no claim? your gain: design of residual value extended ... · design of residual value extended...

No Claim? Your Gain:Design of Residual Value Extended Warranties under

Strategic Claim Behavior

Guillermo GallegoIEOR Department, Columbia University, New York, NY 10027, [email protected]

Ming HuRotman School of Management, University of Toronto, Toronto, ON M5S3E6, [email protected]

Ruxian WangIEOR Department, Columbia University, New York, NY 10027, [email protected]

Julie Ward, Jose L. Beltran, Shailendra JainHewlett-Packard Labs, Palo Alto, CA 94304, {jward, jose-luis.beltran, shelen.jain}@hp.com

November 2010

Traditional extended warranties for IT products do not differentiate customers according to their usage

rates or operating environment. These warranties are priced to cover the costs of high-usage customers who

tend to experience more failures and are therefore more costly to support. This makes traditional warranties

economically unattractive to low-usage customers. To address this issue, residual value warranties have been

introduced in industry practice. These warranties refund a part of the upfront price to customers who have

zero or few claims according to a pre-determined refund schedule. By design, the net cost of these warranties

is lower for light users than for heavy users. As a result, a residual value warranty enables the provider

to price discriminate based on usage rates or operating conditions without the need to monitor individual

customers’ usage. However, a warranty with residual values may induce strategic claim behavior as customers

may prefer to repair small failures out of pocket rather than giving up potential refunds. Taking this strategic

claim behavior into account, we introduce, design and price residual value warranties. We identify conditions

under which residual value warranties are strictly more profitable than traditional warranties. We show that

if either failure costs or usage rates are highly variable, then the strategic claim behavior may undermine

the benefit of residual value warranties to the point where they degenerate into traditional warranties. In

cases of more moderate variability, profits from residual value warranties can be significantly higher than

traditional warranties.

1

mailto:[email protected]



mailto:jward, @hp.com

2 Gallego et al.: Extended Warranties with Residual Values

1. Introduction

As product margins decline in increasingly competitive hardware markets, high-margin, high-

revenue aftermarket services such as extended warranties (EW) are becoming critical to manufac-

turers’ profitability (see Cohen et al. 2006). Beyond direct profits, post-sales services are a critical

lever for influencing customer loyalty, and in commodity product businesses, service quality and

variety are important competitive differentiators. Through the sale of innovative and attractive

post-sales services, a manufacturer can enable its customers to extract more use from their prod-

ucts, and increase customers’ loyalty. Not surprisingly, improving service attach rates to hardware

sales is a top strategic priority to many original equipment manufacturers (OEMs).

OEMs face three important challenges to improve the sales of EW. The first is competition

from its own sales channel. Channel partners usually sell their own or third-party extended service

plans on OEM’s products, since they earn much higher margins on services than on hardware (see

Gallego et al. 2010). A second challenge is a perception among some customers that EWs are not

a good value (see ConsumerReports.org 2006). This perception may be due in part to the fact that

most warranties are offered at a uniform price regardless of usage. The price may be too high for

low-usage customers, and less than the support cost of some high-usage customers. This leads us

to the third challenge, which is how to price discriminate customers based on expected support

when it is not possible, or it is too costly, to measure usage or to monitor the environment where

the product is used. For some products, such as PCs, usage is an elusive concept to define. For

others where natural definitions of usage exist, such as printers or storage products, there may be

technical challenges to verifying usage.

To overcome these challenges, an OEM must design a compelling service offering that differen-

tiates its own EWs from competitors’ services, attracts customers from different market segments

and retains healthy profit margins. One way to achieve market segmentation with a single EW

product without having to verify usage is through a residual value warranty (RVW). It is a finite-

length EW in which customers may receive a full or partial refund at expiration, in the form of

cash or a credit towards future purchases, where the refund size depends on the number of claims

made against the warranty. Customers who are not entitled to a refund are still covered by the

warranty. Like auto insurance policies, this type of EW rewards customers for having few or no

claims. It also provides “peace-of-mind” protection to customers, without requiring customers to

make extra payments at each time of a claim. Since 2007, Hewlett-Packard has implemented the

idea of a “risk-free” care pack with full refund for its commercial customers in the PC business

(see HP 2010).

Gallego et al.: Extended Warranties with Residual Values 3

The idea is to design the price and the refund schedule so that the service product can better price

discriminate among existing users, or be attractive to a broader set of users than the traditional

warranty and at the same time be more profitable to the warranty provider. Low-usage customers

will typically experience a lower net cost because they make fewer claims on average and are

therefore entitled to larger refunds. Through intrinsically equitable pricing, an RVW may attract a

much broader range of customers than a comparable traditional warranty with no refunds, thereby

capturing greater market share and greater profits for the warranty provider – all through a single

service product. Moreover, it can effectively discriminate not only according to usage rates, but

according to any other factor that affects their failure propensity, such as operating conditions.

Significantly, an RVW achieves this market segmentation without any need to verify usage rates

or operating conditions.

A warranty provider offering RVWs may choose to make it incumbent upon the customer to

request the refund, if eligible, at the end of the warranty. Under this choice, customers may forget

or lose the paperwork needed to claim refunds or they may not have the time to do so. While low

redemption rates can benefit the warranty provider in the short run, a customer who experiences

difficulties claiming a refund may decide against buying this service in the future, negatively affect-

ing the lifetime value of the customer to the provider. For this reason, we will assume that the

warranty provider will automatically credit the customer’s account with the appropriate refund

at the end of the warranty. This is consistent with current practices (e.g., Officemax, Staples and

Dell) of making it easier for customers to obtain rebates. This also helps avoid the problem of

modeling customers that are rational at the time of purchase but forgetful at the time of claiming

refunds, although mismatch between intent and execution is a common phenomenon (see DellaV-

igna and Malmendier 2006). More importantly, unlike rebates on physical products which may

price discriminate customers depending on their post-purchase redemption behavior (see Chen

et al. 2005), a residual value warranty with automatic refund credit can still price discriminate

among customers within the same usage group who may have different ex-post experiences with

failures. This is a unique feature of combining rebates with warranty products due to the ex-ante

uncertainty embedded in the failure process.

The price and refund schedule of an RVW must be carefully designed if these benefits are to

be realized. One must consider the expected support costs to the manufacturer for a customer

who buys the RVW; this is in turn affected by the distribution of product usage rates in the

customer population, as well as the strategic claim behavior that may be induced in customers by

the prospect of a refund. We have created an analytical framework for designing and pricing RVWs


that considers all of these factors. We begin with a basic model, in which we assume there is at

most one failure during the warranty period. We investigate the benefits of offering an RVW and

the implications of strategic claim behavior – a strategic customer may decide to pay out of pocket

for an inexpensive repair rather than losing the residual value – on the optimal warranty design.

We show that if the random repair cost has large variability or the segments are distant from

each other in usage rates, the strategic claim behavior induced by residual value warranties may

significantly undermine the benefit of price discrimination among a broader range of customers.

In the general framework, we allow for multiple failures and formulate a continuous-time optimal

control model to study how customers can strategically decide whether or not to place a claim over

time. We characterize the strategic customer’s optimal claim strategy. For the cases of exponentially

distributed and constant repair costs, we obtain closed-form solutions for the customer’s expected

refund, net of self-defrayed repair costs. We also develop a very good approximation to the expected

net refund that is valid for all failure distributions based on a constant threshold policy for claiming

failures. Finally, we design and price residual value warranties based on the fully characterized

strategic claim behavior of customers. To our knowledge this is the first research to design and

price post-sales services by applying ideas from stochastic optimal control theory.

The rest of the paper is structured as follows. Section 2 describes the related literature. Section

3 defines a general RVW and presents modeling assumptions. Section 4 considers the basic model

where there is assumed to be at most one failure. Section 5 considers a general framework that

allows for multiple failures. Section 6 makes conclusions, provides ideas of innovative extended

warranty products and points out future research directions. Most proofs can be found in the

appendix. The rest of proofs and numerical examples are provided in the online appendix.

2. Literature Review

There are several veins of research related to this work. One of these is the design and pricing of

warranties. In the warranty literature, a few papers illustrate how heterogeneity among customers

can enable segmentation of the extended warranty market. For example, Padmanabhan and Rao

(1993) consider pricing strategies in the presence of heterogeneous risk preferences and consumer

moral hazard. Lutz and Padmanabhan (1994) consider income variation among customers, whereas

Lutz and Padmanabhan (1998) examine how customers’ differing utility of a functioning product

enables market segmentation. Three papers discuss usage heterogeneity in the context of warranty

pricing. Padmanabhan (1995) shows how manufacturers can design and price a menu of warranty

options in the presence of consumer moral hazard and usage heterogeneity, and satisfy the warranty


demands of various customer segments. Hollis (1999) examines consumer welfare in the extended

warranty market when customers vary in usage, but the manufacturer cannot verify usage (and so

cannot use usage as part of its warranty terms). Moskowitz and Chun (1994) study the design and

pricing of a menu of usage and time based warranties in the presence of usage variation among

customers. Our paper differs from the prior work in that, with the exception of Moskowitz and

Chun (1994), the prior work takes a simplistic view of product reliability: items either work or

do not work. To model failures in a way that realistically reflects support costs over the warranty

period, one must allow for multiple failures. The intertemporal decision making of customers during

the warranty coverage, which is not considered in the prior art of warranties, is essential in our

modeling of customers’ behavior under RVWs where state-dependent claim decisions may be made

over time. A stream of works studies the customers’ optimal strategy at the expiration of warranty

and designs warranties taking consideration of the customers’ optimal strategy. Jack and Murthy

(2007) consider a setting where the manufacturer sets the price per unit time of the EW and the

fixed markup on each out-of-warranty repair, and customers choose the start time for the EW and

replacement time for the product. Hartman and Laksana (2009) study the optimal strategy for

customers to make warranty renewal and product replacement decisions as the best response to

the providers service terms and pricing.

A second related vein of research is in customers’ strategic claim behavior. The concept of an

RVW is related to the practice in the insurance industry of offering a discounted premium for

policyholders with good claim records. These programs are used in the automobile insurance market

under various names, such as “no claims bonus”, “bonus-malus” and “merit rating” systems.

Several papers have studied customer’s optimal claim strategies and expected costs under such

systems. We do not attempt to provide a comprehensive review of this research area here, but

instead discuss a few selected papers and refer the reader to Kliger and Levikson (2002) for a recent

survey of this stream of research.

Von Lanzenauer (1974) and De Pril (1979) consider a merit rating system in which policyholders

are classified into multiple risk categories in each period of the finite insurance horizon. In a given

period, a policyholder can experience one or more accidents, where the damage associated with each

accident is a random variable, and he must decide whether to file a claim for each such accident. At

the end of each period, customers move from one risk category into another based on the number

of claims they filed in the preceding period. These papers show that the policyholder’s optimal

claim strategy is a threshold-type policy, and they characterize the policyholder’s critical claim

thresholds and expected costs. Moreover, De Pril (1979) shows how to find the optimal critical


claim sizes in closed form for the case of exponentially distributed repair costs. The findings in

this paper for a customer’s optimal claim behavior under an RVW are related to those in De Pril

(1979), although the customer’s optimal claim decision is somewhat different. For example, once

a customer has made enough claims to preclude a refund from their RVW, they will claim every

subsequent failure because there is no advantage to doing otherwise. In the insurance context,

there is always some incentive to avoid making claims, since future premia are determined by the

length of time since the last claim. Neither of these papers discusses the optimal design of insurance

contracts as our model does.

Among papers about merit rating systems, Kliger and Levikson (2002) is the only one to our

knowledge that considers the interplay of the insurer and the insured in a game theoretic framework.

The authors consider a special case of the merit rating system in which a policyholder receives a

discount on future annual premia if they file no claims for a certain number of years. The authors

give the optimal decisions of both the insured and the insurer in this framework, using discrete-time

Markov decision processes and game theory. Their model has several important differences from

ours, including that heterogeneity among users is not considered, nor do the customers have any

reservation value for incentive compatibility.

Warranty design in the supply chain context is the third related area of research. Other than the

insurance rationale of extended warranties as segmentation instruments, Desai and Padmanabhan

(2004) highlight the role of extended warranty in channel coordination. Gallego et al. (2010) study

a supply chain where a supplier sells a basic product and an ancillary service such as extended

warranties through a retailer, and investigate channel coordination mechanisms in the coexistence

of basic products and ancillary services. Given both the manufacturer and retailer can offer the

extended warranties, Hsiao et al. (2010) investigate the profitability of three strategies from the

manufacturer’s perspective in a distribution channel: deter, acquiesce, or foster.

3. Notation and Assumptions

We introduce the definition of the residual value extended warranty as follows.

Definition 1 (Residual Value Warranty (RVW)). The RVW has a refund schedule 0¤

r0 ¤ r1 ¤ � � � ¤ rn for some non-negative integer n. A customer who makes 0 ¤ j ¤ n claims over

the warranty period will receive a positive refund rn�j. A customer who makes more than n claims

will not receive a refund but still be covered for failures that occur during the warranty period.

For convenience, we define rj :� 0 for all integers j 0. Notice that the RVW can be implemented

by giving customers n�1 coupons with values ∆rj :� rj�rj�1, j � 0,1, . . . , n. To file the jth claim,


customers surrender the jth coupon. At the end of the warranty period the customer can redeem

any remaining coupons for their face value. An important special case is when ∆rj is a constant

for every j � 0,1, . . . , n. Once the coupons are exhausted, (rational) customers will file all claims.

Of course, the refunds can be automatically delivered without the need for physical coupons and

mail-in redemptions, but the coupons may help customers conceptualize the idea behind RVWs.

The RVW reduces to the following defined traditional warranty if the refund is zero.

Definition 2 (Traditional Warranty (TW)). We call a warranty a traditional one if it

has a single price for all customers regardless of their usage types or operating conditions.

Customers of TWs have an incentive to file a claim for every product failure regardless of the

repair cost. However, customers who buy an RVW may have different claim behavior. One extreme

is to follow the claim behavior under a TW.

Definition 3 (Claim-all-failures Behavior). We call claim behavior claim-all-failures for

an RVW if customers always file a claim with the service provider as long as there is a failure.

This can be reasonable behavior in industries where repairs can only be performed by the service

provider or a repair performed by a third party will invalidate the warranty or refund.

With an RVW, a customer has the option of paying an out-of-pocket, possibly random cost Ct

for a repair at time t, if he or she decides not to claim the failure. A hyper-rational customer will

dynamically and optimally decide whether to pay out of pocket for a repair or to claim a failure.

Definition 4 (Strategic Claim Behavior). We call a claim behavior strategic for an RVW

if customers follow an optimal claim policy.

Intuitively, this strategic claim policy will depend on the time-to-go, the number of claims already

made, the failure probabilities and the random repair costs.

Throughout the paper, we make the following assumptions.

Assumption 1 (Risk Neutrality). Customers are risk-neutral.

By abstracting away from issues of insurance provision through warranties with risk-averse cus-

tomers, we focus on the ability of the RVW to increase customers’ willingness-to-pay ex-ante and

empower better price discrimination through the refund schedule. Another rationale behind risk

neutrality assumption is that losses related to hardware malfunction are of a magnitude for which

there is little curvature on most people’s utility function. A linear approximation of utility may

work well within the range of losses that would be covered by a typical warranty. See Balachander

(2001) for a similar assumption.

Assumption 2 (Margin on Service Cost). If a repair cost for customers is c, then the

actual service cost for the service provider is βc, 0 β 1.


Assumption 2 reflects that the service provider enjoys economies of scale in providing service. We

assume that the margin on service cost is known to customers.

It is natural to assume that customers’ reservation price for warranty protection is less than

the expected costs of total pay-as-you-go repair service. However, an EW can provide additional

convenience in terms of product servicing that customers may also value. Without loss of generality,

we make the following assumption; all of the results can be reproduced for lower reservation price

than the expected costs of pay-as-you-go repairs.

Assumption 3 (Reservation Value on Warranty Protection). Customers’ reserva-

tion value on warranty protection is equal to the expected pay-as-you-go repair cost during the

warranty period.

We assume away possible replacement decision customers may make. This is an assumption that is

commonly seen in most warranty literature. The rationale behind it may be that a working product

is needed during the entire warranty period, repair costs are usually bounded above by a certain

fraction of replacement costs, and customers value the product at least as much as such a fraction.

Moreover, the true cost of replacement also involves additional costs that may be hard to measure,

such as down time, learning costs, and migration of software. See Balachander (2001) for a similar

assumption.

We assume customers’ willingness-to-pay for an EW with refund takes an additive form of two

components.

Assumption 4 (Willingness-To-Pay for RVW). The customers’ willingness-to-pay for an

RVW is the sum of the reservation price for warranty protection and the expected refund.

This assumption is consistent with risk neutrality and with a zero opportunity cost for money. The

assumption is made mainly for convenience as it is possible to reproduce results with the present

value of the expected refunds.

Finally, in the spirit of the idea of robust optimization, we consider the most conservative scenario

in calculating the service provider’s expected costs by making the following two assumptions:

Assumption 5 (Automatic Refund Redemption). At the end of the warranty period, cus-

tomers receive an automatic refund credit.

Without automatic redemptions, some customers would forget to redeem their refunds so redemp-

tion costs would be lower and profits higher. On the other hand, automatic redemptions make

RVWs more credible and customers are more likely to buy them again.


Assumption 6 (Out-Of-Pocket Repair). Customers take all out-of-pocket repairs to third-

party service providers.

If the service provider captures some of the market share for out-of-pocket repairs, its profits will

be higher than under Assumption 6. Thus, Assumptions 5 and 6 are conservative from the point

of view of the service provider’s profits.

4. A Single Failure Model

We first consider a basic model with two market segments and one Bernoulli distributed failure to

investigate the benefits of offering an RVW and the implications of strategic claim behavior on the

optimal warranty format. Our assumption of at most one failure is reasonable if the product is fairly

reliable and the horizon is sufficiently short. In the extension, we consider a general framework

that allows multiple failures and multiple market segments.

Suppose there are two segments of customers that differ in their failure probabilities: ρi, i�L,H,

with 0 ρL ρH 1. The market is comprised of a γ P p0,1q fraction of high-usage customers; the

remaining 1� γ fraction is the low-usage segment. The service provider sells an RVW, denoted by

pp, rq, at price p with an agreement that a customer gets a refund rp¤ pq if no claims are made.

For a TW, customers always claim the failure. By Assumptions 1 and 3, a customer’s reservation

value for protection from a one-price-for-all TW equals the customer’s expected valuation of repairs,

which we denote by wi. Hence the reservation value of a risk-neutral customer in segment i for

a TW is wi :� ρic, i�H,L, where c :�EC is the mean of the random repair cost. The expected

refund customers get from an RVW with refund r depends on their claim behavior. Under claim-

all-failures behavior, the expected value for protection is ρic and the expected refund is p1� ρiqr

for customer segments i � H,L. By Assumption 4, for an RVW pp, rq, the expected surplus of

customers in segment i who follow claim-all-failures behavior is U iCpp, rq � ρic� p1� ρiqr� p. We

characterize the optimal RVW under claim-all-failures behavior as follows.

Proposition 1 (Optimal RVW under Claim-all-failures Behavior). Under claim-

all-failures behavior,

(i) the optimal RVW is a full refund strategy with p� � r� � c;

(ii) the optimal RVW is strictly more profitable than the optimal TW.

Proof. See the appendix. �

Under claim-all-failures behavior, the service provider would like to set the refund as high as

possible to capture both segments. As a result, the up-front price is equal to the the refund and is


equal to the expected repair cost, hence both segments have the same ex-ante willingness-to-pay

and are both captured in the optimal RVW. Note that the resulting full refund strategy is the very

“risk-free” warranty that has been implemented by Hewlett-Packard in practice.

Under strategic claim behavior, if the repair cost C r, the customer pays out of pocket and

claims the repair otherwise. Then the expected refund net of out-of-pocket costs is giprq � ρiErpr�

Cq�s � p1 � ρiqr, and the expected refund is ziprq � ρiPrpC rqr � p1 � ρiqr, i � H,L. Hence,

the expected out-of-pocket cost is ziprq � giprq � ρiEpC1tC ruq, i�H,L, which is the repair cost

shifted to third parties under strategic claim behavior. By Assumption 4, for an RVW pp, rq, the

expected surplus of customers in segment i who follow strategic claim behavior is

U iSpp, rq � ρ

i�rPrpC rq�EpC1tC¥ruq

��p1� ρiqr� p.

The surplus customers gain under strategic claim behavior is U iSpp, rq�U

iCpp, rq � ρ

iErpr�Cq�s.

Customers from segment i � H,L have a maximum willingness-to-pay wiprq �

ρi�rPrpC rq�EpC1tC¥ruq

��p1� ρiqr�wi� giprq for an RVW with a refund r under strategic

claim behavior. Hence the expected profit to the warranty provider from such a customer, if this

willingness-to-pay wiprq is charged as the up-front price, is p1�βq twi�rziprq� giprqsu, i�H,L.

If an RVW or a TW is self-selected by the low-usage segment, it must also capture the high-usage

segment as it is easy to see that wH ¡ wL and wHprq ¡ wLprq for all r. Moreover, an RVW with

the refund size equal to zero reduces to a TW. Hence, to investigate the optimal warranty format

under strategic claim behavior, we only need to consider and compare the following four cases: the

optimal TW or non-degenerate RVW (with a positive refund) that serves both segments or only

serves the high-usage segment. We denote the profit of the optimal RVW with a refund r and the

optimal TW that serve two segments by πHLR prq and πHL

T respectively. We denote the profit of the

optimal RVW with a refund r and the optimal TW that only serves high-usage segment by πHR prq

and πHT respectively. We have the following relationships among the four cases.

Lemma 1. Under strategic claim behavior, πHR prq ¤ π

HT for all r¥ 0 and maxr¥0 π

HLR prq ¥ πHL

T .


From these relationships, we can see that the optimal warranty contract under strategic claim

behavior is either the optimal (maybe degenerate) RVW that serves two segments or the optimal

TW that only serves the high-usage segment. We characterize the optimal non-degenerate RVW

under strategic claim behavior as follows.


Proposition 2 (Optimal Non-Degenerate RVW under Strategic Claim). Under

strategic claim behavior, the optimal non-degenerate RVW is a refund strategy with p� �

r�� ρLE rpC � r�q�s and r� � arg maxr¡0 θprq, where

θprq :� γpρH � ρLqPrpC ¥ rqr��βγpρH � ρLq� p1�βqρL

�EpC1tC ruq

is the profit improvement over the optimal TW that serves two segments.


We have shown that under claim-all-failures behavior, the optimal RVW is always strictly more

profitable than the optimal TW. If customers are strategic, one implication of offering an RVW is

that it may shift away relatively small repairs to third-party service providers. Under strategic claim

behavior, we have characterized the optimal non-degenerate RVW that serves two segments, which

is a candidate for the optimal RVW. In light of Lemma 1, if maxr¡0 πHLR prq ¡ πH

T , then the optimal

RVW is a non-degenerate one and is strictly more profitable than the optimal TW; otherwise, the

optimal RVW degenerates to a TW. We use this idea to identify the optimal warranty contract for

two special cases of repair cost.

Proposition 3 (Optimal RVW for Constant Cost under Strategic Claim).

Suppose C � c. Under strategic claim behavior,

(i) the optimal RVW is a full refund strategy with p� � r� � c;

(ii) the optimal RVW is strictly more profitable than the optimal TW.


Proposition 4 (Optimal RVW for Exponential Cost under Strategic Claim).

Suppose C � expp1{cq. Under strategic claim behavior,

(i) the optimal non-degenerate RVW that serves two segments is a partial refund strategy with

r� �γpρH � ρLq

p1�βqφc,

p� � r�� ρL expp�r�{cqc,

where φ� γρH �p1� γqρL;

(ii) the optimal RVW may degenerate to a TW.



Figure 1 Single Failure: Profit Improvement of the Optimal RVW over the Optimal TW under Strategic Claim

0.2

0.25

0.3

0.35

0.4

0.20.25

0.30.35

0.4

0

0.5

1

1.5

H

Prof

itabi

lity

of R

VW o

ver T

W

Note. C � expp1{100q, ρL � 0.2, β � 0.7. The profitability of RVW over TW is maxtπHLR �maxtπHT , πHLT u,0u.

Combining Propositions 1 and 3, we can immediately see that if the repair cost is constant, the

“risk-free” RVW with full refund p� � r� � c is the optimal warranty contract and is strictly more

profitable than the optimal TW no matter what the claim behavior is. When there exists more

variability in the repair cost, the RVW with a full refund p� � r� � c is still strictly more profitable

than the optimal TW under claim-all-failures behavior, but the optimal RVW may degenerate to a

TW under strategic claim behavior. The following result demonstrates the implication of strategic

claim behavior and repair cost randomness on the profitability of RVW.

Proposition 5 (Repair Cost Variability on Profitability under Strategic Claim).

Under strategic claim behavior, the optimal RVW is less profitable if the repair cost is random

than if it is constant with the same mean.


Proposition 5 says that strategic customers benefit from (the provider is hurt by) the randomness

of repair cost. The scenario of constant repair cost gives an upper bound on the profitability of the

optimal RVW for any generally distributed random repair cost.

Figure 1 illustrates the profitability of the optimal RVW over the optimal TW for exponential

repair cost under customers’ strategic claim behavior. The repair cost is exponentially distributed

with mean c� 100. The failure experienced by each segment is assumed to be a Bernoulli random

variable. The failure probability of the low-usage segment is ρL � 0.2. The cost fraction to the


service provider is β � 0.7. When the failure probability ρH and the proportion γ of the high-usage

segment is large, e.g., in the neighborhood of pγ, ρHq � p0.4,0.4q, it is not optimal to offer an RVW

and the optimal warranty contract remains a TW: when the proportion of the high-usage segment

is large, the optimal TW captures the high-usage segment only; when the two segments are distant

from each other in usage rates, a large refund needs to be given to capture the low-usage segment;

if customers claim strategically, the profit loss in small claims of the high-usage segment shifted to

third-parties dominates the profit increase in capturing the low-usage segment, hence it would be

better off for the service provider not to offer any refund at all. In the parameter area corresponding

to the wedge shape (when the two segments are close to each other in usage rates or the proportion

of the high-usage segment is small), the optimal RVW captures two segments and is strictly more

profitable than the optimal TW that captures one or two segments: to the left of the wedge’s edge,

the optimal TW captures only the high-usage segment; to the right of it, the optimal TW captures

both segments.

5. A General Multiple Failure Framework

In this section, we extend the basic model to allow for multiple market segments, multiple failures

and time-varying failure rates. The warranty’s coverage period is of length T . We measure time

backwards with t being the time-to-go until the end of the warranty period, and assume that failures

arrive according to a non-homogeneous Poisson process with instantaneous failure rate λut where

u is the index of usage segments. We will suppress the usage superscript in subsections 5.1-5.3,

and recover it when considering multiple usage segments in subsection 5.4. A failure at time t

has random repair cost Ct with publicly known cumulative distribution function (c.d.f.) Ftp�q, tail

distribution Ftp�q and probability density function (p.d.f.) ftp�q. We denote the mean of random

cost Ct by ct :�ECt. Costs are assumed to be time-invariant if we drop the subscript t. For ease

of notation, we will develop formulas suppressing the usage index and recall it when introducing

heterogenous customers with different usage rates once the basic formulas are developed. We define

the expected aggregated failure rate over horizon r0, ts as Λptq :�³t0λsds, 0¤ t¤ T.

5.1. Strategic Claim Behavior

We will see that an optimal claim policy is a function of the state pt, kq, where t is the time-to-go,

and k is the number of remaining claims the customer can place. If k 0, the customer has already

placed more than n claims and is no longer entitled to a refund. As a result, when k 0, it is

optimal for the customer to file a claim for each failure. For k ¥ 0, we expect the optimal policy

to be of a threshold type, where a failure with out-of-pocket cost Ct will be claimed at state pt, kq


if and only if it exceeds a threshold apt, kq. We also intuitively expect apt, kq to be decreasing in t

and increasing in k, which is confirmed by our results under the assumption that the refund rk is

increasing convex in k.

We now formulate the optimal claim policy as a stochastic optimal control problem. Let gpt, kq

be the maximum expected refund, net of out-of-pocket costs, at state pt, kq (i.e., n � k claims

have already been made at time t). We derive the Hamilton-Jacobi-Bellman (HJB) differential and

difference equation that must be satisfied by the value function gpt, kq as follows (see the appendix

for derivation):Bgpt, kq

Bt��λtEminpCt,∆gpt, kqq. (1)

The boundary conditions are gp0, kq � rk, k¥ 0 and gpt, kq � 0, k 0, for all 0¤ t¤ T .

The following result confirms that a threshold policy is optimal with threshold apt, kq :�∆gpt, kq

equal to the marginal value, at time t, of the pn�kqth claim. The result also presents monotonicity

and convexity properties of the function gpt, kq.

Proposition 6 (Properties of Optimal Claim Policy). The following structural proper-

ties hold.

(i) (Optimal Policy) When a failure occurs at state pt, kq, it is optimal to place a claim if and

only if Ct ¥ apt, kq and to pay out of pocket for the claim if Ct apt, kq.

(ii) (Monotonicity) (a) gpt, kq is decreasing in t and increasing in k; (b) gpt, kq is increasing

in rj; (c) gpt, kq is decreasing in Λptq �³t0λs ds.

(iii) (Convexity) (a) If rk is increasing convex in k, apt, kq is increasing in k and decreasing in

t; (b) If λtFtpxq is decreasing in t for all x, gpt, kq is convex in t.


As a direct application of properties of gpt, kq, we can obtain closed-form solutions for the dis-

crete random repair cost and constant repair cost. Moreover, we can also derive the closed-form

solution to the HJB equation when the repair cost is exponentially distributed without imposing

any assumption on the refund schedule. We summarize these results in the following proposition.

Proposition 7 (Closed-Form Optimal Claim Policy). We have the following closed-

form characterizations of the optimal claim policy.

(i) (Discrete Random Repair Cost) Suppose rk is increasing convex in k. If Ct for all t is

discretely distributed taking value cj (8� c0 ¡ c1 ¡ c2 ¡ � � � ¡ cJ ¡ 0) with probability qjt for

j � 1, . . . , J , then jkptq :�max tj P t0,1, . . . , Ju : cj ¥ apt, kqu is increasing in t and decreasing

in k, and it is optimal to claim all failures in the set Sjkptq when a failure occurs at state

pt, kq, where Sj :� t1, . . . , ju for all j and S0 :�H.


(ii) (Constant Repair Cost) Suppose rk is increasing convex in k. If Ct � c for all t and the

refund schedule is increasing convex with r0 ¡ c, then there exists a series of time thresholds

0 t0 t1 � � � tn such that it is optimal to claim a failure at state pt, kq if and only if

t¥ tk, and

gpt, kq �

$&%rk�Λptqc if 0¤ t tk,°k

j�0 gje�ΛjptqΛj,kptq if t¥ tk,

where gj :� rj �Λptjqc, Λjptq :�³ttjλs ds for all t¥ tj, Λj,kptq :�

³ttk

Λj,k�1psqλsds for all t¥

tk, j � 0, . . . , k� 1 and Λk,kptq � 1 for all t¥ tk and k.

(iii) (Exponential Repair Cost) If Ct � expp1{cq for all t, then for all t and k,

gpt, kq � ln

�1�

k

j�0

Rk�jP pt, jq

�c, (2)

where Rj :� erj{c � 1, P pt, jq :� PrpNptq � jq, and Nptq is a Poisson random variable with

parameter Λptq.

Proof. See the online appendix. �

Jensen’s inequality implies that ceteris paribus, gpt, kq has a lower value when out-of-pocket

repair costs are constant than when repair costs incurred by the customer are random with the

same mean. The intuition behind is that random repair costs afford the customers the opportunity

to make strategic claim decisions – self-defraying the costs of inexpensive repairs – to improve their

expected net refund (see Example 4 in the online appendix). Consistent with the basic model, it

is expected that the variability in the repair cost is less favorable to the RVW provider.

5.2. Bounded Rationality: Static Threshold with Claim Limits Policy

While a dynamic threshold policy is shown to be optimal, it may be quite difficult, even for the

most strategic customers, to conceptualize and compute it. More likely, customers will use a simple

heuristic to determine which failures to claim. At one extreme they will make all claims, essentially

using constant threshold a� 0, or, more likely, they will use a combination of a constant threshold

policy and a limit on the number of claims, i.e., they will file up to jpaq failures with repair

costs larger than a. We will show that this heuristic policy is indeed easier to solve and that its

performance is very close to that of the optimal dynamic policy. In a similar setting, Leider and

Sahin (2010) use laboratory experiments to show that a majority of individuals indeed correctly

use a threshold policy.


With few exceptions, such as constant and exponential failure costs, gpt, kq and therefore apt, kq

must be computed numerically in designing the optimal RVW. We now explore heuristics to approx-

imate gpT,nq, for any stationary repair cost distribution, based on the use of a static threshold

policy apt, kq � a for all pt, kq, computed at state pT,nq. For any a ¥ 0, the number of claims

over the life of the warranty is a Poisson random variable, say Na, with parameter ΛpT qF paq.

If the customer uses threshold policy a and is willing to make up to j ¤ n claims, her expected

refund is rpa, jq �Erpn�minpj,Naqq, where rpkq � rk; her expected out-of-pocket cost is mpa, jq �

ΛpT qc� ErC|C ¡ asEminpj,Naq, where C is the random variable of the stationary repair cost.

Let lpa, jq :� rpa, jq �mpa, jq be the expected refund net of out-of-pocket costs. If j � 0, then

lpa,0q � rn�ΛpT qc. For j � 1, . . . , n, notice that

lpa, jq � rpa, jq�mpa, jq

� Errpn�minpj,Naqqs�ErC|C ¡ asEminpj,Naq�ΛpT qc

� rn�ΛpT qc�j

i�1

pErC|C ¡ as�∆rn�1�iqPrpNa ¥ iq.

For fixed a, lpa, jq is increasing in j as long as ErC|C ¡ as ¡ ∆rn�1�j. The only sensible choices

for j are t0,1 . . . , n,8u. For j �8, lpa,8q can be approximated by

lpa,8q�Erpn�Naq�ΛpT qErC|C ¡ asF paq�ΛpT qc.

We can also write

lpa,8q�Erpn�Naq�Eminpn� 1,NaqErC|C ¤ asF paq. (3)

The rigorous derivation to Equation (3) can be found in the online appendix. We can also show

that lpa,8q ¤ lpa,8q. For any a, we will denote by jpaq the smallest choice of j P t0,1, . . . , n,8u

that maximizes lpa, jq and let lpaq :� lpa, jpaqq. Clearly

gpT,nq ¥ gspT,nq :�maxalpaq, (4)

where gspT,nq is the value of the best static threshold policy. The approximation of gspT,nq for

gpT,nq is very good, as suggested by the numerical examples (see Example 1 in the online appendix)

based on the approximation of exponential distribution and normal distribution.


5.3. Expected Costs of the Service Provider

By Assumption 2, when a customer chooses to pay out of pocket for a repair, his cost c is higher

than the repair costs βc that would be incurred by the warranty provider for the same repair. Let

hpt, kq represent the warranty provider’s expected costs of repairs and refunds for a given RVW at

state pt, kq.

The partial differential and difference equation that hpt, kq satisfies under Assumption 6 and

strategic claim behavior is

Bhpt, kq

Bt� λtPrpCt ¡∆gpt, kqq tβErCt|Ct ¡∆gpt, kqs�∆hpt, kqu , (5)

where ∆hpt, kq :� hpt, kq�hpt, k�1q. The boundary conditions are hp0, kq � rk for all k� 0,1, . . . , n

and hpt, kq � βEr³t0λsCsdss for k 0 and 0¤ t¤ T (see the online appendix for derivation).

We now explore the relationship between hpt, kq and gpt, kq by introducing zpt, kq, the expected

refund at state pt, kq under strategic claim behavior. Following the construction for gpt, kq and

hpt, kq, it is easy to verify that zpt, kq satisfies the following differential and difference equation

Bzpt, kq

Bt��λtPrpCt ¡∆gpt, kqq∆zpt, kq, (6)

where ∆zpt, kq :� zpt, kq�zpt, k�1q. The boundary conditions are zp0, kq � rk for all k� 0,1, . . . , n

and zpt, kq � 0, for k 0 and 0 ¤ t ¤ T . The following proposition expresses hpt, kq in terms of

gpt, kq and zpt, kq.

Proposition 8 (Expected Cost under Strategic Claim). For any 0 ¤ t ¤ T and k P

t0,1, . . . , nu,

hpt, kq � β�gpt, kq�Ep

³t0λsCsdsq

��p1�βqzpt, kq. (7)


As special cases, we have the following closed-form solutions to function zpt, kq for constant and

exponential repair costs. For these two scenarios, the characterization of function hpt, kq can follow

immediately from Propositions 7 and 8.

Proposition 9 (Closed-Form Expected Refund). We have the following closed-form

expressions for customers’ expected refund under strategic claim behavior.

(i) (Constant Repair Cost) If Ct � c for all t and the refund schedule is increasing convex

with r0 ¡ c, then

zpt, kq �

$&%rk if 0¤ t tk,°k

j�0 rje�ΛjptqΛj,kptq if t¥ tk,

where tk,Λj and Λj,k are defined in Proposition 7.


(ii) (Exponential Repair Cost) If Ct � expp1{cq for all t, then for any 0 ¤ t ¤ T and k P

t0,1, . . . , nu,

zpt, kq �

°k

j�0Qk�jP pt, jq

1�°k

j�0Rk�jP pt, jq, (8)

where Qj :� rjerj{c and Rj :� erj{c� 1 for j P t0,1, . . . , nu.


5.4. Design and Pricing of Residual Value Warranties

In this subsection, we consider the warranty provider’s problem of how to design and price an

RVW to maximize profitability in a market of heterogeneous usage segments with random usage

U under customers’ strategic claim behavior. We assume that U has a discrete distribution qi �

PrpU � uiq, i� 1, . . . , I with 0 u1 � � � uI 8 and°I

i�1 qi � 1. We also assume that customers

with usage ui experience failure rates λi that are increasing in i, and that customers know their

usage rates and the corresponding failure rates. The warranty provider is assumed to know the

failure rates and the distribution of U , which can be estimated from historic data in practice. We

develop a neoclassical model where customers can compute the expected benefit of an RVW and

compare their calculation with their willingness-to-pay. This is consistent with the assumption that

the decision maker is risk neutral, but assumes that customers can correctly evaluate net benefit

without error.

Let wi be the reservation price that a customer with usage ui will pay for a TW. By Assumptions

1 and 3, wi � λiTc. For any refund schedule ~r� pr0, r1, . . . , rnq, we let gip~rq denote the value gipT,nq

and zip~rq denote the value zipT,nq under schedule ~r for market segment i. This is simply the

expected refund, net of out-of-pocket costs, at the beginning of the EW period. By Assumption

4, a customer from market segment i has maximum willingness-to-pay wip~rq �wi� gip~rq and the

expected profit to the warranty provider at this price from such customer is πip~rq �wip~rq�hip~rq.

Lemma 2 (Monotonicity of willingness-to-pay). wip~rq is increasing in i for any ~r.


If we offer price p and refund schedule ~r, then market segment i will purchase the RVW if and only

if p¤wip~rq and the profit from market segment i will be p�hip~rq. Thus, the objective is to select p

and the vector ~r to maximize the provider’s expected profit πpp,~rq �°ti:p¤wip~rqu q

ipp�hip~rqq. For

any fixed ~r, πpp,~rq as a function of p is increasing and left continuous between consecutive values of

wip~rq and it follows that the function attains its maximum at one of the wip~rq values, i� 1, . . . , I.


By Lemma 2, an RVW with price p� wjp~rq captures all market segments j and above. For any

given market segment j, let πjp~rq :�°I

i�j qipwjp~rq � hip~rqq be the expected profit under schedule

~r at price p�wjp~rq. For any given n, our objective is to select the optimal ~r for every j and then

to pick the optimal j to solve maxj max~r πjp~rq. We have therefore I optimization problems, each

over n� 1 variables.

If the vector ~r is constructed from fixed coupons of size ∆r, i.e., rk � pk� 1q∆r, k � 0,1, . . . , n,

the problem of maximizing πjp~rq reduces to a single dimensional problem over the parameter ∆r.

We will abuse notation slightly and will denote by πjp∆rq the expected profit of offering such

a schedule with n constant coupons of size ∆r. It is easy to numerically search for the value of

∆r that maximizes πjp∆rq. This allows us to then select the market segment j� that maximizes

πj � max∆r πjp∆rq. Experiments maximizing πjp~rq for general refund schedules using a general

purpose non-linear optimizer for exponentially and normally distributed repair costs suggest that

max∆r πjp∆rq is as high as max~r π

jp~rq for large values of n. In other words, constant coupon sizes

perform as well as arbitrary refund schedules when the number of allowable claims to yield a refund

is large (see Examples 2 and 3 in the online appendix).

Recall that after we optimize over the design of RVWs, the expected profit is at least as good

as the optimal TW. The following result identifies sufficient conditions for a refund schedule to be

strictly more profitable than the optimal TW.

Proposition 10 (Profitability of RVW over TW under Strategic Claim). Suppose

the optimal TW, with price p�T , captures all market segments j� and above. Under strategic claim

behavior,

(i) (Contingency discrimination within each segment) if there exists a refund schedule ~r such

that for all i¥ j�, we have

gj�

p~rq ¥ βgip~rq� p1�βqzip~rq, (9)

and at least one of the inequalities holds strongly, then the RVW pp,~rq ��wj�p~rq,~r

is strictly

more profitable than the optimal TW while capturing the same market segments;

(ii) (Price discrimination across segments) if there exists a segment l j� and refund schedule ~r

such that

j��1¸i�l

qi�wlp~rq�hip~rq

�¡

I

i�j�

qi�wj� �wlp~rq�βgip~rq� p1�βqzip~rq

, (10)

then the RVW pp,~rq ��wlp~rq,~r

�is strictly more profitable than the optimal TW by capturing

more market segments l, . . . , j�� 1.



In part (i), the convex combination βgip~rq � p1� βqzip~rq P rgip~rq, zip~rqs. By Proposition 6 part

(ii)(c), we can see that gip~rq gj�p~rq, i¡ j� for any ~r�~0. If there exists ~r such that zip~rq ¤ gj

�p~rq

for all i¡ j�, then inequality (9) is guaranteed for any β P p0,1q. It illustrates the ability of an RVW

as a contingent contract to improve profitability over the optimal TW within a usage group by

segmenting customers according to their realized experience with failures, resulting in an increase

in their up-front willingness-to-pay. In part (ii), the designated RVW strictly improves profitability

over the optimal TW if the profit gain (the left-hand-side of inequality (10)) from capturing more

market segments i� l, . . . , j�� 1 by switching to the RVW strictly dominates the profit loss (the

right-hand-side of inequality (10)) in shifting away relatively small repairs in the initially captured

segments i� j�, . . . , I. This is the case in which an RVW can better price discriminate customers

over the optimal TW across broader market segments in spite of shifting away small repairs, which

is more likely to happen when there is a large population of low-usage groups that is priced out

by the optimal TW or the segments are close to each other in usage rates. If the above sufficient

conditions fail to be satisfied, the optimal RVW may degenerate to a TW (see Example 5 in the

online appendix).

6. Discussions and Conclusions

We consider a general framework of RVW where one or more of the following conditions hold: 1)

there are multiple market segments, 2) the size of the refund depends on the number of claims

made during the warranty period, 3) the failure rates can be time-varying, 4) the repair cost for

any failure has uncertainty and 5) upon a failure customers can choose to either pay out of pocket

or make a claim against the RVW.

When correctly designed, RVWs are especially helpful dealing with heterogenous market seg-

ments as they present a mechanism to attract market segments that would be priced out by

traditional warranties. The mechanism does not require the monitoring of usage or of the environ-

ment where the product is used as customers can influence their net cost by taking into account

their failure rates and the given refund schedule. Our analysis assumes 1) strategic customer claim

behavior, 2) 100% refund redemption and 3) that all out-of-pocket repairs go to third parties.

These assumptions are conservative from the perspective of provider’s profits; when they do not

hold, the provider can expect higher profits than those predicted by the model. In addition, moral

hazard issues excluded from the current model may help reduce claims and in turn allow for a

lower upfront price of the RVW since refunds may induce customers to be more cautious and hence

decrease the failure rates.


Though the RVW as a contingent contract has the ability to improve profitability within the

existing market segments and also to engage broader market segments, customers’ strategic claim

behavior may shift away repairs to third-party service providers. We show that if the random repair

cost has large variability or the segments are distant from each other in usage rates, the induced

strategic claim behavior may significantly undermine the RVW’s benefit from price discrimination

among a broader range of customers. As a result, the optimal RVW may degenerate to a TW.

Moral hazard behavior has the same effect as strategic claim behavior on the service provider,

since it also restricts the ability of the RVW to attract broader market segments through nonzero

refunds: when refunds are offered, the market segments initially captured by TW may reduce their

failure probability under moral hazard hence decrease their up-front willingness-to-pay for warranty

protection.

We have shown via numerical experiments that restricting the RVWs to have constant coupon

sizes does not materially reduce their effectiveness. This is a positive finding, as constant coupon

warranties are easier for customers to understand and simpler to act upon. Moreover, our findings

also indicate that it is sufficient to consider refund schedules of modest length to obtain most of the

benefits that general RVWs can provide. This is another positive finding, as the refund schedule

can be kept simple and easy to understand.

We have assumed that the warranty provider will automatically pay all refunds due to the

customer. We think this maximizes customers’ valuation of RVW and the lifetime value of customers

as automatic refunds will entice them to repurchase RVWs. In practice, it is not uncommon for

providers to observe low redemption rates that result from customers not claiming eligible refunds.

This happens when eligible refunds are relatively small and the process of claiming them is time

consuming. Customers who anticipate not claiming eligible refunds may discount the upfront value

of RVWs. Modeling redemption rates and how such customers value RVWs is a topic of future

research.

We have assumed that customers know their usage and failure rates, and we can also add some

uncertainty in their valuations by introducing idiosyncratic errors in valuations, which we leave

for future research. These errors can also account for suboptimal valuations that result from using

heuristic claim policies, such as the static threshold policy described in the paper. The assumption

that the warranty provider knows the failure rate distribution can be defended by observing the

claim behavior of high-usage customers who buy traditional warranties and lower usage customers

who pay for repairs out of pocket. Nevertheless, the warranty provider may not be able to accurately

estimate the failure rate distribution. In this case, it would be important to design RVWs that are


robust to such errors and that allow for learning so that the provider can refine estimates over

time.

A service provider can design a profitable RVW for which strategic claim behavior results in

customers’ bringing all repairs to the provider. In particular, if the incremental refund sizes (coupon

sizes) are lower than the single repair cost offered by third parties, then customers are induced

to make claims for every failure, and their strategic claim behavior and claim-all-failures behavior

coincide.

Based on the concept of RVW, we introduce several new extended warranty service products as

variants of RVW.

Flexible RVW. We call an EW a flexible RVW if customers pay an upfront price p for a

service “debit card” with an initial balance nr; the customer pays mintCt, ru for a failure with cost

Ct from the card; once debit card funds are exhausted, all repairs are covered; and any positive

balance remaining on the card at the end of warranty coverage is refunded to the customer.

Capped copayment EW. We call an EW a capped copayment EW if customers pay an upfront

price p; a customer pays a copayment r each time for the first n repairs that are brought to the

service provider, regardless of repair costs; all repairs beyond the first n repairs are completely

covered by the warranty.

Capped deductible EW. We call an EW a capped deductible EW if customers pay an upfront

price p; a customer pays a minimum value mintCt, ru between cost Ct for a failure and a deductible

r; all repairs beyond the cap nr on the total out-of-pocket expenses are completely covered by the

warranty.

Claim limited RVW. The claim limited RVWs are similar to RVWs except that they allow

only for a pre-determined finite number of claims. With the same notation as in Definition 1, once

a customer makes more than n claims, he will not receive a refund, nor will he be eligible for future

coverage under the warranty.

In terms of optimal customers’ claim behavior, customers’ expected net refund and provider’s

expected profit under the warranty, an RVW with an upfront price p and n coupons of value r

is equivalent to a capped copayment EW with an upfront price p� nr, a copayment r and a cap

n. Moreover, a flexible RVW with an upfront price p, a debit card value nr and a per-incident

claim cap r is equivalent to a capped deductible EW with an upfront price p�nr, a per-incident

deductible r and a total claim cap nr. It is easy to realize such equivalencies in the absence of cash

flow discounting. The equivalence results are independent of failure rates and can be easily extended

to general refund schedules. Compared to the RVW, the flexible RVW has more fungible coupon


Table 1 Comparison of Extended Warranty Products.

To customers To the service provider

Extendedwarrantytype

Simplecustomers’optimalclaimpolicy

Requirescustomerpayment atthe time ofclaim

Lowupfrontprice

Tracknumber ofclaims

Track costof eachclaim

Benefitsfrom sub-optimalclaimpolicy

Benefitsfrom lowredemptionrates

RVW No No No Yes No Yes Yes

FlexibleRVW

Yes No No No Yes No Yes

CappedcopaymentEW

No Yes Yes Yes No Yes No

CappeddeductibleEW

Yes Yes Yes No Yes No No

Claim limitedRVW

No No Yes Yes No Yes Yes

values, inducing the rational customer to claim every failure against the warranty. This property of

inducing claims for all failures is shared by the capped deductible warranty due to the equivalence.

Compared to the RVW and flexible RVW, the capped copayment EW and capped deductible EW

have lower up-front prices, and are more flexible. A claim limited RVW can enable the provider to

have even broader market coverage by further reducing the upfront price. Nevertheless, the RVW

as the simplest extension of the TW has its own merits in 1) providing “peace-of-mind” protection

to customers, without requiring customers to make extra payments at each time of a claim, 2)

avoiding the necessity for the provider to track and communicate the cost of each claim to the

customer, and 3) providing the service provider with benefits from possible customers’ heuristic

claim behavior and low redemption rates. We compare RVW with its variants and summarize the

pros and cons in Table 1.

In addition to further studying innovative extended warranty products, another future research

direction is to investigate an oligopolistic market where competing service providers offer RVWs.

In particular, it would be interesting to see whether the benefits of RVWs will prevail under

competition.

Appendix. Proofs.

Proof of Proposition 1. An RVW can outperform the optimal TW only if the RVW serves both segments;

otherwise if it only serves the high-usage segment, its profit πHR prq � γp1� βqwH is the same as the profit


πHT � γp1�βqwH of the optimal TW that only serves the high-usage segment. Hence we can restrict ourselves

to the non-degenerate RVWs that serve both segments for the candidate of RVWs that may outperform the

optimal TW. For segment i�H,L with claim-all-failures behavior, the willingness-to-pay of an RVW with

a refund r is wiprq � ρic� p1 � ρiqr, i � H,L. Suppose r ¡ c, which is equivalent to wLprq ¡ wHprq. The

optimal strategy for the service provider to capture both segments is to set p� wHprq � ρHc�p1�ρHqr r.

But this violates the constraint r¤ p. Hence we only need to consider r¤ c.

Suppose r ¤ c, which is equivalent to wLprq ¤ wHprq. The optimal strategy for the service provider to

capture both segments is to set p � wLprq � ρLc� p1 � ρLqr ¥ r and the expected profit is πHLR prq � p1 �

βqρLc � γpρH � ρLqpr � βcq. Since ρH ¡ ρL, the expected profit is maximized at r� � c where wLpr�q �

wHpr�q � p� � c. Obviously, πHLR pr�q ¡ πHLR p0q � πHLT , namely, under claim-all-failures behavior the optimal

RVW is strictly more profitable than the optimal TW that serves two segments.

If the service provider uses a TW to only serve the high-usage segment, it is optimal to set p� ρHc and the

expected profit is πHT � γp1� βqρHc. Note that πHLR pr�q ¡ πHT is equivalent to p1� γqρL ¡ 0, which always

holds. Hence the RVW with p� � r� � c is indeed the optimal one and is strictly more profitable than the

optimal TW. �

Proof of Lemma 1. Note that ziprq� giprq � ρiEpC1tC ruq ¥ 0 for all r¥ 0, i�H,L. Then for all r¥ 0,

πHR prq � γp1�βq twH �rzHprq� gHprqsu ¤ γp1�βqwH � πHT .

Furthermore, to serve two segments, it is optimal to price the RVW at the willingness-to-pay of the low-

usage segment, namely, p�wLprq �wL� gLprq � ρLc� ρLErpr�Cq�s � p1� ρLqr� r� ρLE rpC � rq�s ¥ r,

where the constraint r ¤ p is automatically ensured. The expected profit of this RVW, which is priced at

wLprq, is

πHLR prq � p1� γqp1�βqpwL� gL� zLq� γ wL� gL�

�βgH �βwH �p1�βqzH

�(

� p1� γqp1�βqwL� γpwL�βwHq� θprq,

where

θprq � γpgL� gHq� p1� γqp1�βqpzL� gLq� γp1�βqpzH � gHq

� γpρH � ρLq rPrpC ¥ rqr�EpC1tC ruqs� p1�βq�p1� γqρL� γρH

�EpC1tC ruq

� γpρH � ρLqPrpC ¥ rqr��βγpρH � ρLq� p1�βqρL

�EpC1tC ruq.

Hence, maxr¥0 πHLR prq � p1�γqp1�βqwL�γpwL�βwHq�maxr¥0 θprq ¥ p1�γqp1�βqwL�γpwL�βwHq �

πHLT , where maxr¥0 θprq ¥ 0 since θprq|r�0 � 0. �


Proof of Proposition 2. In view of the proof of Lemma 1, to optimize the expected profit πHLR prq over

r¥ 0, it is equivalent to optimize θprq over r¥ 0. To show the optimal RVW that serves two segments is non-

degenerate, it suffices to show that there exists ro ¡ 0 such that θproq ¡ 0. If A :� βγpρH�ρLq�p1�βqρL ¥ 0,

then θprq ¥ γpρH � ρLqPrpC ¥ rqr; there must exist ro ¡ 0 such that PrpC ¥ roqro ¡ 0, hence θproq ¡ 0. If

A 0, then let ro � sup tr¡ 0 | PrpC ¥ rq ¥ p�Aq{rγpρH � ρLq�Asu. Note that ro must exist and is non-

zero, hence θproq � γpρH�ρLqPrpC ¥ roqro�A �EpC1tC rouq ¡ γpρH�ρLqPrpC ¥ roqro�A �roPrpC roq �

A � ro�rγpρH � ρLq�AsPrpC ¥ roqro ¥ 0. �

Proof of Proposition 3. Note that if r ¤ c, θprq � γpρH � ρLqr ¤ γpρH � ρLqc and if r ¡ c, θprq �

rβγpρH � ρLq� p1�βqρLs c βγpρH�ρLqc. By Proposition 2, r� � arg maxr¡0 θprq � c and p� � r��ρLpc�

r�q � c. Given wi � ρic, i � H,L, and θpcq � γpρH � ρLqc, the profit of this RVW with p� � r� � c is

πHLR � p1�βqtp1� γqρL� γρHuc¡ γp1�βqρHc� πHT . �

Proof of Proposition 4. Since C � expp1{cq, the p.d.f. fpxq � expp�x{cq{c, the tail distribution F pxq �

expp�x{cq and EpC1tC ruq � c�pc� rq expp�r{cq. Taking derivative of θprq with respect to r, we have

Bθprq

Br� γpρH � ρLqF prq� p1�βq

�γρH �p1� γqρL

�rfprq.

The unique optimizer r� of θprq can be characterized by the first order condition Bθprq{Br� 0, which yields

the described expression. Plugging in the expression of r� and wi � ρic, i�H,L, and simplifying terms, we

have πHLR � p1 � γqp1 � βqwL � γpwL � βwHq � θpr�q � expp�r�{cqp1 � βqφc. Since πHT � γp1 � βqρHc, we

have πHLR � πHT � texpp�r�{cqφ� γρHu p1 � βqc. Figure 1 presents examples where the optimal RVW is a

degenerate one, e.g., ρH � γ � 0.4. �

Proof of Proposition 5. If the repair cost is a constant c, the optimal RVW captures both market seg-

ments; by the proof of Proposition 3, the corresponding expected profit is p1 � βqφc. If the repair cost is

random with mean c, by the proof of Proposition 2, the optimal non-degenerate RVW that captures both

market segments has an expected profit equal to φrrPrpC ¥ rq�βEpC1tC¥ruqs�ρLErpC�rq�s ¤ φtrPrpC ¥

rq � βEpC1tC¥ruq �ErpC � rq�su � p1� βqφEpC1tC¥ruq ¤ p1� βqφc, where φ� γρH � p1� γqρL. The first

inequality holds because ρL ¤ φ and the second inequality holds because EpC1tC¥ruq ¤EC � c. Similarly, it

is easy to show that the optimal RVW that captures only the high-usage segment is also less profitable if

the repair cost is random than if it is constant with the same mean. �

Proof of Equation (1). The HJB equation arises as the limit of a discrete-time dynamic program as the

time increment goes to zero. Since the failure process is as a non-homogeneous Poisson process, for small


δ, the probability that only one failure occurs in the interval pt, t� δs is approximately e�λtδλtδ � λtδ, the

probability of no failures is approximately 1�λtδ and the probability of two or more failures is opδq. By the

principle of optimality, for k¥ 0, we have

gpt, kq � λtδEmaxpgpt� δ, kq�Ct, gpt� δ, k� 1qq� p1�λtδqgpt� δ, kq� opδq.

Rearranging terms, we obtain

gpt, kq� gpt� δ, kq

δ��λtEminpCt,∆gpt� δ, kqq�

opδq

δ,

where ∆gpt, kq :� gpt, kq� gpt, k� 1q. Taking the limit at both sides as δ goes to zero, we obtain the desired

differential and difference equation. �

Proof of Proposition 6. (i) Consider the construction of the partial differential and difference equation

(1) and notice that if a claim of size Ct occurs at time t, the choice is between paying out of pocket obtaining

expected payoff of gpt, kq�Ct or making a claim and obtaining expected payoff of gpt, k�1q, so it is optimal

to make the claim if and only if Ct ¥∆gpt, kq.

(ii) (a) To see that gpt, kq ¥ gpt, k� 1q, notice that we can follow the optimal policy as if k was k� 1 and

obtain at least a large a payoff so ∆gpt, kq ¥ 0. By Equation (1), Bgpt, kq{Bt¤ 0 so g is decreasing in t P p0, T s.

(b) Let gpt, k,~rq be the maximum expected refund, net of out-of-pocket costs, at state pt, kq for a refund

schedule ~r � pr0, r1, . . . , rnq. Assume the customer still uses the same threshold policy, when rj increases to

rj� ε. As a result, the out-of-pocket cost is the same, but the refund is larger for any realization of the failure

process. Under this policy, the expected refund net out-of-pocket costs is larger, so the maximum expected

refund net out-of-pocket costs, i.e., gpt, k, pr0, r1, . . . , rj � ε, . . . , rnqq ¥ gpt, k, pr0, r1, . . . , rj , . . . , rnqq. (c) We

obtain the desired result by noting that Bgpt, kq{BΛptq � pBgpt, kq{Btq{pBΛptq{Btq ��EminpCt,∆gpt, kqq ¤ 0.

(iii) (a) By Equation (1),

B∆gpt, kq

Bt��λtE rminpCt,∆gpt, kqq�minpCt,∆gpt, k� 1qqs , (11)

so the convexity of gpt, kq in k, i.e., ∆gpt, kq ¥ ∆gpt, k � 1q, implies that B∆gpt, kq{Bt ¤ 0. Therefore, to

establish part (a) we only need to show the convexity of gpt, kq in k. We will show it by induction. As the

initial step, for k 0, gpt, kq � 0 and the claimed convexity of gpt, kq holds. Suppose there exists some k,

such that the convexity of gpt, kq holds for all j ¤ k. Due to the convexity of rk, we have the convexity of


gpt, kq holds for t� 0. By continuity, there exists t, such that ∆gpt, k� 1q ¥∆gpt, kq for any 0¤ t¤ t. For a

sufficiently small δ¡ 0, by Equation (11) we obtain the following difference equation

∆gpt� δ, k� 1q �∆gpt, k� 1q� δλtE�minpCt,∆gpt, k� 1qq�minpCt,∆gpt, kqq

�� opδq.

Subtracting ∆gpt� δ, kq from both sides gives

∆gpt� δ, k� 1q�∆gpt� δ, kq

�∆gpt, k� 1q�∆gpt� δ, kq� δλtE�

minpCt,∆gpt, k� 1qq�minpCt,∆gpt, kqq�� opδq

��

∆gpt, k� 1q�∆gpt, kq��

∆gpt� δ, kq�∆gpt, kq

�δλtE�

minpCt,∆gpt, k� 1qq�minpCt,∆gpt, kqq�� opδq

¥�

1� δλt

�∆gpt, k� 1q�∆gpt, kq

��∆gpt� δ, kq�∆gpt, kq

�

¥ 0.

The first inequality holds because E�minpCt,∆gpt, k� 1qq�minpCt,∆gpt, kqq

�¤∆gpt, k� 1q�∆gpt, kq; the

second inequality holds because of the assumptions that ∆gpt, k� 1q ¥ ∆gpt, kq holds for any 0 ¤ t¤ t and

that ∆gpt, kq is decreasing in t. This establishes the convexity of gpt, k � 1q for any 0 ¤ t ¤ t� δ. We can

repeat the same procedure to extend the boundary and show that the convexity of gpt, k� 1q holds for any

0¤ t¤ T . Then the desired result is proved by induction.

(b) By Equation (1),

B2gpt, kq

Bt2� �

BpλtErminpCt,∆gpt, kqqqs

Bt

� �B

Bt

�» ∆gpt,kq

0

λtFtpxqdx

� �λtFtp∆gpt, kqqB∆gpt, kq

Bt�

» ∆gpt,kq

0

BpλtFtpxqq

Btdx¥ 0

The last inequality holds because each term is nonnegative by the assumption and part (a). �

Proof of Lemma 2. Note that Bwp~rq{BΛpT q �ErC�minpC,∆gp~rqqs ¥ 0 for any ~r, where Bgp~rq{BΛpT q �

�ErminpC,∆gp~rqqs by the proof of Proposition 6 part (ii)(c). �

Proof of Proposition 10. (i) The optimal TW that captures segment j� and all above should have price

p�T � wj�

. By Lemma 2, to capture the same market segments, the highest price an RVW can charge is

p�wj�

p~rq �wj�

� gj�

p~rq. The profit improvement for the service provider of providing an RVW pp,~rq with

profit GRpp,~rq over the optimal TW with profit GT pp�T q is

GRpp,~rq�GT pp�T q �

I

i�j�

qi�p�hip~rq

��

I

i�j�

qi�pT �βEp

» t

0

λisCs dsq


�I

i�j�

qi�gj�

p~rq�βgip~rq� p1�βqzip~rq.

Under the existence of a refund schedule ~r satisfying (9) with at least one inequality holding strongly,

GRpp,~rq ¡GT pp�T q.

(ii) The optimal TW that captures segment j� and all above should have price p�T � wj�

and profit

GT pp�T q �

°I

i�j�

�wj

�

�βEp³t0λisCs dsq

. For an RVW pp,~rq that captures segment l j� and all above

should have price p � wlp~rq and profit GRpp,~rq �°I

i�lpwlp~rq�hip~rqq. Under the existence of a market

segment l and a refund schedule ~r satisfying (10), GRpp,~rq ¡GT pp�T q. �

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Online Appendix to the Paper“No Claim? Your Gain:

Design of Residual Value Extended Warrantiesunder Strategic Claim Behavior”

Proofs and Numerical Examples

1. Proofs.

Proof of Proposition 7. (i) As a direct application of the structural properties of gpt, kq, we

obtain the optimal claim policy for discrete random repair costs. The set of differential equations

governing gpt, kq is given by

Bgpt, kq

Bt��λt

J

j�1

qjt minpcj,∆gpt, kqq,

with boundary conditions gp0, kq � rk, k ¥ 0 and gpt, kq � 0, k 0, for all t. Since the refund

schedule ~r � pr0, r1, ..., rnq is increasing convex, by Proposition 6 part (iii-a), the threshold apt, kq

is increasing in k and decreasing in t. Note that jkptq is the index of the lowest discrete cost cj

that is greater than the threshold apt, kq.For convenience, we define c0 :�8, so jkptq � 0 whenever

apt, kq ¡ c1. The optimal claim policy based on claiming failures in the set Sjkptq follows directly

from the properties of apt, kq.

(ii) Notice that tk � 0 when rk � rk�1 �∆gp0, kq c. Since we have assumed that the sequence

rk is increasing convex, a sufficient condition to ensure that all tk’s are positive is r0 ¡ c since then

t0 ¡ 0. Moreover, ∆gp0, kq ¡∆gp0, k� 1q for all k¥ 1 so tk ¡ tk�1.

For t tk, the differential and difference equation has the simple form

Bgpt, kq

Bt��λtc, (12)

so the solution is of the form gpt, kq � Ak �Λptqc, where Λptq �³t0λsds. The boundary condition

forces rk � gp0, kq forces Ak � rk.

For t¥ tk the differential and difference equation is of the form

Bgpt, kq

Bt��λtrgpt, kq� gpt, k� 1qs. (13)

We can solve this equation for state pt, kq, t¥ tk by using the threshold policy and conditioning

on the first failure. If the first failure occurs at time t� s¥ tk then a claim is made and the state

drops to pt� s, k�1q. If the failure occurs at time t� s tk then at tk we switch to the policy that

never claims. To express the results more succinctly we will introduce the notation Λps, tq �³tsλv dv

for all s¤ t and also the compact notation Λkptq �³ttkλs ds for all t¥ tk.


This results in

gpt, kq �

» t

tk

λse�Λps,tqgps, k� 1qds� e�Λkptqgk. (14)

Let

gpt, kq �k

j�0

gje�ΛjptqΛj,kptq, t¥ tk. (15)

We will now show that gpt, kq � gpt, kq. The derivative with respect to t of (15) is given by

λt

k�1

j�0

gje�ΛjptqΛj,k�1ptq�λt

k

j�0

gje�ΛjptqΛj,kptq.

Notice that the first term is λtgpt, k� 1q and the second term is λtgpt, kq, so we have

Bgpt, kq

Bt��λtrgpt, kq� gpt, k� 1qs.

Moreover, gptk, kq � gk. We have shown that gpt, kq solves the differential and difference equation

(13) and the boundary conditions at tk, thus gpt, kq � gpt, kq.

(iii) If Ct is exponentially distributed with parameter 1{c then

EminpCt,∆gpt, kqq � r1� expp�∆gpt, kq{cqsc.

To prove the result first notice that the left-hand-side of p1q is

Bgpt, kq

Bt��

°k

j�0Rk�jP pt, jq�°k�1

j�0 Rk�1�jP pt, jq

1�°k

j�0Rk�jP pt, jqλtc.

On the other hand,

expp�∆gpt, kq{cq �1�

°k�1


1�°k

j�0Rk�jP pt, jq,

so the right-hand-side of p1q is

�λtEminpCt,∆gpt, kqq ��

°k

j�0Rk�jP pt, jq�°k�1


1�°k

j�0Rk�jP pt, jqλtc.

Thus, Equation p1q follows. At t � 0, we have P p0,0q � 1 and P p0, jq � 0 for j ¡ 0, so gp0, kq �

lnp1�Rkqc� rk as required. �

Proof of Equation (3). Let gpt, k, aq be the refund, net out-of-pocket cost by using the con-

stant threshold policy with threshold a. Similarly, we have the following differential and difference

equation:

gpt, k, aq � λtδ�PrpC ¡ aqgpt� δ, k� 1, aq�PrpC ¤ aq

�gpt� δ, k, aq�ErC|C ¤ as

��p1�λtδqgpt� δ, k, aq


Rearranging and taking δÑ 0 gives us

Bgpt, k, aq

Bt��λt

�PrpC ¡ aq∆gpt, k, aq�PrpC ¤ aqErC|C ¤ as

(16)

where ∆gpt, k, aq � gpt, k, aq � gpt, k � 1, aq. The boundary conditions are gp0, k, aq � rk for all

k� 0,1,2, . . . , n and gpt, k, aq � 0, k 0, for all 0¤ t¤ T .

For notation simplicity, let λt � λtPrpC ¡ aq and A� PrpC ¤ aqErC|C ¤ as{PrpC ¡ aq. Then,

(16) becomes

Bgpt, k, aq

Bt��λt

�∆gpt, k, aq�A

(17)

Then, we claim for any 0¤ t¤ T and k� 0,1,2, . . . , n, the closed form solution to (17) is

gpt, k, aq �Erpk�Naptqq�Eminpk� 1,NaptqqA (18)

First, we know

BEminpk� 1,Naptqq

Bt�B°k�1

j�1 PrpNaptq ¥ jq

Bt� λt

k

j�0

PrpNaptq � jq

Then, left-hand-side of (17) is

left�hand� side � �λt

k

j�0

rjPrpNaptq � k� jq� λt

k�1

j�0

rjPrpNaptq � k� 1� jq� λt

k

j�0

PrpNaptq � jqA

� �λt

� k

j�0

rjPrpNaptq � k� jq�k�1

j�0

rjPrpNaptq � k� 1� jq�k

j�0

PrpNaptq � jqA

while right-hand-side of (17) is

right�hand� side � �λt

� k

j�0


j�0

rjPrpNaptq � k� 1� jq

��1�Eminpk� 1,Naptqq�Eminpk,Naptqq

�A

� �λt

� k

j�0


j�0

rjPrpNaptq � k� 1� jq

��1�

k�1

j�1

PrpNaptq ¥ jq�k

j�1

PrpNaptq ¥ jq�A

� �λt

� k

j�0


j�0

rjPrpNaptq � k� 1� jq�PrpNaptq ¤ kqA

� left�hand� side

It is easy to very (18) satisfies the boundary conditions. In particular,

lpa,8q � gpT,n,aq �Erpn�Naq�Eminpn� 1,NaqA


� Erpn�Naq�Eminpn� 1,NqPrpC ¤ aqErC|C ¤ as

¥ Erpn�Naq�ΛpT qPrpC ¤ aqErC|C ¤ as � lpa,8q. �

Proof of Equation (5). To derive the partial differential equation that hpt, kq satisfies, we write

hpt, kq � p1�λtδqhpt� δ, kq�λtδ

"PrpCt ¡∆gpt, kqq pErβCt|Ct ¡∆gpt, kqs�hpt� δ, k� 1qq

�PrpCt ¤∆gpt, kqqhpt� δ, kq

*� opδq

� hpt� δ, kq�λtδPrpCt ¡∆gpt, kqq tβErCt|Ct ¡∆gpt, kqs�∆hpt� δ, kqu� opδq.

Rearranging and taking limit δ Ó 0, we obtain Equation (5). �

Proof of Proposition 8. Equation (7) holds for t� 0, k ¥ 0 and for t¡ 0, k 0 because of the

boundary conditions. By Equation (7),

Bhpt, kq

Bt� βλtct�β

Bgpt, kq

Bt�p1�βq

Bzpt, kq

Bt

� βλtErCt�∆gpt, kqs��p1�βqλtPrpCt ¡∆gpt, kqq∆zpt, kq, (19)

where the last equality follows from the differential and difference equations defining gpt, kq and

zpt, kq. By Equation (7), ∆hpt, kq � β∆gpt, kq � p1� βq∆zpt, kq. Expanding the right-hand-side of

Equation (5), we obtain

βλtPrpCt ¡∆gpt, kqqErCt�∆gpt, kq|Ct ¡∆gpt, kqs� p1�βqλtPrpCt ¡∆gpt, kqq∆zpt, kq. (20)

Now, note that the right-hand-side of Equation (19) is equal to expression (20) since for any random

variable X and any scalar x, P pX ¡ xqEpX �x|X ¡ xq �EpX �xq�. �

Proof of Proposition 9. (i) Substituting the expression of gpt, kq in Proposition 7(ii) to (6),

left-hand-side��λt

k

j�0

rje�ΛjptqΛj,kptq�λt

k�1

j�0

rje�ΛjptqΛj,k�1ptq � right-hand-side.

(ii) Substituting (2) into (6) gives us the following differential and difference equation:

Bzpt, kq

Bt��λt

�zpt, kq� zpt, k� 1q

�1�°k�1


1�°k

j�0Rk�jP pt, jq. (21)

Substituting (8) into (21),

left-hand-side ��λt

°k

j�0Qk�jP pt, jq�λt

°k�1

j�0 Qk�1�jP pt, jq

1�°k

j�0Rk�jP pt, jq

�

°k

j�0Qk�jP pt, jq�1�

°k

j�0Rk�jP pt, jq2

��λt

� k

j�0

Rk�jP pt, jq��λt

� k�1

j�0

Rk�1�jP pt, jq�


� �λt

1�°k�1


1�°k

j�0Rk�jP pt, jq

� °k

j�0Qk�jP pt, jq

1�°k

j�0Rk�jP pt, jq�

°k�1

j�0 Qk�1�jP pt, jq

1�°k�1


which equals the right-hand-side of (8). It is easy to show the boundary condition also holds, i.e.,

zp0, kq � rk. �

2. Numerical Examples.

Example 1. Consider a 5-segment market with failure rates λ1 � 1.22, λ2 � 1.27, λ3 � 1.32, λ4 �

1.38 and λ5 � 1.43. We will assume that time has been normalized to T � 1. The cost to repair

a failure is assumed to be an exponential random variable with 1{c � 0.02. The refund schedule

is given by ~r � p$30,$60,$90q. Table 2(a) gives the optimal constant threshold, the approximate

expected refund net of out-of-pocket costs based on approximation (4) and denoted by gisp1,3q,

and the optimal expected refund net of out-of-pocket costs gip1,3q based on formula (2). Notice

that the maximum error among all segments is 0.27% for market segment 5. Moreover, the optimal

threshold values ai are all close to $29 � ∆rk � $30. For this example jpaiq � 8 for all market

segments, so once ai is selected all failures with repair costs exceeding ai are claimed. This is

consistent with our analysis since ErC|C ¡ as � c� a¡∆rn�1�j for all j. The functions lipaq are

fairly flat around the optimal value of a. For example, if we use a � 30 � ∆r for all i then the

largest error increases from 0.27% to just 0.30%.

Table 2 Static Threshold Approximation.

i ai gisp1,3q gip1,3q error(%)

1 $29.13 $63.68 $63.75 0.12%

2 $28.93 $61.53 $61.63 0.17%

3 $28.81 $60.39 $60.51 0.20%

4 $28.68 $59.21 $59.34 0.23%

5 $28.53 $57.98 $58.14 0.27%

i ai gisp1,3q gidp1,3q error(%)

1 $67.62 $114.16 $114.35 0.17%

2 $67.61 $111.68 $111.89 0.19%

3 $67.60 $109.10 $109.34 0.22%

4 $67.59 $106.41 $106.69 0.26%

5 $67.58 $103.61 $103.95 0.33%

(a) Exponential Repair Cost; (b) Normal Repair Cost.

Suppose instead that the repair cost C is a normal random variable with mean µ � 50 and

standard deviation σ� 10 and assume that the refund schedule is ~r� p$50,$110,$175q. Table 2(b)

gives the optimal constant thresholds ai, the approximate expected refund gisp1,3q net of out-of-

pocket costs based on approximation (4), and the expected refund net of out-of-pocket costs gidp1,3q

based on numerical approximation using discrete-time dynamic programming. Again, the optimal

constant threshold approximation is very close to the numerical solution.


Example 2. We revisit Example 1 where there are five market segments. We further assume

that customers are equally likely to be of any of the five types. The fraction of the repair cost to the

provider is β � 0.9 in this and the following examples except where otherwise stated. The optimal

TW will be priced at p3 � w3 � $66.16. This prices out market segments 1 and 2 and results in

profits equal to $6.62, $4.19 and $1.66 from segments 3, 4 and 5 respectively, with expected overall

profit of $2.49.

We first consider an RVW with n � 8 and restrict the warranty to have constant coupons of

size ∆r. We do not pre-restrict ourselves within RVWs with n� 1 � 5 for five market segments

is because a longer refund schedule length may help better price-discriminate within each market

segment, though having the refund schedule length equal to the number of market segments may be

enough to price-discriminate across segments. We maximize πjp∆rq over the five market segments

and find j� � 2, coupon size ∆r� 30 and price p� $304.88 result in expected profits equal to $2.99,

which is 20.03% higher than what the optimal TW can offer. When we compare these expected

profits to those obtained from using a non-linear optimization software to maximize πjp~rq over a

general refund schedule with n� 8, we find we cannot improve the expected profit relative to the

constant coupon schedule. This suggests the restriction to constant coupons does not lead to a

significant loss of optimality. We further see that if we restrict the constant coupon to n� 3, the

optimal expected profit is $2.98 with j� � 2, ∆r� $31 and p� $127.49. This suggests that limiting

the number of coupons gives up little optimality but makes the entry price $127.49 (versus $304.88)

more palatable to customers. Table 3 illustrates how the optimal coupon ∆r, the price wj�p~rq, the

profit Gj�p∆rq and the smallest markets segment served j� vary with the schedule length n� 1.

The last instance corresponds to the optimal TW.

Table 4 repeats the experiment with the same data except that now the repair cost is exponen-

tially distributed with mean c� $100 and the failure rates are one half of what they are before (as

in Table 3). The optimal TW remains the same but the value of the coupons essentially doubles

and the expected profit grows slightly faster in the size of schedule.

Example 3. We assume the repair cost in the previous example is normally distributed with

mean µ� 50 and standard deviation σ � 10. Because the repair cost has the same mean as in the

exponential case, the optimal TW stays the same. We report the optimal RVW design and its

improvement over the optimal TW in Table 5.

The price of the optimal TW only depends on the mean of the failure cost, while the price of the

RVW depends on the repair cost distribution. Table 6 illustrates how the improvements over the

optimal TW change as a function of the standard deviation of the normal distribution, assuming


Table 3 Performance of RVW as a Function of Schedule

Length.

n� 1 ∆r p�wj�

p~rq Gj�p∆rq j� Gain

1 $54 $85.33 $2.73 2 9.62%

2 $36 $104.76 $2.93 2 17.41%

3 $31 $127.49 $2.98 2 19.57%

4 $30 $154.92 $2.99 2 19.98%

5 $30 $184.89 $2.99 2 20.03%

6 $30 $214.88 $2.99 2 20.03%

7 $30 $244.88 $2.99 2 20.03%

8 $30 $274.88 $2.99 2 20.03%

9 $30 $304.88 $2.99 2 20.03%

the optimal TW $66.16 $2.49 3

Exponentially Distributed Repair Costs: the Base Case.


Length.

n� 1 ∆r p�wj�


1 $81 $114.30 $2.86 2 14.58%

2 $63 $160.37 $2.97 2 19.35%

3 $60 $214.93 $2.99 2 19.98%

4 $60 $274.89 $2.99 2 20.03%


Exponentially Distributed Repair Costs with Mean Dou-

bled.

the mean stays constant. With the length of the refund schedule fixed, the more random the repair

cost, the less the profit lift of the optimal RVW over the optimal TW. To better price-discriminate

within each usage segment, a longer length of the refund schedule is needed if repair costs have

more variability.

Example 4. In the context of Example 2, we perform sensitivity analysis for the provider’s

profit with respect to his cost advantage measured by β and the uncertainty in repair costs. We

restrict the refund schedule length to n � 1 � 4 as Example 2 shows an RVW with n � 1 � 4

constant coupons is nearly optimal. Figure 2(a) shows profits of the optimal TW and RVW for an

exponentially distributed repair cost with mean c� 50. Figure 2(b) shows profits of the optimal

TW and RVW for a normally distributed repair cost with mean µ � 50 and standard deviation



Length.

n� 1 ∆r p�wj�


1 $53 $79.34 $2.99 2 19.88%

2 $46 $105.43 $4.03 1 61.74%

3 $41 $136.56 $4.60 1 84.44%

4 $39 $170.51 $4.77 1 91.50%

5 $38 $205.34 $4.81 1 93.17%

6 $38 $243.30 $4.82 1 93.42%


Normally Distributed Repair Costs.

Table 6 Performance of RVW as a Function

of Standard Deviation.

σ ∆r p�wj�


0 $49 $197.66 $6.31 1 153.14%

10 $39 $170.51 $4.77 1 91.50%

20 $37 $167.78 $4.08 1 64.00%

30 $40 $181.57 $3.84 1 54.24%

40 $45 $202.70 $3.81 1 52.92%

Normally Distributed Repair Costs, n� 3.

σ� 10 or σ� 20. The profits in all cases are decreasing in the cost advantage parameter β. When

the repair cost has more randomness, the profit to the provider goes down as under strategic claim

behavior customers are more likely to self-defray the costs of inexpensive repairs to improve their

expected net refund in an environment with more uncertainty in repair costs.

Example 5. We consider a warranty service provider facing two market segments in the setting

of the general framework. The repair cost is exponentially distributed with mean c � 100. The

failures experienced by each segment is assumed to follow a Poisson process. The length of warranty

coverage is T � 10. The failure rate of the low-usage segment is λL � 0.2, namely, the low-usage

segment is expected to experience two failures on average over the warranty coverage. The cost

fraction to the service provider is β � 0.7. Figure 3 illustrates the profitability of the optimal RVW

with a single refund over the optimal TW under customers’ strategic claim behavior for various

combinations of the failure rate and proportion of the high-usage segment. The failure rate λH of

the high segment varies from 0.2 to 0.4, and the proportion γ of the high segment varies from 0.2 to

0.4. When the failure rate λH and the proportion γ of the high-usage segment is large, e.g., in the


Figure 2 Profit Improvement as a Function of β

0.7 0.75 0.8 0.85 0.9 0.95 10

2

4

6

8

10

12

14

Expe

cted

Pro

fit ($

)

Profit of TWProfit of RVW

(a) Exponential Repair Cost

0.7 0.75 0.8 0.85 0.9 0.95 10

2

4

6

8

10

12

14

Expe

cted

Pro

fit ($

)

Profit of TWProfit of RVW ( =10)Profit of RVW ( =20)

(b) Normal Repair Cost

Figure 3 Multiple Failures: Profit Improvement of the Optimal RVW with a Single Refund over the Optimal TW

under Strategic Claim Behavior

0.20.25

0.30.35

0.4

0.2

0.25

0.3

0.35

0.4

0

1

2

3

4

5

6

λH

γ

Pro

fitab

ility

of R

VW

ove

r T

W

Note. C � expp1{100q, T � 10, λL � 0.2, β � 0.7. The profitability of RVW over TW is maxtπHLR pr�q �

maxtπHT , πHLT u,0u.

neighborhood of pγ,λHq � p0.4,0.4q, it is not optimal to offer an RVW and the optimal warranty

contract remains the optimal TW that captures only the high-usage segment. On the wedge shape,

the optimal RVW of capturing two segments is strictly more profitable than the optimal TW,

except along the edge of λH � λL � 0.2, where the two segments degenerate to one.

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