Solution :-
Total Portfolio Amount = $1 million
Target Duration = 23 years
ZCB ( Zero Coupon Bond ) Maturity = 5 yrs
Yield of ZCB = 2%
Yield of Perpetuity = 2%
As we know Duration of ZCB is always equal to yield to maturity So Duration of ZCB = 5 yrs
and Duration of perpetuity Bond = (1 + y) / y where y is yield
= (1 + 0.02 ) / 0.02 = 51 years
let Share of ZCB be x and therefore perpetuity = (1 - x )
now x * 5 + ( 1 - x )*51 = 23
= 5x + 51 - 51x = 23
= 28 = 46x
x = 28 / 46 = 14 / 23
Bond = $1000000 * 14 / 23 = $608695.7 = 60.87%
now Therefore Perpetual bond = $1000000 - $608695.7 = $391304.3 = 39.13%
(b)
Now after 1 year
Duration of ZCB = 4 years as Time to maturity is 4 years now
But duration of Perpetual bond is same = 51 years ( As no change in its yield )
Now target Duration = 22 years
Now
x * 4 + ( 1 - x )*51 = 22
= 4x + 51 - 51x = 22
= 29 = 47x
x = 29 / 47
therefore share of ZCB in $1000000 = $1000000 * 29 / 47 = $617021.3 = 61.7%
now share of Perpetual Bond = $1000000 - $617021.3 = $382978.7 = 38.3%
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