Class 7, Chap 9 - Appendix B. Purpose: Gain a deeper understanding of duration and its properties and weaknesses Properties of duration Hedging with

Class 7, Chap 9 - Appendix B

Purpose: Gain a deeper understanding of duration and its properties and weaknesses

Properties of duration

Hedging with duration

Weaknesses of duration Convexity

1. Duration increases with maturity but at a decreasing rate

2. Duration decreases as the yield to maturity increases

3. Duration decreases as the coupon payments or interest rate increases

Calculate the duration for bonds of several maturities with an 8% coupon paid semiannually, $1,000 face value and yield to maturity of 12%.

When we add a year to a long maturity bond it changes the duration much less than when we add a year to a short maturity bond

Adding a year means:• The big payment occurs 1 year later• Adds 1 year to the weighted average •Because there is not a lot of discounting, the weight on the additional year is large

925.6037.74

0 0.5 1

37.74

0 0.5 1 1.5 2

35.60 33.58

823.78

Calculate the duration for bonds of several maturities with an 8% coupon paid semiannually, $1,000 face value and yield to maturity of 12%.

When we add a year to a long maturity bond it changes the duration much less than when we add a year to a short maturity bond

Adding a year means:• The big payment occurs 1 year later• Adds 1 year to the weighted average • There is a lot of discounting so the weight on the additional year is small compared to other years

37.74

0 0.5 1 28 28.5 29

35.60 1.53

33.42

1.44 37.74

0 0.5 1 29 29.5 30

35.60 1.36

31.53

1.29

6

Lets just look at what happens to the present value of cash flows as the maturity increases

1,040

0 1 2 3 4 5

40

Time to Maturity = 5 years

Duration = 4.14

7


1,040

0 1 2 3 4 5 6 7 8 9 10

40


Duration = 6.61

8


1,040

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

40


Duration = 7.91

9


1,040

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

40


Duration = 8.53

10


1,040

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

40


Duration = 8.53

Total weight (sum) = 48%

A large percent of the bond value has been received early-on !!!

Total weight (sum) = 75% Total weight (sum) = 86%

Conclusion: Duration increases with maturity but at a decreasing rate

because of two effects:

1. Increasing the maturity adds more years to the bond, which increases duration

2. As we increase the time to maturity (TTM), a smaller and smaller fraction of bond value is being received at a later date. This is because later payments are highly discounted. As a result, a large fraction of bond value is received early on, which stabilizes the duration.

1,000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Lets just look at what happens to the present value of cash flows as the YTM increases

40 40 40 40 40 40 40 40 40 40

1,000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


40 40 40 40 40 40 40 40 40 40

YTM = 10%

Duration = 4.18

1,000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


40 40 40 40 40 40 40 40 40 40

YTM = 30%

Duration = 3.74

1,000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


40 40 40 40 40 40 40 40 40 40

YTM = 50%

Duration = 3.23

1,000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


40 40 40 40 40 40 40 40 40 40

YTM = 70%

Duration = 2.71

1,000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


40 40 40 40 40 40 40 40 40 40

YTM = 90%

Duration = 2.26

1,000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


40 40 40 40 40 40 40 40 40 40

As we increase the yield to maturity, the present value (and as a result the duration weights) of the earlier payments increase relative to the PV (duration weights) of the later payments

That is, the percentage of value [PV(future cash flows)] received early in the bond’s life increases – so the later payments (more interest rate sensitive) are not as important

1,000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Again, this has to do with how the present value of cash flows is distributed over time and how that changes when we change the coupon rate.

50 50 50 50 50 50 50 50 50 50

Coupon = 10%

Duration = 4.04

1,000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


200 200 200 200 200 200 200 200 200 200

Coupon = 40%

Duration = 3.29

1,000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


Coupon = 70%

350 350 350 350 350 350 350 350 350 350

Duration = 3.07

1,000

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5


500 500 500 500 500 500 500 500 500 500

Coupon = 100%

Duration = 2.97


As we increase the coupon rate the present value of early cash flows (duration weights) increases relative to later payments

That is, the percentage of value [PV(future cash flows)] received early in the bonds life increases – so the later payments (more interest rate sensitive) are not as important

1. Duration increases with maturity but at a decreasing rate

2. Duration decreases as the yield to maturity increases

3. Duration decreases as the coupon payments increase

4. You need to have a basic understanding of why duration behaves this way

Hedge With Duration

25

We have seen that duration measures the sensitivity of assets to changes in interest rates

Now lets see how we can use that to manage interest rate risk

Basic idea: by taking an offsetting position in an asset/liability with a matched duration an investor can hedge interest rate risk

26

Suppose a company has 5 years left on a loan: The company wants to pay back the loan today but there are stiff

prepayment penalties. So, the company decides to offset the loan with another asset.

The loan is a balloon payment loan - it is paid back in one lump sum payment in five years – no interim interest payments

Current value of the loan is $1,000 at 8% = $1469.33 due in 5 years

The company wants to hedge against changes in interest rates and can choose from the following instruments: A 3 year 3% coupon bond with $1,000 face value A five year zero coupon bond with an 8% YTM and face value = 1000 A six year bond with an 8% coupon paid annually and face value =

1,000 and YTM = 8%27

The company can manage its interest rate risk by matching durations

The duration of the 3 year bond will definitely be too short

The five year zero coupon bond has a duration of 5 years

The six year coupon can not be ruled out so we need to calculate the duration

28

Step#1 Find the couponCoupon = (1,000)*.08 = $80

Draw the cash flows

Step#2 Find present values

Step#4 Find duration weights

Step#5 Find duration

1,000

80 80 80 80 80 80

1 2 3 4 5 6

65432 08.1

1080

08.1

80

08.1

80

08.1

80

08.1

80

08.1

80P

000,158.68045.5480.5851.6359.6804.74 P

1000

58.680

1000

45.54

1000

80.58

1000

51.63

1000

59.68

1000

04.74654321 wwwwww

9927.4)6)(68058.0()5)(05445.0()4)(05880.0()3)(06351.0()2)(06859.0()1)(07404.0( D

The 6 year bond is also a viable option for the hedge

29

The company will owe 1469.33 in 5 years

So the company wants to receive $1469.33 (for sure) in 5 years to be completely hedged

Each bond pays 1000 in 5 years so they need to buy 1469.33/1000 = 1.46933 zero coupon bonds

Cost: The price of the zero coupon = 1000/(1.08)5 = 680.58 The company needs 1.46933 of them so the total cost is (1.46933)

(680.58) = $1,000 The full amount of their loan

30

The company is perfectly hedged!!!!

After purchasing the zero coupon bonds, the company has locked-in a positive1469.33 cash flow in five years no matter what interest rates do!!!

31

0 1 2 3 4 5

1,469.33

- 1,469.33

Bond

Loan

We saw that the 6 year bond had a duration of 5 years so lets try using it to hedge.

To hedge the company can buy one 5 year duration bond for a cost of $1,000

Consider three cases :a. The YTM stays at 8%

b. The YTM instantaneously increases to 9%

c. The YTM instantaneously decreases to 7%

Why 1 bond? – if we find the value of all cash flows at time 5 years (1000)(1.085) =$1,469.33.•If this was not the case, we would need to buy more or less than one bond •But if this was not the case the bond would not have a 5 year duration

32

Base case: Show that the company is hedged if the YTM = 8% The company will hold the bond for 5 years The coupon will be reinvested at the YTM

1,000

80 80 80 80 80 80

1 2 3 4 5 6

Reinvest for 4 years

4)08.1)(80( 3)08.1)(80( 2)08.1)(80( )08.1)(80(08.1

108080




Collect coupon & sell bond

33.1469)100080(40.8631.9378.10084.1085 yrCF

33

Case 1 YTM increases to 9% The company will hold the bond for 5 years The coupon will be reinvested at the YTM

1,000

80 80 80 80 80 80

1 2 3 4 5 6


4)09.1)(80( 3)09.1)(80( 2)09.1)(80( )09.1)(80(09.1

108080





33.1469)83.99080(20.8705.9560.10393.1125 yrCF

34

Case 2 YTM decreases to 7% The company will hold the bond for 5 years The coupon will be reinvested at the YTM

1,000

80 80 80 80 80 80

1 2 3 4 5 6


4)07.1)(80( 3)07.1)(80( 2)07.1)(80( )07.1)(80(07.1

108080





33.1469)35.100980(60.8559.9100.9886.1045 yrCF

35

If the company offsets its assets or liabilities with an instrument of the same duration the position will be immune to changes in interest rates

Do you think this really works?

It could, but we run into two problems1. The duration of the bond (used to hedge) will change

2. The YTM of the bond used to hedge could change

36

What kind of risk would the coupons be subject to?

Lets calculate the duration of the bond right after the second coupon is paid – there are four years (coupons) left. Assume they YTM = 8%

Weights:

Duration

Loan: the loan still has 3 years to maturity so the durations no longer match – this is ok as long as the coupons have and can continue to be reinvested at 8%

100008.1

1080

08.1

80

08.1

80

08.1

80432P

000,183.79351.6359.6804.74 P

1000

83.793

1000

51.63

1000

59.68

1000

04.744321 wwww

yearsD 577.34)079383.0()3)(06351.0()2)(06859.0()1)(07404.0(

1. Duration Change

37

Suppose that after the first two payments the interest rate increases to 9%

So what’s the point? This seems really ineffective – why am I not teaching you how to fully resolve this problem?

1,000

80 80 80 80 80 80

1 2 3 4 5 64)08.1)(80( 3)08.1)(80( 2)09.1)(80( )09.1)(80(

09.1

108080

69.1462)83.99080(20.8705.957.10084.1085 yrCF

The company no longer has enough money to

repay its loan of $1469.33

IT IS HARD!!!

2. Reinvestment risk

38

Difficulties with Duration

39

1. Reallocating large quantities of assets or liabilities to attain the needed durations for assets and liabilities can be very costly

2. Immunization is a dynamic problem1. Every time the interest rate changes the hedging portfolio must be rebalanced

2. One decision managers have to make is how often to rebalance and weigh the cost of doing so

3. Convexity- Duration only works for small changes in the interest rate 1. For large changes in rates duration will not accurately predict the percent

change in the price of a security

40

Convexity refers to the curvature in the relationship between bond prices and interest rates. What does that mean? – watch

41

Price a 20 year bond with coupon of 30% and semiannual

payments


42

69.908,5$

)2/015.1(

1000

)2/015.1(

150

%5.1

40

40

1

t

t

YTM


payments


43

21.556,4$

)2/04.1(

1000

)2/04.1(

150

%4

40

40

1

t

t

YTM


payments


44

21.177,3$

)2/08.1(

1000

)2/08.1(

150

%8

40

40

1

t

t

YTM


payments


45

37.202,2$

)2/13.1(

1000

)2/13.1(

150

%13

40

40

1

t

t

YTM


payments


46

95.488,1$

)2/20.1(

1000

)2/20.1(

150

%20

40

40

1

t

t

YTM


payments


47

000,1$

)2/30.1(

1000

)2/30.1(

150

%30

40

40

1

t

t

YTM


payments


48

$77.666

)2/45.1(

1000

)2/45.1(

150

%45

40

40

1

t

t

YTM


payments


49

What does duration say about this relation?

Duration is the derivative of the bond pricing formula with respect to the interest rate at a specific point on the graph

What does the derivative look like on the graph?

50


51

13%

• This is what duration says the graph (relationship) should look like

• When we do the duration calculation, we find a point on this line

Duration is the derivative. It is the slope of the tangent line

Calculate the duration of the bond if the YTM is 13%

D = 7.23 years

If the YTM dropped to 3% what price would the duration predict?

What is the actual price?

52

7880.02/13.01

1.23.7

)1(

R

RD

P

P

47.1735)37.2202)(7880.0( P

84.393747.173513.22021 PPP tt

64.5038)2/03.1(

1000

)2/03.1(

15040

40

1

tt

80.1100

84.3937

64.5038


53

13%3%

80.110084.3937

64.038,5

Calculate the duration of the bond if the YTM is 13%

D = 7.23 years

If the YTM jumped to 23% what price would the duration predict?

What is the actual price?

54

7880.02/13.01

1.23.7

)1(

R

RD

P

P

47.735,1)37.2202)(7880.0( P

39.46647.173513.22021 PPP tt

44.1300)2/23.1(

1000

)2/23.1(

15040

40

1

tt

834.05

44.1300

39.466


55

13%3%

80.1100

23%

05.834

Asymmetric Pricing Errors!

39.666

44.1300

The larger the convexity the more curvature there is in the line Duration will work better for bonds with low convexity We will calculate convexity next

56

1. Duration is only accurate for small changes in interest rates

2. Duration will predict lower than actual values

3. The under prediction error is greater when interest rates fall then when they increase

4. Duration will change depending on the interest rate!!!!!!!!!

57

1. Calculate the duration weights

2. Multiply the weights by the time period squared plus and the same time period

3. Sum values and divide by (1+ YTM)2 to get convexity

58

nnn ttWttWttWC 22

2221

211 ...

59

80 80 80 80 80 80

Consider a 6 year bond with an 8% coupon paid annually the YTM is 6%. Face value of 1000. Calculate the convexity of the bond

1000

Step #1 find the present value of payments

66554433221 06.1

1080)(

06.1

80)(

06.1

80)(

06.1

80)(

06.1

80)(

06.1

80)( CFpvCFpvCFpvCFpvCFpvCFpv

36.761)( 78.59)( 37.63)( 17.67)( 20.71)( 47.75)( 654321 CFpvCFpvCFpvCFpvCFpvCFpv

Step #2 calculate weights

35.1098

36.761

35.1098

78.59

35.1098

37.63

35.1098

17.67

35.1098

20.71

35.1098

47.75654321 wwwwww

693.0 054.0 058.0 061.0 065.0 069.0 654321 wwwwww

Consider a 6 year bond with an 8% coupon paid annually the YTM is 6%. Face value of 1000. Calculate the convexity of the bond

60

)66)(0.6932()55)(0.0544()44)(0.0577(

)33)(0.0612()22)(0.0648()11)(0687.0(

)06.1(

1222

222

2C

54.29C

16.3306.1

12

C

Measures the curvature of the YTM bond price relationship – larger values = more curvature

Step #3 calculate the convexity

Calculate the convexity of a 1.5 year 4% coupon bond with semiannual payments and face value of 5,000 if the risk free rate is currently 5% and the YTM is 9%

61

We can use it to adjust the accuracy of the duration calculation!!

Example:

Estimate the expected percent change in the price of the bond from the previous example (FV = 5000, coupon = 4%, TTM = 1.5yrs, semiannual compounding) if interest rates are expected to increases from 9% to 11.4% (the duration of the bond 1.47yrs).

62

R

RD

P

P

1

209.01

024.047.1

P

P

)(2

1 2RC

03279.0)024)(.346.3(2

1 2 03375.0

63

Price implied by Duration

Price implied by Duration & Convexity

3 properties of duration Duration increases with maturity but at a decreasing rate Duration decreases as the yield to maturity increases Duration decreases as the coupon payments or interest rate increases

Hedging by matching duration The hedge is only perfect if YTM remains constant over the life of the hedge

Convexity Concept Calculation

64

Documents

Class 7, Chap 9 - Appendix B. Purpose: Gain a deeper understanding of duration and its properties and weaknesses Properties of duration Hedging with