### Neighborly Loan:

Jacob Marley, recently promoted to manager of the newest branch of the
Neighborly Loan Company, is eager to please his new boss by loaning his
annual $15 million budget profitably. Each local branch office at
Neighborly generates profit by the interest income from three types of
loans: first Mortgage loans on real estate at 7% annual interest,
Automobile loans collateralized by liens on automobiles at 12% annual
interest, and Signature loans with no collateral at 15% annual
interest. Having the highest risk, Signature loans carry the
highest loan interest rate.

Neighborly's home office has set loan limits to guide branch managers
and to protect the company from excessive amounts of high-risk loans.
Neighborly requires each branch manager to place at least 60% of its
loans into First Mortgages and no more than 10% of its loans in risky
Signature loans.

If you make the mistake of expressing the mortgage and signatue
constraints as percentage of budget rather than percentage of amount
loaned, you get
the right solutiion but an incorrect sensitivity analysis The
sensitivity analysis gives a dual price of 12 cents per dollar change
in the amount budgeted, but actually changing the amount yields only an
extra 9.3 cents on the dollar!.

Click here for spreadsheet

The reason is that changing the amount budgeted changes the maximum
dollar amount for signature loans and the minimum dolar amount for
first mortgages, which violates the "change just one constraint"
assumption of the sensitivity analysis.

If you express the constraints as perecentages of the amount loaned,
you get the correct solution and also the correct sensitivity analysis
with the correct dual price of 9.3 cents per dollar change in amount
budgeted.

Click here for spreadsheet
Note that the sensitivity report shows the RHS of the mortgage and
signature loans equal to zero.

The constraints on signature loans and first mortgages are examples of
a very important class of "__proportionality constraints__"
in which a decision variable is constrained to be at least or at most
some proportion of a total of which the variable itself is a
part. These will play a
big role in "blending problems."

In fact, Neighborly Loan can be viewed
as a "blending problem" in which financial assets are blended into a
portfolio.

These constraints are not expressed in standard form since there's a
function on the right hand side.

Click here to see the problem expressed in
standard form.

Note how much less human-friendly it is than the form on this and the
previous page.

To Excel, the two are exactly the same.