It is August 29, 2003 and you hold a $1mm (market value) long position in the 1-

Question

It is August 29, 2003 and you hold a $1mm (market value) long position in the 1-yr zero-coupon bond. Using modified 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. This is known as a steepener trade – you profit if the yield curve becomes steeper.

What actually happens during the following month? Calculate the value of your portfo- lio. A 1-yr bond is now an 11-month bond and a 5-yr bond becomes a 4-year 11-month bond. Assume for pricing purposes that the 5-yr rate applies to the 4-yr 11-month bond and the 1-yr rate applies to an 11-month bond when calculating bond prices. Why did your portfolio value change from before?

What would the change in your portfolio value have been if the 1-yr rate had stayed the same and the 5-yr rate had gone up by 0.25 percentage points?

Explanation / Answer

Part(i) of the answer

On August 29,2003

YTM of 1 year Zero coupon bond r = 1.283

Assuming annual compounding for the purpose of this question

The Mecaulay Duration of the Zero Coupon Bond is equal to its time to maturity.

Hence Mecaulay Duration of 1 year Zero Coupon Bond = 1

Modified Duration D can be calculated using the formula

D = Mecaulay Duration / (1+r/k)

where k is equal to the frequeny of compounding which is equal to 2

Therefore Modified Duration of 1 Year Zero Coupon Bond D(1) = 1/(1+0.01283)

D(1) = 1/(1.01283) = 0.9873 (approx)

That is for a 1% or 100 bps change in YTM the price of the bond decreases by 0.9873%

Similarly for a 5 year Zero Coupon Bond

Yield to Maturity r = 3.511%

Mecaulay Duration = 5

Modified Duration D(5) = 5/(1+0.03511) = 5/(1.03511) = 4.8304 (approx)

That is for a 100 bps change in Yield will reduce the price of the bond by 4.8304%

Yield difference between 1 year and 5 year bonds = 3.511 - 1.283 = 2.228%

For a change of 2.228% the change in the price of 5 year zero coupon bond will change by 4.8304*2.228 = 10.7621%

If the trader is holding a long position of $ 1 Million of 1 year Zero Coupon bonds then he should short 1 Million *(1+10.7621%) = 1,107,621 of 1,108,000 worth (market value) of 5 year zero coupond bonds so that the portfolio approximately remain unchanged.

Assuming a face value of 100

Market Price of a 1 year bond = 100/1.01283 = 98.73
Market Price of a 5 year bond = 100/(1+0.3511)^5 = 100/1.18832 = 84.15

Part(ii) of the answer

On 29 September 2003

YTM for 1 year Zero coupon bond = 1.283% (same as earlier)
Time to Maturity = 11 months or 0.92 years

Market price M(1) = 100/(1.01283)^0.92 = 100/1.0260068 = 97.47

Change in Market value over one month = 97.47 - 98.73 = - $ 1.26 or -1.2762%

YTM for 5 year Zero coupon bond = 3.511%

Time to maturity = 4 years 11 months or 4.92 years

Market Price M(5) = 100/(1.03511)^4.92 = 100/1.18504 = 84.39

Change in market value over one month = 84.39 - 84.15 = 0.24 or 0.2852%

Change of the value of the portfolio

Long Position = 1,000,000 * (100%-1.2762%) = 987, 238

Short Position = 1,108,000 * (100%+0.2852%) = 1,111,160

Part(iii) of the answer

Market price of 1 year bond as on 29th September 2003 = 97.47
Change in Market value over one month = 97.47 - 98.73 = - $ 1.26 or -1.2762%

Change in value of long position = 1,000,000 * (100%-1.2762%) = 987, 238

YTM of 5 year coupon bond = 3.511% + 0.25% = 3.761% (due to an increase in rate by 0.25%)

Time to maturity = 4 years 11 months or 4.92 years
Market price of 5 year bond = 100/(1.03761)^4.92 = 100/1.19919 = 83.38

Change in market price over one month = 83.38 - 84.15 = - 0.77 or - 0.9150%

Change in value of short position = 1,108,000 * (100%-0.9150%) = 1,097,862

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