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The best mathematical models for Discounts for Lack of Marketability (DLOM) (Part 3)

Today's post is by Andrew Strickland of Scrutton Bland, Chartered Accountants. Andrew specialises in corporate finance and business valuation. Andrew is a member of the Valuation Committee of the ICAEW and its subject matter expert. Andrew has undertaken a large number of business valuation assignments in respect of shareholder disputes and also in respect of divorce. He also works on fiscal valuations for a variety of tax purposes.

The Finnerty Average Strike Model

The model by John Finnerty of 2012 is a model that was changed after it was identified by Stillian Ghaidarov that there was a problem with the maths of an earlier version. This now seems to be the most popular with practitioners.

With Chaffe the price fixes at the start of the period of illiquidity – with Longstaff the price fixes at the highest price during that period. With Finnerty the price fixes at the average price during the period of illiquidity. The assumption is that the stockholder would have been equally as likely to sell on any day during the illiquidity period if the stock had been liquid. This is therefore an elegant means of addressing the concerns with both the Chaffe and Longstaff models.

John Finnerty says that the model is OK for one year and may be up to two-year restriction periods. The model is therefore not considered suitable by its originator for periods of more than 2 years. So, we have a strong health warning from the man himself: this model is not intended for use when considering periods of illiquidity beyond two years at the most

We also need to be aware that the model has an unexpected ceiling of 32.28% - it then follows a straight-line. A DLOM cannot exceed this figure if using this model. The ceiling is unexpected as it appears to be just a function of the math that was used for the average price.

This is another model that does not comply with expected model behaviour. There is nothing within business valuation theory or practice that states that DLOMs cannot exceed 32.28%.

Ghaidarov Average Strike Model

The Ghaidarov Average Strike Model was produced in response to the Finnerty model.

Stillian Ghaidarov says that one is not innately better than the other in terms of the math relating to the average – they are just two ways of calculating that value for the period of illiquidity.

Neither model has the status of a mathematical proof – the calculation of an average price requires the importation of a function to estimate what that average price might be.

This is a model that exhibits expected model behaviour. It starts at nil and increases to close to 100% over time but does not exceed 100%. It also increases at a decreasing rate as we would expect.

Therefore, although the Finnerty model is seemingly more popular, the Ghaidarov average strike model looks like a compelling alternative, at least for periods of more than 2 years,

If the business valuer considers that the average strike concept is one with which he is comfortable, then the Ghaidarov Average Strike Model must be worth a look. Again, better details can be obtained from Business Valuation Resources.

Ghaidarov Forward Start Model

The forward start option is an exotic option. It is an option that is purchased and paid for at the start but is activated at a date in the future. The option is then like a vanilla option from the activation date to the expiry date. All of the inputs are fixed at the start except the strike price.

What does that mean? We will explain by use of an example:

Let us imagine an underlying security with a current value of $10.

The option contract has an expiry date in 2 years 6 months’ time.

The activation date is in two years’ time.

The strike price is the value of the underlying at the activation date.

It is then a vanilla option for the last 6 months of its life to expiry.

Ghaidarov has taken this exotic option and has come up with a wonderfully simple model.

The cost of illiquidity is the absence of a choice as to when to sell. This is what the forward start model tries to replicate.

This is a version of Black Scholes, so it has a solid intellectual base.

This model is identical to Chaffe in the following circumstances:

The stock price and the strike price are identical;

There are no dividends.

The risk-free rate is nil.

If the risk-free rate is nil, then the Chaffe Protective Put Model and the forward start are identical. That is why we think that Chaffe was so close to having the right answer.

If you'd like to learn more about this topic, register to our upcoming free Fiscal Valuations Webinar, where Andrew will continue discussing these - and related issues.

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