Cash Flow Matching

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RoccoLee

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The pension plan actuary has projected pension payments of 6 million, 5 million and 8 million in years 1 - 3, respectively. The company is considerting buying one-year, two-year and three-year annual pay bonds to match the cash flows of payments. The bonds yield 6% and are selling at par.
What is the purchase cost of the two-year bond? Answer: 4.5 million
Could someone kindly tell me the calculations for the question above, please?
Many thanks!
 
5,000,000 / (1.06)^2
But this assumes that the bond is a zero bond.
 
S2000magician wrote:
MrSmart wrote:5,000,000 / (1.06)^2
But this assumes that the bond is a zero bond.
Click to expand...
It’s not a zero; it pays a 6% coupon (given).
Click to expand...
I read that, maybe they used the sentence to show market yields. and not nessecarily the bonds used for immunization. How else would you explain the answer?
Edit: Using your assumption, another correct answer would be 4,520,000
 
The 3-year bond will have a par value of $8,000,000 / 1.06 = $7,547,170, and a coupon of $452,830 (= $7,547,170 × 6%).
The 2-year bond will have a par value of ($5,000,000 − $452,830) / 1.06 = $4,289,783, and a coupon of $257,387 (= $4,289,783 × 6%).
 
^ Oh right, forgot to include final coupon as part of the payment
That means the 2-year bond should cost it’s par of 4.3m, not 4.5m.
 
S2000magician wrote:
The 3-year bond will have a par value of $8,000,000 / 1.06 = $7,547,170, and a coupon of $452,830 (= $7,547,170 × 6%).
The 2-year bond will have a par value of ($5,000,000 − $452,830) / 1.06 = $4,289,783, and a coupon of $257,387 (= $4,289,783 × 6%).
Click to expand...
What is the logic behind?
why we include the coupon value in the 3-year bond and 2 -year bond?
 
RoccoLee wrote:
S2000magician wrote:
The 3-year bond will have a par value of $8,000,000 / 1.06 = $7,547,170, and a coupon of $452,830 (= $7,547,170 × 6%).
The 2-year bond will have a par value of ($5,000,000 − $452,830) / 1.06 = $4,289,783, and a coupon of $257,387 (= $4,289,783 × 6%).
Click to expand...
What is the logic behind?
why we include the coupon value in the 3-year bond and 2 -year bond?
Click to expand...
It’s a cashflow used to pay off the liability.
 
RoccoLee,
I had to use a spreadsheet to assist my understanding of the conversation between MrSmart and S2000magician above. I’m having trouble uploading an image of the spreadsheet, but here it is in prose:
Year 1 Cash Flow = Bond 1 Interest + Bond 1 Principal + Bond 2 Interest + Bond 3 Interest
Year 2 Cash Flow = Bond 2 Interest + Bond 2 Principal + Bond 3 Interest

Year 3 Cash Flow = Bond 3 Interest + Bond 3 Principal

Sum the cash flows from each year and use the “What If” function to set them equal to the required pension payments each year. Make your interest payments for each bond a function of 6% x the principal payment, but keep the principal payment cells hard-coded.

Without the use of a spreadsheet on the exam, the only way to solve this that I can see is to do exactly as S2000magician did and work backwards from the final year. In the final year, you know you only have the interest and principal payment from Bond 3 left, so the sum of these two amounts must be equal to $8,000,000, thus ($8,000,000 / 1.06) = $7,547,169 must be the Bond 3 Principal and ($7,547,169 * .06) = $452,830 must be the Bond 3 Interest Payment.

If $452,830 is the Bond 3 Interest Payment in Year 3, it must also be the Bond 3 Interest Payment in Year 2, so the principal on Bond 2 must account for a 6% interest payment on Bond 2 and the fact Bond 3’s Year 2 interest payment is also received, giving you ($5,000,000 - $452,830 / 1.06) = $4,289,783 as the Bond 2 Principal Amount and ($4,289,783 x .06) = $257,387 as the Bond 2 Interest Payment, which when summed with the Bond 3 Interest Payment gets you to $5,000,000 in Year 2 cash flow.
 
Curmudgeon wrote:
RoccoLee,
I had to use a spreadsheet to assist my understanding of the conversation between MrSmart and S2000magician above. I’m having trouble uploading an image of the spreadsheet, but here it is in prose:
Year 1 Cash Flow = Bond 1 Interest + Bond 1 Principal + Bond 2 Interest + Bond 3 Interest
Year 2 Cash Flow = Bond 2 Interest + Bond 2 Principal + Bond 3 Interest

Year 3 Cash Flow = Bond 3 Interest + Bond 3 Principal

Sum the cash flows from each year and use the “What If” function to set them equal to the required pension payments each year. Make your interest payments for each bond a function of 6% x the principal payment, but keep the principal payment cells hard-coded.

Without the use of a spreadsheet on the exam, the only way to solve this that I can see is to do exactly as S2000magician did and work backwards from the final year. In the final year, you know you only have the interest and principal payment from Bond 3 left, so the sum of these two amounts must be equal to $8,000,000, thus ($8,000,000 / 1.06) = $7,547,169 must be the Bond 3 Principal and ($7,547,169 * .06) = $452,830 must be the Bond 3 Interest Payment.

If $452,830 is the Bond 3 Interest Payment in Year 3, it must also be the Bond 3 Interest Payment in Year 2, so the principal on Bond 2 must account for a 6% interest payment on Bond 2 and the fact Bond 3’s Year 2 interest payment is also received, giving you ($5,000,000 - $452,830 / 1.06) = $4,289,783 as the Bond 2 Principal Amount and ($4,289,783 x .06) = $257,387 as the Bond 2 Interest Payment, which when summed with the Bond 3 Interest Payment gets you to $5,000,000 in Year 2 cash flow.
Click to expand...
Thank you very much!
 
S2000magician,
Thank you - that explanation helps tremendously. Initially I was confused when you were saying (from your example at the link) to buy €6,576,923 ÷ 1.03 of par value 3% bonds to cover the year 3 cash flow because intuitively I was thinking that you would need to discount that by three periods (so 1.03^3) since it is a year 3 principal repayment. However, now I think about it differently, namely that the principal you need in any year is simply the plug between the total cash you need to match for that year less interest from other bonds less the interest from the bond in question.
Does that make sense at all? Alternatively, can you see why I was confused initially?
 
You got it, Curmudgeon.
I’m glad that the explanation was helpful. And, yes: I can see why it can be confusing.
 
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