金融风险Durationmodel
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It is a simple summary statistic of the effective average maturity of the portfolio
It is also the first order derivative of the bond price with respect to interest rate
1 year
2 years
t:
F: (—$931) $40
$40 $40
$1040
11
Duration of 2-year, 8% bond: Face value = $1,000, YTM = 12%
12
Time :
0
0.5
1
1.5
2 ys
CF: (—$931) $40
$40 $40
So, maturity can not serve well as an accurate measure of interest rate sensitivity of coupon bonds.
4
Maturity effect vs. Coupon effect on bond value
The longer maturity bonds experience greater price changes in response to any change in the discount rate. (Maturity effect)
The range of prices is greater when the coupon is lower. (Coupon effect) • The 6% bond shows greater changes in price in response to a 2% change than the 8% bond. The first bond is has greater interest rate risk.
With higher coupons, more of the bond’s value is generated by cash flows which take place sooner in time. Consequently, less sensitive to changes in R.
$1040
PV:
37.7 35.6 33.6
823.8 ∑= $ 931
Weight: W × time:
0.041 0.038 0.036 0.885 ∑=1
0.020
0.038 0.054
1.770 ∑=1.883ys
( Duration)
13
Duration of Zero-coupon Bond
wt=
CFt×DFt = CFt / (1+R)t = PVt
P
P
P
N
N
P =tΣ=1PVt =Σt=C1 Ft×DFt ;
DFt = 1/ (1+R)t
For better and easier understanding, take a coupon
Bond as a bundle or a portfolio of “zero-coupon” bonds.
be taken into account.
3
Shortcoming of Maturity model: Ignoring coupon effect
Bonds with identical maturities but different coupon payments responds differently to interest rate changes. Coupon effect does exist.
7
Exploring the ambiguity of maturity
What does maturity of a bond exactly mean in a coupon bond case? An ambiguous term!
In the case of coupon bonds, the maturity of a bond is not the maturity of all the cash flows generated in the bond, only that of the last payment, the last coupon plus face value, while other cash flows have shorter “maturities”.
9
The integrated formula
N
Σ t×CFt×DFt
D= t=1
=
N
Σ CFt×DFt
t=1
N
Σt=1PVt × t
N
Σ PVt
t=1
N
= Σt=1PVt × t P
N
= Σ (PVt / P)× t
t=1
10
Computing duration
Consider a 2-year, 8% coupon bond, with a face value of $1,000 and yield-to-maturity of 12%. Coupons are paid semi-annually.
2
Reflection on the Shortcoming of Maturity model: Ignoring coupon effect
Maturity model tries to take advantage of the maturity effect on bond value and use its maturity as an indicator of its interest rate sensitivity.
金融风险 Durationmodel
2024年2月2日星期五
Introduction
First developed in 1938 by Frederick Macaulay Taking into account both leverage and timing of
cash flow of assets and liabilities More accuracy in interest rate measurement Better for interest rate risk immunization Regulatory requirement
But strictly speaking, it is a good case only when the bond generates no coupon, i.e., it is a zero coupon bond.
Coupon effect is ignored in maturity model. More bonds pay coupons, and coupon effect must
So, maturity model can serve as a special case of duration model only when in the case of zero coupon bonds.
14
Duration of a consol bond(Perpetuities)
Therefore, each coupon payment is $40 and the per period YTM is (1/2) × 12% = 6%.
Present value of each cash flow equals CFt ÷ (1+ 0.06)t where t is the period number.
Mc= ∞ (Infinite) Dc=1+1/R (Finite) Proof to be conducted by those who know
calculus. Example: at a 10%yield, the duration of a
perpetuity that pays $100 once a year forever will equal 1.10/0.10=11years, but at an 8%yield it will equal 1.08/0.08=13.5years
(Interest Rate Sensitivity of 6% Coupon Bond)
5
Maturity effect vs. Coupon effect on bond value
(Interest Rate Sensitivity of 8% Coupon Bond)
6
Remarks on Preceding Slides
Consol bond is a bond that pays a fixed coupon each year forever.
Consol bonds that were issued by the British government in the 1890s to finance the Boer Wars in South Africa are still outstanding.
Maturities
0.5
1
1.5
2 years
of CFs
t:
0
1
2
3
4
CF: (—$931) $40
$40 $40
$1040
8
Definition and Calculation of duration
N
D=tΣ=1t×wt
Duration: The average life of a bond,or more technically, the weighted-average time to maturity of all cash flows, using relative present values of cash flows as weights.
15
Interpreting duration
A tool of interest rate risk management (measurement) for fixed income portfolio
Measure the sensitivity of a portfolio to interest rate change
For a zero coupon bond, duration = maturity since 100% of its present value is generated by the payment of the face value, at maturity.
For all other bonds: duration < maturity
Duration Gap and IR risk immunization strategy
16
Effective average maturity
--Interpret duration as a time concept
The ambiguity of “maturity”: life of the contract, life of the last cash flow
The weight: market value The weighted average, life of all the cash flows
involved in the contract Time concept working as a sensitivity measure:
the longer time the cash flows are exposed in interest rate risk(The larger the duration is), the more sensitive the contract(bond) is to interest rate change For better and easier understanding, take a coupon Bond as a bundle or a portfolio of zerocoupon bonds.
It is also the first order derivative of the bond price with respect to interest rate
1 year
2 years
t:
F: (—$931) $40
$40 $40
$1040
11
Duration of 2-year, 8% bond: Face value = $1,000, YTM = 12%
12
Time :
0
0.5
1
1.5
2 ys
CF: (—$931) $40
$40 $40
So, maturity can not serve well as an accurate measure of interest rate sensitivity of coupon bonds.
4
Maturity effect vs. Coupon effect on bond value
The longer maturity bonds experience greater price changes in response to any change in the discount rate. (Maturity effect)
The range of prices is greater when the coupon is lower. (Coupon effect) • The 6% bond shows greater changes in price in response to a 2% change than the 8% bond. The first bond is has greater interest rate risk.
With higher coupons, more of the bond’s value is generated by cash flows which take place sooner in time. Consequently, less sensitive to changes in R.
$1040
PV:
37.7 35.6 33.6
823.8 ∑= $ 931
Weight: W × time:
0.041 0.038 0.036 0.885 ∑=1
0.020
0.038 0.054
1.770 ∑=1.883ys
( Duration)
13
Duration of Zero-coupon Bond
wt=
CFt×DFt = CFt / (1+R)t = PVt
P
P
P
N
N
P =tΣ=1PVt =Σt=C1 Ft×DFt ;
DFt = 1/ (1+R)t
For better and easier understanding, take a coupon
Bond as a bundle or a portfolio of “zero-coupon” bonds.
be taken into account.
3
Shortcoming of Maturity model: Ignoring coupon effect
Bonds with identical maturities but different coupon payments responds differently to interest rate changes. Coupon effect does exist.
7
Exploring the ambiguity of maturity
What does maturity of a bond exactly mean in a coupon bond case? An ambiguous term!
In the case of coupon bonds, the maturity of a bond is not the maturity of all the cash flows generated in the bond, only that of the last payment, the last coupon plus face value, while other cash flows have shorter “maturities”.
9
The integrated formula
N
Σ t×CFt×DFt
D= t=1
=
N
Σ CFt×DFt
t=1
N
Σt=1PVt × t
N
Σ PVt
t=1
N
= Σt=1PVt × t P
N
= Σ (PVt / P)× t
t=1
10
Computing duration
Consider a 2-year, 8% coupon bond, with a face value of $1,000 and yield-to-maturity of 12%. Coupons are paid semi-annually.
2
Reflection on the Shortcoming of Maturity model: Ignoring coupon effect
Maturity model tries to take advantage of the maturity effect on bond value and use its maturity as an indicator of its interest rate sensitivity.
金融风险 Durationmodel
2024年2月2日星期五
Introduction
First developed in 1938 by Frederick Macaulay Taking into account both leverage and timing of
cash flow of assets and liabilities More accuracy in interest rate measurement Better for interest rate risk immunization Regulatory requirement
But strictly speaking, it is a good case only when the bond generates no coupon, i.e., it is a zero coupon bond.
Coupon effect is ignored in maturity model. More bonds pay coupons, and coupon effect must
So, maturity model can serve as a special case of duration model only when in the case of zero coupon bonds.
14
Duration of a consol bond(Perpetuities)
Therefore, each coupon payment is $40 and the per period YTM is (1/2) × 12% = 6%.
Present value of each cash flow equals CFt ÷ (1+ 0.06)t where t is the period number.
Mc= ∞ (Infinite) Dc=1+1/R (Finite) Proof to be conducted by those who know
calculus. Example: at a 10%yield, the duration of a
perpetuity that pays $100 once a year forever will equal 1.10/0.10=11years, but at an 8%yield it will equal 1.08/0.08=13.5years
(Interest Rate Sensitivity of 6% Coupon Bond)
5
Maturity effect vs. Coupon effect on bond value
(Interest Rate Sensitivity of 8% Coupon Bond)
6
Remarks on Preceding Slides
Consol bond is a bond that pays a fixed coupon each year forever.
Consol bonds that were issued by the British government in the 1890s to finance the Boer Wars in South Africa are still outstanding.
Maturities
0.5
1
1.5
2 years
of CFs
t:
0
1
2
3
4
CF: (—$931) $40
$40 $40
$1040
8
Definition and Calculation of duration
N
D=tΣ=1t×wt
Duration: The average life of a bond,or more technically, the weighted-average time to maturity of all cash flows, using relative present values of cash flows as weights.
15
Interpreting duration
A tool of interest rate risk management (measurement) for fixed income portfolio
Measure the sensitivity of a portfolio to interest rate change
For a zero coupon bond, duration = maturity since 100% of its present value is generated by the payment of the face value, at maturity.
For all other bonds: duration < maturity
Duration Gap and IR risk immunization strategy
16
Effective average maturity
--Interpret duration as a time concept
The ambiguity of “maturity”: life of the contract, life of the last cash flow
The weight: market value The weighted average, life of all the cash flows
involved in the contract Time concept working as a sensitivity measure:
the longer time the cash flows are exposed in interest rate risk(The larger the duration is), the more sensitive the contract(bond) is to interest rate change For better and easier understanding, take a coupon Bond as a bundle or a portfolio of zerocoupon bonds.