1 fin 2802, spring 08 - tang chapter 16: managing bond portfolios fina2802: investments and...

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



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?



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.



Interest Rate RiskInterest Rate Risk

Interest rate sensitivity:

•Time to maturity

•Coupon rate



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



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



Cash Flows of 8-yr Bond with 9% annual coupon and 10% YTM



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



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



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



Spreadsheet 16.1 Calculating the Duration of Two BondsSpreadsheet 16.1 Calculating the Duration of Two Bonds



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.



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.



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 *



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?



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)?



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)



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.



Figure 16.3 Duration as a Function of Maturity



Table 16.3 Bond Duration (Initial Bond Yield 8% APR)Table 16.3 Bond Duration (Initial Bond Yield 8% APR)



Price Approximation Using Modified DurationPrice Approximation Using Modified Duration



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%)



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



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%



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:



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.



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% ?



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)




ReinvestmentValue of Coupon Bond

Obligation

D*=5yrs

Value of Coupon Bondr = 8%

Value of Coupon Bondr = 9%

Time

$10,000




• 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



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?



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.



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



Figure 16.12 Contingent ImmunizationFigure 16.12 Contingent Immunization



Active Bond ManagementActive Bond Management

Sources of potential profits:

•Interest rate forecasts

•Identification of mispriced bonds



Maturity

Yield to Maturity %

3 mon 6 mon 9 mon

1.5 1.25 .75

Yield Curve RideYield Curve Ride



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



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

1 fin 2802, spring 08 - tang chapter 16: managing bond portfolios fina2802: investments and...

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