Bond
Model - Exact Matching with Cash Carryforward |
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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 |
$175 |
$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 |
$1,355.66 |
$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 |
50 |
50 |
50 |
50 |
50 |
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$303,253 |
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| Cash Flow |
Bond 1 |
Bond 2 |
Bond 3 |
Bond 4 |
Bond 5 |
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Total w/Int |
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Liability |
| Year 1 |
$5,000 |
$6,250 |
$7,500 |
$8,750 |
$3,750 |
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$31,250 |
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$32,000 |
| Year 2 |
$5,000 |
$6,250 |
$7,500 |
$8,750 |
$3,750 |
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$30,448 |
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$25,000 |
| Year 3 |
$5,000 |
$6,250 |
$7,500 |
$8,750 |
$3,750 |
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$37,079 |
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$22,000 |
| Year 4 |
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$6,250 |
$7,500 |
$8,750 |
$3,750 |
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$42,384 |
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$28,000 |
| Year 5 |
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$6,250 |
$7,500 |
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$3,750 |
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$32,891 |
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$25,000 |
| Year 6 |
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$7,500 |
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$3,750 |
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$19,694 |
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$20,000 |
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| Problem |
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| An investor wants to put together a portfolio consisting of
up to 6 different bonds. He has certain cash-flow requirements in the future |
| that the coupons of the bonds should cover. (For example, a pension fund must meet
requirements for future pension payments.) |
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payments are independent of interest rate changes. Excess payments in a period can be reinvested, to be available
in the |
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period, at a certain interest rate. How should the investor choose his portfolio to minimize the cost of
the bonds, while making |
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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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| The solution is similar to the one in BOND3. The difference
is that the cash-flow takes into account the reinvestment of excess |
| funds in one period for the next period. Remember that the
original idea behind exact matching was to minimize the influence |
| of interest rate changes. In this model, however, we are again more dependent on the interest
rate, since a shift in the future |
| rate
would affect the solution to the model. Thus, the market value of the portfolio may fluctuate to a greater
extent than if we |
| ignored
reinvestment opportunities. |
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