Pricing Tranches in a CAPM-like World

In buy-and-hold scenarios, slicing a loss distribution into tranches creates `slice' risk in the sense that the original pool may have superior Sharpe ratio to its individual tranche components. We aim to determine the equilibrium price to tranches in an illiquid CAPM-like world where continuous trading is impossible and neither slice risk nor idiosyncratic risk is compensated. In our simple world there is a constant excess return only to systematic risk. Systematic risk cannot potentially be eliminated by either completing the capital structure (as for slice risk) or diversification (as for idiosyncratic risk). First the case without idiosyncratic risk is considered: a perfectly diversified portfolio is tranched, and the equilibrium price is derived. Next, the approach is generalised to tranches of pools containing idiosyncratic risk.