Managing Disruption Risk: The Interplay Between Operations andInsurance

Lingxiu Dong • Brian Tomlin

Olin Business School, Washington University in Saint Louis, Saint Louis, MO 63130, USATuck School of Business at Dartmouth, 100 Tuck Hall, Hanover, NH 03755, USA

dong@wustl.edu • brian.tomlin@tuck.dartmouth.edu

December 1, 2010

Disruptive events that halt production can have severe business consequences if not appropriatelymanaged. Business interruption (BI) insurance offers firms a financial mechanism for managingtheir exposure to disruption risk. Firms can also avail of operational measures to manage the risk.In this paper we explore the relationship between BI insurance and operational measures. Taking aMarkov Decision Process approach, we model a manufacturing firm that can purchase BI insurancefrom a competitive market and that can invest in inventory and avail of emergency supply. Wecharacterize the optimal disruption strategy, including the inventory level and the BI insurancedeductible and coverage limit. We prove that insurance and operational measures are not alwayssubstitutes, and establish conditions under which they can be complements, e.g., insurance leads toa higher inventory investment. We find that the appropriate disruption strategy depends heavilyon the disruption profile and on the consequences of financially significant disruptions. Inventoryis augmented by BI insurance as disruptions become longer/rarer; however, emergency sourcingdisplaces inventory as disruptions become even longer and rarer. The value of insurance is higherfor those firms less able to absorb the consequences of financially significant disruptions.

1 Introduction

The purpose of business interruption (BI) insurance is to protect against losses incurred when a firm

cannot operate normally because of a disruption to one of its facilities.1 Firms can also purchase

contingent BI insurance to protect against losses resulting from a disruption to a supplier’s facility.

BI insurance is an important line of business in the commercial insurance industry:

The largest single source of insurance payout in the [Sept. 11] terrorist attacks was

not the property claims, but business interruption insurance. . . . In today’s dollars,

business interruption payouts made up $12.1 billion, or 33 percent of the $35.6 billion

in total insured losses. . . . Business interruption claims from Katrina are expected to

cost about half of the $20.8 billion in commercial losses. (Mowbray, 2006).1We are grateful to Frank Teterus and Tom Skwarek of Swiss Re and Errol Geller and Jan Hanson of IBM for

sharing their expertise on business interruption insurance.

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As with all insurance products, the principle of indemnity applies: an insurance payment should

not make the firm better off than it would have been if the event had not occurred. BI insurance

reimburses the firm for lost net income, standing charges, i.e., those expenses that are still incurred

during the disruption, and extra expenses incurred to minimize lost income, e.g., sourcing from an

emergency facility at a higher than normal variable cost.

Two high-profile disruptions serve to illustrate the benefits and limitations of BI insurance.

Ericsson received an insurance settlement in the region of $200 million resulting from the Philips

plant failure in 2000 that disrupted supply of chips used in its mobile phones (Norrman and Jansson,

2004). However, Ericsson lost substantial market share to Nokia whose operational measures,

including emergency sourcing, successfully mitigated the impact of the Philips plant failure (Latour,

2001). Philips itself received an insurance settlement of approximately e39 million due to lost

income and physical damage (Philips Annual Report 2001 p.53).

Genzyme, a large biopharmaceutical company, temporarily stopped production in its Allston,

Massachusetts plant in June 2009 when a virus was detected in the equipment. Genzyme’s in-

ventory stockpile was insufficient to meet patient demand for the affected drugs, and when it

announced that the shutdown would last longer than initially anticipated, “Mark Schoenebaum,

senior biotechnology analyst at Deutsche Bank, yesterday boosted his forecast of Genzyme’s rev-

enue shortfall stemming from the virus to $245 million, up from an earlier forecast of $100 million”

(Weisman, 2009). According to Genzyme’s 2009 Annual Report, “last year was the most challeng-

ing in our 28-year history. Setbacks in our manufacturing operations hindered our ability to fully

supply Cerezyme and Fabrazyme, two of our largest products. As a result, we did not meet our

commitment to patients or to shareholders.” The financial implications of the disruption were very

large, not because Genzyme did not have BI Insurance (they did) but because BI insurance does

not cover disruption losses due to viral contamination; that, and the fact that Genzyme did not

have adequate operational measures in place to prevent the loss of sales.

There must be three elements in place for BI insurance to take effect. According to Torpey

et al. (2004) p.25, these elements are “direct physical damage to or loss of property at the premises

described in the policy declarations that was caused by a covered cause of loss, a necessary sus-

pension of operations during the period of restoration, and an actual loss of business income.”

We elaborate on each of these in turn. First, the disruption must be directly caused by property

damage (PD) which resulted from a peril covered by the firm’s PD insurance policy - fire being the

most common (but not only) covered peril. A firm must have a PD insurance policy in order to

purchase BI insurance. Second, coverage is provided for a limited time. In U.S. polices, coverage is

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provided until the facility achieves technical operational readiness, i.e., the point in time at which

the facility “should be repaired, rebuilt or replaced with reasonable speed and similar quality,”

Torpey et al. (2004) p.29.2 Third, the business must actually suffer a loss due to the disruption.

Being a legal document, a BI insurance policy can have a multitude of clauses. However, three

key parameters are the premium, the coverage limit, and the deductible. The premium is the

price the firm pays the insurer to obtain the insurance coverage. The coverage limit specifies the

maximum the insurer will pay the firm in the event of a loss. Insurers typically do not want

to process frequent small claims and such claims are precluded by the use of a deductible. The

deductible can be a monetary or time deductible. In a monetary deductible, the policy specifies a

monetary value and the firm absorbs this amount of any BI related loss. With a time deductible,

the policy specifies a duration such that the firm absorbs any loss incurred during that initial

duration of a disruption. While the firm can choose different coverage limits and deductibles, its

choice may be constrained by what the insurer allows. Insurers may, for example, set a minimum

allowed deductible and may not be willing to insure losses above some maximum value.

In this paper we explore the use of BI insurance and operational measures for managing dis-

ruption risk. We consider a manufacturing firm that purchases BI insurance from a competitive

insurance marketplace. We take the actuarial approach to insurance pricing (Schlesinger, 1983) in

which the firm pays a premium equal to its expected reimbursement plus a loading factor to cover

the insurer’s administrative costs. In addition to or in lieu of insurance, the firm may stockpile

inventory and/or avail of an emergency supplier. We characterize the optimal strategy, i.e., the

combination of measures to deploy, and the optimal implementation, i.e., the insurance deductible

and coverage and the inventory level. While intuition might suggest that insurance and operational

measures are substitutes, e.g., the firm invests in less inventory if it purchases insurance, we show

that this intuition does not always hold. We prove that insurance and inventory are substitutes un-

der additive loading but can be complements under proportional loading. Insurance and emergency

sourcing can be substitutes or complements under both loading models. Inventory and emergency

sourcing are substitutes. The appropriate disruption strategy is heavily dependent on the dis-

ruption profile (rare/long versus frequent/short) and on the consequences of financially significant

disruptions. Inventory is augmented by BI insurance as disruptions become longer/rarer; however,

emergency sourcing displaces inventory as disruptions become even longer and rarer. The value2In U.K. policies coverage is provided until commercial operating readiness or the maximum indemnity period is

reached, whichever is first. Commercial operating readiness “is when the company is once again able to produce itsnormal financial results” (SwissRe, 2004) p. 19. This may differ from technical operational readiness if, for example,demand after a disruption is temporarily reduced because the firm has yet to win back customers who switched to acompetitor or because the economy was impacted by the disruptive event, e.g., Hurricane Katrina.

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of insurance is higher for those firms less able to absorb the consequences of financially significant

disruptions.

The rest of the paper is organized as follows. Section 2 discusses the relevant literature. Section

3 describes the model. The case of no BI insurance is analyzed in §4. We analyze the case of

BI insurance with additive loading in §5. Section 6 examines the interplay between operational

and insurance strategies, and also explores strategy preference. The proportional loading model is

analyzed in §7. Conclusions and directions for future research are presented in §8.

2 Literature

The commercial importance of BI insurance is reflected by practitioner books dedicated to the topic,

e.g., Cloughton (1999) and Torpey et al. (2004), that give in-depth coverage of the accounting and

legal considerations. Despite its commercial importance, BI insurance has received little attention

in the academic insurance literature. A search for the phrase “interruption insurance” in the Journal

of Risk and Insurance (1964-date) revealed only one paper with the phrase in the title or abstract.

That paper (Zajdenweber, 1996) is an empirical investigation of the distribution of total annual BI

claims in France. A wider search in the JSTOR journal database (which includes the Journal of Risk

and Insurance) uncovered one additional paper (Kahler, 1932) that gives a practical introduction

to BI insurance.3 While the phrase “interruption insurance” appears in the main body of 15 papers

in the Journal of Risk and Insurance, these papers only mention BI insurance in passing for the

most part.

The academic insurance literature has extensively studied policy design for individual insurance

products, e.g., Schlesinger (1983), Eeckhoudt et al. (1988), and Kaplow (1992), and non-BI com-

mercial insurance products, e.g., Kahane and Kroll (1985). An actuarial approach is often used for

pricing insurance. A premium which equals the expected insurance payout is said to be actuarially

fairly priced (Schlesinger, 1983), and reflects a competitive insurance marketplace in which the

insurers have no costs associated with writing policies and processing claims. To account for these

administrative costs, models apply a “loading” that inflates the actuarially fair price. The loading

is often proportional to the expected insurance payout, e.g., Kahler (1979) and Schlesinger (2006),

or a constant added to the expected payout, e.g., Kaplow (1992) and Kliger and Levikson (1998).

The operations literature has typically modeled disruptions in one of two manners: infinite

horizon models or single-period models. In the infinite horizon models, a facility alternates between3BI insurance is sometimes referred to as business income insurance. No papers in the JSTOR database had this

term in the title or abstract.

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up and down phases with the status of the facility being known at the time of production, e.g.,

Tomlin (2006) and references therein. Single-period models have been used to explore disruption

management for products with short life cycles and long lead times. Production is uncertain and

typically modeled as a Bernoulli random variable in which production either succeeds in full or

completely fails, e.g., Babich (2006) and Chaturvedi and Martınez-de-Albeniz (2008).

The literature has investigated a variety of operational strategies for managing disruption risk:

inventory (e.g., Tomlin (2006) and Schmitt et al. (2009)), emergency sourcing (e.g., Chopra et al.

(2007) and Yang et al. (2009)), dual sourcing (e.g., Parlar and Perry (1996), Dada et al. (2007) and

Babich et al. (2007)), demand management (Tomlin, 2009), and process improvement (Wang et al.

(2010) and Bakshi and Kelindorfer (2009)). Financial mechanisms for managing supply risk, e.g.,

supplier subsidies to reduce the likelihood of bankruptcy-induced disruptions, have been considered

by Swinney and Netessine (2009) and Babich (2010). The use of operational and financial hedging

for managing demand risk has been explored by Gaur and Seshadri (2005), Chod et al. (2010) and

references therein.

BI insurance has not been explored in the disruption literature to the best of our knowledge.

Addressing that gap is the primary objective of this paper; although our model of inventory and

emergency supply also extends the literature by allowing for multiple types of disruptions, the

possibility that inventory may be destroyed by a disruption, and a penalty for financially significant

disruptions.

3 Model

The firm has three disruption management options at its disposal: inventory, emergency supply

and BI insurance. We describe the operational and insurance elements of the model, and then

formulate the firm’s problem.

3.1 Operational Elements

We consider a firm that sells a single product over an infinite (discrete-time) horizon. It operates

a single facility that is subject to disruptions. In any given period, the facility is either operational

(up) or non-operational (down). When the facility is up it can produce any quantity, but it can

produce nothing when down. Disruptive events occur only when the facility is up and cause the

facility to remain down for an uncertain duration. Disruptive events are either insurable (e.g.,

a fire damages the facility as in the Philips example) or non-insurable (e.g., a labor strike or

viral contamination as in the Genzyme example). We assume that inventory is either completely

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destroyed by the disruptive event or survives completely intact. As such, there are four types of

disruptions depending on the cause (insurable or non-insurable) and whether inventory is destroyed

or not. We assume a constant probability of each disruption type. The disruption probability is

denoted by θXY , where X ∈ I,N denotes whether the disruptive event is insurable (I) or non-

insurable (N) and Y ∈ S, F denotes whether the disruptive event destroys the inventory (F for

failure) or not (S for survival). Let θ =∑

X∈I,N∑

Y ∈S,F θXY denote the total probability of

a disruption occurring. When down, the probability of recovery (i.e., disruption ending) depends

only on (i) the number of periods n the disruption has lasted and (ii) the type of disruption (XY ).

The recovery probability is denoted by λXY (n).

The firm incurs a standing (fixed) cost of f per period irrespective of the production quantity

or whether the facility is up or down. Production (including any raw material procurement) incurs

a variable cost of v per unit and has a lead time of zero. The firm receives a revenue of r per unit

sold. We assume that r > v+f to avoid trivial cases in which the firm is never profitable. Demand

in each period is deterministic, and set equal to 1 without loss of generality. Unfilled demand is

lost, and the firms incurs an intangible goodwill cost of g per unit lost sale. The goodwill cost can

be thought of as a proxy for the future impact of customer dissatisfaction.

We assume the firm operates a base stock policy in which the ending inventory is brought to a

level B in any period that the facility is up. The firm incurs intangible inventory costs (opportunity,

etc.) in each period based on the current inventory level at a cost of hi per unit. The firm incurs

tangible inventory costs (physical storage, etc.) in each period based on the maximum possible

inventory level, i.e., B, at a cost of ht per unit. If the firm adopts emergency sourcing as a strategy,

it pays a reservation fee b every period to ensure supply availability. During a disruption the firm

can purchase units from this supplier at a variable cost of e, with v < e ≤ r reflecting the fact that

emergency supply has a higher variable cost than regular production.

3.2 Insurance Elements

We assume that, as is almost universally the case, the firm has PD insurance. So as to focus our

attention on BI insurance, we assume that PD insurance fully covers any insurable-cause property

damage loss and that non-insurable events do not result in property damage losses.

As discussed in the introduction, a BI policy has three key parameters: the premium, the

deductible, and the coverage limit.4 While the premium is typically paid annually, we assume that4We focus on the U.S. BI insurance format. The U.K. policy differs in two regards: the distinction between technical

and commercial readiness and the inclusion of a maximum indemnity period (often chosen to be 12 months). Thefirst distinction is moot here as demand is not impacted by a disruption. The second distinction is relevant only in

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payments are spread out evenly over the year, and so the firm pays the insurer p in each period.

We focus on a monetary deductible, that is, the policy specifies a value D and the firm absorbs

this amount of any BI related loss. (We do not include our analysis of the time deductible as

the findings are similar to the monetary deductible.) The coverage limit L specifies the maximum

the insurer will pay the firm in the event of a loss.5 The firm chooses the deductible D and limit

L. Insurers may impose a minimum deductible to prevent frequent small claims and may not be

willing to cover all loss sizes. As such, the firm’s deductible and limit choices are constrained by a

minimum deductible D and a maximum limit L, which are exogenously given. If D = 0 (L = ∞)

then there is no minimum deductible (maximum limit).

BI insurance covers only tangible losses, that is, intangible costs, e.g., goodwill and opportu-

nity, are not relevant to the BI loss calculation. Let πD(n) denote the net tangible income during

a disruption of n periods including the first period back up, and let πU (n) denote the net tangible

income that the firm would have earned during that same duration if there had not been a disrup-

tion. For the moment we suppress the notational dependence of these profits on the base-stock level

B and whether the firm has an emergency source or not. Expressions for πD(n) and πU (n) will be

developed later. Define the interruption loss as IL(n) = πU (n)− πD(n). The BI policy reimburses

the firm R(n) = min[IL(n)−D]+ , L. We normalize the firm’s claim preparation cost to zero

and assume that claims are processed and reimbursed in full in the first period after a disruption

ends.6 We note that the insurance company can verify the firm’s historic inventory level during

a claims process through an audit of the firm’s enterprise planning systems. In effect, then, the

base-stock level is observable to all parties for the purpose of contract execution.

We adopt the actuarial pricing model that is widely used in the insurance literature, and

consider both additive and proportional loading models. Suppressing the notational dependence

on insurance / inventory / emergency sourcing choices made by the firm, let R denote the long-run

average BI payout from the insurer to the firm. The actuarially fair premium with additive loading

is p = R+m, where m is the insurer’s markup to cover administrative expenses (and possibly some

notion of a “fair” profit). The actuarially fair premium with proportional loading is p = (1 +m)R.

very long disruptions, i.e., longer than 12 months.5Separate deductibles and limits apply to BI and PD losses in some policies. In other policies the deductible

and/or limit apply to the combined BI and PD loss. We consider the former case as our focus is on BI insurance.6Because businesses are more complex than reflected in this model, in practice there can be substantial disagree-

ment between the firm and the insurer as to the actual business loss and what is covered by the policy. The claimsprocess can be arduous and may result in delayed and partial settlements.

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3.3 The firm’s problem

There is one more cost to consider in addition to the revenues and costs already discussed. A

disruption that significantly impairs the firm’s financial performance can result in a loss of “rep-

utation and credibility [making] it more expensive to raise capital; [and to] restore credibility top

management may have to spend more time meeting and talking to investors, a costly activity,”

Hendricks and Singhal (2008). Investor lawsuits are also a concern (Hendricks and Singhal, 2005);

investors filed an $8bn lawsuit following Genzyme’s 2009 disruption (Taylor, 2009). We assume

the firm incurs a penalty K ≥ 0 if the lost income during a disruption (factoring in any insurance

reimbursement) exceeds a threshold T , i.e., K is incurred in the first up-period after a disruption of

length n if IL(n)−R(n) > T . This penalty represents the cost associated with impaired financial

performance. Alternatively, it may reflect a firm’s aversion to large disruption-related financial

losses.

The firm chooses the base stock B ≥ 0, whether or not to have an emergency supplier, and

whether or not to purchase BI insurance. If the firm chooses to purchase BI insurance, it chooses

a deductible D ≥ D and a coverage limit L ≤ L. These choices are made with the objective

of maximizing its long-run average profit. Let Π (B) [Π (B,D,L)] denote the firm’s long run

average profit without [with] BI insurance when it has no emergency supplier. Similarly, let ΠE (B)

[ΠE (B,D,L)] denote the firm’s long run average profit without [with] BI insurance when it has an

emergency supplier.

The firm’s problem can be formulated as a Markov Decision Process with the following states:

• (U, 0): Facility is up (U) and was up in the previous period (i.e., the current up period was

preceded by 0 down periods).

• (D,n, I, S): Facility has been down (D) for n ≥ 1 periods (including the current period)

because of an insurable (I) event in which the inventory survived (S).

• (D,n, I, F ): Facility has been down (D) for n ≥1 periods (including the current period)

because of an insurable (I) event in which the inventory was destroyed (failed) (F).

• (D,n,N, S): Facility has been down (D) for n ≥ 1 periods (including the current period)

because of a non-insurable (N) event in which the inventory survived (S).

• (D,n,N, F ): Facility has been down (D) for n ≥ 1 periods (including the current period)

because of a non-insurable (N) event in which the inventory was destroyed (F).

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• (U, n, I, S): Facility is up (U) for the first period after being down for n ≥ 1 periods because

of an insurable (I) event in which the inventory survived (S).

• (U, n, I, F ): Facility is up (U) for the first period after being down for n ≥ 1 periods because

of an insurable (I) event in which the inventory was destroyed (F).

• (U, n,N, S): Facility is up (U) for the first period after being down for n ≥ 1 periods because

of a non-insurable (N) event in which the inventory survived (S).

• (U, n,N, F ): Facility is up (U) for the first period after being down for n ≥ 1 periods because

of a non-insurable (N) event in which the inventory was destroyed (F).

Let ρ(s) denote the steady state probability of being in state s. Expressions for ρ(s) are given in

Appendix A. Let w(s) denote the profit in state s. For expositional ease we suppress the notational

dependence of w(s) on the firm’s decision variables. (Expression for w(s) will be developed later.)

Applying Theorem 3.3.3 in Tjims (2003), the long-run average profit Π is given by the steady-state

expected profit, i.e., Π =∑

s∈S ρ(s)w(s) where S is the collection of all states.

We now introduce a condition that precludes the disruption recovery probabilities λXY (n) from

growing too rapidly in the disruption length n.

Condition 1 λXY (n+ 1) ≤ λXY (n)1−λXY (n) for all n.

This is a relatively mild condition that allows for many disruption-length distributions, including

the commonly assumed geometric distribution. The condition leads to the following useful property

that is required for many (but not all) results in the paper.

Property 1 ρ(U, n,X, Y ) is decreasing in n if and only if Condition 1 holds.

Proof All proofs are contained in Appendix C.

For ease of exposition we assume that Condition 1 holds throughout. It will be clear from theorem

proofs which results require the condition.

In closing this section, we note that the terms increasing, decreasing, larger than, and smaller

than are used in the weak sense in this paper. Also, [x]+ = maxx, 0 and bxc is the smallest

integer less than or equal to x. Because inventory levels are integer, we work with first- and

second differences rather than derivatives. The first and second differences with respect to B are

OΠ(B) = Π(B + 1)−Π(B) and O2Π(B) = OΠ(B + 1)− OΠ(B).

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4 No Insurance

In this section we analyze a firm that does not have BI insurance. We first consider the inventory

strategy in isolation and then consider it in combination with emergency sourcing.

4.1 Inventory

To develop the long-run average profit expression Π(B) we need to specify the profit w(s) in each

state s. It is convenient to first analyze the “no-penalty” case in which K = 0.

• (U, 0): The firm sells one unit for a revenue r and produces one unit at a variable cost v. It

incurs the standing charge f . The inventory level is B and the firm incurs inventory costs

hiB + htB. The profit is w(s) = r − v − f − hiB − htB.

• (U, n, I, S): The firm sells one unit for a revenue r. To obtain the base-stock level B it

must produce minn,B + 1 units, each at a variable cost v. It incurs the standing charge

f . The inventory level is B and the firm incurs inventory costs hiB + htB. The profit is

w(s) = r − v (minn,B+ 1)− f − hiB − htB.

• (U, n, I, F ): The firm sells one unit for a revenue r. Because the inventory failed, to obtain

the base-stock level B it must produce B + 1 units (each at a variable cost v.) However,

the cost vB of replacing the damaged inventory is covered by PD insurance in the case of an

insurable event. It incurs the standing charge f . The inventory level is B and the firm incurs

inventory costs hiB + htB. The profit is w(s) = r − v − f − hiB − htB.

• (U, n,N, S): Similar to (U, n, I, S) as insurable and non-insurable events do not influence the

associated profit. The profit is w(s) = r − v (minn,B+ 1)− f − hiB − htB.

• (U, n,N, F ): The firm sells one unit for a revenue r. Because the inventory failed, to obtain

the base-stock level B it must produce B + 1 units (each at a variable cost v.) It incurs the

standing charge f . The inventory level is B and the firm incurs inventory costs hiB + htB.

The profit is w(s) = r − v (B + 1)− f − hiB − htB.

• (D,n, I, S): The firm sells a unit for a revenue r if and only if n ≤ B, i.e., the firm has

inventory. Otherwise it incurs the lost sale penalty g. It produces nothing but incurs the

standing charge f . The inventory level is (B − n) if n ≤ B and 0 otherwise. The profit is

w(s) = r − f − hi (B − n)− htB if n ≤ B, and w(s) = −g − f − htB otherwise.

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• (D,n, I, F ): The firm has no inventory as it was destroyed by the disruption. The profit is

w(s) = −g − f − htB.

• (D,n,N, S): Similar to (D,n, I, S) as insurable and non-insurable events do not influence

the associated profit. The profit is w(s) = r − f − hi (B − n) − htB if n ≤ B, and w(s) =

−g − f − htB otherwise.

• (D,n,N, F ): Similar to (D,n, I, F ) as insurable and non-insurable events do not influence

the associated profit. The profit is w(s) = −g − f − htB.

Substituting these w(s) expressions into Π0(B) =∑

s∈S ρ(s)w(s), we obtain

Π0(B) = (r − v − f − hiB − htB) ρ(U, 0)

+B∑n=1

(r − v (n+ 1)− f − hiB − htB) (ρ(U, n, I, S) + ρ(U, n,N, S))

+∞∑

n=B+1

(r − v (B + 1)− f − hiB − htB) (ρ(U, n, I, S) + ρ(U, n,N, S))

+∞∑n=1

(r − v − f − hiB − htB) ρ(U, n, I, F )

+∞∑n=1

(r − v (B + 1)− f − hiB − htB) ρ(U, n,N, F )

+B∑n=1

(r − f − hi (B − n)− htB) (ρ(D,n, I, S) + ρ(D,n,N, S))

+∞∑

n=B+1

(−g − f − htB) (ρ(D,n, I, S) + ρ(D,n,N, S))

+∞∑n=1

(−g − f − htB) (ρ(D,n, I, F ) + ρ(D,n,N, F )) , (1)

where the subscript 0 on Π0(B) denotes the no-penalty case, i.e., K = 0. Let B∗0 denote the optimal

B for this case.

Proposition 1 (i) Π0(B) is concave in B. (ii) B∗0 is the smallest B such that OΠ0(B) ≤ 0.

An expression for OΠ0(B) is given in Appendix C, see equation (C.1).

We now consider the penalty case in which K > 0. This differs from the no-penalty case in that

an additional cost K is incurred in the first up period after a disruption of length n if the tangible

interruption loss IL(B,n) = πU (B,n)− πD(B,n) exceeds the threshold T .7 We need to determine7Because there is no insurance IL(B,n) − R(B,n) > T becomes IL(B,n) > T . Also, note that we now include

the notational dependence on B.

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the states (U, n,X, Y ) in which the penalty K applies. The tangible net income when up πU (B,n)

is given by πU (B,n) = (n+ 1) (r − v − f − htB) for all disruption types. The tangible net income

when down πD(B,n) depends on the disruption type, and we consider each of the four types in

turn:

• (U, n, I, S): the first period back up after an insurable event in which the inventory sur-

vived and the firm was down for n periods. During the disruption the firm incurs the

standing charge f and the tangible inventory cost htB in all n periods. It also earns a

revenue of rminn,B. In the first period back up the firm earns r and incurs f and

htB. It also incurs the variable cost v (minn,B+ 1) to obtain the base-stock level B.

Thus, πDIS(B,n) = (r − v) (minn,B+ 1)− (n+ 1) (f + htB), and ILIS(B,n) = πU (B,n)−

πDIS(B,n) = (r − v) [n−B]+. Now ILIS(B,n) > T if and only if n > B + Tr−v , and so the

penalty K is incurred if and only if n > B + Tr−v .

• (U, n,N, S): the first period back up after a non-insurable event in which the inventory

survived and the firm was down for n periods. This proceeds in an identical manner to the

(U, n, I, S) analysis, and so the penalty K is incurred if and only if n > B + Tr−v .

• (U, n, I, F ): the first period back up after an insurable event in which the inventory was

destroyed and the firm was down for n periods. During the disruption the firm incurs the

standing charge f and the tangible inventory cost htB in all n periods. It earns no revenue

as there is no inventory. In the first period back up the firm earns r and incurs f and htB. It

also incurs the variable cost v (B + 1) to obtain the base-stock level B but the PD insurance

covers the cost vB for the destroyed inventory. Thus, πDIF (B,n) = r− v − (n+ 1) (f + htB),

and ILIF (B,n) = πU (B,n) − πDIF (B,n) = (r − v)n. Now ILIF (B,n) > T if and only if

n > Tr−v , and so the penalty K is incurred if and only if n > T

r−v .

• (U, n,N, F ): the first period back up after a non-insurable event in which the inventory was

destroyed and the firm was down for n periods. During the disruption the firm incurs the

standing charge f and the tangible inventory cost htB in all n periods. It earns no revenue

as there is no inventory. In the first period back up the firm earns r and incurs f and htB. It

also incurs the variable cost v (B + 1) to obtain the base-stock level B. Thus, πDNF (B,n) =

r− (B + 1) v− (n+ 1) (f + htB), and ILNF (B,n) = πU (B,n)−πDNF (B,n) = n (r − v) +Bv.

Now ILNF (B,n) > T if and only if n > Tr−v −

Bvr−v , and so the penalty K is incurred if and

only if n > T−Bvr−v .

12

The profits in all states are the same as those for the no-penalty case, except that the additional

cost K in incurred in states (U, n,X, Y ) when IL(B,n) > T . Letting ΠK(B) denote the long-run

average cost in the penalty case, it then follows that

ΠK(B) = Π0(B)−K (ΩIS(B) + ΩNS(B) + ΩIF + ΩNF (B)) (2)

where

ΩIS(B) =∞∑

n=B+b Tr−vc+1

ρ(U, n, I, S) ; ΩNS(B) =∞∑

n=B+b Tr−vc+1

ρ(U, n,N, S)

ΩIF =∞∑

n=b Tr−vc+1

ρ(U, n, I, F ) ; ΩNF (B) =∞∑

n=b [T−Bv]+

r−vc+1

ρ(U, n,N, F ). (3)

Equation (2) reflects the fact that the penalty-case profit is lower than the no-penalty case profit,

and the Ω functions capture the states in which the penalty K is incurred. Note that ΩIF is constant

in B. Let Π∗K and Π∗0 denote the optimal profits for the penalty and no-penalty cases respectively.

As one would expect the presence of a penalty decreases the profit, i.e., Π∗K ≤ Π∗0 (proof omitted).

We now turn our attention to the optimal base-stock level.

Proposition 2 If θNF = 0 then (i) ΠK(B) is concave in B, and (ii) B∗K is the smallest B such

that OΠK(B) ≤ 0, where OΠK(B) = OΠ0(B)−K (OΩIS(B) + OΩNS(B)).

Thus, a sufficient condition for concavity is that non-insurable events do not destroy the inventory.

Expressions for OΩIS(B) and OΩNS(B) are given in Appendix C, see equations (C.3) and (C.5).

Theorem 1 B∗K ≥ B∗0 if θNF = 0.

In other words, the penalty case results in a higher optimal base-stock level because inventory

reduces the probability of incurring the penalty K. We note that B∗K increases in K.

4.2 Emergency supply and inventory

We now consider the case in which the firm uses emergency supply in addition to holding inventory.

The firm pays a reservation fee b (every period) to ensure availability and pays e per unit sourced

from the emergency supplier, where v < e ≤ r. The firm only sources from the emergency supplier

during a disruption (because e > v) and then only when it runs out of inventory (because filling

demand from inventory is preferable as it reduces the intangible holding cost.)

The development of the long run average profit expressions follows in a similar manner to the

inventory-only expressions developed above, except that instead of incurring the goodwill cost g

13

when inventory runs out, the firm incurs the variable emergency sourcing cost e. It also incurs b

every period. Accounting for these differences, it can be shown that

ΠE0 (B) = Π0(B)− b

+ (r + g − e)

[ ∞∑n=B+1

(ρ(D,n, I, S) + ρ(D,n,N, S)) +∞∑n=1

(ρ(D,n, I, F ) + ρ(D,n,N, F ))

](4)

and

ΠEK(B) = ΠE

0 (B)−K(ΩEIS(B) + ΩE

NS(B) + ΩEIF + ΩE

NF (B))

(5)

where

ΩEIS(B) =

∞∑n=B+b T

e−vc+1

ρ(U, n, I, S) ; ΩENS(B) =

∞∑n=B+b T

e−vc+1

ρ(U, n,N, S)

ΩEIF =

∞∑n=b T

e−vc+1

ρ(U, n, I, F ) ; ΩENF (B) =

∞∑n=b [T−Bv]+

e−vc+1

ρ(U, n,N, F ). (6)

Expression (4) shows that in the no-penalty case (for a given B), emergency supply prevents lost

revenue and goodwill, but at a cost e. Comparing expressions (5) and (6) with expressions (2) and

(6), in the penalty case (for a given B), emergency supply also reduces the interruption loss (because

e ≤ r) and, therefore, decreases the probability of incurring the penalty K. These benefits may

be more than offset by the reservation fee b, and so the firm may be better off without emergency

supply. It is readily shown that for a given e there is a threshold value b above which emergency

sourcing will not be part of the firm’s strategy.

Let BE∗0 (BE∗

K ) denote the optimal B for the no-penalty (penalty) case when there is an emer-

gency supplier.

Proposition 3 (i) ΠE0 (B) is concave in B. (ii) BE∗

0 is the smallest B such that OΠE0 (B) ≤ 0.

Proposition 4 If θNF = 0 then (i) ΠEK(B) is concave in B, and (ii) BE∗

K is the smallest B such

that OΠEK(B) ≤ 0, where OΠE

K(B) = OΠE0 (B)−K

(OΩE

IS(B) + OΩENS(B)

).

Expressions for OΠE0 (B), OΩE

IS(B) and OΩENS(B) are given in the proofs. Let ΠE∗

0 and ΠE∗K denote

the optimal profits. As one would expect, ΠE∗K ≤ ΠE∗

0 (proof omitted). Also, the penalty results in

a higher optimal base-stock level.

Theorem 2 BE∗K ≥ BE∗

0 if θNF = 0.

14

5 BI Insurance with Additive Loading

We now consider the case in which the firm carries BI insurance. The firm chooses a deductible

D ≥ D and a coverage limit L ≤ L. It pays a premium in every period. The premium p =

R(B) +m in the additive loading model, where R(B) is the long-run average reimbursement. We

explicitly denote the dependence on B but suppress (for the moment) the dependence on whether

the firm uses emergency supply. We also suppress the dependence on the deductible and coverage.

Reimbursements only occur in the fist period back up after an insurable-event disruption. Therefore,

p =∞∑n=1

(RIS(B,n)ρ(U, n, I, S) +RIF (B,n)ρ(U, n, I, F )) +m. (7)

where RIS(B,n) (RIF (B,n)) is the reimbursement in the event of an n-period disruption in which

the inventory survived (failed).

It is again convenient to first consider the no-penalty case. Recall that notation with a hat ()

represents the case of BI insurance and an E superscript denotes emergency supply.

Theorem 3 For any D and L, (i) Π0(D,L,B) = Π0(B)−m; (ii) ΠE0 (D,L,B) = ΠE

0 (B)−m.

In other words, the profit with BI insurance equals the profit without BI insurance less the margin

m. This is true regardless of the D and L values as these get “priced out” of the profit function,

and so the firm is indifferent among choices for the deductible and limit. Let Π∗0 denote the optimal

profit with BI insurance and let B∗0 denote the optimal base stock level. The following results follow

immediately from Theorem 3.

Corollary 1

(i) Π∗0 < Π∗0 and ΠE∗0 < ΠE∗

0 if m > 0. Π∗0 = Π∗0 and ΠE∗0 = ΠE∗

0 if m = 0.

(ii) Π0(B) is concave in B and B∗0 = B∗0 . ΠE0 (B) is concave in B and BE∗

0 = BE∗0 .

The optimal base-stock level is independent of the insurance decision. More importantly, BI

insurance adds no value and the firm will purchase it only if m = 0, in which case it is indifferent.

This is not a surprising result. The non-BI insurance literature justifies the purchase of actuarially

fair priced insurance (with a positive loading) by appealing to risk aversion (on the part of individ-

uals) or bankruptcy, tax, and accounting considerations (on the part of businesses) to ensure that

the benefit of insurance is not fully captured by the price (Mayers and Smith, 1982).

We now turn our attention to the penalty model and will see that BI insurance can add positive

value in this case. We first examine the inventory strategy in isolation and then in combination

with emergency sourcing.

15

5.1 Inventory

The firm rebuilds its inventory to the base-stock level after a disruption ends. PD insurance covers

this expense when the inventory is destroyed by an insurable event. Stock must also be rebuilt even

if the inventory is not destroyed (as it is used to fill demand during the disruption). Legal cases

suggest that whether this rebuild expense (after an insurable event) is reimbursed by BI insurance

depends on the precise policy language. We examined both cases in our research. As one would

expect the firm’s profit is lower in the no-reimbursement case, but otherwise the results are not

materially different from the reimbursement case. In this paper we present results only for the

reimbursement case.

Theorem 4

(i) If D > T then ΠK(D,L,B) = ΠK(B)−m.

(ii) If D ≤ T then ΠK(D,L,B) = ΠK(B) +K (ΓIS(L,B) + ΓIF (L))−m where

ΓIS(L,B) =B+bT+L

r−vc∑

n=B+b Tr−vc+1

ρ(U, n, I, S) ; ΓIF (L) =bT+L

r−vc∑

n=b Tr−vc+1

ρ(U, n, I, F ). (8)

Insurance adds no value if the chosen deductible D exceeds the penalty threshold T . To understand

this, it is important to recognize that BI insurance potentially adds value by reducing the probability

of incurring the penalty K. If D > T , then the penalty is incurred in the exact same states with or

without BI insurance, and so insurance adds no value. In the case of D ≤ T , the penalty is incurred

in fewer states with BI insurance. Without BI insurance [see eqn. (2)] the penalty K is incurred

after disruptions that exceed B+ b Tr−v c in length if the inventory is not destroyed and b T

r−v c if the

inventory is destroyed. BI insurance reimbursements, up to the limit L, reduce the tangible losses,

with the result that the penalty K is not incurred unless disruptions exceed B + bT+Lr−v c (inventory

succeeds) and bT+Lr−v c (inventory fails). The penalty avoidance in these states is reflected by the Γ

functions in expression (8). If the coverage limit L is infinite then the penalty K is avoided in all

states.

Let D∗ and L∗ denote the optimal deductible and coverage limit respectively. If there is more

than one optimal D (L), then we define D∗ (L∗) to be the smallest (largest) of the optimal D (L).

The following proposition shows that (if the firm purchases insurances) it is optimal to choose the

minimum allowed deductible and the maximum allowed coverage limit.8

8The firm is in fact indifferent between deductibles as long as D ≤ T , so any deductible in (D, T ) is optimal.

16

Proposition 5 D∗ = D and L∗ = L for any B.

With this result in hand, the firm’s problem is to choose the base-stock level B to maximize

ΠK(B), where we suppress the D and L dependency with the understanding that they are set to

the optimal values. Let Π∗K denote the optimal profit with BI insurance and let B∗K denote the

optimal base stock level.

Theorem 5 There exists a markup m ≥ 0 such that Π∗K ≥ Π∗K if and only if m ≤ m.

In other words, BI insurance adds value (if the markup is not too large); a result which directly con-

trasts with the no-penalty case in which BI insurance adds no value. For later use, we characterize

the optimal base-stock level.

Proposition 6 If θNF = 0 then (i) ΠK(B) is concave in B, and (ii) B∗K is the smallest B such

that OΠK(B) ≤ 0, where OΠK(B) = OΠK(B) + K OΓIS(B) if D ≤ T , and OΠK(B) = OΠK(B)

otherwise.

An expression for OΓIS(B) is given in the proof; see equation (C.19).

5.2 Emergency supply and inventory

Emergency supply is used if a disruption lasts long enough to completely deplete the firm’s inventory.

In this case the firm does not lose revenue but its tangible net income is reduced (vis-a-vis no

disruption) because it pays a higher-than-usual variable cost, i.e., e > v. The additional variable

cost is reimbursed by BI insurance as it reduces the overall interruption loss because e ≤ r.

The analysis of the combined strategy follows in a similar manner to the inventory-only strategy,

and analogous results hold. See Appendix B. We note that the firm’s per-period tangible loss (after

any inventory is exhausted) is e−v with emergency supply as compared to r−v without emergency

supply. The tangible loss is thus lower with emergency supply as v < e ≤ r by assumption.

Therefore, for a given base-stock level B, disruptions must be longer for (i) the penalty threshold

T to be reached and (ii) the coverage limit L to be exceeded. This observation has interesting

implications for the value of insurance in the presence of emergency supply, as will be shown in

Section 6.1.2.

We note the firm would still use an emergency supplier if r < e < r+ g even though we restrict

attention to v < e ≤ r. A standard BI insurance policy would not reimburse the portion of the

emergency cost that exceeds r. The firm could, however, purchase “extra expenses” coverage as

part of a BI policy and this would cover the total additional variable cost.

17

6 Comparing strategies under additive loading

In this section we answer two questions: (i) are inventory, emergency supply, and BI insurance

substitutes or complements?, and (ii) what circumstances favor one strategy over another?

6.1 Substitutes or Complements?

We first compare inventory and BI insurance, then emergency supply and BI insurance, and finally

inventory and emergency supply.

6.1.1 Inventory and BI insurance

There are two reasonable definitions of substitutes and complements in our context. Insurance and

inventory are substitutes (complements) if

1. Insurance reduces (increases) the incremental value of inventory, i.e., OΠ(B) ≤ (≥) OΠ(B).

2. Insurance reduces (increases) the value of inventory, and vice versa, i.e., ∆V ≤ (≥) 0, where

∆V = Π(B∗)− Π(0)− (Π(B∗)−Π(0)).9

Definition 1 is grounded in the economics literature, and implies B∗ ≤ (≥) B∗, i.e., insurance

reduces (increases) the optimal base-stock level, if Π(B) and Π(B) are both concave. Definition 2

has precedent in the operations literature; Chod et al. (2010) define flexibility and financial hedging

to be complements if “flexibility increases or decreases the value of financial hedging” (p. 1037). We

say that insurance and inventory are independent for a given definition if the relevant expression

holds with equality. Insurance and inventory are independent according to both definitions in

the no-penalty case (see Theorem 3 and Corollary 1) or in the penalty case with D > T (using

Proposition 1). We, therefore, focus our attention on the penalty case with D ≤ T in what follows.

Theorem 6 Inventory and insurance are substitutes (in the presence of the emergency supply or

not) according to Definitions 1 and 2.

BI insurance and inventory both serve to reduce the probability of the penalty K being in-

curred, and so intuitively one might expect that they would be substitutes. A complicating factor,

potentially, is that an increase in inventory reduces the insurance premium because the long-run

average reimbursement decreases in the inventory level (assuming θIS > 0, i.e., insurable events

9The value of inventory with (without) insurance is given by Π(B∗) − Π(0) (Π(B∗) − Π(0)), and ∆V equals thedifference. Likewise, the value of insurance with (without) inventory is given by Π(B∗)− Π(B∗) (Π(0)− Π(0)), and∆V also equals this difference.

18

don’t always destroy the inventory.) However, this premium reduction is exactly matched by the

reimbursement reduction, and so there is no “net” premium-effect of inventory. This leaves only

the penalty-reduction effect, with the result that inventory and BI insurance are substitutes.

We note (proof omitted) that the following ranking of inventory levels is easily established given

our earlier results if θNF = 0: B∗0 = B∗0 ≤ B∗K ≤ B∗K . BI insurance (in the penalty case) reduces

the optimal base-stock level (vis-a-vis no insurance) but it never reduces it below the optimal level

for the no-penalty case. An equivalent ranking holds in the presence of an emergency supplier.

6.1.2 Emergency supply and BI insurance

Unlike the inventory level, insurance and emergency supply are both binary decisions; the firm

either uses the strategy or not. As such, Definition 2 is the appropriate substitute/complement

definition. That is, insurance and emergency supply are substitutes (complements) if insurance

reduces (increases) the value of emergency supply, and vice versa. It is readily shown that they are

independent in the no-penalty case or in the penalty case with D > T . We, therefore, focus our

attention on the penalty case with D ≤ T in what follows. The substitute/complement question can

be explored either in the absence of inventory (all strategies use zero inventory) or in its presence

(i.e., at the appropriate optimal base-stock levels). We discuss the case of zero inventory, but we

numerically observed that the main insights carried over to the case with inventory.

Emergency supply and BI insurance both reduce the interruption loss, and so one might expect

that the use of one of the two instruments would reduce the need for and the value of the other.

That is, BI insurance and emergency supply would be substitutes. As shown in the following

theorem, this is not always the case.

Theorem 7 Assume that (i) the recovery probability is geometric and is independent of disruption

type, i.e., λXY (n) = λXY and λXY = λ for X ∈ I,N and Y ∈ S, F, and (ii) all strategies use

zero inventory. If λ < 1 −(

TT+L

) r−vL then there exists a threshold e (λ) < r such that insurance

and emergency supply are substitutes (complements) for e ≤ e (λ) (e > e (λ)); otherwise, insurance

and emergency supply are substitutes.

Without emergency supply, the value of BI insurance is reflected by the Γ functions in expression

(8). With emergency supply, the Γ functions are similar except that the revenue r is replaced by

the variable emergency cost e; see (B.1). Recall that the Γ functions capture the states in which

the penalty K is avoided due to the use of BI insurance. The value of BI insurance is driven by

two effects (i) the number of states in which the penalty is avoided due to BI insurance, and (ii)

19

the probability of being in these states. Emergency supply increases (i) [assuming L <∞] because

the coverage limit protects against longer disruptions as e ≤ r, but decreases (ii) because it also

takes longer disruptions to exceed the penalty threshold, again because e ≤ r. Whether the value

of BI insurance is lower or higher with emergency supply (substitutes or compliments) depends

on which effect dominates. Effect (ii) dominates, resulting in BI insurance and emergency supply

being substitutes, unless the recovery probability is low and the variable emergency sourcing cost

is high in which case (i) dominates and BI insurance and emergency supply are complements.

6.1.3 Inventory and emergency supply

As with the inventory and insurance case, both substitute/complement definitions are appropriate

- but replacing insurance with emergency supply in the definitions.

Theorem 8 For both the no-penalty case and the penalty case inventory and emergency supply are

substitutes (in the presence of insurance or not) according to Definitions 1 and 2.

In the no-penalty case, the availability of emergency supply decreases the marginal value of

inventory. In the penalty case, inventory and emergency supply both help to reduce the probability

of incurring the penalty K. It is intuitive that inventory and emergency supply are substitutes.

6.2 Strategy Preference

We now examine how the firm’s optimal strategy is influenced by model inputs. Because the di-

rectional effect of strategy-specific parameters (inventory costs, emergency costs, and the insurance

markup) are obvious, we focus our attention on two key drivers that influence the performance of

all three strategies; these being the disruption profile and the disruption penalty. The disruption

profile refers to the concept that, for a given percentage uptime, disruptions can range from short-

but-frequent to long-but-rare. The consequences of financially-significant disruptions, as captured

by the penalty K, may differ across firms; for example, financially distressed companies may incur

a higher penalty for large disruption-related income losses.

In this subsection we assume geometric disruptions, i.e., there is a constant probability of a

disruption ending regardless of how long it has lasted. For simplicity we assume that the recovery

probability does not depend on the disruption type, but that is easily relaxed. The disruption

profile is then captured by the recovery probability λ, and the expected disruption length is given

by 1λ . Holding the steady-state probability of being up constant, disruptions become shorter and

more frequent as λ increases.

20

Define the value of BI insurance as the increase in profit gained by adding insurance to an

existing strategy. In Figure 1, we plot the value of insurance for three existing strategies [(a)

nothing, (b) inventory, (c) emergency] as a function of the disruption profile, measured by λ,

and the disruption penalty K.10 Insurance has a nonnegative value because we set the insurance

markup m = 0; a positive markup simply reduces the value of insurance by m. Two observations

are apparent in Figure 1(a); the value of insurance (i) increases in the penalty K, and (ii) increases

and then decreases in λ. Both of these observations can be proven analytically (proofs omitted).

We discuss each observation in turn.

BI insurance adds value by reducing the probability of incurring the penalty K; and the benefit

from this probability reduction is thus larger for higher penalties. Firms facing higher penalties

therefore have more to gain from purchasing insurance.

With longer disruptions more likely to incur the disruption penalty K, one might expect that

insurance would become more valuable as disruptions become rarer-but-longer. This intuition is

only partially correct. Moving from right to left on the x-axis, i.e., as λ decreases and disruptions

become rarer-but-longer, the value of insurance increases to a peak but then decreases. The peak

occurs at 1λ = 1+ T

r−v (proof omitted), i.e., when the expected disruption length exceeds the penalty

threshold scaled by the variable profit margin. (The peak is very close to zero when the penalty

threshold is high; we chose a moderate threshold here to ensure the peak is easily seen.) The

reason for the peak is that the probability of a disruption exceeding the threshold is not monotonic

decreasing in λ (when the percentage uptime is held constant); and the value of insurance attains a

maximum at the point at which this threshold-exceeding probability is maximized. If the probability

of a disruption θ is held constant instead of the percentage uptime, then the value of insurance

monotonically decreases in λ.

Similar behaviors with regard to K and λ are observed in Figure 1(b) and 1(c), but the value

of insurance is significantly diminished by the presence of an operational strategy. The peak occurs

at 1λ = 1 + T

e−v when there is an emergency supplier.

In Figure 2 we plot the value of each strategy [(a) insurance, (b) inventory, and (c) emergency],

where a strategy’s value is measured by the increase in profit gained by adding it to an existing

portfolio of the other two strategies. For example Figure2(a) measures the value of adding insurance10The following revenue and costs were used for all figures in this subsection: r = 1, v = 0.1, ht = 0.0001835,

hi = 0.0002691, f = 0.05, g = 0.5, b = 0.01 and e = 0.5. With periods being interpreted as weeks, these holdingcosts represent an annual holding cost rate of 25%; 10% for tangible and 15% for intangible. The penalty thresholdT = 10(r − v), i.e., the penalty is incurred if the interruption loss exceeds 10 weeks of variable net income. Thesteady-state probability of being up is 0.99. Insurable events account for 80% of disruptions. Inventory fails in 50%of insurable disruptions. The insurance markup m = 0, the minimum deductible D = 2, and maximum coveragelimit L = 52(r − v).

21

if the firm already has inventory and emergency supply in its portfolio. Inventory cannot have

negative value as the base-stock level can be set as 0. Emergency supply can have a negative value

because the firm pays a reservation fee b to add it to its portfolio. The value of inventory increases

in λ [see Figure 2(b)] whereas the value of emergency supply decreases [see Figure2(c)].

The above observations lead to the following recommendations. When disruptions are frequent-

but-short; the firm can rely on inventory alone to manage disruption risk. However, inventory

should be augmented by BI insurance as disruptions become longer/rarer; and emergency sourcing

displaces inventory as disruptions become even longer and rarer. Firms facing a significant penalty

for financially significant disruptions may need to deploy all three strategies.

0.0 0.1 0.2 0.3 0.4 0.50.0000

0.0005

0.0010

0.0015

0.0020

0.0025

Λ

DP

HaL Insur-Nothing

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0.010

0.015

0.020

0.025

Λ

DP

HbL InvInsur-Inv

0.0 0.1 0.2 0.3 0.4 0.50.000

0.005

0.010

0.015

0.020

0.025

Λ

DP

HcL EmInsur-Em

K ­

K ­K ­

Figure 1: The incremental value of insurance over (a) nothing, (b) inventory, (c) emergency supply

0.0 0.1 0.2 0.3 0.4 0.5-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0.001

Λ

DP

HaL All-InvEm HValue of InsuranceL

0.0 0.1 0.2 0.3 0.4 0.5-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0.001

Λ

DP

HbL All-EmIns HValue of InventoryL

0.0 0.1 0.2 0.3 0.4 0.5-0.005

-0.004

-0.003

-0.002

-0.001

0.000

0.001

Λ

DP

HcL All-InvIns HValue of Emergency SupplyL

K ­ K ­

K ­

Figure 2: The incremental value of each strategy over a combination of the other strategies

7 BI Insurance with Proportional Loading

The total markup is proportional to the expected reimbursement in proportional loading. The

premium is set as p = (1 +m)R, i.e.,

p = (1 +m)

( ∞∑n=1

(RIS(B,n)ρ(U, n, I, S) +RIF (B,n)ρ(U, n, I, F ))

), (9)

where R =∑∞

n=1 (RIS(B,n)ρ(U, n, I, S) +RIF (B,n)ρ(U, n, I, F )) is the long-run average reim-

bursement. In what follows we use R(D,L,B) to explicitly note the dependence of R on the

deductible D, coverage limit L, and base-stock level B.

22

We focus on the inventory strategy and ignore the emergency supply strategy in this section.

In an analogous manner to the proofs of Theorems 3 and 4, but using (9) instead of (7), one can

show that

Π0(D,L,B) = Π0(B)−mR(D,L,B)

ΠK(D,L,B) = ΠK(B)−mR(D,L,B), T < D

ΠK(D,L,B) = ΠK(B) +K (ΓIS(L,B) + ΓIF (L))−mR(D,L,B), T ≥ D (10)

These expressions differ from the additive loading model in that the markup m is scaled by the long-

run average reimbursement. The total markup is m in the additive model but it is mR(D,L,B) in

the proportional model, and so the markup parameter m has a somewhat different interpretation

in the proportional model.

It follows from (10) that the firm will not purchase BI insurance in the no-penalty case (or in the

penalty case with D > T ) unless m = 0, at which point the firm is indifferent between purchasing or

not. The same result held for additive loading. We focus our attention on the penalty model with

D ≤ T in what follows. We also impose the mild condition that the penalty-threshold T = nT (r−v)

where nT is integer.11

Proposition 7 D∗ = T for all L and B.

With additive loading any deductible in (D, T ) was optimal. With proportional loading it is

optimal for the firm to choose a deductible equal to the penalty threshold, and the firm is not

indifferent among deductibles in (D, T ). The firm prefers a higher deductible because it reduces

the total margin paid on the premium but, if insurance is to add value, the deductible cannot

exceed the penalty threshold. With this result in hand, we turn our attention to the coverage limit

decision, and for the remainder of this section we assume geometric recoveries, i.e., λXY (n) = λXY

for all n.

Proposition 8 For any given B,

(A): If K > m(r−v)λIS

and K > m(r−v)λIF

then L∗ = L.

(B): If K ≤ m(r−v)λIS

and K ≤ m(r−v)λIF

then L∗ = 0.

(C): If K > m(r−v)λIS

and K ≤ m(r−v)λIF

then L∗ = n∗L(r − v)

11Recall that T is the threshold above which an interruption loss triggers the penalty K. Restricting T to be aninteger multiple of r− v does not cause any significant fidelity loss in setting this exogenous parameter, as reasonablechoices for T are likely to be significant multiples of r − v.

23

(D): If K ≤ m(r−v)λIS

and K > m(r−v)λIF

then L∗ = n∗L(r − v)

where n∗L is the smallest12 nL such that(1− λIS1− λIF

)nL(K − m(r − v)

λIS

)θISλIS (1− λIS)B+nT +

(K − m(r − v)

λIF

)θIFλIF (1− λIF )nT ≤ 0.

Recall that BI insurance adds value by reducing the probability that the penalty K is incurred.

In the additive loading model, i.e., a constant markup, the firm selects the maximum possible

coverage limit. This does not always hold for proportional loading because the total markup

increases in the selected coverage limit. In case (A) the premium markup m is low enough (relative

to the penalty K) that the firm purchases the highest possible coverage limit. In case (B), the

markup is high enough to render insurance unattractive, and the firm does not purchase it; zero

coverage is equivalent to not purchasing. In the intermediate markup cases (C and D) the firm

chooses an intermediate coverage that (in some sense) balances the penalty reduction and markup

tradeoff; although the extreme coverages of 0 and L are also possible in these regions.

Theorem 9

(A): If K > m(r−v)λIS

and K > m(r−v)λIF

then insurance and inventory are substitutes (defns. 1,2).

(B): If K ≤ m(r−v)λIS

and K ≤ m(r−v)λIF

then insurance is not purchased.

(C): If K > m(r−v)λIS

and K ≤ m(r−v)λIF

then insurance and inventory are substitutes (defn. 1).

(D): If K ≤ m(r−v)λIS

and K > m(r−v)λIF

then insurance and inventory are complements (defns. 1,2)

if and only if(K − m(r − v)

λIS

)θISλIS (1− λIS)nT +

(K − m(r − v)

λIF

)θIFλIF (1− λIF )nT > 0. (11)

Insurance and inventory can be complements under proportional loading; a direct contrast to

our earlier result for additive loading. We first explain why this can happen, and then discuss

when it happens. Inventory has a larger insurance-premium effect in proportional loading than in

additive loading because it reduces the expected reimbursement and the total markup. Because the

total markup decreases in the inventory level, the “net” inventory premium effect is positive (rather

than zero as in the additive case.) Therefore, unlike in additive loading, the insurance-premium

benefit of inventory can offset the penalty-reduction effect of insurance, with the overall result that

insurance increases the inventory investment (i.e., they are complements).12If the inequality does not hold at the boundary nL = b L

r−vc then n∗L = b L

r−vc

24

The insurance-premium / penalty-reduction tradeoff is relevant only for insurable disruptions in

which inventory survives, i.e., type IS. If K > m(r−v)λIS

(cases A and C) then the penalty-reduction

effect dominates because the markup is low, and so insurance and inventory are substitutes. If

K ≤ m(r−v)λIS

, then the markup is high enough so that the penalty-reduction effect might not

dominate. However, if in addition K ≤ m(r−v)λIF

(case B), then the markup is so high that insurance

is unattractive. That leaves case D in which the markup is low enough to make insurance attractive

but high enough so that penalty-reduction effect of insurance does not dominate the insurance-

premium effect of inventory.

Case D can only occur if λIF > λIS . This implies that disruptions in which inventory fails

are shorter (on average) than one in which inventory does not fail. (For geometric recoveries,

the average disruption length is 1λ .) At first glance, this necessary condition for complementarity

might seem unlikely to hold in practice. For example, recovering from a larger fire that causes

more damage (including inventory) would likely take longer than recovering from a smaller fire

that caused less damage and did not destroy the inventory. There are reasonable circumstances,

however, in which the necessary condition would hold. We offer the following example. Consider

the two-step process, such as pharmaceutical production followed by filling/packing, illustrated in

Figure 3. Short disruptions can be associated with inventory destruction if the inventory is not

stored in the immediate vicinity of the difficult-to-repair processing technology.

Zone A Zone B

Step 1 Step 2

Zone A Zone B

pDifficult to Repair/Replace Production Technology

pEasy to Repair/Replace Production Technology FGI

i i d ff i i d G bFire in Zone A does not effect FGI but disruption is long.

Fire in Zone B destroys FGI but disruption is short

Figure 3: Short disruptions associated with inventory destruction

Insurance and inventory are not always complements in case D; (11) must also hold. This

condition is more likely to hold when the penalty threshold nT is low and θIS < θIF , i.e., there is

a higher probability of (insurable) disruptions that destroy inventory than ones that don’t.

25

8 Conclusions

In this research we have examined BI insurance, inventory, and emergency supply as strategies for

managing a firm’s disruption risk. We model a manufacturing firm that purchases BI insurance from

a competitive market and that can invest in inventory and avail of emergency supply. Adopting an

actuarial approach to insurance pricing (and considering both additive and proportional loading),

we characterized the optimal disruption strategy, including the inventory level and BI insurance

deductible and coverage limit. We proved that inventory and insurance are always substitutes

in an additive loading model because of the penalty-reduction effect of insurance. Inventory and

insurance can, however, be complements in a proportional loading model because inventory has a net

premium effect that can outweigh the penalty reduction effect of insurance. We also established that

insurance and emergency supply can be substitutes or complements with either loading model, and

that inventory and emergency supply are substitutes. We observed that the appropriate disruption

strategy depends heavily on the disruption profile and on the consequences of financially significant

disruptions. Inventory is augmented by BI insurance as disruptions become longer/rarer; however,

emergency sourcing displaces inventory as disruptions become even longer and rarer. The value

of insurance is higher for those firms less able to absorb the consequences of financially significant

disruptions.

While our model focused on an internal plant, the results would carry over to a supplier dis-

ruption (with BI insurance now being augmented by “contingent” BI insurance) but with a caveat.

Our model does not reflect any strategic interactions or misaligned incentives between the party

investing in protection (insurance, inventory, and emergency supply) and the party that owns and

operates the plant. Whether interactions and incentive issues would alter the paper’s findings is a

subject for future research.

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29

Appendices

A Steady State Probabilities

The steady state probability of being in state (·) is denoted by ρ(·). The relevant states are

(U, 0), (D,n,X, Y ) and (U, n,X, Y ) where X ∈ I,N and Y ∈ S, F. It is convenient to

specify the steady state probability of being up, which we denote by ρ(UP ). This is given by

ρ(UP ) = ρ(U, 0) +∑

X∈I,N∑

Y ∈S,F (∑∞

n=1 ρ(U, n,X, Y )). Conducting probability balances for

states (D,n,X, Y ) and (U, n,X, Y ) results in

ρ(D,n,X, Y ) = (1− λXY (n− 1)) ρ(D,n− 1, X, Y ) , n > 1, (A.12)

ρ(U, n,X, Y ) = λXY (n)ρ(D,n,X, Y ) , n ≥ 1. (A.13)

Using (A.12) and (A.13) recursively, and conducting a probability balance for (D, 1, X, Y ) yields:

ρ(D,n,X, Y ) =

n∏j=2

(1− λXY (j − 1))

θXY ρ(UP ) , n ≥ 1, (A.14)

ρ(U, n,X, Y ) = λXY (n)

n∏j=2

(1− λXY (j − 1))

θXY ρ(UP ) , n ≥ 1. (A.15)

Note that∏1j=2 (1− λXY (j − 1)) = 1. Conducting a probability balance for (U, 0) yields:

ρ(U, 0) = ρ(UP )(

1− θθ

) ∑X∈I,N

∑Y ∈S,F

θXY ∞∑n=1

λXY (n)n∏j=2

(1− λXY (j − 1))

.

(A.16)

(A.14), (A.15), and (A.16) specify the steady state probabilities in terms of one unknown ρ(UP ).

Solving for ρ(UP ) by applying the fact that the sum of all steady state probabilities equals 1 yields

ρ(UP ) =1∑

X∈I,N∑

Y ∈S,F

(θXY

∑∞n=1

((1 + λXY (n)

θ

)(∏nj=2 (1− λXY (j − 1))

))) . (A.17)

For geometric recoveries, i.e., λXY (n) = λXY for all n, (A.14), (A.15), (A.16) and (A.17) become

ρ(D,n,X, Y ) = (1− λXY )n−1 θXY ρ(UP ) , n ≥ 1, (A.18)

ρ(U, n,X, Y ) = λXY (1− λXY )n−1 θXY ρ(UP ) , n ≥ 1, (A.19)

ρ(U, 0) = ρ(UP )(1− θ

). (A.20)

ρ(UP ) =1

1 + θISλIS

+ θNSλNS

+ θIFλIF

+ θNFλNF

. (A.21)

30

B Inventory and emergency supply with BI insurance

The results in this appendix are direct analogues to the results for the inventory with BI insurance

section; see §4.2.

Theorem B.1

(i) If D > T then ΠEK(D,L,B) = ΠE

K(B)−m.

(ii) If D ≤ T then ΠEK(D,L,B) = ΠE

K(B) +K(ΓEIS(L,B) + ΓEIF (L)

)−m where

ΓEIS(L,B) =B+bT+L

e−vc∑

n=B+b Te−vc+1

ρ(U, n, I, S) ; ΓEIF (L) =bT+L

e−vc∑

n=b Te−vc+1

ρ(U, n, I, F ). (B.1)

Proposition B.1 D∗ = D and L∗ = L for any B.

Theorem B.2 There exists a markup m ≥ 0 such that ΠE∗K ≥ ΠE∗

K if and only if m ≤ mE.

Proposition B.2 If θNF = 0 then (i) ΠEK(B) is concave in B, and (ii) BE∗

K is the smallest B such

that OΠEK(B) ≤ 0, where OΠE

K(B) = OΠEK(B) + K OΓEIS(B) if D ≤ T , and OΠE

K(B) = OΠK(B)

otherwise.

C Proofs

Proof of Property 1: Using equations (A.12) and (A.13), ρ(U, n,X, Y ) = λXY (n)ρ(D,n,X, Y )

and ρ(U, n + 1, X, Y ) = λXY (n + 1) (1− λXY (n)) ρ(D,n,X, Y ). Therefore, ρ(U, n + 1, X, Y ) ≤

ρ(U, n,X, Y )⇔ λXY (n+1) (1− λXY (n)) ≤ λXY (n), which can be rearranged to give the statement

condition.

Proof of Proposition 1: (i) Using (1) to take the first difference of Π0(B) with respect to (w.r.t.)

B, and rearranging terms, gives

OΠ0(B) = (r + g) (ρ(D,B + 1, I, S) + ρ(D,B + 1, N, S))− ht

− hi

(ρ(UP ) +

B∑n=1

(ρ(D,n, I, S) + ρ(D,n,N, S))

)

− v

( ∞∑n=1

ρ(U, n,N, F ) +∞∑

n=B+1

(ρ(U, n, I, S) + ρ(U, n,N, S))

). (C.1)

31

Using (A.12) and (A.13), the second difference of Π0(B) w.r.t. B can be shown to equal

O2Π0(B) = (v − r − g) (ρ(U,B + 1, I, S) + ρ(U,B + 1, N, S))

− hi (ρ(D,B + 1, I, S) + ρ(D,B + 1, N, S)) . (C.2)

The ρ(·) are all non-negative. Further hi ≥ 0 and v− r−g < 0 because r > v by model assumption

and g ≥ 0. Therefore, O2Π0(B) ≤ 0, i.e., Π0(B) is concave in B. (ii) Proof follows from (i).

Proof of Proposition 2: (i) ΩNF (B) = 0 because θNF = 0. Therefore, using (2), ΠK(B) =

Π0(B)−K (ΩIS(B) + ΩNS(B) + ΩIF ). Therefore

OΠK(B) = OΠ0(B)−K (OΩIS(B) + OΩNS(B))

O2ΠK(B) = O2Π0(B)−K(O2ΩIS(B) + O2ΩNS(B)

),

where,

OΩIS(B) = −ρ(U,B + b T

r − vc+ 1, I, S

)≤ 0. (C.3)

O2ΩIS(B) = −ρ(U,B + b T

r − vc+ 2, I, S

)+ ρ

(U,B + b T

r − vc+ 1, I, S

). (C.4)

Using Condition 1 and the associated Property 1, O2ΩIS(B) ≥ 0. In a similar manner,

OΩNS(B) = −ρ(U,B + b T

r − vc+ 1, N, S

)≤ 0. (C.5)

O2ΩNS(B) = −ρ(U,B + b T

r − vc+ 2, N, S

)+ ρ

(U,B + b T

r − vc+ 1, N, S

), (C.6)

and O2ΩNS(B) ≥ 0 for all B. Now, O2Π0(B) ≤ 0 using Proposition 1 and K ≥ 0. Therefore

O2ΠK(B) ≤ 0 i.e., ΠK(B) is concave in B. (ii) Proof follows from (i).

Proof of Theorem 1: Follows from the fact that OΠK(B) = OΠ0(B)−K (OΩIS(B) + OΩNS(B))

and OΩIS(B) ≤ 0 and OΩNS(B) ≤ 0; see (C.3) and (C.5). Because K ≥ 0 it then follows

that OΠK(B) ≥ OΠ0(B). Proof then follows because Π0(B) and ΠK(B) are concave in B; see

Propositions 1 and 2. (This theorem can also be proven if Condition 1 and the associated Property

1 does not hold, in which case ΠK(B) might not be concave.)

Proof of Proposition 3: (i) Using (4) to take the first and second differences of ΠE0 (B), it can

be shown that

OΠE0 (B) = OΠ0(B)− (r + g − e) ((ρ(D,B + 1, I, S) + ρ(D,B + 1, N, S))) , (C.7)

32

and

O2ΠE0 (B) = (v − e) (ρ(U,B + 1, I, S) + ρ(U,B + 1, N, S))

− hi (ρ(D,B + 1, I, S) + ρ(D,B + 1, N, S)) . (C.8)

The ρ(·) are all non-negative. Further hi ≥ 0 and v − e < 0 because e > v by model assumption.

Therefore, O2ΠE0 (B) ≤ 0, i.e., ΠE

0 (B) is concave in B. (ii) Proof follows from (i).

Proof of Proposition 4: (i) Follows in an analogous manner to proof of Proposition 2 but with

e replacing r in the first- and second differences of the Ω functions, i.e.,

OΩIS(B) = −ρ(U,B + b T

e− vc+ 1, I, S

)≤ 0. (C.9)

O2ΩIS(B) = −ρ(U,B + b T

e− vc+ 2, I, S

)+ ρ

(U,B + b T

e− vc+ 1, I, S

). (C.10)

OΩNS(B) = −ρ(U,B + b T

e− vc+ 1, N, S

)≤ 0. (C.11)

O2ΩNS(B) = −ρ(U,B + b T

e− vc+ 2, N, S

)+ ρ

(U,B + b T

e− vc+ 1, N, S

), (C.12)

(ii) Proof follows immediately from (i).

Proof of Theorem 2: Analogous to proof of Theorem 1 but using the emergency-case expressions

and propositions.

Proof of Theorem 3: (i) The profit w(s) associated with each state s is the same as in the

no-insurance case except that (i) the premium cost p is incurred in every state, and (ii) the firm

receives the reimbursement RIS(B,n) in states (U, n, I, S) and RIF (B,n) in states (U, n, I, F ) for

n = 1, . . . ,∞. Therefore,

Π0(D,L,B) = Π0(B)− p+∞∑n=1

(RIS(B,n)ρ(U, n, I, S) +RIF (B,n)ρ(U, n, I, F )) . (C.13)

Using (7), we then have Π0(B) = Π0(B)−m. (ii) Analogous to (i) but for the case where there is

an emergency supplier.

Proof of Theorem 4: The profit w(s) associated with each state s is the same as in the no-

insurance case except that (a) the premium cost p is incurred in every state, and (b) the firm

receives the reimbursement RIS(B,n) in states (U, n, I, S) and RIF (B,n) in states (U, n, I, F ) for

n = 1, . . . ,∞. Because of (b), the cost K is incurred in the first up period after a insurable-event

33

disruption of length n if the interruption loss IL(B,n)− R(B,n) > T . We need to determine the

values of n for which the cost K is incurred in states (U, n, I, S) and (U, n, I, F ). The reimbursement

function is given by RIY (B,n) = min[ILIY (B,n)−D]+ , L where ILIY (B,n) = πU (B,n) −

πDIY (B,n) and Y ∈ S, F denotes inventory success or failure. Expressions for IL(B,n) were

developed in §4.1 for all four disruption types XY .13 For the insurable-event disruptions,

ILIS(B,n) = (r − v) [n−B]+ ,

ILIF (B,n) = (r − v)n. (C.14)

and therefore

RIS(B,n) =

0, n < B + D

r−v(n−B) (r − v)−D, B + D

r−v ≤ n < B + D+Lr−v

L, n ≥ B + D+Lr−v ,

(C.15)

RIF (B,n) =

0, n < D

r−vn (r − v)−D, D

r−v ≤ n <D+Lr−v

L, n ≥ D+Lr−v .

(C.16)

Using (C.14) and (C.15) it can be shown that the additional K cost is incurred in (U, n, I, S) if and

only if n > B+ Tr−v when D < T and if and only if n > B+ T+L

r−v when D ≤ T .14 Using (C.14) and

(C.16) it can be shown that the additional K cost is incurred in (U, n, I, F ) if and only if n > Tr−v

when D < T and if and only if n > T+Lr−v when D ≤ T .

(i) D > T : The penalty K is incurred in the same states with BI insurance as without. Therefore,

ΠK(D,L,B) = ΠK(B)− p+∞∑n=1

(RIS(B,n)ρ(U, n, I, S) +RIF (B,n)ρ(U, n, I, F )) .

and proof follows by applying (7).

(ii) D ≤ T : As compared to the case of no BI insurance, the penalty K is incurred in (U, n, I, S)

if and only if n > B + T+Lr−v (instead of n > B + T

r−v ) and is incurred in (U, n, I, F ) if and only if

n > T+Lr−v (instead of n > T

r−v ). Therefore,

ΠK(D,L,B) = ΠK(B) +K (ΓIS(L,B) + ΓIF (L))

− p+∞∑n=1

(RIS(B,n)ρ(U, n, I, S) +RIF (B,n)ρ(U, n, I, F ))

13The premium p was not relevant in the no-insurance analysis. However, because p is a tangible cost incurred inevery period regardless of the state, we then have (n+ 1) p appearing in both πU (B,n) and πD(B,n). This termcancels out in the IL(B,n) expressions, which therefore remain the same as before.

14This analysis will differ if BI insurance does not reimburse stockpile usage.

34

where ΓIS(L,B) and ΓIF (L) are given by (8). Proof follows by applying (7).

Proof of Proposition 5: First consider the case in which T < D. We then have T < D for all

feasible D. Using Proposition 4, ΠK(D,L,B) = ΠK(B)−m. This is independent of D and L, and

so all feasible D and L are optimal. Thus D∗ = D and L∗ = L. Next consider the case in which

T ≥ D. Using Proposition 4, ΠK(D,L,B) = ΠK(B)+K (ΓIS(L,B) + ΓIF (L))−m for D ≤ D ≤ T

and ΠK(D,L,B) = ΠK(B)−m for D > T . Now, K (ΓIS(L,B) + ΓIF (L)) ≥ 0 and so there must

exist an optimal D ≤ T for any L and B. For D ≤ T , ΠK(D,L,B) is independent of D, and so all

D ≤ D ≤ T are optimal. Therefore D∗ = D. Also, ΠK(D,L,B) increases in L for D ≤ T because

ΓIS(L,B) and ΓIF (L,B) both increase in L; see (8). Therefore L∗ = L.

Proof of Proposition 6: First consider the case in which D > T . ΠK(B) = ΠK(B) − m

using Propositions 4 and 5. Proof follows from Proposition 2. Next consider the case in which

D ≤ T . Using Propositions 4 and 5, ΠK(B) = ΠK(B) + K(ΓIS(L, B) + ΓIF (L)

)− m. Using

(2), ΠK(B) = Π0(B) −K (ΩIS(B) + ΩNS(B) + ΩIF ) because ΩNF (B) = 0 when θNF = 0. Now

ΓIS(L, B) = ΩIS(B)− ΩIS(B) and ΓIF (L) = ΩIF − ΩIF , where

ΩIS(B) =∞∑

n=B+bT+Lr−vc+1

ρ(U, n, I, S) ; ΩNF =∞∑

n=bT+Lr−vc+1

ρ(U, n, I, F )

and ΩIS(B) and ΩIF are given by (3). Therefore, ΠK(B) = Π0(B)−K(

ΩIS(B) + ΩNS(B) + ΩIF

).

The first and second order differences with respect to B are then given by

OΠK(B) = OΠ0(B)−K(OΩIS(B) + OΩNS(B)

)O2ΠK(B) = O2Π0(B)−K

(O2ΩIS(B) + O2ΩNS(B)

)Now O2Π0(B) ≤ 0 using Proposition 1, and O2ΩNS(B) ≥ 0 (see proof of Proposition 2.) Now,

OΩIS(B) = −ρ(U,B + bT + L

r − vc+ 1, I, S

)≤ 0. (C.17)

O2ΩIS(B) = −ρ(U,B + bT + L

r − vc+ 2, I, S

)+ ρ

(U,B + bT + L

r − vc+ 1, I, S

). (C.18)

O2ΩIS(B) ≥ 0 for all B using Condition 1 and the associated Property 1. We then have OΠK(B) ≤

0 because O2Π0(B) ≤ 0, O2ΩNS(B) ≥ 0, O2ΩIS(B) ≥ 0 and K ≥ 0. This proves (i). Proof of

(ii) follows from (i). Note that OΠK(B) = OΠK(B) + KOΓIS(B) because ΠK(B) = ΠK(B) +

K(ΓIS(L, B) + ΓIF (L)

)−m. For later use we also note that using (8),

OΓIS(B) = ρ(U,B + 1 + bT + L

r − vc, I, S)− ρ(U,B + b T

r − vc+ 1, I, S). (C.19)

35

Proof of Theorem 5: First consider D > T . Π∗K = Π∗K − m using Propositions 4 and

5. Statement is true with m = 0. Next consider D ≤ T . Π∗K = ΠK(B∗K) = ΠK(B∗K) +

K(

ΓIS(L, B∗K) + ΓIF (L))−m, where the last equality follows from Propositions 4 and 5. Π∗K =

ΠK(B∗K). Therefore, Π∗K − Π∗K = ΠK(B∗K) − ΠK(B∗K) + K(

ΓIS(L, B∗K) + ΓIF (L))− m, which

strictly decreases in m. Therefore, if Π∗K < Π∗K for some m then Π∗K < Π∗K for all larger

m. It therefore remains to prove that there exists some m ≥ 0 such that Π∗K ≥ Π∗K . Π∗K =

ΠK(B∗K) ≥ ΠK(B∗K) = ΠK(B∗K) + K(ΓIS(L, B∗K) + ΓIF (L)

)−m. Now, Π∗K = ΠK(B∗K) and so

Π∗K ≥ Π∗K +K(ΓIS(L, B∗K) + ΓIF (L)

)−m. Proof then follows because K ≥ 0 and ΓIS(L, B∗K) ≥ 0

and ΓIF (L) ≥ 0; see (8).

Proof of Theorem B.1: Analogous to proof of Theorem 4 but with ILEIS(B,n) = (e− v) [n−B]+

and ILEIF (B,n) = (e− v)n.

Proof of Proposition B.1: Analogous to proof of Proposition 5.

Proof of Proposition B.2: Analogous to proof of Proposition 6.

Proof of Theorem B.2: Analogous to proof of Theorem 5.

Proof of Theorem 6: We present the proof for the case in which there is no emergency supplier.

The proofs for the case of an emergency supplier are directly analogous.

Definition 1: Using Propositions 4 and 5, OΠK(B) = OΠK(B) + K OΓIS(B). Using Condition 1

and the associated Property 1, OΓIS(B) ≤ 0; see (C.19). Therefore OΠK(B) ≤ OΠK(B) as K ≥ 0.

Definition 2: ∆VK = ΠK(B∗K)−ΠK(B∗K)−(

ΠK(0)−ΠK(0))≤ ΠK(B∗K)−ΠK(B∗K)−

(ΠK(0)−ΠK(0)

)because ΠK(B∗K) ≤ ΠK(B∗K). Using Theorem 4 and Proposition 5, it then follows that ∆VK ≤

K(

ΓIS(L, B∗K)− ΓIS(L, 0)))

. Now ΓIS(L,B), given by (8), is decreasing in B using Condition 1

and the associated Property 1. Therefore ∆VK ≤ 0 as K ≥ 0.

Proof of Theorem 7: Using Definition 2 insurance and emergency supply are substitutes (com-

plements) if ∆V E .= ΠEK (0) − ΠK (0) −

(ΠEK (0)−ΠK (0)

)< (>)0. Using Theorem 4 and its

emergency supply counterpart we have

∆V E (e) = K(ΓEMIS

(L, 0

)+ ΓEMIF

(L)− ΓIS

(L, 0

)− ΓIF

(L))

= K

bT+Le−vc∑

n=b Te−vc+1

ρ(U, n, I, S) +bT+L

e−vc∑

n=b Me−vc+1

ρ(U, n, I, F )−bT+L

r−vc∑

n=b Tr−vc+1

ρ(U, n, I, S)

−bT+L

r−vc∑

n=b Tr−vc+1

ρ(U, n, I, F )

.

36

Tailoring the above expression to geometric recoveries and identical recovery probability λXY = λ,

∆V E (e) = ρ (UP ) (θIS + θIF )((

(1− λ)T

e−v − (1− λ)T+Le−v

)−(

(1− λ)T

r−v − (1− λ)T+Lr−v

)).

Taking derivative of ∆V E with respect to e,

(∆V E (e)

)′= ρ (UP ) (θIS + θIF ) (1− λ)

Te−v

ln (1− λ)(e− v)2 χ (e, λ) .

where χ (e, λ) .=(−T + (1− λ)

Le−v(T + L

)). Since ln (1− λ) < 0, the sign of

(∆V E (e)

)′ is

determined by the sign of χ (e, λ). It is straightforward to show that χ (e, λ) is a decreasing function

of e and λ, and establish the following:

λ < 1−(

TT+L

) r−vL ∃e (λ) such that χ (e, λ) < (>)0 for e ≤ e (λ) (e > e (λ))

λ ≥ 1−(

TT+L

) r−vL

χ (e, λ) < 0, which implies

λ < 1−(

TT+L

) r−vL ∃e (λ) such that ∆V E (e)′ > (<)0 for e ≤ e (λ) (e > e (λ))

λ ≥ 1−(

TT+L

) r−vL (

∆V E (e))′> 0

.

Because ∆V E (v) < 0 and ∆V E (r) = 0, it follows that if λ < 1−(

TT+L

) r−vL then there exists

e (λ) such that ∆V E (e) ≤ (>)0 for e ≤ e (λ) (e > e (λ)); if λ ≥ 1−(

TT+L

) r−vL then ∆V E (e) < 0.

Proof completes.

Proof of Theorem 8: We show the statement holds for two definitions of substitute/complement.

Emergency supply and inventory are substitutes (complements) if

Definition 1: Emergency supply reduces (increases) the incremental value of inventory, i.e.,

OΠE(B) ≤ (≥) OΠ(B).

Definition 2: Emergency supply reduces (increases) the value of inventory, and vice versa, i.e.,

∆W ≤ (≥) 0, where ∆W = ΠE(BE∗)−Π(B∗)−(ΠE(0)−Π(0)

).

We present the proof for the case without insurance. The proofs for the case with insurance are

directly analogous.

Definition 1: For the no-penalty case, OΠE0 (B)− OΠ0(B) = − (r + g − e) (ρ(D,B + 1, I, S)+

ρ(D,B + 1, N, S) < 0. For the penalty case, from Propositions 2 and 4, OΠEK(B) = OΠE

0 (B)−

K(OΩE

IS(B) + OΩENS(B)

)and OΠK(B) = OΠ0(B)−K (OΩIS(B) + OΩNS(B)) . Then, by (3) and

(6), we have OΠEK(B)− OΠK(B) = OΠE

0 (B)− OΠ0(B)−K(−ρ(U,B + b T

e−v c, I, S)

+ρ(U,B + b Tr−v c, I, S)− ρ(U,B + b T

e−v c, N, S) + ρ(U,B + b Tr−v c, N, S)

).

Because b Te−v c ≥b

Tr−v c and ρ(U,B + b T

e−v c, X, S) ≤ ρ(U,B + b Tr−v c, X, S), X ∈ I,N by

Property 1, OΠEK(B)− OΠK(B) ≤ 0.

37

Definition 2: For the no-penalty case, ∆W0 = ΠE0 (BE∗

0 )−Π(B∗0)−(ΠE

0 (0)−Π0(0))≤ ΠE

0 (BE∗0 )−

Π(BE∗0 )−

(ΠE

0 (0)−Π0(0))

because Π0(BE∗0 ) ≤ Π0(B∗0). Using (4), ∆W0 ≤ − (r + g − e)(∑BE∗

0n=1 (ρ(D,n, I, S) + ρ(D,n,N, S))

)< 0. For the penalty case, ∆WK = ΠE

K(BE∗K )−ΠK(B∗K)−(

ΠEK(0)−ΠK(0)

)≤ ΠE

K(BE∗K )−ΠK(BE∗

K )−(ΠEK(0)−ΠK(0)

). Using (5), ∆WK ≤ − (r + g − e)(∑BE∗

Kn=1 (ρ(D,n, I, S) + ρ(D,n,N, S))

)−K

(∑b Te−vc

n=b Tr−vc+1

ρ(U, n, I, S) +∑b T

e−vc

n=b Tr−vc+1

ρ(U, n,N, S))

+K(∑BE∗

K +b Te−vc

n=BE∗K +b T

r−vc+1

ρ(U, n, I, S) +∑BE∗

K +b Te−vc

n=BE∗K +b T

r−vc+1

ρ(U, n,N, S))< 0, where the second

inequality follows Property 1.

Proof of Proposition 7: BI Insurance adds no value if the deductible D exceeds the penalty

threshold T . We can therefore restrict attention to D ≤ D ≤ T . It is convenient to define the

following expressions

ZIS(D,L,B) =∞∑n=1

(RIS(B,n)ρ(U, n, I, S)) ≥ 0

ZIF (D,L,B) =∞∑n=1

(RIF (B,n)ρ(U, n, I, F )) ≥ 0.

We then have R(D,L,B) = ZIS(D,L,B) + ZIF (D,L,B). Using the reimbursement expressions

(C.15) and (C.16) developed in the proof of Theorem 4,

ZIS(D,L,B) =B+bD+L

r−vc∑

n=B+b Dr−vc+1

((n−B) (r − v)−D) ρ(U, n, I, S) + L

∞∑n=B+bD+L

r−vc+1

ρ(U, n, I, S)

ZIF (D,L,B) =bD+L

r−vc∑

n=b Dr−vc+1

(n (r − v)−D) ρ(U, n, I, F ) + L∞∑

n=bD+Lr−vc+1

ρ(U, n, I, F ). (C.20)

As ZIF (D,L,B) does not depend on B we write it as ZIF (D,L) hereafter. Using R(D,L,B) =

ZIS(D,L,B) + ZIF (D,L), it follows from expression (10) that

ΠK(D,L,B) = ΠK(B) +K (ΓIS(L,B) + ΓIF (L))−m (ZIS(D,L,B) + ZIF (D,L)) . (C.21)

The optimal L and D are integer multiples of r − v for any B. See Proposition C.1 below.

We therefore write D = nD(r − v) and L = nL(r − v), and refer to nD and nL in what follows

with the understanding that D = nD(r − v) and L = nL(r − v). We then have ZIS(nD, nL, B) =

(r − v) zIS(nD, nL, B) and ZIF (nD, nL) = (r − v) zIF (nD, nL) where

zIS(nD, nL, B) =nL∑n=1

nρ(U, n+B + nD, I, S) + nL

∞∑n=nL+1

ρ(U, n+B + nD, I, S) (C.22)

38

zIF (nD, nL) =nL∑n=1

nρ(U, n+ nD, I, F ) + nL

∞∑n=nL+1

ρ(U, n+ nD, I, F ). (C.23)

We can then write (C.21) as

ΠK(nD, nL, B) = ΠK(B) +K (ΓIS(nL, B) + ΓIF (nL))−m(r − v) (zIS(nD, nL, B) + zIF (nD, nL)) ,(C.24)

where tailoring (8) to L = nL(r − v) and T = nT (r − v),

ΓIS(nL, B) =B+nT +nL∑n=B+nT +1

ρ(U, n, I, S) ; ΓIF (nL) =nT +nL∑n=nT +1

ρ(U, n, I, F ), (C.25)

It follows from (C.24) that the nD that minimizes zIS(nD, nL, B)+zIF (nD, nL) maximizes ΠK(nD, nL, B).

Let OnDzIS(nD, nL, B) denote the first difference of zIS(nD, nL, B) w.r.t. nD Then

OnDzIS(nD, nL, B) =nL∑n=1

n (ρ(U, n+B + nD + 1, I, S)− ρ(U, n+B + nD, I, S))

+ nL

∞∑n=nL+1

(ρ(U, n+B + nD + 1, I, S)− ρ(U, n+B + nD, I, S)) . (C.26)

We also have

OnDzIF (nD, nL) =nL∑n=1

n (ρ(U, n+ nD + 1, I, S)− ρ(U, n+ nD, I, S))

+ nL

∞∑n=nL+1

(ρ(U, n+ nD + 1, I, S)− ρ(U, n+ nD, I, S)) . (C.27)

It follows from condition 1 and the associated Property 1 that OnDzIS(nD, nL, B) ≤ 0 and OnDzIF (nD, nL) ≤

0. Therefore, it is optimal to set nD at the maximum possible value, which is nT = Tr−v because

we are in the region D ≤ D < T . Therefore n∗D = nT or equivalently D∗ = T .

Proposition C.1 The optimal L and D are integer multiples of r − v for any B.

Proof of Proposition C.1: No-penalty case: Π0(D,L,B) = Π0(B)−m (ZIS(D,L,B) + ZIF (D,L,B))

so the optimal D and L (for any given B) minimize ZIS(D,L,B)+ZIF (D,L). If D+Lr−v is not integer,

then ZIS(D,L,B) and ZIF (D,L) both decrease in D until an integer is reached; see (C.20). We

can, therefore, restrict attention to D and L such that D + L is an integer multiples of r − v. IfDr−v is not integer, then decreasing L by epsilon while increasing D by epsilon so D + L remains

constant (at a multiple of r− v) reduces both ZIS(D,L,B) and ZIF (D,L,B). Therefore, D∗ must

be an integer multiple of r − v. L∗ must also be an integer multiple of r − v because we already

proved that D∗ + L∗ is an integer multiples of r − v

39

Penalty case with D > T : Identical to proof of no-penalty case.

Penalty case with D <= T : ΠK(D,L,B) = ΠK(B)+K (ΓIS(L,B) + ΓIF (L))−m (ZIS(D,L,B) + ZIF (D,L)).

Therefore, the optimal D (for any given L and B) minimizes ZIS(D,L,B) +ZIF (D,L). As above,

we can, therefore, restrict attention to D and L such that D+L is an integer multiples of r−v. De-

creasing L by epsilon while increasing D to preserve D+L constant (at a multiple of r−v) reduces

both ZIS(D,L,B) and ZIF (D,L,B). Because T is an integer multiple of r − v (by assumption in

the proportional loading section), L can be decreased (until an integer multiple of r− v is reached)

without altering ΓIS(L,B) or ΓIF (L); see (8). These two observations imply that L∗ must be an

integer multiple of r − v, which (from above) implies D∗ must be an integer multiple of r − v

Proof of Proposition 8: Using Proposition 7 and (C.25)

ΠK(nL, B) = ΠK(B) +K (ΓIS(nL, B) + ΓIF (nL))−m(r − v) (zIS(nT , nL, B) + zIF (nT , nL)) ,(C.28)

where ΓIS(nL, B) and ΓIF (nL) are given by (C.25) and (C.27). Tailoring (C.22), (C.22), and (C.25)

to geometric recoveries,

zIS(nT , nL, B) = ρ(UP )(θISλIS

)(1− λIS)B+nT (1− (1− λIS)nL)

zIF (nD, nL) = ρ(UP )(θIFλIF

)(1− λIF )nT (1− (1− λIF )nL)

ΓIS(nL, B) = ρ(UP )θIS (1− λIS)B+nT (1− (1− λIS)nL)

ΓIF (nL) = ρ(UP )θIF (1− λIF )nT (1− (1− λIF )nL) .

It follows that zIS(nT , nL, B) = ΓIS(nL,B)λIS

and zIF (nT , nL) = ΓIF (nL)λIF

. Therefore,

ΠK(nL, B) = ΠK(B) +(K − m(r − v)

λIS

)ΓIS(nL, B) +

(K − m(r − v)

λIF

)ΓIF (nL) (C.29)

For any given B, the first and second differences of ΠK(nL, B) w.r.t. nL are

OnLΠK(nL, B) =(K − m(r − v)

λIS

)OnLΓIS(nL, B) +

(K − m(r − v)

λIF

)OnLΓIF (nL) (C.30)

where

OnLΓIS(nL, B) = ρ(UP )θISλIS (1− λIS)B+nT +nL ≥ 0

OnLΓIF (nL) = ρ(UP )θIFλIF (1− λIF )nT +nL ≥ 0, (C.31)

and

O2nL

ΠK(nL, B) =(K − m(r − v)

λIS

)O2nL

ΓIS(nL, B) +(K − m(r − v)

λIF

)O2nL

ΓIF (nL),

40

where

O2nL

ΓIS(nL, B) = −ρ(UP )θISλ2IS (1− λIS)B+nT +nL ≤ 0

O2nL

ΓIF (nL) = −ρ(UP )θIFλ2IF (1− λIF )nT +nL ≤ 0.

O2nL

ΠK(nL, B) can be less than or greater than 0, and so ΠK(nL, B) may not be concave in nL.

Case A: K > m(r−v)λIS

and K > m(r−v)λIF

. Using (C.30) and (C.31), OnLΠK(nL, B) > 0 for all B, and

so n∗L = nL. Therefore, L∗ = nL(r − v) = L.

Case B: K ≤ m(r−v)λIS

and K ≤ m(r−v)λIF

. Using (C.30) and (C.31), OnLΠK(nL, B) ≤ 0 for all B, and

so n∗L = 0. Therefore, L∗ = 0(r − v) = 0.

Case C: K > m(r−v)λIS

andK ≤ m(r−v)λIF

. Using (C.30) and (C.31) and rearranging terms, OnLΠK(nL, B) ≤

0 for any given B if and only if(1− λIS1− λIF

)nL(K − m(r − v)

λIS

)θISλIS (1− λIS)B+nT +

(K − m(r − v)

λIF

)θIFλIF (1− λIF )nT ≤ 0,

K > m(r−v)λIS

and K ≤ m(r−v)λIF

in case C, which implies λIS > λIF . Therefore, 1−λIS1−λIF

< 1, and(1−λIS1−λIF

)nL

decreases in nL. Therefore, the left-hand side (lhs) above decreases in nL, and if

OnLΠK(nL, B) ≤ 0 for some nL = n, then OnLΠK(nL, B) ≤ 0 for all larger nL = n. Thus

ΠK(nL, B) is quasiconcave in nL for any B, and so the first difference condition above is necessary

sufficient for optimality, i.e., n∗L is given by the smallest nL such that OnLΠK(nL, B) ≤ 0. By

definition L∗ = n∗L(r − v) = L.

Case D: K ≤ m(r−v)λIS

andK > m(r−v)λIF

. Using (C.30) and (C.31) and rearranging terms, OnLΠK(nL, B) ≤

0 for any given B if and only if(1− λIS1− λIF

)nL(K − m(r − v)

λIS

)θISλIS (1− λIS)B+nT +

(K − m(r − v)

λIF

)θIFλIF (1− λIF )nT ≤ 0,

K ≤ m(r−v)λIS

and K > m(r−v)λIF

in case D, which implies λIS < λIF . Therefore, 1−λIS1−λIF

> 1, and(1−λIS1−λIF

)nL

increases in nL. Therefore, the lhs above decreases in nL, and if OnLΠK(nL, B) ≤ 0 for

some nL = n, then OnLΠK(nL, B) ≤ 0 for all larger nL = n. Thus ΠK(nL, B) is quasiconcave in

nL for any B, and so the first difference condition above is necessary sufficient for optimality, i.e.,

n∗L is given by the smallest nL such that OnLΠK(nL, B) ≤ 0. By definition L∗ = n∗L(r− v) = L.

Proof of Theorem 9: The firm does not purchase insurance in case B (see Proposition 8). We

therefore only need to consider cases A, C, and D. We first consider definition 1, i.e., insurance and

inventory are substitutes (complements) if OΠ(B) ≤ (≥) OΠ(B). Using (C.29)

OBΠK(nL, B) = OBΠK(B) +(K − m(r − v)

λIS

)OBΓIS(nL, B),

41

where using (C.19), tailored to the geometric case,

OBΓIS(nL, B) = ρ(UP )θISλIS (1− λIS)B+nT ((1− λIS)nL − 1) ≤ 0

Thus, for any given coverage limit nL, OBΠK(nL, B) ≤ (≥)OBΠK(B) if and only K ≥ (≤)m(r−v)λIS

.

It immediately follows that insurance and inventory are definition 1 substitutes in cases A and

C and complements in case D. We now consider definition 2, i.e., insurance and inventory are

substitutes (complements) if ∆V ≤ (≥) 0, where ∆V = Π(B∗)−Π(B∗)−(

Π(0)−Π(0))

.

Case A: K > m(r−v)λIS

andK > m(r−v)λIF

. Recall that n∗L = nL; see proof of Proposition 8. Now ∆VK =

ΠK(nL, B∗K)−ΠK(B∗K)−(

ΠK(nL, 0)−ΠK(0))≤ ΠK(nL, B∗K)−ΠK(B∗K)−

(ΠK(nL, 0)−ΠK(0)

)because ΠK(nL, B∗K) ≤ ΠK(nL, B∗K). Using (C.29) it follows that

∆VK ≤(K − m(r − v)

λIS

)(ΓIS(nL, B∗K)− ΓIS(nL, 0)

).

The first term in parenthesis is positive because K > m(r−v)λIS

. The second term is negative because

ΓIS(nL, B) decreases in B; see proof of Theorem 9. Therefore ∆VK ≤ 0.

Case D: K ≤ m(r−v)λIS

and K > m(r−v)λIF

. Let n∗L(B) denote the optimal nL for a given B.

∆VK = ΠK(n∗L(B∗K), B∗K) − ΠK(B∗K) −(

ΠK(n∗L(0), 0)−ΠK(0))≥ ΠK(n∗L(0), B∗K) − ΠK(B∗K) −(

ΠK(n∗L(0), 0)−ΠK(0))

because ΠK(n∗L(B∗K), B∗K) ≥ ΠK(n∗L(0), B∗K). Using (C.29) it follows that

∆VK ≥(K − m(r − v)

λIS

)(ΓIS(n∗L(0), B∗K)− ΓIS(n∗L(0), 0)

).

The first term in parenthesis is negative because K ≤ m(r−v)λIS

. The second term is negative because

ΓIS(nL, B) decreases in B; see proof of Theorem 9. Therefore ∆VK ≥ 0.

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