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An insurance company must pay liabilities of 99 at end of year 1, 102 at end of year 2, and 100 at end of year 3. The only investments available to the company are the following 3 bonds.

Bond A - 1 year to maturity, yield rate 6% coupon rate 7%

Bond B - 2 years to maturity, yield rate 7% coup rate 0

Bond C - 3 years to maturity, yield rate 9% coupon rate 5%

How many units of each bond will the insurance company need to purchase in order to match liabilities exactly?

I do not really understand this question.. As far as the cash flow is concerned.. it seems bond C is the only one that will offset its balance at par value ...

swordlordswamplord
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2 Answers2

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$\begin{array}{}\text{Bond}&\text {Flow in year 1}&\text {Flow in year 2}&\text {Flow in year 3}\\A&1.07\\B&0&1\\C&0.05&0.05&1.05\end{array}$

These are the flows for each dollar of par value of the bonds purchased. The yield will affect the price of the bonds, but you haven't been asked anything about that. So, we will ignore that part of the question.

If you buy par value of $x,y,z$ of bonds A,B,C respectively,

$1.05 z = 100\\ 1y + .05z = 102\\ 1.07x + 0.05z = 99$

Now you have a system of equations with 3 unknowns. Solve for $x,y,z$

Doug M
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  • I think this is just preference but i let a, b and c express the units of each bond for A, B and C respectively. And I also used face value of each bond rather than using percentage to make it easier for instance liability at end of year 1 is 99 = 107a + 5c and liability at end of year 2 is 102 = 100b + 5c and lastly 100 = 105c so c is the first one we solve which is around 95% and then we can plug it in to the other 2 equations to find the units of bond A and B – swordlordswamplord Oct 26 '20 at 22:32
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You just have $3$ linear equations. At the end of year $1$, the company will receive the face amount of bond A plus the coupons on bonds A and C. At the end of year $2$, it get the face amount of $B$ plus the coupon on $C$. At the end of year $3$ it get the face amount and coupon on bond C. (I'm assuming the bonds all bear annual coupons, for otherwise it seems impossible that cash inflows could exactly match the outflows.)

saulspatz
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  • yes. it seems i just figured it out.. i just had no idea how to evaluate this kind of cash flow.. i thought i had to first figure out the present value at t=0 and see how much coupon payments i receive and reduce it from the liabilities for that year then try to figure out the percentage of bond i would need but that just wasn't feasible and i kept hitting a dead-end. Just now I used face amount at redemption for each bond and put an A B and C next to the respective bonds which will represent units of them and it worked out seamlessly – swordlordswamplord Oct 26 '20 at 22:09