Draft Evaluation of equity-based debt obligations Alexander Fromm - - PowerPoint PPT Presentation

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Evaluation of equity-based debt obligations

Alexander Fromm

• 28. March 2019

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Table of Contents

The Problem

EbDOs

Discrete time model

Continuous time model

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Participation rights

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.

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Participation rights

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?

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Participation rights

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

• n 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

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What do we expect from a good investment vehicle?

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.

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What do we expect from a good investment vehicle?

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.

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What do we expect from a good investment vehicle?

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.

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What do we expect from a good investment vehicle?

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).

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What do we expect from a good investment vehicle?

Invariance w.r.t. jurisdiction: The nature of the legal arrangement should not depend on the legal system or current legislation.

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What do we expect from a good investment vehicle?

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

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What do we expect from a good investment vehicle?

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.

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Definition

In the following, we use the term Equity-Based Debt Obligation (EbDO) for an arrangement according to which a company pays h(YT) to an investor, where T is the maturity, i.e. time of payment, YT 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.

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Example

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 X0 > 0, h(y) := α · y, where α > 0. Then: Y0 = X0 − h(Y0) = X0 − αY0,

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Example

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 X0 > 0, h(y) := α · y, where α > 0. Then: Y0 = X0 − h(Y0) = X0 − αY0, which results in Y0 =

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Example

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 X0 > 0, h(y) := α · y, where α > 0. Then: Y0 = X0 − h(Y0) = X0 − αY0, which results in Y0 =

Furthermore: h(Y0) = X0 ·

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Example

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 X0 > 0, h(y) := α · y, where α > 0. Then: Y0 = X0 − h(Y0) = X0 − αY0, which results in Y0 =

Furthermore: h(Y0) = X0 ·

In other words, the total money X0 is divided between the investor and the company with the ratio α : 1, where Y0 =

is the money, which remains with the company after the pay-off.

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Example

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 X0 > 0, h(y) := α · y, where α > 0. Then: Y0 = X0 − h(Y0) = X0 − αY0, which results in Y0 =

Furthermore: h(Y0) = X0 ·

In other words, the total money X0 is divided between the investor and the company with the ratio α : 1, where Y0 =

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.

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Properties

EbDOs Common Shares Predictability

• ✗

Incentives

• ✗

Flexibility

• ✗

Invariance

• ✗

Tax efficiency

• ✗

Simple issuance

• ✗

Neutrality

• ✗

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Fundamental Problem

Given the current equity before EbDO-debt X0 and n EbDOs h1(YT1), h2(YT2), . . . , hn(YTn), with non-negative maturities 0 ≤ T1 < T2 < . . . < Tn, calculate the current equity Y0, as well as the current fair value of every of the n EbDOs.

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Fundamental Problem

Given the current equity before EbDO-debt X0 and n EbDOs h1(YT1), h2(YT2), . . . , hn(YTn), with non-negative maturities 0 ≤ T1 < T2 < . . . < Tn, calculate the current equity Y0, 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.

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First Case

Assume an EbDO matures right now, i.e. at time 0: Y0 = X0 − h(Y0)

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First Case

Assume an EbDO matures right now, i.e. at time 0: Y0 = X0 − h(Y0) ⇐ ⇒ Y0 = (Id + h)−1(X0). Furthermore: h(Y0) = h ◦ (Id + h)−1(X0).

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First Case

Assume an EbDO matures right now, i.e. at time 0: Y0 = X0 − h(Y0) ⇐ ⇒ Y0 = (Id + h)−1(X0). Furthermore: h(Y0) = h ◦ (Id + h)−1(X0). If we have several, e.g. two, EbDOs, given by h and g maturing at time 0, the total pay-off is (h + g)(Y0) and we have a reduction to the previous situation: Y0 = (Id + h + g)−1(X0), h(Y0) = h ◦ (Id + h + g)−1(X0), g(Y0) = g ◦ (Id + h + g)−1(X0)

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Second Case

Assume T > 0 and we have only one EbDO given by h(YT). Again, only X0 is known. We postulate Xt = X0 + t

t

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Second Case

Assume T > 0 and we have only one EbDO given by h(YT). Again, only X0 is known. We postulate Xt = X0 + t

t

XT = X0 · Z, where Z ∼ LN

• µ − 1

2σ2

• · T, σ2 · T
• .

µ reflects how quickly assets inside the company grow (per unit

• f time) and σ how volatile this growth is.

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Second Case

Assume T > 0 and we have only one EbDO given by h(YT). Again, only X0 is known. We postulate Xt = X0 + t

t

XT = X0 · Z, where Z ∼ LN

• µ − 1

2σ2

• · T, σ2 · T
• .

µ reflects how quickly assets inside the company grow (per unit

• f time) and σ how volatile this growth is.

If there is no inflation or deflation µ = 0 should be set for book-keeping purposes.

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Second Case

Assume T > 0 and we have only one EbDO given by h(YT). Again, only X0 is known. We postulate Xt = X0 + t

t

XT = X0 · Z, where Z ∼ LN

• µ − 1

2σ2

• · T, σ2 · T
• .

µ reflects how quickly assets inside the company grow (per unit

• f time) and σ how volatile this growth is.

If there is no inflation or deflation µ = 0 should be set for book-keeping purposes. We get: YT = (Id + h)−1(XT) , h(YT) = h ◦ (Id + h)−1(XT),

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Second Case

Assume T > 0 and we have only one EbDO given by h(YT). Again, only X0 is known. We postulate Xt = X0 + t

t

XT = X0 · Z, where Z ∼ LN

• µ − 1

2σ2

• · T, σ2 · T
• .

µ reflects how quickly assets inside the company grow (per unit

• f time) and σ how volatile this growth is.

If there is no inflation or deflation µ = 0 should be set for book-keeping purposes. We get: YT = (Id + h)−1(XT) , h(YT) = h ◦ (Id + h)−1(XT), Y0 = E

• (Id + h)−1(XT)
• , E[h(YT)] = E
• h ◦ (Id + h)−1(XT)
• .

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Third Case

We have two EbDOs given by g(Y0) and h(YT), where T > 0. Again, X0 is known. To model XT, first define X ′

company’s net assets after paying out g(Y0), but before h(YT). This value is unknown, however: XT := X ′

• µ − 1

2σ2

• · T, σ2 · T
• .

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Third Case

We have two EbDOs given by g(Y0) and h(YT), where T > 0. Again, X0 is known. To model XT, first define X ′

company’s net assets after paying out g(Y0), but before h(YT). This value is unknown, however: XT := X ′

• µ − 1

2σ2

• · T, σ2 · T
• .

We get: Y0 = E

• (Id + h)−1(XT)
• =: f(X ′

Furthermore: X ′

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Third Case

We have two EbDOs given by g(Y0) and h(YT), where T > 0. Again, X0 is known. To model XT, first define X ′

company’s net assets after paying out g(Y0), but before h(YT). This value is unknown, however: XT := X ′

• µ − 1

2σ2

• · T, σ2 · T
• .

We get: Y0 = E

• (Id + h)−1(XT)
• =: f(X ′

Furthermore: X ′

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Third Case

We have two EbDOs given by g(Y0) and h(YT), where T > 0. Again, X0 is known. To model XT, first define X ′

company’s net assets after paying out g(Y0), but before h(YT). This value is unknown, however: XT := X ′

• µ − 1

2σ2

• · T, σ2 · T
• .

We get: Y0 = E

• (Id + h)−1(XT)
• =: f(X ′

Furthermore: X ′

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General Case

We have n EbDOs hi(YTi) with increasing Ti and T1 = 0. Let us now write Yi instead of YTi for short etc. Again, define X ′

EbDOs, but before subtracting the others.

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General Case

We have n EbDOs hi(YTi) with increasing Ti and T1 = 0. Let us now write Yi instead of YTi for short etc. Again, define X ′

EbDOs, but before subtracting the others. Also, define Xi = X ′

where Zi has a log-normal distribution.

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General Case

We have n EbDOs hi(YTi) with increasing Ti and T1 = 0. Let us now write Yi instead of YTi for short etc. Again, define X ′

EbDOs, but before subtracting the others. Also, define Xi = X ′

where Zi has a log-normal distribution. We define fn−1(X ′

• (Id + hn)−1(Xn)
• = Yn−1,

which we use to get Yn−1 = (f −1

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General Case

We have n EbDOs hi(YTi) with increasing Ti and T1 = 0. Let us now write Yi instead of YTi for short etc. Again, define X ′

EbDOs, but before subtracting the others. Also, define Xi = X ′

where Zi has a log-normal distribution. We define fn−1(X ′

• (Id + hn)−1(Xn)
• = Yn−1,

which we use to get Yn−1 = (f −1

Yn−2 = En−2

• (f −1

• = fn−2(X ′

Similarly, construct fn−3 from fn−2 etc. Finally, we obtain Y0 = (f −1

+ h1)−1(X0).

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General Case

To sum up, we have constructed non-negative random variables Xi, Yi, X ′

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General Case

To sum up, we have constructed non-negative random variables Xi, Yi, X ′

The filtration (Fi)i∈{0,...,n} is generated by independent Zi ∼ LN

• µ − 1

2σ2

• · (Ti − Ti−1) , σ2 · (Ti − Ti−1)
• ,

i = 1, . . . , n.

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Pros and Cons

Advantages: straight-forward numerics processes X, X ′, Y can be simulated using XTi = X ′

+ hi)−1(XTi) → get all expected pay-offs using Monte-Carlo-simulation We can simulate with a different µ than 0. → Investors can calculate how much they will get depending on their profitability assumptions.

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Pros and Cons

Advantages: straight-forward numerics processes X, X ′, Y can be simulated using XTi = X ′

+ hi)−1(XTi) → get all expected pay-offs using Monte-Carlo-simulation We can simulate with a different µ than 0. → Investors can calculate how much they will get depending on their profitability assumptions. Disadvantages: possibly slow calculation for large number of maturities All performance-based debt must be modelled via EbDOs.

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Continuous time model

Assume we have so many different EbDOs that they effectively mature continuously in time with a certain rate function h : [0, T] × [0, ∞) → [0, ∞). Then, we may assume that X, Y satisfy Xs = X0− s h(r, Yr) dr + s Xr · σ dWr Ys = XT − T

Zr dWr, s ∈ [0, T]

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Continuous time model

Assume we have so many different EbDOs that they effectively mature continuously in time with a certain rate function h : [0, T] × [0, ∞) → [0, ∞). Then, we may assume that X, Y satisfy Xs = X0− s h(r, Yr) dr + s Xr · σ dWr Ys = XT − T

Zr dWr, s ∈ [0, T] Under the condition

above FBSDE can be shown using the method of decoupling fields.

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Continuous time model

Assume that h is uniformly Lipschitz continuous in y such that h(·, 0) = 0.

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Continuous time model

Assume that h is uniformly Lipschitz continuous in y such that h(·, 0) = 0. Theorem For all X0 ≥ 0 the FBSDE has a unique square integrable solution (X, Y, Z) and we have 0 ≤ Y ≤ X.

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Continuous time model

Assume that h is uniformly Lipschitz continuous in y such that h(·, 0) = 0. Theorem For all X0 ≥ 0 the FBSDE has a unique square integrable solution (X, Y, Z) and we have 0 ≤ Y ≤ X. Moreover, there exists a decoupling field u : [0, T] × [0, ∞) → [0, ∞) s.t. u(s, Xs) = Ys and s.t.: u(s, ·) is Lipschitz continuous uniformly in s, u(s, 0) = 0 and u(s, ·) is monotonically increasing for every s.

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Continuous time model

Assume that h is uniformly Lipschitz continuous in y such that h(·, 0) = 0. Theorem For all X0 ≥ 0 the FBSDE has a unique square integrable solution (X, Y, Z) and we have 0 ≤ Y ≤ X. Moreover, there exists a decoupling field u : [0, T] × [0, ∞) → [0, ∞) s.t. u(s, Xs) = Ys and s.t.: u(s, ·) is Lipschitz continuous uniformly in s, u(s, 0) = 0 and u(s, ·) is monotonically increasing for every s. Once u is obtained we can simulate the whole system and calculate all expectations (also w.r.t. non-martingale measures).

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Pros and Cons

Advantages: existing theory to show existence and uniqueness FBSDEs are a highly flexible modelling tool:

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Pros and Cons

Advantages: existing theory to show existence and uniqueness FBSDEs are a highly flexible modelling tool:

Disadvantages: have to replace/approximate classic EbDOs with EbDOs which ”mature” gradually over time with some rate, need fast and easy to use numerical methods for coupled FBSDEs: e.g. automatic choice of discretization, ”black-box” type algorithm/package.