In: Finance
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 |
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 | |||||||