class #6, chap 9 1. purpose: to understand what duration is, how to calculate it and how to use it....
TRANSCRIPT
Class #6, Chap 9
1
Purpose: to understand what duration is, how to calculate it and how to use it.
Toolbox: Bond Pricing Review
Duration Concept Interpretation Calculation Examples
2
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%
4
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
6
• 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
7
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)
8
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?
9
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
10
Concept of Duration
11
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
12
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
13
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
14
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
19
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.
21
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
22
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
23
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
24
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
25
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
26
Calculating Duration
27
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)
28
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
29
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
31
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)
33
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
34
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
37
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
39
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
40
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)
43