The Variables

In the pervious chapter the volatility was estimated using simulation. Before the binomial lattice can be set up, the other parameters need to be estimated.

Underlying asset

The underlying asset in the PE options is the PE target, but as mentioned above it is the underlying asset’s market value which is relevant not the value it has to the PE firm. Nevertheless, the underlying asset is assumed to have different characteristics in the holding period and post exit period. The different characteristics are caused by active ownership of the PE firm. It could be argued that these changes made by the PE firm does not influence the value to the potential buyer, as they are specific actions made to increase the value for the PE firm, and should therefore not be included in valuation of the option. But since the vast majority of the return of an PE investment comes from the exit of the portfolio company, I believe, it is a fair assumption that the changes made by the PE firm in the holding period are focused on increasing the market value of the target, as this would increase the return. Therefore, the assumption that the underlying asset has different characteristics in the holding period and post exit period is valid.

In the binomial model, as I have chosen to use it, the changes made by the PE firm are indirectly implemented via the volatility (Standard deviation), and hence the different volatilities. Note the development of the underlying asset is estimated using the up and down factors, which in turn are determinate by the volatility. Because of that, and because the volatility has great influence on the value of the option, the estimation of the volatility is essential to the value of the underlying asset and thereby the option and potential improvements should be implemented in the volatility (chapter 7.3). In the Post exit period the company is assumed to enter a more steady state with no activities from the PE firm (Chapter7.2).

Since the evolution of the underlying asset is determined by the u and d factors the value at of S0

is essential for the value of the option. In chapter 7.3 I established that the DCF model was the most appropriate to use as valuation tool for the underlying asset. Therefore, it is also the DCF value which should be used as S0 and not the value from the LBO model.

Exercise Price

The exercise price is the costs that the firm has when exercising their option. In many real options this is an actual cost associated with for example investing in new business, product or other investments needed in order to get a return from exercising the option. The PE firm’s options do not have any significant direct cost of exercising the option (exiting the target). Of cause there are some transaction costs, but they are not large enough to influence the value of the option, and are therefore assumed to be zero. This does not mean that the exercise price is zero on the option. The cost for PE firms when exiting is the required return from all stakeholders in the investment. As described in chapter 7.2.4 the IRR contains all stakeholders required return, and when the equity is discounted using the IRR we get the equity value needed to give the required return. Therefore, in the Holding option, the PE firm should only exercise the option, if the value of the underlying asset is higher than the firm value at the given point in time that gives the stakeholders their required return (required return value). Or said in another way, by exiting before maturity (expected exit) the PE firm gives up its right to sell at maturity which the cost of exercising the option i.e. exercise price. Hence the exercise price is calculated as the equity extrapolated with the IRR plus the remaining debt cf. the LBO model.

!

= B\

+ A]

= B\

∙ 1 + ^__

+ A]

− `

∙ 

Source: Own construction4

The same general principle goes for the Post Exit option. The return should at least be the IRR before exiting. The difference here is that the characteristic of the company has changed and the capital structure can be assumed to be constant (chapter 7.2).

The debt obligations which consumed most of the cash flow in the holding period would now be reduced considerably, and the assumption that all cash flow goes to debt repayment can now be relaxed, hence, the company has cash flow available to pay out to investors. The PE firm is keeping the target longer than expected and unless any excess cash flow can be invested in positive NPV projects, they should be paid out to investors. Because the stakeholders are now receiving dividends they are realising some of their expected return (IRR), which in turn means that the required return value of the company is correspondingly lower. Thus, the equity is extrapolated with the IRR and each period’s dividends are subtracted and the periods debt level is added to get the required return value in each period for the Post Exit option.

!

= B\

− cd

+ ]

= B\

∙ 1 + ^__

− cd

+ ]

Source: Own construction

The dividends can be calculated directly from the financial statement using the expected dividend policy, while the debt is calculated using the assumption of the development of the debt. What the dividends policy and debt assumptions are will obviously vary from target to target. I believe though, that there are one scenario which can be assumed to be applicable in most cases: pay all net earnings in dividends and a constant debt-equity ratio. The dividend payments is due to the investors wanting some return realised, and the debt because, as the argumentation for using DCF as valuation tool, it is a reasonable assumption when the debt future debt structure is unknown. Hence, it is only the equity that changes according to the expected return and the total value is found using the debt-equity ratio and the formula can be written as:

!

= B\

− cd

+ ]

= )B\

∙ 1 + ^__

− cd

/ ∙A]_B\

Source: Own construction

Alternatively the debt level could be assumed to be constant, as a constant debt-equity ratio assumes that the debt raises and falls proportional with equity, causing a rapidly increasing/decreasing exercise price.

The above of cause gives a changing exercise price in both options, which needs to be incorporate in the Option value lattice.

Time to Expiration

In many real options it is hard to determine the exact time to maturity. For example, it can be hard to determine for how long a business opportunity will be there, which depends on whether the option is proprietary or shared. When the option is shared, the value of the option is not necessary increasing with time to expiration, as competitors can also exercise the option. The PE firm’s Exit options on the other hand are completely proprietary as the PE firm are the only one who can exercise the option and sell the portfolio company, which as a starting point entails an increase in the value when time to expiration increases.

The Holding option follows the estimated holding period and the time to expiration is then simply the same as the estimated holding period. It is more complicated when dealing with the PE Post Exit option. In theory, there is no upper bound on the time horizon for the PE Post Exit option, as the PE firm in principle can keep the firm as long as it wants. This is no problem in the short run, but unless the PE firm neglect the fact that they are an on-going concern that needs investors for future investments, they need to keep their obligations to investors. One of those obligations is the time horizon of their investments. As mentioned earlier, because of the illiquidity of investments in PE firms, investors are highly dependent on the PE firm’s ability to sell its portfolio companies and time value of money is essential. Therefore, because investors are crucial for the existence of PE firm, they need to show both current but also future investors that they fulfil their obligations. Hence, I find it reasonable to assume that the time horizon of the Post Exit option is determined by each fund’s agreed lifespan, meaning that the length of the option on each portfolio company depends on when it was acquired in the life of the fund. (See

figure 4). As stated earlier, the lifespan of PE funds is 10 years with the possibility of extending it with up to a further three years to a total of 13 years. So a firm acquired in the third year with exit planned six years later, will have the option to postpone exit for four years. This is of cause very case dependant, and in some cases it could maybe be possible to extend it further into the future. But in the further analysis I will assume that life plus three years is the time horizon of the option.

Another consequence of on going concern arises when the Post Exit option is at expiration. Normally when using real options, the holder can chose not to exercise the option at expiration, in this case exiting the portfolio company. But based on the same arguments as above, this is not necessarily the case for PE firms, as they again will not fulfil their obligations, and it is therefore questionable weather the fact that the PE firm can actually choose not to exit at expiration in realistic. A reasonably assumption here is that if the option goes all the way to maturity the PE firm will have to exit. This means that the equation for the value at the terminal node is no longer a max formula but changes to simply5:



=

− B

Source: Own construction

The reasoning behind this is that the PE firm has to incorporate the fact that because the choice to exit is not valid, the risk associated with the option is higher, and the option could have a negative value at expiration which should be discounted back like any positive value. Weather this results in a negative total value depends on the characteristics of the option.

Of cause the two assumptions above can be relaxed in some situations, but as assumptions for a general case I find them reasonable and realistic.

Risk-free rate of return

Because of the use of risk-neutral probabilities, the time value of money for the underlying asset is the risk-free rate of return. The job here is to find an acceptable measure of what the risk free rate of return is. The closest to a risk free interest rate is a government bond or bond guaranteed by a government, and because the interest rate is used over the entire life of the option, a bond with the same maturity as the option is the most accurate to use.

5

Dividends

As mentioned in section 2.2, most PE firms do not pay dividends from their portfolio companies in the holding period, and it is therefore reasonable to assume that the PE Holding option has no dividends. As mentioned above, it is likely that dividends are paid in the Post Exit Option. Besides using the dividends to determine the exercise price, they should also be incorporated in the valuation of the option through the risk-neutral probabilities. In order to do that the dividends must be given as a continuous percentage (chapter 5.1).