Acquisition risk swap

David Dewhurst *

September 24, 2024

*Defense Advanced Research Projects Agency (DARPA), Arlington, VA 22203, USA. david.dewhurst@darpa.mil

Abstract

We introduce an acquisition risk swap (ARS), a type of subordinated risk swap that could be used to hedge the risk of being unable to acquire a capability during an uncertain future time interval. Examples of such scenarios include a nation being unable to procure specific military capabilities in the event of a future conflict or a firm being unable to procure a product component in the event of a future pivot to a new market. We describe the contractual structure of an ARS, pricing formulae, and possible market structures that could facilitate the efficient exchange of ARS. Simulations of ARS price distributions under a variety of example event probability and capability value models indicate typical positive skew, implying the need for a writer of ARS to hedge their position. We outline a market structure that decouples pricing and sale of ARS from creation of the capability to be acquired. Participation in this market enables the writer of ARS to hedge its risk and is a positive-sum outcome for both the writer of the ARS and the creator of the capability.

1 Introduction

Swaps are financial instrument that enable two or more parties to exchange value flows at pre-defined times or upon the occurrence of precisely defined events. Subordinated risk swaps are a generic class of swaps that trade a firm’s idiosyncratic risk (e.g., managerial or event risk) for a series of cash flows. We introduce an acquisition risk swap, a type of subordinated risk swap designed to enable the acquiring agent (“acquirer”) to hedge the risk of being unable to acquire a capability at an uncertain time in the future. In the case of defense acquisition, which is the motivating driver of this work, an example of such a risk is that associated with the widely reported shortage of 155mm shells used in modern artillery pieces1 .

We argue that adoption of ARS by users of a capability might be desirable for at least three interrelated reasons – risk reduction, efficiency, and transparency:

In Section 2 we outline the basic contract structure of an ARS and derive explicit formulae for pricing it under multiple assumptions, some of which we then relax. Section 3 describes possible market structures that could facilitate pricing and exchange of ARS, while Section 4 describes empirical distributions of ARS prices under multiple different modeling assumptions. We close by describing possible generalizations of our work in Section 5.

2 Asset definition and pricing

We outline the asset definition and pricing in discrete time; an extension to continuous time should be straightforward. An agent that needs a capability during a future time interval {t,...,T} holds the ARS, paying a coupon of mτ∈{0,...,t - 1}, where t ≤∞ (the ARS may never be exercised, in which case the ARS is a perpetual obligation). When the capability is needed – the a priori unknown future time t – the holder of the ARS ceases to pay the coupon and is delivered a capability that has value vτ∈{t,...,T}.

We solve for the coupon rate mτ by finding such a value that makes the cash flows equivalent at time zero. Defining the valuation V (t1,t2) = τ=t1t2-1βτ[mτδ(τ < t) + vτδ(τ t)], where β = -1-
1+r and r is the risk free rate, the coupon payments are those values mτ that satisfy V (0,t) = V (t,T). With the simplifying assumption that mτ = m, τ ∈{0,...,t - 1}, the coupon payment is m = β-t1-
β-1 τ=tT-1βτvτ, or under the additional assumption that the value of the acquired capability is equal at all times τ ∈{t,...,T - 1}, m = vβT-βt
βt-1. Here, m is a random variable, as it depends on the random time interval {t,...,T - 1} over which the capability is required. Modeling the time interval with the joint distribution (t,T) ~ p(t,T), we take the coupon payment as the expected value of m, ˆm = E(t,T)~p(t,T)[v T  t
ββt--β1-].

More generally, the interest rate rτ and the valuation of the capability vτ may vary in time. Interest rates may be correlated with the existence or nature of the conditioning event; for example, conflict could lead to shortages of basic necessities such as food and fuel, raising inflation which could cause a central bank to raise interest rates [BK84]. The value of one capability immediately after the inception of a conflict could be very high but decrease as time progresses, while another capability could increase in importance as a conflict progresses. With these considerations in mind, a more general valuation of the ARS is m = E(r,v,t,T)~p(r,v,t,T)[∑T -1 τ
-∑τt=-t1 βτvττ
   τ=0βτ], where analogously βτ = --1-
1+rτ.

3 Market structure

We undertake a brief discussion of naive and more nuanced market structures for ARS, and discuss methods by which price discovery for ARS is likely to occur. We leave in-depth analysis of possible market structures for future work.

The mechanics of an ARS contract are specified in Section 2; a graphical depiction of these simple mechanics is displayed in Fig. 1,


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Figure 1: Basic mechanics of an acquisition risk swap (ARS) contract: until a event that may or may not occur at an unspecified future time, the buyer of the ARS (“acquirer”) pays a stream of cash flows mτ to the writer of the ARS, shown in panel (a). During the time window specified by the start and end dates of the event, the acquirer ceases to pay the cash flows and the writer provides the acquirer with a capability valued at vτ in each time period, shown in panel (b).


in which panel (a) refers to before the ARS is exercised by the holder and panel (b) describes the structure after it is exercised. We display a notional, more-developed market structure in Fig. 2.


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Figure 2: A more realistic market structure for ARS than that presented in Fig. 1 decomposes the specializations of estimating the risk of exercise and computation of ARS value from the specialization of actually producing the capability. In this structure, the writer of the ARS effectively puts a capability creator (e.g., a manufacturer) on retainer in case the conditioning event occurs, depicted in panel (a). If the event does occur, the writer pays the manufacturer an income stream (e.g., equal to the marginal cost of producing the capability) and the creator delivers the capability to the writer, displayed in panel (b).


In this example, the writer sells the ARS to the acquirer and hedges the risk it takes on in doing so by contracting with a creator of the capability (e.g., in the 155mm shell example, a manufacturer capable of rapidly retooling to create shell casings), paying that creator a fixed-fee retainer R at the initial time period. If the triggering event occurs, the writer pays a further stream of cashflows rτ (which are possibly zero) to the creator; the creator creates the capability and delivers it to the acquirer.

A more complex market structure, such as the one outlined in the previous paragraph and displayed in Fig. 2, has at least two benefits over the naive market structure displayed in Fig. 1:

  1. Decoupling of financial valuation from physical delivery. In general, there is no reason to believe that a firm with the expertise valuing an ARS – i.e., a firm that believes it can accurately and precisely estimate the probability distribution of the window over which a capability would be needed – would also have the expertise required to rapidly create that capability. For example, a geopolitical risk firm that believes it can assess the likelihood and duration of a conflict between two nations is probably not the same firm that can rapidly manufacture a large quantity of shell casings.

  2. Incentive to participate in the market due to speculation and innovation. The writer of an ARS might believe that the conditioning event will occur in the very distant future, or has a very low probability of occurrence at all, and therefore believe that the negotiated stream of income mτ represents a low-risk profit opportunity. The creator of the capability, on the other hand, may have no opinion about the likelihood of the event at all, but assesses it can very rapidly create the required capability with almost no advance warning and at a cost exactly equal to rτ in perpetuity; in this case, the retaining fee R is essentially a risk-free profit for the creator.

How might the price of an ARS be discovered? As with other swaps, this would likely depend on the liquidity of the underlying capability that the acquirer is hedging and the relative market power of the acquirer(s) compared with the writer(s). In the monopsonistic case, and when the capability is relatively liquid or even standardized, it would seem likely that the acquirer would run an English reverse auction (or similar commonly practiced single reverse auction). When the capability is less liquid the contracted price could be negotiated over-the-counter between the writer(s) and acquirer. We explore other mechanisms for price discovery in Section 5.

4 Simulation

We outline some empirical statistical properties of the ARS under simple statistical models and an example market structure.2

4.1 Statistics of price distributions

We briefly compare empirical distributions of ARS price under different generative models of event time interval {t,...,T - 1}, value v, and interest rate r. We emphasize that these distributions are rough indicators of behavior only as the price of ARS will depend crucially on estimation of value and event time interval probability that will very likely be highly context-dependent. Specializations to specific operational contexts are not in scope for this work.

Each pricing example uses the naive event probability distribution p(t,T) = p(t)p(δt) = Geometric(t|ρt) Geometric(δt|ρδt), where T = t + δt. Each example begins with a notional risk-free rate of 4%, capability value of v = 100 (abstract numeraire), and ρt = ρδt = 1
4. Empirical density functions of price are displayed in Figure 3.


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Figure 3: Empirical ARS price distributions under multiple simple pricing assumptions generally have positive skew. The parameterizations of each scenario are outlined in Section 4.1.


We describe scenarios in terms of changes to the default parameters listed above.

In these examples, ARS price distributions generally have excess mass in their right tails, meaning that there is substantial tail risk that the actual value of the cash flows to the right of the event start date have higher value than what would be suggested by the expected value pricing formulae presented earlier. We will discuss hedging strategies in Section 5.

4.2 Market structure

We construct a simulation of a simple form of the market structure presented in Fig. 2. There is a single acquirer who has a mean reservation price ˆm ~ Normal(m,σ2), where ˆm is computed under the distribution p(t,T) = Geometric(t|ρt*) Geometric(δt|ρδt*), where T = t + δt and the estimates ρt* and ρδt* are publicly known. The acquirer’s observed price of the ARS is given by m ~ Normal(mˆˆm2). There are n = 1,...,N writers of ARS who compete to sell a single ARS to the acquirer via an English reverse auction. They have heterogeneous beliefs about the start and stop probability of the event, modeling it as ρt(n) ~ Normal(ρt*ρt*2) and ρδt(n) ~ Normal(ρδt*ρδt*2) conditioned to lie in (0,1). There are = 1,...L manufacturers who can create the capability desired by the acquirer. Each manufacturer has a private reserve value R ~ Normal(m,σR) they require to accept a contract for manufacture at an unknown future date, and each has a marginal production cost of c ~ Normal(m∕2c). Before time τ = 0, the acquirer conducts an English reverse auction to purchase the ARS from a writer. Then, the successful writer conducts an English reverse auction on reserve price with the manufacturers. If this market clears, meaning that the the writer who wins the auction has a price for the ARS is less than or equal to the acquirer’s maximum willingness to pay and, similarly, that the manufacturer who wins the auction has a reserve price for manufacturing that is less than or equal to the writer’s maximum willingness to pay, the model advances in discrete time, with contract mechanics as specified in Section 2.3


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Figure 4: Example wealth distributions for writers and manufacturers conditioned on contract award. In this example, N = 2 writers competed for a contract with a single acquirer and subsequently M = 2 manufacturers competed for a contract with the sole successful writer.


We display wealth distributions of successfully matched writers and manufacturers in Figure 4 and market clearing probability as a function of number of writers and manufacturers in Figure 5. Monte Carlo simulations of this market structure suggest that the market clears with high probability even with relatively low participation. For example, even with only two writers and two manufacturers participating in the marketplace, the market clears more than half of the time. (Under this model, a market with only one potential writer and 10 manufacturers clears almost half of the time – 44% – while an ARS market with a healthy five writers and 20 manufacturers clears a full 92% of the time!)


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Figure 5: A Monte Carlo computation of the probability of market clearing – here, meaning that the acquirer actually receives delivery of the acquired capability in the future when required – suggests that the market clears with high probability even with relatively low participation.


5 Generalizations and structural considerations

We constructed a pricing and market structure framework for acquisition risk swaps (ARS), a type of subordinated risk swap that eliminate the risk that an agent faces from needing to acquire a costly capability during an unknown future time interval. However, we made multiple simplifying assumptions that, in our judgement, would be eliminated in a commercial implementation of an ARS marketplace. We outline several unaddressed shortcomings of the present work each of which could be ground for further research.

Acknowledgements

The views stated in this paper are those of the author and do not necessarily represent the position of the U.S. Government or the Department of Defense. Distribution Statement A: approved for public release, distribution is unlimited.

References

[BK84]    Daniel K Benjamin and Levis A Kochin. War, Prices, and Interest Rates: A Martial Solution to Gibson’s Paradox. 1984.

[SHGS15]   Tom Schaul, Daniel Horgan, Karol Gregor, and David Silver. Universal value function approximators. In International conference on machine learning, pages 1312–1320. PMLR, 2015.