1. Equity and debt in terms of options

Depending on the enterprise value when the debt matures, two outcomes are possible:

• The enterprise value is higher than the amount of debt to be redeemed (e.g. EV = 120). In this case, the shareholders let the company repay the lenders and take the residual value of 20.
• The enterprise value is lower than the amount of debt to be redeemed (e.g. EV = 70). The shareholders may then invoke their limited liability clause, forfeiting only their investment, and transfer the company to the lenders who will bear the difference between the enterprise value and their claim.

Now let us analyse this situation in terms of options. From an economic standpoint, shareholders have a call option (known as a European call if it can only be exercised at the end of its life) on the firm’s assets. Its features are:

• Underlying asset = capital employed.
• Exercise price = amount of debt to be reimbursed (100).
• Volatility = volatility of the underlying assets, i.e. the capital employed.
• Maturity = expiration date.
• Interest rate = risk-free rate corresponding to the maturity of the option.

At the expiration date, shareholders exercise their call option and repay the lenders, or they abandon it. The value of the option is none other than the value of equity (V _E).

The lender, on the other hand, who has invested in the firm at no risk, has sold the shareholders a put option on the capital employed. We have just seen that in the event of default, the creditors may find themselves the unwilling owners of the company. Rather than recouping the amount they lent, they get only the value of the company back. In other words, they have “bought” the company in exchange for the outstanding amount of debt.

The sale of this (European-style) put option results in additional remuneration for the debtholder, which, together with the risk-free rate, constitutes the total return. This is only fair, since the debtholder runs the risk that the shareholders will exercise their put option; in other words, that the company will not pay back the debt.

The features of the put option are:

• Underlying asset = capital employed.
• Exercise price = amount of debt redeemable upon maturity (100).
• Volatility = volatility of the underlying asset, i.e. the capital employed.
• Maturity = maturity of the debt.
• Interest rate = risk-free rate corresponding to the maturity of the option.

The value of this option is equal to the difference between the value of the loan computed by discounting its cash flows at the risk-free rate and its market value (discounted at a rate that takes into account the default risk, i.e. the cost of debt k _D). This is the risk premium that arises between any loan and its risk-free equivalent.

All this means is that the debtholder has lent the company 103 at an interest rate equal to the risk-free rate. The company should have received 103, but the value of the loan is only 100 after discounting the flows at the normal rate of return required in view of the company’s risk, rather than the risk-free rate.

The company uses the balance of 3, which represents the price of the credit risk, to buy a put option on the capital employed. In short, the company receives 100 while the bank pays 100 for a risky claim since it has sold a put option for capital employed that the company, and therefore the shareholders, will exercise if its value is lower than that of the outstanding date at maturity. By exercising the option, the company, and thus its shareholders, discharges its debt by transferring ownership of the capital employed to the creditors.

In conclusion, we see that, depending on the situation at the redemption date, one of the following two will apply:

• if V _D < V, the value of the call option is higher than 0, the value of the put option is zero and equity is positive;
• if V _D > V, the value of the call option is zero, the value of the put option is higher than 0 and the equity is worthless.

2. An options approach to financial securities

We have already seen that the additivity rule for equity and debt applies and that there is no connection between enterprise value and the type of financing:

Based on the preceding developments, we deduce that:

This brings us back to the fundamental equality between put and call options we examined in Chapter 23:

This underscores the relationship between the value of a call on capital employed and the value of a put on the same capital employed:

Section 34.2 Contribution of options theory to the valuation of equity

We have demonstrated that the value of a firm’s equity is comparable to the value of a call option on its capital employed. The option’s exercise price is the amount of debt to be repaid at maturity, the life of the option is that of the debt, and its underlying asset is the firm’s capital employed.

This means that, at the valuation date, the value of equity is made up of an intrinsic value and a time value. The intrinsic value of the call option is the difference between the present value of capital employed and the debt to be repaid upon maturity. The time value corresponds to the difference between the total value of equity and the intrinsic value.

Take, for example, a company where the return on capital employed is lower than that required by investors in view of the related risk. The market value is thus lower than the book value.

If the debt were to mature today, the shareholders would exercise their put option since the capital employed is worth only 70 while the outstanding debt is 80. The company would have to file for bankruptcy. Fortunately, the debt is not redeemable today but only in, say, two years’ time. By then, the enterprise value may have risen to over 80. In that case, equity will have an intrinsic value equal to the difference between the enterprise value at the redemption date and the amount to be redeemed (in our case, 80).

Today, however, the intrinsic value is zero and the present value of equity (8) can only be explained by the time value, which represents the hope that, when the debt matures two years hence, enterprise value will have risen enough to exceed the amount of debt to be repaid, giving the equity an intrinsic value.

As seen in the following graphs, a company’s financial position can be considered from either the shareholders’ or the creditors’ standpoint.

By now you must be eager to apply your newfound knowledge of options to corporate finance!

• The time value of an option increases with the volatility of the underlying asset.

The more economic or industrial risk on a company, the higher the volatility of its capital employed and the higher the time value of its equity.

The options method is thus used to value large, risky projects financed by debt, such as the Channel Tunnel, leisure parks, etc., or those with inherent volatility, such as biotech start-ups.

• The time value of an option depends on the position of the strike price relative to the market value of the underlying asset.

When the call option is out-of-the-money (enterprise value lower than outstanding debt), the company’s equity has only time value. Shareholders hope for an improvement in the company, whose equity has no intrinsic value.

When the call option is at-the-money (enterprise value equal to debt at maturity), the time value of equity is at its highest and anything can happen. Using the options method to value equity is now particularly relevant, since it can quantify shareholders’ anticipation.

When-the-money (enterprise value higher than outstanding debt at maturity), the intrinsic value of equity quickly outweighs the time value. The risk on the debt held by the lenders decreases and becomes nearly non-existent when the enterprise value tends towards infinity. This brings us back to the traditional idea that the higher the enterprise value, the less risk creditors have of a default, and the more the cost of debt approaches the risk-free rate.

The options method is therefore applied to companies that carry heavy debt or are very risky.

• The time value of an option increases with its maturity.

This is why it is so important for companies in distress to reschedule debt payments, preferably at very long maturities.

The example below illustrates the use of options to value equity.

Take a company that has both debt and equity financing and let us assume its debt is 100, redeemable in one year. If, based on its degree of risk, the debt carries 6% interest, then the amount to be repaid to creditors one year later is 106.

Traditional theory tells us that if the firm’s value is 150 at the time of calculation, then the value of equity - defined as the difference between enterprise value and the value of debt - will be 150 − 100 = 50.

What happens if we apply options theory to this value?

We shall assume the risk-free rate is 5%. The discounted value of the debt + interest payment at the risk-free rate is 106/1.05, or 100.95.

The value of debt can be expressed as:

i.e. the value of the put = 100.95 − 100 = 0.95.

We know that the value of equity breaks down into its intrinsic and time value:

You can see that, for this company with limited risk, the time value measuring the actual risk is far lower than the intrinsic value. Similarly, the value of the put, which acts as a risk premium, is very low as well.

Now, let’s increase the risk to the capital employed and assume that the interest rate required by the creditors is 15% rather than 6%, corresponding to a 10% risk premium. The amount to be repaid in one year is thus 115.

The value of the debt discounted at the risk-free rate is 115/1.05, or 109.52. The value of the put is thus 109.52 − 100 = 9.52.

Note that the risk premium for this company is much higher than in the preceding example, reflecting the increasing probability that the company will default on its debt.

The value of equity, which is still 50, breaks down into an intrinsic value of 35 
(150 − 115) and a time value of 15 (50 − 35). Since there is more risk than in our previous example, the time value accounts for a higher share of the equity value.

Section 34.3 Using options theory to analyse a company’s financial decisions

Options theory helps us understand how major corporate financial decisions (choice of capital structure, dividend payout, investment decisions, etc.) affect shareholders and creditors differently, and how they can result in a transfer of value between the two.

The table below lists the closing prices for a call option on a Daughter plc share at various exercise prices:

The enterprise value of Holding plc is equal to the number of Daughter plc shares multiplied by their market price, i.e. £223 000.

Consider each of the 100 shares booked under liabilities at Holding plc as being an option on its capital employed (the shares of Daughter plc), i.e. £223 000, with an exercise price that is equal to the amount of Holding plc debt outstanding, giving 300 bonds × £1000 = £300 000.

Each Holding plc share can thus be considered to be a call option with an exercise price of £300 000/100 shares = £3000, and a maturity of three years.

According to the table above, Holding plc’s equity value is thus £45 × 100 shares = £4500.

One bond is therefore worth £728.3 (£218 500/300), corresponding to an implied yield of 11.1% (in fact: 728.3 = 1000/(1 + 0.111)³).

We will now discuss a few major financing or investment decisions in a context of equilibrium – that is, where the debt, shares and assets held are bought or sold at their fair value, without the market having anticipated the decision.

1. Increasing debt

Suppose the shareholders of Holding plc decide to issue 20 additional bonds and use the proceeds to reduce the company’s equity by distributing an exceptional dividend. The overall exercise price corresponding to the redemption value of the debt at maturity is:

A look at the listed prices of the options shows us that at an exercise price of £3200, Holding plc’s equity is valued as £31 × 100 shares = £3100, indicating that the value of its debt at the same date is £219 900 (£223 000 − £3100).

The new bondholders will thus pay £13 744 (20 bonds × £219 900/320 bonds), which will go to reduce the equity of Holding plc.

The shareholders consequently have £13 744 in cash and £3100 in shares, i.e. a total of £16 844 compared with the previous £4500. They have gained £12 344 to the detriment of the former creditors, who have seen the value of their claim fall from £218 500 to 
300 bonds × £687.19, or £206 156.

Their loss (£218 500 − £206 156 = £12 344) exactly mirrors the shareholders’ gain. The implicit yield to maturity has risen to 13.3%, reflecting the fact that the borrowing has become riskier since it now finances a larger share of the same amount of capital employed.

Increasing the risk to creditors has enhanced the value of the shares, thereby reducing that of the bonds. The existing creditors have lost out because they were not able to anticipate the change in corporate structure and have been harmed by the dividend distribution.

Common (accounting) sense seems to indicate that distributing £13 744 in cash to shareholders should translate into an equivalent decrease in the value of their Holding plc shares. According to this reasoning, after the buy-back the Holding plc shares should have been revalued at −£9244 (£4500 − £13 744), but that cannot be!

Options theory solves this apparent paradox. It shows that when new debt is issued to reduce equity, the time value of the shares decreases less than the amount received by shareholders and remains positive. True, the likelihood that the value of Daughter plc shares will be higher than that of the redeemable debt upon maturity has lessened (since debt has increased), but it is still not nil, giving a time value that, while lower, is still positive.

Of course, this example is exaggerated. Such a decision would have catastrophic consequences for shareholders, who would be taken to court by the creditors and lose all credibility in the eyes of the market. But it effectively illustrates the contribution of options theory to equity valuations.

Increasing debt increases the value of shareholders’ investment, to the detriment of the claims held by existing creditors. Thus, value is transferred from creditors to shareholders.

Conversely, when debt is reduced by a capital increase, the overall value of shares does not increase by the value of the shares issued. The old debt, which has become less risky, has, in fact, “confiscated” some of the value, to the benefit of creditors and the detriment of shareholders.

Section 34.4 Resolving conflicts between shareholders and creditors

Creditors have a number of means at their disposal to protect themselves and overcome the asymmetry from which they suffer. They can be grouped under two main headings:

• hybrid financial securities;
• restrictive covenants.

Section 34.5 Analysing the firm’s liquidity

In most cases, the company pays off part of its debt with its free cash flows and refinances the balance of its debt by taking out a new loan. Most of the time, the sum of free cash flows is higher than the amount of debt to be repaid, but the flows generally are further off in time than the due date for the debt, and so are insufficient in the short term. The duration (see Chapter 20) of cash flows is generally longer than the duration of debt flows, which rarely exceed six to seven years.

The firm is then exposed to a double risk:

• the risk of the interest rate at which it will refinance part of its current debt in the future;
• a liquidity risk since, at the time the firm has to take out a new loan, market conditions may not allow it to if there is a major liquidity crisis under way (as was the case in late 2008/early 2009).

It is possible to hedge against these two risks, as we shall see in Chapter 50. Frequently, however, the liquidity risk is unhedged, either because it is not always possible to hedge against it, or because the cost of hedging is seen as prohibitive, or possibly because severe liquidity crises are so rare that it is not deemed necessary to hedge against this risk.

The difference between the duration of a firm’s free cash flows and the duration of its debt (often a shorter period) constitutes an asset liability refinancing gap (ALRG). 
Aït-Mokhtar (2008) has shown that it is the same as a liability for a firm, as if it had put itself in the position of selling a borrower FRA (see Chapter 50). On maturity of its debt, the firm will only be able to make the repayment if it is able to find lenders that are prepared to lend to it, since the free cash flows it receives will be insufficient to pay off the whole of the debt. So what it has done is undertaken to take out future debt at an unknown interest rate in order to continue its activity. In normal times, this liability is worth a negligible amount as it is reasonable to expect that a healthy firm will have no problems in refinancing in the future. But in the event of a liquidity crisis and for firms with imminent debt repayment deadlines (a few months or quarters), this ALRG has a very high value. It is equal to the existing uncertainty as to the possibility of the company being able to find the necessary financing.

So we can say:

When investors start to worry about the ability of the company to refinance in the near future, the value of the ALRG increases, pushing down the value of equity. And the phenomenon can pick up speed if the current lenders try and hedge their risks by selling short the firm’s shares, hoping to gain on this short-selling what they will lose as a result of the decline in the value of their debt.

When the firm is able to find refinancing for its debt, for example through a share issue, we see in some cases (Lafarge in 2009) an increase in the share price, which contradicts what we have seen up to now. On the one hand, the value of the share is negatively impacted by the transfer of value to the creditors, but on the other hand, it benefits fully from the disappearance of the ALRG. And if the latter were worth more than the discount on the debt, the net impact would be positive and the value of the share would rise.

Section 34.6 Conclusion

The concept of time value for equity is the main added value of the application of option theory to corporate finance.

We are now quite far away from the simple book leverage effect that seemed to prove that shareholders could create value by investing funds at a higher rate than the interest rate. The relationship between shareholders and lenders is in practice quite different. Their interests can actually diverge significantly due to a change in the risk profile of the firm, even if there are no cash flow exchanges between them and the enterprise value remains constant.

We hope that our readers will have understood the importance of reasoning in value terms and now have the reflex of assessing any financial decision in terms of return, but also risk. The use of options may have been overwhelming. We hope so, as readers will now always remember to assess risk and value transfers in financial decisions.

Summary

Questions

Exercises

Answers