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.
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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
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.
This problem can be solved using in the FS the market value of A/L instead of the historical value
Dec. 31, 2009
The same result could be obtained using the duration gap
Let’s go back to our bank
For an interest rates increase of 100 bp the market value of shareholders’ equity would decrease by 6 m€ (60% of the original value)
The result (6) is different form what we got before (3.63) for three main reasons:
Even if rates do not change, duration decreases: linearly with “jumps” related to coupon payments
Problem 1:duration changesDuration
Coupon payments
t1
t2
t3
time
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
Second derivative of VMA to i
Modified convexity MC
Dividing both terms by MVA :
Convexity, C
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
First order proxy, 6.23
convexity gap,equal to 61.6
Very close to the true change
(5.94)
Similar to standardized repricing gap. For each asset (liability) estimate:
Then substitute in the change of the value of the bank
betaduration gap
The impact of an interest rate change depends on 4 factors:
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
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).
3. Which of the following statements is NOT correct?
A. The convexity gap makes it possible to improve the precision of an interestrate risk measure based on duration gap
B. The convexity gap is a secondorder 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.
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
1. From prices of zcb we extract the corresponding rt
Ex.
2. We use these zerocoupon 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
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
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 cashflow mapping techniquesModified
analytical
principal method
Requires M nodes
Extremely simplified
Does not consider coupons reinvestment risk
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 principalconsider 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 principalWhat 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: clumpingdates
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
Market Value and Modified Duration for the real cash flows
Modified Duration for the two virtual cash flows
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
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)
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)
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.
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 (“cashflow 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.
3. Bank X adopts a zerocoupon 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 threemonth, sixmonth, oneyear and twoyear nodes.
4. Based on the following market prices and using the bootstrapping method, compute the yearlycompounded zerocoupon rate for a maturity of 2.5 years