeroes (reimbursable at par of
  1,000)
• T Price YTM
• 1 925.93 8
• 2 841.75 8.995
• 3 758.33 9.66
• 4 683.18 9.993

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Bootstrapping Fwd Rates from Zeroes 3

• Forward interest rate for year 1
• Forward interest rate for year 2

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Bootstrapping Fwd Rates from Zeroes 4

• Short rate for years 3 and 4
• keep applying the method
• you find f3 11 f4
• General Formula
• f1 YTM1

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Yield, Maturity and Period Return

• Data
• 2 bonds, both zeroes (reimbursable at par of
  1,000)
• T Price YTM
• 1 925.93 8
• 2 841.75 8.995
• Question
• investor has 1-period horizon no uncertainty
• does bond 2 (higher YTM) dominate bond 1?

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Yield, Maturity and Period Return 2

• Answer Nope
• Bond 1 HPR
• Bond 2 HPR
• f2 10
• price in 1 year Par/(1 f2) 909.09
• capital gain at year-1 end

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Fwd Rate Expected Future Short Rate

• Interpreting the term structure
• Short perspective
• liquidity preference theory (investors)
• liquidity premium theory (issuer)
• Expectations hypothesis
• Long perspective
• Market Segmentation vs. Preferred Habitat
• Examples

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Fwd Rate Exp. Future Short Rate 2

• Short perspective
• liquidity preference theory (short investors)
• investors need to be induced to buy LT securities
• example 1-year zero at 8 vs. 2-year zero at
  8.995
• liquidity premium theory (issuer)
• issuers prefer to lock in interest rates
• f2 ? Er2
• f2 Er2 risk premium

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Fwd Rate Exp. Future Short Rate 3

• Long perspective
• long investors wish to lock in rates
• roll over a 1-year zero at 8
• or lock in via a 2-year zero at 8.995
• Er2 ? f2
• f2 Er2 - risk premium

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Fwd Rate Exp. Future Short Rate 4

• Expectation Hypothesis
• risk premium 0 and Er2 f2
• idea arbitrage
• Market segmentation theory
• idea clienteles
• ST and LT bonds are not substitutes
• reasonable?
• Preferred Habitat Theory
• investors do prefer some maturities
• temptations exist

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Fwd Rate Exp. Future Short Rate 5

• In practice
• liquidity preference preferred habitat
• hypotheses have the edge
• Example
• BKM Fig. 15.5

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Fwd Rate Exp. Future Short Rate 6

• Example 2
• short term rates r1 r2 r3 10
• liquidity premium constant 1 per year
• YTM

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Measurement Zeroes vs. Coupon Bonds

• Zeroes
• ideal
• lack of data may exist (need zeroes for all
  maturities)
• Coupon Bonds
• plentiful
• coupons and their reinvestment
• low coupon rate vs. high coupon rate
• short term rates -gt they may have different YTM

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Short Rates, Coupons and YTM

• Example
• short rates are 8 10 for years 1 2
  certainty
• 2-year bonds Par 1,000 coupon 3 or 12
• Bond 1
• Bond 2

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Measurements with Coupon Bonds 2

• Example
• 2-year bonds Par 1,000 coupon 3 or 12
• Prices 894.78 (coupon 3) 1,053.87 (coupon
  12)
• Year-1 and Year-2 short rates
• 894.78 d1 x 30 d2 x 1,030
• 1,053.87 d1 x 120 d2 x 1,120
• Solve the system d2 0.8417, d1 0.9259
• Conclude ...

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Measurements with Coupon Bonds 3

• Example (continued)

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Measurements with Coupon Bonds 4

• Practical problems
• pricing errors
• taxes
• are investors homogenous?
• investors can sell bonds prior to maturity
• bonds can be called, put or converted
• prices quotes can be stale
• market liquidity
• Estimation
• statistical approach

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Rising yield curves

• Causes
• either short rates are expected to climb Ern ?
  Ern-1
• or the liquidity premium is positive
• Fig. 15.5a
• Interpretative assumptions
• estimate the liquidity premium
• assume the liquidity premium is constant
• empirical evidence
• liquidity premium is not constant past -gt
  future?!

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Inverted yield curve

• Easy interpretation
• if there is a liquidity premium
• then inversion ? expectations of falling short
  rates
• why would interest rates fall?
• inflation vs. real rates
• inverted curve ? recession?
• Example
• current yield curve The Economist

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Arbitrage Strategies
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Arbitrage Strategies
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Fixed Income Portfolio Management

• In general
• bonds are securities just like other
• -gt use the CAPM
• Bond Index Funds
• Immunization
• net worth immunization
• contingent immunization

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Bond Index Funds

• Idea
• US indices
• Solomon Bros. Broad Investment Grade (BIG)
• Lehman Bros. Aggregate
• Merrill Lynch Domestic Master
• composition
• government, corporate, mortgage, Yankee
• bond maturities more than 1 year
• Canada ScotiaMcLeod (esp. Universe Index)

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Bond Index Funds 2

• Problems
• lots of securities in each index
• portfolio rebalancing
• market liquidity
• bonds are dropped (maturities, calls, defaults, )

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Bond Index Funds 3

• Solution
• cellular approach
• idea
• classify by maturity/risk/category/
• compute percentages in each cell
• match portfolio weights
• effectiveness
• average absolute tracking error 2 to 16 b.p. /
  month

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Special risks for bond portfolios

• cash-flow risk
• call, default, sinking funds, early repayments,
• solution select high quality bonds
• interest rate risk
• bond prices are sensitive to YTM
• solution
• measure interest rate risk
• immunize

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Interest Rate Risk

• Equation
• P PV(annuity) PV(final payment)
• Yield sensitivity of bond Prices
• P ? ? yield ?
• Measure?

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Interest Rate Risk 2

• Determinants of a bonds yield sensitivity
• time to maturity
• maturity ? ? sensitivity ? (concave function)
• coupon rate
• coupon ? ? sensitivity ?
• discount bond vs. premium bond
• zeroes have the highest sensitivity
• intuition coupon bonds average of zeroes
• YTM
• initial YTM ? ? sensitivity ?

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Duration

• Idea
• maturity ? ? sensitivity ?
• ? to measure a bonds yield sensitivity,
• measure its effective maturity
• Measure
• Macaulay duration

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Duration 2

• Duration effective measure of elasticity
• Proof
• Modified duration with

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Duration 4

• Interpretation 1
• average time until bond payment
• Interpretation 2
• price change of coupon bond of a given
  duration
• price change of zero with maturity to
  duration

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Duration 4

• Example (BKM Table 15.3)
• suppose YTM changes by 1 basis point (0.01)
• zero coupon bond with 1.8853 years to maturity
• old price
• new price

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Duration 5

• Example BKM4 Table 15.3
• suppose YTM changes by 1 basis point (0.01)
• coupon bond
• either compare the bonds price with YTM 5.01
  relative to the bonds price with YTM 5
• or simply compute the price change from the
  duration

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Duration 6

• Properties of duration (other things constant)
• zero coupon bond duration maturity
• time to maturity
• maturity ? ? duration ?
• exception deep discount bonds
• coupon rate
• coupon ? ? duration ?
• YTM
• YTM ? ? duration ?
• exception zeroes (unchanged)

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Duration 7

• Properties of duration
• duration of perpetuity
• less than infinity!
• coupon bonds (annuities zero)
• see book
• simplifies if par bond

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Duration 8

• Importance
• simple measure
• essential to implement portfolio immunization
• measures interest rate sensitivity effectively

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Possible Caveats to Duration

• 1. Assumptions on term structure
• Macaulay duration uses YTM
• only valid for level changes in flat term
  structure
• Fisher-Weil duration measure

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Possible Caveats to Duration 2

• problems with the Fisher-Weil duration
• assumes a parallel shift in term structure
• need forecast of future interest rates
• bottom line same problem as realized compound
  yield
• Cox-Ingersoll-Ross duration
• bottom line lets keep Macaulay

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Possible Caveats to Duration 3

• 2. Convexity
• Macaulay duration
• first-order approximation
• small changes vs. large changes
• duration point estimate
• for larger changes, an arc estimate is needed
• solution add convexity

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Possible Caveats to Duration 4

• Convexity (continued)
• second-order approximation

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Possible Caveats to Duration 5

• Convexity numerical example
• P Par 1,000 T 30 years 8 annual coupon
• computations give D11.26 years convexity
  212.4 years
• suppose YTM 8 -gt YTM 10

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Bottom Line on Duration

• Very useful
• But take it with a grain of salt for large
  changes

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Immunization

• Why?
• obligation to meet promises (pension funds)
• protect future value of portfolio
• ratios, regulation, solvency (banks)
• protect current net worth of institution
• How?
• measure interest rate risk duration
• match duration of elements to be immunized

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Immunization

• What?
• net worth immunization
• match duration of assets and liabilities
• target date immunization
• match inflows and outflows
• immunize the net flows
• Who?
• insurance companies, pension funds
• target date immunization
• banks
• net worth immunization

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Net Worth Immunization

• Gap management
• assets vs. liabilities
• long term (mortgages, loans, ) vs. short term
  (deposits, )
• match duration of assets and liabilities
• decrease duration of assets (ex. ARM)
• increase duration of liabilities (ex. term
  deposits)
• condition for success
• portfolio duration 0 (assets liabilities)

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Target Date Immunization

• Idea
• Example suppose interest rates fall
• good for the pension fund
• price risk
• existing (fixed rate) assets increase in value
• bad for the pension fund
• reinvestment risk
• PV of future liabilities increases
• so more must be invested now

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Target Date Immunization 2

• Solution
• match duration of portfolio and funds horizon
• single bond
• bond portfolio
• duration of portfolio
• weighted average of components duration
• condition assets have equal yields

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Target Date Immunization 3
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Target Date Immunization 4
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Target Date Immunization 5
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Dangers with Immunization

• 1. Portfolio rebalancing is needed
• Time passes ? duration changes
• bonds mature, sinking funds,
• YTM changes ? duration changes
• example BKM4 Table 15.7
• duration YTM 5 8
  4.97 7 5.02 9

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Dangers with Immunization 2

• 2. Duration nominal concept
• immunization only for nominal liabilities
• counter example
• childrens tuition
• why?
• solution
• do not immunize
• buy assets

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An Alternative? Cash-Flow Dedication

• Buy zeroes
• to match all liabilities
• Problems
• difficult to get underpriced zeroes
• zeroes not available for all maturities
• ex. perpetuity

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Contingent Immunization

• Idea
• try to beat the market
• while limiting the downside risk
• Procedure (BKM6 Fig. 16.10 BKM4 Fig. 15.6)
• compute the PV of the obligation at current rates
• assess available funds
• play the difference
• immunize if trigger point is hit