Valuations failed Northern Rock

Why is it that the value of Northern Rock's equity in September 2007 was trading at around £6 (Dh33) per share, while the balance sheet put the equity value at zero? There was no hidden intangible missing – the media mauling the previous month had effectively destroyed any residual brand value. What was missing was the additional value that accrues to the shareholder through the benefit of limited liability.

Investors coming to the market in September 2007 knew that the spot value of the bank's net asset value (NAV) was zero, but that there was the potential for those assets to appreciate if market conditions improved. If market conditions deteriorated and the bank went into negative equity, investors knew that they could walk away. Limited liability allows investors to shred their share certificates if the value of the bank's assets falls.

As Northern Rock entered the last few months of its life, the value of its future cash flow in the hands of the shareholder would have been based solely upon the expected value of those assets that occurred on the upside of the distribution of possible outcomes. The asymmetric claim that limited liability gives is at a maximum when a business is 'at the money' - ie, when its assets and liabilities are in exact balance. However, the valuation of asymmetric claims is tricky and requires an understanding of option pricing.

The idea that option pricing can be used to value equity is well understood in the area of default prediction, which is in essence the art of determining how close a company is to the money. It is not so well understood when dealing with highly-leveraged businesses such as those that dominate the private equity market, leveraged buy-outs and financial institutions that tread the thin line between success and failure. Banks carry very high debt-to-equity ratios.

They have spent much lobbying effort in persuading regulators to let them reduce their capital adequacy requirements below the 8-10 per cent range dictated by the Basel agreements.

Looking at Northern Rock, the balance sheet allows us to pinpoint the value of the bank's assets. It also gives a good measure of the level of its outstanding liabilities and, by making some reasonable assumptions, we can estimate their average term to maturity. Deploying the option-pricing model requires two further inputs: the risk-free rate of interest (this comes from the mechanics of how options are valued, assuming zero arbitrage opportunities) and the volatility of the bank's assets. It is this last figure that caused the biggest problem when attempting to apply the Black-Scholes Model to Northern Rock.

Using historical data corroborated by up-to-date measures of the implied volatility of the company's equity, I was able to reverse engineer the underlying asset volatility using the Black-Scholes Model. However, on the premiss that it is not a good practice to use a model to test itself, I then had to compare my results with other banks, predominantly in the mortgage business. My estimate was that Northern Rock's asset volatility was about five per cent.

In conclusion, in March 2007 the reported asset value of Northern Rock was £109.31 billion, with liabilities of £106.60bn. The average maturity on the liabilities was estimated to be six months and the prevailing risk-free rate attaching to UK Treasury bonds was 3.5 per cent. With 421 million of shares in issue, the balance sheet value of the equity was £6.41 per share against a share price trading close to £11.20. Using the rough 'guesstimates' outlined above, the Black-Scholes Model returned a value of £11.31.

By September, the evidence was that the bank's asset values had contracted, and the equity account stood at zero. Although the evidence from the bank was rather thin during this fraught period, taking the downgrading at face value the Black-Scholes Model (using an asset value of £106.6bn) predicted a share price of £6.16 – very close to the early September price of around £6.20 per share.

This analysis seems to suggest that when we are dealing with a company trading 'near the money', fair value calculations – whether on a mark-to-market or a mark-to-model basis – are only part of the explanation of equity value. It also warns us that even the most sophisticated flow valuation techniques will not capture the full value of highly leveraged entities.

The analysis also points out an often-forgotten key truth. Options of any sort increase in value the riskier the underlying assets. This is quite a counterintuitive result. Normally, value falls with risk, but if we assume an asset volatility of 10 per cent instead of my estimate of five per cent, the value of the equity leaps to £9.49 per share! The reason for this is that when a business is in danger of failure, the limited liability shareholders have very little to lose. Their preference switches in favour of risk taking – a persuasive explanation of why banks regularly fail.

- The writer is Professor (Emeritus), Manchester Business School

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