Transcript Duration Gap & Clumping
Mafinrisk – 2010 Market Risk
The duration gap model and clumping
Session 2 Andrea Sironi
Agenda
Market value versus historical cost accounting
The duration gap model
The Clumping Model
2
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 3
Market Value vs Historical Value
ASSETS
Fixed rate (5%) 10 Y Mortgages
Dec. 31, 2008 € m LIABILITIES
100 Fixed Rate (3%) 2 y Notes Shareholders’ Equity
Total 100 Total € m
90 10
100
NII
2009
II
2009
IE
2009 5 % 100 3 % 90 5 2 .
7 2 .
3
ASSETS
Cash Fixed rate (5%) 10 Y Mortgages
Total Dec. 31, 2009 € m LIABILITIES
2.3
100 Fixed Rate (3%) 2 y Notes Shareholders’ Equity
102.3 Total € m
90 12.3
102.3
4
follows
ASSETS
Cash Fixed rate (5%) 10 Y Mortgages
Total Dec. 31, 2010 € m LIABILITIES
4.6
100 Fixed Rate (3%) 2 y Notes Shareholders’ Equity
104.6 Total
On the 1/1/2009 the ECB increase the interest rates of 100 bp
€ m
90 14.6
104.6
Nothing changes in the FS of the bank
ROE
2009 2 .
3 23 % 10
ROE
2010 2 .
3 12 .
3 18 .
7 % 5
follows
In 2011 the bank has to finance the 10Y Mortgages with a new fixed rate note issued at the new market rate: 4%
NII
2011
II
2011
IE
2011 5 % 100 4 % 90 5 3 .
6 1 .
4
ROE
2011 1 .
4 14 .
6 9 .
59 % The effect of the increase of the interest rates on the profitability of the bank appears only two years after the variation itself. 6
follows
This problem can be solved using in the FS the market value of A/L instead of the historical value
MV Mortgage
t
9 1 1 5 6 %
t
1 100 6 % 9 93 .
2
MV Note
1 92 .
7 4 % 89 .
13
ASSETS
Cash Fixed rate (5%) 10 Y Mortgages
Total Dec. 31, 2009 € m LIBILITIES
2.3
93.2
Fixed Rate (3%) 2 y Notes Shareholders’ Equity
95.5 Total € m
89.13
6.37
95.5
P
/
L
MV A
MV L
MV SE P
/
L
2009 93 .
2 5 100 89 .
13 2 .
7 90 1 .
8 1 .
83 3 .
7 63
follows
Next Year (2010)
MV Mortgage
t
8 1 1 5 6 %
t
1 100 6 % 8 93 .
79 Notes (maturity) 90
ASSETS
Cash Fixed rate (5%) 10 Y Mortgages
Total Dec. 31, 2010 € m LIBILITIES
4,6 93,79 Fixed Rate (3%) 2 y Notes Shareholders’ Equity
98,39 Total € m
90,00 8,39
98,39
P
/
L
2010 5 93 , 79 93 .
2 90 2 .
7 89 .
13 5 .
59 3 .
57 2 .
02 8
Agenda
Market value versus historical cost accounting
The duration gap model
The Clumping Model
9
The Duration gap
The same result could be obtained using the duration gap
MVA MVA
1
D
A i A
i A
MVL MVL
1
D L
i L
i L
MVA
MVA
1
D A
i A
i A
MVA
MD A
i A
MVL
MVL
1
D L
i L
i L
MVL
MV L
i L
MVE
MVA
MVL
MVA
MD A
i A
MVL
MD L
i L
MVE
MVA
MD A
MVL
MD L
i
MVE
MD A
LEV
MD L
i MVA
MVE
MD A
LEV
MD L
MVA
i
10
The Duration gap
MVE
MD A
LEV
MD L
MVA
i
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
MVE
DG
MVA
i
11
•
Immunization
If MV A = MV L changes if MD A MVE is not sensitive to interest rates = MD L .
• If MV A > MV L MVE is not sensitive to interest rates changes if DG=0, i.e. MD A < MD L . 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.
MVE MVE
MD A
LEV
MD P
MVA
i MVE
12
The example again
D Mortgage
Let’s go back to our bank
t
10 1
t
CF t i t MV
t
9 1
t
5 1 .
05
t
100 10 105 1 .
05 10 100 8 .
108
MD Mortgage
D Mortgage
8 .
108 1 .
05 7 .
722
D Note
t t
1
t
CF t
t MV
1 2 .
7 1 .
90 2 1 92 .
.
03 7 2 90 1 .
971
MD Note
D Note
1 .
971 1 .
03 1 .
914 13
follows
DG
MD A
LEV
MD L
7 .
722 0 .
90 1 .
914 6
MVE
DG
MVA
i
6 100 1 % 6 For an interest rates increase of 100 bp the market value of shareholders’ equity would decrease by 6 m€ (60% of the original value) 14
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 15
Duration gap: problems and limits
1.
2.
3.
D uration (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 D uration (and duration gap linear approximation ) is based on a Impact not estimated precisely The model assumes uniform interest rate changes ( rates i ) of assets and liabilities interest 16
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
t
1
t
2
t
3
Coupon payments
time 17
Answer to problem 2: convexity
Rather than proxying % change in value with the first derivative only
MV A MV A
dMV A di MV A
i
…we could add the second term in Taylor (or McLaurin) including second derivative:
MV A MV A
dMV A di MV A
i
d
2
MV A di
2
MV A
(
i
) 2 2
See following slides
18
Answer to problem 2: convexity
Second derivative of
VM
A to
i d
2
MV A di
2
d di t N
1
t
CF t
t
1
t N
1
t
(
t
1 )
CF t
t
2 1 1
i
2
t N
1 (
t
2
t
)
CF t
t
Dividing both terms by
MV
A :
d
2
MV A di
2
MV A
1 1
i
2
t N
1 (
t
2
t
)
CF t
MV A t
Modified convexity MC
Convexity, C
19
Answer to problem 2: duration gap and convexity gap
Substituting duration and convexity in the second order expansion
MV MV A A
MD A
i
MC A
( 2
i
) 2 Multiplying both terms by
MV
A :
MV A
MD A
i
MV A
MC A
(
i
) 2 2
MV A
Same for liabilities:
MV L
MD L
i
MV L
MC L
(
i
) 2
MV L
2 The change in market value of the bank’s equity can now be better estimated:
MV E
MD A
L
MD L
i
MV A
MC A
L
CM L
(
i
) 2 2
MV A
duration gap convexity gap
20
Duration gap and convexity gap: our example
MC A
MC mortgage
1 1 5 % 2
t
8 1 (
t
2
t
) 5 1 5 %
t
( 81 9 ) 105 100 1 5 % 9 100 69 .
79
CM L
CM CD
1 1 3 % 2 92 .
7 1 3 % 0 .
97 90
VM B
DG
1 % 100 69 .
79 0 .
9 0 .
97 ( 1 %) 2 2 100 5 .
93
First order proxy, -6.23
convexity gap, equal to 61.6
Very close to the true change (-5.94)
21
Answer to problem 3: beta-duration gap
Similar to standardized repricing gap. For each asset (liability) estimate:
i A
A
i
i P
P
i
Then substitute in the change of the value of the bank
MV B
MV MD
MV A A A
MV
MD MD
A A
A A
L
i
MV L A A
MD L
MV L
MV L L
L
MD L MD L
L
i L
i
MV A
i
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 ( MV A ) 22
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 23
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 24
Questions & Exercises
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).
25
Questions & Exercises
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 effect C. 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.
26
Questions & Exercises
4. Using the data in the table below i) 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 results
Assets
Open credit lines Floating rate securities Fixed rate loans Fixed rate mortgages
Liabilities
Checking accounts Fixed rate CDs Fixed rate bonds
Value
1000 600 800 1200
Value
1200 600 1000
Modified duration
0 0.25 3.00 8.50
Modified duration
0 0.5 3
Modified convexity
0 0.1 8.50 45
Modified convexity
0 0.3 6.7 27
Agenda
Market value versus historical cost accounting
The duration gap model
The Clumping Model
28
A common problem and a possible solution
Repricing gap and duration gap assumption uniform change of interest rates for different maturities of The Clumping o cash-bucketing model different maturities a model with independent changes of interest rates at The model is built upon the repricing gap and the duration gap model were focused on the yield curve).
zero-coupon curve (both the The model works trough the mapping of single cash flows on a predetermined number of nodes (or maturities) on the term structure.
29
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 ( coupon rate coupon paying bond with a price of 100.
r 2,5 ) zero For this maturity we only have a 4.5% (semi-annual) 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 ( r t )) 30
How to estimate zero coupon rates: bootstrapping
1. From prices of
zcb
we extract the corresponding
r
t
Zero Coupon Bond
6 months 1 year 18 months 24 months
Maturity
0.5
1 1.5
2
Price
98 96 94 92
Rate
4.12% 4.17% 4.21% 4.26% Ex.
r t
t
100
VM Zt
1
r
2 2 100 1 4 .
26 % 92 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
102.25
0
2.25
0.5
2.21
2.25
1
2.16
8.55
2.25
1,5
2.12
2.25
2
2.07
2.5
Es.
( 1 2 .
25 4 .
26 31 %) 2
How to estimate zero coupon rates: bootstrapping
3. Find the rate that equates the present value of 102.5 to the residual value of the bond which has not been explained by the PV of the four coupons 0
2.25
0.5
2.21
2.25
1
2.16
8.55
2.25
1.5
2.12
100
r
2 .
5
r
102 .
25 ( 1
r
) 2 .
5 91 .
45
102.25
2.25
2
2.07
2.5
= 91.45
r
2 .
5 2 .
5 102 .
25 91 .
45 1 4 .
57 % 32
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 dates, into a simplified portfolio, based on a limited number q (< p 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).
33
Cash-flow mapping
We can get an interest rate curve with different rates for every individual maturity Do I really need to consider M x N 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 34
Some simplifying cash-flow mapping techniques
Analytical principal Given M securities, “maps” each of them to the “principal” maturity node
Modified analytical principal
Analytical duration Given M securities, it maps each of them to its duration
method
Synthetic principal Given M securities, it only considers the maturity of principal (computes an average) Does not consider coupons reinvestment risk Synthetic duration Given M securities, it only considers the duration (computes an average) 35
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 10 7.5
5 2.5
0 0 2.5
5
Time to maturity
Coupon =0% Coupon =2% Coupon =5% Coupon =15% 7.5
10 36
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%
Up to 1 month 1 - 3 months 3 - 6 months 6 - 12 months Up to 1 month 1 - 3 months 3 - 6 months 6 -12 months
Average modified duration (MD
i
)
0.00 0.20 0.40 0.70 37
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
0,5 0,5 0,75 1,25 1,75 2,25 Clumping: 1 0,75 1,25 1,75 2,25 1 2,5 2,75
dates nodes
2,75
dates
2,5
38
nodes
Clumping
In the clumping model a large number of cash flows, maturing in curve.
p different dates are reduced to virtual cash flows on q q (with q
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 39
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 the nodes have: into two virtual cash flows with maturities on n and n+1 (with n < t < n+1 ), we must The same market value The same modified duration 40
Mapping in practice
We have two unknowns and two equations
MV t
1
NV t
r t
t
MV n
MV n
1 1
MV n
r n
1
MV
r n n
1 1
n
1
MD t
MD n MV n MV n
MV n
1
MD n
1
MV n MV n
1
MV n
1
MD n MV n MV t
MD n
1
MV n
1
MV t
NV n NV n
1
MV n MV n
1 1 1
r n
n
r n
1
MV t
n
1
MD t MD
MV t n
MD n
1
MD n
1 1
r n
n
MD n MD n
MD t MD n
1 1
r n
1
n
1 41
An example
A cash flow with a nominal value of 50,000 € and maturity 3y and 3m.
Zero-coupon IR : 3.55%
Maturity
1 month 2 months 3 months 6 months 9 months 12 months 18 months 2 years
3 years Zero-Coupon Rate
2.80% 2.85% 2.90% 3.00% 3.10% 3.15% 3.25% 3.35%
3.50% 4 years
5 years 7 years
3.70%
3.80% 3.90% 4.00% 10 years 15 years 4.10% 30 years 4.25%
r
3 , 25
r
3 (
r
4
r
3 ) ( 3 .
25 3 ) ( 4 3 ) 3 .
5 % ( 3 .
7 % 3 .
5 %) 0 .
25 1 3 .
55 % 42
follows
Market Value and Modified Duration for the real cash flows
MV t
1
NV t
r t t
MD t
1
D t
r t
50 , 000 1 .
0355 3 .
25 1 .
0355 3 .
25 44 , 640 .
82 3 .
139 Modified Duration for the two virtual cash flows
MD n
1
D n
r n
MD n
1 1
D n
r
1
n
1 3 1 .
035 2 .
899 4 1 .
037 3 .
857 43
follows
Market value for the two virtual cash flows
MV n
MV n
1 44 , 640 .
82 44 , 640 .
82 3 .
139 2 .
899 2 .
899 2 .
899 3 .
857 3 .
857 3 .
139 3 .
857 33 , 464 .
45 11 , 176 .
37 Nominal value for the two virtual cash flows
NV n NV n
1
MV n
1
MV n
1 1
r n
n
r n
1 37 , 102 .
63
n
1 12 , 924 .
56 44
follows
Real Cash Flow 3Y Virtual Cash Flow 4Y Virtual Cash Flow
T 3.25
3.00
4.00
NV 50,000.00
37,102.63
12,924.56
MV
44,640.82
33,464.45
11,176.37
r
3.55%
3.50% 3.70% D
3.25
3 4
MD
3.139
2.899
3.857
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 45
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. VM t e VM t+1 are chosen in such a way that: 2
s
2
VM VM t s
2
t
2
VM t
1
s
t
2 1 2
VM VM t s VM VM t
1
s
t
2 ,
t
1 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 0 1 46
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 47
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 48
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) 49
The Basel Committee Approach
Time Band Revocable or sight Up to 1 month from 1 to 3 months from3 to 6 months from 6 months to 1 year from 1 year to 2 years from 2 to 3 years from 3 to 4 years from 4 to 5 years from 5 to 7 years from 7 to 10 years from 10 to 15 years from 15 to 20 years beyond 20 years Average maturity (D i ) 0 0.5 month 2 months 4.5 months 9 months 1.5 years 2.5 years 3.5 years 4.5 years 6 years 8.5 years 12.5 years 17.5 years 22.5 years Band 9 10 11 12 1 2 3 4 5 6 7 8 13 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, NP i 50
1 2 3 4 5 6 7 8 9 10 11 12 13 14
The Basel Committee Approach
Band Modified duration MD i = D i /(1+5%) 0 0.04 years 0.16 years 0.36 years 0.71 years 1.38 years 2.25 years 3.07 years 3.85 years 5.08 years 6.63 years 8.92 years 11.21 years 13.01 years Weighting factor MD i y i (with y i =2%) 0.00 % 0.08 % 0.32 % 0.72 % 1.43 % 2.77 % 4.49 % 6.14 % 7.71 % 10.15 % 13.26 % 17.84 % 22.43 % 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%
NP i
NP i
MD i
y i
Total risk is computed as the sum of all these NP i 51
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 52
The Basel Committee Approach: cons
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
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) 54
Questions & Exercises
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.
55
Questions & Exercises
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. 56
Questions & Exercises
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. 57
Questions & Exercises
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 6-month T-bill, zero coupon 12-month T-bill, zero coupon 18-month T-bill, zero coupon 24-month T-bill, zero coupon 30-month T-bond with a 2% coupon every 6 months Maturity 0.5 1 1.5 2 2.5 Price 98 96 94 92 99 58