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FIN 2802, Spring 08 - Tang
Chapter 16: Managing Bond Portfolios1
Fina2802: Investments and Portfolio Analysis
Spring, 2008Dragon Tang
Fina2802: Investments and Portfolio Analysis
Spring, 2008Dragon Tang
Lecture 12Managing Bond Portfolios
February 28/29, 2008
Readings: Chapter 16Practice Problem Sets: 1-13
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Chapter 16: Managing Bond Portfolios2
Wall Street Interview QuestionWall Street Interview Question
You strongly believe that the yield curve is going to steepen very soon. It may be a fall in short-term rates, a rise in long-term rates, or some combination of these. What strategy should you pursue in the bond market to position yourself to profit from your beliefs?
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Chapter 16: Managing Bond Portfolios3
Managing Bond PortfoliosManaging Bond Portfolios
Objectives:
• Analyze the features of a bond that affect the sensitivity of its price to interest rates.
• Compute the duration of bonds.
• Formulate fixed-income immunization strategies for various investment horizons.
• Analyze the choices to be made in an actively managed fixed-income portfolio.
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Chapter 16: Managing Bond Portfolios4
Interest Rate RiskInterest Rate Risk
Interest rate sensitivity:
•Time to maturity
•Coupon rate
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Change in Bond Price as a Function of Change in Yield to MaturityChange in Bond Price as a Function of Change in Yield to Maturity
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DurationDuration
Measures the effective maturity by weighting the payments by their proportion of the bond value.
where t =1, 2, 3, ... T are the times to maturity of payments
Price Bond
1/CF
Price Bond
CFPV ttt
t
yw
y is the bond's yield to maturity (current market rate)
D t wtt
T
1
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Cash Flows of 8-yr Bond with 9% annual coupon and 10% YTM
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Calculating DurationCalculating DurationExample: What is the duration of a 6% semiannual coupon bond with par of $1,000 and maturity in two years if market interest rates are currently 5% (semi-annual)?
(1) (2) (3) (4) (5) Time to Payment Payment Column (1) Payment Payment discounted Weight Times (years) Amount at 5% (3)/Sum Column (4)
0.5 $ 30 $ 28.57 .03075 .015371.0 $ 30 $ 27.21 .02929 .02929 1.5 $ 30 $ 25.92 .02790 .04183
2.0 $ 1,030 $ 847.38 .91206 1.82412
$ 929.08 1.0000 1.91061
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Calculating DurationCalculating Duration
28.847$030,1$82270.82270.
05.1
1
92.25$30$86384.86384.05.1
1
21.27$30$90703.90703.05.1
1
57.28$30$9524.95238.05.1
1
4
3
2
1
5% semiannual
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Calculating DurationCalculating DurationExample: What is the duration of a zero-coupon bond which matures in two years if market interest rates are currently 5% (semi-annual)?
(1) (2) (3) (4) (5) Time to Payment Column (1) Payment Payment discounted Payment Times (years) Amount at 5% Weight Column (4)
0.5 $ 0 $ 0.00 .0 .01.0 $ 0 $ 0.00 .0 .01.5 $ 0 $ 0.00 .0 .02.0 $ 1000 $ 822.70 1.0 2.0
$ 822.70 1.0 2.0
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Spreadsheet 16.1 Calculating the Duration of Two BondsSpreadsheet 16.1 Calculating the Duration of Two Bonds
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ExampleExample
A pension plan is obligated to make disbursements of $1 million, $2 million, and $1 million at the end of each of the next three years, respectively. Find the duration of the plan’s obligations if the interest rate is 10% annually.
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DurationDuration
Duration measure does three things:
• It measures the effective average maturity of a bond.
• It measures interest rate sensitivity correctly.
• It provides the necessary information for immunization.
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Chapter 16: Managing Bond Portfolios14
Duration and Interest Rate SensitivityDuration and Interest Rate Sensitivity
Sensitivity of prices to interest rate changes:
y
yD
P
P
1
where y is the yield to maturity
yDP
P
*
y
DD
1 and *
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Duration Duration
Example: The duration for a bond, currently priced at $929.08, with a yield-to-maturity (YTM) of 10% is 1.91061 years. If interest rates rise by 0.5 percentage points (50 basis points), what will be the dollar change in the price of the bond?
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Chapter 16: Managing Bond Portfolios16
ExampleExample
You own a fixed-income asset with a duration of five years. If the level of interest rates, which is currently 8%, goes down by 10 basis points, how much do you expect the price of the asset to go up (in percentage terms)?
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Bond Price SensitivityBond Price Sensitivity
Determinants of a bond’s price sensitivity to interest rate changes:
•the time to maturity (Duration not always increasing in time to Maturity)
•the coupon rate(Duration always decrease with high Coupon)
•the yield to maturity(Duration always decrease if YTM increase)
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Duration Rules & ResultsDuration Rules & Results
• The duration of a zero-coupon bond is equal to its time to maturity.
• Other things equal, a lower coupon rate results in a higher duration.
• Other things equal, a longer time to maturity increases duration (not always but usually)
• Other things equal, a lower yield to maturity increases duration.
• The duration of a perpetuity is equal to (1+y)/y.
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Figure 16.3 Duration as a Function of Maturity
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Table 16.3 Bond Duration (Initial Bond Yield 8% APR)Table 16.3 Bond Duration (Initial Bond Yield 8% APR)
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Price Approximation Using Modified DurationPrice Approximation Using Modified Duration
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Figure 16.4 Bond Price Convexity (30-Year Maturity, 8% Coupon; Initial Yield to Maturity = 8%)
Figure 16.4 Bond Price Convexity (30-Year Maturity, 8% Coupon; Initial Yield to Maturity = 8%)
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Correction for ConvexityCorrection for Convexity
n
tt
t tty
CF
yPConvexity
1
22
)()1()1(
1
All else equal, a higher coupon corresponds to a smaller convexityAll else equal, a longer maturity entails a larger convexityAll else equal, convexity is larger at a lower yield
Correction for Convexity:
])([21 2yConveixityyD
P
P
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An ExampleAn Example
• 18-year 12% coupon bond @ 9% YTM, priced at 126 ½.
– Modified duration = 8.38, convexity = 107.70
– 1% decline in yield (price increase 8.92%)
» Percentage increase in price due to duration: 8.38%
» Percentage increase in price due to convexity: 0.54%
– 3% increase in yield (price decline 20.56%)
» Percentage decline in price due to duration: 25.41%
» Percentage increase in price due to convexity: 4.85%
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Passive Bond ManagementPassive Bond Management
1. Net Worth Immunization (Present) (e.g. Banks: Asset/Liability Management)
2. Target Date Immunization (Future) (e.g. Pension Funds: meet future obligations)
Takes prices as given and tries to control the risk of the fixed-income portfolio.
Measures:
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Main Idea Behind ImmunizationMain Idea Behind Immunization
• Net Worth Immunization:
Match duration of asset and liabilities by adjusting their maturity structure (Gap Management)
• Target Date Immunization:Set the duration of a portfolio equal to the target
date. This guarantees that at this date reinvestment risk and price risk exactly cancel out.
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Target Date ImmunizationTarget Date Immunization
Example. An insurance company issue a 5-years Guaranteed investment contract (GIC) at 8%, nominal value $10,000. The insurance company decides to meet this obligation by investing $10,000 in 8% annual coupon bonds with maturity in 6yrs.
Can the firm meets its obligation at time 5?
What if interest rate drops to 7% ?
What if interest rate increases to 9% ?
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Table 16.4 Terminal value of a Bond Portfolio After 5 Years (All Proceeds Reinvested)
Table 16.4 Terminal value of a Bond Portfolio After 5 Years (All Proceeds Reinvested)
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Chapter 16: Managing Bond Portfolios29
Target Date ImmunizationTarget Date Immunization
ReinvestmentValue of Coupon Bond
Obligation
D*=5yrs
Value of Coupon Bondr = 8%
Value of Coupon Bondr = 9%
Time
$10,000
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Target Date ImmunizationTarget Date Immunization
• Given a future obligation X to be met in D* years
• Match it with a portfolio with Duration D* and worth X at time D*
• This guarantees that the value of the portfolio at time D* will be always be approximately X for any relatively small change in the interest rate
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ExampleExampleYou are managing a portfolio of $1 million. Your target duration is 10 years, and you can choose from two bonds: a zero-coupon bond with maturity 5 years, and a perpetuity, each currently yielding 5%.
a. How much of each bond will you hold in your portfolio?
b. How will these fractions change next year if target duration is now nine years?
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Net Worth ImmunizationNet Worth Immunization
• Given a liability currently worth L and with duration DL
• Match it with an asset currently worth L and with duration DL.
• This guarantees that, for small changes in the interest rate the net worth will always be approximately zero.
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Chapter 16: Managing Bond Portfolios33
Net Worth ImmunizationNet Worth Immunization
Current Value of Asset and Liabilities
Interest rate
Current of Coupon Bond (Asset)(YTM=8%)
Present Value of CIG (Liability)(YTM=8%)
8%=YTM
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Figure 16.12 Contingent ImmunizationFigure 16.12 Contingent Immunization
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Active Bond ManagementActive Bond Management
Sources of potential profits:
•Interest rate forecasts
•Identification of mispriced bonds
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Chapter 16: Managing Bond Portfolios36
Maturity
Yield to Maturity %
3 mon 6 mon 9 mon
1.5 1.25 .75
Yield Curve RideYield Curve Ride
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Current “Hot” StrategiesCurrent “Hot” Strategies
• Convertible arbitrage
–Sell stock, buy convertible bond of the same company
–Ken Griffin, founder of Citadel, made a fortune as a sophomore
• Capital structure arbitrage
–Trade stock and bond in opposite direction
–Hurt by the GM/Ford event
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Chapter 16: Managing Bond Portfolios38
SummarySummary
•Interest rate risk and default risk•Duration as a measure of the average life of a bond •Sensitivity of a bond's price to changes in yield•Passive Bond Management
• Immunization (Net Worth and Target date) makes the individual or firm immune from interest rate movements
• Portfolio must be rebalanced periodically
•Active Bond Management• Adjusting portfolio based on interest rate forecasts
•Next Class: Equity Valuation