Dr. Sudhakar Raju
FN 6700
ANSWERS TO ASSIGNMENT 5: ASSET LIABILITY MANAGEMENT (ALM)
1.) The repricing gap is a measure of the difference between the dollar value of assets that will
reprice and the dollar value of liabilities that will reprice within a specific time period, where
reprice means the potential to receive a new interest rate. Rate sensitivity represents the time
interval where repricing can occur. The model focuses on the potential changes in the net
interest income variable. In effect, if interest rates change, interest income and interest
expense will change as the various assets and liabilities are repriced, that is, receive new
interest rates.
2.) The maturity bucket is the time window over which the dollar amounts of assets and
liabilities are measured. The length of the repricing period determines which of the securities in
a portfolio are rate-sensitive. The longer the repricing period, the more securities either mature
or need to be repriced, and, therefore, the more the interest rate exposure. An excessively
short repricing period omits consideration of the interest rate risk exposure of assets and
liabilities that are repriced in the period immediately following the end of the repricing period.
That is, it understates the rate sensitivity of the balance sheet. An excessively long repricing
period includes many securities that are repriced at different times within the repricing period,
thereby overstating the rate sensitivity of the balance sheet.
3.) Impact of an 1% interest rate increase:
Repricing gap = RSA - RSL = $200 - $100 million = +$100 million.
NII = ($100 million)(+.01) = +$1.0 million, or $1,000,000.
Repricing gap = RSA - RSL = $100 - $150 million = -$50 million.
NII = (-$50 million)(+.01) = -$0.5 million, or -$500,000.
Repricing gap = RSA - RSL = $150 - $140 million = +$10 million.
NII = ($10 million)(+.01) = +$0.1 million, or $100,000.
1
Impact of an 1% interest rate decease:
NII = ($100 million) (-.01) = -$1.0 million, or -$1,000,000.
NII = (-$50 million) (-.01) = +$0.5 million, or $500,000.
NII = ($10 million) (-.01) = -$0.1 million, or -$100,000.
4.) The gap ratio is the ratio of the cumulative gap position to the total assets of the bank. The
cumulative gap position is the sum of the individual gaps over several time buckets. The value
of this ratio is that it tells the direction of the interest rate exposure and the scale of that
exposure relative to the size of the bank.
5.) Book value accounting reports assets and liabilities at the original issue values. Current
market values may be different from book values because they reflect current market
conditions, such as interest rates or prices. This is especially a problem if an asset or liability has
to be liquidated immediately. If the asset or liability is held until maturity then the reporting of
book values does not pose a problem.
For an FI, a major factor affecting asset and liability values is interest rate changes. If interest
rates increase, the value of both loans (assets) and deposits and debt (liabilities) fall. If assets
and liabilities are held until maturity, it does not affect the book valuation of the FI. However, if
deposits or loans have to be refinanced, then market value accounting presents a better picture
of the condition of the FI.
The process by which changes in the economic value of assets and liabilities are accounted is
called marking to market. The changes can be beneficial as well as detrimental to the total
economic health of the FI.
6.) Book values represent historical costs of securities purchased, loans made, and liabilities
sold. They do not reflect current values as determined by market values. Effective financial
decision-making requires up-to-date information that incorporates current expectations about
future events. Market values provide the best estimate of the present condition of an FI and
serve as an effective signal to managers for future strategies.
Book values are clearly measured and not subject to valuation errors, unlike market values.
Moreover, if the FI intends to hold the security until maturity, then the security's current
liquidation value will not be relevant. That is, the paper gains and losses resulting from market
value changes will never be realized if the FI holds the security until maturity. Thus, the
changes in market value will not impact the FI's profitability unless the security is sold prior to
maturity.
2
7.) The following is a simplified FI (Financial Institution) balance sheet:
Assets
Loans
Total Assets
$1,000
0
$1,000
Liabilities and Equity
Deposits
Equity
Total Liabilities & Equity
$850
$150
$1,000
Note that the average maturity of loans is 4 years and the average maturity of deposits is 2
years. Both loans and deposits are reported as book value, zero-coupon items. The market
value (as opposed to the book value) of equity can be determined thus:
a.) Market Value of Loans = $1,000 / (1 + .09)4 = $708.43
Market Value of Deposits = $850 / (1 + .09)2 = $715.43.
Market Value of Equity (Net Worth) = $708.43 - $715.43 = -$7.0028.
Thus, even though the book value of equity is $150, the market value of equity is negative
implying that the financial institution is insolvent.
b.) For the market value of equity to be zero, market value of loans must equal the market
value of deposits. Let x be the interest rate that equates the market value of loans to the
market value of deposits. Thus:
$708.43 = $850 / (1 + x)2
Note that the above is in the form of PV = FV / (1 + r)n. Solving for x:
[$850/$708.43] = (1 + x)2
[1.1998]1/2 = (1+x)
[1.1998].50 = (1+x)
1.0954 = (1+x)
X = .0954 or 9.54%
That is, deposit rates will have to increase more because they have a shorter maturity. You can
also solve this on your calculator thus:
3
1 Shift P/Yr
850 FV
708.43 +/- PV
2N
I/YR = 9.54%
c.) For the market value of equity to be zero, market value of loans must equal the market
value of deposits. Let n be the average maturity of deposits that sets the market value of equity
to be zero. Thus:
$708.43 = $850 / (1 + .09)n
Note that the above is in the form of PV = FV / (1 + r)n. Solve for n. Thus:
1 Shift P/Yr
850 FV
708.43 +/- PV
9 I/Yr
N = 2.11 years
The result is 2.11 years. If interest rates remain at 9%, then the average maturity of deposits
has to be higher in order to match the value of a 4-year loan.
8.)
Assets
Total Assets [5]
$100
Total Assets
$100
Liabilities and Equity
Liabilities [3]
Equity
Total Liabilities & Equity
$90
$10
$100
The Macaulay duration of the assets and liabilities is within parentheses. Assume that
assets are fixed rate assets and liabilities are fixed rate liabilities. Suppose rates rise to
11%. Then:
Dollar Change in Total Assets = (-DTA) (∆ Interest Rate) (Total Assets)
= (-5) (+.01) ($100)
= -$5 million
4
Dollar Change in Total Liabilities = (-DTL) (∆ Interest Rate) (Total Liabilities)
= (-3) (+.01) ($90)
= -$2.70 million
The new balance sheet is then given by:
NEW BALANCE SHEET AFTER INTEREST RATE INCREASE
Assets
Total Assets
$95
Total Assets
$95
Liabilities and Equity
Liabilities
Equity
Total Liabilities & Equity
$87.30
?
$95
Note that both assets and liabilities have decreased. However, the decrease in assets is
greater than the decrease in liabilities since assets have a greater duration (more interest
rate risk) as compared to liabilities. The new value of equity is then equal to:
New Value of Equity = $7.70
∆ Equity = $7.70 - $10 = -$2.30
The decrease in equity is equal to ([$7.70 - $10] / [$10]) or a drop of 23% in net worth.
(Note that if we were to use Modified duration rather than Macaulay duration the
dollar change in equity would be given by: -$2.30 / (1 + .10) = -$2.09). It does not
make much difference whether one uses Macaulay or Modified duration so long as
one measure is consistently used.)
b.) The Duration Gap (i.e. leverage adjusted duration gap) is given by:
DUR GAP
=
(DURTA)
=
(5)
=
2.30
– (TL/TA) (DURTL)
– ($90/$100) (3)
5
(Note: This can be easily adjusted for modified duration as: 2.30 / (1 + .10) = 2.09).
This implies that if for a 1% increase in interest rates, the value of equity will decrease by
2.30%. For a 1% increase in interest rates, the dollar change in equity (as opposed to the
percentage change in equity) is given by:
∆ Equity
= (-DUR GAP) (∆ Interest Rate) (Total Assets)
= (-2.30) (+.01) ($100 million)
= - $2.30 million
If one uses Modified Duration, the corresponding value above is -$2.09 million
9.)
The balance sheet for GBI Bank, Inc. is presented below ($ millions):
Assets
Cash
Federal funds
Loans (floating)
Loans (fixed)
Total assets
Liabilities and Equity
Core deposits
Federal funds
Euro CDs
Equity
Total liabilities & equity
$30
20
105
65
$220
$20
50
130
20
$220
NOTES TO THE BALANCE SHEET: The Fed funds rate is 8.50%, the floating loan rate is LIBOR + 4% and currently
LIBOR is 11% . Fixed rate loans have five-year maturities, are priced at par, and pay 12% annual interest. Core
deposits are fixed-rate for 2 years at 8% paid annually. Euros currently yield 9%.
a.) Face Value of Fixed Loans = $65; Annual Interest Payment = 12% x $65 = $7.80; Maturity = 5
years; CIR = YTM = 12% (priced at par). The modified duration or D* can be determined to be:
1 Shift P/Yr
65 FV
7.80 PMT
5N
12 I/Yr
PV → 65
12.10 I/Yr
PV → 64.7663
11.90 I/Yr
PV → 65.2349
6
Y1
Y0
Y2
Yield
11.90%
12%
12.10%
D*=
P1
P0
P2
PB
$65.2349
$65
$64.7663
P2  P1
64.7663  65.2349  .007209231
=
=
P0
65
.002
.1210  .1190
Y2  Y1
D* = – 3.60
[Note: Since D* 
MacDur
, one can solve for Mac Duration thus: (-3.60) x (1 + .12) = 1  PeriodicYT M
4.03].
b.) DURTA = ( [30 (0) + 20 (.36) + 105 (.36) + 65 (4.03)] / [220] ) = 1.40 years
Note: Since cash has no cash flows, it has a duration of zero.
c). Face Value of Core Deposits = $20; Annual Interest Payment = 8% x $20 = $1.60; Maturity =
2 years; CIR = YTM = 8% (priced at par). The modified duration or D * can be determined to be:
1 Shift P/Yr
20 FV
1.60 PMT
2N
8 I/Yr
PV → 20
8.10 I/Yr
PV → 19.9644
7.90 I/Yr
PV → 20.0357
7
Yield
Y1
Y0
Y2
D*=
7.90%
8%
8.10%
PB
P1
P0
P2
$20.0357
$20
$19.9644
P2  P1 19.9644  20.0357  .003565
=
=
20
P0
.002
.0810  .0790
Y2  Y1
D* = – 1.78
The Mac Duration is : (-1.78) x (1 + .08) = - 1.93.
d).
DURTL = [20*(1.93) + 180*(.401)] / 200 = .55 years
Note: Total Liabilities sum to $200.
e.)
Duration Gap = (DURTA) – (TL/TA) (DURTL)
= (1.40) - ($200/220)(.55)
= (1.40) - (.9091) (.55)
= .90 years
The duration gap (leverage adjusted duration gap) is positive implying that if interest rates go
up by 1%, bank equity will decline by .90%.
f.) Since GBI’s Duration Gap is positive, an increase in interest rates will lead to a decline in net
worth. For a 1% increase, the change in equity or net worth is:
∆ Equity
= (-Dur GAP) (∆ Interest Rate) (Total Assets)
= (-.90) (+.01) ($220 million)
= - $1,980,000
Thus, net worth will decline by $1,980,000 (approximately) if interest rates go up by 1%. The
new equity will then be equal to: $20,000,000 - $1,980,000 = $18,020,000.
8
g.) Since GBI’s duration gap is positive, a decrease in interest rates will lead to an increase in
net worth. For a 0.50% (50 basis point) decrease, the change in net worth is:
∆ Equity
= (-Dur GAP) (∆ Interest Rate) (Total Assets)
= (-.90) (-.0050) ($220 million)
= + $990,000
The new net worth will be higher at: $20,000,000 + $990,000 = $20,990,000.
h.) Immunization requires the bank to have a Dur GAP of 0. In other words:
∆ Equity = 0 requires that WTA DURTA = WTL DURTL
Assume that the weights of TA and TL cannot be changed. The bank then has two choices. It can
manipulate either the duration of total assets or the duration of total liabilities. Suppose it
decides to manipulate the duration of its TA. The new duration of TA that is required to
immunize the bank’s balance sheet is given by:
WTA DURTA = WTL DURTL
(1) (x) = (.9091) (.55)
DURTA = .50
Thus, the DURTA needs to be decreased to .50 years from 1.40 years to immunize the bank’s
balance sheet. GBI could reduce the duration of its assets by using more fed funds or issuing
more floating rate loans.
Now, suppose that GBI decides to manipulate the duration of its TL instead. The new duration
of TL that is required to immunize the bank’s balance sheet is given by:
WTA DURTA = WTL DURTL
(1)(1.40) = (.9091) (x)
DURTL = 1.54
9
Thus, the duration of GBI’s total liabilities needs to be substantially increased from its current
.55 years to 1.54 years to immunize the bank’s balance sheet. A final option for GBI is to use
some combination of reducing asset duration and increasing liability duration in such a manner
that the duration gap is forced to be equal to zero.
10.) a) The structure of the liability (issued zero coupon bond) is as follows: Current Value =
$90; Face Value = $97.20; Initial Interest Rate = 8%, N = 1 year.
The PV of the liability after the interest rate increases by 1.50% (150 bps) is given by:
PV of Liability = [$97.20 / (1 + .0950)1] = $88.77 million
The structure of the asset (purchased bond) is as follows: Current Value = $100 million; Initial
Interest Rate = 10%, Coupon Payment = $10 million; N =2 years. Suppose the interest rate
increases by 150 bps to 11.50%. The value of the bond is then given by:
1 Shift P/Yr
100 FV
10 PMT
2N
11.50 I/Yr
PV → $97.45
Thus, the market value of the liability declines from an initial value of $90 million to $88.77 - a
decline of $1.23 million. This is a favorable outcome for the insurance company. However, this
is offset by the fact that the value of the asset also declines (from $100 million to $97.45 – a
decline of $2.55 million) much more than the liability.
b).  Equity = TA - TL = -$2.55 - (-$1.23 million) = -$1.32 million
c.) Macaulay Duration of Liability = 1 year. The Modified Duration at the initial interest rate is
given by 1/1.08 = .9259.
The Modified Duration of the Asset can be determined thus:
Face Value = $100; Annual Interest Payment = $10 million; Maturity = 2 years; YTM = 10%
(priced at par). The modified duration or D* can be determined to be:
10
1 Shift P/Yr
100 FV
10 PMT
2N
10 I/Yr
PV → 100
10.10 I/Yr
PV → 99.8267
9.90 I/Yr
PV → 100.1738
Y1
Y0
Y2
D*=
Yield
9.90%
10%
10.10%
P1
P0
P2
PB
$100.1738
$100
$99.8267
P2  P1 99.8267  100.1738  .003471
=
=
100
P0
.002
.1010  .0990
Y2  Y1
D* = – 1.74
(Note: The Mac Duration is : (-1.74) x (1 + .10) = - 1.91).
d.)
∆ in Liability Value
= (-D*) (∆ Interest Rate) (Total Liability)
= (-.9259) (+.0150) ($90 million)
= - $1.25 million
∆ in Asset Value
= (-D*) (∆ Interest Rate) (Total Liability)
= (-1.74) (+.0150) ($100 million)
= - $2.61 million
∆ in Equity
= (∆ Total Assets) - (∆Total Liability)
= (-$2.61 million) - (-$1.25 million)
= - $1.36 million
11
e.)
Duration Gap
= (TA/TA) (DURTA) – (TL/TA) (DURTL)
= WTA DURTA - WTL DURTL
= ($100/ $100)(1.74) - ($90/100)(.9259)
= 1.74 - .8333
= .91
f.) ∆ Equity
= (-Dur GAP) (∆ Interest Rate) (Total Asset)
= (-.91) (+.0150) ($100 million)
= - $1.37 million
Note that this estimated change in net worth is the same (except for approximation errors) as
the estimate above in part (d).
g.) The duration of the loan could be shortened relative to the liability or alternatively the
duration of the liability could be lengthened relative to the loan, or some combination of both.
Shortening the loan duration would mean the possible use of variable rates. The duration of the
liability cannot be lengthened without extending its maturity. In either case, the loan officer
may have been up against market or competitive constraints in that the borrower or investor
may have had other options. Other methods to reduce the interest rate risk under conditions
of this nature include using derivatives such as options, futures, and swaps.
11.) a). For Bank A, an increase of 100 basis points in interest rate will cause the market value
of it’s assets to change as follows:
1 Shift P/Yr
1,000,000 FV
120,000 PMT
10 N
12 I/Yr
PV → $1,000,000
13 I/Yr
PV → $945,738
Thus, assets decrease from $1 million to $945,738 a decline of $54,262
Bank A’s liabilities (CD) will change as follows:
1 Shift P/Yr
1,000,000 FV
100,000 PMT
10 N
12
10 I/Yr
PV → $1,000,000
11 I/Yr
PV → $941,108
Thus, liabilities decrease from $1 million to $941,108 – a decline of $58,892.
The net change in equity for Bank A is given by:
∆ in Equity
= (∆ Total Assets) - (∆Total Liability)
= (-$54,262) - (-$58,892)
= + $4630
For Bank B, an increase of 100 basis points in interest rate will cause the market value of it’s
assets to change as follows:
1 Shift P/Yr
1,976,362.88 FV
0 PMT
7N
12 I/Yr
PV → $894,006
13 I/Yr
PV → $840,074
Thus, assets decrease from $894,006 to $840,074 a decline of $53,932.
Bank A’s liabilities (CD) will change as follows:
1 Shift P/Yr
1,000,000 FV
82,750 PMT
10 N
10 I/Yr
PV → $894,006
11 I/Yr
PV → $839,518
Thus, liabilities decreased from $894,006 to $839,518 – a decline of $54,488.
The net change in equity for Bank B is given by:
∆ in Equity
= (∆ Total Assets) - (∆Total Liability)
= (-$53,932) - (-$54,488)
= + $556
13
b.)
The assets and liabilities of Bank A change in value by different amounts even though
the face values and maturities are the same since the durations of the assets and
liabilities are not the same. For Bank B, the maturities of the assets and liabilities are
different but the current market values and durations are the same. Thus the change
in interest rates causes the same (approximate) change in value for both liabilities and
assets.
c.) The Macaulay duration of the Bank A’s assets is 6.33. The Modified Duration is, therefore,
[6.33 / (1 + .12)] = 5.65. The Macaulay duration of the Bank A’s liabilities is 6.76. The Modified
Duration is therefore [6.76 / (1 + .10)] = 6.15.
Note: Compute duration at the initial interest rate – that is, the interest rate that prevailed
before rates went up.
∆ in Asset Value
= (-D*) (∆ Interest Rate) (Total Liability)
= (-5.65) (+.01) ($1 million)
= - $56,500
∆ in Liability Value
= (-D*) (∆ Interest Rate) (Total Liability)
= (-6.15) (+.01) ($1 million)
= - $61,500
∆ in Equity
= (∆ Total Assets) - (∆Total Liability)
= (-$56,500) - (-$61,500)
= $5000
The drop in equity for Bank A is about $5000 using the duration estimation model. The small
difference in this estimate and the estimate found in part (a) above is due to the convexity of
the two financial assets.
Similarly, the Macaulay duration of the Bank B’s assets is 7. The Modified Duration is, therefore,
[7 / (1 + .12)] = 6.25. The Macaulay duration of the Bank A’s liabilities is also 7. The Modified
Duration is thus [7 / (1 + .10)] = 6.36.
∆ in Asset Value
= (-D*) (∆ Interest Rate) (Total Liability)
= (-6.25) (+.01) ($894,006)
= - $55,875
14
∆ in Liability Value
= (-D*) (∆ Interest Rate) (Total Liability)
= (-6.36) (+.01) ($894,006)
= - $56,859
∆ in Equity
= (∆ Total Assets) - (∆Total Liability)
= (-$55,875) - (-$56,859)
= + $984
The change in value of the assets and liabilities for Bank B is $984 using the duration estimation
model. The small difference in this estimate and the estimate found in part (a) above is due to
the convexity of the two financial assets. Note also that the reason the change in asset values
for Bank A is considerably larger than for Bank B is because of the difference in the durations of
the loan and CD for Bank A.
12.)
a.) DURATIONTOTAL ASSETS = [(.50)(90) + (.90)(55) + (4.39)(176) + (7)(2724)] / [3045] = 6.55 years
b.) DURATIONTOTAL LIABILITIES = [(1.00)(2092) + (0.01)(238)]/ [2330] = 0.90 years
c.) (Leverage Adjusted) Duration Gap = DURTA – (TL/TA) (DURTL)
= 6.55 - ($2330/$3045)(.90)
= 6.55 - .69
= 5.86
The duration gap is positive implying that if interest rates go up by 1%, bank equity will decline
by about 5.86%.
d.) Since the Duration Gap is positive, an increase in interest rates will lead to a decline in net
worth. For a 1/2% (50 bps) increase, the change in equity or net worth is:
∆ Equity
= (-Dur GAP) (∆ Interest Rate) (Total Assets)
= (-5.86) (+.005) ($3045)
= - $89.22
The loss in equity of $89.22 (or $89,220) will reduce the equity from $715,000 to $625,780.
15
e. ) Since the Duration Gap is positive, a decrease in interest rates will lead to an increase in
equity. For a 1/4% (25 bps) decrease, the change in equity or net worth is:
∆ Equity
= (-Dur GAP) (∆ Interest Rate) (Total Assets)
= (-5.86) (- .0025) ($3045)
= + $44.61
The market value of equity will increase by $44.61 ( $44,610) from$715,000 to $759,610.
f). Immunization requires the bank to have a leverage-adjusted duration gap of zero. Therefore,
the FI could reduce the duration of its assets to 0.69 by using more T-bills and floating rate
loans. Or the FI could try to increase the duration of its deposits by using fixed-rate CDs with a
maturity of 3 or 4 years. Finally, the FI could use a combination of reducing asset duration and
increasing liability duration in such a manner that the duration gap is set equal to zero. The
duration gap of 5.86 years is quite large and it is not likely that the FI will be able to reduce it to
zero by using only balance sheet adjustments. For example, even if the FI moved all of its loans
into T-bills, the duration of the assets still would exceed the duration of the liabilities after
adjusting for leverage. This adjustment in asset mix would imply foregoing a large yield
advantage from the loan portfolio relative to the T-bill yields in most economic environments.
13.)
The three criticisms of the Duration GAP model are:
a.) Immunization is a dynamic problem because duration changes over time. Thus, it is
necessary to rebalance the portfolio as the duration of the assets and liabilities change over
time.
b.) Duration matching can be costly because it is not easy to restructure the balance sheet
periodically, especially for large FIs.
c.) Duration is not an appropriate tool for immunizing portfolios when the expected interest
rate changes are large because of the existence of convexity. Convexity exists because the
relationship between bond price changes and interest rate changes is not linear, which is
assumed in the estimation of duration. Using convexity to immunize a portfolio will reduce the
problem.
14.)
The growth of the derivatives markets, asset securitization, and loan sales markets have
increased considerably the speed of major balance sheet restructurings. Further, as these
markets have developed, the cost of the necessary transactions has also decreased. Finally, the
growth and development of the derivative markets provides significant alternatives to
managing the risk of interest rate movements only with on-balance sheet adjustments.
16
Assets approach maturity at a different rate of speed than the duration of the same assets
approaches zero. Thus, after a period of time, a portfolio or asset that was immunized against
interest rate risk will no longer be immunized. In fact, portfolio duration will exceed the
remaining time in the investment or target horizon, and changes in interest rates could prove
costly to the institution.
15.)
a.) The duration of the bonds is as follows:
Face Value = $1000; Annual Interest Payment = $80; Maturity = 3 years; YTM = 10% . The
modified duration or D* can be determined to be:
1 Shift P/Yr
1000 FV
80 PMT
3N
10 I/Yr
PV → $950.26
10.10 I/Yr
PV → $947.87
9.90 I/Yr
PV → $952.67
Y1
Y0
Y2
D*=
Yield
9.90%
10%
10.10%
P1
P0
P2
PB
$952.67
$950.26
$947.87
P2  P1 947.87  952.67  .00505
=
=
P0
950.26
.002
.1010  .0990
Y2  Y1
D* = – 2.53
The Mac Duration is : (-2.53) x (1 + .10) = - 2.78.
17
The investment horizon is 2 year, 9.50 months or 2 + (9.50/12) = 2.79 years. Since the Macaulay
duration of the asset is approximately equal to the investment horizon (or holding period), for
all practical purposes, the bond investment horizon was immunized at the time of purchase.
b.) The Macaulay duration of the bonds one year later can be determined thus:
1 Shift P/Yr
1000 FV
80 PMT
2N
10 I/Yr
PV → $965.29
10.10 I/Yr
PV → $963.60
9.90 I/Yr
PV → $966.98
Y1
Y0
Y2
D*=
Yield
9.90%
10%
10.10%
P1
P0
P2
PB
$966.98
$965.29
$963.60
P2  P1 963.60  966.98  .003502
=
=
P0
965.29
.002
.1010  .0990
Y2  Y1
D* = –1.75
The Mac Duration is : (-1.75) x (1 + .10) = -1.93.
After one year, the investment horizon is 1 year, 9.50 months or 1.79 years. At this time, the
bonds will have a duration of 1.93 years. Thus, the bonds will no longer be immunized.
c). The investment horizon is 1.79 years. The Macaulay duration of the zero coupon bonds is 1.
The Macaulay duration of the bonds after 1 year is 1.93 years. The proportion of bonds that
should be placed in the zero coupon bonds (WC) can be determined thus:
1.79
=
(WZC) (DURZC) + (1 - WZC) (DURBonds)
18
1.79
1.79
1.79 – 1.93
-.14
Wzc
Wzc
=
=
=
=
=
=
(WZC) (1.00) + (1 - WZC) (1.93)
1 WZC + 1.93 - 1.93 WZC
-.93 WZC
-.93 WZC
-.14/ -.93
.15 or 15%
Thus, about 15% of the bond portfolio should be placed in the zero coupon bonds after one
year.
19