A corporation is going to have to pay a debt of $1,200,000 after one year, a deb

Question

A corporation is going to have to pay a debt of $1,200,000 after one year, a debt of $1,500,000 after 2 years and $2,000,000 after 3 years. In order to set up an absolute matching strategy to pay these debts, the corporation may purchase:

A one year bond with 4% annual coupons

A two year bond with 2% annual coupons

A three year bond with 3% annual coupons

The spot rates are s1=5%, s2=4% and s3=3%. Find the amount the corporation will have to invest now in each of these bonds to set up this strategy. Find the yield Rate of this investment strategy.

Explanation / Answer

The amount in one year bond with 4% annual coupons be A,amount in two year bond with 2% annual coupons be B,amount in three year bond with 3% annual coupons be C.

For one year bond,Amount=A=1.04FV1/1.05=>FV1=(1.05/1.04)A

For two year bond,Amount=B=0.02*FV2/1.05+1.02FV2/1.04^2=>FV2=B/(0.02/1.05+1.02/1.04^2)

For three year bond ,Amount=C=0.03*FV3/1.05+0.03*FV3/1.04^2+1.03FV3/1.03^3

=>FV3=C/(0.03/1.05+0.03/1.04^2+1.03/1.03^3),

FV1,FV2,FV3 are face values of bonds with 1,2,3 year maturities respectively.

therefore to match pay of $1,200,000 after one year,

1.04*FV1+0.02*FV2+0.03*FV3=1,200,000 ...1) (1.04*FV1 is pay from 1 year bond ,0.02*FV2 is pay from 2 year bond,0.03*FV3 is pay from 3 year bond)

to match pay of $1,500,000 after 2year,

1.02*FV2+0.03*FV3=1,500,000 ....2)

to match pay of $2,000,000 after 3 year,

1.03FV3=2,000,000 ...3)

FRom 3=>FV3=2,000,000 /1.03=C/(0.03/1.05+0.03/1.04^2+1.03/1.03^3)

=>C=(2,000,000 /1.03)*(0.03/1.05+0.03/1.04^2+1.03/1.03^3)

=>C=$ 1939619.46

from 2=>1.02*FV2+0.03*FV3=1,500,000 =>1.02*FV2+0.03*(2,000,000 /1.03)=1,500,000

=>FV2=(1/1.02)*(1,500,000-0.03*(2,000,000 /1.03))=1413478=B/(0.02/1.05+1.02/1.04^2)

=>B=1413478*(0.02/1.05+1.02/1.04^2)

=>B=$ 1359900.05

from 1=>1.04*FV1+0.02*FV2+0.03*FV3=1,200,000

=>1.04*FV1+0.02*1413478+0.03*(2,000,000 /1.03)=1,200,000

=>FV1=(1,200,000-0.02*1413478-0.03*(2,000,000 /1.03))/1.04=1070651.9354=(1.05/1.04)A

=>A=(1.04/1.05)*1070651.9354

=>A=$ 1060455.25

Total Amount invested=A+B+C=1060455.25+1359900.05+1939619.46 =4359974.76

Let y be the yield rate of this investment strategy, therefore

4359974.76=1,200,000/(1+y) + 1,500,000/(1+y)^2 + 2,000,000/(1+y)^3

RHS=1,200,000/(1+y) + 1,500,000/(1+y)^2 + 2,000,000/(1+y)^3

for y=3%,RHS=1200000/(1+.03)+1500000/(1+.03)^2+2000000/(1+.03)^3=4409225.73

for y=3.5%,RHS=1200000/(1+.035)+1500000/(1+.035)^2+2000000/(1+.035)^3=4363571.75

for y=3.53%,RHS=1200000/(1+.0353)+1500000/(1+.0353)^2+2000000/(1+.0353)^3=4360856.7

for y=3.55%,RHS=1200000/(1+.0355)+1500000/(1+.0355)^2+2000000/(1+.0355)^3=4359048.19

for y=3.54%,RHS=1200000/(1+.0354)+1500000/(1+.0354)^2+2000000/(1+.0354)^3=4359952.29~4359974.76=LHS

therefore the yield Rate of this investment strategy is approximately 3.54%.

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