Draft Evaluation of equity-based debt obligations Alexander Fromm University of Jena CMS-MMEI-2019, TU Chemnitz 28. March 2019 1 / 18
Table of Contents Draft The Problem 1 EbDOs 2 Discrete time model 3 Continuous time model 4 2 / 18
Participation rights Draft Assume a company wants to obtain goods, services, money, information, software or other assets which are of value for or are needed by the company from a given investor in return for a certain share of future profits. 3 / 18
Participation rights Draft Assume a company wants to obtain goods, services, money, information, software or other assets which are of value for or are needed by the company from a given investor in return for a certain share of future profits. Fundamental question: What does ”share of future profits” mean exactly? 3 / 18
Participation rights Draft Assume a company wants to obtain goods, services, money, information, software or other assets which are of value for or are needed by the company from a given investor in return for a certain share of future profits. Fundamental question: What does ”share of future profits” mean exactly? There is a whole spectrum of different arrangements depending on how close the instrument/obligation issued by the company is to the company’s equity (e.g. common shares) or to fixed income debt (e.g. bonds). � Mezzanine capital 3 / 18
What do we expect from a good investment vehicle? Draft Predictability: Being able to tell in advance how much the investor will get depending on the performance of the company, but also when he/she will get it. 4 / 18
What do we expect from a good investment vehicle? Draft Predictability: Being able to tell in advance how much the investor will get depending on the performance of the company, but also when he/she will get it. Good incentives: Investor and company should be all in the same boat in terms of profits, s.t. investors cannot benefit at the expense of the company and vice versa. This is especially important if executives are compensated with participation rights. 4 / 18
What do we expect from a good investment vehicle? Draft Predictability: Being able to tell in advance how much the investor will get depending on the performance of the company, but also when he/she will get it. Good incentives: Investor and company should be all in the same boat in terms of profits, s.t. investors cannot benefit at the expense of the company and vice versa. This is especially important if executives are compensated with participation rights. Flexibility: Tailor made conditions in order to fit a given investor, e.g. in terms of maturity, minimal or maximal pay-off etc. 4 / 18
What do we expect from a good investment vehicle? Draft Predictability: Being able to tell in advance how much the investor will get depending on the performance of the company, but also when he/she will get it. Good incentives: Investor and company should be all in the same boat in terms of profits, s.t. investors cannot benefit at the expense of the company and vice versa. This is especially important if executives are compensated with participation rights. Flexibility: Tailor made conditions in order to fit a given investor, e.g. in terms of maturity, minimal or maximal pay-off etc. Easy issuance: Issuance by the management subject to simple criteria without necessary approval by other investors (including shareholders). 4 / 18
What do we expect from a good investment vehicle? Draft Invariance w.r.t. jurisdiction: The nature of the legal arrangement should not depend on the legal system or current legislation. 5 / 18
What do we expect from a good investment vehicle? Draft Invariance w.r.t. jurisdiction: The nature of the legal arrangement should not depend on the legal system or current legislation. Tax efficiency: The company’s estimated debt from participation rights should be tax deductible. � lower corporate taxes 5 / 18
What do we expect from a good investment vehicle? Draft Invariance w.r.t. jurisdiction: The nature of the legal arrangement should not depend on the legal system or current legislation. Tax efficiency: The company’s estimated debt from participation rights should be tax deductible. � lower corporate taxes Neutrality in terms of corporate governance: Issuance of new participation rights should not have an effect on existing corporate architecture and how (and by whom) control is exercised. Also, it should be consistent with the managements commitment to maximize long term profits. 5 / 18
Definition Draft In the following, we use the term Equity-Based Debt Obligation (EbDO) for an arrangement according to which a company pays h ( Y T ) to an investor, where T is the maturity, i.e. time of payment, Y T is the company’s equity (= assets - all debts) at time T , h is the pay-off-function, which must be monotonically increasing and non-negative. W.l.o.g. we assume h ( 0 ) = 0. For practical purposes it is sufficient to consider h which are piecewise linear and continuous. 6 / 18
Example Draft Assume the company has issued only one EbDO, which matures now, i.e. T = 0, the equity without the EbDO would have been some value X 0 > 0, h ( y ) := α · y , where α > 0. Then: Y 0 = X 0 − h ( Y 0 ) = X 0 − α Y 0 , 7 / 18
Example Draft Assume the company has issued only one EbDO, which matures now, i.e. T = 0, the equity without the EbDO would have been some value X 0 > 0, h ( y ) := α · y , where α > 0. X 0 Then: Y 0 = X 0 − h ( Y 0 ) = X 0 − α Y 0 , which results in Y 0 = 1 + α . 7 / 18
Example Draft Assume the company has issued only one EbDO, which matures now, i.e. T = 0, the equity without the EbDO would have been some value X 0 > 0, h ( y ) := α · y , where α > 0. X 0 Then: Y 0 = X 0 − h ( Y 0 ) = X 0 − α Y 0 , which results in Y 0 = 1 + α . Furthermore: h ( Y 0 ) = X 0 · 1 + α . α 7 / 18
Example Draft Assume the company has issued only one EbDO, which matures now, i.e. T = 0, the equity without the EbDO would have been some value X 0 > 0, h ( y ) := α · y , where α > 0. X 0 Then: Y 0 = X 0 − h ( Y 0 ) = X 0 − α Y 0 , which results in Y 0 = 1 + α . Furthermore: h ( Y 0 ) = X 0 · 1 + α . α In other words, the total money X 0 is divided between the X 0 investor and the company with the ratio α : 1, where Y 0 = 1 + α is the money, which remains with the company after the pay-off. 7 / 18
Example Draft Assume the company has issued only one EbDO, which matures now, i.e. T = 0, the equity without the EbDO would have been some value X 0 > 0, h ( y ) := α · y , where α > 0. X 0 Then: Y 0 = X 0 − h ( Y 0 ) = X 0 − α Y 0 , which results in Y 0 = 1 + α . Furthermore: h ( Y 0 ) = X 0 · 1 + α . α In other words, the total money X 0 is divided between the X 0 investor and the company with the ratio α : 1, where Y 0 = 1 + α is the money, which remains with the company after the pay-off. However, a more typical pay-off function is h ( y ) := α · ( y − r ) + , where r > 0 is some reference value, e.g. the company’s (past) equity at the moment the EbDO was issued. 7 / 18
Properties Draft Common EbDOs Shares Predictability � ✗ Incentives � ✗ Flexibility � ✗ Invariance � ✗ Tax efficiency � ✗ Simple issuance � ✗ Neutrality � ✗ 8 / 18
Fundamental Problem Draft Given the current equity before EbDO-debt X 0 and n EbDOs h 1 ( Y T 1 ) , h 2 ( Y T 2 ) , . . . , h n ( Y T n ) , with non-negative maturities 0 ≤ T 1 < T 2 < . . . < T n , calculate the current equity Y 0 , as well as the current fair value of every of the n EbDOs. 9 / 18
Fundamental Problem Draft Given the current equity before EbDO-debt X 0 and n EbDOs h 1 ( Y T 1 ) , h 2 ( Y T 2 ) , . . . , h n ( Y T n ) , with non-negative maturities 0 ≤ T 1 < T 2 < . . . < T n , calculate the current equity Y 0 , as well as the current fair value of every of the n EbDOs. Furthermore, these calculations should be based on reasonable assumptions about the future evolution of X and Y , which is subject of uncertainty. 9 / 18
First Case Draft Assume an EbDO matures right now, i.e. at time 0: Y 0 = X 0 − h ( Y 0 ) 10 / 18
First Case Draft Assume an EbDO matures right now, i.e. at time 0: Y 0 = ( Id + h ) − 1 ( X 0 ) . Y 0 = X 0 − h ( Y 0 ) ⇐ ⇒ Furthermore: h ( Y 0 ) = h ◦ ( Id + h ) − 1 ( X 0 ) . 10 / 18
First Case Draft Assume an EbDO matures right now, i.e. at time 0: Y 0 = ( Id + h ) − 1 ( X 0 ) . Y 0 = X 0 − h ( Y 0 ) ⇐ ⇒ Furthermore: h ( Y 0 ) = h ◦ ( Id + h ) − 1 ( X 0 ) . If we have several, e.g. two, EbDOs, given by h and g maturing at time 0, the total pay-off is ( h + g )( Y 0 ) and we have a reduction to the previous situation: Y 0 = ( Id + h + g ) − 1 ( X 0 ) , h ( Y 0 ) = h ◦ ( Id + h + g ) − 1 ( X 0 ) , g ( Y 0 ) = g ◦ ( Id + h + g ) − 1 ( X 0 ) 10 / 18
Second Case Draft Assume T > 0 and we have only one EbDO given by h ( Y T ) . Again, only X 0 is known. � t � t We postulate X t = X 0 + 0 µ X s d s + 0 σ X s d W s for t ∈ [ 0 , T ] : 11 / 18
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