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Bond Prices and Yields

CHAPTER 14

Bond Characteristics

- Face or par value
- Coupon rate
- Zero coupon bond
- Compounding and payments
- Accrued Interest
- Indenture

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Different Issuers of Bonds

- U.S. Treasury
- Notes and Bonds
- Corporations
- Municipalities
- International Governments and Corporations
- Innovative Bonds
- Floaters and Inverse Floaters
- Asset-Backed
- Catastrophe

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Figure 14.1 Listing of Treasury Issues

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Figure 14.2 Listing of Corporate Bonds

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Provisions of Bonds

- Secured or unsecured
- Call provision
- Convertible provision
- Put provision (putable bonds)
- Floating rate bonds
- Preferred Stock

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Convertible Bonds

- Give bondholders an option to exchange each bond for a specified nb of shares of common stock
- Conversion ratio
- = number of shares per convertible bond
- Market conversion value
- = conversion ratio * current market value per share
- Conversion premium
- = bond value - conversion value
- intuitively: extra amount to pay so as to become a shareholder

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Conversion Example

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Conversion Example

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Innovation in the Bond Market

- Inverse Floaters
- Asset-Backed Bonds
- Catastrophe Bonds
- Indexed Bonds

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Table 14.1 Principal and Interest Payments for a Treasury Inflation Protected Security

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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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PB= Price of the bond

Ct = interest or coupon payments

T = number of periods to maturity

y = semi-annual discount rate or the semi-annual yield to maturity

Bond PricingBahattin Buyuksahin, Derivatives Pricing

Price: 10-yr, 8% Coupon, Face = $1,000

Ct = 40 (SA)

P = 1000

T = 20 periods

r = 3% (SA)

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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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Bond Prices and Yields

- Prices and Yields (required rates of return) have an inverse relationship
- When yields get very high the value of the bond will be very low
- When yields approach zero, the value of the bond approaches the sum of the cash flows

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Prices vs. Yields

- P yield
- intuition
- convexity
- Fig 14.3
- intuition: yield P price impact

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Figure 14.3 The Inverse Relationship Between Bond Prices and Yields

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Table 14.2 Bond Prices at Different Interest Rates (8% Coupon Bond, Coupons Paid Semiannually)

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Yield to Maturity

- Interest rate that makes the present value of the bond’s payments equal to its price

Solve the bond formula for r

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Yield to Maturity Example

10 yr Maturity Coupon Rate = 7%

Price = $950

Solve for r = semiannual rate

r = 3.8635%

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Yield Measures

Bond Equivalent Yield

7.72% = 3.86% x 2

Effective Annual Yield

(1.0386)2 - 1 = 7.88%

Current Yield

Annual Interest / Market Price

$70 / $950 = 7.37 %

Yield to Call

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Pure Discount Bonds (Zero-Coupon Bonds)

A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T

- Discount bonds, also called zero-coupon bonds, are securities which “make a single payment at a date in the future known as maturity date. The size of this payment is the face value of the bond. The length of time to the maturity date is the maturity of the bond” (Campbell, Lo, MacKinley (1996)).

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Pure Discount Bond

- The promised cash payment on a pure discount bond is called its face value or par value. Yield (interest rate) on a pure discount bond is the annualized rate of return to investors who buy it and hold it until it matures.

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Example

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Bond Pricing

- To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate
- The theoretical price of a two-year bond providing a 6% coupon semiannually is

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Bond Yield

- The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond
- Suppose that the market price of the bond in our example equals its theoretical price of 98.39
- The bond yield is given by solving

to get y = 0.0676 or 6.76%.

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Par Yield

- The par yield for a certain maturity is the coupon rate that causes the bond price to equal its face value.
- In our example we solve

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Par Yield (continued)

In general if m is the number of coupon payments per year, d is the present value of $1 received at maturity and A is the present value of an annuity of $1 on each coupon date

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Bootstrap Method to calculate discount factor

- A discount function is a set of discount factors, where each discount factor is just a present value multiplier. For example, d(1.0) is the present value of $1 dollar received in one year. The key idea is that each d(x) can be solved as one variable under one equation because we already solved for shorter-term discount factors.
- The most popular approach is to use bootstrap method

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Bootstrap : Example

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Discount Factor

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Determining Treasury Zero Rates

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Treasury Zero Rate Curve

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Figure 14.4 Bond Prices: Callable and Straight Debt

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Example 14.4 Yield to Call

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Realized Yield versus YTM

- Reinvestment Assumptions
- Holding Period Return
- Changes in rates affect returns
- Reinvestment of coupon payments
- Change in price of the bond

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Figure 14.5 Growth of Invested Funds

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Figure 14.6 Prices over Time of 30-Year Maturity, 6.5% Coupon Bonds

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Holding-Period Return: Single Period

HPR = [ I + ( P0 - P1 )] / P0

where

I = interest payment

P1= price in one period

P0 = purchase price

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Holding-Period Return Example

CR = 8% YTM = 8% N=10 years

Semiannual Compounding P0 = $1000

In six months the rate falls to 7%

P1 = $1068.55

HPR = [40 + ( 1068.55 - 1000)] / 1000

HPR = 10.85% (semiannual)

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Figure 14.7 The Price of a 30-Year Zero-Coupon Bond over Time at a Yield to Maturity of 10%

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Default Risk and Ratings

- Rating companies
- Moody’s Investor Service
- Standard & Poor’s
- Fitch
- Rating Categories
- Investment grade
- Speculative grade/Junk Bonds

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Figure 14.8 Definitions of Each Bond Rating Class

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Factors Used by Rating Companies

- Coverage ratios
- Leverage ratios
- Liquidity ratios
- Profitability ratios
- Cash flow to debt

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Table 14.3 Financial Ratios and Default Risk by Rating Class, Long-Term Debt

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Figure 14.9 Discriminant Analysis

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Protection Against Default

- Sinking funds
- Subordination of future debt
- Dividend restrictions
- Collateral

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Figure 14.10 Callable Bond Issued by Mobil

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Default Risk and Yield

- Risk structure of interest rates
- Default premiums
- Yields compared to ratings
- Yield spreads over business cycles

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Figure 14.11 Yields on Long-Term Bonds, 1954 – 2006

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Credit Risk and Collateralized Debt Obligations (CDOs)

- Major mechanism to reallocate credit risk in the fixed-income markets
- Structured Investment Vehicle (SIV) often used to create the CDO
- Mortgage-backed CDOs were an investment disaster in 2007

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Figure 14.12 Collateralized Debt Obligations

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Overview of Term Structure

- Information on expected future short term rates can be implied from the yield curve
- The yield curve is a graph that displays the relationship between yield and maturity
- Three major theories are proposed to explain the observed yield curve

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Figure 15.1 Treasury Yield Curves

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Bond Pricing

- Yields on different maturity bonds are not all equal
- Need to consider each bond cash flow as a stand-alone zero-coupon bond when valuing coupon bonds

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Table 15.1 Yields and Prices to Maturities on Zero-Coupon Bonds ($1,000 Face Value)

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Yield Curve Under Certainty

- An upward sloping yield curve is evidence that short-term rates are going to be higher next year
- When next year’s short rate is greater than this year’s short rate, the average of the two rates is higher than today’s rate

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Figure 15.2 Two 2-Year Investment Programs

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Figure 15.3 Short Rates versus Spot Rates

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Forward Rates from Observed Rates

fn = one-year forward rate for period n

yn = yield for a security with a maturity of n

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Example 15.4 Forward Rates

4 yr = 8.00% 3yr = 7.00% fn = ?

(1.08)4 = (1.07)3 (1+fn)

(1.3605) / (1.2250) = (1+fn)

fn = .1106 or 11.06%

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Downward Sloping Spot Yield CurveExample

Zero-Coupon RatesBond Maturity

12% 1

11.75% 2

11.25% 3

10.00% 4

9.25% 5

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Forward Rates for Downward Sloping Y C Example

1yr Forward Rates

1yr [(1.1175)2 / 1.12] - 1 = 0.115006

2yrs [(1.1125)3 / (1.1175)2] - 1 = 0.102567

3yrs [(1.1)4 / (1.1125)3] - 1 = 0.063336

4yrs [(1.0925)5 / (1.1)4] - 1 = 0.063008

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Interest Rate Uncertainty

- What can we say when future interest rates are not known today
- Suppose that today’s rate is 5% and the expected short rate for the following year is E(r2) = 6% then:
- The rate of return on the 2-year bond is risky for if next year’s interest rate turns out to be above expectations, the price will lower and vice versa

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Interest Rate Uncertainty Continued

- Investors require a risk premium to hold a longer-term bond
- This liquidity premium compensates short-term investors for the uncertainty about future prices

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Term Structure of Interest Rates

- The term structure of interest rates (or yield curve) is the relationship of the yield to maturity against bond term (maturity).
- Typical shapes are: increasing (normal), decreasing, humped and flat.

Yield

Maturity

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Upward vs Downward SlopingYield Curve

- For an upward sloping yield curve:

Fwd Rate > Zero Rate > Par Yield

- For a downward sloping yield curve

Par Yield > Zero Rate > Fwd Rate

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Theories of the Term Structure

- A number of theory have been proposed: Expectation Hypothesis, Liquidity Preference Theory, Preferred Habitats Theory, Segmentation Hypothesis.
- Fabozzi (1998): PureExpectation Hypothesis, Liquidity Preference Theory, Preferred Habitats Theory are different forms of the expectation theory ==> two major theories: expectation theory and market segmentation theory.

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Theories of the Term Structure of Interest Rates (1)

- The Pure Expectation Hypothesis: Implied forward rates are unbiased expectations of future spot rates ==> a rising term structure indicate that market expects short-term rates to rise in the future; a flat term structure reflects expectations that the future short term structure will be constant; and so on; Hicks (1937). Problems: It neglects the risks inherent in investing in bonds: if forward rates were perfect predictors of future interest rates then the future prices of bonds will be known with certainty.
- The Liquidity Preference Theory (Keynes): Given that there is uncertainty, long bonds should have higher returns than short bonds ==> we should expect a risk premium arising out from investors liquidity preferences. It is consistent with the empirical results that yield curves are upward sloping ==> positive risk premium.

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Theories of the Term Structure of Interest Rates (2)

- The Preferred Habitat Theory: It adopts the view that the term structure is composed by two components: Expectations plus risk premium (= liquidity preference theory). However, the risk premium might be negative as well as positive to induce market participants to shift out of their preferred habitat (Modigliani & Sutch (1966)).
- The Segmentation Hypothesis (Culbertson (1957)): It also recognises that investors have preferred habitat (= preferred habitat theory) ==> individuals have strong maturity preferences ==> there need be no relationship between bonds with different maturities ==> bonds with different maturities are traded in different markets.

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Expectations Theory

- Observed long-term rate is a function of today’s short-term rate and expected future short-term rates
- Long-term and short-term securities are perfect substitutes
- Forward rates that are calculated from the yield on long-term securities are market consensus expected future short-term rates

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Liquidity Premium Theory

- Long-term bonds are more risky
- Investors will demand a premium for the risk associated with long-term bonds
- The yield curve has an upward bias built into the long-term rates because of the risk premium
- Forward rates contain a liquidity premium and are not equal to expected future short-term rates

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Figure 15.4 Yield Curves

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Figure 15.4 Yield Curves (Concluded)

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Interpreting the Term Structure

- If the yield curve is to rise as one moves to longer maturities
- A longer maturity results in the inclusion of a new forward rate that is higher than the average of the previously observed rates
- Reason:
- Higher expectations for forward rates or
- Liquidity premium

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Figure 15.5 Price Volatility of Long-Term Treasury Bonds

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Figure 15.6 Term Spread: Yields on 10-Year Versus 90-Day Treasury Securities

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Forward Rates as Forward Contracts

- In general, forward rates will not equal the eventually realized short rate
- Still an important consideration when trying to make decisions :
- Locking in loan rates

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Figure 15.7 Engineering a Synthetic Forward Loan

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Bond Pricing Relationships

- Inverse relationship between price and yield
- An increase in a bond’s yield to maturity results in a smaller price decline than the gain associated with a decrease in yield
- Long-term bonds tend to be more price sensitive than short-term bonds

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Figure 16.1 Change in Bond Price as a Function of Change in Yield to Maturity

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Bond Pricing Relationships Continued

- As maturity increases, price sensitivity increases at a decreasing rate
- Price sensitivity is inversely related to a bond’s coupon rate
- Price sensitivity is inversely related to the yield to maturity at which the bond is selling

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Table 16.1 Prices of 8% Coupon Bond (Coupons Paid Semiannually)

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Table 16.2 Prices of Zero-Coupon Bond (Semiannually Compounding)

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Duration

- A measure of the effective maturity of a bond
- The weighted average of the times until each payment is received, with the weights proportional to the present value of the payment
- Duration is shorter than maturity for all bonds except zero coupon bonds
- Duration is equal to maturity for zero coupon bonds

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Duration: Calculation

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Spreadsheet 16.1 Calculating the Duration of Two Bonds

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Duration/Price Relationship

Price change is proportional to duration and not to maturity

D*= modified duration

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Rules for Duration

Rule 1 The duration of a zero-coupon bond equals its time to maturity

Rule 2 Holding maturity constant, a bond’s duration is higher when the coupon rate is lower

Rule 3 Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity

Rule 4 Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower

Rules 5 The duration of a level perpetuity is equal to: (1+y) / y

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Figure 16.2 Bond Duration versus Bond Maturity

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Table 16.3 Bond Durations (Yield to Maturity = 8% APR; Semiannual Coupons)

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Convexity

- The relationship between bond prices and yields is not linear
- Duration rule is a good approximation for only small changes in bond yields

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Figure 16.3 Bond Price Convexity: 30-Year Maturity, 8% Coupon; Initial Yield to Maturity = 8%

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Figure 16.4 Convexity of Two Bonds

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Callable Bonds

- As rates fall, there is a ceiling on possible prices
- The bond cannot be worth more than its call price
- Negative convexity
- Use effective duration:

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Figure 16.5 Price –Yield Curve for a Callable Bond

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Mortgage-Backed Securities

- Among the most successful examples of financial engineering
- Subject to negative convexity
- Often sell for more than their principal balance
- Homeowners do not refinance their loans as soon as interest rates drop

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Figure 16.6 Price -Yield Curve for a Mortgage-Backed Security

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Mortgage-Backed Securities Continued

- They have given rise to many derivatives including the CMO (collateralized mortgage obligation)
- Use of tranches

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Figure 16.7 Panel A: Cash Flows to Whole Mortgage Pool; Panels B–D Cash Flows to Three Tranches

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Passive Management

- Bond-Index Funds
- Immunization of interest rate risk:
- Net worth immunization

Duration of assets = Duration of liabilities

- Target date immunization

Holding Period matches Duration

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Figure 16.8 Stratification of Bonds into Cells

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Table 16.4 Terminal value of a Bond Portfolio After 5 Years (All Proceeds Reinvested)

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Figure 16.9 Growth of Invested Funds

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Figure 16.10 Immunization

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Table 16.5 Market Value Balance Sheet

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Cash Flow Matching and Dedication

- Automatically immunize the portfolio from interest rate movement
- Cash flow and obligation exactly offset each other
- i.e. Zero-coupon bond
- Not widely used because of constraints associated with bond choices
- Sometimes it simply is not possible to do

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Active Management: Swapping Strategies

- Substitution swap
- Intermarket swap
- Rate anticipation swap
- Pure yield pickup
- Tax swap

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Horizon Analysis

- Select a particular holding period and predict the yield curve at end of period
- Given a bond’s time to maturity at the end of the holding period
- Its yield can be read from the predicted yield curve and the end-of-period price can be calculated

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Contingent Immunization

- A combination of active and passive management
- The strategy involves active management with a floor rate of return
- As long as the rate earned exceeds the floor, the portfolio is actively managed
- Once the floor rate or trigger rate is reached, the portfolio is immunized

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Figure 16.11 Contingent Immunization

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