**Valuation of share-based remuneration: importance of underlying assumptions**

There are particular circumstances when a company needs to calculate a fair value of share options or performance share awards. As the majority of performance shares in the UK are structured as nil-cost options this article refers throughout to options only. The most common circumstances are:

1. To recognise an accounting expense under IFRS2 or FRS102 (the Finance Director’s nightmare)

2. To agree the taxable value of the grant with HMRC – this can be needed, for example, to determine the taxable value (if any) on the acquisition of restricted securities, including growth shares or JSOP interests.

3. To ‘benchmark’ share-based rewards against competitive practice more precisely than would be possible using the ‘face value’ of the shares involved, for example where there are different performance conditions for the company’s own share-based rewards and for those of a comparator company.

4. To compare the value of share-based incentives with other parts of the remuneration package, where a trade-off between elements is being considered: such as a choice between share-based incentives and cash payments, or between different forms of long-term incentive.

In each case, another party has to be satisfied that the resulting fair value is indeed fair – the auditor on behalf of the shareholders, HMRC, the remuneration committee and the executives receiving the grants. Executives frequently prove to be the hardest to convince.

Nevertheless, it is probable that very few of these parties really understand the mathematics involved, and most take the calculation on trust or apply some standard formula. In fact, the final value is surprisingly sensitive to the valuation assumptions, such as share price volatility and the expected period before an option-holder chooses to exercise (“option life”).

Volatility is the key to calculating the value of share options and performance share awards with market-based vesting hurdles. The future pay-off from an option is a positive value or zero, depending on whether the share price at the time of exercise is higher than the exercise price (which is usually, but not necessarily, the share price at grant). There are two components to the price increase: the underlying drift of the share price (a function of market expectations) and the extent to which the seemingly random daily changes add up to produce a resultant increase or decrease. If a share price experiences large daily fluctuations, we say that it has high volatility. With a more volatile share price there is more chance of a high gain at exercise. There is also more chance of a low downside in the share price, but because the pay-off cannot be less than zero (the option holder just would not exercise), this does not cancel out the extra value from the possibility of a high upside.

Before we consider how different volatility assumptions affect the value of a share option, we need to find a workable definition of volatility. To calculate the daily volatility, we look at the standard deviation of the logarithm of the ratio of each day’s share price to that of the previous day. We then multiple this daily volatility by the square root of the number of trading days in the year to get the annualised volatility, which is the measure used in valuing options. We take the natural logarithm because it results in a normal “bell curve” for compounding returns – which makes it possible, later in the valuation process, to model future outcomes randomly in our valuation model. A key assumption in share price forecasting is that returns are normally distributed.

Not surprisingly, the value of a share option is highly sensitive to the assumption about share price volatility. The graph below shows how the fair value of an option varies with volatility in a typical company. For clarity, we have expressed the fair value as a percentage of the face value of the shares under option.

At 10% volatility, the fair value is 15% of the face value of the share. At 40% volatility, the fair value is 43% of the face value, with close to a straight line relationship in between. The fair value per share is almost three times as much at 40% volatility as it is at 10% volatility.

We can see that this variation matters when we make assumptions about likely future volatilities. The reality is we do not know what the future volatility will be or how it will vary. Most companies rely on the past as a predictor of the future. If the company issues traded options, we can work out the implied volatility (ie the volatility assumed by market makers) but this will not apply for most smaller companies.

The situation is even more difficult if we are dealing with a private company. The company is probably valued once a year for tax purposes or for internal share transfers. The valuation methodology typically uses a profit multiple, or maybe a projection of future profits. In this circumstance, there is no measurable “wiggle” in the share price. The company has to estimate its volatility, for example by using an average of the observed volatilities of listed peer companies in its sector to provide a proxy.

The table below shows how the volatility of one listed company’s shares has fluctuated, depending on the quarter over which it is measured. The volatility varied by a factor of three, depending on the period chosen, ie Q1 2017 vs Q2 2015. Neither historical period has a superior claim to representing the future. There may be industry characteristics for 2017 which suggest that figure is a better predictor because it is more recent, but it depends largely on judgement. As shown in the graph above, the volatility assumption has a crucial impact on the value of an option or performance share award.

**Conclusion**

Depending on the purpose of valuation, the company has a lot of opportunity to choose volatility assumptions which suit its own purpose, provided it can persuade the interested parties, HMRC, shareholders or executives, that the final result is reasonable – one might say “fair”. This persuasion/ negotiation is more important than the mathematical result. It could well be easier to take a rule of thumb of, say, 30% of face value (MM&K uses 30% of face value for share options in surveys and this is often talked about as a market norm) and agree with the relevant parties that that is a fair figure for the particular purpose in hand. Unfortunately HMRC is currently insisting that a Black-Scholes or similar option-pricing model is used for valuation of growth shares and JSOP interests, which therefore requires the use of a volatility assumption, even though we have shown this is effectively arbitrary.

Volatility is not the only assumption that introduces a large degree of imprecision. We also have to decide the likely behaviour of participants in exercising their options – in order to determine the option life (grant to exercise period). The graph below shows the impact of different option lives for the value of the option in a typical company. This is yet another reason for agreeing a rule of thumb.

For further information contact Damien Knight