Question

In: Finance

You are managing a portfolio of $2.1 million. Your target duration is 10 years, and you...

You are managing a portfolio of $2.1 million. Your target duration is 10 years, and you can choose from two bonds: a zero-coupon bond with maturity 5 years, and a perpetuity, each currently yielding 5%.

Required:
(a)

How much of each bond will you hold in your portfolio? (Round your answers to 4 decimal places.)

  Zero-coupon bond   
  Perpetuity bond   
(b)

How will these fractions change next year if target duration is now nine years? (Round your answers to 4 decimal places.)

  Zero-coupon bond   
  Perpetuity bond   

Solutions

Expert Solution

Given,
Value of Portfolio = $2100000
Target Duration = 10 Years
Maturity of Zero Coupon Bonds = 5 Years
Current Yield of Zero Coupon Bonds = 5%
Current Yield of Perpetuity Bond = r = 5%
Now,
Duration of Perpetuity Bond
= (1+r)/r
= (1+5%)/5%
= 1.05/0.05
= 21 Years
Let "w" be the weight of zero coupon bond.
Therefore,
Weight of Perpetuity Bond = 1-w
a) So,
Target Duration
= Weight of zero Coupon Bond*Duration of Zero Coupon Bond
+ Weight of Perpetuity Bond*Duration of Perpetuity Bond
10 = w*5 + (1-w)*21
10 = 5w + 21 - 21w
10 = 21 - 16w
16w = 11
w = 0.6875
Weight of Zero Coupon Bond = w = 0.6875
Weight of Perpetuity Bond = 1-w = 1-0.6875 = 0.3125
Therefore,
Portfolio hold by
Zero Coupon Bond = $2100000*0.6875 = $1443750
Perpetuity Bond = $2100000*0.3125 = $656250
b) Duration of Zero Coupn Bond after 1 year
= 5-1 = 4 years
Target Duration
= Weight of zero Coupon Bond*Duration of Zero Coupon Bond
+ Weight of Perpetuity Bond*Duration of Perpetuity Bond
9 = w*4 + (1-w)*21
9 = 4w + 21 - 21w
9 = 21 - 17w
17w = 12
w = 0.7059
Weight of Zero Coupon Bond = w = 0.7059
Weight of Perpetuity Bond = 1-w = 1-0.7059 = 0.2941
Therefore,
Portfolio hold by
Zero Coupon Bond = $2100000*0.7059 = $1482390
Perpetuity Bond = $2100000*0.2941 = $617610

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