Duration model
IntroductionIntroduction
Calculating durationCalculating duration
Interpreting durationInterpreting duration
Understanding more features of durationUnderstanding more features of duration
Using duration for interest rate risk immunizationUsing duration for interest rate risk immunization
Limitations of duration immunizationLimitations of duration immunization
Introduction
First developed in 1938 by Frederick MacaulayFirst developed in 1938 by Frederick Macaulay
Taking into account both leverage and timing of Taking into account both leverage and timing of
cash flow of assets and liabilitiescash flow of assets and liabilities
More accuracy in interest rate measurementMore accuracy in interest rate measurement
Better for interest rate risk immunizationBetter for interest rate risk immunization
Regulatory requirementRegulatory requirement
Reflection on the Shortcoming of Maturity
model: Ignoring coupon effect
Maturity model tries to take advantage of the Maturity model tries to take advantage of the
maturity effect on bond value and use its maturity effect on bond value and use its
maturity as an indicator of its interest rate maturity as an indicator of its interest rate
.
But strictly speaking, it is a good case only when But strictly speaking, it is a good case only when
the bond generates no coupon, ., it is a zero the bond generates no coupon, ., it is a zero
coupon bond. coupon bond.
Coupon effect is ignored in maturity effect is ignored in maturity model.
More bonds pay coupons, and coupon effect must More bonds pay coupons, and coupon effect must
be taken into account. be taken into account.
Shortcoming of Maturity model:
Ignoring coupon effect
Bonds with identical maturities but different Bonds with identical maturities but different
coupon payments responds differently to interest coupon payments responds differently to interest
rate changes. Coupon effect does exist. rate changes. Coupon effect does exist.
With higher coupons, more of the bond’s value is With higher coupons, more of the bond’s value is
generated by cash flows which take place sooner generated by cash flows which take place sooner
in time. Consequently, less sensitive to changes in time. Consequently, less sensitive to changes
in R.
So, maturity can not serve well as an accurate So, maturity can not serve well as an accurate
measure of interest rate sensitivity of coupon measure of interest rate sensitivity of coupon
.
Maturity effect vs. Coupon effect on bond valueMaturity effect vs. Coupon effect on bond value
(Interest Rate Sensitivity of 6% Coupon Bond)(Interest Rate Sensitivity of 6% Coupon Bond)
Maturity effect vs. Coupon effect on bond valueMaturity effect vs. Coupon effect on bond value
(Interest Rate Sensitivity of 8% Coupon Bond)(Interest Rate Sensitivity of 8% Coupon Bond)
Remarks on Preceding Slides
The longer maturity bonds experience greater The longer maturity bonds experience greater
price changes in response to any change in the price changes in response to any change in the
discount rate. (Maturity effect)discount rate. (Maturity effect)
The range of prices is greater when the coupon is The range of prices is greater when the coupon is
lower. (Coupon effect)lower. (Coupon effect)
•• The 6% bond shows greater changes in price The 6% bond shows greater changes in price
in response to a 2% change than the 8% bond. in response to a 2% change than the 8% bond.
The first bond is has greater interest rate first bond is has greater interest rate risk.
Exploring the ambiguity of maturity
What does maturity of a bond exactly mean in a What does maturity of a bond exactly mean in a
coupon bond case? An ambiguous term!coupon bond case? An ambiguous term!
In the case of coupon bonds, the maturity of a In the case of coupon bonds, the maturity of a
bond is not the maturity of all the cash flows bond is not the maturity of all the cash flows
generated in the bond, only that of the last generated in the bond, only that of the last
payment, the last coupon plus face value, while payment, the last coupon plus face value, while
other cash flows have shorter “maturities”.other cash flows have shorter “maturities”.
t : 0 1 2 3 4
1 2 years
CF: (—$931) $40 $40 $40 $1040
Maturities
of CFs
Definition and Calculation of duration
NN
DD=Σ =Σ tt×w×wtt
tt=1=1
CFCFtt××DFDFt t CFCFt t / (1+/ (1+RR))
t t PVPVtt
wwtt= = == = =
PP PP PP
N NN N
PP =Σ =Σ PVPVt t =Σ CF=Σ CFtt××DFDFtt ; DF ; DFt t == 1 1/ (1+/ (1+RR))
t t
tt=1 =1 tt=1=1
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.
For better and easier understanding, take a coupon For better and easier understanding, take a coupon
Bond as a bundle or a portfolio of “zero-coupon” bonds. Bond as a bundle or a portfolio of “zero-coupon” bonds.
The integrated formula
D= =D= =
= = = =
N N
Σ CF Σ CFtt××DFDFt t
tt=1=1
N N
Σ Σ tt×CF×CFtt××DFDFt t
tt=1=1
N N
Σ PVΣ PVt t
tt=1=1
N N
Σ PVΣ PVt t ×× tt
tt=1=1
N N
Σ PVΣ PVt t ×× tt
tt=1=1
PP
N N
Σ (PVΣ (PVt t // P)×P)× tt
tt=1=1
Computing duration
Consider a 2-year, 8% coupon bond, with a face Consider a 2-year, 8% coupon bond, with a face
value of $1,000 and yield-to-maturity of 12%. value of $1,000 and yield-to-maturity of 12%.
Coupons are paid are paid semi-annually.
Therefore, each coupon payment is $40 and the Therefore, each coupon payment is $40 and the
per period YTM is (1/2) × 12% = 6%.per period YTM is (1/2) × 12% = 6%.
Present value of each cash flow equals CFPresent value of each cash flow equals CF tt ÷ (1+ ÷ (1+
))tt where where tt is the period number. is the period number.
t : 0 1 2 3 4
1 year 2 years
CF: (—$931) $40 $40 $40 $1040
Duration of 2-year, 8% bond:
Face value = $1,000, YTM = 12%
Time : 0 1 2 ys
CF: (—$931) $40 $40 $40 $1040
PV: ∑= $ 931
Weight: ∑=1
W × time: ∑=
( Duration)
Duration of Zero-coupon Bond
For a zero coupon bond, duration = maturity For a zero coupon bond, duration = maturity
since 100% of its present value is generated by since 100% of its present value is generated by
the payment of the face value, at payment of the face value, at maturity.
For all other bonds:For all other bonds:
duration < maturity duration < maturity
So, maturity model can serve as a special case of So, maturity model can serve as a special case of
duration model only when in the case of zero duration model only when in the case of zero
coupon bonds.
Duration of a consol bond(Perpetuities)
Consol bond is a bond that pays a fixed coupon Consol bond is a bond that pays a fixed coupon
each year year forever.
Consol bonds that were issued by the British Consol bonds that were issued by the British
government in the 1890s to finance the Boer government in the 1890s to finance the Boer
Wars in South Africa are still in South Africa are still outstanding.
MMcc= = ∞ (∞ (Infinite)Infinite)
DDcc=1+1/R (Finite)=1+1/R (Finite)
Proof to be conducted by those who know Proof to be conducted by those who know
calculus. calculus.
Example: at a 10%yield, the duration of a Example: at a 10%yield, the duration of a
perpetuity that pays $100 once a year forever perpetuity that pays $100 once a year forever
will equal but at an 8%yield will equal but at an 8%yield
it will equal will equal
Interpreting duration
A tool of interest rate risk management A tool of interest rate risk management
(measurement) for fixed income portfolio(measurement) for fixed income portfolio
Measure the sensitivity of a portfolio to interest Measure the sensitivity of a portfolio to interest
rate changerate change
It is a simple summary statistic of the effective It is a simple summary statistic of the effective
average maturity of the portfolioaverage maturity of the portfolio
It is also the first order derivative of the bond It is also the first order derivative of the bond
price with respect to interest rate price with respect to interest rate
Duration Gap and IR risk immunization strategyDuration Gap and IR risk immunization strategy
Effective average maturity
--Interpret duration as a time concept--Interpret duration as a time concept
The ambiguity of “maturity”: life of the contract, The ambiguity of “maturity”: life of the contract,
life of the last cash flowlife of the last cash flow
The weight: market value The weight: market value
The weighted average, life of all the cash flows The weighted average, life of all the cash flows
involved in the contractinvolved in the contract
Time concept working as a sensitivity measure: Time concept working as a sensitivity measure:
the longer time the cash flows are exposed in the longer time the cash flows are exposed in
interest rate risk(The larger the duration is), the interest rate risk(The larger the duration is), the
more sensitive the contract(bond) is to interest more sensitive the contract(bond) is to interest
rate changerate change
For better and easier understanding, take a For better and easier understanding, take a
coupon Bond as a bundle or a portfolio of zero-coupon Bond as a bundle or a portfolio of zero-
coupon bonds.
The first order derivative
--Interpret duration as risk measure concept --Interpret duration as risk measure concept
( Interest sensitivity, or elasticity) ( Interest sensitivity, or elasticity)
T T CFCFtt
PP = Σ = Σ
tt=1 =1 (1+(1+RR))tt
Dp Dp T T t ·CFt ·CFt t DDPP
==－－Σ = Σ = －－
d dR R t=t=11 (1+ (1+RR))tt+1 +1 1+1+R R
ddPP //PP
= = －－DD
d dR/R/(1+(1+RR))
Interest Elasticity, or sensitivity
of a bond is defined as
the percentage change in
the price of a bond for any
given change in interest rates.
Modified Duration
ddPP//PP ddPP d dRR
D= D= －－ = = －－D D
d dRR/ (1+/ (1+RR) ) PP 1+ 1+RR
Let MD=D/(1+R), Let MD=D/(1+R), where MD is modified MD is modified duration.
Then:Then: dP/P = -D[dR/(1+R)] = -MD × dRdP/P = -D[dR/(1+R)] = -MD × dR
(in percentage)(in percentage)
To estimate the change in price, rewrite this as:To estimate the change in price, rewrite this as:
dP = -D[dR/(1+R)]P = -(MD) × (dR) × (P)dP = -D[dR/(1+R)]P = -(MD) × (dR) × (P)
(in dollar value)(in dollar value)
Note the direct Note the direct linearlinear relationship between dP and -D. relationship between dP and -D.
An example
Consider a 6-year Eurobond (face value $1000 Consider a 6-year Eurobond (face value $1000
and coupon paid annually)with an 8% coupon and coupon paid annually)with an 8% coupon
and 8% yield. Its durations was and 8% yield. Its durations was
D=.(See Textbook Page 197)D=.(See Textbook Page 197)
Suppose that yields were to rise by one bp from Suppose that yields were to rise by one bp from
8% to %, how much will the Eurobond lose 8% to %, how much will the Eurobond lose
its value?its value?
dP/P = -D[dR/(1+R)] dP/P = -D[dR/(1+R)]
dP = -D[dR/(1+R)]P dP = -D[dR/(1+R)]P
= -()× [ ] ×1000 = -()× [ ] ×1000
= =
So, the duration model predicts that the price of So, the duration model predicts that the price of
the bond would fall to $ after the the bond would fall to $ after the
increase in yield by in yield by 1bp.
If MD is given, the calculation is even easier. If MD is given, the calculation is even easier.
Notice print error in Textbook 203
Relationship between dP and -D.
-D
Yield changes in percentage
(dR/(1+R))
Price change Price change
in percentagein percentage
dP/PdP/P
Understanding more features of duration
Duration and maturity:Duration and maturity:
•• D increases with M, but at a decreasing increases with M, but at a decreasing rate.
Duration and yield-to-maturity:Duration and yield-to-maturity:
•• D decreases as yield decreases as yield increases.
Duration and coupon interest:Duration and coupon interest:
•• D decreases as coupon increasesD decreases as coupon increases
Duration of a portfolio
Duration of portfolio of assets (liabilities) equals Duration of portfolio of assets (liabilities) equals
weighted average of durations of individual weighted average of durations of individual
components of the portfolio, with the weights components of the portfolio, with the weights
being their values relative to the entire portfolio being their values relative to the entire portfolio
.
Duration Gap
In the case of a 2-year, 8% coupon bond In the case of a 2-year, 8% coupon bond
(Coupons are paid semi-annually), with a face (Coupons are paid semi-annually), with a face
value of $1,000 and yield-to-maturity of 12%. value of $1,000 and yield-to-maturity of 12%.
Suppose the bond is the only loan asset (L) of an Suppose the bond is the only loan asset (L) of an
FI, funded by a 2-year certificate of deposit (D). FI, funded by a 2-year certificate of deposit (D).
Maturity gap: MMaturity gap: MLL - M - MDD = 2 -2 = 0 = 2 -2 = 0
Duration Gap: DDuration Gap: DLL - D - DDD = - = = - =
•• Deposit has greater interest rate sensitivity Deposit has greater interest rate sensitivity
than the loan, so DGAP is negative. than the loan, so DGAP is negative.
•• FI exposed to declining interest exposed to declining interest rates.
Immunizing the
Balance Sheet of an FI
Duration gap vs. Leverage-adjusted duration gap Duration gap vs. Leverage-adjusted duration gap
From the balance sheet, E= the balance sheet, E=A-L.
Therefore, Therefore, E=E=A-A-L. L.
In the same manner to determine the change in In the same manner to determine the change in
bond prices, we can find the change in value of bond prices, we can find the change in value of
equity using durationequity using duration..
A/A=-DA/A=-DA A [[ R/(1+R)]; R/(1+R)]; L/L=-DL/L=-DL L [[ R/(1+R)]R/(1+R)]
E = [-DE = [-DAAA + DA + DLLL] L] R/(1+R) orR/(1+R) or
E = -[E = -[DDAA -- D DLLk] k] ×× A A ××[ [ R/(1+R) ] R/(1+R) ]
Let k=L/A, the leverage ratio of the FI. Let k=L/A, the leverage ratio of the FI.
Duration and Immunization
E = -[DA - DLk] ×× A ××[ R/(1+R) ]
The formula shows 3 effects of interest rate The formula shows 3 effects of interest rate
changes on the market value of an FI’s Equity or changes on the market value of an FI’s Equity or
net worth(net worth(E E ) :) :
•• Leverage adjusted Duration Gap= DLeverage adjusted Duration Gap= DAA -- D DLLk k
•• The size of the FI=AThe size of the FI=A
•• The size of the interest rate shock= The size of the interest rate shock= R/(1+R) R/(1+R)
An example:
Suppose DSuppose DAA = 5 years, D = 5 years, DLL = 3 years and rates = 3 years and rates
are expected to rise from 10% to 11%. (Rates are expected to rise from 10% to 11%. (Rates
change by 1%). Also, A = 100, L = 90 and E = change by 1%). Also, A = 100, L = 90 and E =
10. Find change in . Find change in E.
E = -[E = -[DDAA - D - DLLk]A[k]A[R/(1+R)]R/(1+R)]
= -[5 - 3(90/100)]100[.01/] = -[5 - 3(90/100)]100[.01/]
= - $(millions) = - $(millions)
Methods of immunizing balance sheet
If DIf DAA=D=DLL, ,
E = -[ E = -[DDAA - D - DLLk]A[k]A[R/(1+R)]R/(1+R)]
= -[5 - 5(90/100)]100[.01/] = -[5 - 5(90/100)]100[.01/]
= - $(millions) = - $(millions)
So, DSo, DAA=D=DL L match does not work for a leveraged match does not work for a leveraged
FI immunization.
Importantly, DImportantly, DAA = D = DLLkk
E = -[E = -[DDAA - D - DLLk]A[k]A[R/(1+R)]R/(1+R)]
= = [0[0]A[]A[R/(1+R)]=0R/(1+R)]=0
To achieve DTo achieve DAA = D = DLLk, adjust Dk, adjust DAA , D , DLL or k. or k.
Immunization and
Regulatory Concerns
As we know from previous slides, FI’s As we know from previous slides, FI’s
shareholders focus on shareholders focus on E, E, and target on and target on E = 0, E = 0,
the immunization strategy is the immunization strategy is DDAA = kD = kDL L ..
While regulators focus on While regulators focus on (E/A), because they(E/A), because they
set target ratios for a bank’s capital (net worth): set target ratios for a bank’s capital (net worth):
Capital (Net worth) ratio = E/ (Net worth) ratio = E/A.
If target is to set If target is to set (E/A) = 0, rather than (E/A) = 0, rather than E = 0, E = 0,
the immunization strategy is the immunization strategy is DDAA = D = DLL,, rather than rather than
DDAA = kD = kDLL
Proof to be conducted by yourself. A key clue is Proof to be conducted by yourself. A key clue is
given as:given as:
(E/A) = (1/A) (E/A) = (1/A) E – (E/AE – (E/A2 2 ) ) A=…?A=…?
*Limitations of Duration
Immunizing the entire balance sheet through Immunizing the entire balance sheet through
duration match can be costly in restructuring. duration match can be costly in restructuring.
However, easing factors include Growth of purchased funds, asset However, easing factors include Growth of purchased funds, asset
securitization, and loan sales market; taking hedging poisitions in securitization, and loan sales market; taking hedging poisitions in
the markets for markets for derivatives.
Immunization is a dynamic process since duration Immunization is a dynamic process since duration
depends on instantaneous on instantaneous R.
Large interest rate change effects not accurately Large interest rate change effects not accurately
.
Convexity: Non-linear relationship between bond price and Convexity: Non-linear relationship between bond price and
interest rate change(yield curve )interest rate change(yield curve )
The problem of the flat yield curve The problem of the flat yield curve
complex if nonparallel shift in yield if nonparallel shift in yield curve.
The problem of default riskThe problem of default risk
…….…….
beta coefficient of stocks
Stock valuation modelsStock valuation models
Calculating beta coefficientCalculating beta coefficient
Taking beta as a sensitivity measureTaking beta as a sensitivity measure
Extension to multi-factor pricing modelExtension to multi-factor pricing model
The capital asset pricing formula
E E ((rrii))
== rrf f + β+ βii [ [ E E ((rrMM) – ) – rrff ]]
The risk premium of an individual asset is The risk premium of an individual asset is
proportional to its proportional to its ββ coefficient and the risk coefficient and the risk
premium of the market portfoliopremium of the market portfolio
ββi i = σ= σiMiM / / σσMM
22
More than one factor in APT
APT Pricing EquationAPT Pricing Equation
EEii=λ=λ00+λ +λ 11ββii11+λ +λ 22ββii22+…+λ +…+λ kkββikik
λ λ0 :0 :risk-free raterisk-free rate
β: sensitive coefficient β: sensitive coefficient
λi: risk premium of common factor I λi: risk premium of common factor I
Common factorsCommon factors
GDP, Inflation rate, Interest rate, Oil price… GDP, Inflation rate, Interest rate, Oil price…
Important Terms
DurationDuration
Interest elasticity Interest elasticity
Modified durationModified duration
Duration gapDuration gap
Leverage-adjusted duration gapLeverage-adjusted duration gap
** Convexity Convexity
Questions
Why is duration considered a more complete Why is duration considered a more complete
measure of an asset or liability’s interest rate measure of an asset or liability’s interest rate
sensitivity than maturity?sensitivity than maturity?
When is the duration of an asset equal to its When is the duration of an asset equal to its
maturity?maturity?
How to calculate a duration of a coupon bond? How to calculate a duration of a coupon bond?
What is the duration of a one-year,8 percent What is the duration of a one-year,8 percent
coupon, 10 percent yield bond that pays coupons coupon, 10 percent yield bond that pays coupons
quarterly?quarterly?
What is the duration of a zero-coupon bond? What is the duration of a zero-coupon bond?
What is the duration of a consol bond?What is the duration of a consol bond?
Which has the longest duration, a 30-year, 8%, Which has the longest duration, a 30-year, 8%,
zero-coupon bond or a an 8% infinite maturity zero-coupon bond or a an 8% infinite maturity
consol bond?consol bond?
What is the relationship between duration and What is the relationship between duration and
yield to maturity on a financial security?yield to maturity on a financial security?
Do high-coupon bonds have high or low duration?Do high-coupon bonds have high or low duration?
What is the relation between the duration of a What is the relation between the duration of a
bond and the interest elasticity of a bond?bond and the interest elasticity of a bond?
How can a manager use information on an FI’s How can a manager use information on an FI’s
duration gap to restructure , and thereby duration gap to restructure , and thereby
immunize, the balance sheet against interest rate immunize, the balance sheet against interest rate
risk?risk?
Suppose DSuppose DAA= 3 years,D= 3 years,DL L = 6 years, K=.8, and = 6 years, K=.8, and
A=$100 million. What is the effect on owners’net A=$100 million. What is the effect on owners’net
worth if worth if △△RR/(1+/(1+RR) rises 1 percent?) rises 1 percent?
What minimum target ratio is typically used by What minimum target ratio is typically used by
regulators to measure a bank’s net worth relative regulators to measure a bank’s net worth relative
to its assets?to its assets?
Is immunizing a bank’s net worth the same as Is immunizing a bank’s net worth the same as
immunizing its net worth-assets ratio? If not, immunizing its net worth-assets ratio? If not,
why not?why not?
Suggest Reading and Problem Set
Suggested Reading: Suggested Reading:
Chapter 9, Financial Institutions ManagementChapter 9, Financial Institutions Management
Problem set 1, part 2 Problem set 1, part 2
Questions and Problems: 2 and 20 Questions and Problems: 2 and 20
See Page213 and 217, Financial Institutions See Page213 and 217, Financial Institutions
ManagementManagement
