Measuring Unit Prices for Communications Bucket Plans Using the Black-Scholes Model for Valuing Options
Telecommunications network operators have traditionally charged on a unit basis for messages sent and received. Dating back to the heyday of telegraphy, operators charged by the word. With the rise of the Bell System telephony was charged on a per minute basis. The 1984 Divesture of AT&T saw the rise of two part tariffs, to separate costs related to access and long lines networks. However, the basic premise was a per minute charge (oftentimes including a connection or first minute charge). This arrangement was simple and allowed for easy comparisons of unit charges, dividing total revenues by total minutes of use (MoUs). This price per minute is an easily comparable metric - a voice minute has changed little since Alexander Graham Bell famously whispered, “Watson, come here I need you.” Regulators could employ price per minute performance as a gauge on the efficacy of policies, and consumers could understand the value of what they are getting.
However, now, driven by competition in mobile telephony, communications carriers have offered an alternative set of arrangements (the same is now true for fixed line carriers and even some data networks). These have been variant of the flat rate plan, mixing flat rate and per unit pricing plans. Under such “buckets of minutes” plans, customers pay a flat rate for a huge number of minutes whether they use them or not. If their usage is over the allotted amount, then they pay on a per minute basis at a rate much higher. The per-minute charges are not intended to be used on a routine basis; rather, they are set at high or punitive levels, so as to enforce the need to upgrade to the next higher band as the customer’s usage increases over time. These plans are most appropriately viewed as representing banded flat rate arrangements. There are different bands, representing different numbers of total MoUs per month. In the United States, these banded flat rate plans have been well accepted by consumers and industry. They probably track underlying costs of mobile telephone service reasonably well, to the extent that usage-based average incremental costs in the United States are primarily a function of air time. Customers appreciate the predictability and the relative simplicity of the plans. So long as the consumer does not exceed the maximum number of minutes in the band, the consumer will tend to think of the plan as being purely flat rate.
The introduction of bucket plans, however, frustrates any calculation of per minute unit price. Plans of this type usually incorporate a per-minute charge for minutes, but it would be a mistake to analyse a banded flat rate plan as if it were a simple two-part tariff. In calculating per minute charges it is unclear whether to utilize actual minutes of use consumed or plan minutes sold, but not used. The results would be quite different. The former would yield a higher price, but would not indicate the value “left on the table” by the consumer. While the latter would tend to under represent the actual cost by making implicit the value of the option of further minutes of use at zero incremental price.
Insight: The appropriate way to regard bucket plans, and hence, to value per minute unit prices, is as a permutation of a financial call option. A call option affords the right, but not the obligation of use at a specified price, called the exercise or strike price, on or before a specified date. The intuition begins with a graphical analysis, see inserted figures
. The graphic on the right displays the payoff of a financial option at various prices. The x-axis represents the price of the underlying security and the y-axis represents the potential profit or loss. The value of the option stays flat until the market price exceeds the strike price. At this point, its value increases and becomes positive once it exceeds the cost of the acquiring the option. Now consider the figure on the left. It shows the cost of two bucket plans. The x-axis is still the independent variable: minutes of use. T
he value (or price) of the plan is flat until exceeds the flat-rate portion of minutes. It then increases at a linear per minute rate, until at some point it is preferable for the customer to take the larger bucket. These two graphics look virtually identical, suggesting that the option valuation method holds promise as an effective metric.
The value of options can be captured by a complicated formula developed by Fischer Black and Myron Sholes in the early 1970s (the two shared the Nobel Prize for this observation). The option value is based, inter alia, on the current market price, the strike price, the variability of market price with regard to the strike price, the risk free cost of capital, and the maturity of the option. By finding equivalents for these variables in bucket plans, we can easily derive per minute values for under bucket plans. I realize that this may be a bit of an oranges-to-tangerines comparison. There are certain items not present in the derivatives markets in these service plans. Nonetheless, such a calculus would allow for straight forward comparisons of the price of services plans across firms, countries and over time series where such plans might not exist.