The duration of the bond is the component of the time at which the bond gives the interest payments intermittently. In case of zero coupon bonds they are issued at discount and no such payments are made in between the maturity of the bond. This is why the duration is reset at zero in the beginning.

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Bond prices change inversely with interest rates so there is interest rate risk with bonds. The magnitude of bonds should always be positive, regardless of the direction of change and duration is a tangent line to the price-yield curve at the calculated point, and the difference between the duration tangent line and the price-yield curve increases as the yield moves farther away in either direction from the point of tangency.

Q U E S T I O N 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?

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collateral.

Arbitrage opportunities exist when the price of similar assets are set at different level. This opportunity allows an investor to achieve a profit with zero risk and limited funds by simply selling the assets in the overpriced market and simultaneously buying it in the cheaper market.

If there is no arbitrage opportunity, one reason is the individual investors may have the different view on how, why and to what degree market prices are off. Also, Investors are reluctant to believe that there are no arbitrage opportunity and so they append a good deal of time watching price movements, fermenting out

Q U E S T I O N 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?

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Q U E S T I O N 4 15 points SavedSaved

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Because of lower duration, Z (today for 10 year maturity) < Z (today for 1 year maturity), that is Z (0,10) < Z (0,1).

Interest rates and bond yields are highly correlated, and sometimes the terms are used interchangeably. An interest rate might be thought of as the rate at which money can be borrowed in the form of a loan and, while most bonds have an interest rate that determines their coupon payments, the true cost of borrowing or investing in bonds is determined by their current yields.

A bond's yield is simply the discount rate that can be used to make the present value of all of a bond's cash flows equal to its price. A bond's price is the sum of the present value of

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)?

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Factor neutrality refers to making the portfolio immune to the changes in the factor. The portfolio value depends on the a set of factors F1 and F2,...Fn then the portfolio value changes when any of the factor Fi value changes, the change in the value of the portfolio with a change in any factor Fi value(other factors constant) is called Factor Duration. When the portfolio is constituted such that the portfolio value do not change when any of the factors Fi changes we say that the portfolio has become neutral to the factor Fi this is referred to as factor neutrality. Factor neutrality makes the portfolio immune to the changes in the factor values that are the portfolio value remains the same whatever value the factor assumes.

Q U E S T I O N 5

What is factor neutrality? How does it help beyond calculations based on duration and convexity alone?

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Yes, because of bond rating. If the credit rating is lowered, the bond price declines. If the rating is upgraded, the price goes up. The change in price corresponds to the amount necessary to bring the price of a bond in line with other bonds rated at the same level.

Q U E S T I O N 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?

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Q U E S T I O N 7

con$nuously compounded annual spot interest rates, semiannually compounded annual spot interest rates.

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

Use these discount rates to calculate equivalent: Use these discount rates to calculate equivalent:

Plot the resul$ng yield curves.

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Q U E S T I O N 8

8.25 year coupon bond paying a semiannual coupon of 4.85% annually 4.5 year floa$ng rate coupon bond paying a semiannual coupon with a spread of 75 basis points 0.75% 7.25 year floa$ng rate coupon bond paying a semiannual coupon with a spread of 75 basis

Using the following yield curve, calculate the price of:

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points 0.75% . Thr 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

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Q U E S T I O N 9

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

Using the following yield curve, calculate the Dura%on of:

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100 4.85% t Z b.CF b.DCF w w*t T

0.25 0.9891 2.425 2.3986 2.26% 0.005644425 0.25 0.5 0.9798 0.0000 0.00% 0 0.5

0.75 0.9713 2.425 2.3554 2.22% 0.016628542 0.75 1 0.9633 0.0000 0.00% 0

1.25 0.9553 2.425 2.3166 2.18% 0.027257707 1.25 1.5 0.9473 0.0000 0.00% 0

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

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Q U E S T I O N 1 0

$90 million long posi$on in 3.25 year coupon bond paying a quarterly coupon of 7.30% annually $130 million short posi$on in 9.5 year floa$ng rate coupon bond paying a quarterly coupon.

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

What is the dollar dura$on of a porNolio composed of:

Use the following yield curve:

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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

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Q U E S T I O N 1 1

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

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.

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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

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Q U E S T I O N 1 2

8.25 year coupon bond paying a semiannual coupon of 4.85% annually 4.5 year floa$ng rate coupon bond paying a semiannual coupon with a spread of 75 basis points

Calculate the convexity of:

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7.25 year floa$ng rate coupon bond paying a semiannual coupon coupon determined at the last reset date 4.50% annual with a spread of 75 basis points.

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

Use the following yield curve:

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t Z 8.25yr @ 4.85% annual 4.5yr float 75bps semi 0.25 0.9891 CF DCF w*(t,T) w*(t,T)2 CF

0.5 0.9798 2.425 2.40 0.01 0.00 0.75 0.9713 0 0.00 0.00 0.00 0.375 0.367425

1 0.9633 2.425 2.36 0.02 0.01 1.25 0.9553 0 0.00 0.00 0.00 0.375 0.3612375

1.5 0.9473 2.425 2.32 0.03 0.03 1.75 0.9392 0 0.00 0.00 0.00 0.375 0.3552375

Q U E S T I O N 1 3

$75 million long posi$on in 6.5 year coupon bond paying a semiannual coupon of 9.60% $120 million short posi$on in 9.5 year floa$ng rate coupon bond paying a quarterly coupon

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

What will the dollar change in the value of this porNolio according to the dura%on and convexity method of es%ma%ng price change:

Use the following yield curve:

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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

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Q U E S T I O N 1 4

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

What is the 95% one-month Expected Shor:all on a porNolio of: $90 million long posi$on 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:

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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

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Q U E S T I O N 1 5

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

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 dura$on hedging? The current yield curve is given by:

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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

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