On Comparison of Cash Flow Streams

In problems where planning over time plays a role, natural objects for consideration are cash flow streams. In order to compare different policies we require some preference ordering over these streams. A customary approach to this problem is to assume that money can be borrowed and lent freely at some fixed interest rate---i.e., that there exists a perfect capital market---and to compare streams with respect to such borrowing and lending. The induced ordering is based on the size of a single quantity, termed the discounted value, obtainable from the stream by a linear calculation. In this note we consider preference rankings of cash streams in imperfect capital markets; in particular, we suppose that the interest rate may depend on the size of the loan. We introduce an ordering based, as before, on the ability to convert one stream into another by borrowing and lending, and we describe its structural features: salient here is the fact that there generally exist incomparable streams. We then specialize to the case that the interest rate is constant but different for borrowing and lending, and repayment is either short-term or long-term. There we show that, when one stream is preferred to another, it has a larger discounted value with respect to each interest rate in the range between the borrowing and lending rates. The present model therefore provides a simple example of a situation in which discounting a stream over a variety of interest rates enters naturally into the preference ranking.