2005 FIXED INCOME FINAL CORRECTION
FIXED INCOME
FINAL EXAM CORRECTION
APRIL 2005
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
A
B
C
D
total
MV
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 (-17% + 1.73%) that is –
15.27% using duration and convexity. The portfolio will then be worth : 155 x (1 –
15.27%) = $131.3 million
8.548 x 2 = 17%
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. Both IBM bonds have the same coupon, maturity and rating. Their price
(yield) differs as one must be of different seniority compared to the other. The
higher yielding bond must be an unsecured bond as opposed to a secured bond
or a senior bond.
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 investor expects rates to go increase he would rather lower his
duration and swap his Coca Cola 5.75 2011 bonds for the Citigroup 5
2007 as it is shorter in maturity.
b) $DKO = (MDC x MVC)/100.
You need to calculate the modified duration of the Citigroup bond:
0.5* (5/(1+0.0310)0.5) + 1*(5/(1+0.0385)1) + 1.5*(5/(1+0.0405)1.5)+
2*(105/(1+0.043)2 )= 2.03 (Mac Caulley) ModC = 2.03/(1+0.0405)=1.95
MDC= 1.95
$DKo= 5.026% x 4000 x 1050 = $211,092
MVC= $DKO/ (MDC/100) = $10,825,000 to be invested in
Citigroup
Citigroup trades at $1017, so he should buy 10,825,000/1017 =10,665 bonds
$10.66 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.) 300 000 000 x 0.5 x 0.0022 = 330 000 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 :
(330 000 x 6) + (330 000/2) = 2,145,000
9) T T T T
10) P (default) of IBM5.676% after 5 years= [1- (1+5.015%)5/(1+5.676%)5] = 3.08%
P (default) of IBM5.676% after 6 years= [1- (1+5.015%)6/(1+5.676%)6] = 3.69%
P(default) between year 5 and year 6 = 3.69 – 3.08 = 0.61%
P (default) of IBM5.536% after 5 years= [1- (1+5.015%)5/(1+5.536%)5] = 2.44%
P (default) of IBM5.536% after 6 years= [1- (1+5.015%)6/(1+5.536%)6] = 2.92%
P(default) between year 5 and year 6 = 3.69 – 3.08 = 0.48%
The calculations show that the higher yielding bond has a higher probability of default
which confirms our answer on question 4 above that the higher yielding bond must be the
“riskier” one as it is an unsecured bond compared to the lower yielding bond which must be
secured.
11) The Citigroup bond yields around 4.10% and the libor rate is 3.38% so an asset swap
using this bond would require a spread of 4.30 – 3.38 = 92bp
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 2005
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
A
B
C
D
total
MV
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 (-17% + 1.73%) that is –
15.27% using duration and convexity. The portfolio will then be worth : 155 x (1 –
15.27%) = $131.3 million
8.548 x 2 = 17%
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. Both IBM bonds have the same coupon, maturity and rating. Their price
(yield) differs as one must be of different seniority compared to the other. The
higher yielding bond must be an unsecured bond as opposed to a secured bond
or a senior bond.
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 investor expects rates to go increase he would rather lower his
duration and swap his Coca Cola 5.75 2011 bonds for the Citigroup 5
2007 as it is shorter in maturity.
b) $DKO = (MDC x MVC)/100.
You need to calculate the modified duration of the Citigroup bond:
0.5* (5/(1+0.0310)0.5) + 1*(5/(1+0.0385)1) + 1.5*(5/(1+0.0405)1.5)+
2*(105/(1+0.043)2 )= 2.03 (Mac Caulley) ModC = 2.03/(1+0.0405)=1.95
MDC= 1.95
$DKo= 5.026% x 4000 x 1050 = $211,092
MVC= $DKO/ (MDC/100) = $10,825,000 to be invested in
Citigroup
Citigroup trades at $1017, so he should buy 10,825,000/1017 =10,665 bonds
$10.66 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.) 300 000 000 x 0.5 x 0.0022 = 330 000 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 :
(330 000 x 6) + (330 000/2) = 2,145,000
9) T T T T
10) P (default) of IBM5.676% after 5 years= [1- (1+5.015%)5/(1+5.676%)5] = 3.08%
P (default) of IBM5.676% after 6 years= [1- (1+5.015%)6/(1+5.676%)6] = 3.69%
P(default) between year 5 and year 6 = 3.69 – 3.08 = 0.61%
P (default) of IBM5.536% after 5 years= [1- (1+5.015%)5/(1+5.536%)5] = 2.44%
P (default) of IBM5.536% after 6 years= [1- (1+5.015%)6/(1+5.536%)6] = 2.92%
P(default) between year 5 and year 6 = 3.69 – 3.08 = 0.48%
The calculations show that the higher yielding bond has a higher probability of default
which confirms our answer on question 4 above that the higher yielding bond must be the
“riskier” one as it is an unsecured bond compared to the lower yielding bond which must be
secured.
11) The Citigroup bond yields around 4.10% and the libor rate is 3.38% so an asset swap
using this bond would require a spread of 4.30 – 3.38 = 92bp
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)