7FNCE026W

Seminar 4: Interest rate gap and

duration

1. Suppose that a bank has $5 billion of one-year loans and $20 billion of five-year
loans. These are financed by $15 billion of one-year deposits and $10 billion of five-
year deposits. Explain the impact on the bank’s net interest income of interest
rates increasing by 1% every year for the next three years.

In this case, the interest rate mismatch is $10 billion. The bank’s net interest income
declines by $100 million each year for the next three years.

2. What does duration tell you about the sensitivity of a bond portfolio to interest
rates? What are the limitations of the duration measure?

Duration provides information about the effect of a small parallel shift in the yield curve on
the value of a bond portfolio. The percentage decrease in the value of the portfolio equals
the duration of the portfolio multiplied by the amount by which interest rates are increased
in the small parallel shift. Its main limitation is that it applies only to parallel shifts in the
yield curve that are small.

3. A five-year bond with a yield of 11% pays an 8% coupon at the end of each year.
(a) What is the bond’s price?
(b) What is the bond’s duration and modified duration?
(c) Use the modified duration to calculate the effect on the bond’s price of a 0.2%
decrease in its yield.
(d) Recalculate the bond’s price on the basis of a 10.8% per annum yield and verify
that the result is in agreement with your answer to (c).

(a) The bond’s price is:

8 8 8 8 108
P 0= + + + + =88.91
( 1.11) (1.11) (1.11) (1.11) (1.11)5
2 3 4
(b) Duration:

8 ×1 8× 2 8× 3 8 × 4 108 ×5
+ + + +
(1.11) (1.11) (1.11) (1.11)4 (1.11)5
2 3
D= =4.27
P0

Modified duration:

−4.27
V= =−3.84
1.11

(c) When the yield decreases by 0.2% the bond’s price increases by 0.68:

∆P ∆i ⇒
=V ×∆ i=−D× =0.768 % ❑ ∆ P=0.68
P ( 1+i )

(d) recomputing the bond’s price with a yield of 10.8% gives a price of 89.60, which is
approximately consistent with (a) and (c).

4. When is a firm asset sensitive? Liability sensitive?

A firm is asset sensitive when it has more interest-rate sensitive assets maturing or subject
to repricing during a specific time period than rate-sensitive liabilities. A liability sensitive
position, in contrast, would find the institution having more interest-rate sensitive deposits
and other liabilities than rate-sensitive assets for a particular planning period.

5. Suppose Carroll Bank and Trust reports interest-sensitive assets of $570 million
and interest-sensitive liabilities of $685 million. What is the bank’s dollar interest-
sensitive gap? Its relative interest-sensitive gap and interest-sensitivity ratio?

Dollar Interest-Sensitive Gap = Interest-Sensitive Assets – Interest Sensitive Liabilities

= $570 mill. − $685 mill. = − $115 mill.

Relative Gap = IS Gap = − $115 = − 0.2018

Total assets $570

ISR = Interest-Sensitive Assets = $570 = 0.8321

Interest-Sensitive $685
Liabilities
 6. New Comers State Bank has recorded the following financial data for the past
three years (dollars in millions):

Current Previous Year Two Years Ago

Year
Interest revenues $82 $80 $78
Interest expenses 64 66 68
Loans 450 425 400
Investments 200 195 200

a. Calculate the bank’s net interest margin for the past three years.
b. What do you think caused the changes you observe?
c. Do you have any recommendations for New Comers’ management team?

Net interest margin (NIM) = Net interest income/Total earning assets

Net interest income = Net interest revenues - Net interest expenses
Total earning assets = Loans + Investments

NIM (Current) = ($82-64)/ (450 + 200) = 18/650 = 0.028 or 2.77%

NIM (Previous) = ($80-66)/ (425 + 195) = 14/620 = 0.0226 or 2.26%
NIM (Two years ago) = ($78-68)/ (400 + 200) = 10/600 = 0.0167 or 1.67%

The net interest margin has been increasing over the years. As interest revenues and
expenses as well as the bank’s assets have increased consistently over the years, there has
been a constant increase in the net interest margin. If the bank can further cut down on its
interest expenses and increase its assets in the next years, the net interest margin will
increase at a higher rate.

7. A government bond currently carries a yield to maturity of 6% and a market price

of $1,168.49. If the bond promises to pay $100 in interest annually for five years,
what is its current duration?

The duration of the bond is computed as follows:

= 4.23 years
 8. Blue Moon National Bank holds assets and liabilities whose average durations and
dollar amounts are as shown in this table:

Avg. Duration
Asset and Liability Items (years) Dollar Amount (millions)

Investment Grade Bonds 15.00 $65.00

Commercial Loans 3.00 $400.00
Consumer Loans 7.00 $250.00
Deposits 1.25 $600.00
Nondeposit Borrowings 0.50 $50.00

a. What is the weighted average duration of Blue Moon’s asset portfolio and
liability portfolio?
b. What is the leverage-adjusted duration gap?

The weighted average duration of Blue Moon’s asset portfolio is calculated as follows:

The weighted average duration of the liability portfolio is calculated as follows:

9. Carter National Bank holds $15 million in government bonds having a duration of
12 years. If interest rates suddenly rise from 6% to 7%, what percentage change
should occur in the bonds’ market price?

The percentage change in market price is computed as follows:

10. Conway Thrift Association reports an average asset duration of 7 years and an
average liability duration of 4 years. In its latest financial report, the association
recorded total assets of $1.8 billion and total liabilities of $1.5 billion.
 a. If interest rates began at 5% and then suddenly climbed to 6%, what change
will occur in the value of Conway’s net worth?
b. By how much would Conway’s net worth change if, instead of rising,
interest rates fell from 5% to 4.5%?

The key formula is: Change in net worth NW =

a) For the change in interest rates from 5 to 6%, change in net worth will be:

[
¿ −7 ×
0.01
( 1+0.05 ) ][
×1.8 − −4 ×
0.01
( 1+ 0.05 )
×1.5
]
= – $0.12 billion – (–$0.05714 billion) = – $0.06286 billion

b) On the other hand, if interest rates decline from 5 to 4.5%, change in net worth will
be:

[
¿ −7 ×
−0.005
( 1+0.05 ) ][
×1.8 − −4 ×
−0.005
( 1+ 0.05 )
×1.5
]
= + $0.06 billion – $0.02857 billion = + $0.03143 billion