The U.S. Treasury has just publicized a review of the Treasury’s methodologies for valuation warrants issued to the Treasury as compensation for the government rescues of distressed financial institutions. Kamakura’s Managing Director for Research Robert A. Jarrow was retained by the Treasury to author this review. We summarize Professor Jarrow’s insights and add comments in this blog.

The full text of Professor Jarrow’s study for the U.S. Treasury is available at this link:

http://www.financialstability.gov/docs/Jarrow%20TARP%20Warrants%20Valuation%20Method.pdf

The U.S. Department of the Treasury purchases the warrants under what is known as the “Capital Purchase Program” in conjunction with its investments in the preferred stock of “qualified financial institutions.” Valuation of the warrants by the Treasury is essential in insuring that the Treasury receives “fair value” when the warrants are repurchased by the issuing bank holding company. www.financialstability.gov has a description of the process by which the warrant repurchase price is negotiated.

Professor Jarrow outlines the key analytical assumptions and methodologies employed by the Treasury in warrant valuation. We summarize those points here:

**
Valuation technology:** Modified Cox-Ross binomial lattice

**The Treasury explicitly recognizes the difference between warrants and options. Warrants are sold by the issuer of common shares and the number of common shares will increase upon exercise. This is not the case with traditional put and call options analyzed by Black and Scholes.**

Warrants vs. options:

Warrants vs. options:

**Stock price input:** The Treasury uses a 20 day moving average of observable stock prices but is conscious of current stock prices in doing the valuation.

**Dividends on common stock: **The Treasury approximates the discrete, stochastic nature of dividends by creating an average constant dividend yield over the 9 or 10 year life of the warrants.

**
American vs. European option:** The Treasury recognizes that the dividend payments can trigger early exercise and that the options should be analyzed as American, not European, options.

**Volatility:** The Treasury uses implied volatility whenever it is observable in the traded options market for the issuer.

**
Adjustment for stochastic interest rates:** Professor Jarrow notes that two modifications are necessary to recognize that interest rates are stochastic, not constant as assumed by Black and Scholes. The first modification is to use the risk free interest rate with a maturity that matches the 9 or 10 year life of the warrant. The Treasury makes this modification. The second modification, if one is using historical volatility, is an adjustment that recognizes the increase in stock price randomness that stems from interest rate changes. Professor Jarrow observes that this adjustment is not necessary because the Treasury is using implied volatility, not historical volatility.

**The Treasury estimates the 9 or 10 year forward average volatility as follows, according to Professor Jarrow:**

Stochastic volatility:

Stochastic volatility:

The Treasury “uses both observable implied volatility and historical volatility to construct a 10-year forward volatility curve. The initial segment of the curve consists of the observed implied volatilities for traded options. The last segment of the curve consists of a ‘normalized’ 10-year average historical volatility. The volatility is normalized by removing any abnormally high recent volatilities from the estimate. The middle segment is determined using straight-line interpolation between the initial and terminal segments. The estimated forward volatility curve is typically downward-sloping, consistent with a reversion in volatilities to a long-run value.”

Professor Jarrow adds that the Treasury stress tests valuations around this projected 10 year forward volatility curve.

**
Market imperfections-illiquidity:** The Treasury considers the impact of large purchases and sales of warrants by making a scale adjustment after the basic valuation is completed.

**Differences in funding costs for the banking firm issuing the warrant and for the Treasury create a range of “fair values” and Professor Jarrow shows how the modified Black-Scholes valuation used by the Treasury falls within this range.**

Market imperfections-funding costs:

Market imperfections-funding costs:

**The Treasury makes no explicit adjustment for this on the grounds that the current market price correctly anticipates the potential for dilution.**

Dilution upon Warrant Exercise:

Dilution upon Warrant Exercise:

Professor Jarrow goes on to note that the warrant contract contains dividend-related and corporate combination-related protections.

Professor Jarrow concludes his review of the Treasury’s valuation procedures as follows:

“As documented above, it is my belief that the Treasury’s modeling methodology for valuing the warrants is consistent with industry best practice and the highest academic standards. The methodology uses the industry standard model for pricing options, the Black-Scholes model, in a modified form to account for the size of the warrant position, stochastic interest rates, stochastic volatility, as well as numerous market imperfections.

“Furthermore, as previously detailed, the Treasury’s financial model is only one component of a robust valuation procedure. For warrant positions that are evaluated, the Treasury also collects market prices (where available) or indications from market participants and valuations from outside consultants/financial agents. All valuation information is considered in the determination of an appropriate fair market value for the warrants of a specific institution.

“The valuation process results in a warrant valuation that is fair to both the participating banks and the U.S. taxpayers.”

**Additional Considerations
**

As the economic recovery proceeds, the taxpayers would surely be pleased if the Treasury liquidated its warrant positions at substantial gains. In that happy scenario, the questions of valuation are moot. Until that time, there are two other possible modifications that may be useful in pinpointing “fair value” with additional precision:

**Re-examining the dilution issue: **As shareholders in Kamakura Corporation, Professor Jarrow and I recognize that the exercise of stock options in Kamakura triggers an increase in shares outstanding, exactly like the TARP warrants. It is well known that one can capture the impact of this dilution by using a slightly modified version of the Black-Scholes formula that assumes that the value of the assets of the firm, instead of the stock price, has a normally distributed return. This modification becomes more and more important in the TARP context when the firm’s rescue takes place (and the warrants are issued) when the “qualified financial institution” is extremely distressed. The dilution impact, of course, is much larger if the warrants are issued at lower and lower stock prices.

**
Default risk:** Default risk can have a very large potential impact on valuation of a warrant or call option on the stock of a firm that can go bankrupt. Merton’s 1976 analysis of the impact of default risk on options valuation assumed interest rates are constant. Jarrow and Turnbull in 1995 generalized this result to the random interest rates case. For a firm with 0% bankruptcy risk, of course, the warrant valuation is unchanged. For a firm with 100% bankruptcy risk, the obvious result is that the warrant has no value at all. In the case of Citigroup, the graph below shows that the 1 year version 4.1 Jarrow Chava default probability for the firm exceeded 50%.

Generalizing the TARP warrant valuation methodology for the impact of default risk is an extension that could lead to significant differences in valuation. Obviously the terms of the total TARP bailout package take the risk and return to the taxpayers of the United States into account, but it is still necessary to value each component of the package as the bailout program unwinds.

Donald R. van Deventer

Kamakura Corporation

Honolulu, October 27, 2009