Valuing Hybrid Securities and Noncontrolling Interests  353 • Option-pricing model. If the company’s enterprise value has changed since the last financial filing, estimate the value using option valuation models such as Black-Scholes or more advanced binomial (lattice) mod- els. Under U.S. GAAP and IFRS, the notes to the balance sheet report the total value of all employee stock options outstanding, as estimated by such option-pricing models. Note that the balance sheet value is a good approximation only if your estimate of share price is close to the one underlying the option values in the annual report. Otherwise, you need to create a new valuation using an option-pricing model.28 The notes disclose the information required for valuation. • Exercise value approach. The exercise value approach provides only a lower bound for the value of employee options, the smallest value that would round up to the estimated value. It assumes that all options are exercised immediately and thereby ignores the time value of the op- tions. The resulting valuation error increases as options have longer time to maturity, the company’s stock has higher volatility, and the company’s share price is closer to the exercise price. Given that a more accurate valuation is already disclosed in the annual report, we do not recommend this method. However, it is still quite common among practitioners. Exhibit 16.5 provides an example of the three valuation methods. The first data column is based on the fair value reported by the company, which it calls “aggregate intrinsic value.” The second and third data columns use the Black-Scholes option-pricing model to value first the outstanding options and second the options that can be currently exercised. The value of outstanding options will be less than that of options that can be exercised, because out- standing options include some options that will be lost if the employee leaves the company. To estimate the value of employee stock options, you need six inputs: the current stock price, the average strike price, the stock’s volatility, the risk-free rate, the time to maturity, and the stock’s dividend rate. Square’s current share price equals $56.09. The other inputs are disclosed in Square’s 10-K for both outstanding and exercisable options. For outstanding options, the weighted average strike price equals $9.52, the volatility of Square’s shares equals 30.9 percent, and the average time to maturity is reported at 5.45 years. The cur- rent risk-free rate over five years is 2.51 percent, and the expected dividend rate is zero. The Black-Scholes estimator prices the average option at $47.81.29 28 For more on the valuation of employee stock options, see, for example, J. Hull and A. White, “How to Value Employee Stock Options,” Financial Analysts Journal 60, no. 1 (January/February 2004): 114–119. 29 Using Black-Scholes to determine the value of a single option on an average strike price will undervalue a portfolio of options with a spread of strike prices. Unless you know the spread of strike prices, you cannot measure the bias. 354  Moving from Enterprise Value to Value per Share With 33.2 million options outstanding, the aggregate value of options is val- ued at $1.58 billion. To estimate share price, deduct the aggregate value from enterprise value, and divide by the number of undiluted shares. Since some outstanding options will go unclaimed, repeat the process for just the options that can be exercised. The actual value will fall somewhere between the two. Under the exercise value approach, employee options are assumed to be exercised immediately. According to Square’s 2018 10-K, 33,152,881 shares can be exercised at an average strike price of $9.52, for total proceeds of $315.6 million. Exercise of employee options generates cash for the company and in- creases shares outstanding from 419.7 million to 452.8 million. Dividing equity value by diluted shares leads to a value of $56.1, slightly lower than the value under the Black-Scholes method. Noncontrolling Interests by Other Companies When a company controls a subsidiary but does not fully own it, the subsid- iary’s financial statements must be fully consolidated in the group accounts. The subsidiary’s assets and liabilities will be indistinguishable from the parent company’s accounts, but the portion of the subsidiary’s equity not owned by the parent company will be separated from other equity accounts as noncon- trolling interest.30 Since the full value of the subsidiary will be incorporated EXHIBIT 16.5  Square Employee Options, December 2018 $ million Capital structure Using Black-Scholes2 Aggregate intrinsic value1 Value of outstanding options Value of exercisable options Exercise value approach Enterprise value 26,300.0 26,300.0 26,300.0 26,300.0 Convertible debt at 0.375% due 2022 (515.7) (515.7) (515.7) (515.7) Convertible debt at 0.5% due 2023 (901.5) (901.5) (901.5) (901.5) Convertible note hedge 230.9 230.9 230.9 230.9 Employee options: value (1,543.8) (1,584.9) (1,507.9) – Employee options: exercise proceeds – – – 315.6 Equity value 23,570.0 23,528.9 23,605.8 25,429.4 Number of shares, millions Number of nondiluted shares 419.7 419.7 419.7 419.7 New shares issued – – – 33.2 Number of diluted shares 419.7 419.7 419.7 452.8 Value per share, $ 56.1 56.0 56.2 56.1 1 Value of options reported in 2018 10-K in note 15, “Shareholder’s Equity,” under “Aggregate Intrinsic Value.” 2 Value estimated using Black-Scholes option-pricing model and company-disclosed inputs. 30 For example, Berkshire Hathaway reported $3.8 billion in noncontrolling interests in 2018. This amount can be found on the company’s balance sheet under shareholders’ equity.