Consider a self-financed convexity trade. Three zero couple bonds: i. 2Y zero at 1.60%; ii. 10Y...

Consider a self-financed convexity trade.

Three zero couple bonds:

i. 2Y zero at 1.60%; ii. 10Y zero at 1.85%; iii. 30Y zero at 2.30%

a. If you want to combine 2Y and 30Y zero to match the $100M bullet in 10Y zero for dollar duration, what is percentage weights in 2Y and 30Y respectively? (note: combined value in 2Y and 30Y is also $100M, ie weights sum up to 100%)

Expert Solution


	Duration of a Zero Coupon Bond is equals to its maturity.
	Also duration of portfolio bond is equal to weighted average
	duration of all bonds in the portfolio.

	Duration of 2 year Zero Coupon Bond = 2 Years
	Duration of 10 year Zero Coupon Bond = 10 Years
	Duration of 30 year Zero Coupon Bond = 30 Years

	Let the weight of 2 Year Zero Coupon Bond be "x"
	then weight of 30 Year bond will become "1-x".

	This should be matched with duration of 10 year zero
	coupon bond.
	So,

	Duration of 10 Year Zero Coupon Bond
	= Duration of 2 Year Zero Coupon Bond * Weight of 2 Year Bond
	+ Duration of 30 Year Zero Coupon Bond * Weight of 30 Year Bond
	10 = 2 * x + 30 * (1-x)
	10 = 2 * x + 30 - 30*x
	10 = 30 - 28*x
	28 * x = 30-10
	28 * x = 20
	x = 0.7143

	Weight of 2 Year Zero Coupon Bond = x = 0.7143 = 71.43%
	Weight of 30 Year Zero Coupon Bond = 1-x = 1-0.7143 = 0.2857 = 28.57%

jeff jeffy answered 2 years ago

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