Discounts for illiquidity: where are we standing now? Part 4

This post series started three weeks past with the key concepts of illiquidity discount – DLOM in valuing restrictedly traded assets. Two posts followed with the earliest theoretical quantitative models to estimate the size of DLOM postulated by Chaffe and Longstaff. Those models assume DLOM as European-style put options on the corresponding unrestricted asset, which can be exercised at the end of the restriction time, with an exercise (strike) price that is either equal to the current price of the asset or equal to the maximum price of the asset obtainable during the restriction time. The underlying assumptions are that the investor has either none or full knowledge of the optimal exercise timing. They indicate lower and upper bounds for DLOM.

We will now examine the next steps in the evolution of sizing DLOM which try to approach a middle ground between those extreme estimates. For the sake of brevity, in this post we will look at three closely related models: two of them were proposed by one author and the other was a critique to the first of them. Therefore, the three models share many assumptions and allow us to see them as a pack.

In 2002, following Chaffe’s and Longstaff’s reasoning, Finnerty[1] suggested that DLOM could be emulated by a put option, but that the investor has no ability to choose the optimal exercise price, and that he/she would “be equally likely to sell the shares anytime during the restriction period”. That assumption implies that DLOM behaves like an American-style put options, but there are no closed-end formulas to evaluate such options. As a remedy, he proposed to approximate the evenly probable exercise timing by assuming the strike price is the arithmetic average of the asset’s future prices during the restriction period. From that approximation, he derived the following equations:

on what:

DLOM: Discount for illiquidity as a percentage of the price of the unrestricted asset.

D: Duration of the restriction on the asset.

r: Interest rate for no or low-risk assets, e.g., US Govern bonds, for the duration (D).

q: Dividend yield of the asset for the duration (D), that is the future dividends divided by the market price.

s: Constant expected volatility – the standard deviation of the asset’s prices, for the duration (D).

v: Volatility of the average of the future prices.

N(): Cumulative standard normal probability distribution, mean equals to zero, standard deviation equals to 1.

e: Euler’s constant.

ln(): Natural logarithm.

Here are some key insights from Finnerty 2002:

-??????? The size of the DLOM is positively correlated to the volatility and to the restriction time. Unlike Chaffe’s model, DLOM is positively correlated to the interest rate, which intuitively makes more sense: DLOM is a measure of the opportunity cost of missed trades during the illiquidity period, and other measures of opportunity cost also grow with increasing interest rate; for instance, the cost of capital of companies and projects. Therefore, Finnerty 2002 seems superior to Chaffe in this issue.

-??????? As with Longstaff, in combinations of extreme volatility and/or illiquidity time and/or interest rates, Finnerty 2002 will produce DLOMs north of 100%, which makes no economic sense – a restricted asset can have a very low value but not a negative one. Indeed, Finnerty 2002 can generate DLOM values higher than Longstaff, which is statedly an upper bound.

This second point is highlighted in critique made by Ghaidarov in 2009[2], as he provided mathematical proof that DLOM cannot exceed 100%. He extended Finnerty’s model by including geometric average strike prices. Since it is well known that geometric averages never exceed arithmetic ones, Ghaidarov used that relationship to derive lower and upper limits for Finnerty’s DLOM:

Furthermore, Ghaidarov proposes adjustments to Finnerty’s formula in line with the lower and upper bounds:

The variables have the same meaning as those of Finnerty’s model.

Note that Ghaidarov’s model is also very sensitive to the volatility and restriction time of the asset, but lacks connection with interest rates. Therefore, it is mathematically closer to Longstaff than to Finnerty 2002, although not allowing discounts superior to 100%.

Finnerty revisited his model in 2012 [3] and made important changes to it by the way of deriving the value of the average strike put option, that resulted in the following formula:

Finnerty 2012 does solve the problem of excessive DLOM in his previous model but at the following cost:

-??????? The size of the DLOM loses the connection with the interest rate.

-??????? The formula now can never be greater than 33%.

There is no good economic reason to accept that limit to DLOM, which disagrees with empirical evidence as hinted in the first post of this series.

The next two articles will show some improved models and some alternative numerical algorithms for estimating DLOM.

In the eighth and very last post of this series, we will compare the performance of these and other models with the market data.

I look forward to seeing your comments.

Many thanks to Wulaia Consultoria ’s team for their reviews and to the readers who are sharing our ideas.

[1]?FINNERTY, John D. The impact of transfer restrictions on stock prices. Analysis group/economics. New York, 2002.

[2]?GHAIDAROV, S. Analysis and Critique of the Average Strike Put Option Marketability Discount Model. Available at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1478266.

[3]?FINNERTY, John D. An Average-Strike Put Option Model of the Marketability Discount. The Journal of Derivatives, Vol. 19, Issue 4, Summer 2012, p. 53-69. New York, 2012.