The sub-prime mortgage debacle and what we can learn from mathematical programs.

Abstract

In this paper we discuss mathematical programming methods for insurance companies, mutual fund managers, and banks to match cash flow and liabilities. We focus on mortgage-backed securities, and methods for using them for asset allocation. Given the recent and ongoing sub-prime mortgage debacle, it is imperative that the conscientious and conservative investor use robust mathematical models to decide whether to hold or sell their current investments, or to invest in other portfolios, instead of making such decisions without careful consideration. We first discuss the difference between a deterministic and stochastic instrument, then describe a linear programming model for allocating bonds deterministically, and then, as in the case of mortgage-backed securities, stochastically.

Introduction

The objective of this paper is to learn from our current mortgage crisis to avoid investment without careful consideration and without proper valuation. In particular, this paper is directed at large investors such as insurance companies and mutual funds whose investments are supposed to be conservative and with low risk.

Methodology

We begin by describing the mortgage debacle, how it developed and who was affected. We then focus on the financial objectives of mutual funds and insurance companies. Insurance companies collect large sums of money as premiums. Those premiums are held in financial funds, which require asset allocation decisions. These funds are often governed by laws that aim at keeping the investments conservative and low risk. After describing the general asset allocation problem we explain the difference between a deterministic and stochastic cash flow.

* Section 3: A Deterministic Linear Program: Cash Flow/Liability Matching with Deterministic Bonds, describes a mathematical (linear) program that assumes investment in a bond that has a deterministic (known) cash flow. The linear program matches cash inflow with liabilities.

* Section 5: A Stochastic Linear Program: Cash Flow/Liability Matching with Mortgage Backed Securities (MBS) describes a linear program that allows investment in a Mortgage-Backed-Security (MBS), a security that has stochastic (unknown) cash flow. Stochastic instruments do have problems with their past performance, and a probability distribution associated with them.

* The purpose of Section 4: Pricing a Stochastic Instrument, is to introduce the reader to the way a stochastic instrument is valued. This is accomplished by generating many scenarios, Monte Carlo style, based on the probability distribution of each one, and by then taking the Expected Value with respect to all of those scenarios.

* Section 4 can thus be viewed as an introduction to section 5.

* The mathematical programs of sections 3 and 5 are specifically suited to insurance companies and banks that invest their capital but yet have specific obligations that must be met.

* Section 6: More Examples of Stochastic Linear Programs, discusses some examples where mathematical programming has been utilized to match cash inflow and liabilities of complicated real-world situations.

The Mortgage Debacle

We are in the midst of a mortgage debacle spurred on by rising property values and reckless lending practices. As property values rose, lenders, desirous of capitalizing on the buying frenzy, offered loans to borrowers even if their credit-worthiness was sub-prime.

In order to entice possible borrowers, variable-rate loans with low initial interest rates, also known as "teaser rates," were offered. These loans offered the borrower low interest payments for the first few years, which would then jump after a number of years. It was recognized that the only way these borrowers would be able to continue with the payments would be through constantly rising property values. Indeed, these easy loans with teaser rates helped drive property values up even higher. The generous lending policies brought many more people into the home-buying market.

In order for banks to offset their risk and to raise immediate cash, lending banks would package these loans into mortgage-backed securities and sell them. In this manner the inherent risk was passed on to investors.

As long as property values rose, the market was able to sustain repayment of these loans. The home-owner was able to either refinance based on higher home values, or he or she would find a buyer at a higher purchase price and thus pay off the loan.

But once the upward swing of property values stopped, the defaults began. A scheme such as this ultimately tumbles in the same manner as any pyramid scheme.

There are a number of losers in this game:

1. Home buyers who lose their homes after buying homes that are beyond their means

2. Investors who bought up the mortgage-backed securities for homes, some of which are moving towards default and some of which have already defaulted and

3. Banks that are left holding many mortgage-backed securities with no buyers.

It is important to note that it was not only risk-taking investors who bought mortgage-backed securities, but also conservative investors; mutual funds, retirement funds, and others. These securities were often given high credit ratings by the agencies upon whom those investors relied. Investors trusted banks to make loans only to credit-worthy borrowers, and they assumed that this package of collateralized securities must be a very safe instrument.

With this debacle in mind it is imperative that mutual funds, retirement funds, insurance companies and other conservative investors use robust methods, such as mathematical models, to help in asset allocation decisions.

1. The Asset Allocation Problem

The financial Asset Allocation Problem considers the question of how a portfolio should be weighted with different security assets in order to satisfy an investor's objective (Markowitz, 1959). Investors' objectives may differ. For instance, an individual who is saving money for retirement will want to be quite sure that the money will be there when it is needed; thus the risk must be minimized.

An expansion of this idea occurs when we build liability and other constraints into the model. For example, an insurance company that sells insurance on a long term basis will want to invest the initial money culled from sales to earn a substantial enough return to enable it to charge competitive prices, while still maintaining confidence in its ability to meet its obligations. A company having obligations due over many years may want to invest, but may not wish to risk the amount necessary for its future obligations.

2. Deterministic vs. Stochastic Instruments

A deterministic instrument is an instrument in which the cash flows are known with certainty. An example of this is a conventional bond. The investor knows that over the course of some time, a set amount, known as the coupon, will be paid for every period specified. This assumes that the cash flow does not come from the sale of the bond. The value of the bond itself does depend on interest rates, and the cash from its sale would not be deterministic.

A stochastic obligation is an obligation in which the cash flows are not known with certainty. An example of this is any stock or a bond that pays a floating rate instead of a fixed rate. That is, the payment is pegged to an outside rate that changes in a non-deterministic way (such as LIBOR). Clearly, if interest rates go down the bond will yield less.

Another example of a stochastic obligation is a Mortgage Backed Security (MBS). When a bank wishes to raise capital to lend to homeowners, it may create a security called an MBS, which can be traded. A number of mortgages are associated with one MBS. An investor buys the MBS and the bank shifts the mortgage payments to the investor when they are paid. These payments are stochastic due to the homeowner's refinancing option. Suppose, for example, that interest rates were to go down so that the homeowner can now borrow at a lower rate of interest. The homeowner may prepay the principal, and as a result, the investor's cash flow is not deterministic. There are, of course other sources of risk, most notably in the current crisis: risk of default.

3. A Deterministic Linear Program: Cash Flow/Liability Matching with Deterministic Bonds

As just mentioned, a deterministic portfolio of bonds does not depend on changes in interest rates. This may be seen from the following scenario (Schrage, 1994). A pension fund has a list of people to whom pensions must be paid. The fund would collate all the numbers so that they have one aggregate liability due each year for a set number of years. To fund these liabilities, the fund may choose from a specific group of bonds. The question is how to best balance a portfolio of bonds from among their set of bonds.

Let n = the number of types of bonds in their possible set.

Let p = the number of periods (in our case, years) to which the liabilities extend.

Let [C.sub.ij] = the coupon rate bond i pays in year j. For some years j, bond i may be past maturity and [C.sub.ij] = 0. [C.sub.ij] for all j's less than bond i's maturity year are equal.

Let [L.sub.j] = the liability due in year j.

Let [B.sub.i] = the number of bonds of type i that we will purchase for our portfolio in time 0. These are the variables for which we need to solve.

Let R = the lower bound of the short term interest rate.

Let [S.sub.j] = the amount of extra cash reinvested in year j.

Let [price.sub.i] = the cost of bond i.

Note that j = 0 is before the first year; at the time we run the mathematical program and make our decision.

The idea is to minimize the cost of the portfolio, given that the liabilities will be funded.

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