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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 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%; T=60

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**Prices vs. Yields P yield convexity intuition**

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 NPV=0 interpretation (geometric) average return to maturity Example: Ct = $40; Par = $1,000; T=60; sells at par

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**Measuring Rates of Return on Bonds 3**

Yield To Call definition discount rate s.t. NPV=0, 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; T=60; P = $1,025; callable in 10 years (TC=20), call price = $1,000

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**Measuring Rates of Return on Bonds 4**

Yield To Default definition discount rate s.t. NPV=0, with TD= expected default date default premium and business cycle economic difficulties and “flight to quality” Example: Ct = $50, semi; Par = $1,000; T=10; P = $200; expected to default in 2 years (TC=4), 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 < market interest rates built-in capital gain (discount bond) coupon rate > market interest rates built-in capital loss (premium bond) Behavior of prices over time BKM6 Fig. 14.6; BKM4 Fig Tax treatment capital gains vs. interest income

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**Discount Bonds OID vs. par bonds Zeroes**

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 )

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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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**“Term”inology 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 re:Short 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 $ % 2 $ % 3 $ % 4 $ %

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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 $ % 2 $ % 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) = $ 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 E[r2] f2 = E[r2] + 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% E[r2] f2 f2 = E[r2] - risk “premium”

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**Fwd Rate & Exp. Future Short Rate 4**

Expectation Hypothesis risk premium = 0 and E[r2] = 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**

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 -> 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: $ (coupon = 3%); $1, (coupon = 12%) Year-1 and Year-2 short rates $ = d1 x d2 x 1,030 $ 1, = d1 x d2 x 1,120 Solve the system: d2 = , d1 = 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 Interpretative assumptions**

either short rates are expected to climb: E[rn] E[rn-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 -> future?!

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**Inverted yield curve Easy interpretation Example**

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 -> use the CAPM Bond Index Funds Immunization net worth immunization contingent immunization

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**Bond Index Funds Idea US indices composition**

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 effectiveness**

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: Yield sensitivity of bond Prices:**

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 bond’s 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 Measure maturity sensitivity **

to measure a bond’s 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 Interpretation 2**

= 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 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 bond’s price with YTM = 5.01% relative to the bond’s 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: let’s 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 D*=11.26 years; convexity = years suppose YTM = 8% -> 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? How? 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? Who? 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 banks

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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 fund’s 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 % % %

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**Dangers with Immunization 2**

2. Duration = nominal concept immunization only for nominal liabilities counter example children’s 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 ; 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

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