Investment decisions in the rent-to-own industry in the absence of inventory

Abstract

This study addresses the product investment decision faced by firms in the rent-to-own industry. In this setting, a customer arrives according to a random process and requests one unit of a product to rent (and eventually own should he/she choose to make all the required payments). At the time of request, if the product is available in inventory, the firm enters into a contractual agreement (by accepting the customer's offer) and rents the merchandise. More interesting and the case considered here, if the requested item is not in inventory, the firm must decide whether to purchase the item in order to rent it out or to simply reject the request. The customer's offer specifies the desired maximum contract length and the payment frequency—from which the firm determines the fixed periodic payment charged. The firm makes its investment decision based on the characteristics of the offer as well as those of the product (eg, initial and resale values, useful life and carrying costs) in essence performing a complicated cost benefit analysis. An extension is also considered whereby instead of simply rejecting the request the firm can adjust the required payment amount. Dynamic programming techniques are used to address the problem and to solve for the firm's optimal decision.

Keywords:

• dynamic programming
• inventory level
• consumer credit
• payment schedule
• transactions data
• empirical estimation

^†The names are listed solely in alphabetical order, the authors made equal contributions.

^†The names are listed solely in alphabetical order, the authors made equal contributions.

Notes

1 See, for example, Swagler et al (1995) and Stegman and Faris (2003).

2 This high cost has been argued in, for example, Freedman (1993) and Hill et al (1998).

3 Looking at the transactions that end in a skip, it appears that this occurs due to circumstance rather than an attempt by customers to defraud. In particular, considering median behaviour, in a skip only 27.34% of the rent due (if all payments are made) is received, further, the skip occurs 45.75% into the contract.

4 $K represents product cost plus delivery, set up and some allocation of store overhead.

5 The size of this period depends on a variety of factors including consumer and product characteristics and store location. While its precise length is an empirical question, all our model requires is that it be finer than the interval between customer payments. For concreteness, one can think of it as being one day.

6 In our numerical study the units are years, k={WK, BW, SM, MO} and c_k∈{52, 26, 24, 12} ,where the elements of C refer to weekly, bi-weekly, semi-monthly and monthly payment schedules, respectively and the elements of c_k refer to the number of payments per time period for the corresponding k. For example, for contract type k=WK, we have c_k=52, that is, 52 (weekly) payments per year.

7 One motivation for this choice would be to match up the payment frequency with his/her cash flow stream. Another could be to match up his/her expectation over the outcome. That is, those expecting to rent might prefer a weekly arrangement for the greater flexibility offered while those expecting to purchase might prefer monthly for the greater convenience. See Anderson and Jaggia (2009a) for additional discussion.

8 Over its useful life, a product can be rented under a variety of payment schedules and so number of payments, d, will not necessarily map directly to age. The variable d̂ yields a common integer measure of age expressed in the units of the finest time partition employed—in our numerical example this is weeks.

9 This still represents a degree of aggregation as agreements are written on several brands and models and we also aggregate agreements on washers with those on dryers. However, the data suggest this is a good balance of the conflicting needs for sample size and merchandise homogeneity.

10 See Anderson and Jaggia (2009b) for more analysis of this point.

11 For instance, for a contract running 18 months and requiring bi-weekly payments, ∑_i=r,b,wq_i=64.17+33.33+2.50=1.0.