51±¾É«

Before the world economy underwent a catastrophic 2007-2008 subprime mortgage crisis, investment banks, including Lehman Brothers, sought an opportunity to increase their profits by securitizing more mortgages (i.e., issuing mortgage-backed securities). At the same time, Lehman Brothers acquired a large share of credit default swaps to insure against the credit risk exposure from mortgage-backed securities. Then the US Federal Reserve started to raise interest rates to cool down the economy. This eventually led to a massive default and the fall of Lehman Brothers. In the aftermath of the collapse, Lehman Brothers’ share value plummeted, and its credit rating was downgraded. Worst of all, Lehman Brothers brought down the sellers of the swaps, to whom they were unable to cover the losses completely. Ultimately, in 2008, Lehman Brothers filed for bankruptcy, which included the possibility of immediate liquidation of its assets. The US Federal Reserve intervened to reorganize the firm’s business and emerged from bankruptcy a few years later.  

This experience sowed investors that monitoring and maintaining sufficient equity or capital balance to finance its debts is important for solvent firms, as well as to sustain high credit ratings. Firms that fail to maintain an equity balance may lead to a degradation of the firm’s creditworthiness. One common way firms can increase their capital (equity) is by issuing corporate bonds. The firm can invest in credit default swaps (CDSs) to minimize the loss arising from credit events. A CDS is a financial derivative that allows an investor to swap or offset their credit risk with that of another investor. When a firm defaults on the underlying bond, the owner of the CDS receives recovery payments from the issuer of the CDS (i.e., the default risk is transferred to the issuer of the CDS), but premium (usually periodic payments) must be paid to the issuer of the CDS.  

My research aims to find reliable CDS prices. CDS prices are positively related to credit risks from the buyer side. Since credit ratings are revised infrequently, monitoring the total value of the firm’s assets is a good source for credit crisis predictions. In my study, the firm’s value follows some diffusion process (e.g., ). Diffusion processes capture the natural phenomenon of the firm’s value and have become the main tool among modern mathematicians in credit risk modelling. Structural models (e.g., ) attempt to find the likelihood of credit events completely determined by the movement of the firm’s value. However, structural models tend to underprice a CDS because little information is incorporated into its price. Including more relevant information helps determine a meaningful CDS price and prevents the economy from huge losses impacted by a credit crisis. Charging a CDS more than its underprice gives the firm issuing bonds to debtors less incentive to buying an excess amount of the CDS and reduces losses inflicted on the firm, as well as on the seller of the CDS.  

One way to improve the structural models is by incorporating hazard rates. A hazard rate measures the rate at which the firm liquidates its assets at a specific point in time, and it may or may not be related to the firm’s value. In the new model, a stochastic hazard rate is imposed; that is, the firm liquidates its assets at a low hazard rate, and the rate soars up during the reorganization process. If the reorganization process fails, the firm may be forced to liquidate and pay out as many outstanding debts as possible. The main advantage of the new model is that CDS prices can be derived mathematically, and the calculations are often carried out in an efficient manner. The new model shows a significant improvement over structural models, gives a near-perfect fit to typical market CDS price data, and better reflects actual market activity.  

About Hiro

Hiro Kato is a third-year PhD student in the Mathematical and Statistical Modelling program offered by the Department of Mathematics (51±¾É«). He is working under the supervision of Giuseppe (Joe) Campolieti and Roman Makarov.

He completed a Bachelor of Arts in Economics and Financial Mathematics (51±¾É«), and a Master of Science in Mathematics (51±¾É«). His research has been supported by multiple awards, including the Ontario Graduate Scholarship and the Queen Elizabeth II Scholarship. 

Additional Reading

• Mathematical and Statistical Modelling (PhD) 

Related Publication

• Campolieti, G., Kato, H., and Makarov, R. (2022). . Risks, 10(12), 228.