Linear beta pricing with efficient/inefficient benchmarks and short-selling restrictions

We provide two security pricing models that can be used when short sales of risky securities are not permitted. The first model uses a benchmark located on the expected return-standard deviation efficient frontier without short sales and presents security expected returns as a weighted linear function of two betas, one induced by the benchmark and the other adjusting for the short-sale constraints. The second model uses a benchmark that is inefficient relative to the efficient frontier, does not allow short sales, and expresses security returns as a weighted linear function of three betas:one induced by the inefficient benchmark, and the other two adjusting simultaneously for the short-sales restrictions and the benchmark's inefficiency.These results complement the linear pricing models that link expected returns and betas by allowing, for efficient or inefficient expected return-standard deviation, benchmarks with restricted short sales.