You have the following information on two portfolios. Portfolio A consists of a

Question

You have the following information on two portfolios.

Portfolio A consists of a 1000 par-value 4-year bond with 7% annual coupons and a 5-year zero-coupon bond with a par-value of X. Both bonds redeem at par.

Portfolio B consists of a single 4-year zero-coupon bond with maturity value of 10000. All bonds yield an annual effective rate of 7%. The portfolios both have the same volatility (i.e. modified duration).

Find the X. Give your answer rounded to the nearest whole number.

NOTE: THE CURRENT ANSWERS ON CHEGG ARE WRONG and PLEASE SHOW YOUR WORK

Explanation / Answer

Macaulay Duration of Zero coupon bond is equal to maturity

Modified duration of a zero coupon bond=Macaulay Duration/(1+ytm)

Modified duration of 4 year Zero coupon bond=4/1.07=3.738318

Modified duration of 5 year zero coupon bond=5/1.07=4.672897

Price of 4 year zero coupon bond=X/1.07^5

Price of coupon bond will be par value that is 1000 as yield is equal to coupon

Modified duration of coupon bond=((1*70/1.07+2*70/1.07^2+3*70/1.07^3+4*1070/1.07^4)/1000)/(1+7%)=3.387211

Portfolio duration will be=(1000*3.387211+X/1.07^5*4.672897)/(1000+X/1.07^5)

This should equal 3.738318

Hence, (1000*3.387211+X/1.07^5*4.672897)/(1000+X/1.07^5)=3.738318

=>X=526.61

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