Extending the concept of this example, suppose a pension plan has liabilities that are expected to grow over time and wants to invest in bonds to provide the necessary cash flows against these future obligations. Suppose it does so at a time when yields are near historic lows. What, specifically, should the bond portfolio manager be concerned about with respect to interest rate risk beyond duration? (Hint: think convexity.)

When regulation Q was lifted, savings & loans (S&Ls) suddenly found themselves competing with commercial banks for savings deposits. S&Ls’ balance sheets were dominated by long-term mortgage loans. Detail the changing landscape for S&L risk as a result of this deregulation. (Hint: the duration of liabilities and assets diverged.)

Cash matching: let's add some realism to the pension problem. Suppose, as given, there are 10 bonds, ranging in maturity from a one-year zero to a six-year bond.

Go to the companion website for more details (see Pension under Chapter 2 Examples).

The coupons and cash flows are given in the following table.

The market prices of the bonds are given in the second-to-last row. The pension has estimated its retirement obligations for the next six years, which are given in the next-to-last column (Req'd). We wish to invest in a portfolio of bonds whose cash flows will be sufficient to honor these obligations. Letting the last row, x, denote the quantity of each bond and the last column (Actual) denote the realized cash flow in dollars for each year, we wish to invest an amount, px, that is just enough to cover the obligations. That is, we want a solution to the problem:

Here, c_ij refers to the coupon in year i on bond j.

This is a linear programming problem that we solve in the Pension tab of Chapter 2 Examples.xlsx, using Excel's Solver (click on Excel's Data tab at the top of the sheet). Cell L9 is the sum of our expenditure, which we want to minimize, and we are searching for a combination of bonds (x), zero amounts or better (first constraint), whose total value in each year is sufficient to cover the necessary obligation (second constraint). The problem is set up in the spreadsheet as this:

The Solver Parameters window used to solve this problem looks like this:

Solving this problem gives us the solution:

Notice that in some cases, the actual cash flow exceeds the required cash flow, suggesting that this particular set of bonds will not match our objective perfectly and that we have therefore overallocated to this portfolio. Nevertheless, we have satisfied our obligation.