Transcript Topic 1. Introduction to financial derivatives

Topic 2. Measuring Interest Rate Risk
2.1 Repricing model
2.2 Duration
2.3 Convexity
1
2.1 Repricing model
Repricing model is based on the book (historic) value of
assets and liabilities in the FI’s balance sheet.
 Under the repricing model, FI is required to divide its
asset and liability portfolio into different maturity
intervals (maturity bucket).
 Rate sensitive assets of the ith maturity bucket (RSAi) is
defined as the total book value of all FI’s assets which the
first repricing at current market interest rates is within ith
maturity bucket.
 Rate sensitive liabilities of the ith maturity bucket (RSLi)
is defined in the same way as RSAi for the FI’s liabilities.

2
2.1 Repricing model


The repricing of an asset or a liability in ith maturity bucket
can be the result of a rollover of an asset or a liability in that
bucket or the corresponding asset or liability is variable-rate
instrument in which the rate is reset within that bucket.
Change in net interest income in the ith bucket NIIi is
defined as
NIIi  RSAi  RSLi Ri
 GAPi Ri
(2.1)
where
Ri is the change in the level of interest rates impacting
assets and liabilities in the ith bucket.
GAPi = RSAi – RSLi
3
2.1 Repricing model

Under Eq. (2.1), we assume the same interest rate Ri is
applied to all assets and liabilities in the ith bucket. In
reality, different interest rates should be used for assets
and liabilities with different maturities within the bucket.
4
2.1 Repricing model

Examples of RSAi and RSLi:
Let the i-th maturity bucket be (3 months, 12 months].
•
•
•
•
•
Fixed rate 1-month loan is NOT RSAi or RSLi since the first
repricing is not within i-th maturity bucket.
Fixed rate 9-month loan is RSAi or RSLi since the first repricing
is within i-th maturity bucket.
Fixed rate 2-year loan is NOT RSAi or RSLi since the first
repricing is not within i-th maturity bucket.
Floating rate 10-year loan (interest rate reset every 1 month) is
NOT RSAi or RSLi since the first repricing is not within i-th
maturity bucket.
Floating rate 30-year loan (interest rate reset every 6 months) is
in RSAi or RSLi since the first repricing is within i-th maturity 5
bucket.
2.1 Repricing model
Table 2-1
ith
Maturity bucket
Assets
(million)
Liabilities
(million)
Gaps
(million)
1
1 day
$ 20
$30
$ -10
2
>1 day to 3 mths
30
40
-10
3
> 3 mths to 6 mths
70
85
-15
4
> 6 mths to 12 mths
90
70
+20
5
> 1 yr to 5 yrs
40
30
+10
6
> 5 yrs
10
5
+5
260
260
Total
6
2.1 Repricing model

In first bucket (1-day),
GAP1 = $10 million
If the 1-day interest rates (fed or overnight repo rate) rise
1% per annum, i.e. R1 = 0.01, then from Eq. (2.1)
NII1   $10 million 0.01  $100,000(annualized)
That is, the negative gap (GAP1 < 0) resulted in a loss
(gain) of $100,000 in net interest income in the first
bucket for the FI when the 1-day interest rate increases
(decreases) 1%.
7
2.1 Repricing model

Cumulative 1-year gap (CGAP1-year)
CGAP1year  RSA1year  RSL1year
(2.2)
where RSA1-year and RSL1-year are the total value of assets
and liabilities in which they are repriced within 1 year.
Taking [0, 1 year] as the first maturity bucket and from
Eq. (2.1), we have
NII1-year  CGAP1-year R1year
(2.3)
where R1-year is the average interest rate change affecting
assets and liabilities that can be repriced within a year.
8
2.1 Repricing model
From Table 2-1 with R1-year = 0.01, we have
NII1- year   $10M  $10M  $15M  $20M(0.01)
 $150,000
For R1-year =1%, the 1-year net interest income drops by
$150,000.
9
2.1 Repricing model

CGAPt
gap ratio (t ) 
A
(2.4)
where A is the total asset value.
The gap ratio in (2.4) is useful to tell us that


the direction of interest rate risk exposure (+ve or –ve CGAPt)
the scale of that exposure relative to the total asset size of the
financial institution.
10
2.1 Repricing model

Example 2.1
Assets
$ (M)
Liabilities
$ (M)
1. Short-term consumer loans
(1-year maturity)
50
1. Equity capital (fixed)
60
2. Long-term consumer loans
(2-year maturity)
25
2. Three-month bankers
acceptances
20
3. Three-month Treasury bills
30
3. Passport savings
30
4. Six-month Treasury notes
35
4. Three-month CDs
40
5. Three-year Treasury bonds
70
6. Six-month commercial paper
60
6. 10-year, fixed-rate mortgages
20
7. One-year time deposits
20
7. 30-year, floating rate mortgages
(rate adjusted every 9 months).
40
8. Two-year time deposits
40
270
270
11
2.1 Repricing model
Determine CGAP1-year.
RSA1-year consists:
• Short-term consumer loans: repriced at the end of one year.
• Three-month T-bills: repriced on maturity (rollover) every 3
months.
• Six-month T-notes: repriced on maturity (rollover) every 6
months.
• 30-year floating rate mortgages: repriced (mortgage rate is reset)
every 9 months.
So, RSA1-year = $155M.
12
2.1 Repricing model
RSL1-year consists:
• Three-month CDs: repriced on maturity (rollover) every 3 months.
• Three-month bankers acceptances: repriced on maturity (rollover) every
3 months.
• Six-month commercial paper: repriced on maturity every 6 months.
• One-year time deposits: repriced at the end of the year.
So, RSL1-year = $140M.
$15M
 5.6%
CGAP1-year=$15M and gap ratio(1 - year)
$270M
From the calculated gap ratio, RSA1-year is more than RSL1-year by
5.6% of the total asset value.
13
2.1 Repricing model

Under equal changes in interest rates on RSA and RSL,
CGAP effects refer to the relation between changes in
interest rates (R) and changes in net interest income
(NII).
14
2.1 Repricing model
CGAP > 0  ΔR and ΔNII are positively correlated.
CGAP < 0  ΔR and ΔNII are negatively correlated.
15
2.1 Repricing model

Under unequal changes in interest rates on RSA and RSL,
spread RA  RL
spread RA  RL
(2.5)
where RA and RL are the interest rates applying on asset
and liability respectively.
NII  RSA RA  RSL RL
 CGAP RA  RSL spread
(2.6)
Spread effect: For all else being fixed, there is a positive
relation between spread and NII.
16
2.1 Repricing model
From Eq. (2.6), when both CGAP effect and spread effect
play their role to ΔNII, we observe the following
17
2.1 Repricing model
Weakness of repricing model
 Ignores market value effect
The repricing model does not consider the change of the
present value of cash flows on asset and liability as
interest rates change. (rates-asset has corr, duration profile)
 Overaggregative
The actual maturities distributions of assets and liabilities
within individual maturity bucket is not considered.
Mismatching of maturity could cause refinancing or
reinvestment risk. (maturies not enough granularity)
18
2.1 Repricing model
Although GAP3-mth to 6-mth = 0, the
assets’ maturity < liabilities’ maturity (reinvestment risk).
19
2.1 Repricing model
Ignores effects of runoffs
The periodic cash flow of interest and principal
amortization payments on long-term asset being classified
as rate insensitive and ignored under the repricing model.
Actually, these cash flows can be reinvested at market
interest rates.
 Ignores off-balance-sheet activities

20
2.2 Duration
Duration of a bond
 Given a N-year coupon bond with annual coupon rate of c
and principal (face) value of F. Suppose the coupon
frequency is m and the bond yield is R (per annum) with
the compounding frequency as the same as the coupon
frequency.
The price of the bond at the commencing date is given by
1
c  F  
Nm
F
m

P( R)  

i
i 1 1  R
1 R
m
m

 

Nm
(2.7)
21
2.2 Duration

The duration of a bond D is defined as the weighted
average of cash flow dates of a bond
Nm
D   wi ti
(2.8)
i 1
where
ti  i
m

1
c

F

 
 1
m



1  i  Nm  1
i
 P ( R ) 1  R
m
wi  
1

c  F    F
1

m

i  Nm
Nm
 P( R)
1 R

m




22
2.2 Duration

D is also called Macaulay duration.

Example 2.2
Consider a 2-year coupon bond with coupon rate of 4%
per annum and principal value of $100. The coupon
frequency is 2 and the bond yield is 4%.
m = 2; ti = i/2; R = 4%
Let CFi be the cash flow at ti.
Let DFi be the discount factor at ti.
1
DFi 
1  R mi
23
2.2 Duration
CFi
DFi
CFi DFi
CFi  DFi  ti
0.5
2
0.9804
1.96
0.98
1
2
0.9612
1.92
1.92
1.5
2
0.9423
1.88
2.83
102
0.9238
94.23
188.46
100
194.19
ti
2
D = 194.19/100 = 1.9419
24
2.2 Duration

On the other hand, it can be shown that (refer to the
lecture notes of SEEM 2520)
dP( R)
P( R)
 D
dR
1 R m
D
  MD  P( R) where MD 
1 R m
(2.9)
MD is called the modified duration.
25
2.2 Duration model
If the changes of R (R) is small, then Eq. (2.9) can be
approximately written as
P
  MD  P
R
P
  MD  R
P
(2.10)
Eq. (2.10), MD measures a bond’s price sensitivity
as a percentage change of its current price to the small
changes in its yield.
 From
26
2.2 Duration
Properties of duration
 Duration of a zero coupon bond is equal to its maturity.
  coupon
  maturity
  Duration (D)
  yield
27
2.2 Duration
Duration of a portfolio of bond
Consider a Portfolio, A, consists of N bonds.
Notation:
For i =1, …, N,
Pi
: Price of the bond i
Fi
: Face value of the bond i
mi
: Coupon frequency of bond i
Ri
: Bond yield of bond i
Di
: Duration of bond i
MDi : Modified duration of bond i
ni
: Number of units of bond i in Portfolio A
(ni > 0: Long position; ni < 0: short position)
28
2.2 Duration
The value of Portfolio A is given by
N
PA   ni Pi
i 1
For small Ri, i=1, …, N, we have
N
PA   ni Pi
i 1
N
  ni  MDi Pi  Ri 
(2.11)
i 1
29
2.2 Duration
If R1= R2=…= RN= R (parallel yield shift), then
Eq. (2.11) becomes
PA
 N

   xi MDi R
PA
 i 1

n i Pi
where xi 
PA
(2.12)
The modified duration of Portfolio A, MDA, is then given by
N
MDA   xi MDi
(2.13)
i 1
30
2.2 Duration
In particular, if R1=R2=…=RN (flat yield curve) and
m1=m2=…=mN , the duration of Portfolio A, DA, is given by
N
DA   xi Di
(2.14)
i 1
31
2.2 Duration
 From
Eq. (2.13) and Eq. (2.14), we observe that the
modified duration and also the duration of a portfolio are
the weighted average of the modified duration and
duration of its components respectively.
 Portfolio A is called duration neutral if MDA =0.
 The value of a duration neutral portfolio is insensitive to
the small change of the interest rates.
32
2.2 Duration
 Example
2.3
Suppose Portfolio A consists of 3 bonds: B1, B2 and B3.
Position
Price (per $100 face value)
MD (years)
B1 Long 200 units $90
1.4
B2 Short 140 units $105
2.5
B3 Long 130 units $103
3.6
Here, 1 unit of bond corresponds to $100 in bond’s face value.
PA  200 $90  140 $105 130 $103  $16,690
200 $90
140 $105
130 $103
MDA 
1.4 
 2.5 
 3.6
$16,690
$16,690
$16,690
33
 2.1962years
2.2 Duration
Duration model
Let MDA be the duration of asset portfolio of a FI which
consists of NA bonds.
Let MDL be the duration of liability portfolio of a FI which
consists of NL bonds.
Assume parallel yield shift.
From Eq. (2.13), we have
NA
MDA   x iA MDiA
i 1
NL
MDL   x iL MDiL
i 1
34
2.2 Duration
Let A and L be the value to asset and liability portfolio of the
FI respectively.
Assume R to be small and the same for both asset and
liability portfolio.
From Eq. (2.12) and Eq. (2.13), we have
A
  MDA R
A
L
  MDL R
L
35
2.2 Duration
Let E be the equity value of the FI.
E  A L
E  A  L
L
 MDA  k  MDL  A  R where k 
A
(2.15)
k is a measure of the FI’s financial leverage.
36
2.2 Duration
(MDA – kMDL) is the leverage adjusted modified duration
gap (modified duration gap for short) which reflect the
degree of duration mismatch in an FI’s balance sheet.
To immunize the equity or net worth (E) from interest rate
shocks, FI should set the modified duration gap to 0.
Immunization: The procedure to protect against the interest
rate risk.
37
2.2 Duration
If all the bonds in the asset and liability portfolio have the
same coupon frequency and equal to 1 and also the yield
curve is flat, then Eq. (2.15) becomes
R
E  DA  k  DL  A 
1 R
(2.16)
In Eq. (2.16), (DA – kDL) is called the leverage adjusted
duration gap (duration gap for short).
38
2.2 Duration
Example 2.4
The FI’s initial balance is assumed to be
Assets ($ millions)
Liabilities ($ millions)
A = 100*
L = 90*
E = 10
100
100
* bond’s price
A: 5-year zero coupon bond
L: 3-year zero coupon bond
Suppose R = 10%, R = 1%, DA = 5 years and DL = 3 years.
0.01
E  5  0.9  3 $100 million 
 $2.09 million
1.1
39
2.2 Duration
For 1% increase in interest rate, the equity value drops
$2.09 millions.
The insolvency occurs when E  $10 million. This
corresponds to R  4.783%.
40
2.2 Duration


Financial leverage (gearing) of a firm is the extent to
which debts are used to finance a firm’s assets. If a
high percentage of a firm’s assets are financed by
debts, then the firm is said to have a high degree of
financial leverage.
The higher the financial leverage of a firm, the higher
the chance for it to bankrupt.
41
2.2 Duration

The following are common ratios which are used to
measure the financial leverage:
D
1. Debt - to - asset ratio 
A
D
2. Debt - to - equit y ratio 
E
E
3. Capit alasset ratio 
A

Ratio (1) and (2) are increasing as the level of
leverage increases, while the ratio (3) is decreasing as
the level of leverage decreases.
42
2.2 Duration
Limitations of duration model
 It is costly and time consuming for the FI to restructure
the balance sheet to achieve the immunization against the
interest rate shock. The growth of asset securitization has
eased the speed and lowered the transaction costs of the
balance sheet restructuring.
 Immunization is a dynamic process since duration
depends on instantaneous R.
43
2.2 Duration
Duration model performs poorly under non-parallel shift
of yield curve. In reality, points in a yield curve do not
often shift by same amount and they sometimes do not
even move in same direction.
 Large interest rate change effects are not accurately
captured. This can be improved by introducing the
convexity.

44
2.2 Convexity
Convexity
 The duration measure is a linear approximation of a
nonlinear function. If there is a large changes in R, the
approximation is much less accurate.
45
2.2 Convexity
P(R)
Duration model
Error
R*  ΔR
R*
R* + ΔR
R (%)
46
2.2 Convexity

P(R+R) – P(R) can be better approximated by using
Taylor Expansion with 2nd order as follows
dP( R)
1 d 2 P( R)
2


P( R  R)  P( R) 
R 
R
2
dR
2 dR
P( R  R)  P( R)
R 1 1 d 2 P( R)
2


 D
 


R
P( R)
1  R 2 P( R) dR2
P( R)
1
2
  MD  R  CX R 
(2.18)
P( R)
2
47
2.2 Convexity

In Eq. (2.18), CX is defined as
1 d 2 P( R)
CX 
P( R) dR2

(2.19)
CX is called the convexity of a bond.
48
2.2 Convexity

Example 2.5
Consider a 6-year maturity coupon par bond with annual
coupon of 8% and principal value of $1,000. The coupon is
paid annually.
Since the bond is par, the annual yield is also 8%.
From Eq.(2.8), D = 4.9927.
From Eq. (2.19),
1 d 2 P( R)
CX 
P(8%) dR2 R 8%
 28.0484
49
2.2 Convexity
Suppose R = 2%. P(10%) = $912.8948 (Exact value)
Without convexity adjustment (Eq. (2.10)):
4.9927
P  
(0.02)1000
1  8%
 92.4574
EstimatedP(10%)  $907.5426
With convexity adjustment (Eq. (2.18)):
4.9927
P  
(0.02)1000
1  8%
 0.528.0484(0.02) 2 (1000)
 86.8477
EstimatedP(10%)  $913.1523
50
2.2 Convexity

Taking the convexity into account, Eq. (2.12) can be
modified as
PA
1 N
 N


2
   xi MDi R    xi CX i R 
PA
2  i 1
 i 1


(2.20)
where CXi is the convexity of bond i.
The convexity of Portfolio A, CXA, is given by
N
CX A   xi CX i
(2.21)
i 1
51
2.2 Convexity
Taylor Expansion
Let f(y1, y2, …, yn) be a function of n variables and has
continuous partial derivatives of all orders. Given two points,
namely a = (a1, a2, …, an) and b = (b1, b2, …, bn).
f(a) – f(b) can be approximated by Taylor expansion with mth
order as follows
1

f (a)  f (b)     ak  bk 
yk
i 1 i!  k 1
m
n
i

 f b 

(2.17)
52
2.2 Convexity
The term
 n

  ak  bk 
yk
 k 1



i
in (2.17) is found by the usual binomial
expansion. For example, when n = i = 2,
 2

  ak  bk 
yk
 k 1
2
2
 

 
   a1  b1 

 a2  b2 
y1
y2 
 
2
2
2
2 
2 
 a1  b1 
 2a1  b1 a2  b2 
 a2  b2 
2
y1
y1y2
y22
53
2.2 Convexity

In risk management perspective, a portfolio manager can
limit his/her interest rate risk by making the portfolio to
be duration neutral and making the portfolio to have
positive convexity.
54
2.2 Convexity

With convexity adjustment, Eq. (2.15) becomes
1
2
E  MDA  k  MDL  A  R  CX A  k  CX L  A  R 
2
(2.22)

Under the condition of (MDA – kMDL)= 0.
If (CXA – kCXL) > 0, then the value of the equity will be
increased irrelevant to the direction of the change of R.
If (CXA – kCXL) < 0, then the value of the equity will be
decreased irrelevant to the direction of the change of R.
55
2.2 Convexity

Example 2.6
Referring to Example 2.4,
CXA = 24.7934; CXL = 9.9174
CXA – kCXL =15.8678
From Eq. (2.22),
0.01

2
E    5  0.9  3
 0.5 15.8678 0.01  100M
1.1


 2.01M
(In Example 2.3, E = 2.09M.)
56
2.2 Beyond duration and convexity

Liner rates product – static yield curve
-- Deterministic CF (certain payoffs)
-- zero-cpn bond, fixed-cpn bond
-- today’s discount curve is enough
-- Stochastic cash flows (uncertain payoffs)
-- floating-cpn bond, fixed-float IR swap
-- need today’s discount curve & forward curve
57
2.2 Beyond duration and convexity

Nonlinear rates product -- dynamic yield curve
-- Volatility of Libor Rates
-- Libor futures, e.g. Eurodollar futures on LIBOR rates;
-- Libor forward, e.g. Forward Rate Agreement(FRA):
-- Libor options, e.g. ED Fut Opt, Cap/Floor
-- Correlation among Libor Rates
-- Options on Libor Swap, e.g. payer swaptions
-- Constant Maturity Swap, e.g. USD CMS
-- Vol & Corr among IR vols
-- Bermudan swaptions, callable swaps,
-- Structured products, e.g. Range Accruals,
58