International Fixed Income Topic IB: Fixed Income Basics - Risk.

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International Fixed Income Topic IB: Fixed Income Basics - Risk
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Transcript of International Fixed Income Topic IB: Fixed Income Basics - Risk.

Page 1: International Fixed Income Topic IB: Fixed Income Basics - Risk.

International Fixed Income

Topic IB: Fixed Income Basics - Risk

Page 2: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Readings

• Duration: An Introduction to the Concept and Its Uses, (Dym & Garbade, Bankers Trust (1984))

• Convexity: An Introduction, (Yawitz, Goldman Sachs)

Page 3: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Outline

II. Interest rate risk

A. Interest rate sensitivity - Summary

B. Duration

C. Convexity

D. Hedging

Page 4: International Fixed Income Topic IB: Fixed Income Basics - Risk.

A. Interest Rate Sensitivity

• Values of fixed income securities change as economic conditions change.

• Even though bond prices are not perfectly correlated, they tend to move together. People try and relate bond prices to a single variable, “level of interest rates”.

• They want simple answers to questions like: "How much will the value of my portfolio change if interest rates go up 10 basis points?"

Page 5: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Price-Yield Relation• For zeroes, there is a very explicit formula relating the

price to its discount rate or yield.• For coupon bonds, or portfolios with fixed cash flows,

we have a formula that gives the price as a function of all the discount rates associated with the cash flows. Alternatively, we have a formula that gives price as a function of yield.

• For other instruments, there is no explicit formula relating price to interest rates. Instead, they require a model which incorporates both interest rates and estimates of volatility.

Page 6: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Parallel Shifts

Maturity (years)

Yield

0 3010 20

Increase in interest rates

Decrease in interest rates

Page 7: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Price-Yield relation: Illustration

In general, the price of a bond is given by

P

Mc

r

M

rt

tt

T

T

T

/

( / ) ( / )/

2

1 2 1 221

2

2

P

Mc

y

M

yt

t

T

T

/

( / ) ( / )

2

1 2 1 21

2

2

But, if the yield curve is flat, then each of the spot rates must equal the bond’s yield y:

Result: y provides a complete description of the term structure

Page 8: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Zero Prices as a Function of Yield

0

20

40

60

80

100

0 2 4 6 8 10 12 14 16

Yield (%)

Price

30-year

10-year

5-year

Consider three zeros with maturities of 5,10 and 30 years. What do their prices look like as a function of their yields?

Page 9: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Terminology

• Delta - measures how the price (i.e., bond) changes as the underlying (i.e., interest rate) changes.

• Gamma - measures how the Delta changes as the underlying (i.e., interest rate) changes.

Page 10: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Characteristics of the Price/Yield Relation

• The higher the yield, the lower the price (Delta is negative)

• The higher the yield, the smaller the magnitude of delta (Prices are convex in the yield, i.e., Gamma is positive)

• The longer the maturity, the higher the magnitude of delta (longer maturity bonds are more sensitive to interest rate changes than shorter maturity bonds)

Page 11: International Fixed Income Topic IB: Fixed Income Basics - Risk.

The Effect of Convexity

Price

Yield

P*

Y* Y Y**

P**P

Y-Y**=Y*-Y, butP-P**<P*-P

Page 12: International Fixed Income Topic IB: Fixed Income Basics - Risk.

The Effect of MaturityPercent decrease in price for a 100 basis point increase

25.17

24.53

24.01

23.35

9.21

8.97

8.74

8.5

4.72

4.59

4.46

4.34

0 5 10 15 20 25 30

3%

6%

9%

12%

Leve

l of

y

% change

30-year 10-year 5-year

Page 13: International Fixed Income Topic IB: Fixed Income Basics - Risk.

The Effect of the Coupon RatePercent decrease price for a 100 basis point increase

25.17

24.53

24.01

23.35

15.74

12.83

10.25

8.21

14.1

11.5

9.31

7.61

0 5 10 15 20 25 30

3%

6%

9%

12%

Level of y

% change

30-yr zero 30-yr 5% 30-yr 10%

Page 14: International Fixed Income Topic IB: Fixed Income Basics - Risk.

General Conclusions

• Level effect: Regardless of the coupon rate, the magnitude of delta is decreasing in yield (sensitivity is greater when yields are low)

• Maturity effect: Regardless of the coupon rate, the magnitude of delta is increasing in maturity (longer maturities are more sensitive)

• Coupon effect: The lower the coupon, the more sensitive the price to changes in interest rates

Page 15: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Outline

II. Interest rate risk

A. Interest rate sensitivity - Summary

B. Duration

C. Convexity

D. Hedging

Page 16: International Fixed Income Topic IB: Fixed Income Basics - Risk.

B. Duration

• Loose Definition: The duration of a bond is an approximation of the percent change in its price given a 100 basis point change in interest rates.

• For example, a bond with a duration of 7 will gain about 7% in value if interest rates fall 100 bp.

• For zeroes, this measure is easy to define and compute with a formula.

• For securities with fixed cash flows, we must make assumptions about how rates shift together.

• To compute duration for other instruments requires further assumptions and numerical estimation.

Page 17: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Dollar Duration of Zeroes• Definition: The dollar duration of a zero-coupon

bond is a linear approximation of the dollar change in its price divided by the change in its discount rate. Because $dur is essentially the derivative of the bond price with respect to the interest rate, we often call it the bond's Delta.

rdurP

dur

$

i.e. ,$ ratein ChangePricein Change

Page 18: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.2

0.4

0.6

0.8

1

1.2

30-year spot rate

Price of 30-year zero

At a rate of 5%, the price is 0.2273If rates fall to 4%,the price is 0.3048

The actual change is 0.077

dr30

3060

1

1 2

( / )

100 bp

Using a linear approximation, the change is about 0.0665

Page 19: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example: Dollar Duration

• The dollar duration of $1 par of a 30-year zero at an interest rate of 5% is 6.65, as illustrated in the last slide.

• -0.0665/(-0.01)=0.0665/0.01=6.65.• The illustration shows that the dollar duration is

related to the slope of the price-rate function.• We can use calculus to get an explicit formula for

the dollar duration of any zero.

Page 20: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Dollar Duration: Formula

65.6)2/05.01(

30

)2/1()('

2273.0)2/05.01(

1

)2/1(

1)(

6112

602

tt

tt

tt

tt

r

trd

rrd

To avoid working with negative numbers,the dollar duration is quoted in positive terms,that is, 6.65.

Page 21: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Dollar Duration: Example

What's the dollar duration of $1 par of a 1.5-year zero if the 1.5-year discount rate is 5.47%?

actual price rise is 0.0009432

346535.142

0547.122

)1(5.1

)1(

ttrt

Page 22: International Fixed Income Topic IB: Fixed Income Basics - Risk.

The Approximation

3 6 9 120

0.020.040.060.08

Bond price

change

3 6 9 12

Level of y

Decrease in bond price per 1 basis point increase

ActualChange

Delta Approx

3 6 9 1202468

Bond price

change

3 6 9 12

Level of y

Decrease in bond price per 100 basis points increase

Actualchange

Delta approx

3 6 9 120

0.20.40.60.8

Bond price

change

3 6 9 12

Level of y

Decrease in bond price per 10 basis points increase

Actualchange

Delta approx

3 6 9 1206

121824

Bond price

change

3 6 9 12

Level of y

Decrease in bond price per 300 basis points increase

Actualchange

Delta approx

Page 23: International Fixed Income Topic IB: Fixed Income Basics - Risk.

DurationDuration is a measure of the interest ratesensitivity of the bond that does not depend onscale or size. It is defined as the dollar durationscaled by the value of the bond:

For a t-year zero, we have

)1(Duration

2)1(1

)1(

22

122

tttr

ttr

r

tt

Page 24: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Duration: Example

• Duration approximates the percent change in price for a 100 basis point change in rates

• For example, at an interest rate of 5.47%, the duration of the 1.5-year zero is

46.12/0547.01

5.1

)2/1(duration

46.19222.0

3465.1

price

durationdollar =duration

tr

t

Page 25: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Duration: Example Continued...

26.292/05.01

30

)2/1(duration

2273.0

65.6

price

durationdollar =duration

tr

t

At an interest rate of 5%, the duration of a30-year zero in our example is

Page 26: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Macaulay DurationNote that the duration of a zero is just slightlyless than its maturity. This measure of durationis known as MODIFIED duration.

This is to distinguish itself from another measure of duration, MACAULAY duration, which equals: MODIFIED(1+r/2)=t years.

Macaulay duration is popular because it allowsus to describe duration in terms of the years thecash flows of the bond will be around.

Page 27: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Duration of a Portfolio of Cash Flows• Definition: The dollar duration of a portfolio approximates

the dollar change in portfolio value divided by the change in interest rates, assuming all rates change by the same amount.

• It follows that the dollar duration of a portfolio is the sum of the dollar durations of each of the cash flows in the portfolio.

• Why? The change in the portfolio value is the sum of the changes in the value of each cash flow. – The dollar duration of each cash flow describes its value change.

– The sum of all the dollar durations describes the total change.

Page 28: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Formula

• Suppose the portfolio has cash flows K1, K2, K3,... at times t1, t2, t3,.... Then its dollar duration would be

n

jt

t

jj

j

jr

tK

112)2/1(

Page 29: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

• What is the dollar duration of a portfolio of consisting of $500 par of the 1.5-year zero and $100 par of the 30-year zero?– (500 x 1.35) + (100 x 6.65) = 1340– This means the portfolio value will change

about $13.40 for every 100 basis point shift in interest rates.

Page 30: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Portfolio Value as a Function of Interest Rate Shifts

Instead of thinking of the portfolio value as a function of all of the different discount rates, we can think of it as a function of just the change or shift s in all rates:

603 )2/)0500.0(1(

100

)2/)0547.0(1(

500)(

sssV

At s=0, V=483.85. At s=.005 (50 bp increase), V=477.41.

Page 31: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Dollar Duration and Its Derivative

n

jt

t

jjs

n

jt

t

j

j

j

j

j

r

tKsV

sr

KsV

1120

12

)2/1(|)('$dur

)2/)(1()(

The dollar duration of a portfolio is the sum of the dollar durations of the component zeroes.Just as with zeroes, the dollar duration of the portfolio value is related to its derivative. We take the derivative with respect to the change in interest rates, s, and evaluate it at level, s = 0:

Page 32: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Duration of a Portfolio

• Just as with a zero, the duration of a portfolio is its dollar duration divided by its market value.

• The duration gives the percent change in value for each 100 basis point change in all rates.

Page 33: International Fixed Income Topic IB: Fixed Income Basics - Risk.

ExampleThe duration of the portfolio consisting of $500par of the 1.5-yr zero and $100 par of the 30-yrzero is

8.285.483

1340

uemarket val

durationdollar =duration

This means that the portfolio will change 2.8%for every 100 basis points change in rates.

Page 34: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Formula: Duration of a Portfolio

n

jtj

n

j t

jtj

j

j

j

dK

r

tdK

1

1 )2/1()(

=duration

The duration of the portfolio is the average duration of the component zeroes, weighted bytheir market values.

Page 35: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

• Recall the portfolio consisting of $500 par of the 1.5-year zero and $100 par of the 30-year zero.– The market value of the 1.5-year zero is 500 x 0.92224

= $461.12. Its duration is 1.46.

– The market value of the 30-year zero is 100 x 0.2273 = $22.73. Its duration is 29.26.

• The duration of the portfolio is

8.273.22$12.461$

)26.2973.22($)46.112.461($

Page 36: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Macaulay Duration

n

jt

j

n

jjt

j

j

j

y

K

ty

K

12

12

)2/1(

)2/1(=durationMacaulay

The Macaulay duration of a portfolio is the averagematurity of each cash flow, weighted by its presentvalue at the yield on each security. [It is theModified Duration times (1+y/2)].

Page 37: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Coupon and Maturity Effects

5-year 10-year 30-year 15%10%

5%Zero

05

101520253035

Duration

5-year 10-year 30-year 15%10%

5%Zero

Maturity of the Bond

Macaulay Duration at current yield of 10%

15%

10%

5%

Zero

Page 38: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Problems with Duration

• Accuracy: duration is accurate only for small yield changes.

• Applicability: duration begins to break down for nonparallel shifts in the yield curve.

• Generality: duration is only valid for option-free bonds.

Page 39: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Outline

II. Interest rate risk

A. Interest rate sensitivity - Summary

B. Duration

C. Convexity

D. Hedging

Page 40: International Fixed Income Topic IB: Fixed Income Basics - Risk.

C. Convexity• Convexity is a measure of the curvature of the value of

a security or portfolio as a function of interest rates. It tells you how the duration changes as interest rates change.

• As its name suggests, convexity is related to the second derivative if the price function. As such, it is often called a bond's Gamma.

• Using convexity together with duration gives a better approximation of the change in value given a change in interest rates than using duration alone.

Page 41: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Illustration

Yieldyy** y*

Steeplysloped

Mildlyslope Almost

flat

As y changes to y** (y*), the slope of the bond pricing function increases (decreases). This slope is simply the dollar duration of the bond.

Price

Page 42: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Illustration

Yield

Actual price

y

Error in estimating pricebased only on duration

Slope at y,( i.e., dollar duration)

Price

Page 43: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

4.50% 5.00% 5.50% 6.00% 6.50% 7.00% 7.50% 8.00% 8.50%0.15

0.2

0.25

0.3

0.35

0.4

0.45

20-Year Discount Rate

20-Year Zero Price

d rr20 20

2040

1

1 2( )

( / )

approximation of price function around the point r = 6.5%

error

Page 44: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Correcting the Duration Error

• The price-rate function is not linear.

• Duration and dollar duration use a linear approximation to the price rate function to measure the change in price given a change in rates.

• The error in the approximation can be substantially reduced by making a convexity correction.

Page 45: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Taylor Series

Recall from calculus that the value of a function can be approximated near a given point using its Taylor series around that point. Using only the first two derivatives, the Taylor series approximation is:

200000 )()(''

2

1)()(')()( xxxfxxxfxfxf

Page 46: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Derivatives!

convexitydollar )2/1(

2/)(''

durationdollar -)2/1(

)('

price)2/1(

1)(

22

2

12

2

tt

tt

tt

tt

tt

tt

r

ttrd

r

trd

rrd

Page 47: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

0043.107)2/065.01(

410

)2/1(

2/)(''

389364.5)2/065.01(

20

)2/1()('

278226.0)2/065.01(

1

)2/1(

1)(

4222

2

4112

402

tt

tt

tt

tt

tt

tt

r

ttrd

r

trd

rrd

For the 20-yr at 6.5%, we get:

Page 48: International Fixed Income Topic IB: Fixed Income Basics - Risk.

The Convexity Correction

20,0,2

10,0,0, )()()()()()( tttttttttttt rrrdrrrdrdrd

Applying the Taylor series approximation, thechange in the zero price given a change in rates:

Change in price = -dollar duration x change in rates (1/2) x dollar convexity x change in rates squared

Page 49: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

How does the 20-year zero price change as its discount rate changes from 6.5% to 7.5%? The actual change is:

048888.0

278226.0229038.0

)065.0()075.0( 4040 )2/065.01(1

)2/075.01(1

2020

dd

Page 50: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example continued...

Change in price = -dollar duration x change in rates (1/2) x dollar convexity x change in rates squared-0.538964 + [(1/2) x 107.0043 x 0.0001]=-0.048543

Change in price = -dollar duration x change in rates -5.38964 x 0.01 = -0.538964

Duration approximation is far off

Duration/Convexity approximation does muchbetter

Page 51: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Summary

Rate (%) 20-Year Price Actual Change

Duration Approximation

Duration and Convexity

5.50 0.337852 0.059626 0.053894 0.0592446.50 0.2782267.50 0.229338 -0.048888 -0.053894 -0.048543

Page 52: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Duration/Convexity Approximations for 10-year zero

Decrease in bond price per 1 basis point increase in y

3% 6% 9% 12%0

0.02

0.04

0.06

0.08

Level of y

Bond price change

Actual ChangeGamma ApproxDelta Approx

Decrease in bond price per 100 basis points increase in y

3% 6% 9% 12%0

2

4

6

8

Level of y

Bond price change

Actual changeGamma approxDelta approx

Decrease in bond price per 10 basis points increase in y

3% 6% 9% 12%0

0.2

0.4

0.6

0.8

Level of y

Bond price change

Actual changeGamma approxDelta approx

Decrease in bond price per 300 basis points increase in y

3% 6% 9% 12%0

5

10

15

20

25

Level of y

Bond price change

Actual changeGamma approxDelta approx

Page 53: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Estimating Price Movements

3%

6%

9%

12%

6.85 7.22

6.84 5.00

5.27 4.99

3.69 3.88

3.68 2.74 2.87

2.73

0 2 4 6 8 10

Bond price change

3%

6%

9%

12%

Leve

l of

y

Decrease in bond price per 100 basis points increase in yield

Actual change Delta approx Gamma approx

Page 54: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Estimating Price Movements

3%

6%

9%

12%

18.57 21.67

18.20 13.60

15.80 13.34

10.04 11.63

9.87 7.48

8.62 7.35

0 5 10 15 20 25

Bond price change

3%

6%

9%

12%

Level of y

Decrease in bond price per 300 basis points increase in yield

Actual change Delta approx Gamma approx

Page 55: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Convexity

2

2

2

22

2

)2/1(

2/

)2/1(1

)2/1(2/

=convexity

value

convexitydollar =convexity

tt

t

tt

r

tt

r

rtt

To get a scale-free measure of curvature, convexityis defined as

Convexity of a zero is its maturity squared.

Page 56: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

Maturity Rate Price Dollar Duration

Duration Dollar Convexity

Convexity

10 6.00% 0.553676 5.375493 9.70874 54.7987 98.972620 6.50% 0.278226 5.389364 19.37046 107.0043 384.595130 6.40% 0.151084 4.391974 29.06977 129.8015 859.1356

10-, 20-, 30-yr zero:

Page 57: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Dollar Convexity of a Portfolio

n

jt

t

jjj j

jr

ttK

122

2

)2/1(

2/

Suppose the portfolio has cash flows K1,K2,K3,…at times t1,t2,t3,… then the dollar convexity is

Page 58: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

• Consider a portfolio consisting of – $25,174 par value of the 10-year zero– $91,898 par value of the 30-year zero.

• The dollar convexity of the portfolio is– (25,174 x 54.7987) + (91,898 x 129.8015) =

13,307,997

Page 59: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Convexity of a Portfolio

The convexity of a portfolio is dollar convexitydivided by its value

n

jt

t

j

n

jt

t

jjj

j

j

j

j

r

K

r

ttK

12

122

2

)2/1(

)2/1(

2/

=convexity

Page 60: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Convexity of a Portfolio

Alternatively, market-weighted averageof convexities of zeroes

n

jt

t

j

n

j t

jjt

t

j

j

j

j

j

j

r

K

r

tt

r

K

12

12

2

2

)2/1(

)2/1(

2/

)2/1(=convexity

Page 61: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

• Consider the portfolio of 10- and 30-year zeroes.– The 10-year zeroes have market value

• $25,174 x 0.553676 = $13,938.

– The 30-year zeroes have market value • $91,898 x 0.151084 = $13,884.

– The market value of the portfolio is $27,822.

• The convexity of the portfolio is– 13,307,997/27,822 = 478.32.

Page 62: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example continued...• Alternatively, the convexity of the portfolio

is the average convexity of each zero weighted by market value:

32.478884,13938,13

)1356.859884,13()9726.98938,13(

Page 63: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Effects of Maturity, Coupons, and Yields

5yr 10yr 30yr0

200

400

600

800

1000

Convexity

5yr 10yr 30yr

Maturity of the Bond

Convexity at current yield of 10%

15%10%5%Zero

5yr 10yr 30yr0

200

400

600

800

1000

Convexity

5yr 10yr 30yr

Maturity of the Bond

Convexity at current yield of 5%

15%10%5%Zero

Page 64: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Barbells and Bullets

• We can construct a portfolio of a long-term and short-term zero (a barbell) that has the same market value and duration as an intermediate-term zero (a bullet).

• The barbell will have more convexity.

Page 65: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example• Bullet portfolio: $100,000 par of 20-year zeroes

– market value = $100,000 x 0.27822 = 27,822

– duration = 19.37

• Barbell portfolio: from previous example – $25,174 par value of the 10-year zero

– $91,898 par value of the 30-year zero.

– market value = 27,822

37.19884,13938,13

)06977.29884,13()70874.9938,13(=duration

Page 66: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

• The convexity of the bullet is 385.

• The convexity of the barbell is 478.

Page 67: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Value of Barbell and Bullet

-0.03 -0.02 -0.01 0 0.01 0.02 0.0310,000

20,000

30,000

40,000

50,000

60,000

Shift in Rates

Portfolio Value

Bullet

Barbell

bullet:

barbell:

V ss

V ss s

1 40

2 20 60

100 000

1 0 065 2

25174

1 0 06 2

91898

1 0 064 2

( ),

( ( . ) / )

( ),

( ( . ) / )

,

( ( . ) / )

Page 68: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Does the Barbell Always Do Better?

If there is an immediate parallel shift in interest rates, either up or down, then the barbell will outperform the bullet.If the shift is not parallel, anything could happen.If the rates on the bonds stay exactly the same, then as time passes the bullet will actually outperform the barbell:

the bullet will return 6.5%the barbell will return about 6.2%, a blend of the 6% and 6.4% on the 10- and 30-year zeroes.

Page 69: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Outline

II. Interest rate risk

A. Interest rate sensitivity - Summary

B. Duration

C. Convexity

D. Hedging

Page 70: International Fixed Income Topic IB: Fixed Income Basics - Risk.

D. Hedging Interest Rate Risk

• Suppose you have liabilities or obligations consisting of a stream of fixed cash flows you must pay in the future.

• How can you structure an asset portfolio to fund these liabilities?

Page 71: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Dedication

• The only completely riskless approach is to construct an asset portfolio with cash flows that exactly match the liability cash flows.

• This funding method is called dedication.

• This approach may be infeasible or excessively costly.

• In some situations, risk managers may want more flexibility.

Page 72: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Immunization

• Consider a more flexible but more risky approach, called immunization.– The liabilities have a certain market value.– That market value changes over time as interest

rates change.– Construct an asset portfolio with the same

market value and the same interest rate sensitivity as the liabilities so that the asset value tracks the liability value over time.

Page 73: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Immunization continued...

• If the assets and liabilities have the same market value and interest rate sensitivity, the net position is said to be hedged or immunized against interest rate risk.

• The approach can be extended to settings with debt instruments that do not have fixed cash flows.

Page 74: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Duration Matching

• The most common form of immunization matches the duration and market value of the assets and liabilities.

• This hedges the net position against small parallel shifts in the yield curve.

Page 75: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

• Suppose the liabilities consist of $1,000,000 par value of a 7.5%-coupon 29-year bond.

• This liability has a duration of 12.58.

Page 76: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Mkt. Val. Of Liabilities

0.52.0

3.55.0

6.58

9.511

12.514

15.517

18.520

21.523

24.526

27.529

0

50,000

100,000

150,000

200,000

Maturity

Market Value

Page 77: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

• Construct an asset portfolio that has the same market value and duration as the liabilities using– a 12-year zero and– a 15-year zero.

Page 78: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

Coupon (%) Maturity (Years)

Par Value ($)

Market Value ($)

Dollar Duration

Duration

7.5 29 1000000 1151802 14486304 12.580 12 1 0.4784 5.5668 11.640 15 1 0.3881 5.6412 14.53

The table gives relevant information on the marketvalue and duration of the securities (using class’discount rates):

Page 79: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

Note that if the assets have the same market valueand dollar duration as the liability, then they hagethe same duration as the liability:

To construct the hedge portfolio, solve twoequations:Asset mkt. Val. = Liability mkt. ValAsset $ duration = Liability $ duration

Page 80: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

304,486,146412.55668.5

802,151,13881.04784.0

1512

1512

NN

NN

With N12 and N15 representing the par amountsof the 12- and 15-year zero, we have

Page 81: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example Solution

• In other words, the immunizing asset portfolio consists of $1,626,424 face value of 12-year zeroes and $962,969 face value of 15-year zeroes. By construction it has– the same market value ($1,151,802) and – the same dollar duration (14,486,304), and

therefore – the same duration (12.58), as the liability.

SOLUTION: N12=1,626,424; N15=962,969

Page 82: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Mkt. Val. Of Duration-Matched Portfolio

0.52.0

3.55.0

6.58

9.511

12.514

15.517

18.520

21.523

24.526

27.529

0

200,000

400,000

600,000

800,000

Maturity

Market Value

Page 83: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Performance of Hedge-100 bp -10 bp 0 +10 bp +100 bp

MARKET VALUEAssets 1306689 1166384 1151802 1137411 1016080Liabilities 1312293 1166433 1151802 1137458 1020267Net Equity -5604 -49 0 -47 -4188

DOLLAR DURATIONAssets 16538729 14678971 14486304 14296286 12699204Liabilities 17742932 14777840 14486304 14201643 11921307Net Equity -1204203 -98869 0 96462 777898

DURATIONAssets 12.66 12.59 12.58 12.57 12.50Liabilities 13.52 12.67 12.58 12.49 11.68

Page 84: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Duration/Convexity Hedge• The duration match performed well for small parallel

shifts in the yield curve, but not for large shifts.

• Also the durations and dollar durations of the assets changed with interest rates by different amounts.

• For large interest rate changes, the duration-matched hedge has to be rebalanced.

• A way to mitigate this problem is to match the convexity of assets and liabilities as well as duration and market value.

Page 85: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

• Consider structuring an asset portfolio that matches the convexity of the liabilities as well as their duration and market value.

• Use the following instruments for the asset portfolio.– a 2-year zero– a 15-year zero– a 25-year zero

Page 86: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

Note that if the assets have the same market value$ duration and $convexity as the liability, then they have the same duration and convexity as the liability:

To construct the hedge portfolio, solve threeequations:Asset mkt. Val. = Liability mkt. ValAsset $ duration = Liability $ durationAsset $ convexity = Liability $ convexity

Page 87: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

Coupon (%)

Maturity (Years)

Par Value ($)

Market Value ($)

Dollar Duration

Dollar Convexity

Duration Convexity

7.5 29 1000000 1151802 14486304 288068417 12.58 250.10

0 2 1 0.8972 1.7463 4.2489 1.95 4.74

0 15 1 0.3881 5.6412 84.7226 14.53 218.28

0 25 1 0.1977 4.7852 118.1290 24.20 597.48

Numbers are from class discount rates:

Page 88: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Example

For our example, the three equations become:

417,068,288129.1187226.842489.4

304,486,147852.46412.57463.1

802,151,11977.03881.08972.0

25152

25152

25152

NNN

NNN

NNN

The solution is:N2=497,576; N15=920,680; N25=1,760,379

Page 89: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Mkt. Val. Of Duraation/Convexity Matched Portfolios

0.52.0

3.55.0

6.58

9.511

12.514

15.517

18.520

21.523

24.526

27.529

0

100,000

200,000

300,000

400,000

500,000

Maturity

Market Value

Page 90: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Duration/Convexity Performance-100 bp -10 bp 0 +10 bp +100 bp

MARKET VALUEAssets 1312210 1166433 1151802 1137458 1020328Liabilities 1312293 1166433 1151802 1137458 1020267Net Equity -82 -0.07 0 0.07 60

DOLLAR DURATIONAssets 17716763 14777623 14486304 14201435 11904016Liabilities 17742932 14777840 14486304 14201643 11921307Net Equity -26169 -217 0 -208 -17290

DURATIONAssets 13.50 12.67 12.58 12.49 11.67Liabilities 13.52 12.67 12.58 12.49 11.68

Page 91: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Yield Curve Shift One Day Later

0.52.0

3.55.0

6.58

9.511

12.514

15.517

18.520

21.523

24.526

27.529

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Maturity

Change in Discount Rate

Page 92: International Fixed Income Topic IB: Fixed Income Basics - Risk.

$ Duration Liabilities

0.52.0

3.55.0

6.58

9.511

12.514

15.517

18.520

21.523

24.526

27.529

0

1,000,000

2,000,000

3,000,000

4,000,000

5,000,000

Maturity

Dollar Duration

Page 93: International Fixed Income Topic IB: Fixed Income Basics - Risk.

$ Duration of Duration Matched

0.52.0

3.55.0

6.58

9.511

12.514

15.517

18.520

21.523

24.526

27.529

0

2,000,000

4,000,000

6,000,000

8,000,000

10,000,000

Maturity

Dollar Duration

Page 94: International Fixed Income Topic IB: Fixed Income Basics - Risk.

$ Duration of Dur/Conv. Matched

0.52.0

3.55.0

6.58

9.511

12.514

15.517

18.520

21.523

24.526

27.529

0

2,000,000

4,000,000

6,000,000

8,000,000

10,000,000

Maturity

Dollar Duration

Page 95: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Effect of Yield Curve Shift

• The average change in rates was +1 bp.• If the interest rate shift had been parallel, dollar

duration of 14,486,304 would have predicted a change of -14,486,304 x 0.0001 = -$1449 in the value of the liability and each asset portfolio.

• The actual change in the liability was -$2126• The dollar duration of the liability is concentrated

on year 29. The 29-year discount rate increased 2 bp.

Page 96: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Effect of Actual Yield Curve Shift

• The value of the duration-matched portfolio changed by only $-889. – The 12-year discount rate did not change at all.

– The 15-year discount rate rose 2 bp.

• Net equity under this immunization would have increased to $1237.

Page 97: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Effect of Actual Yield Curve Shift

• The value of the duration-convexity-matched portfolio changed by $-3365. – Most of its dollar duration was on year 25.

The 25-year discount rate rose 3 bp.

• Net equity under this immunization would have fallen to -$1239.

Page 98: International Fixed Income Topic IB: Fixed Income Basics - Risk.

Lesson

• Duration or duration-convexity matching hedges against parallel shifts of the yield curve.

• To hedge against other shifts, the cash flows of the assets and liabilities must have similar exposure to different parts of the yield curve.