0 portfolio management 3-228-07 albert lee chun duration, convexity and bond portfolio management...

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Portfolio ManagementPortfolio Management3-228-073-228-07

Albert Lee ChunAlbert Lee Chun

Duration, Convexity and Duration, Convexity and Bond Portfolio Management Bond Portfolio Management

StrategiesStrategies

Lecture 10Lecture 10

27 Nov 2008

Albert Lee Chun Portfolio Management 2

Bond Portfolio ManagementBond Portfolio Management

The major source of risk facing bond portfolio The major source of risk facing bond portfolio managers has to do with managers has to do with shifts in interest ratesshifts in interest rates. .

In this session:In this session:

1. We examine how bond prices respond to changes in 1. We examine how bond prices respond to changes in this source of risk.this source of risk.

2. We discuss ways of constructing bond portfolios to 2. We discuss ways of constructing bond portfolios to insulate against this risk.insulate against this risk.


TodayToday

Review of Bond FundamentalsReview of Bond Fundamentals Term Structure of Interest RatesTerm Structure of Interest Rates Interest Rate RiskInterest Rate Risk Sensitivity of Bond Prices to Interest RatesSensitivity of Bond Prices to Interest Rates - Duration - Duration - Convexity- Convexity Portfolio ImmunizationPortfolio Immunization Yield Curve StrategiesYield Curve Strategies Some Fun ExamplesSome Fun Examples


Review of Bond FundamentalsReview of Bond Fundamentals


Bond DefinitionBond Definition

Definition:Definition: A bond is a debt security that a A bond is a debt security that a corporation or a government issues to borrow on a corporation or a government issues to borrow on a long-term basis.long-term basis.

Normally an interest-only loan (when issued at Normally an interest-only loan (when issued at parpar))

- Borrower pays only the interest but none of the - Borrower pays only the interest but none of the

principle is paid until the end of the loan.principle is paid until the end of the loan. Interest is paid in the form of a periodic Interest is paid in the form of a periodic coupon.coupon.


Components of Bond Components of Bond

Bond prices depend on 4 factorsBond prices depend on 4 factors

1.1. Par or Face ValuePar or Face Value

2.2. Coupon RateCoupon Rate

3.3. Number of years to maturityNumber of years to maturity

4.4. Yield to Maturity Yield to Maturity ((YTMYTM) The “interest rate” that ) The “interest rate” that makes the discounted present value of the bond’s makes the discounted present value of the bond’s coupons equal to its market price (also called coupons equal to its market price (also called internal rate of return of a bondinternal rate of return of a bond).).


Yield to MaturityYield to Maturity

The The yield to maturityyield to maturity (YTM) (YTM) is an interest rate such is an interest rate such that present value of the bond’s coupons and the face that present value of the bond’s coupons and the face value equals the current market price.value equals the current market price.

Often simply called the bond’s Often simply called the bond’s yieldyield as in “The as in “The yieldyield on the 10 year bond is 5%.”on the 10 year bond is 5%.”

Important:Important: YTM is quoted as an YTM is quoted as an Annual percentage Annual percentage rate (APR).rate (APR).

Bond prices are Bond prices are inverselyinversely related to yields. related to yields.


Suppose a bond pays 1 coupon per yearSuppose a bond pays 1 coupon per year

Present Value = Present Value = PV(coupon payments)PV(coupon payments) + + PV(face value)PV(face value) = = PV of an annuityPV of an annuity + + PV of FPV of F

F = face value C = annual coupon amount

= F × coupon rate r = yield to maturity t = number of years until maturity

Equation for BondsEquation for Bonds

tt r

F

rr

C

)1()1(

11PV



For a bond paying a semi-annual coupon (2 times a year)For a bond paying a semi-annual coupon (2 times a year)

F = face value C = yearly coupon amount

= F × coupon rate r = yield to maturity t = number of periods until maturity

22 )2/1()2/1(

11

2/

2/PV

tt r

F

rr

C


More generally, if a bond pays m coupons per year More generally, if a bond pays m coupons per year

F = face value C = yearly coupon amount = F × coupon rate r = yield to maturity m = number coupons per year t = number of years until maturity

mtmt mr

F

mrmr

mC

)/1()/1(

11

/

/PV



Bond DetailsBond Details

The The coupon ratecoupon rate, the , the face valueface value (or (or parpar value) and the value) and the maturitymaturity date are all determined by the bond issuer date are all determined by the bond issuer (corporation or government).(corporation or government).

Treasury Bills – One year and less; no couponsTreasury Bills – One year and less; no coupons Treasury Notes – Between 2 and 10 years; coupons.Treasury Notes – Between 2 and 10 years; coupons. Treasury Bonds – Longer than 10 years; coupons.Treasury Bonds – Longer than 10 years; coupons.


Bond ExampleBond Example

Simple Example:Simple Example: Eggbert’s Egg Co. issues a bond with time Eggbert’s Egg Co. issues a bond with time to maturity of 7 years; the yield to maturity is 10% APR. The to maturity of 7 years; the yield to maturity is 10% APR. The firm pays an interest rate in the form of $40 every 6 months firm pays an interest rate in the form of $40 every 6 months for 7 years, and a $1,000 principal at the end of 7 years.for 7 years, and a $1,000 principal at the end of 7 years.

Eggbert’s Egg Company


Example: Valuing a BondExample: Valuing a Bond

Eggbert’s Egg Co. issues a semiannual coupon paying Eggbert’s Egg Co. issues a semiannual coupon paying bond with a coupon rate of 8% and a face value of bond with a coupon rate of 8% and a face value of $1,000 that matures in 7 years. If we assume, that the $1,000 that matures in 7 years. If we assume, that the yield to maturity is 10%, what is the price of this bond?yield to maturity is 10%, what is the price of this bond?

The bondholder receives a payment of $40 every six The bondholder receives a payment of $40 every six months (a total of $80 per year or 8% per year)months (a total of $80 per year or 8% per year)

01.90105.1

000,1

05.01.05

1-1

40 Price Bond14

14


PAR BondsPAR Bonds

The price of a par bond is equal to F.The price of a par bond is equal to F. A par bond has YTM = coupon rate.A par bond has YTM = coupon rate.

Why?Why? Let’s take YTM = coupon rate = r. F = 1000. Let’s take YTM = coupon rate = r. F = 1000. Suppose at time t, P(t) = 1000(1+r). At time t-1, the value of the Suppose at time t, P(t) = 1000(1+r). At time t-1, the value of the

bond is:bond is: =1000(1+r)/(1+r)=1000(1+r)/(1+r) =1000 =1000 By induction, since this is true at time T-1, this is true for all t. By induction, since this is true at time T-1, this is true for all t.

Hence, when YTM = coupon rate, PV = 1000. Hence, when YTM = coupon rate, PV = 1000. Most bonds are issued at parMost bonds are issued at par, with the coupon rate set , with the coupon rate set

equal to the prevailing market yield.equal to the prevailing market yield.


Example: Valuing a “Par Bond”Example: Valuing a “Par Bond”

Suppose you are looking at a bond that has a 10% Suppose you are looking at a bond that has a 10% coupon rate and a face value of $100. There are 10 coupon rate and a face value of $100. There are 10 years to maturity and the yield to maturity is 10%. years to maturity and the yield to maturity is 10%. What is the price of this bond?What is the price of this bond? Using the formula:

P = PV of annuity + PV of FP = 5[1 – 1/(1.05)20] / .05 + 100/ (1.05)20

P = 100.00

Price = F=100.

It’s a par bond.


Example: Yield to MaturityExample: Yield to Maturity

The price of EggbertThe price of Eggbert’’s Egg Co. is currently trading at 1,200. s Egg Co. is currently trading at 1,200. The time to maturity is 4-years with a 14% annual coupon. The time to maturity is 4-years with a 14% annual coupon. What is the yield to maturity?What is the yield to maturity?

The equation to solve is: The equation to solve is:

Using Goal Seek in Excel we get YTM = 2 x r = 7.96%.

44 )1(

1000,1

)1(

11

1401,200

rrr


Term Structure of Interest RatesTerm Structure of Interest Rates


The World is Not FlatThe World is Not Flat

In earlier courses, you were presented with a rather In earlier courses, you were presented with a rather simplified view of interest rates, with a constant simplified view of interest rates, with a constant single interest rate used to discount all cash flows.single interest rate used to discount all cash flows.

The yield to maturity was assumed to be the same for all The yield to maturity was assumed to be the same for all bonds.bonds.

In reality, bond of different maturities will have In reality, bond of different maturities will have different yields to maturity.different yields to maturity.

The yield curve is not flat.The yield curve is not flat.


Term Structure of Interest RatesTerm Structure of Interest Rates The term structure of interest rates gives the relation The term structure of interest rates gives the relation

between the time to maturity and the yield to between the time to maturity and the yield to maturity of a bond.maturity of a bond.

Yield curve – graphical representation of the term Yield curve – graphical representation of the term structure.structure. Normal – upward-sloping, long-term yields are

higher than short-term yields Inverted – downward-sloping, long-term yields are

lower than short-term yields.


Canada Yield CurveCanada Yield CurveNovember 2002November 2002


Canada Yield CurveCanada Yield CurveMay 2006May 2006

Canada Yield Curve as of May 2006

3.90%

4.00%

4.10%

4.20%

4.30%

4.40%

4.50%

4.60%

0.25 2 3.75 5.5 7.25 9 10.75 12.5 14.25 16 17.75 19.5 21.25 23 24.75 26.5 28.25 30

Maturity

Inte

rest

Rat

e


Interest Rate Risk and the Yield CurveInterest Rate Risk and the Yield Curve

The U.S. The U.S. central bankcentral bank sets the short term interest rate – sets the short term interest rate – the fed funds rate.the fed funds rate.

Long term interest rates are the functions of Long term interest rates are the functions of expected expected short-term interest ratesshort-term interest rates, in addition there is a risk-, in addition there is a risk-premia associated with uncertainty about the premia associated with uncertainty about the underlying economy forces, such as the growth rate underlying economy forces, such as the growth rate of the economy, inflation, etc.of the economy, inflation, etc.

Fed Chairman:

Ben Bernanke


Links to Macro-economyLinks to Macro-economy

The central bank will respond to macroeconomic The central bank will respond to macroeconomic forces in setting the interest rate. forces in setting the interest rate.

Thus, the yield curve embeds information about Thus, the yield curve embeds information about current and current and futurefuture macroeconomic conditions. macroeconomic conditions.

Conversely, it can serve as a barometer of the Conversely, it can serve as a barometer of the economy.economy.

Inverted yield curves are a leading indicator of Inverted yield curves are a leading indicator of recessions.recessions.


Price Sensitivity to Interest RatesPrice Sensitivity to Interest Rates


Interest Rate RiskInterest Rate Risk

Interest Rate Risk ↑ as Time to Maturity ↑ Interest Rate Risk ↑ as Coupon Rate ↓ Interest Rate Risk ↑ as Yield to Maturity ↓


Interest Rate Risk and MaturityInterest Rate Risk and Maturity

Longer maturityLonger maturity bond prices are more sensitive to bond prices are more sensitive to changes in yields than changes in yields than shorter maturityshorter maturity bonds. bonds. Interest rate risk is Interest rate risk is largerlarger for for longer maturity bondslonger maturity bonds


IntuitionIntuition

We can see that the slope of the price-yield curve as a We can see that the slope of the price-yield curve as a function of the interest rate is much steeper for the function of the interest rate is much steeper for the 30-year bond than for the 1-year bond. 30-year bond than for the 1-year bond.

The steeper price-yield curve, the more sensitive the The steeper price-yield curve, the more sensitive the bond is to interest rate changes.bond is to interest rate changes.

A large portion of a bond’s value is due to the A large portion of a bond’s value is due to the payment of the face value. payment of the face value.

If the bond has a longer maturity, the cash flow from If the bond has a longer maturity, the cash flow from the payment of the face value is realized further in the the payment of the face value is realized further in the future, this makes it more sensitive to changes in future, this makes it more sensitive to changes in interest rates. interest rates.


IntuitionIntuition

Why?Why? Because even a small change in the interest rate can Because even a small change in the interest rate can have a significant effect if it is have a significant effect if it is compoundedcompounded over a longer time. over a longer time.

Let’s see an example:Let’s see an example:

20 year bond: Price = 1000x(1+r)20 year bond: Price = 1000x(1+r)-20-20

10 year bond: Price = 1000x(1+r)10 year bond: Price = 1000x(1+r)-10-10

10 years 20 years

7% 508.3493 258.419

8% 463.1935 214.5482

% Change 0.097488 0.20448


Interest Rate Risk and the Coupon RateInterest Rate Risk and the Coupon Rate

Bonds with Bonds with higher coupon rateshigher coupon rates are less sensitive to are less sensitive to changes in yields. Interest rate risk ischanges in yields. Interest rate risk is inversely inversely related to the related to the coupon ratecoupon rate..

Why?Why? because their value depends less on the because their value depends less on the discounted face value.discounted face value.

Higher coupon payments move the Higher coupon payments move the average maturityaverage maturity of the bond’s cash flows forward in time, making the of the bond’s cash flows forward in time, making the bond less sensitive to the face value payment.bond less sensitive to the face value payment.


Coupon Rates and SensitivityCoupon Rates and Sensitivity

Zero coupon bonds are more sensitive to yield changes.Zero coupon bonds are more sensitive to yield changes.

Sensitivity increases with maturity.Sensitivity increases with maturity.


Changes in Bond Prices Changes in Bond Prices

ABCD

Change in yield to maturity (%)

Perc

en

tag

e c

han

ge in

bon

d p

rice

BondBond CouponCoupon MaturityMaturity Initial Initial YTMYTM

AA 12%12% 5 years5 years 10%10%

BB 12%12% 30 years30 years 10%10%

CC 3%3% 30 years30 years 10%10%

DD 3%3% 30 years30 years 6%6%


Sensitivity of PricesSensitivity of Prices

As the maturity increases from A to B, the bond As the maturity increases from A to B, the bond becomes more sensitive.becomes more sensitive.

As the coupon rate decreases from B to C, the bond As the coupon rate decreases from B to C, the bond becomes more sensitive.becomes more sensitive.

As the initial yield to maturity decreases from C to D, As the initial yield to maturity decreases from C to D, the bond becomes more sensitive (for coupon paying the bond becomes more sensitive (for coupon paying bonds). At lower yields, the more distant payments bonds). At lower yields, the more distant payments have relatively greater present values and account for have relatively greater present values and account for a greater share of the bond’s total value. a greater share of the bond’s total value.


Trading StrategiesTrading Strategies

Suppose you expect that the Federal Reserve will Suppose you expect that the Federal Reserve will lower interest rates at the next FOMC meeting. You lower interest rates at the next FOMC meeting. You want to build a portfolio of bonds such that you want to build a portfolio of bonds such that you maximize the gain in the value of your portfolio. maximize the gain in the value of your portfolio. Would you construct a portfolio with Would you construct a portfolio with

A. Maximum interest rate sensitivity?A. Maximum interest rate sensitivity? B. Minimum interest rate sensitivity?B. Minimum interest rate sensitivity?

Answer: A.Answer: A. You want to build a portfolio that will You want to build a portfolio that will maximize the price appreciation for a negative change maximize the price appreciation for a negative change in interest rates, that is one with in interest rates, that is one with maximum maximum interest interest rate sensitivity. rate sensitivity.



So how can we construct a portfolio that is most So how can we construct a portfolio that is most sensitive to interest rates movement?sensitive to interest rates movement?

We know that we want to choose:We know that we want to choose:

- Portfolio of long maturity bonds over a portfolio of - Portfolio of long maturity bonds over a portfolio of short maturity bondsshort maturity bonds

- Portfolio of low coupon bonds over a portfolio of - Portfolio of low coupon bonds over a portfolio of high coupon bondshigh coupon bonds

In other words, ideally you would want to hold a In other words, ideally you would want to hold a portfolio of portfolio of long maturity zero-coupon bondslong maturity zero-coupon bonds!!



Suppose Suppose wwe want to hold a bond portfolio but suspect e want to hold a bond portfolio but suspect that interest rates are about to that interest rates are about to riserise. What should we . What should we do?do?

We would want to hold a portfolio with We would want to hold a portfolio with minimum minimum interest rate sensitivityinterest rate sensitivity..

Short-maturity bonds with high coupons.Short-maturity bonds with high coupons.


DurationDuration


The Duration MeasureThe Duration Measure

Developed by Frederick R. MacaulayDeveloped by Frederick R. Macaulay tt = = time at which coupon or principal payment occurs time at which coupon or principal payment occurs

CCtt = = interest or principal payment that occurs at time interest or principal payment that occurs at time tt

rr = = yield to maturity on the bondyield to maturity on the bond

Price

)(

)1(

)1( 1

1

1

n

tt

n

tt

t

n

tt

t tCPV

r

C

tr

C

D


Characteristics of Macaulay DurationCharacteristics of Macaulay Duration

A zero-coupon bond’s duration equals its A zero-coupon bond’s duration equals its maturitymaturity

Duration of a bond with coupons is always less Duration of a bond with coupons is always less than its term to maturity because duration gives than its term to maturity because duration gives weight to these interim paymentsweight to these interim payments

There is an inverse relationship between duration There is an inverse relationship between duration and coupon size.and coupon size.

There is a positive relationship between term to There is a positive relationship between term to maturity and duration, but duration increases at a maturity and duration, but duration increases at a decreasing rate with maturity.decreasing rate with maturity.

There is an inverse relationship between YTM and There is an inverse relationship between YTM and durationduration..


Bond Pricing with Continuous CompoundingBond Pricing with Continuous Compounding

is the price of the i-th cash flow

at time ,

given the yield to maturity

Continuous Compounding

Semi-annualCompounding

Note: This is not done in practice, used mostly in academic models, but it will be useful to illustrate the idea of duration.


Continuous CompoundingContinuous Compounding

• If we compound more and more frequently...

• Every minute, every second, every millisecond...

• Then in the limit, we have ccontinuous compounding...ontinuous compounding...

Letting x = YTM and taking the inverse, that is 1/exp(x), we have our 1-year continuous

compounding discount factor.


Bond PricingBond Pricing

Continuous Compounding

In General

where

Price of a bond with n

coupon payments.


DurationDuration is the weighted average maturity of a bond's cash is the weighted average maturity of a bond's cash flows.flows.

Note that for Note that for continously compounded yieldscontinously compounded yields we have: we have:

Price

)exp(1

n

tt trtC

D

Duration (Continuous Compounding)Duration (Continuous Compounding)

where Ct is the cash flow

at time t, and r is the yield to maturity

Cash flow times t weighted by

present value of cash flow.


More GenerallyMore Generally

is the price, the i-th cash flow

at time and

is the yield to maturity

For the case of a fixed income security with continuous compounding with n cash flows Ci each occurring at time ti


Sensitivity of Price to YieldSensitivity of Price to Yield

Differentiate price with respect to the yield to maturity.

So given continuously compounded yields, duration is the (negative) percentage chance in price to change in the yield.


Modified DurationModified Duration

For yields compounded m time per year, an adjusted For yields compounded m time per year, an adjusted measure of duration can be used to approximate measure of duration can be used to approximate the percentage change in price to a change in the percentage change in price to a change in yield:yield:

m

y1

durationMacaulay duration modified mod

D



1 2

2 3

We know that

...1 1 1 1

where C is the coupon, F is the face value,

y is the yield of the bond, and T the maturity.

Differentiating with respect to y we get that

1 2P..

y 1 1

T T

C C C FP

y y y y

C C

y y

1 1

1 2

1 2

.1 1

1 1* 2* * *...

1 1 1 1 1

Dividing with P on both sides we get that

P 1 1 1* 2* * * 1...

y 1 1 1 1 1

Cash flow at time t / 11*

1

T T

T T

T T

t

T C T F

y y

C C T C T F

y y y y y

C C T C T F

P y Py y y y

yt

y

*

Bond Price

1*

1D D

y


Modified Duration and Bond Price VolatilityModified Duration and Bond Price Volatility

Bond price movements will vary proportionally with Bond price movements will vary proportionally with modified duration for small changes in yieldsmodified duration for small changes in yields

An estimate of the percentage change in bond prices equals An estimate of the percentage change in bond prices equals the change in yield multiplied by modified durationthe change in yield multiplied by modified duration

yDP

P

mod

P = change in price for the bond

P = beginning price for the bond

Dmod = the modified duration of the bond

y = yield change


Test your Intuition AgainTest your Intuition Again

Suppose you are hired to manage Eggbert Kapital Suppose you are hired to manage Eggbert Kapital Management’s portfolio of bonds, and you expect a Management’s portfolio of bonds, and you expect a significant decrease in interest rates. Should you significant decrease in interest rates. Should you invest in long maturity bonds with low coupons, or invest in long maturity bonds with low coupons, or low maturity bonds with high coupons? High low maturity bonds with high coupons? High duration bond or low duration bonds?duration bond or low duration bonds?

Answer: You should invest bonds whose price will Answer: You should invest bonds whose price will go up the most if interest rates fell, so pick bonds go up the most if interest rates fell, so pick bonds with with long maturities and low coupon rateslong maturities and low coupon rates. You . You would want to pick would want to pick high duration bondshigh duration bonds!!


Interpretation of DurationInterpretation of Duration

0

200

400

600

800

1000

1200

1 2 3 4 5 6 7 8

Year

Cas

h flo

w Bond Duration = 5.97 years

Blue: present value of each cash flow. Fulcrum

Point

8-year, 9% annual coupon bond8-year, 9% annual coupon bond


Bond Duration in Years Under Different TermsBond Duration in Years Under Different Terms

COUPON RATES

Years toMaturity 0.02 0.04 0.06 0.08

1 0.995 0.990 0.985 0.9815 4.756 4.558 4.393 4.254

10 8.891 8.169 7.662 7.28620 14.981 12.980 11.904 11.23250 19.452 17.129 16.273 15.829

100 17.567 17.232 17.120 17.064

8 17.167 17.167 17.167 17.167

Source: L. Fisher and R. L. Weil, "Coping with the Risk of Interest Rate Fluctuations:

Returns to Bondholders from Naïve and Optimal Strategies," Journal of Business 44, no. 4

(October 1971): 418. Copyright 1971, University of Chicago Press.

YTM=6%


Bond Duration vs Maturity Bond Duration vs Maturity

15-5115-51



This is the first derivative of price with respect to the This is the first derivative of price with respect to the yield.yield.

For small changes this will give a good estimate, but For small changes this will give a good estimate, but this is a linear estimate on the tangent linethis is a linear estimate on the tangent line

dy

dPPD mod


ExampleExample

Eggbert’s Egg Company issued an 8% bond with 3 Eggbert’s Egg Company issued an 8% bond with 3 years to maturity. The yield to maturity is 8% and the years to maturity. The yield to maturity is 8% and the bond pays semi-annual coupons.bond pays semi-annual coupons.

What is the Macaulay duration? What is the Modified What is the Macaulay duration? What is the Modified Duration?Duration?

(See Spreadsheet)(See Spreadsheet)


ExampleExample

Year Payment PV Weight Year*Weight

0.5 4 3.846154 0.038462 0.019231

1 4 3.698225 0.036982 0.036982

1.5 4 3.555985 0.03556 0.05334

2 4 3.419217 0.034192 0.068384

2.5 4 3.287708 0.032877 0.082193

3 104 82.19271 0.821927 2.465781

Price 100 1 2.725911

D = 2.5726 and Dm = 2.621D = 2.5726 and Dm = 2.621


Short Cut FormulasShort Cut Formulas

Macaulay Duration “Short Cut” FormulasMacaulay Duration “Short Cut” Formulas

Modified Duration “Short Cut” FormulaModified Duration “Short Cut” Formula

where c = coupon rate per periodwhere c = coupon rate per period

y = yield per periody = yield per period

T = periods remaining T = periods remaining

yyc

yycT

y T

11

1/11D*

yyc

ycTy

y

yT

11

11D

yyc

ycTy

y

yT

12/1

2/2/22/12/1D

2

Semi-annual coupons


Price-Yield CurvePrice-Yield Curve

Price

Yield to Maturity


Duration provides a first-order approximation of the price-yield curve.

Approximating a curve with a straight line

First-order ApproximationFirst-order Approximation

Approximation Error

Price

Yield

PDslope mod


ConvexityConvexity


ConvexityConvexity

The The convexity convexity is the measure of the curvature and is is the measure of the curvature and is the second derivative of price with respect to yield the second derivative of price with respect to yield ((dd22P/dyP/dy22) divided by price) divided by price

Convexity is the Convexity is the percentage changepercentage change in in dP/dydP/dy for a for a given change in yieldgiven change in yield

P

dy

Pd2

2

Convexity


We can better approximate a curve using a quadratic function.

Correction for ConvexityCorrection for Convexity

2)(21 yConvexityyD

P

P

T

tt

t tty

CF

yPConvexity

1

22

)()1()1(

1

T

t t

t tty

CFy

PConvexity

2

1

2/

2

)2

1

2)(

2(

)2

1()2

1(

1Semi-annual

coupons


Duration plus ConvexityDuration plus Convexity

The Taylor Series for a function P around y is given byThe Taylor Series for a function P around y is given by

P(y+P(y+ΔΔy) = Py) = P(y) + P’(y) (y) + P’(y) ΔΔy + ½ P’’(y) (y + ½ P’’(y) (ΔΔy)y)2 2 + ...+ ...

Thus, as a second order approximationThus, as a second order approximation

P(y+P(y+ΔΔy) - Py) - P(y) (y) ≈≈ P’(y) P’(y) ΔΔy + ½ P’’(y) (y + ½ P’’(y) (ΔΔy)y)22

Therefore,Therefore,

ΔΔP ≈ - Dm P P ≈ - Dm P ΔΔy + ½ P C (y + ½ P C (ΔΔy)y)22 P C2

2

dy

Pd


Duration plus ConvexityDuration plus Convexity

Thus for a small change in the yield Thus for a small change in the yield ΔΔy, the modified y, the modified duration and the convexity give the second-order duration and the convexity give the second-order approximation to the price-yield curve. approximation to the price-yield curve.

ΔΔP ≈ P ≈ - Dm P - Dm P ΔΔyy + + ½ P C (½ P C (ΔΔy)y)22 Price change due to durationPrice change due to duration

- Dm P - Dm P ΔΔyy Price change due to convexity Price change due to convexity

½ P C (Δy)2


Convexity of BondsConvexity of Bonds

0

Change in yield to maturity (%)

Perc

en

tag

e c

han

ge in

bon

d

pri

ce Portfolio

Duration + ConvexityDuration


Duration-Convexity EffectsDuration-Convexity Effects

Changes in a bond’s price resulting from a change in Changes in a bond’s price resulting from a change in yields are due to:yields are due to: Bond’s modified duration Bond’s convexity

Relative effect of these two factors depends on the Relative effect of these two factors depends on the characteristics of the bond (its convexity) and the size characteristics of the bond (its convexity) and the size of the yield changeof the yield change

Convexity is desirableConvexity is desirable


Portfolio ImmunizationPortfolio Immunization


Protection from Interest Rate RiskProtection from Interest Rate Risk

Pension funds, insurance companies and other Pension funds, insurance companies and other financial institutions hold billions of dollars in fixed financial institutions hold billions of dollars in fixed income securities. income securities.

One of the most widely used analytical techniques is One of the most widely used analytical techniques is known as immunization, as it protects a bond known as immunization, as it protects a bond portfolio from interest rate movements.portfolio from interest rate movements.

This is of major practical value so let’s learn how to This is of major practical value so let’s learn how to structure a portfolio to protect it against interest rate structure a portfolio to protect it against interest rate risk!risk!


2 Scenarios2 Scenarios

Scenario 1Scenario 1: You want to save money for a major : You want to save money for a major expense one year from now. expense one year from now.

Scenario 2Scenario 2: You want to save money to pay for your : You want to save money to pay for your kid’s college tuition 20 years from now.kid’s college tuition 20 years from now.

Given scenario 1Given scenario 1: If you invest in one year treasury : If you invest in one year treasury bills, there is very little risk. Investing in 20 year bills, there is very little risk. Investing in 20 year treasuries exposes you to treasuries exposes you to interest rate riskinterest rate risk..

Given scenario 2Given scenario 2: Holding 20 year treasury bonds : Holding 20 year treasury bonds would provide predictable results, investing in one would provide predictable results, investing in one year T-bills would result in year T-bills would result in reinvestment riskreinvestment risk. .


Life Insurance CompanyLife Insurance Company

Suppose you work for a life insurance company, and Suppose you work for a life insurance company, and you expect to make a series of cash payments. you expect to make a series of cash payments.

One thing you can do is to purchase zero-coupon One thing you can do is to purchase zero-coupon bonds, having different maturities so that the bonds, having different maturities so that the principal exactly matches each separate obligation.principal exactly matches each separate obligation.

This may not be the best option as corporate bonds This may not be the best option as corporate bonds offer higher yields and zero-coupon corporate bonds offer higher yields and zero-coupon corporate bonds are rare.are rare.


Matching DurationsMatching Durations

Match the duration of your portfolio with the duration Match the duration of your portfolio with the duration of your obligations.of your obligations.

Intuition:Intuition: If the duration of your bond portfolio If the duration of your bond portfolio matches the duration of your obligation stream, then matches the duration of your obligation stream, then the present value of both your bond portfolio and the present value of both your bond portfolio and your obligation stream will respond (to a first-order your obligation stream will respond (to a first-order approximation) to a change in the underlying yield.approximation) to a change in the underlying yield.

SpecificallySpecifically, if yields decrease, the present value of , if yields decrease, the present value of your obligations will increase, but the value of your your obligations will increase, but the value of your bond portfolio will increase (approximately) by the bond portfolio will increase (approximately) by the same amount – so the value of your portfolio will be same amount – so the value of your portfolio will be enough to cover the obligation!enough to cover the obligation!


Duration for a PortfolioDuration for a Portfolio

If the portfolio has price PVIf the portfolio has price PV

PV = V1 + V2 + V3 + ... + Vm PV = V1 + V2 + V3 + ... + Vm

D = w1D1 + w2D2 + ... + wmDmD = w1D1 + w2D2 + ... + wmDm

where wi = Vi/ PV in a portfolio of m bonds. Vi is where wi = Vi/ PV in a portfolio of m bonds. Vi is the value of Bond i in the portfolio.the value of Bond i in the portfolio.


ExampleExample

Suppose Eggbert’s Egg Company has to pay for a Suppose Eggbert’s Egg Company has to pay for a new chocolate factory 10 years from now. The cost of new chocolate factory 10 years from now. The cost of the factory is 1 million dollars and it wishes to invest the factory is 1 million dollars and it wishes to invest that money now. that money now.

Suppose no zero-coupon bonds of that maturity are Suppose no zero-coupon bonds of that maturity are available. Suppose there are 3 corporate bonds available. Suppose there are 3 corporate bonds (FV=100) to choose from:(FV=100) to choose from:

Coupon Maturity Price Yield 1 1 6%6% 3030 69.0469.04 9%9%

22 11%11% 1010 113.01113.01 9%9%3 3 9%9% 2020 100.00100.00 9%9%


ExampleExample

Duration of each Bond:Duration of each Bond:D1 = 11.44D1 = 11.44

D2 = 6.54D2 = 6.54 D3 = 9.61D3 = 9.61

Since Bond 1 has duration greater than 10, we Since Bond 1 has duration greater than 10, we mustmust use use this bond in our portfolio.this bond in our portfolio.

Let’s use bonds 1 and 2. (See Spreadsheet)Let’s use bonds 1 and 2. (See Spreadsheet)


ExampleExample

Present Value of the obligation is: $414, 642.86Present Value of the obligation is: $414, 642.86 The duration of the obligation is 10 years.The duration of the obligation is 10 years.

To immunize the portfolio we need to To immunize the portfolio we need to 1. Equate the present value of the portfolio with the 1. Equate the present value of the portfolio with the

present value of the obligation.present value of the obligation. 2. Equate the duration of the portfolio with the 2. Equate the duration of the portfolio with the

duration of the obligation.duration of the obligation.


ExampleExample

The value of total invested in the Bonds 1 and Bond 2 has to The value of total invested in the Bonds 1 and Bond 2 has to equal to PV of the obligationequal to PV of the obligation

V1 + V2 = PV = $414, 642.86V1 + V2 = PV = $414, 642.86

And the duration of the bond portfolio has to equal the And the duration of the bond portfolio has to equal the duration of the obligationduration of the obligation

V1/PV * D1 + V2/PV* D2 = 10V1/PV * D1 + V2/PV* D2 = 10

We get: We get: V1/PV * D1 + (PV – V1) /PV* D2 = 10V1/PV * D1 + (PV – V1) /PV* D2 = 10

=> One equation and 1 unknown.=> One equation and 1 unknown.


ExampleExample

From the spreadsheet:From the spreadsheet:

V1 = 292617.60V1 = 292617.60V2 = 122025.26V2 = 122025.26

We want to purchase We want to purchase

V1/P1 = 4238.20 shares of Bond 1V1/P1 = 4238.20 shares of Bond 1V2/P2 = 1079.80 shares of Bond 2V2/P2 = 1079.80 shares of Bond 2

Of, course, in practice, you will have to round these numbers.Of, course, in practice, you will have to round these numbers.


ExampleExample

Suppose there is a sudden shift in the yield to either Suppose there is a sudden shift in the yield to either 8% or 10%. 8% or 10%.

The value of the bond portfolio is:The value of the bond portfolio is:

Portfolio Obligation Error 8%8% 457928.3 457928.3 456386.9456386.9 +1541.33 +1541.33 10% 378075.210% 378075.2 376889.5376889.5 +1185.69 +1185.69

Due to convexity, the portfolio is always worth more Due to convexity, the portfolio is always worth more than the obligation.than the obligation.


HomeworkHomework

See if you can replicate the previous example at See if you can replicate the previous example at home, without looking at the spreadsheet. home, without looking at the spreadsheet.

If you look at the spreadsheet answer first, you will If you look at the spreadsheet answer first, you will think that this is very easy.think that this is very easy.

Try doing it from scratch.Try doing it from scratch. Replicate the immunization problem using only Replicate the immunization problem using only

Bonds 1 and 3.Bonds 1 and 3.


Yield Curve StrategiesYield Curve Strategies

Bullet strategyBullet strategy: maturity of the securities are highly : maturity of the securities are highly concentrated at one point on the curve.concentrated at one point on the curve.

Barbell strategyBarbell strategy: maturity of securities included in the : maturity of securities included in the portfolio are concentrated at two extreme maturities.portfolio are concentrated at two extreme maturities.

Ladder strategyLadder strategy: the portfolio is constructed to have : the portfolio is constructed to have approximately equal amounts of each maturity.approximately equal amounts of each maturity.



Bond Coupon Maturity Price YTM Duration_ Convexity

A 0.085 5 100 0.085 4.0054435 19.81635

B 0.095 20 100 0.095 8.8815081 124.1702

C 0.0925 10 100 0.0925 6.43409 55.45054

Bullet Portfolio: 100% bond C

Barbell Portfolio: 50.2% bond A and 49.8% bond B



Duration of Bullet: 6.43409

Duration of Barbell: 6.433724

Convexity of Bullet: 55.45054

Convexity of Barbell: 71.78459

Yield of Bullet: 0.0925

Yield of Barbell: 0.08998

This is the convexity yield

0.00252

Give up yield to get better convexity.


Barbell vs. BulletBarbell vs. Bullet

The choice between these strategies depends on the The choice between these strategies depends on the magnitude of the shift in yields.magnitude of the shift in yields.

Although convexity is preferred, there is the Although convexity is preferred, there is the disadvantage of a disadvantage of a convexity yieldconvexity yield, as the market , as the market charges a higher price and offers a lower yield.charges a higher price and offers a lower yield.

Thus the benefits from convexity are only realized for Thus the benefits from convexity are only realized for large shifts in yields.large shifts in yields.

See the article in the course reader for details.See the article in the course reader for details.

0 portfolio management 3-228-07 albert lee chun duration, convexity and bond portfolio management...

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