Assignment: Fixed Income Securities
QUESTION 1
Why is the duration of a floating rate coupon zero at the reset date?
QUESTION 2
When interest rates go up, duration-based calculation shows that the value of the bond will go down and vice-versa. Why is the convexity adjustment always a positive amount regardless of the direction of the interest rate change?
QUESTION 3
When a bond goes on special, the repo rate for borrowing against that bond goes below the General Collateral Rate (GCR) which applies to all other Treasury bonds. Why does that not lead to arbitrage opportunities?
QUESTION 4
Why does an inverted yield curve (long rates lower than short rates) not (for example) result in Z(today for 10 year maturity) < Z(today for 1 year maturity), that is Z(0,10) < Z(0,1)?
QUESTION 5
What is factor neutrality? How does it help beyond calculations based on duration and convexity alone?
QUESTION 6
If the yield curve did not change (interest rates in the economy did not change at all) and the supply and demand for your bond in the market did not change, would the price of the bond you own still change from one day to another? Why?
QUESTION 7
Use these discount rates to calculate equivalent: Use these discount rates to calculate equivalent:
continuously compounded annual spot interest rates, semiannually compounded annual spot interest rates.
Plot the resulting yield curves.
|
t
|
Z
|
|
0.25
|
0.9891
|
|
0.5
|
0.9798
|
|
0.75
|
0.9713
|
|
1
|
0.9633
|
|
1.25
|
0.9553
|
|
1.5
|
0.9473
|
|
1.75
|
0.9392
|
|
2
|
0.931
|
|
2.25
|
0.9227
|
|
2.5
|
0.9143
|
|
2.75
|
0.9059
|
|
3
|
0.8973
|
|
3.25
|
0.8888
|
|
3.5
|
0.8801
|
|
3.75
|
0.8714
|
|
4
|
0.8627
|
|
4.25
|
0.8538
|
|
4.5
|
0.845
|
|
4.75
|
0.8361
|
|
5
|
0.8272
|
QUESTION 8
Using the following yield curve, calculate the price of:
8.25 year coupon bond paying a semiannual coupon of 4.85% annually
4.5 year floating rate coupon bond paying a semiannual coupon with a spread of 75 basis points 0.75%
7.25 year floating rate coupon bond paying a semiannual coupon with a spread of 75 basis points 0.75%. The coupon determined at the last reset date was 4.50% annually.
|
t
|
Z
|
|
0.25
|
0.9891
|
|
0.5
|
0.9798
|
|
0.75
|
0.9713
|
|
1
|
0.9633
|
|
1.25
|
0.9553
|
|
1.5
|
0.9473
|
|
1.75
|
0.9392
|
|
2
|
0.931
|
|
2.25
|
0.9227
|
|
2.5
|
0.9143
|
|
2.75
|
0.9059
|
|
3
|
0.8973
|
|
3.25
|
0.8888
|
|
3.5
|
0.8801
|
|
3.75
|
0.8714
|
|
4
|
0.8627
|
|
4.25
|
0.8538
|
|
4.5
|
0.845
|
|
4.75
|
0.8361
|
|
5
|
0.8272
|
|
5.25
|
0.8182
|
|
5.5
|
0.8093
|
|
5.75
|
0.8003
|
|
6
|
0.7913
|
|
6.25
|
0.7823
|
|
6.5
|
0.7733
|
|
6.75
|
0.7643
|
|
7
|
0.7554
|
|
7.25
|
0.7465
|
|
7.5
|
0.7376
|
|
7.75
|
0.7287
|
|
8
|
0.7199
|
|
8.25
|
0.7111
|
|
8.5
|
0.7024
|
|
8.75
|
0.6938
|
|
9
|
0.6852
|
|
9.25
|
0.6767
|
|
9.5
|
0.6683
|
|
9.75
|
0.6599
|
|
10
|
0.6516
|
QUESTION 9
Using the following yield curve, calculate the Duration of:
8.25 year coupon bond paying a semiannual coupon of 4.85% annually
4.5 year floating rate coupon bond paying a semiannual coupon with a spread of 75 basis points 0.75%
7.25 year floating rate coupon bond paying a semiannual coupon with a spread of 75 basis points 0.75%. The coupon determined at the last reset date was 4.50% annually.
|
t
|
Z
|
|
0.25
|
0.9891
|
|
0.5
|
0.9798
|
|
0.75
|
0.9713
|
|
1
|
0.9633
|
|
1.25
|
0.9553
|
|
1.5
|
0.9473
|
|
1.75
|
0.9392
|
|
2
|
0.931
|
|
2.25
|
0.9227
|
|
2.5
|
0.9143
|
|
2.75
|
0.9059
|
|
3
|
0.8973
|
|
3.25
|
0.8888
|
|
3.5
|
0.8801
|
|
3.75
|
0.8714
|
|
4
|
0.8627
|
|
4.25
|
0.8538
|
|
4.5
|
0.845
|
|
4.75
|
0.8361
|
|
5
|
0.8272
|
|
5.25
|
0.8182
|
|
5.5
|
0.8093
|
|
5.75
|
0.8003
|
|
6
|
0.7913
|
|
6.25
|
0.7823
|
|
6.5
|
0.7733
|
|
6.75
|
0.7643
|
|
7
|
0.7554
|
|
7.25
|
0.7465
|
|
7.5
|
0.7376
|
|
7.75
|
0.7287
|
|
8
|
0.7199
|
|
8.25
|
0.7111
|
|
8.5
|
0.7024
|
|
8.75
|
0.6938
|
|
9
|
0.6852
|
|
9.25
|
0.6767
|
|
9.5
|
0.6683
|
|
9.75
|
0.6599
|
|
10
|
0.6516
|
QUESTION 10
What is the dollar duration of a portfolio composed of:
$90 million long position in 3.25 year coupon bond paying a quarterly coupon of 7.30% annually
$130 million short position in 9.5 year floating rate coupon bond paying a quarterly coupon. Use the following yield curve:
|
t
|
Z
|
|
0.25
|
0.9891
|
|
0.5
|
0.9798
|
|
0.75
|
0.9713
|
|
1
|
0.9633
|
|
1.25
|
0.9553
|
|
1.5
|
0.9473
|
|
1.75
|
0.9392
|
|
2
|
0.931
|
|
2.25
|
0.9227
|
|
2.5
|
0.9143
|
|
2.75
|
0.9059
|
|
3
|
0.8973
|
|
3.25
|
0.8888
|
|
3.5
|
0.8801
|
|
3.75
|
0.8714
|
|
4
|
0.8627
|
|
4.25
|
0.8538
|
|
4.5
|
0.845
|
|
4.75
|
0.8361
|
|
5
|
0.8272
|
|
5.25
|
0.8182
|
|
5.5
|
0.8093
|
|
5.75
|
0.8003
|
|
6
|
0.7913
|
|
6.25
|
0.7823
|
|
6.5
|
0.7733
|
|
6.75
|
0.7643
|
|
7
|
0.7554
|
|
7.25
|
0.7465
|
|
7.5
|
0.7376
|
|
7.75
|
0.7287
|
|
8
|
0.7199
|
|
8.25
|
0.7111
|
|
8.5
|
0.7024
|
|
8.75
|
0.6938
|
|
9
|
0.6852
|
|
9.25
|
0.6767
|
|
9.5
|
0.6683
|
|
9.75
|
0.6599
|
|
10
|
0.6516
|
QUESTION 11
What is the price value of one basis point of: 3.25 year coupon bond paying a quarterly coupon of 7.30% annually. Use the following yield curve.
|
t
|
Z
|
|
0.25
|
0.9891
|
|
0.5
|
0.9798
|
|
0.75
|
0.9713
|
|
1
|
0.9633
|
|
1.25
|
0.9553
|
|
1.5
|
0.9473
|
|
1.75
|
0.9392
|
|
2
|
0.931
|
|
2.25
|
0.9227
|
|
2.5
|
0.9143
|
|
2.75
|
0.9059
|
|
3
|
0.8973
|
|
3.25
|
0.8888
|
|
3.5
|
0.8801
|
|
3.75
|
0.8714
|
|
4
|
0.8627
|
|
4.25
|
0.8538
|
|
4.5
|
0.845
|
|
4.75
|
0.8361
|
|
5
|
0.8272
|
|
5.25
|
0.8182
|
|
5.5
|
0.8093
|
|
5.75
|
0.8003
|
|
6
|
0.7913
|
|
6.25
|
0.7823
|
|
6.5
|
0.7733
|
|
6.75
|
0.7643
|
|
7
|
0.7554
|
|
7.25
|
0.7465
|
|
7.5
|
0.7376
|
|
7.75
|
0.7287
|
|
8
|
0.7199
|
|
8.25
|
0.7111
|
|
8.5
|
0.7024
|
|
8.75
|
0.6938
|
|
9
|
0.6852
|
|
9.25
|
0.6767
|
|
9.5
|
0.6683
|
|
9.75
|
0.6599
|
|
10
|
0.6516
|
QUESTION 12
Calculate the convexity of:
8.25 year coupon bond paying a semiannual coupon of 4.85% annually
4.5 year floating rate coupon bond paying a semiannual coupon with a spread of 75 basis points
7.25 year floating rate coupon bond paying a semiannual coupon determined at the last reset date 4.50% annual with a spread of 75 basis points.
Use the following yield curve:
|
t
|
Z
|
|
0.25
|
0.9891
|
|
0.5
|
0.9798
|
|
0.75
|
0.9713
|
|
1
|
0.9633
|
|
1.25
|
0.9553
|
|
1.5
|
0.9473
|
|
1.75
|
0.9392
|
|
2
|
0.931
|
|
2.25
|
0.9227
|
|
2.5
|
0.9143
|
|
2.75
|
0.9059
|
|
3
|
0.8973
|
|
3.25
|
0.8888
|
|
3.5
|
0.8801
|
|
3.75
|
0.8714
|
|
4
|
0.8627
|
|
4.25
|
0.8538
|
|
4.5
|
0.845
|
|
4.75
|
0.8361
|
|
5
|
0.8272
|
|
5.25
|
0.8182
|
|
5.5
|
0.8093
|
|
5.75
|
0.8003
|
|
6
|
0.7913
|
|
6.25
|
0.7823
|
|
6.5
|
0.7733
|
|
6.75
|
0.7643
|
|
7
|
0.7554
|
|
7.25
|
0.7465
|
|
7.5
|
0.7376
|
|
7.75
|
0.7287
|
|
8
|
0.7199
|
|
8.25
|
0.7111
|
|
8.5
|
0.7024
|
|
8.75
|
0.6938
|
|
9
|
0.6852
|
|
9.25
|
0.6767
|
|
9.5
|
0.6683
|
|
9.75
|
0.6599
|
|
10
|
0.6516
|
QUESTION 13
What will the dollar change in the value of this portfolio according to the duration and convexity method of estimating price change:
$75 million long position in 6.5 year coupon bond paying a semiannual coupon of 9.60%
$120 million short position in 9.5 year floating rate coupon bond paying a quarterly coupon Use the following yield curve:
|
t
|
Z
|
|
0.25
|
0.9891
|
|
0.5
|
0.9798
|
|
0.75
|
0.9713
|
|
1
|
0.9633
|
|
1.25
|
0.9553
|
|
1.5
|
0.9473
|
|
1.75
|
0.9392
|
|
2
|
0.931
|
|
2.25
|
0.9227
|
|
2.5
|
0.9143
|
|
2.75
|
0.9059
|
|
3
|
0.8973
|
|
3.25
|
0.8888
|
|
3.5
|
0.8801
|
|
3.75
|
0.8714
|
|
4
|
0.8627
|
|
4.25
|
0.8538
|
|
4.5
|
0.845
|
|
4.75
|
0.8361
|
|
5
|
0.8272
|
|
5.25
|
0.8182
|
|
5.5
|
0.8093
|
|
5.75
|
0.8003
|
|
6
|
0.7913
|
|
6.25
|
0.7823
|
|
6.5
|
0.7733
|
|
6.75
|
0.7643
|
|
7
|
0.7554
|
|
7.25
|
0.7465
|
|
7.5
|
0.7376
|
|
7.75
|
0.7287
|
|
8
|
0.7199
|
|
8.25
|
0.7111
|
|
8.5
|
0.7024
|
|
8.75
|
0.6938
|
|
9
|
0.6852
|
|
9.25
|
0.6767
|
|
9.5
|
0.6683
|
|
9.75
|
0.6599
|
|
10
|
0.6516
|
QUESTION 14
What is the 95% one-month Expected Shortfall on a portfolio of: $90 million long position in 3.25 year coupon bond paying a quarterly coupon of 7.30% Monthly Mu(dr) = 0.00065% Monthly Sigma(dr) = 0.415300%
Note: you need to calculate Mu(p) and Sigma(p) yourself. The current yield curve is given by:
|
t
|
Z
|
|
0.25
|
0.9891
|
|
0.5
|
0.9798
|
|
0.75
|
0.9713
|
|
1
|
0.9633
|
|
1.25
|
0.9553
|
|
1.5
|
0.9473
|
|
1.75
|
0.9392
|
|
2
|
0.931
|
|
2.25
|
0.9227
|
|
2.5
|
0.9143
|
|
2.75
|
0.9059
|
|
3
|
0.8973
|
|
3.25
|
0.8888
|
|
3.5
|
0.8801
|
|
3.75
|
0.8714
|
|
4
|
0.8627
|
|
4.25
|
0.8538
|
|
4.5
|
0.845
|
|
4.75
|
0.8361
|
|
5
|
0.8272
|
|
5.25
|
0.8182
|
|
5.5
|
0.8093
|
|
5.75
|
0.8003
|
|
6
|
0.7913
|
|
6.25
|
0.7823
|
|
6.5
|
0.7733
|
|
6.75
|
0.7643
|
|
7
|
0.7554
|
|
7.25
|
0.7465
|
|
7.5
|
0.7376
|
|
7.75
|
0.7287
|
|
8
|
0.7199
|
|
8.25
|
0.7111
|
|
8.5
|
0.7024
|
|
8.75
|
0.6938
|
|
9
|
0.6852
|
|
9.25
|
0.6767
|
|
9.5
|
0.6683
|
|
9.75
|
0.6599
|
|
10
|
0.6516
|
QUESTION 15
You need to hedge an 8.25 year coupon bond paying a semiannual coupon of 4.85% annual with 3.25 year coupon bond paying a quarterly coupon of 7.30% How many 3.25-year would you need for this hedge according to duration hedging? The current yield curve is given by:
|
t
|
Z
|
|
0.25
|
0.9891
|
|
0.5
|
0.9798
|
|
0.75
|
0.9713
|
|
1
|
0.9633
|
|
1.25
|
0.9553
|
|
1.5
|
0.9473
|
|
1.75
|
0.9392
|
|
2
|
0.931
|
|
2.25
|
0.9227
|
|
2.5
|
0.9143
|
|
2.75
|
0.9059
|
|
3
|
0.8973
|
|
3.25
|
0.8888
|
|
3.5
|
0.8801
|
|
3.75
|
0.8714
|
|
4
|
0.8627
|
|
4.25
|
0.8538
|
|
4.5
|
0.845
|
|
4.75
|
0.8361
|
|
5
|
0.8272
|
|
5.25
|
0.8182
|
|
5.5
|
0.8093
|
|
5.75
|
0.8003
|
|
6
|
0.7913
|
|
6.25
|
0.7823
|
|
6.5
|
0.7733
|
|
6.75
|
0.7643
|
|
7
|
0.7554
|
|
7.25
|
0.7465
|
|
7.5
|
0.7376
|
|
7.75
|
0.7287
|
|
8
|
0.7199
|
|
8.25
|
0.7111
|
|
8.5
|
0.7024
|
|
8.75
|
0.6938
|
|
9
|
0.6852
|
|
9.25
|
0.6767
|
|
9.5
|
0.6683
|
|
9.75
|
0.6599
|
|
10
|
0.6516
|
Format your assignment according to the following formatting requirements:
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