Bond Model - Exact Matching |
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| What
is the minimum cost portfolio, consisting of up to 6 bonds, that provides
enough |
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| cash flow to cover
liabilities in each period? |
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| Interest Rate |
7% |
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| Characteristics of bonds |
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Bond 1 |
Bond 2 |
Bond 3 |
Bond 4 |
Bond 5 |
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| Face Value |
$1,000 |
$1,000 |
$1,000 |
$1,000 |
$1,000 |
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| Coupon Payment |
$100 |
$125 |
$150 |
$200 |
$75 |
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| Years to Maturity |
3 |
5 |
6 |
4 |
6 |
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| Price |
$1,078.73 |
$1,225.51 |
$1,381.32 |
$1,440.34 |
$1,023.83 |
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Bond 1 |
Bond 2 |
Bond 3 |
Bond 4 |
Bond 5 |
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Cost |
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| Number
Purchased |
10 |
10 |
10 |
10 |
10 |
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$61,497 |
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| Cash Flow |
Bond 1 |
Bond 2 |
Bond 3 |
Bond 4 |
Bond 5 |
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Total |
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Liability |
| Year 1 |
$1,000 |
$1,250 |
$1,500 |
$2,000 |
$750 |
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$6,500 |
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$32,000 |
| Year 2 |
$1,000 |
$1,250 |
$1,500 |
$2,000 |
$750 |
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$6,500 |
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$25,000 |
| Year 3 |
$1,000 |
$1,250 |
$1,500 |
$2,000 |
$750 |
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$6,500 |
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$22,000 |
| Year 4 |
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$1,250 |
$1,500 |
$2,000 |
$750 |
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$5,500 |
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$28,000 |
| Year 5 |
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$1,250 |
$1,500 |
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$750 |
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$3,500 |
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$25,000 |
| Year 6 |
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$1,500 |
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$750 |
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$2,250 |
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$20,000 |
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| Problem |
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| In
models BOND1 and BOND2 we saw a way for an investor to protect against
interest rate fluctuations. Here, we'll look |
| at another method. An investor wants to put together a portfolio consisting of up to 6
different bonds. He has certain cash- |
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requirements in the future that the coupons of the bonds should cover. (For example, a pension fund must meet |
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for future pension payments.) These
payments are independent of interest rate changes. How should the |
| investor choose his portfolio to minimize the cost of the
bonds, while making sure that the payments cover his future cash- |
| flow requirements? |
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| Solution |
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| 1)
The variables are the number of each bond to include in the portfolio. In
worksheet BOND3 these are given the name |
| Purchased_bonds. |
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| 2) The
constraints are very simple. First we have the logical constraints: |
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Purchased_bonds >= 0 via the Assume Non-Negative option |
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Purchased_bonds = integer (We can not buy fractions of a bond) |
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| Then there is
the constraint to make sure that the cash-flow requirements are met: |
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Cash_flow >= Liabilities |
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| 3) The
objective is to minimize the portfolio cost. This is given the name
Total_cost. |
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| Remarks |
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| In this model we assume that money coming in from maturing
bonds can not be used to cover the cash-flow requirements. |
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we do not account for excess money in one period that may be transferred to
the next period. In model BOND4 we |
| will account for this. |
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