Transcript Slide 1
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%.
Duration & Time to Maturity
10
9
8
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
Duration
7
6
5
4
823.78
925.60
3
37.74
37.74
35.60 33.58
2
1
0
0.5
0
1
0.5
1
1.5
2
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Years
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
27
28
29
30
Calculate the duration for bonds of several maturities with an 8% coupon paid semiannually,
$1,000 face value and yield to maturity of 12%.
Duration & Time to Maturity
10
9
8
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
Duration
7
6
5
4
33.42
3
37.74
2
1
0
0.5
35.60
1.53
1.44
1
28
28.5
31.53
37.74
29
0
0.5
35.60
1.36
1.29
1
29
29.5
30
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Years
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
27
28
29
30
0
Lets just look at what happens to the present value of cash flows as the maturity increases
1,040
40
1
2
3
4
5
Time to Maturity = 5 years
Duration = 4.14
6
0
Lets just look at what happens to the present value of cash flows as the maturity increases
1,040
40
1
2
3
4
5
6
7
8
9
10
Time to Maturity = 10 years
Duration = 6.61
7
0
Lets just look at what happens to the present value of cash flows as the maturity increases
1,040
40
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Time to Maturity = 15 years
Duration = 7.91
8
0
Lets just look at what happens to the present value of cash flows as the maturity increases
1,040
40
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Time to Maturity = 20 years
Duration = 8.53
9
0
Lets just look at what happens to the present value of cash flows as the maturity increases
1,040
Duration = 8.53
40
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Time to Maturity = 20 years
A large percent of the bond value has been received early-on !!!
Total weight (sum) = 48%
Total weight (sum) = 75%
Total weight (sum) = 86%
10
20
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.
Lets just look at what happens to the present value of cash flows as the YTM increases
1,000
0
40
40
40
40
40
40
40
40
40
0.5
1
1.5
2
2.5
3
3.5
4
4.5
40
5
Lets just look at what happens to the present value of cash flows as the YTM increases
1,000
0
40
40
40
40
40
40
40
40
40
0.5
1
1.5
2
2.5
3
3.5
4
4.5
YTM = 10%
Duration = 4.18
40
5
Lets just look at what happens to the present value of cash flows as the YTM increases
1,000
0
40
40
40
40
40
40
40
40
40
0.5
1
1.5
2
2.5
3
3.5
4
4.5
YTM = 30%
Duration = 3.74
40
5
Lets just look at what happens to the present value of cash flows as the YTM increases
1,000
0
40
40
40
40
40
40
40
40
40
0.5
1
1.5
2
2.5
3
3.5
4
4.5
YTM = 50%
Duration = 3.23
40
5
Lets just look at what happens to the present value of cash flows as the YTM increases
1,000
0
40
40
40
40
40
40
40
40
40
0.5
1
1.5
2
2.5
3
3.5
4
4.5
YTM = 70%
Duration = 2.71
40
5
Lets just look at what happens to the present value of cash flows as the YTM increases
1,000
0
40
40
40
40
40
40
40
40
40
0.5
1
1.5
2
2.5
3
3.5
4
4.5
YTM = 90%
Duration = 2.26
40
5
Lets just look at what happens to the present value of cash flows as the YTM increases
1,000
0
40
40
40
40
40
40
40
40
40
0.5
1
1.5
2
2.5
3
3.5
4
4.5
40
5
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
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.
1,000
0
50
50
50
50
50
50
50
50
50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Coupon = 10%
Duration = 4.04
50
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.
1,000
0
200
200
200
200
200
200
200
200
200
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Coupon = 40%
Duration = 3.29
200
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.
1,000
0
350
350
350
350
350
350
350
350
350
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Coupon = 70%
Duration = 3.07
350
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.
1,000
0
500
500
500
500
500
500
500
500
500
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Coupon = 100%
Duration = 2.97
500
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.
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 coupon
Coupon = (1,000)*.08 = $80
Draw the cash flows
1,000
80
80
80
80
80
1
2
3
4
5
80
6
Step#2 Find present values
P
80
80
80
80
80
1080
1.08 1.082 1.083 1.084 1.085 1.086
P 74.04 68.59 63.51 58.80 54.45 680.58 1,000
The 6 year bond is also a
viable option for the hedge
Step#4 Find duration weights
w1
74.04
1000
w2
68.59
1000
w3
63.51
1000
w4
58.80
1000
w5
54.45
1000
w6
680 .58
1000
Step#5 Find duration
D (0.07404)(1) (0.06859)(2) (0.06351)(3) (0.05880)(4) (0.05445)(5) (0.68058)(6) 4.9927
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!!!
1,469.33
Bond
0
1
2
3
4
5
Loan
- 1,469.33
31
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.
b.
c.
Why 1 bond? – if we find the value of all cash
The YTM stays at 8%
flows at time 5 years (1000)(1.085) =$1,469.33.
The YTM instantaneously
to 9%
• If this increases
was not the case,
we would need to buy
more or less than one bond
The YTM instantaneously
7%the bond would not
• But if decreases
this was not thetocase
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
Reinvest Reinvest Reinvest ReinvestCollect coupon &
for 4 yearsfor 3 years
for 2 yearsfor 1 years sell bond 1,000
80
80
80
80
80
1
2
3
4
5
(80)(1.08) 4
(80)(1.08)3 (80)(1.08) 2 (80)(1.08)
80
6
1080
80
1.08
CF5 yr 108.84 100.78 93.31 86.40 (80 1000) 1469.33
33
Case 1
YTM increases to 9%
The company will hold the bond for 5 years
The coupon will be reinvested at the YTM
Reinvest Reinvest Reinvest ReinvestCollect coupon &
for 4 yearsfor 3 years
for 2 yearsfor 1 years sell bond 1,000
80
80
80
80
80
1
2
3
4
5
(80)(1.09)4
(80)(1.09)3 (80)(1.09)2
80
6
(80)(1.09) 80 1080
1.09
CF5 yr 112.93103.60 95.05 87.20 (80 990.83) 1469.33
34
Case 2
YTM decreases to 7%
The company will hold the bond for 5 years
The coupon will be reinvested at the YTM
Reinvest Reinvest Reinvest ReinvestCollect coupon &
for 4 yearsfor 3 years
for 2 yearsfor 1 years sell bond 1,000
80
80
80
80
80
1
2
3
4
5
(80)(1.07)4
(80)(1.07)3 (80)(1.07)2 (80)(1.07)
80
6
1080
80
1.07
CF5 yr 104.86 98.00 91.59 85.60 (80 1009.35) 1469.33
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 problems
1.
2.
The duration of the bond (used to hedge) will change
The YTM of the bond used to hedge could change
What kind of risk would the
coupons be subject to?
36
1. Duration Change
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:
P
80
80
80
1080
1000
1.08 1.082 1.083 1.084
P 74.04 68.59 63.51 793.83 1,000
w1
74.04
1000
w2
68.59
1000
w3
63.51
1000
w4
793 .83
1000
Duration
D (0.07404)(1) (0.06859)(2) (0.06351)(3) (0.079383)4 3.577years
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%
37
2. Reinvestment risk
Suppose that after the first two payments the interest rate increases to 9%
1,000
80
80
80
80
80
1
2
3
4
5
(80)(1.08) 4
(80)(1.08)3 (80)(1.09)2
80
The company no longer
has enough money to
repay its loan of $1469.33
6
(80)(1.09) 80 1080
1.09
CF5 yr 108.84 100.7 95.05 87.20 (80 990.83) 1462.69
So what’s the point? This seems really ineffective – why am I not
teaching you how to fully resolve this problem?
IT IS HARD!!!
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 problem
1.
2.
3.
Every time the interest rate changes the hedging portfolio must be rebalanced
One decision managers have to make is how often to rebalance and weigh the
cost of doing so
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
Price a 20 year bond with coupon
of 30% and semiannual payments
6000
5000
Bond Price
4000
3000
2000
1000
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Yield To Maturity
41
Convexity refers to the curvature in the relationship between bond prices
and interest rates. What does that mean? – watch
Price a 20 year bond with coupon
of 30% and semiannual payments
6000
YTM 1.5%
5000
Bond Price
4000
40
150
1000
t
(1 .015/ 2) 40
t 1 (1 .015/ 2)
3000
$5,908.69
2000
1000
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Yield To Maturity
42
Convexity refers to the curvature in the relationship between bond prices
and interest rates. What does that mean? – watch
Price a 20 year bond with coupon
of 30% and semiannual payments
6000
YTM 4%
5000
Bond Price
4000
40
150
1000
t
(1 .04 / 2) 40
t 1 (1 .04 / 2)
3000
$4,556.21
2000
1000
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Yield To Maturity
43
Convexity refers to the curvature in the relationship between bond prices
and interest rates. What does that mean? – watch
Price a 20 year bond with coupon
of 30% and semiannual payments
6000
YTM 8%
5000
Bond Price
4000
40
150
1000
t
(1 .08 / 2) 40
t 1 (1 .08 / 2)
3000
$3,177.21
2000
1000
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Yield To Maturity
44
Convexity refers to the curvature in the relationship between bond prices
and interest rates. What does that mean? – watch
Price a 20 year bond with coupon
of 30% and semiannual payments
6000
YTM 13%
5000
Bond Price
4000
40
150
1000
t
(1 .13 / 2) 40
t 1 (1 .13 / 2)
3000
$2,202.37
2000
1000
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Yield To Maturity
45
Convexity refers to the curvature in the relationship between bond prices
and interest rates. What does that mean? – watch
Price a 20 year bond with coupon
of 30% and semiannual payments
6000
YTM 20%
5000
Bond Price
4000
40
150
1000
t
(1 .20 / 2) 40
t 1 (1 .20 / 2)
3000
$1,488.95
2000
1000
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Yield To Maturity
46
Convexity refers to the curvature in the relationship between bond prices
and interest rates. What does that mean? – watch
Price a 20 year bond with coupon
of 30% and semiannual payments
6000
YTM 30%
5000
Bond Price
4000
40
150
1000
t
(1 .30 / 2) 40
t 1 (1 .30 / 2)
3000
$1,000
2000
1000
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Yield To Maturity
47
Convexity refers to the curvature in the relationship between bond prices
and interest rates. What does that mean? – watch
Price a 20 year bond with coupon
of 30% and semiannual payments
6000
YTM 45%
5000
Bond Price
4000
40
150
1000
t
(1 .45 / 2) 40
t 1 (1 .45 / 2)
3000
666.77$
2000
1000
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Yield To Maturity
48
Convexity refers to the curvature in the relationship between bond prices
and interest rates. What does that mean? – watch
6000
5000
4000
Bond Price
3000
2000
1000
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Yield To Maturity
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
Convexity refers to the curvature in the relationship between bond prices
and interest rates. What does that mean? – watch
Duration is the derivative. It is
the slope of the tangent line
• This is what duration says the graph
(relationship) should look like
• When we do the duration calculation,
we find a point on this line
13%
51
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?
P
R
.1
D
7.23
0.7880
P
(1 R)
1 0.13 / 2
P (0.7880)(2202.37) 1735.47
Pt 1 Pt P 2202.13 1735.47 3937.84
What is the actual price?
40
150
1000
5038.64
t
40
(1 .03/ 2)
t 1 (1 .03 / 2)
5038.64
3937.84
1100.80
52
Convexity refers to the curvature in the relationship between bond prices
and interest rates. What does that mean? – watch
5,038.64
1100 .80
3937 .84
3%
13%
53
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?
P
R
.1
D
7.23
0.7880
P
(1 R)
1 0.13/ 2
P (0.7880)(2202.37) 1,735.47
Pt 1 Pt P 2202.131735.47 466.39
What is the actual price?
40
150
1000
1300.44
t
40
(1 .23/ 2)
t 1 (1 .23 / 2)
466.39
1300.44
834.05
54
Convexity refers to the curvature in the relationship between bond prices
and interest rates. What does that mean? – watch
Asymmetric
Pricing Errors!
1100 .80
1300 .44
834 .05
666 .39
3%
13%
23%
55
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
C W1 t12 t1 W2 t 22 t 2 ... Wn t n2 t n
3.
Sum values and divide by (1+ YTM)2 to get
convexity
58
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
80
80
80
80
80
80
Step #1 find the present value of payments
pv (CF1 )
80
1.06
pv (CF2 )
80
1.06 2
pv (CF3 )
80
1.06 3
pv (CF4 )
80
1.06 4
pv (CF5 )
80
1.06 5
pv (CF6 )
1080
1.06 6
pv(CF1 ) 75.47 pv(CF2 ) 71.20 pv(CF3 ) 67.17 pv(CF4 ) 63.37 pv(CF5 ) 59.78 pv(CF6 ) 761.36
Step #2 calculate weights
w1
75.47
1098 .35
w1 0.069
w2
71.20
1098 .35
w2 0.065
w3
67.17
1098 .35
w3 0.061
w4
63.37
1098 .35
w4 0.058
w5
59.78
1098 .35
w5 0.054
w6
761 .36
1098 .35
w6 0.693
59
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
Step #3 calculate the convexity
2
2
2
1 (0.0687)(1 1) (0.0648)(2 2) (0.0612)(3 3)
C
2
2
2
2
(1.06 ) (0.0577)(4 4) (0.0544)(5 5) (0.6932)(6 6)
C
1
33 .16
2
1.06
C 29.54
Measures the curvature of the
YTM bond price relationship –
larger values = more curvature
60
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!!
R 1
P
D
C ( R 2 )
P
1 R 2
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).
0.024 1
P
0(.03375
3.346 )(. 024 2 ) 0.03279
1.47
P
1 0.09 2 2
62
Price implied
by Duration
Price implied by
Duration & Convexity
63
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