2006 Correction Fixed Income Midterm
Correction Fixed Income Midterm 2006
1. The adequate FRA to use would be the June 2007 which is tells you what the markets
expects the 3-month rate to be 15 months from now. The future june 2007 eurodollar
contract trades at 94.885 which means the market expects the 3 month rate next June
to be at 5.115% (100 – 94.885). We need to sell the FRA to hedge our mismatch.
The initial 1MM deposited will grow to 1MM (1+5.20%)15/12 = $1,065,000
After 15 months, if the FRA trades at 6.09/6.20, we can settle at 6.19%. The loss on
the FRA position is therefore:
6.19 - 5.115 = 1.075%
The amount dealt was 1,065,000 sop the loss amounts to :
1,065,000 (0.01075)/(4)=$2,691.
The amount roll over for the last 3 month is therefore : 1,065,000 – 2,691 =
$1,062,300 invested for 3 month @6.09% = $1,078,100
Amount of the loan: 1MM (1 + 5.50%)18/12= $1,083,624
Total loss = 1,078,100 - 1,083,624 = -5,500
2. If R1,L= 5.70% then R1,H= 5.70 e2 x 0.10= 6.96%
V2,L = (103.25 + 8)/(1+0.057) = 105.25
V2,R = (103.25 + 8)/(1+0.0696) = 104.01
V’1 = (105.25 + 8)/(1+5.30%) = 107.54
V1 = (104.01 + 8)/(1+5.30%) = 106.37
(V’1 + V1 ) / 2= 106.95
The value of the option free bond is :
8/(1+0.053)1 + 8/(1+0.054)2 + 108/(1+0.055)3= 106.77 (close enough to the
value found above).
If the bond is callable at 101, V2,L becomes 101 and V2,R becomes 101 as well.
Then
V2,L = (101 + 8)/(1+0.057) = 103.12
V2,R = (101 + 8)/(1+0.0696) = 101.90
V’1 = (101 + 8)/(1+5.30%) = 103.51
V1 = (101 + 8)/(1+5.30%) = 103.51
(V’1 + V1) / 2= 103.51
The option price is therefore 106.77 - 103.51 =3.25 or $32.5
3. (7 + (100 – 101)/20 ) / (101/100) = 6.88%
To determine the dirty price, one must calculate the accrued interest:
Number of days since last coupon (it’s a corp. bond so 30/360): 142 days
Accrued interest = (142/360) x 7 = 2.76 or $27.60
Dirty price is therefore 1010 + 27.60 = $1037.60
120 bonds would cost me 120 x 1037.60 = $124,512
4. Price of the bond = 5/(1+5.5)1 + 5/(1+5.5)2 + 5/(1+5.5)3 + 5/(1+5.5)4 + 105/(1+5.5)5=
97.86
Weighted price :
5/(1+5.5)1 + 5x2/(1+5.5)2 + 5x3/(1+5.5)3 + 5x4/(1+5.5)4 + 105x5/(1+5.5)5 = 444.33
Modified duration = (444.33/97.86) /( 1+0.055)= 4.30
If the bond has a convexity of 120, a 50BP increase in yield will lower the price by :
-(4.30 x 0.5) + (1/2 x 120 x (0.005)2 = -2%
The new bond price is therefore = 95.90
5. The fed requires 20 billion so there is $50 billion left. The highest bidder would be the
bidder with the lowest rate.
GS gets its 15 billion
JPM gets its 20 billion
MS gets 15 of the requested 22 billion
ML gets nothing.
Fed’s price is the weighted average of the bidders that is n:
(15 x 4.63 + 20 x 4.65 + 15 x 4.66 ) / 50 = 4.647%
6.
Portfolio A with 78% of the 5-year note and 22% of the 10-year note, you get : (0.78 x
4.34) + (0.22 x 7.38) = 5.00
Portfolio B with 28% of the 30-year note and 72% of the 2-year bill, you get : (0.27 x
13.29) + (0.73 x 1.91) = 5.00
You would chose the portfolio with the highest convexity which is Portfolio B with a
convexity of 74. (0.27 x 264.667) + (0.73 x 4.426)
1 x 12.84 + z x 1.85 = 0
z=-6.94
For each 30 year bond you buy, you must sell 6.94 2-year bond or 69.2 million
face value of the 2-year bond. 69.2 million face means 69.2M/1001.30 = 69 110
bonds.
Convexity of the portfolio = 264.667 x 1 - 6.94 x 4.426 = 233.498
7. 3.5
1. The adequate FRA to use would be the June 2007 which is tells you what the markets
expects the 3-month rate to be 15 months from now. The future june 2007 eurodollar
contract trades at 94.885 which means the market expects the 3 month rate next June
to be at 5.115% (100 – 94.885). We need to sell the FRA to hedge our mismatch.
The initial 1MM deposited will grow to 1MM (1+5.20%)15/12 = $1,065,000
After 15 months, if the FRA trades at 6.09/6.20, we can settle at 6.19%. The loss on
the FRA position is therefore:
6.19 - 5.115 = 1.075%
The amount dealt was 1,065,000 sop the loss amounts to :
1,065,000 (0.01075)/(4)=$2,691.
The amount roll over for the last 3 month is therefore : 1,065,000 – 2,691 =
$1,062,300 invested for 3 month @6.09% = $1,078,100
Amount of the loan: 1MM (1 + 5.50%)18/12= $1,083,624
Total loss = 1,078,100 - 1,083,624 = -5,500
2. If R1,L= 5.70% then R1,H= 5.70 e2 x 0.10= 6.96%
V2,L = (103.25 + 8)/(1+0.057) = 105.25
V2,R = (103.25 + 8)/(1+0.0696) = 104.01
V’1 = (105.25 + 8)/(1+5.30%) = 107.54
V1 = (104.01 + 8)/(1+5.30%) = 106.37
(V’1 + V1 ) / 2= 106.95
The value of the option free bond is :
8/(1+0.053)1 + 8/(1+0.054)2 + 108/(1+0.055)3= 106.77 (close enough to the
value found above).
If the bond is callable at 101, V2,L becomes 101 and V2,R becomes 101 as well.
Then
V2,L = (101 + 8)/(1+0.057) = 103.12
V2,R = (101 + 8)/(1+0.0696) = 101.90
V’1 = (101 + 8)/(1+5.30%) = 103.51
V1 = (101 + 8)/(1+5.30%) = 103.51
(V’1 + V1) / 2= 103.51
The option price is therefore 106.77 - 103.51 =3.25 or $32.5
3. (7 + (100 – 101)/20 ) / (101/100) = 6.88%
To determine the dirty price, one must calculate the accrued interest:
Number of days since last coupon (it’s a corp. bond so 30/360): 142 days
Accrued interest = (142/360) x 7 = 2.76 or $27.60
Dirty price is therefore 1010 + 27.60 = $1037.60
120 bonds would cost me 120 x 1037.60 = $124,512
4. Price of the bond = 5/(1+5.5)1 + 5/(1+5.5)2 + 5/(1+5.5)3 + 5/(1+5.5)4 + 105/(1+5.5)5=
97.86
Weighted price :
5/(1+5.5)1 + 5x2/(1+5.5)2 + 5x3/(1+5.5)3 + 5x4/(1+5.5)4 + 105x5/(1+5.5)5 = 444.33
Modified duration = (444.33/97.86) /( 1+0.055)= 4.30
If the bond has a convexity of 120, a 50BP increase in yield will lower the price by :
-(4.30 x 0.5) + (1/2 x 120 x (0.005)2 = -2%
The new bond price is therefore = 95.90
5. The fed requires 20 billion so there is $50 billion left. The highest bidder would be the
bidder with the lowest rate.
GS gets its 15 billion
JPM gets its 20 billion
MS gets 15 of the requested 22 billion
ML gets nothing.
Fed’s price is the weighted average of the bidders that is n:
(15 x 4.63 + 20 x 4.65 + 15 x 4.66 ) / 50 = 4.647%
6.
Portfolio A with 78% of the 5-year note and 22% of the 10-year note, you get : (0.78 x
4.34) + (0.22 x 7.38) = 5.00
Portfolio B with 28% of the 30-year note and 72% of the 2-year bill, you get : (0.27 x
13.29) + (0.73 x 1.91) = 5.00
You would chose the portfolio with the highest convexity which is Portfolio B with a
convexity of 74. (0.27 x 264.667) + (0.73 x 4.426)
1 x 12.84 + z x 1.85 = 0
z=-6.94
For each 30 year bond you buy, you must sell 6.94 2-year bond or 69.2 million
face value of the 2-year bond. 69.2 million face means 69.2M/1001.30 = 69 110
bonds.
Convexity of the portfolio = 264.667 x 1 - 6.94 x 4.426 = 233.498
7. 3.5