Question

3/26

1 mo 0.01

6 mo 0.04

1 yr 0.13

2 yr 0.3

3 yr 0.36

5 yr 0.51

3/27

1 mo 0.01

6 mo 0.02

1 yr 0.11

2 yr 0.25

3 yr 0.3

5 yr 0.41

1. It is March 26 and you hold a $1mm (market value) long position in the 1-yr zero-coupon bond. Using modied durations, determine how much of the 5-yr zero-coupon bond you need to short so that your portfolio remains approximately unchanged if the 1-yr and 5-yr zero rates move in parallel.

2. What would the change in your portfolio value be if both the 1-yr rate and the 5-yr rate go up by 2 basis points?

3. What would the change in your portfolio value be if both the 1-yr rate and the 5-yr rate go down by 2 basis points?

4. What would the change in your portfolio value be if the 1-yr rate goes down by 2 basis points but the 5-yr rate stays the same?

5. What actually happens the following day? Calculate the value of your portfolio. Why did your portfolio value change from before?

Answer 1

1.

Macaulay Duration of the ZCB is the duration of the Bond.

Modified Duration is Macaulay Duration / (1+r).

Therefore, Modified Duration for each bond is:

1 Year ZCB: 1/ (1+.13) = 0.8850

5 Year ZCB: 1/ (1+.51) = 3.3113

Investor Have $1mm (market value) long position in the 1-yr zero-coupon bond. Therefore, 5-yr zero-coupon bond you need to short are:

= (Market Value of 1 Year ZCB * Modified Duration of 1Year ZCB) / Modified Duration of 5 Year ZCB

= (1mm*0.8850) / 3.3112 = 0.2673 mm (Market Value of Bond)

2.

Notional Value of each ZCB is:

1 Year ZCB: 1 * (1+0.13) = 1.13

5 Year ZCB: 0.267257 * (1+.51) = 0.4036

Changes in Bond Market Value due to Changes in Rate by 2BP

New 1 Year ZCB Rate: 0.13 + 0.0002 = 0.1302

New 5 Year ZCB Rate: 0.51 + 0.0002 = 0.5102

Market Value 1 Year ZCB: 1.13 / (1+ 0.1302) = 0.9998

Market Value 5 Year ZCB: 0.403558 / (1+ 0.5102) = 0.2672

Loss 1 Year ZCB (Long Position): 1 - 0.999823 = 0.0002mm

Profit on 5 Year ZCB (Short Position): 0.2673 – 0.2672 = 0.0001

Net Loss: 0.0001 (Almost 0, Difference due to Rounding Off

3.

Notional Value of each ZCB is:

1 Year ZCB: 1 * (1+0.13) = 1.13

5 Year ZCB: 0.267257 * (1+.51) = 0.4036

Changes in Bond Market Value due to Changes in Rate by 2BP

New 1 Year ZCB Rate: 0.13 - 0.0002 = 0.1298

New 5 Year ZCB Rate: 0.51 - 0.0002 = 0.5098

Market Value 1 Year ZCB: 1.13 / (1+ 0.1302) = 0.9998

Market Value 5 Year ZCB: 0.403558 / (1+ 0.5102) = 0.2672

Profit on 1 Year ZCB (Long Position): 1 – 1.0002 = 0.0002mm

Loss on 5 Year ZCB (Short Position): 0.2673 – 0.2672 = 0.0001

Net Loss: 0.0001 (Almost 0, Difference due to Rounding Off)

4. Notional Value of each ZCB is:

1 Year ZCB: 1 * (1+0.13) = 1.13

5 Year ZCB: 0.267257 * (1+.51) = 0.4036

Changes in Bond Market Value due to Changes in Rate by 2BP

New 1 Year ZCB Rate: 0.13 - 0.0002 = 0.1298

New 5 Year ZCB Rate: Remain Same

Market Value 1 Year ZCB: 1.13 / (1+ 0.1302) = 0.9998

Market Value 5 Year ZCB: 0.403558 / (1+ 0.51) = 0.2672

Profit 1 Year ZCB (Long Position): 1 - 0.999823 = 0.0002mm

No Changes in 5 Year ZCB (Short Position)

Net Profit: 0.0002

5.

Notional Value of each ZCB is:

1 Year ZCB: 1 * (1+0.13) = 1.13

5 Year ZCB: 0.267257 * (1+.51) = 0.4036

Changes in Bond Market Value due to Changes in Rate by 2BP

New 1 Year ZCB Rate: 0.11

New 5 Year ZCB Rate: 0.41

Market Value 1 Year ZCB: 1.13 / (1+ 0.11) = 1.0180

Market Value 5 Year ZCB: 0.403558 / (1+ 0.41) = 0.2862

Profit on 1 Year ZCB (Long Position): 1 – 1.0180 = 0.0180mm

Loss on 5 Year ZCB (Short Position): 0.2673 – 02862 = 0.0190

Net Loss: 0.001

There are loss due to Convexity, Modified Duration do well in short span of Time.

Note:

1. Language in Question is Very Vague.

2. Some Figure is missing or Difficult to Interperet

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