VIGRE2 Vertical Intergration of Research and Education Department of Mathematics, University of Utah

Cory Steffen








Mentor: Jingyi Zhu
Major: Mathematics

Spring 2007 project description:

Mortgage-Backed Securities and the Black-Scholes Model

A mortgage-backed security gives its owner a share in the cash flows from a pool of mortgages. To value and hedge these securities requires a model of mortgage prepayment behavior, since this determines the monthly cash flow. The components of an individual mortgage-backed security can be represented by the combination of a long position in a noncallable coupon bond and a short position in a series of prepayment call options.

The valuation of a bond is based on two sources of uncertainty; default risk (credit risk) and interest rate risk. A mortgage-backed security adds a third risk; prepayment risk. Most mortgages allow the borrower to prepay all or part of the unpaid mortgage balance ahead of schedule, almost always without penalty. These prepayments introduce an uncertainty into the cash flow of mortgage securities that isnÕt present in many other fixed-income securities. Prepayment is classified as a risk for investors because it tends to occur when interest rates drop and the fixed income of the bond becomes more valuable. If the investors money is returned to them at this time, they will be forced to reinvest it at a lower rate of return.

The goal of our project is to build a prepayment model and combine it with option-based pricing methodology. Following we will apply our model to the valuation of mortgage-backed securities and observed data in the local market.

Summer 2007 project description:

Modern Portfolio Theory and Investment Risk

Modern Portfolio Theory (MPT) is the economist's way of looking at the market as a whole, and is the opposite of traditional stock picking. It proposes how rational investors can use diversification to optimize their portfolios by considering how securities contribute to the overall risk of the portfolio, and how all the investments in the portfolio can be expected to move together in price under the same circumstances.

MPT models an asset's return as a random variable and the portfolio’s return as a weighted combination of assets. The portfolio's return consequently has an expected value and a variance. Risk is modeled as the standard deviation of the portfolio's return.

MPT is limited by measures of risk and return that do not always represent the realities of the market due to the above assumptions. Using standard deviation to measure investment risk implies that better-than-expected returns are just as risky as those returns that are worse-than-expected.

I plan to derive a more robust model to measure investment risk and apply it in the context of MPT in order to create an optimal portfolio of local companies. I will then track this portfolio against multiple benchmarks from the stock market.