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Liabilities of 10,000 each mature in 2 years and 4 years, and liabilities of 20,000 each mature in 5 years and 8 years
Liabilities of 10,000 each mature in 2 years and 4 years, and liabilities of 20,000 each mature in 5 years and 8 years. Assets of amount A in one year and amount B in 7 years have the same present value and Macaulay duration as liabilities. Find A and B if the annual effective rate is 10%.
Expert Solution
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1. As per Macualay duration concept calcualting duration of liability
Year (A) Liabilites to be meet(A) Present value @ 10%(1/1.10)^n (B) Present value of liability(A*B) Proportion in total present value Weighted time average 1 0 =1/(1.10)^1= 0.91 - =0/36843= - - 2 10,000 =1/(1.10)^2= 0.83 8,264 =8264/36843= 0.22 0.45 3 - = 1/(1.10)^3= 0.75 - =0/36843= - - 4 10,000 =1/(1.10)^4= 0.68 6,830 =6830/36843= 0.19 0.74 5 20,000 =1/(1.10)^5= 0.62 12,418 =12418/36843= 0.34 1.69 6 - =1/(1.10)^6= 0.56 - =0/36843= - - 7 - =1/(1.10)^7= 0.51 - =0/36843= - - 8 20,000 =1/(1.10)^8= 0.47 9,330 =9330/36843=0.25 2.03 Total 60000 36,843 4.9014 So present value of the liability is 36843
Duration of liability =4.9014 years
Given present value of assets = present value of liabilities
= present value of assets i.e wA+wB==36843
=wA of asset=x and wB of asset =1-x
Given duration of one asset is 1 year so DA=1
Given duration of second asset is 8 is so DB=7 years
as per immunization theory duration of liability=duration of asset A(DA)*Weight of Asset A(wA)+duration of asset B*Weight of Asset B(wB)
=DL=DA*wA+DB*wB
=4.9014=1*wA+7*wB
=4.9014=1*x+7*(1-x)
=6x=2.0989
=x=35%
So weight of asset A= 35%*36843=12895.11
Weight of asset B=36843-weight of asset A=23948.06
So Asset A value-12895.11
asset B value= 23948.06
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