Transcript Difficulties with Duration Convexity

Class 8, Chap 22 & 24

Duration & Convexity Review/Discussion

http://vimeo.com/31788176

http://video.cnbc.com/gallery/?video=3000114010

Where would you put your money if you wanted the
lowest interest rate risk exposure?
a)
b)
c)
d)
e)
5 year Greek bond
10 year UK bond
30 year US Government Treasury
20 year German bond
Well diversified equity portfolio
Purpose: To understand how banks can use duration to
hedge interest rate risk exposure on the
balance sheet
1. Duration Gap
▪
▪
▪
Calculate duration of assets and liabilities
Calculate equity duration
Immunize the equity capital to changes in interest rates
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DURATION GAP
IMMUNIZING THE ENTIRE BALANCE SHEET

We know that matching durations can immunize an asset or a
portfolio against changes in interest rates

We can now apply this logic to measure how exposed a bank
is to interest rate risk

Question: what component of the balance sheet indicates how
risky or safe a bank is?
Equity capital: it is the buffer between assets and liabilities – if the bank
looses too much equity capital the bank becomes insolvent
We know:
A LE
Rewrite:
E  A L
We want to know how equity will
change with a change in interest rates:
E  A  L
6

Question: If we had the duration for the portfolio of assets and liabilities
on the balance sheet could we find ΔA and ΔL?
Remember:
P
R
 D
P
(1  R)
A
R
  DA
A
(1  R)
A   DA  A 
R
(1  R)
L
R
  DL
L
(1  R)
L   DL  L 
R
(1  R)
E  A  L
E   DA  A 

R
R 
  DL  L 
(1  R) 
(1  R) 
E  DL  L  DA  A
R
(1  R)
7

The book has a slightly different form
E  DL  L  DA  A
R
(1  R)
E  DA  DL k  A
R
(1  R)
k = leverage L/A
Leverage adjusted
This expression which includes
the duration gap tells us how equity will
duration gap
change when interest rates change
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DA  DL k

Tells us – on a leverage adjusted basis, how much more/less sensitive are
assets than liabilities to a change in interest rates.

What does k do – why is it there?
Balance Sheet
Assets
100M D = 5yrs
Total Assets 100M
Liabilities
50M D = 5yrs
Equity
50M
Total Equity & Liabilities 100M
If the interest rate (on both assets and Liab.) decreased from 7% to 3% would you expect
equity capital to increase, decrease or stay the same? (think intuitively)
• Both A & L have the same sensitivity to interest rates so the % change will be the same
• What about the change in dollar value? It will be larger for Assets (the denominator is larger)
∆E = ∆A – ∆L so ∆E > 0
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DA  DL k

Tells us – on a leverage adjusted basis, how much more/less sensitive are
assets than liabilities to a change in interest rates.

What does k do – why is it there?
k – The value of both assets and liabilities will change with interest rates.
However, the change is asset value will be greater – just because there
are more assets than liabilities. We need to know how much more and
that is what leverage tells us (ie. the value of liabilities will change by k%
of the change in asset values)
k – In our example, the value of L ↑ by ½ as much as the value of A
AND LEVERAGE = 50M/100M = 1/2
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DA  DL k

Tells us – on a leverage adjusted basis, how much more/less sensitive are
assets than liabilities to a change in interest rates.

What does k do – why is it there?

Measured in years – but it is not a duration

The larger the gap, either positive or negative, the more exposed the bank
is to interest rate risk


possibility that equity capital will increase/decrease with interest rate changes
Shows the degree of maturity mismatch at a bank
Would you expect the leverage adjusted duration gap to be positive or negative?
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E  DA  DL k  A
R
(1  R)
Other factors that contribute to interest rate risk
 FI Size: in terms of total assets “A”:
The larger the bank is (more assets they hold) the larger the
dollar gain or loss will be from a change in interest rates
 Size of the rate change r/(1+r):
The larger the shock to interest rates, the greater the gain or loss
will be to the FI.
These are external, and uncontrollable by the FI
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Example: Suppose the duration of a financial institutions assets is 5 years and the duration of their liabilities
is 3 years. Suppose the interest rate is currently 10%, and is expected to increase to 11%
a. Calculate the change in equity capital
Assets
Liabilities
b. Calculate the change in the equity capital ratio
A = 100m
L= 90m
E=10m
100m
100m
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
A 1% increase in interest rates would decrease the FIs equity
capital ratio from:
10
 10%
100

7.91
 8.29%
95.45
Because of the effect on equity capital, the bank may want to
immunize their balance sheet to changes in interest rates
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Example: readjust the banks asset duration so that the firm’s balance sheet will not be effected by changes
in interest rates. Repeat the exercise for the liability duration.
15

To immunize the balance sheet to changes in the interest rate
the bank must set its leverage adjusted duration gap to zero
E  DA  DL k  A
R
(1  R)
DA  DL k   0
DA  DL k
Key point about leverage:
leverage affects the way that duration of assets and liabilities
should be weighted to eliminate interest rate risk.
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To this point we have been given asset and liability durations. However,
these can also be calculated from the durations of individual securities
Asset Duration: it is simply a weighted average of individual asset
durations
DA 
A
A
A1
A
DA1  2 DA2  3 DA3  ...  n DAn
AT
AT
AT
AT
Liability Duration: it is simply a weighted average of individual
liability durations
DL 
L
L
L1
L
DL1  2 DL 2  3 DL3  ...  n DLn
LT
LT
LT
LT
To this point we have been given asset and liability durations. However,
these can also be calculated from the durations of individual securities
Asset Duration: it is simply a weighted average of individual asset
durations
DA  A1DA1  A2 DA2  A3 DA3  ...  An DAn 
1
AT
Liability Duration: it is simply a weighted average of individual
liability durations
DL  L1 DL1  L2 DL 2  L3 DL3  ...  Ln DLn 
1
LT
To this point we have been given asset and liability durations. However,
these can also be calculated from the durations of individual securities
Equity Duration:
DE 
A
L
DA  DL
E
E
Where does it come from?
E  DA  DL k  A
R
(1  R)
Example: Use the following balance sheet to answer the questions below:
A)
B)
Assets (Millions)
Cash
$39.00
Marketable securities
$167.00
Mortgages
$449.00
C&I Loans
$268.00
Consumer loans
$254.00
$1,177.00
D
0
0.34
22.4
15.4
7.5
Liabilities & Equity (Millions) D
Deposits
$490.00
1.3
Commercial paper
$130.00
0.9
Long-term debt
$158.00 10.38
Short-term debt
$264.00
2.4
Equity
$135.00
$1,177.00
Calculate the banks equity duration
Adjust the banks equity duration to 25 years by Selling Mortgages for cash at 100% of their book value

Duration Gap
 Can adjust the sensitivity of equity capital to interest rates
 Immunization is the extreme case

Leverage adjusted duration gap
 Measures sensitivity of equity capital to interest rate changes

Calculate:
 Asset Duration
 Liability Duration
 Equity Duration
Duration Practice Problems
Isaac West Bank currently holds a bond portfolio with duration (Macaulay) of 5.3 years the current yield to
maturity of the portfolio is 6.3% on average. Find the expected change in the value of the portfolio if the
average YTM is expected to increase to 7%. The current market value of the portfolio is $2.8M, total face
value of the portfolio is $142M and the 5 year treasury rate is currently 2.3%.
a)
b)
c)
Calculate the change in value using Macaulay Duration
Calculate the change in value using Modified Duration
Calculate the change in value using Dollar Duration
a)
b)
c)
Calculate the duration of a two year bond with $3,000 face value and 7% coupon paid semiannually if
the YTM is currently 3%.
Suppose that the maximum loss you can afford to suffer on this bond is $1200. Find the approximate
change in interest rate that will result in a $1200 loss on your position.
Do you believe your estimate of the change in interest rate is accurate, too large or too small? Explain
your answer.