2004 Correction Final Exam
FIXED INCOME
FINAL EXAM CORRECTION
APRIL 2004
1. The bond will trade at a discount as YTM >Coupon. The price of the bond
is :
3/(1+0.07)0.5 + 3/(1+0.07)1 + 3/(1+0.07)1.5 + 103/(1+0.07)2= 98.37 or
$983.7
2. Bond MV
A
B
C
D
total
Weight
$20 million
$35 million
$60 million
$40 million
$155
Duration
0.129
0.226
0.387
0.258
1.000
2
7
8
14
8.548
Convx.
25
90
56
160
86.51
If rates increase by 200BP, the portfolio will decrease by (4.27%) + 1.73% that is –2.54%
using duration and convexity. The portfolio will then be worth : 155 x (1 – 2.54%) =
$151.28 million
8.548 x 0.5 = 4.27%
½ x 86.51 x (0.02)2 = 1.73
3.
a. The present value of the $12 million is 12/(1+0.035)6 = $9,762,000
million
b. To guarantee the future amount to its client , the bank must find an
investment which duration is matched to its’ liability’s.
3/(1+7%/2) = 2.89
c. If rates climb by 2% and the duration of the bond found above is
2.89 then the bond decreases in value by 2.89 x 2 = 5.98% that is
$9,762,000 x (1 – 5.98%) = $9,177,000
Coupon income is : 9,762,000 x 9% = $878, 580
Total value of portfolio after 1 year = 9, 177,000 + 878,850 =
$10,055,600
The client’s target is $12 million and we have 2 years left to achieve
that objective with $10,055,600 at our rate of 9%. The FV of
$10,055,600 at our rate of 9% is
$10,055,600 (1 + 9%)2 = $11,947,000
We have reached a level of $53 000 below the safety cushion and
immunization must be immediate.
4. Bond A premium of : 800/20 = 40
40 - 30 = 10
Bond B premium of : 1100/50 = 22
30 - 22 = 8
Bond B is cheaper
10/30
8/30
33%
26%
5. By mixing a zero coupon bond with a call option on the index.
Ex. With $1000 initial investment, one can invest in a zero coupon bond
that will mature at par in 5 years (YTM 8%) at $680… (1000/(1+8%)5).
The reaming $320 is then invested in a call option on a stock index.
Zero coupon =1000
Call option =0
5 years later:
Zero coupon =1000
Call option > 0
In both cases , your capital is guaranteed.
6.
a) If the portfolio manager expects rates to climb, he’d rather lower
his duration and therefore swap his KO 5.75 2011 bonds for the
IBM 2.25 2007 bonds.
b) $DKO = (MDIBM x MVIBM)/100 .
MDIBM= 2.95 $DKo= 5.68% x 5000 x 1076 = $305,580
MVIBM= $DKO/ (MDIBM/100) = $10,358,000 to be invested in IBM
IBM trades at $980, so he should buy 10,358,000/980 =10,570
bonds $10.57 million face value
7.
a) The value of the assets, portfolio of bonds is $1200 million
as
Econ. Surplus = Market value assets - PV of liabilities
b) Duration of liabilities is 5 Duration of assets is 5 x 1.2 = 6
If rates decrease by 50BP, assets will increase by 1200x 0.05 x 6 = 400
Liabilities will increase by 500 x 0.05 x 5 = 125
The new economic surplus will increase by : $375 million
8. Callable bond = non callable band – call option. Therefore the value of the call is 2.
9. 300 000 000 x 0.5 x 0.0028 = 420 000 is paid every 6 month form the buyer to the
seller. If default occurs after 3 years and 3 months , the seller would receive a total of :
(420 000 x 6) + (420 000/2) = 2,730,000
10. T T T T
11. You need to first find the spreads over swap for 3 years and 4 years for the Citigroup
bond . The 3 and 4 year swap are respectively at 3% and 3.45%. The bond yields
6.05%. So the spread is 3% for the 3 year (6% -3%) and 2.6% for the 4 year (6.05% 3.45%) .
Q(3) = 1- [(1+ 0.03)3 / (1+0.0602)3)] / (1-0.4) = 0.1384%
Q(4) = 1- [(1+ 0.0345)4 / (1+0.0602)4)] / (1-0.4) = 0.1561
Q(4) - Q(3) = 0.1561% - 0.1384% = 0.017%
12. Receives Coupon payment : 5.75/2 x $10,000,000 = $287,500
Pays Libor payment : 4% + 50BP = 4.5% that is 2.25% for six months that is
($225,000)
Capital loss : 100 – 98 = 2 0.02 x 10,000,00 = ($200,000)
Net cash flow : ($137,500)
FINAL EXAM CORRECTION
APRIL 2004
1. The bond will trade at a discount as YTM >Coupon. The price of the bond
is :
3/(1+0.07)0.5 + 3/(1+0.07)1 + 3/(1+0.07)1.5 + 103/(1+0.07)2= 98.37 or
$983.7
2. Bond MV
A
B
C
D
total
Weight
$20 million
$35 million
$60 million
$40 million
$155
Duration
0.129
0.226
0.387
0.258
1.000
2
7
8
14
8.548
Convx.
25
90
56
160
86.51
If rates increase by 200BP, the portfolio will decrease by (4.27%) + 1.73% that is –2.54%
using duration and convexity. The portfolio will then be worth : 155 x (1 – 2.54%) =
$151.28 million
8.548 x 0.5 = 4.27%
½ x 86.51 x (0.02)2 = 1.73
3.
a. The present value of the $12 million is 12/(1+0.035)6 = $9,762,000
million
b. To guarantee the future amount to its client , the bank must find an
investment which duration is matched to its’ liability’s.
3/(1+7%/2) = 2.89
c. If rates climb by 2% and the duration of the bond found above is
2.89 then the bond decreases in value by 2.89 x 2 = 5.98% that is
$9,762,000 x (1 – 5.98%) = $9,177,000
Coupon income is : 9,762,000 x 9% = $878, 580
Total value of portfolio after 1 year = 9, 177,000 + 878,850 =
$10,055,600
The client’s target is $12 million and we have 2 years left to achieve
that objective with $10,055,600 at our rate of 9%. The FV of
$10,055,600 at our rate of 9% is
$10,055,600 (1 + 9%)2 = $11,947,000
We have reached a level of $53 000 below the safety cushion and
immunization must be immediate.
4. Bond A premium of : 800/20 = 40
40 - 30 = 10
Bond B premium of : 1100/50 = 22
30 - 22 = 8
Bond B is cheaper
10/30
8/30
33%
26%
5. By mixing a zero coupon bond with a call option on the index.
Ex. With $1000 initial investment, one can invest in a zero coupon bond
that will mature at par in 5 years (YTM 8%) at $680… (1000/(1+8%)5).
The reaming $320 is then invested in a call option on a stock index.
Zero coupon =1000
Call option =0
5 years later:
Zero coupon =1000
Call option > 0
In both cases , your capital is guaranteed.
6.
a) If the portfolio manager expects rates to climb, he’d rather lower
his duration and therefore swap his KO 5.75 2011 bonds for the
IBM 2.25 2007 bonds.
b) $DKO = (MDIBM x MVIBM)/100 .
MDIBM= 2.95 $DKo= 5.68% x 5000 x 1076 = $305,580
MVIBM= $DKO/ (MDIBM/100) = $10,358,000 to be invested in IBM
IBM trades at $980, so he should buy 10,358,000/980 =10,570
bonds $10.57 million face value
7.
a) The value of the assets, portfolio of bonds is $1200 million
as
Econ. Surplus = Market value assets - PV of liabilities
b) Duration of liabilities is 5 Duration of assets is 5 x 1.2 = 6
If rates decrease by 50BP, assets will increase by 1200x 0.05 x 6 = 400
Liabilities will increase by 500 x 0.05 x 5 = 125
The new economic surplus will increase by : $375 million
8. Callable bond = non callable band – call option. Therefore the value of the call is 2.
9. 300 000 000 x 0.5 x 0.0028 = 420 000 is paid every 6 month form the buyer to the
seller. If default occurs after 3 years and 3 months , the seller would receive a total of :
(420 000 x 6) + (420 000/2) = 2,730,000
10. T T T T
11. You need to first find the spreads over swap for 3 years and 4 years for the Citigroup
bond . The 3 and 4 year swap are respectively at 3% and 3.45%. The bond yields
6.05%. So the spread is 3% for the 3 year (6% -3%) and 2.6% for the 4 year (6.05% 3.45%) .
Q(3) = 1- [(1+ 0.03)3 / (1+0.0602)3)] / (1-0.4) = 0.1384%
Q(4) = 1- [(1+ 0.0345)4 / (1+0.0602)4)] / (1-0.4) = 0.1561
Q(4) - Q(3) = 0.1561% - 0.1384% = 0.017%
12. Receives Coupon payment : 5.75/2 x $10,000,000 = $287,500
Pays Libor payment : 4% + 50BP = 4.5% that is 2.25% for six months that is
($225,000)
Capital loss : 100 – 98 = 2 0.02 x 10,000,00 = ($200,000)
Net cash flow : ($137,500)