the duration gap model and clumping session 2 andrea sironi mafinrisk – 2010 market risk

The duration gap model and clumping

Session 2Andrea Sironi

Mafinrisk – 2010Market Risk

2

Agenda

Market value versus historical cost accounting

The duration gap model

The Clumping Model

3

The Duration Gap Model

“Market Value” model target variable = market value of shareholders’ equity

Focus on impact of interest rate changes on the market value of assets and liabilities

Gap = difference between the change in the market value of assets and the market value of liabilities

4

Market Value vs Historical Value

3.27.2590%3100%5200920092009 IEIINII

ASSETS € m LIABILITIES € m

Fixed rate (5%) 10 Y Mortgages 100 Fixed Rate (3%) 2 y Notes Shareholders’ Equity

9010

Total 100 Total 100

Dec. 31, 2008


CashFixed rate (5%) 10 Y Mortgages

2.3100

Fixed Rate (3%) 2 y Notes Shareholders’ Equity

9012.3

Total 102.3 Total 102.3

Dec. 31, 2009

5

follows

On the 1/1/2009 the ECB increase the interest rates of 100 bp

Nothing changes in the FS of the bank



4.6100


9014.6


Dec. 31, 2010

%7.183.12

3.2%23

10

3.220102009 ROEROE

6

follows

In 2011 the bank has to finance the 10Y Mortgages with a new fixed rate note issued at the new market rate: 4%

The effect of the increase of the interest rates on the profitability of the bank appears only two years after the

variation itself.

4.16.3590%4100%5201120112011 IEIINII

%59.96.14

4.12011 ROE

7

followsThis problem can be solved using in the FS the market value of

A/L instead of the historical value

2.93

%61

100

%61

59

19%5

t

tMortgageMV 13.89%41

7.92%3

NoteMV

SELA MVMVMVLP /

63.383.18.1907.213.8910052.93/ 2009 LP

ASSETS € m LIBILITIES € m


2.393.2


89.136.37


Dec. 31, 2009

8

followsNext Year (2010)

79.93

%61

100

%61

58

18%5

t

tMortgageMV Notes (maturity) 90

02.257.359.513.897.2902.9379,935/ 2010 LP

ASSETS € m LIBILITIES € m


4,693,79


90,008,39

Total 98,39 Total 98,39

Dec. 31, 2010

9

Agenda



The Clumping Model

10

The Duration gap

The same result could be obtained using the duration gap

AA

A ii

D

MVA

MVA

1 L

L

L ii

D

MVL

MVL

1

AAAA

A iMDMVAii

DMVAMVA

1 LLLL

L iMVMVLii

DMVLMVL

1

LLAA iMDMVLiMDMVAMVLMVAMVE

iMDMVLMDMVAMVE LA

iMDLEVMDMVA

MVELA

iMVAMDLEVMDMVE LA

11

The Duration gap

The change in the market value of Shareholders’ Equity is a function of three variables:

1. The difference between the modified duration of assets and the modified duration of the liabilities corrected for the bank’s leverage (“leverage adjusted duration gap”) duration gap (DG)

2. The size of the intermediation activity of the bank measured by the market value of total assets

3. The size of the interest rates change

iMVAMDLEVMDMVE LA

iMVADGMVE

12

Immunization• If MVA = MVL MVE is not sensitive to interest

rates changes if MDA = MDL.

• If MVA > MVL MVE is not sensitive to interest rates changes if DG=0, i.e. MDA < MDL. In this case the higher sensitivity of liabilities will compensate the initial lower market value and the change in the absolute value of assets and liabilities will be equal.

iMVE

MVAMDLEVMD

MVE

MVE PA

13

The example againLet’s go back to our bank

108.810005.1

105

1010005.1

5

1 9

1

1010

1%5

t

t

t

tt

Mortgage tMVi

CF

tD

722.705.1

108.8

1%5

%5

i

DMD Mortgage

Mortgage

971.19003.1

7.92

29003.17.2

11 2

1%3

t

t

tt

Note MVi

CF

tD

914.103.1

971.1

1%3

%3

i

DMD Note

Note

14

follows

6914.190.0722.7 LA MDLEVMDDG

6%11006 iMVADGMVE

For an interest rates increase of 100 bp the market value of shareholders’ equity would

decrease by 6 m€ (60% of the original value)

15

Some remarks

The result (-6) is different form what we got before (-3.63) for three main reasons:

-6 m€ is an instantaneous decrease estimated at the time of the int. rates change (January 1st 2004);

In the – 3.63 m€ we also have 2.3 m€ of interest margin

The duration is just a first order approximation

16

Duration gap: problems and limits

1. Duration (and duration gap) changes every instant, when interest rate change, or simply because of the passage of time

Immunization policies based on duration gap should be updated continuously

2. Duration (and duration gap) is based on a linear approximation

Impact not estimated precisely

3. The model assumes uniform interest rate changes (i) of assets and liabilities interest rates

17

Problem 1:duration changes

Every time market interest rate change, duration needs to be computed again wuth new weights (PV of cash flows)

Even if rates do not change, duration decreases: linearly with “jumps” related to coupon payments

Duration

t2 timet3t1

Coupon payments

18

Answer to problem 2:convexity

idi

MVdMV

MV

MV AA

A

A

2

)( 2

2

2 i

di

MVMVdi

di

MVdMV

MV

MV AAAA

A

A

Rather than proxying % change in value with the first derivative only

…we could add the second term in Taylor(or McLaurin) including second derivative:

See following slides

19

Answer to problem 2:convexity

N

tt

t

N

t

tt

N

t

tt

A

i

CFtt

i

iCFttiCFtdi

d

di

MVd

1

22

1

2

1

1

2

2

1)(

1

1

1)1(1

N

t A

ttAA

MV

iCFtt

idi

MVMVd

1

222

2 1)(

1

1 Convexity, C

Modified convexity MC

Second derivative of VMA to i

Dividing both terms by MVA :

20

Answer to problem 2:duration gap and convexity gap

2

)( 2iMCiMD

MV

MVAA

A

A

AAAAA MVi

MCMViMDMV

2

)( 2

LLLLL MVi

MCMViMDMV

2

)( 2

ALAALAE MVi

CMLMCMViMDLMDMV

2

)( 2

duration gap convexity gap

Substituting duration and convexity in the second order expansion

Multiplying both terms by MVA:

The change in market value of the bank’s equity can now be better estimated:

Same for liabilities:

21

Duration gap and convexity gap: our example

79.69100

%51105)981(

100

%515)(

%51

1 98

1

22

t

t

mortgageA

tt

MCMC

97.090

%317.9211

%31

1 22

CDL CMCM

93.51002

%)1(97.09.079.69100%1

2

DGVM B

Very close to the true change

(-5.94)

First order proxy, -6.23

convexity gap,equal to 61.6

22

Answer to problem 3:beta-duration gap

The impact of an interest rate change depends on 4 factors: average MD of assets and liabilities average sensitivity of assets and liabilities interest rates to the base rate (beta) financial leverage L size of the bank (MVA)

ii AA Then substitute in the change of the value of the bank

Similar to standardized repricing gap. For each asset (liability) estimate:

iMVMDLMD

iMDLMVMDMV

iMDMViMDMV

MVMVMV

ALLAA

LLLAAA

LLLAAA

LAB

beta-duration gap

ii PP

23

Residual Problems

Assumption of a uniform change of assets and liabilities’ interest rates.

Assumption of a uniform change of interest rates for different maturities.

The model does not consider the effect of a variation of interest rates on the volume of financial assets and liabilities

24

Questions & Exercises

1. Which of the following does not represent a limitation of the repricing gap model which is overcome by the duration gap model?

A) Not taking into account the impact of interest rates changes on the market value of non sensitive assets and liabilities

B) Delay in recognizing the impact of interest rates changes on the economic results of the bank

C) Not taking into account the impact on profit and loss that will emerge after the gapping period

D) Not taking into account the consequences of interest rate changes on current account deposits

25


2. A bank’s assets have a market value of 100 million euro and a modified duration of 5.5 years. Its liabilities have a market value of 94 million euro and a modified duration of 2.3 years. Calculate the bank’s duration gap and estimate which would be the impact of a 75 basis points interest rate increase on the bank’s equity (market value).

26


3. Which of the following statements is NOT correct?A. The convexity gap makes it possible to improve

the precision of an interest-rate risk measure based on duration gap

B. The convexity gap is a second-order effectC. The convexity gap is an adjustment needed

because the relationship between the interest rate and the value of a bond portfolio is linear

D. The convexity gap is the second derivative of the value function with respect to the interest rate, divided by a constant which expresses the bond portfolio’s current value.

27


4. Using the data in the table belowi) compute the bank’s net equity value, duration gap and convexity

gap;ii) based on the duration gap only, estimate the impact of a 50 basis

points increase in the yield curve on the bank’s net value;iii) based on both duration and convexity gap together, estimate the

impact of a 50 basis points increase in the yield curve on the bank’s net value;

iv) briefly comment the resultsAssets Value Modified duration

Modified convexity

Open credit lines 1000 0 0

Floating rate securities 600 0.25 0.1

Fixed rate loans 800 3.00 8.50

Fixed rate mortgages 1200 8.50 45

Liabilities Value Modified duration

Modified convexity

Checking accounts 1200 0 0

Fixed rate CDs 600 0.5 0.3

Fixed rate bonds 1000 3 6.7

28

Agenda



The Clumping Model

29

A common problem and a possible solution

Repricing gap and duration gap assumption of uniform change of interest rates for different maturities

The Clumping o cash-bucketing model a model with independent changes of interest rates at different maturities

The model is built upon the zero-coupon curve (both the repricing gap and the duration gap model were focused on the yield curve).

The model works trough the mapping of single cash flows on a predetermined number of nodes (or maturities) on the term structure.

30

How to estimate zero coupon rates: bootstrapping

For longer maturities we typically have no zero coupon bonds

We need to extract them from coupon bonds One possibility is through bootstrapping

Assume we want to estimate the 2.5 (r2,5) zero-coupon rate For this maturity we only have a 4.5% (semi-

annual) coupon paying bond with a price of 100. For the preceding maturities (t = 0.5; 1; 1.5; 2) we

have zero coupon bonds (from their prices we can get their yield to maturity (rt))

31


1100

t

Ztt VMr

%26.4192

1002

2 r

2. We use these zero-coupon rates to estimate the present value of the first four cash flows (coupons) of the 4.5% coupon paying bond

1. From prices of zcb we extract the corresponding rt

Es.

102.25

2.50 12.122.162.21 2.07

1,5 20.5

2.252.252.25 2.25

8.552%)26.41(

25.2

Ex.

Zero Coupon Bond Maturity Price Rate6 months 0.5 98 4.12%

1 year 1 96 4.17%18 months 1.5 94 4.21%24 months 2 92 4.26%

32


3. Find the rate that equates the present value of 102.5 to the residual valueof the bond which has not been explained by the PV of the four coupons

2.50 12.122.162.21 2.07

1.5 20.5

2.252.252.25 2.25

8.55

45.91)1(

25.1025.25.2

rrr

102.25

100 - = 91.45

%57.4145.91

25.1025.2

5.2 r

33

What is the mapping for?

The mapping is a procedure to simplify the representation of the financial position of the bank.

Mapping is used to transform a portfolio with real cash flows, associated to an excessive number of p dates, into a simplified portfolio, based on a limited number q (<p) of maturity nodes (standard dates).

After mapping, it’s easier to implement effective risk management policies

Goal: reduce all the banks’ cash flows to a small number of significant nodes (maturities).

34

Cash-flow mapping

We can get an interest rate curve with different rates for every individual maturity

Do I really need to consider MxN nodes? No, cash-flow mapping allows to map a portfolio

of assets and liabilities (with a large number of cash flows associated to a large number of maturities) to a limited number of maturity nodes

It represents a special case of mapping A methodology to map a portfolio to a limited number of

risk factors: e.g. international equity portfolio to S&P500, Dax and MIB 30

35

Some simplifying cash-flow mapping techniques

Analytical principal Given M securities,

“maps” each of them to the “principal” maturity node

Synthetic principal Given M securities, it

only considers the maturity of principal (computes an average)

Analytical duration Given M securities, it

maps each of them to its duration

Synthetic duration Given M securities, it

only considers the duration (computes an average)

Req

uire

s M

no

des

Ext

rem

ely

sim

plifi

ed

Does not consider coupons reinvestment risk

Modified analytical principal method

36

An hybrid technique: modified principal Computing analytic

duration for each asset and liability might be complex

Using principal is not precise as it does not consider the coupons

However, given the level of interest rates (e.g. 5% in the chart), there exists a relationship between principal and duration for bonds with different coupon level

0

2.5

5

7.5

10

0 2.5 5 7.5 10

Time to maturity

Mo

dif

ied

du

rati

on

Coupon =0%

Coupon =2%

Coupon =5%

Coupon =15%

37

Modified principal

To simplify the step from principal to duration consider only two cases e.g., < o > 3%) Divide principal values in few large maturity

buckets Assign an average duration to each maturity

(“modified principal”)

Residual Life Bracket (i) Coupon < 3% Coupon 3%

Average modified

duration (MDi) Up to 1 month Up to 1 month 0.00 1 - 3 months 1 - 3 months 0.20 3 - 6 months 3 - 6 months 0.40 6 - 12 months 6 -12 months 0.70

38

A more refined technique: clumping

The objective is the same: link real cash flows to a number q (<p) of “nodes”

What changes? Rather than compacting flows into a single one at a unique date, each cash flow gets divided into more nodes

How to map cash flows? Building a new security,

identical to the real cash flow in terms of market value and riskiness

1

1,25 1,75 2,250,75 2,75

2,50,5

dates

nodes

1

1,25 1,75 2,250,75 2,75

2,50,5

dates

nodes

Clumping:

39

Clumping In the clumping model a large number of cash flows,

maturing in p different dates are reduced to q (with q<p) virtual cash flows on q different dates called “nodes” on the curve.

In order to choose the number and the position of the nodes we have to remember that: Changes in short term interest rate are more frequent and

larger than changes in long term interest rates. The relationship between volatility and maturity of interest

rates is negative. Usually cash flows with short maturities are more frequent

that cash flows with long maturities

It’s better to have a larger number of nodes on the short term part of the zero coupon curve

40

The nodes

The choice of the node is also influenced by the availability of hedging instruments: FRA, futures, swaps, etc.

When we divide a real cash flow with maturity in date t into two virtual cash flows with maturities on the nodes n and n+1 (with n<t<n+1), we must have:

The same market value

The same modified duration

41

Mapping in practice

We have two unknowns and two equations

t

nn

t

nn

nn

nn

nn

nnt

nn

nn

n

nnnt

t

tt

MV

MVMD

MV

MVMD

MVMV

MVMD

MVMV

MVMDMD

r

MV

r

MVMVMV

r

NVMV

11

1

11

1

11

11

111

11

1

1111

1

1

11

11

nn

nn

tnt

nnnn

nn

nn

ntt

nnnn

rMDMD

MDMDMVrMVNV

rMDMD

MDMDMVrMVNV

42

An example

A cash flow with a nominal value of 50,000 € and maturity 3y and 3m.

Zero-coupon IR: 3.55%

%55.31

25.0%)5.3%7.3(%5.3

)34(

)325.3()( 34325,3

rrrr

Maturity Zero-Coupon Rate

1 month 2.80%

2 months 2.85%

3 months 2.90%

6 months 3.00%

9 months 3.10%

12 months 3.15%

18 months 3.25%

2 years 3.35%

3 years 3.50%

4 years 3.70%

5 years 3.80%

7 years 3.90%

10 years 4.00%

15 years 4.10%

30 years 4.25%

43

follows

139.30355.1

25.3

1

82.640,440355.1

000,50

1 25.3

t

tt

tt

tt

r

DMD

r

NVMV

857.3

037.1

4

1

899.2035.1

3

1

1

11

n

nn

n

nn

r

DMD

r

DMD

Market Value and Modified Duration for the real cash flows

Modified Duration for the two virtual cash flows

44

follows

37.176,11857.3899.2

139.3899.282.640,44

45.464,33857.3899.2

857.3139.382.640,44

1n

n

MV

MV

Market value for the two virtual cash flows

56.924,121

63.102,3711

111n

nnn

nnnn

rMVNV

rMVNV

Nominal value for the two virtual cash flows

45

follows

The sum of the market values of the two virtual flows is equal to the market value of the real cash flow.

The market value of the 3Y cash flow is greater than the MV of the 4Y cash flow. This happens because the real flow maturity is nearer to 3 than to 4

T NV MV r D MD

Real Cash Flow 3.25 50,000.00 44,640.82 3.55% 3.25 3.139

3Y Virtual Cash Flow 3.00 37,102.63 33,464.45 3.50% 3 2.899

4Y Virtual Cash Flow 4.00 12,924.56 11,176.37 3.70% 4 3.857

46

Clumping on the basis of price volatility

Another form of clumping centers on the equivalence between price volatility of the initial flow and the total price volatility of the two new virtual positions

This is calculated by taking into account also the correlations between volatilities associated with price changes for different maturities. VMt e VMt+1 are chosen in such a way that:

Since this is a quadratic equation, we get two solutions for we need to assume that the original position and the two new virtual positions have the same sign 10

21,

121

2

12

2

2 2

tt

s

t

s

tt

s

tt

s

ts VM

VM

VM

VM

VM

VM

VM

VM

47

Clumping

After the mapping of all the bank positions on the nodes it’s possible to:

Evaluate the effect on the market value of the shareholders’ equity of a change of the interest rates for certain maturities

Implement interest risk management activities

Implement hedging activities

48

Residual Problems

Assumption of a uniform change of assets and liabilities’ interest rates.

The model does not consider the effect of a variation of interest rates on the volume of financial assets and liabilities

49

The Basel Committee Approach

Banks are required to allocate their assets and liabilities to 14 maturity buckets based on their residual maturity

For each bucket, they estimate the difference between assets and liabilities (long and short positions, i.e. net position)

The net position is weighted by a coefficient that proxies the potential change in value The product between the average modified

duration and a 2% change in the interest rate (parallel shift of the yield curve)

50

The Basel Committee Approach

Time Band Average maturity

(Di)

Band

Revocable or sight 0 1Up to 1 month 0.5 month 2from 1 to 3 months 2 months 3from3 to 6 months 4.5

months4

from 6 months to 1 year

9 months 5

from 1 year to 2 years 1.5 years 6from 2 to 3 years 2.5 years 7from 3 to 4 years 3.5 years 8from 4 to 5 years 4.5 years 9from 5 to 7 years 6 years 10from 7 to 10 years 8.5 years 11from 10 to 15 years 12.5 years 12from 15 to 20 years 17.5 years 13beyond 20 years 22.5 years 14

• Banks are required to allocate their assets and liabilities to 14 different maturity bands

• For each maturity bucket, the net position must be calculated (difference assets and liabilities)• Net position, NPi

51

The Basel Committee ApproachBand Modified

duration MDi = Di /(1+5%)

Weighting factorMDi yi (with yi=2%)

1 0 0.00 %2 0.04 years 0.08 %3 0.16 years 0.32 %4 0.36 years 0.72 %5 0.71 years 1.43 %6 1.38 years 2.77 %7 2.25 years 4.49 %8 3.07 years 6.14 %9 3.85 years 7.71 %10 5.08 years 10.15 %11 6.63 years 13.26 %12 8.92 years 17.84 %13 11.21

years22.43 %

14 13.01 years

26.03 %

The net position for each maturity bucket is weighted by a risk coefficient espressing the potential change in value Product between

average modified duration and y = 2%

Total risk is computed as the sum of all these NPi

iiii yMDNPNP

52

The Basel Committee Approach: pros

It’s an economic value approach It does not only measure the impact of interest rate

changes on the bank’s income, but also on its equity value

It considers the independence of interest rate curves for different currencies: The risk indicator has to be computed separately for

each currency abosrbing at least 5% of the bank’s balance sheet

It considers the link between risk and capital The sum of all the risk indicators (in absolute value)

related to the different currencies must be computed as a ratio to the bank’s regulatory capital

53

It considers a unique interest rate volatility for both short and long term rates, while the latter are empirically less volatile because of a mean reversion phenomenon

It allows a full netting among the positions of different time buckets, implicitly assuming parallel shifts of the curve

These two drawbacks are overcome by the generic risk indicator for debt securities in

the market risk capital requirement framework (trading portfolio)

The Basel Committee Approach: cons

54

The Basel Committee Approach: cons

It’s an economic value approach, but it uses as inputs the book values of assets and liabilities

It treats rather imprecisely Amortizing items Items with an uncertain rate repricing

date Customer assets & liabilities with no

precise maturity (e.g. demand deposits)

55

1. A bank holds a zero-coupon T-Bill with a time to matuity of 22 months and a face value of one million euros. The bank wants to map this position to two given nodes in its zero-rate curve, with a maturity of 18 and 24 months, respectively. The zero coupon returns associated with those two maturities are 4.2% and 4.5%. Find the face values of the two virtual cash flows associated with the two nodes, based on a clumping technique that leaves both the market value and the modified duration of the portfolio unchanged.


56

2. Cash flow bucketing (clumping) for a bond involves …

A) …each individual bond cash flow gets transformed into an equivalent cash flow with a maturity equal to that of one of the knots;

B) … the different bond cash flows get converted into one unique cash flow;

C) … only those cash flows with maturities equal to the ones of the curve knots are kept while the ones with different maturity get eliminated through compensation (“cash-flow netting”);

D) …each individual bond cash flow gets transformed into one or more equivalent cash flows which are associated to one or more knots of the term structure.


57

3. Bank X adopts a zero-coupon rate curve (term structure) with nodes at one month, three months, six months, one year, two years. The bank hold a security cashing a coupon of 6 million euros in eight months and another payment (coupon plus principal) of 106 million euros in one year and eight months. Using a clumping technique based on the correspondence between present values and modified durations, and assuming that the present term structure is flat at 5% for all maturities between one month and two years, indicate what flows the bank must assign to the three-month, six-month, one-year and two-year nodes.


58

4. Based on the following market prices and using the bootstrapping method, compute the yearly-compounded zero-coupon rate for a maturity of 2.5 years


Security Maturity Price

6-month T-bill, zero coupon 0.5 98 12-month T-bill, zero coupon 1 96 18-month T-bill, zero coupon 1.5 94 24-month T-bill, zero coupon 2 92 30-month T-bond with a 2% coupon every 6 months 2.5 99

the duration gap model and clumping session 2 andrea sironi mafinrisk – 2010 market risk

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