# Estimating an illiquidity discount using option modelling

An investor in a public company can sell their shares instantaneously. In comparison it usually takes time for an investor in a private company to find a buyer and exchange contracts.

According to the Pepperdine 2023 Private Capital Market report, business brokers observe it takes an average of nine months from listing a business to closing the contract[1].

What is the cost of this comparative lack of liquidity when selling a privately held company compared to selling shares in a public company and how does it impact the comparative value of the private company today?

While the owner is waiting to exchange contracts and providing the company continues to be profitable then the investor presumably is still receiving a return. Has the investor lost out? Maybe not.

But what if in the time between the date of listing and closing an event occurs that negatively affects the value of the company and consequently the buyer expects to pay a lower price? How is that risk reflected in the comparative market value today? An option model may provide a possible answer.

# The impact of illiquidity

At 30 June Public Company A’s share are trading at $100 per share. Investor A has one share. The value of Investor A’s share at 30 June would then appear to be $100. Investor A can sell the share for $100 on 30 June.

Investor A, however, has a restraint on their share. They cannot sell the share on 30 June; they must wait nine months to sell and they must accept the market price at that time. How does the nine-month sales restraint impact the value of Investor A’s share at 30 June?

The impact on the value of Investor A’s share at 30 June depends on the expected value of Public Company A’s shares in nine months. If Company A’s share price is expected to fall to $90 in nine months’ time, then it would seem the value of Investor A’s share at 30 June is less than $100. There is a cost to illiquidity.

# Using a put option to value illiquidity

Investor A can protect against the risk of the trading price being less than $100 in nine months by acquiring a European put option. A European put option gives the option holder the right to exercise the option at a pre agreed exercise date and price (the strike price).

If Investor A buys a put option with a pre agreed exercise date being nine months from 30 June and a strike price of $100, then Investor A has essentially protected against having to sell below $100.

# The price of a time restraint

The price of the put option depends on the price volatility, the strike price, the time frame, and the risk-free rate. As volatility and or the time frame increase, all other factors remaining constant, the price of the put option increases. As the risk-free rate increases the price of the put option goes down.

The price of the put option can be calculated using the Black-Scholes model. Under the Black-Scholes model the stock price (price at 30 June) and the strike price (price in nine-months’ time) are the same, $100.

If the price volatility for Company A’s shares is 36%, the risk-free rate is 3% and the tax rate 25%, then according to the Black-Scholes model the price of the put option is approximately $10.

In essence the cost of the illiquidity imposed by the selling restraint is $10. The value of Investor A’s share at 30 June is the market price less the cost of the put option. Investor A’s share is worth $90 at 30 June. There is a 10% illiquidity discount.

# Application to valuing a private company

If a private company is being valued using public company market trading data (representing liquid shares), then the application of an illiquidity discount maybe applicable.

The figure below illustrates how industry price volatility and the sale period potentially increase the implied illiquidity discount. The illustration uses the Black-Scholes model and assumes that the current price equals the strike price, observed industry volatility rates, a sale period of either six, nine or twelve months, a risk-free rate of 3.0%, and a tax rate of 25.0%.[2]

The figure shows that industries with low share price volatility, such as grocery retail, will have an expected lower illiquidity discount compared to industries with a higher share price volatility such as oil and gas production. Further, the longer it is expected to take to sell the company the higher the discount rate.

Figure: Impact of share price volatility and sale period on the implied illiquidity discount

# Limitations

The limitations of the Black-Scholes model include the assumption that the risk-free rate and volatility are known and constant. Is past volatility a reasonable guide to future volatility?

Further, according to Barenbaum, Schubery and Garcia the use of a simple put option model significantly overstates any discount. This is because the put option not only guarantees that the investor will receive no less than the current value but allows the investor to maintain the benefit of any increase in the share price. The authors propose instead quantifying the discount using a loan secured by an at-the-money equity collar.[3]

# Summary

An option model provides a relatively simple method to estimate an illiquidity discount taking into account the price volatility and the time from listing to close. However, an option model may overstate that discount.

[1] 2023 Pepperdine Private Capital Markets Report, figure 111 median observations for deal size $2 million to $50 million.

[2] Industry volatilities per https://pages.stern.nyu.edu/~adamodar/New_Home_Page/dataarchived.html

[3] Barenbaum, Schubert and Garcia, Determining Lack of Marketability Discounts: Employing an Equity Collar, 2015 The Journal of Entrepreneurial Finance