class #6, chap 9 1. purpose: to understand what duration is, how to calculate it and how to use it....

Class #6, Chap 9

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Purpose: to understand what duration is, how to calculate it and how to use it.

Toolbox: Bond Pricing Review

Duration Concept Interpretation Calculation Examples

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3

Bond Pricing Review

Zero coupon bond with the YTM

Coupon bond with the YTM

Coupon bond with Yield Curve

Yield Curve and YTM

Price a zero coupon bond with 10 years left to maturity, face value of $1000 and YTM of 5%

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Step 1: Find coupon payments & draw cash flowsCoupon Payments = $1,000*0 = $0

Step 2: discount cash flows

91.613$)05.1(

100010

P

1000

1 2 3 4 5 6 7 8 9 10

Price a 4 year coupon bond with face value of 1000 and an annual coupon of 7% if the yield to maturity is 13%

5

Step 1: Find coupon payments & draw cash flowsCoupon Payments = $1,000*.07 = $70

Step 2: discount cash flows

53.821$)13.1(

1070

)13.1(

70

)13.1(

70

)13.1(

704321P

1000

1 2 3 4

70 70 70 70

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• Yield curve gives the market rate for a pure discount bond at each maturity• The market price of a coupon bond incorporates several different rates for

different time horizons (1 year money, 2 year money …. )• Every bond has its own yield curve – treasury bonds, Ford bonds, GM bonds• The yield curve will change every day

1 2 3

Current Yield Curve

4.3%

3.1%

1.1%

70 70 70

1000

1 2 3

Example: Find the price of a three year bond that pays an annual coupon of 8%. The following rates are taken from the current yield curve. Face value is $1,000

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Term Rate

6 months 1.2%

1 year 2.1%

2 years 2.5%

3 years 2.7%

We used two different methods what is the difference?

Method 1: we pulled rates off the yield curve

Method 2: we used one constant rate – lets value the bond above with a constant rate of 2.676% (Yield to Maturity)

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54.1151)027.1(

1080

)025.1(

80

021.1

8032

P

54.1151)02676.1(

1080

)02676.1(

80

02676.1

8032P

What does this yield curve look like?

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The yield to maturity (YTM) is basically a weighted average of rates off the yield curve

It is the constant rate, over the full maturity, that gives you the market price of the bond

When you use the YTM you assume that the yield curve is flat!

DURATION Concept Calculation Using duration

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Concept of Duration

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What does duration do? It measures the sensitivity of the asset price to changes in interests rates

How is that going to help us? We have been trying to measure interest rate risk▪ The movement in asset prices in response to a change in interest rates

Repricing gap gave us a rough measure but had several problems Duration improves upon some of these short falls

Advantages of Duration It is a market based measure so it takes into account current values It considers the current time to maturity rather than the defined term

Disadvantages of Duration It requires more information to calculate

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Definition of Duration: Duration is the present value cash flow weighted average time to

maturity of a loan/bond

What the #@&! ? Duration tells us, in terms of present value, the average timing of cash flows

from a loan or bond. (i.e. on average, when do we receive the value of our bond)

Not much help?1. What is duration – work on explaining the definition

2. Why is it important – how does it help measure interest rate sensitivity

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Lets split this discussion into two parts:

This is what we are after. I want you to see that duration tells us: on average when do we receive the value of our bond/loan

Duration is the present value cash flow weighted average time to maturity of a loan/bondDuration is the present value cash flow weighted average time to maturity of a loan/bond

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First thing we want to realize is that any bond can be thought of as a portfolio of zero coupon bonds

Consider a 6 year coupon bond that pays an annual coupon of 4% and has a $1,000 face value

Duration is the present value cash flow weighted average time to maturity of a loan/bond

3

40

0

15

0

40

1

1040

6 0 5

40

0 4

40

0

40

2 0

We can think of the coupon bond as a portfolio of 6 zero coupon bonds. The average maturity

5.36

654321

Average TTM =

Duration is the present value cash flow weighted average time to maturity of a loan/bond

3

40

0

16

0

40

1

1040

6 0 5

40

0 4

40

0

40

2 0

Average TTM = 3.5 yrs

Do you think that this tells you when, on average, you receive the value of you payments?

Duration is going to depend on two things:1. The timing of payments 2. The amount of payments – in terms of present value

Before or after 3.5 years? Why?

Take away: We can calculate the average time to maturity of a bond but that does not always tell us when, on average, we receive the full value of payments

PV = 27.32PV = 30.05PV = 33.06

Duration is the present value cash flow weighted average time to maturity of a loan/bond

3

40

0

17

0

40

1

1040

6 0 5

40

0 4

40

0

40

2 0

For this part, let’s just start with how much of the bond value we receive at each point in time. Assume YTM = 10%

On average, when do you think we receive the full bond value? How can we adjust the average to account for this? What do we use for weights?

Duration is the present value cash flow weighted average time to maturity of a loan/bond

36.3610.1

40PV = 36.36 PV = 24.84

PV = 587.05

weighted average

Present value of cash flows

Somewhere around year 5 or 6 is a good guess

06.33)10.1(

402

05.30)10.1(

403

32.27)10.1(

404

84.24)10.1(

405

05.587)10.1(

10406

Take away: On average we receive the full bond value close to the largest PV(payment). So we need to weight by PV(CFs)

Which pmts is most/ least valuable?

Duration is the present value cash flow weighted average time to maturity of a loan/bond

3

40

0

18

0

40

1

1040

6 0 5

40

0 4

40

0

40

2 0

On average we will receive the full value of our payments 5.35 years from today.

Duration = 5.35 yrs

1. What is Duration – Definition

2. Why is it important – how does it measure interest rate sensitivity

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Duration is important because it tells us the interest rate sensitivity of a bond!!! But how?It turns out that the weighted average time to maturity (duration) gives us the maturity of the equivalent zero coupon bond.

This “equivalent” zero coupon bond will have the same interest rate sensitivity as the coupon bond

Example: consider two bonds:

20

0

1000

5.35Bond 2:

3

40

0

40

1

1040

6 5

40

4

4040

2Bond 1:

Duration = 5.35

Example: Price both bonds with YTM = 10% then again with YTM = 10.5% and compare the price sensitivity.

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3

40

0

40

1

1040

6 5

40

4

4040

2Bond 1:

Duration = 5.35

68.73810.1

1040

10.1

40

10.1

40

10.1

40

10.1

40

10.1

40%)10(

65432V

01.721105.1

1040

105.1

40

105.1

40

105.1

40

105.1

40

105.1

40%)5.10(

65432V

02393.068.738

68.73801.721

Sensitivity

0

1000

5.35Bond 2:

82.60010.1

1000%)10(

35.5V

45.586105.1

1000%)5.10(

35.5V

02395.082.600

82.60045.586

Sensitivity

The two bonds have the same interest rate sensitivity

20,728,012

What if we change the principal amount?

What does this get us? …. Lets see

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0

1000

5Bond 1:

1000

0 10Bond 2:

Which bond is more interest rate sensitive?

What is the duration of this bond?

What is the duration of this bond?

Which bond is more interest rate sensitive? Is it harder to see?

What does this get us? …. Lets see

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1000

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

0 1 2 3 4 5

1000

Bond 1:

$40

$40

Which bond is more interest rate sensitive?

What does this get us? …. Lets see

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500

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10Bond 2:

$40

Which bond is more interest rate sensitive?

1000

0 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10Bond 1:

$90

Duration = 5.96

Duration = 6.95

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Bond 1

Face value

Time to Maturity

Coupon Rate

Compounding

Bond 2

Beginning of Period YTM

End of Period YTM

Bond 1 Interest Rate Sensitivity

Bond 2 Interest Rate Sensitivity

Calculate

Conclusion: Duration tells us:

With respect to interest rate sensitivity, this bond will behave like a zero coupon bond with D years to maturity (where D is the bond duration/maturity)

We know that the zero coupon bond with longer maturity is more interest rate sensitive

Therefore, we also know that a bond with longer duration is more interest rate sensitive

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Calculating Duration

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Step 1: draw out the cash flows

Step 2: take the present value of all the cash flows

Step 3: calculate weights

Step 4: calculate the weighted average time to maturity (duration)

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Calculate the duration of a two year treasury bond with an 8% semiannual coupon, 12% YTM, and $1,000 face value

Step #1: draw out the cash flows

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40 40 40 40

1000

0.5 1 1.5 2

Calculate the duration of a two year treasury bond with an 8% semiannual coupon, 12% YTM, and $1,000 face value

Step #2: Take the present value of cash flows

30

40 40 40 40

1000

0.5 1 1.5 2

78.823)2/12.1(

1040

58.33)2/12.1(

40

60.35)2/12.1(

40

74.37)2/12.1(

40

4

3

2

Calculate the duration of a two year treasury bond with an 8% semiannual coupon, 12% YTM, and $1,000 face value

Step #3: Calculate weights

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40 40 40 40

1000

0.5 1 1.5 2

78.823)2/12.1(

1040

58.33)2/12.1(

40

60.35)2/12.1(

40

74.37)2/12.1(

40

4

3

2

First thing we need to do is sum the present values

70.930

What is this number?

Calculate the duration of a two year treasury bond with an 8% semiannual coupon, 12% YTM, and $1,000 face value

Step #3: Calculate weights

32

40 40 40 40

1000

0.5 1 1.5 2

78.823)2/12.1(

1040

58.33)2/12.1(

40

60.35)2/12.1(

40

74.37)2/12.1(

40

4

3

2

70.930

0405.70.930

74.375. w

What do the weights mean?

They are the percentages of the present value of all cash flows that occur on that time period

Example: 3.61% of the present value of all cash flows is received at year 1.5Why do we take the present value?

We are trying to compare the relative importance of different cash flows so we need to compare them at the same point in time – which is more valuable to an investor $1 today or $1.10 in one year if the one year interest rate is 10%?

0383.70.930

60.351 w

0361.70.930

58.335.1 w

8851.70.930

78.8232 w

8829.1)2( 8851.)5.1( 0361.)1( 0383.)5(. 0405. D

Calculate the duration of a two year treasury bond with an 8% semiannual coupon, 12% YTM, and $1,000 face value

Step #4: Calculate the duration (present value cash flow weighted average time to maturity)

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8851.70.930

78.823

0361.70.930

58.33

0383.70.930

60.35

0405.70.930

74.37

weights nntwtwtwtwD ...332211

4321 8851. 0361. 0383. 0405. ttttD

What are the ts?

What are these?

Years

Macaulay Duration

To this point, we know that duration is a measure of interest rate sensitivity

It turns out that duration is also the interest rate elasticity of a security (bond) price

What more does that tell us? Because it is an elasticity, we can use it to determine how much the bond price will move

in response to a change in interest rates

Elasticity Equation

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YY

XX

Yinchange

Xinchangee

%

%

)1()1()]1()1[( 1 R

RP

P

RRR

PP

D

t

tt

We can rewrite the duration equation:

This gives us an equation for calculating the percent change in the bond price (return) due to a change in the interest rate

35

D

)1( RR

PP

)1( R

RD

P

P

We have been working with a two year 8% coupon treasury bond with 12% YTM and a price of $930.70

Suppose the interest rate decreased to .115 what would you expect the percent change in the price to be?

1. Calculate the percent change in the bond price:

With Duration:

36

12.939)2/115.1(

1040

)2/115.1(

40

)2/115.1(

40

)2/115.1(

40432

p

70.930)2/12.1(

1040

)2/12.1(

40

)2/12.1(

40

)2/12.1(

40432

p

00893.070.930

70.93012.939

0088815.0)2/12.1(

005.882888.1

)1(

R

RD

P

P Not exact but pretty close

Duration gives us a way to measure the sensitivity of an asset price to changes in the interest rate

Duration also gives us a way to calculate the magnitude of the percent change in price in response to a change in interest rates

Alternative forms of duration Bond traders developed more convenient ways to write duration Modified duration (MoD) Dollar duration

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Macaulay Duration: (D)

Modified Duration (MoD)

Dollar Duration

38

)1( R

RD

P

P

R

DMoD

1

PMoDD $

MoD allows you to calculate the %change in the bond price just by multiplying by the change in interest rate

$D dollar duration allows you to calculate the change in bond price from the change in the interest rate

RMoDP

P

RDP $

Example: calculate the duration, modified duration and dollar duration for a bond with: face value = 1000; annual coupon; coupon rate = 3%; YTM = 9%; and four years to maturity

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Examples: Suppose the YTM = 9%i)Find the percent change in the bond price if YTM increases from 9% to 14% the duration is 3.8 yearsii)The percent change in the bond price if the YTM increases to 11% given MoD = 3.492iii)The raw change in the bond price if the YTM decrease to 8.5% if $D = 2813.06

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How would things change if the bond had semiannual coupons?

The bond pricing would change as we have already seen

Modified duration would also change:

41

)09.1(

8.3

)1(

R

DMoD

)2/09.1(

8.3

)2/1(

R

DMoDFor semiannual

coupons we have

Different durations: Macaulay Duration = D Modified duration(MoD) Dollar duration = (MoD)(bond price)

42

)/1( kR

D

D = Macaulay durationR = the yield to maturityk = compounding periods

We learned the meaning of duration (concept)

How to calculate duration (D, MoD, $D)

How to use duration to calculate the expected change (%change in price)

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