the duration gap model and clumping l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
The duration gap model and clumping PowerPoint Presentation
Download Presentation
The duration gap model and clumping

Loading in 2 Seconds...

play fullscreen
1 / 58

The duration gap model and clumping - PowerPoint PPT Presentation


  • 445 Views
  • Uploaded on

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. The Duration Gap Model.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'The duration gap model and clumping' - lovette


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
agenda
Agenda
  • Market value versus historical cost accounting
  • The duration gap model
  • The Clumping Model
the duration gap model
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
market value vs historical value
Market Value vs Historical Value

Dec. 31, 2008

Dec. 31, 2009

follows
follows

Dec. 31, 2010

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

Nothing changes in the FS of the bank

follows6
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.

follows7
follows

This problem can be solved using in the FS the market value of A/L instead of the historical value

Dec. 31, 2009

follows8
follows

Next Year (2010)

Notes (maturity)  90

Dec. 31, 2010

agenda9
Agenda
  • Market value versus historical cost accounting
  • The duration gap model
  • The Clumping Model
the duration gap
The Duration gap

The same result could be obtained using the duration gap

the duration gap11
The Duration gap
  • The change in the market value of Shareholders’ Equity is a function of three variables:
  • 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)
  • The size of the intermediation activity of the bank measured by the market value of total assets
  • The size of the interest rates change
immunization
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.
the example again
The example again

Let’s go back to our bank

follows14
follows

For an interest rates increase of 100 bp the market value of shareholders’ equity would decrease by 6 m€ (60% of the original value)

some remarks
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
duration gap problems and limits
Duration gap: problems and limits
  • 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
  • Duration (and duration gap) is based on a linear approximation
    • Impact not estimated precisely
  • The model assumes uniform interest rate changes (Di) of assets and liabilities interest rates
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

Problem 1:duration changes

Duration

Coupon payments

t1

t2

t3

time

answer to problem 2 convexity
Answer to problem 2:convexity

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

answer to problem 2 convexity19
Answer to problem 2:convexity

Second derivative of VMA to i

Modified convexity MC

Dividing both terms by MVA :

Convexity, C

answer to problem 2 duration gap and convexity gap
Answer to problem 2:duration gap and convexity gap

Substituting duration and convexity in the second order expansion

Multiplying both terms by MVA:

Same for liabilities:

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

duration gap

convexity gap

duration gap and convexity gap our example
Duration gap and convexity gap: our example

First order proxy, -6.23

convexity gap,equal to 61.6

Very close to the true change

(-5.94)

answer to problem 3 beta duration gap
Answer to problem 3:beta-duration gap

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

Then substitute in the change of the value of the bank

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 leverageL
  • size of the bank (MVA)
residual problems
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
questions exercises
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

questions exercises25
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).

questions exercises26
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.

questions exercises27
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

agenda28
Agenda
  • Market value versus historical cost accounting
  • The duration gap model
  • The Clumping Model
a common problem and a possible solution
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.
how to estimate zero coupon rates bootstrapping
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))
how to estimate zero coupon rates bootstrapping31
How to estimate zero coupon rates: bootstrapping

1. From prices of zcb we extract the corresponding rt

Ex.

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

8.55

102.25

2.25

2.25

2.25

2.25

0

0.5

1

1,5

2

2.5

Es.

2.21

2.16

2.12

2.07

how to estimate zero coupon rates bootstrapping32
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

8.55

102.25

2.25

2.25

2.25

2.25

0

0.5

1

1.5

2

2.5

2.21

2.16

2.12

2.07

100

-

= 91.45

what is the mapping for
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).
cash flow mapping
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
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)

Some simplifying cash-flow mapping techniques

Modified

analytical

principal method

Requires M nodes

Extremely simplified

Does not consider coupons reinvestment risk

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

An hybrid technique: modified principal
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”)

Modified principal
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

A more refined technique: clumping

dates

0,75

1,25

1,75

2,25

2,75

nodes

0,5

1

2,5

Clumping:

dates

0,75

1,25

1,75

2,25

2,75

nodes

0,5

1

2,5

clumping
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
the nodes
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
mapping in practice
Mapping in practice
  • We have two unknowns and two equations
an example
An example
  • A cash flow with a nominal value of 50,000 € and maturity 3y and 3m.
  • Zero-coupon IR: 3.55%
follows43
follows

Market Value and Modified Duration for the real cash flows

Modified Duration for the two virtual cash flows

follows44
follows

Market value for the two virtual cash flows

Nominal value for the two virtual cash flows

follows45
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
clumping on the basis of price volatility
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 
clumping47
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
residual problems48
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
the basel committee approach
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)
the basel committee approach50
The Basel Committee Approach
  • 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
the basel committee approach51
The Basel Committee Approach
  • 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 Dy = 2%
  • Total risk is computed as the sum of all these DNPi
the basel committee approach pros
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

slide53
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

The Basel Committee Approach: cons

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
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)

questions exercises55
Questions & Exercises
  • 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.

questions exercises56
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.

questions exercises57
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.

questions exercises58
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